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+MNRAS 000, 1–58 (2022)
+Preprint 13 January 2023
+Compiled using MNRAS LATEX style file v3.0
+QUIJOTE scientific results – IV. A northern sky survey in intensity
+and polarization at 10–20 GHz with the Multi-Frequency Instrument
+J. A. Rubiño-Martín,1,2★ F. Guidi,1,2,3 R. T. Génova-Santos,1,2 S. E. Harper,4 D. Herranz,5
+R. J. Hoyland,1,2 A. N. Lasenby,6,7 F. Poidevin,1,2 R. Rebolo,1,2,8 B. Ruiz-Granados,1,2,9
+F. Vansyngel,1,2 P. Vielva,5 R. A. Watson,4 E. Artal,10 M. Ashdown,6,7 R. B. Barreiro,5
+J. D. Bilbao-Ahedo,5,11 F. J. Casas,5 B. Casaponsa,5 R. Cepeda-Arroita,4 E. de la Hoz,5,11
+C. Dickinson,4 R. Fernández-Cobos,5,12 M. Fernández-Torreiro,1,2 R. González-González,1,2
+C. Hernández-Monteagudo,1,2 M. López-Caniego,13,14 C. López-Caraballo,1,2
+E. Martínez-González,5 M. W. Peel,1,2 A. E. Peláez-Santos,1,2 Y. Perrott,6,15 L. Piccirillo,4
+N. Razavi-Ghods,6 P. Scott,6 D. Titterington,6 D. Tramonte,1,2,16,17 R. Vignaga.1,2
+1Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
+2Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
+3Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France.
+4Jodrell Bank Centre for Astrophysics, Alan Turing Building, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester,
+Oxford Road, Manchester M13 9PL, Manchester, UK
+5Instituto de Física de Cantabria (IFCA), CSIC-Univ. de Cantabria, Avda. los Castros, s/n, E-39005 Santander, Spain
+6Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK
+7Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
+8Consejo Superior de Investigaciones Científicas, E-28006 Madrid, Spain
+9Departamento de Física. Facultad de Ciencias. Universidad de Córdoba. Campus de Rabanales, Edif. C2. Planta Baja. E-14071 Córdoba, Spain.
+10Departamento de Ingenieria de COMunicaciones (DICOM), Laboratorios de I+D de Telecomunicaciones, Plaza de la Ciencia s/n, E-39005 Santander, Spain
+11Departamento de Física Moderna, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain
+12Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. los Castros, s/n, E-39005 Santander, Spain
+13Aurora Technology for the European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692
+Villanueva de la Cañada, Madrid, Spain
+14Universidad Europea de Madrid, 28670, Madrid, Spain
+15School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
+16Purple Mountain Observatory, CAS, No.10 Yuanhua Road, Qixia District, Nanjing 210034, China
+17NAOC-UKZN Computational Astrophysics Center (NUCAC), University of Kwazulu-Natal, Durban 4000, South Africa.
+Accepted 2022 November 11. Received 2022 November 1; in original form 2022 July 29
+ABSTRACT
+We present QUIJOTE intensity and polarization maps in four frequency bands centred around
+11, 13, 17 and 19 GHz, and covering approximately 29 000 deg2, including most of the Northern
+sky region. These maps result from 9 000 h of observations taken between May 2013 and June
+2018 with the first QUIJOTE instrument (MFI), and have angular resolutions of around 1◦,
+and sensitivities in polarization within the range 35–40 𝜇K per 1-degree beam, being a factor
+∼ 2–4 worse in intensity. We discuss the data processing pipeline employed, and the basic
+characteristics of the maps in terms of real space statistics and angular power spectra. A number
+of validation tests have been applied to characterise the accuracy of the calibration and the
+residual level of systematic effects, finding a conservative overall calibration uncertainty of
+5 %. We also discuss flux densities for four bright celestial sources (Tau A, Cas A, Cyg A and
+3C274) which are often used as calibrators at microwave frequencies. The polarization signal
+in our maps is dominated by synchrotron emission. The distribution of spectral index values
+between the 11 GHz and WMAP 23 GHz map peaks at 𝛽 = −3.09 with a standard deviation of
+0.14. The measured BB/EE ratio at scales of ℓ = 80 is 0.26 ± 0.07 for a Galactic cut |𝑏| > 10◦.
+We find a positive TE correlation for 11 GHz at large angular scales (ℓ ≲ 50), while the EB and
+TB signals are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150. The maps discussed
+in this paper are publicly available.
+Key words: cosmology: observations – cosmic microwave background
+★ E-mail: jalberto@iac.es
+© 2022 The Authors
+arXiv:2301.05113v1  [astro-ph.GA]  12 Jan 2023
+
+2
+Rubiño-Martín et al.
+1
+INTRODUCTION
+Measurements of the Cosmic Microwave Background (CMB)
+anisotropies provide one of the most powerful tools in modern cos-
+mology, playing a fundamental role in our current understanding of
+the physics of the early Universe and structure formation (Bennett
+et al. 2013; Planck Collaboration et al. 2020a). Moreover, CMB
+polarization observations open a window to probe the amplitude
+of primordial gravitational waves generated during the inflationary
+epoch (Kamionkowski et al. 1997; Zaldarriaga & Seljak 1997). Fol-
+lowing this scientific motivation, observations of B-modes at large
+angular scales have progressed substantially over the last few years.
+Current best upper limits on the tensor-to-scalar ratio come from
+the BICEP/Keck 2018 CMB polarization data (Ade et al. 2021),
+and give 𝑟 < 0.036 at 95% confidence level, which improves to
+𝑟 < 0.032 when adding the latest Planck PR4 data (Tristram et al.
+2022). Upcoming ground-based experiments like Simons Observa-
+tory (Ade et al. 2019) or CMB-S4 (Abazajian et al. 2022), and space
+missions like LiteBIRD (LiteBIRD Collaboration et al. 2022) will
+improve these constraints in the coming years.
+Due to the low amplitude of this primordial B-mode signal,
+the control and removal of diffuse Galactic foreground contamina-
+tion in polarization is becoming a key challenge for current and
+future CMB experiments. Basically there are two main Galactic
+foregrounds that are known to emit linearly polarized radiation:
+the synchrotron emission resulting from cosmic ray electrons ac-
+celerated around the Galactic magnetic field lines, and the thermal
+radiation from interstellar dust grains also aligned with the mag-
+netic field (Bennett et al. 2013; Planck Collaboration et al. 2016g,
+2020d). Anomalous microwave emission (AME) has been also de-
+tected in intensity, but no polarization has been measured up to date
+(Rubiño-Martín et al. 2012a; Dickinson et al. 2018). Although there
+are theoretical motivations to expect negligible polarization levels if
+AME is produced by spinning dust grains (Draine & Hensley 2016),
+improved low frequency observations will be needed to consolidate
+our understanding of this physical process.
+The Planck satellite (Planck Collaboration et al. 2020a) pro-
+duced seven full sky polarization maps covering the frequency range
+between 30 and 353 GHz. The Wilkinson Microwave Anisotropy
+Probe (WMAP) satellite (Bennett et al. 2013) scanned the full sky
+in polarization in five bands between 23 and 94 GHz. The analysis of
+these data shows that, for a B-mode signal with amplitude 𝑟 = 10−3
+(which is the target of the LiteBIRD space mission), there is no
+frequency domain or sky region where the sum of the synchrotron
+and thermal dust foregrounds is subdominant with respect to the
+expected CMB B-mode signal (Planck Collaboration et al. 2016a;
+Krachmalnicoff et al. 2016). Moreover, further analyses of these
+and other datasets show increasing evidence of complexity in the
+spectral and spatial behaviour of the Galactic dust and synchrotron
+emissions (Choi & Page 2015; Planck Collaboration et al. 2017a;
+Krachmalnicoff et al. 2018; Fuskeland et al. 2021; Weiland et al.
+2022; de Belsunce et al. 2022).
+The situation is particularly complex for the polarized syn-
+chrotron emission. The sensitivity of the low frequency channels
+from Planck and WMAP does not allow the detection of polarized
+synchrotron signal at intermediate and high Galactic latitudes, and
+therefore we are lacking a detailed spectral modelling of this emis-
+sion precisely in the regions of cosmological interest. In this context,
+there is a need for complementing the existing satellite observations
+with measurements at lower frequencies in order to improve our de-
+scription of the foregrounds at the required level for B-mode studies.
+There are only a limited number of radio surveys that preserve the
+large-scale structure of Galactic emission, and most of them pro-
+vide only intensity measurements (Haslam et al. 1982; Berkhuijsen
+1972; Reich et al. 2001; Jonas et al. 1998), but this situation is
+now changing. The S-band Polarization All-Sky Survey (S-PASS;
+Carretti et al. 2019) recently provided the first map of the polar-
+ized radio emission over the southern sky at declinations below −1◦
+taken with the Parkes radio telescope at 2.3 GHz. The C-Band All
+Sky Survey (C-BASS; Jones et al. 2018) will cover the full sky at
+5 GHz, and the maps of the northern sky will be soon available.
+With the aim of providing spectral coverage complementary
+to WMAP and Planck at intermediate frequencies, the Q-U-I JOint
+Tenerife Experiment (QUIJOTE, Rubiño-Martín et al. 2010) is a sci-
+entific collaboration between the Instituto de Astrofisica de Canarias
+(IAC), the Instituto de Fisica de Cantabria (IFCA), the Universities
+of Cantabria, Manchester and Cambridge, and the IDOM company.
+It has the goal of characterising the polarization of the CMB and
+other Galactic and extragalactic physical processes in the frequency
+range 10–40 GHz and at large angular scales ( >∼ 1◦). QUIJOTE has
+been designed to have the required sensitivity to detect a primordial
+gravitational-wave component if the tensor-to-scalar ratio is larger
+than 𝑟 = 0.05. The experiment is located at the Teide Observatory
+(altitude of 2,400 m a.s.l) in Tenerife (Canary Islands), and con-
+sists of two telescopes equipped with three instruments: the Multi-
+Frequency Instrument (hereafter, MFI), operating at 10–20GHz, the
+Thirty-GHz Instrument (TGI) and the Forty-GHz Instrument (FGI).
+The two QUIJOTE telescopes, QT-1 (Gomez et al. 2010) and QT-2
+(Sanquirce et al. 2014; Sanquirce-García et al. 2016), are based on
+an offset crossed-Dragone design with projected apertures of 2.25
+and 1.89 m for the primary and secondary mirrors respectively,
+and provide optimal polarization properties (polarization leakage
+≤ −25dB), low sidelobes (≤ −40 dB) and highly symmetric beams
+(ellipticity ≤ 2 %).
+MFI is a multi-channel instrument that has been operating
+between November 2012 and October 2018 mounted on the first
+QUIJOTE telescope, QT-1. MFI consists of four polarimeters (also
+called here "horns"). Horns 1 and 3 operate in the band 10–14 GHz,
+while horns 2 and 4 operate at 16–20 GHz. Using frequency filters
+in the back-end module (hereafter BEM) of the instrument, each
+horn provides outputs in two frequency sub-bands, each one with
+an approximate bandwidth of Δ𝜈 = 2 GHz. There are a total of 8
+outputs for each polarimeter, and these are then fed into the Data
+Acquisition Electronics (DAE). In total, the MFI provides four fre-
+quency bands centred around 11, 13, 17 and 19 GHz, with each band
+covered by two independent horns. The approximate angular resolu-
+tion, given in terms of the full width at half-maximum, is 52 arcmin
+for the low-frequency bands (11 and 13 GHz), and 38 arcmin for
+the 17 and 19 GHz channels. During the lifetime of the instrument,
+we had basically two instrumental configurations for the MFI. The
+main difference of the second configuration with respect to the first
+one is the integration of 90◦ hybrid couplers in each polarimeter,
+giving correlated outputs in all four detectors. A more detailed de-
+scription of the instrument can be found in Hoyland et al. (2012);
+Pérez-de-Taoro et al. (2016), and will be included in a future paper
+(Hoyland et al., in prep). A complete description of the MFI instru-
+ment characteristics, as well as the MFI data processing pipeline, is
+included in an accompanying paper (Génova-Santos et al. 2023).
+As described in Rubiño-Martín et al. (2010), most of the
+QUIJOTE-MFI observing time was dedicated to two main surveys:
+a shallow Galactic survey (hereafter the "wide survey") covering
+all the visible sky from Tenerife at elevations larger than 30◦, and
+a deep cosmological survey covering approximately 3 000 deg2 in
+three separated sky patches in the northern sky. In addition to those
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+3
+two main surveys, a fraction of the MFI observing time was dedi-
+cated to raster scan observations in some selected Galactic regions.
+Data from some of those MFI raster scan observations were already
+presented in three QUIJOTE collaboration publications (Génova-
+Santos et al. 2015, 2017; Poidevin et al. 2019), where we charac-
+terised the presence of AME towards several Galactic molecular
+complexes, as the Perseus region, W43, W47 or Taurus, and to-
+wards a supernova remnant, W44. In particular, the study of W43
+provides the strongest upper limits to date on the polarization frac-
+tion of the AME (Génova-Santos et al. 2017). Additional raster scan
+observations were carried out in W51, IC443, rho-Ophiucus, and
+M31, among others.
+A preliminary version of the MFI wide survey maps, in com-
+bination with C-BASS North data, were used in the study of the
+𝜆-Orionis region (Cepeda-Arroita et al. 2021). This paper presents
+the final maps of the QUIJOTE-MFI wide survey. Section 2 de-
+scribes the observations and the data processing pipeline. The final
+maps are presented in Section 3. The validation and characterisation
+of these maps is presented in Section 4. An assessment of the overall
+calibration uncertainty of the maps is discussed in Section 5, while
+Section 6 describes the generation of specific noise simulations for
+the QUIJOTE MFI wide survey. Sections 7, 8 and 9 discuss some
+of the basic properties of the maps both in real and harmonic space,
+including the photometry results of some bright radio sources. Fi-
+nally, Section 10 describes the data products and associated scien-
+tific papers accompanying this paper. All of them are devoted to
+the understanding of the low frequency Galactic foregrounds in in-
+tensity and polarization, either in the full QUIJOTE MFI footprint
+or in localised regions, and using various analysis techniques. The
+conclusions of this work are presented in Section 11.
+2
+THE QUIJOTE-MFI WIDE SURVEY DATA
+The QUIJOTE wide survey is a shallow survey which covers all
+the visible sky from the Teide Observatory (latitude +28.3◦) with
+elevations greater than 30◦ (more than 29 000 deg2). This was one
+of the main scientific objectives of QUIJOTE (Rubiño-Martín et al.
+2012b), and in particular, of the MFI instrument. This paper presents
+the QUIJOTE MFI wide survey maps, which were obtained with
+approximately 9 000 h of observing time. The four final maps at
+nominal frequencies 11, 13, 17 and 19 GHz, smoothed to 1 degree
+resolution, are shown in Figs. 1, 2, 3 and 4, respectively. All maps
+were generated using the HEALPix1 pixelization scheme (Górski
+et al. 2005) with 𝑁side = 512. In HEALPix the sphere is divided into
+12𝑁side2 pixels of equal area. In particular, 𝑁side = 512 corresponds
+to a pixel size of approximately 6.9 arcmin on the sky. Figure 5 also
+shows the polarized intensity (𝑃 =
+√︁
+𝑄2 + 𝑈2), the polarization
+angle direction2 (𝛾 = 0.5 arctan(−𝑈/𝑄)), and the direction of mag-
+netic field lines for the 11 GHz map. In the following subsections
+we describe the observations, the data processing pipeline, the map-
+making and the specific post-processing and recalibration applied
+to these maps.
+1 https://healpix.sourceforge.io
+2 QUIJOTE polarization maps use the COSMO convention from HEALPix,
+so we use a minus sign in the definition of 𝛾 to recover the IAU convention
+for the angle.
+2.1
+Observations
+The maps described in this paper are based on MFI observations
+carried out between May 2013 and June 2018 using the so-called
+"nominal mode", which consists of continuous (360◦) azimuth scans
+at a constant telescope elevation. The default azimuth scan speed
+was 𝑣AZ = 6 deg s−1 from the beginning of the survey until January
+9th 2014, but this was increased to 𝑣AZ = 12 deg s−1 after this date,
+in order to reduce the 1/ 𝑓 noise contribution in the intensity maps. In
+this observing mode, every day each MFI horn covers a continuous
+band of 360◦ in right ascension, and a certain declination range
+specified by the elevation of the telescope. As in all QUIJOTE-
+MFI observations, and in order to minimize systematic effects in
+the polarization parameters, observations are carried out in four
+discrete positions of the polar modulators 𝜃pm =(0◦, 22.5◦, 45◦ and
+67.5◦). In the wide survey, each observation at a given elevation
+and modulator angle position has a typical duration of 24 h.
+The combination of multiple elevations allows us to obtain a
+more homogeneous sampling of the sky. Table 1 contains the final
+set of telescope elevations considered here to produce the maps, to-
+gether with the total number of hours observed and used in each case.
+In total, there are approximately 9 200 h of observations, equivalent
+to 383 observing days. Almost all of this observing time was suit-
+able for use in the preparation of the intensity maps. However, the
+final polarization maps only use of the order of 5 700 h, as explained
+below.
+Observations are also separated in periods of several months.
+The definition of each period is usually associated with changes
+either in the MFI instrument configuration, telescope configuration,
+or simply to new observing cycles after instrument maintenance.
+A complete description of those periods, as well as the associated
+instrument changes, can be found in Génova-Santos et al. (2023).
+We note that for the MFI wide survey, we conducted observations
+only during periods 1, 2, 5 and 6. The global dates and effective
+epoch (year) for each of those periods are listed in Table 2.
+As noted in this table, an extended shielding was installed in the
+first QUIJOTE telescope (QT-1) at the beginning of period 2. The
+main reason for this was to minimize the impact of far sidelobes due
+to the emission of geo-stationary satellites, which were particularly
+important for horn 1 (Génova-Santos et al. 2023). In addition, during
+the operations horn 1 was either not operative (periods 5 and 6) or
+had problems with the positioning of the polar modulator (period
+2). Because of these reasons, although wide-survey maps of horn
+1 have been produced for internal consistency tests, they have not
+been used for this paper because they are significantly affected by
+systematic effects.
+2.2
+Data processing pipeline
+A complete description of the MFI data processing pipeline can be
+found in the MFI pipeline paper (Génova-Santos et al. 2023). Here,
+we summarize the basic characteristics of the MFI data, and we
+discuss those aspects which are specific of the MFI wide survey.
+Each MFI polarimeter is divided into a lower and upper band
+of approximately 2 GHz bandwidth which is defined by the band-
+pass filters. Each sub-band has four outputs, which are labelled as
+(𝑉x+y,𝑉x−y,𝑉x,𝑉y). The first two outputs are called "correlated"
+channels because in the first (original) configuration of the instru-
+ment they passed through a 180◦-hybrid, and therefore they have
+correlated (common) 1/ 𝑓 noise properties. The second pair is called
+"uncorrelated" channels, and in the original configuration provided
+two outputs with independent noise. The first instrument configu-
+MNRAS 000, 1–58 (2022)
+
+4
+Rubiño-Martín et al.
+Figure 1. QUIJOTE MFI maps at 11 GHz in Galactic coordinates, smoothed to 1 degree resolution and using 𝑁side = 512. Top: intensity 𝐼. Middle: polarization
+𝑄 component. Bottom: polarization 𝑈 component.
+MNRAS 000, 1–58 (2022)
+
+QUJOTE1H311GHz(1deg)
+5
+mk
+20QUJOTEQH311GHz(1deg)
+mKQUOTEUH311GHz（1deg）
+mKQUIJOTE MFI wide survey
+5
+Figure 2. QUIJOTE MFI maps at 13 GHz smoothed to 1 degree resolution. Top: intensity 𝐼. Middle: polarization 𝑄 component. Bottom: polarization 𝑈
+component.
+MNRAS 000, 1–58 (2022)
+
+QUJOTE1H313GHz(1deg)
+5
+mk
+20QUJOTEQH313GHz(1deg)
+1
+mKQUJOTEUH313GHz(1deg)
+1
+mK6
+Rubiño-Martín et al.
+Figure 3. QUIJOTE MFI maps at 17 GHz smoothed to 1 degree resolution. Top: intensity 𝐼. Middle: polarization 𝑄 component. Bottom: polarization 𝑈
+component.
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE117GHz combined H2+H4(1deg)
+5
+mk
+20QUijOTEQ17GHzcombinedH2+H4(1deg)
+mkQUjOTEU17GHzcombinedH2+H4(1deg)
+mkQUIJOTE MFI wide survey
+7
+Figure 4. QUIJOTE MFI maps at 19 GHz smoothed to 1 degree resolution. Top: intensity 𝐼. Middle: polarization 𝑄 component. Bottom: polarization 𝑈
+component.
+MNRAS 000, 1–58 (2022)
+
+QUJOTE119GHz combinedH2+H4(1deg)
+5
+mk
+20QUljOTEQ19GHzcombinedH2+H4(1deg)
+mkQUljOTEU19GHzcombinedH2+H4(1deg)
+mk8
+Rubiño-Martín et al.
+Figure 5. QUIJOTE MFI maps at 11 GHz smoothed to 1 degree resolution. Top: polarized intensity 𝑃 =
+√︁
+𝑄2 + 𝑈2. Middle: polarization angle. Bottom:
+Polarization angle at 11 GHz, rotated by 90◦ to indicate the direction of the Galactic magnetic field projected on the plane of the sky. The colours represent the
+polarized intensity signal. The "drapery" pattern was obtained with the healpy routine line_integral_convolution, and it is smoothed to 2◦ for display purposes.
+MNRAS 000, 1–58 (2022)
+
+QUUJOTEP11GHz(1deg)
+0
+mk
+1.3QUJOTEang11GHz(1deg)
+-90
+deg
+90MFI 11GHz - LICQUIJOTE MFI wide survey
+9
+Table 1. List of telescope elevations used for the wide survey observations with the QUIJOTE MFI instrument. The second column indicates the total observing
+time (𝑇observed) in hours dedicated to each elevation. Columns 3 to 6 show the total observing time for the actual subset of observations used for the final
+intensity (𝑇used,I) and polarization (𝑇used,P) maps. In the later case, different subsets of data are used for each particular horn. Observations are separated in
+periods (column 7), which correspond to specific epochs (column 8) and instrumental configurations (see text for details).
+Elevation (◦)
+𝑇observed (h)
+𝑇used,I (h)
+𝑇used,P,H2 (h)
+𝑇used,P,H3 (h)
+𝑇used,P,H4 (h)
+Period
+Range of Dates
+30
+121.9
+0.0
+0.0
+0.0
+0.0
+1
+06/2013–07/2013
+60
+986.5
+986.5
+0.0
+0.0
+0.0
+1
+05/2013–03/2014
+65
+665.2
+665.2
+0.0
+0.0
+0.0
+1
+05/2013–03/2014
+70
+394.9
+0.0
+0.0
+0.0
+0.0
+1
+06/2013–03/2014
+30
+829.4
+829.4
+829.4
+829.4
+0.0
+2
+08/2014–03/2015
+40
+489.3
+489.3
+489.3
+489.3
+0.0
+2
+08/2014–01/2015
+50
+564.7
+564.7
+564.7
+564.7
+0.0
+2
+08/2014–10/2015
+60
+91.7
+91.7
+91.7
+91.7
+0.0
+2
+06/2014–09/2014
+65
+128.9
+128.9
+128.9
+128.9
+0.0
+2
+08/2014–10/2014
+30
+200.1
+0.0
+0.0
+0.0
+0.0
+5
+08/2016–10/2016
+40
+324.6
+324.6
+0.0
+324.6
+324.6
+5
+08/2016–10/2016
+50
+488.6
+488.6
+0.0
+488.6
+488.6
+5
+08/2016–10/2016
+60
+198.4
+198.4
+0.0
+198.4
+198.4
+5
+08/2016–09/2016
+35
+1998.6
+1998.6
+1998.6
+1998.6
+1998.6
+6
+12/2017–06/2018
+50
+326.7
+326.7
+326.7
+326.7
+326.7
+6
+03/2017–04/2017
+60
+552.5
+552.5
+552.5
+552.5
+552.5
+6
+12/2016–02/2017
+65
+430.7
+430.7
+430.7
+430.7
+430.7
+6
+03/2017–04/2017
+70
+400.8
+400.8
+400.8
+400.8
+400.8
+6
+02/2017–04/2017
+TOTAL:
+9193.6
+8476.6
+5813.3
+6824.9
+4720.9
+Table 2. Definition of the four observing periods during which we carried out wide survey observations with the QUIJOTE MFI instrument. Last column
+indicates the instrument configuration and main changes. Configuration 1 corresponds to the original MFI design (Hoyland et al. 2012), while configuration 2
+corresponds to the installation of 90◦-hybrids (Pérez-de-Taoro et al. 2016). See text for details.
+Period
+From
+To
+Effective year
+Comments
+(dd/mm/yyyy)
+(dd/mm/yyyy)
+1
+12/11/2012
+10/04/2014
+2013.7
+Configuration 1 for all horns. No extended shielding.
+2
+11/04/2014
+30/11/2015
+2014.9
+Horn 1 in configuration 2. Extended shielding installed.
+5
+01/05/2016
+14/10/2016
+2016.7
+All horns in configuration 2. Horn 1 not operative.
+6
+15/10/2016
+01/11/2018
+2017.8
+All horns in configuration 2. Horn 1 not operative.
+ration (Hoyland et al. 2012) was used during periods 1 and 2 (see
+Table 2), but a new configuration was later implemented using 90◦-
+hybrids (Pérez-de-Taoro et al. 2016). In this second configuration,
+all MFI channels are formally correlated, but for historical reasons
+we maintain the notation of correlated and uncorrelated channels.
+The sum of pairs of channels provides two independent mea-
+surements of the intensity. For example, for the first MFI configu-
+ration, we have
+𝑉x + 𝑟u𝑉y = 𝑠x𝑔2𝐼
+(1)
+𝑉x+y + 𝑟c��x−y = 𝑠x+y𝑔2𝐼,
+(2)
+while the difference of the pairs of channels provides two measure-
+ments of the linear polarization
+𝑉x − 𝑟u𝑉y = 𝑠x𝑔2�
+𝑄 cos(4𝜃pm + 2𝛾p) − 𝑈 sin(4𝜃pm + 2𝛾p)
+�
+(3)
+𝑉x+y − 𝑟c𝑉x−y = 𝑠x+y𝑔2�
+𝑄 sin(4𝜃pm + 2𝛾p) + 𝑈 cos(4𝜃pm + 2𝛾p)
+�
+,
+(4)
+where 𝑉𝑖
+represents the output voltage for channels 𝑖
+∈
+{x, y, x + y, x − y}, 𝑠x and 𝑠x+y are the responsivities of those
+branches in the MFI instrument, 𝑔 represents the voltage gain of
+the two MFI Low Noise Amplifiers (here taken to be the same in
+the two LNAs for simplicity), 𝑟c and 𝑟u are the so-called r-factors
+which measure the possible gain and responsivity imbalance in the
+pair of channels, 𝜃pm is the position angle of the polar modulator,
+and 𝛾p is the parallactic angle (see details in Génova-Santos et al.
+2023). When the two channels in the pair have correlated noise,
+then the difference cancels significantly the 1/ 𝑓 component. In the
+MFI pipeline, maps for correlated and uncorrelated channels are
+produced separately, and combined afterwards. Due to their noise
+properties, in polarization we use only those pair of channels with
+common 1/ 𝑓 properties, i.e. the "correlated" channels during pe-
+riods 1 and 2, and both of them ("correlated" and "uncorrelated"
+channels) for periods 5 and 6.
+The MFI data sampling rate is 1 ms. For the wide survey, all
+time streams (hereafter Time-Ordered Data or TODs) are binned in
+40 ms samples. Note that this is different from the binning scheme
+of 60 ms used for raster scan observations in the past (e.g. Génova-
+Santos et al. 2017), due to the higher azimuth scan speed. The bin-
+ning process allows us to assign a variance 𝜎2
+𝑖 to each binned sam-
+MNRAS 000, 1–58 (2022)
+
+10
+Rubiño-Martín et al.
+Table 3. QUIJOTE-MFI basic peformance parameters. Values for 11 and 13 GHz correspond to horn 3 of MFI. Values for 17 and 19 GHz have been obtained
+as the weighted average of horns 2 and 4, using the relative weights described in Table 9.
+Parameter
+11 GHz
+13 GHz
+17 GHz
+19 GHz
+MFI horns contributing to these bands
+3
+3
+2,4
+2,4
+Centre frequency (nominal), 𝜈0 (GHz)
+11.1
+12.9
+16.8
+18.8
+Effective frequency for 𝛼 = −1, 𝜈𝑒 (𝛼 = −1) (GHz)
+10.98
+12.89
+16.85
+18.85
+Bandwidth (GHz)
+2.17
+2.20
+2.24
+2.34
+Beam FWHM (arcmin)
+55.38
+55.84
+38.95
+40.32
+Main beam solid angle, Ωmb (10−4sr)
+2.748
+2.781
+1.362
+1.428
+Beam ellipticity𝑎, 𝑒
+0.013
+0.040
+0.034
+0.035
+Antenna sensitivity, Γ (𝜇KCMB/Jy)
+961.9
+703.8
+847.0
+645.2
+White-noise level in timelines (𝜇KCMBs1/2)
+858
+697
+773
+866
+Knee frequency 𝑓k in polarization (mHz)
+254
+198
+223
+556
+1/ 𝑓 slope in polarization
+1.95
+1.86
+1.73
+1.34
+Overall calibration uncertainty I (%)
+5
+5
+5
+5
+Overall calibration uncertainty Q,U (%)
+5
+5
+6
+6
+𝑎 The ellipticity is defined here as 𝑒 = 1 − FWHMmin/FWHMmax.
+Table 4. Colour correction coefficients, 𝐶(𝛼, 𝜈0) = 𝑐0 + 𝑐1𝛼 + 𝑐2𝛼2.
+The colour corrected temperature is obtained as 𝐶(𝛼, 𝜈0)𝑇 , being 𝑇 the
+uncorrected one.
+Band
+𝜈0
+𝑐0
+𝑐1
+𝑐2
+11
+11.1
+0.981
+0.0125
+-0.0015
+13
+12.9
+1.001
+0.0018
+-0.0012
+17
+16.8
+1.007
+-0.0022
+-0.0007
+19
+18.8
+1.007
+-0.0020
+-0.0008
+ple 𝑖, which we used to define the associated weights (𝑤𝑖 = 1/𝜎2
+𝑖 ).
+When propagated through the entire pipeline, the resulting weight
+maps are used for the combination of maps from correlated and
+uncorrelated channels, and will be used also in the noise character-
+ization.
+Table 3 contains the summary of basic parameters (central fre-
+quencies, beams, solid angles) for all MFI horns, extracted from
+Génova-Santos et al. (2023). We also include the calibration uncer-
+tainties discussed in Sect. 5, and representative noise characteristics
+(knee frequencies and 1/ 𝑓 slopes) that we have obtained from this
+data. Table 4 also presents the colour corrections for these maps, de-
+rived from the associated bandpasses as explained in Génova-Santos
+et al. (2023). Colour corrections are presented here in terms of sec-
+ond order polynomials as a function of the spectral index 𝛼. For a
+sky emission having a flux density law 𝑆𝜈 ∝ 𝜈𝛼, the coefficients
+𝐶(𝛼, 𝜈0) provide the multiplicative correction factor to the mea-
+sured flux density for the MFI frequency map at nominal frequency
+𝜈0. These corrections are identical for intensity and polarization.
+Throughout the paper, we use the following notation to refer
+to specific MFI maps per horn and frequency. We will use three
+numbers, the first one refering to the horn number (i.e. 2, 3 or 4),
+and the other two indicating the nominal frequency (i.e. 11, 13, 17
+or 19). For example, the 19 GHz map for horn 4 will be cited either
+as 𝑚4,19, 𝑚419, or directly, 419 map. We recall that each map will
+be made, in principle, from the contribution of both the correlated
+and uncorrelated channels. In some case, we use the same notation
+to refer to channels. For example, the correlated channels of 419 are
+obtained from the 𝑉x+y and 𝑉x−y outputs of horn 4 at 19 GHz.
+In the following, we discuss specific additions to the MFI
+pipeline in the case of the wide survey. In particular, we discuss
+the gain model for wide survey data and the specific data flagging
+applied in ”nominal mode". After this, we present our approach
+to correct for Radio Frequency Interference (RFI) signals and at-
+mospheric contamination in the MFI wide survey data. For these
+corrections, the general philosophy adopted in our pipeline follows
+a two step approach. We first implement specific methods to de-
+tect and mitigate the effect of RFI and atmospheric signals both
+at the TOD (see Sect. 2.2.3 and 2.2.4) and at the map-level in the
+post-processing stage (Sect. 2.4). Then, a detailed assessment is
+made later of residual signals in the maps by a variety of techniques
+(Sect. 4). In practice, the values of uncertainties in calibration and
+other error budgets are increased appropriately if there is clear evi-
+dence of residual effects still being present in the maps (Sect. 5).
+2.2.1
+Gain model
+Gain calibration and the associated relative gain factors (𝑟c and 𝑟u)
+between pairs of channels are based on Cas A and Tau A observa-
+tions taken during each period. Relative gain variations with respect
+to the mean gain value 𝐺0 during the full period are traced using
+the signal of a thermally stabilized calibration diode, located at the
+centre of the secondary mirror. Every 30 s, the diode injects a signal
+during 1 s, which is used to measure the relative gain of each chan-
+nel, 𝛿𝐺(𝑡) ≡ 𝐺(𝑡) −𝐺0 (see Génova-Santos et al. 2023, for details).
+Nominal mode observations used for the wide survey usually have
+a duration of one day for each polarimeter position. Specifically
+for this nominal mode data, a smooth (interpolated) gain model is
+obtained by applying a top-hat smoothing kernel on the individual
+gain measurements. The width of this kernel is 30 minutes for low
+frequency channels, and 120 minutes for high frequency ones, due
+to the different signal-to-noise ratio of the diode signal in the differ-
+ent channels. We have checked that the typical MFI gain variations
+occur on timescales longer than those. These interpolated models
+are used to correct the instrument gain as
+𝐺(𝑡) = 𝐺0
+�
+1 + 𝛿𝐺(𝑡)
+𝐺0
+�
+.
+(5)
+Once these interpolated gain models are generated for the entire
+survey, they are inspected in order to find residual features (peaks
+or jumps) in the models. These features are introduced in flagging
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+11
+tables which are later applied during the generation of the calibrated
+TOD.
+2.2.2
+Data flagging
+Génova-Santos et al. (2023) describes the basic data flagging that
+is applied by default to all MFI observations, including flags due
+to voltage ranges, house-keeping parameters, emission of the Sun
+and Moon (using a 10◦ exclusion radius), and also the emission of
+geo-stationary satellites. In particular, this last flagging produces
+the empty strip around declination zero degrees that is seen in the
+11 and 13 GHz maps (Figures 1 and 2), and also the noise increase
+in the same region in the 17 and 19 GHz maps (Figures 3 and 4),
+due to the lower number of independent crossings in the area.
+For the wide survey, a specific flagging based on the root mean
+square (rms) of the data in each scan has been implemented as
+follows. A first version of the wide survey maps is produced with
+the default pipeline. From here, and separately for each period, we
+compute the rms of the data minus the reprojected version of that
+map onto the TOD, in scales of 30 s. This time value corresponds to
+the length of one azimuth scan at the default scanning speed, and to
+the length of half azimuth scan for the scanning speed used in part
+of period 1. Histograms with the distribution of these rms values
+are built for each channel and period, and are used to flag those
+scans with extreme rms values (either above 1.7 times the median
+rms value in the entire period, or below 0.5 times that median rms).
+The fraction of excluded data using this procedure depends on the
+channel, but typically is of the order of 10–20 per cent. Once this
+flagging is applied, no obvious residual spikes or rings are visible in
+the reconstructed maps. Finally, for the final wide survey maps we
+also exclude Jupiter, Venus and Mars, using a 2◦ exclusion radius
+directly in the TOD. Appendix A contains detailed tables with the
+percentage of used (and flagged) data for each MFI channel in every
+observing period. Those fractions of used data apply to the total
+number of used hours in each case, which were listed in Table 1. On
+average we are using 61 % of the data after applying all the different
+flags. Out of the flagged 39 %, most of it (approximately three
+quarters) is excluded in the specific post-processing stage described
+in this subsection. The percentage of used data was slightly lower
+in period 2 (52 %), and higher in period 6 (68 %).
+2.2.3
+RFI correction
+Specifically for the wide survey data, residual random spikes as well
+as possible RFI signals from satellites not identified in our standard
+pipeline are flagged using a dedicated matched-filter code that is
+applied to the one-dimensional TOD. The only assumption is that
+the object to be detected is unresolved, and thus should match the
+beam profile. The code3 excludes the location of the known bright
+radio-sources, which are also easily detected in the TOD.
+Residual RFI signals appear at fixed azimuth (AZ) locations. In
+the case of QUIJOTE MFI, most of these signals are due to the radio
+emission of geo-stationary satellites entering through the beam far
+sidelobes. These signals were particularly visible in period 1 and at
+low frequencies (horns 1 and 3), until the installation of the extended
+shielding of the first QUIJOTE telescope was completed. All other
+periods are much less affected, due to the significant suppression
+of the far sidelobes. Because of this reason, period 1 was used for
+the intensity maps only, and not for polarization. In order to remove
+3 https://gitlab.com/HerranzD/quijote-satdet
+these RFI signals, we generate spatial templates in the azimuth
+direction, by obtaining stacks of the TOD signal as a function of
+AZ, 𝑓 (AZ). These templates are computed for each period and each
+elevation separately, and thus rely on the assumption that the RFI
+signal is stable in time during the whole period. The templates are
+generated both for the sum and difference of MFI channels, and
+thus, they are applied to the intensity and the polarization TOD.
+Finally, a smoothed version of these templates (in scales of 10◦)
+is subtracted from the TOD. Figure 6 shows two examples of the
+global RFI patterns removed using this procedure. These figures are
+obtained as the difference between the end-to-end MFI maps with
+and without applying the RFI correction at the TOD level. We also
+note that once the final maps are produced, any residual RFI signals
+are effectively corrected in the post-postprocessing stage, using a
+function of the declination as described in Sect. 2.4.
+Some remaining RFI features and glitches are removed after
+a careful inspection of the final maps. For this purpose, separate
+maps for each elevation and period are produced. Once a particular
+RFI feature is identified in these maps, the corresponding location is
+introduced in specific flagging tables for each period and elevation,
+which are later applied to the calibrated TOD.
+2.2.4
+Atmospheric correction
+Although the observations are done at (nominal) constant elevation,
+there are still some residual variations due to changes in the atmo-
+spheric contribution along different directions. These variations are
+seen in the data as correlated patterns repeating in azimuth on very
+large angular scales, and with the amplitude increasing strongly with
+frequency, as expected for MFI frequencies due to the proximity of
+the 22 GHz atmospheric water line (see e.g. Paine 2019). It also
+evolves and changes on the scale of several hours, which is expected
+due to varying integrated water vapour content along lines of sight
+as weather systems blow over the site. It is possible to try to remove
+these effects especially at the more troublesome higher frequencies
+by a Principal Component Analysis (PCA) decomposition to look
+for these correlated signals.
+To model this atmospheric component in the MFI intensity
+data, only broad scale features are removed by using baselines up
+to only 5 harmonics over the azimuth scans. A mask is used to
+avoid bright emission from the Galactic plane and strong point
+sources. The baseline atmospheric patterns are generated over an
+hour, as a compromise between good signal to noise and the time
+evolution of the atmosphere. The PCA decomposition method used
+is implemented in Python, using the sklearn module (Pedregosa
+et al. 2011) on all the channels.
+The first most significant component found is one that in-
+creases strongly with frequency, with the spectrum expected for
+water vapour. A histogram of the ratios between 17 and 19 GHz, the
+two most strongly affected frequencies, shows a clear broad peak
+at 0.42 near the values expected from atmospheric models for the
+Teide Observatory of 0.49 (see e.g. Paine 2019, and typical PWV
+conditions of 3–4 mm), although this sits on a smaller but much
+broader distribution. Points outside the range 0.3 to 0.6 appear to
+be for dryer conditions, with the implication that the water vapour
+signal is too weak to be reliably recovered. It was decided to use
+this range ratio of 17 to 19 GHz signal as an indicator of a usable
+atmospheric signal that can be removed. The removal is done by
+subtracting the PCA template with the coefficient found for each
+frequency channel at the TOD level.
+Maps of this atmospheric emission can be produced running
+the full pipeline with and without this atmospheric correction, and
+MNRAS 000, 1–58 (2022)
+
+12
+Rubiño-Martín et al.
+Figure 6. RFI patterns removed from the maps of the QUIJOTE MFI wide survey. Top row corresponds to the RFI emission at 11 GHz (horn 3, labelled as
+"311"), while the bottom row corresponds to 17 GHz (horn 4, labelled as "417"). From left to right, we show the residuals for intensity, Stokes Q and Stokes U
+parameters. The colour scale is the same in all six panels, corresponding to the temperature range ±0.2 mK. For visualization purposes, all maps are smoothed
+to one degree resolution.
+then taking the difference of the two resulting maps. The atmo-
+spheric emission maps for horns 3 and 4 are shown in Figure 7.
+The map for horn 2 is similar to the one for horn 4, so it is omitted
+for clarity. As expected, this atmospheric contribution is more rel-
+evant at higher MFI frequencies, and affects large angular scales.
+As shown below (see Sect. 2.5), when doing a spherical harmonic
+expansion of the maps, this correction is only relevant in the in-
+tensity maps at multipoles ℓ ≲ 15 for 11 GHz, and ℓ ≲ 25 for
+19 GHz. No atmospheric correction is needed in polarization for
+the MFI wide survey maps. When a similar procedure is applied
+to the polarization data, the results are consistent with essentially
+unpolarized atmospheric emission.
+2.3
+Map-making
+The QUIJOTE MFI wide survey maps are produced using the PI-
+CASSO code (Guidi et al. 2021), a destriping algorithm based on
+the MADAM approach (Keihänen et al. 2005, 2010) but specifi-
+cally implemented and optimised for QUIJOTE MFI. The destriping
+technique corrects for a correlated noise component by modelling
+the 1/ 𝑓 drifts in the TOD with a set of consecutive offsets with a
+given time length 𝑡b, the so-called baselines. The PICASSO code
+has been tested extensively using realistic simulations matching the
+actual observations of the MFI wide survey and with realistic noise
+properties (Guidi et al. 2021). In these conditions, the reconstructed
+maps preserve all angular scales with high fidelity, and in particular,
+we expect a signal error better than 0.001 per cent at 20 < ℓ < 200.
+Those realistic simulations were also used to set the reference
+parameters adopted for the production of the final MFI wide survey
+maps. In particular, we use a baseline length of 𝑡b = 2.5 s for the
+entire survey. Maps are generated using the HEALPix pixelization
+scheme with 𝑁side = 512. The specific priors for the 1/ 𝑓 noise
+properties (knee frequency 𝑓𝑘, slope 𝛾, and cutoff frequency 𝑓cut)
+are shown in Table 5, both for the intensity and polarization maps.
+In the later case, the parameters are different depending on the
+noise levels of the corresponding pair of channels (i.e. if they are
+correlated or uncorrelated channels). As discussed in Guidi et al.
+Table 5. Map-making parameters and related information. We consider three
+different cases of use with the PICASSO code: intensity maps, polarization
+maps with correlated channels, and polarization maps with uncorrelated
+channels. For each case, we quote the prior values for the knee frequency
+𝑓k, the slope of the 1/ 𝑓 noise component 𝛾, and the low cut-off frequency
+𝑓cut, as well as the 𝑁side value of the HEALPix map and the baseline length
+𝑡b (in seconds). See text for details.
+Case
+𝑓k
+𝛾
+𝑓cut
+𝑁side
+𝑡b
+[Hz]
+[Hz]
+[s]
+I
+40.0
+1.5
+0.033
+512
+2.5
+Q,U corr
+0.3
+1.8
+0.033
+512
+2.5
+Q,U uncorr
+40.0
+1.5
+0.033
+512
+2.5
+(2021), those priors are assumed to be stationary parameters for the
+whole survey.
+2.4
+Post-processing of MFI wide survey maps
+2.4.1
+Combination of maps
+For each horn and frequency sub-band, maps for the correlated
+and uncorrelated pairs are produced running the PICASSO code
+separately for each one of them. These maps are combined at this
+post-processing stage, using the weight maps which are also pro-
+duced by the map-making code as the propagation of the individual
+weights for each sample in the binned TOD. The combination of
+correlated (𝑥c) and uncorrelated (𝑥u) maps is done with the usual
+formula for the weighted arithmetic mean:
+𝑚 = 𝑤c𝑥c + 𝑤u𝑥u
+𝑤c + 𝑤u
+.
+(6)
+Given that both correlated and uncorrelated channels share the same
+amplifier, we expect a high level of correlation between the two
+intensity measurements. As shown below in Sect. 4.3.4, this cor-
+relation is indeed of the order of 90–95 per cent for the intensity
+channels, and consistent with zero for the polarization ones. Al-
+though in principle it is possible to construct a minimum variance
+MNRAS 000, 1–58 (2022)
+
+RFlmap (311,I)
+-0.2 
+mK
+0.2RFl map (311,Q)
+-0.2
+mk
+0.2RF map (311,U)
+-0.2
+mk
+0.2RFl map (417,I)
+-0.2
+mK
+0.2RFl map (417,Q)
+-0.2
+mk
+0.2RFl map (417,U)
+-0.2
+mk
+0.2QUIJOTE MFI wide survey
+13
+Figure 7. Atmospheric pattern removed from the intensity maps of the
+QUIJOTE MFI wide survey. From top to bottom, we have the atmospheric
+emission at 11 GHz (horn 3), 13 GHz (horn 3), 17 GHz (horn 4) and 19 GHz
+(horn 4). The colour scale is the same in the four panels (±3 mK), in order to
+visualise the increasing contribution of the atmospheric emission at higher
+MFI frequencies. For visualization purposes, all maps are smoothed to one
+degree resolution.
+estimator accounting for these correlations in the intensity pairs, we
+still use equation 6 for the combination of the intensity (correlated
+and uncorrelated) maps, in order to have a more robust estimate of
+the combination (see e.g. Schmelling 1995).
+From equation 6, we can derive the expression for the weight
+map of the linear combination as
+𝑤 =
+(𝑤c + 𝑤u)2
+𝑤c + 𝑤u + 2𝜌√𝑤c𝑤u
+,
+(7)
+where 𝜌 stands for the correlation fraction between correlated and
+uncorrelated channels.
+The map-making code also produces an estimate of the co-
+variance matrix in polarization, 𝑐𝑜𝑣(𝑄,𝑈), as well as the condition
+number (𝑟cond) map (see equations 44 and 45 in Guidi et al. 2021).
+Before forming the combination of the polarization maps in the
+wide survey, those pixels with 𝑟cond > 3 are excluded. In practice,
+this only affects a small number of pixels close to the boundary of
+the satellite strip, as well as to the north celestial pole. In particular,
+for the 419 map (i.e. horn 4 at 19 GHz) the number of affected pixels
+is slightly larger in those areas. Appendix C contains the 𝑟cond maps
+for all the MFI wide survey maps, together with other relevant maps,
+as discussed in Sect. 3. Once the combination of the correlated and
+uncorrelated maps is carried out in polarization, the corresponding
+weight maps (𝑤𝑄, 𝑤𝑈) and covariance matrices 𝑐𝑜𝑣(𝑄,𝑈) are also
+derived. Appendix C also presents images of the 𝑐𝑜𝑣(𝑄,𝑈) maps
+for all horns and frequencies. These maps show that, as expected,
+the normalized covariance 𝑐𝑜𝑣(𝑄,𝑈)/(𝜎𝑄𝜎𝑈) is very small (well
+below 0.01 %), so effectively each pair of 𝑄 and 𝑈 maps are almost
+independent.
+2.4.2
+Residual interference: the FDEC filtering
+After the map-making step, the resulting maps still present some
+residual RFI and large-scale patterns, which are corrected during
+this post-processing stage. As described in Sect. 2.2.3, residual RFI
+signals appear at fixed azimuth locations, so during the map-making
+process these features are projected onto the maps in stripes of con-
+stant declination. This residual RFI is removed using a function of
+the declination, 𝑓 (𝛿) (hereafter FDEC4), which is extracted directly
+from the maps as the median of all pixels with the same declination.
+This template function is built using a |𝑏| < 10◦ mask to exclude the
+Galactic emission, and specific masks in intensity and polarization
+for each frequency channel excluding the 10 per cent of the brightest
+pixels. The procedure is applied both in intensity and polarization.
+In polarization, the maps are first rotated to local (equatorial) coor-
+dinates in order to extract the correction function. In this way, the
+RFI contamination from static sources in local coordinates appears
+as a constant signal in a given declination band.
+Figure 8 shows the correction functions for intensity and po-
+larization for all MFI maps based on correlated channels. Similar
+curves are obtained for uncorrelated channels. Note that in this fig-
+ure, the panel for Stokes Q parameter corresponds to equatorial
+coordinates. As expected for RFI signals, these correction functions
+are larger in the vicinity of the geo-stationary strip (around declina-
+tion zero) and at low declinations (corresponding to low elevation
+values of the telescope, where the RFI is expected to be larger). We
+also note that they are also larger in intensity than in polarization.
+Once these correction functions 𝑓 (𝛿) are derived, they are re-
+projected onto a map in order to produce a RFI template. These
+4 https://github.com/jarubinomartin/sancho.git
+MNRAS 000, 1–58 (2022)
+
+Atmosphere (311,D)
+mk
+-3
+3Atmosphere (313,1)
+mk
+-3
+3Atmosphere (417,)
+mK
+-3
+3Atmosphere (419,D)
+-3
+mk
+314
+Rubiño-Martín et al.
+Figure 8. Examples of 𝑓 ( 𝛿) correction functions (FDEC) to remove resid-
+ual RFI in the MFI maps. Top: Stokes I FDEC for correlated channels.
+Bottom: Stokes Q parameter in equatorial (RADEC) coordinates for corre-
+lated channels.
+templates are subtracted from the data before carrying out the com-
+bination of correlated and uncorrelated maps. Figure 9 illustrates
+the final FDEC correction applied to the maps of horn 3 at 11 GHz,
+after combining the correlated and uncorrelated maps in intensity.
+2.4.3
+Monopole and dipole removal
+Finally, a monopole and a dipole component are subtracted from the
+correlated and uncorrelated maps before their combination, using
+the remove_dipole routine of HEALPix with a Galactic mask
+excluding the region |𝑏| < 10◦. The removed dipole is consistent
+with the expected CMB dipole, as discussed in Sect. 5.3.2.
+2.5
+Effective transfer function
+The PICASSO map-making code essentially preserves all angular
+scales in the MFI wide survey maps. The expected signal error is
+better than 0.001 per cent in the multipole range 20 < ℓ < 200 both
+for intensity and polarization maps, and stays well within one per
+cent down to ℓ = 10 (Guidi et al. 2021, and see also Fig. 10). How-
+ever, some of the specific procedures applied in the MFI pipeline to
+correct for RFI signals and atmospheric contributions might have
+an impact on the effective transfer function of the wide survey. In
+particular, we should consider the impact of the RFI (Sect. 2.2.3)
+Figure 9. Example of the effective correction map based on a function
+declination (FDEC) for the 311 map (horn 3 at 11 GHz). Top: Stokes I, with
+a colour scale in the range ±2 mK. Middle and bottom: Stokes Q and U
+parameters, with a colour scale in the range ±1 mK.
+and atmospheric (Sect. 2.2.4) corrections at the TOD level, and the
+RFI correction at the post-processing stage using a function of the
+declination FDEC (see previous subsection). In terms of their am-
+plitudes at the map level, the largest correction corresponds to the
+third case (subtracting a function of declination), so we discuss the
+transfer function of this case in detail.
+It is important to note that, by construction, after applying this
+FDEC correction, the zero mode at constant declination will be
+missing from the maps. To characterize its impact on the effective
+transfer function of the wide survey, we follow the methodology
+described in Sect. 6.3 of Guidi et al. (2021). Here, we use simulations
+in the ideal case including CMB and foregrounds, but without a
+noise component. The transfer function is then computed in terms of
+the power spectra of the map with residuals 𝐶res
+ℓ
+(i.e. reconstructed
+map minus input sky) and that of the reconstructed map 𝐶map
+ℓ
+, both
+MNRAS 000, 1–58 (2022)
+
+Stokes I, corr
+217c
+219c
+311c
+313c
+417c
+419c
+()[mk]
+-2
+-20
+0
+20
+40
+60
+80
+Declination  [deg]Stokes
+corr
+10
+217c
+219c
+311c
+313c
+0.5
+417c
+419c
+[mk]
+0.0
+-0.5
+-20
+0
+20
+40
+60
+80
+Declination  [deg]FDEC (311, I)
+-2.0
+2.0 mKFDEC (311, Q)
+-1.0
+1.0 mKFDEC (311, U)
+-1.0
+1.0 mKQUIJOTE MFI wide survey
+15
+Figure 10. Transfer function (TF) of the QUIJOTE MFI wide survey map
+at 11 GHz, after accounting for the post-processing stage of a subtraction of
+a function of the declination (FDEC). The TF for TT is marked with circles
+connected by red solid lines; the EE case with triangles and red dashed lines,
+and the BB with diamonds and red dotted lines. As a reference, in green we
+also include the TF of the PICASSO map-making code (Guidi et al. 2021).
+computed within the same mask, using this expression:
+𝑓ℓ =
+1
+1 − 𝐶res
+ℓ /𝐶map
+ℓ
+.
+(8)
+Figure 10 presents the result obtained for the 311 case. As expected,
+we find that the FDEC correction is affecting low multipoles (ℓ ≲
+15). The reconstruction of the sky signal is better than one per cent
+down to ℓ ≈ 10 in intensity. In polarization, the correction stays
+within one per cent down to ℓ ≈ 30, being at ℓ = 10 of the order of
+20 % for BB, and 5 % for EE. Because of this reason, and although
+we are able to reconstruct the sky signal to lower multipoles, as a
+conservative approach the power spectra analyses in this paper will
+be restricted to ℓ ≥ 30, so no transfer function correction will be
+needed. Appendix B contains a more detailed discussion on how
+a given map is affected by the FDEC filtering. The impact of the
+RFI and atmospheric corrections at the TOD level is discussed in
+detail in Sect. 4.4, although we anticipate that their impact is lower
+than the 𝑓 (𝛿) discussed here (except maybe for 19 GHz, where
+the atmospheric contribution becomes comparable to the FDEC
+correction).
+2.6
+Recalibration of the wide survey maps using Tau A
+Once the MFI wide survey maps are produced using the pipeline
+described above, we re-evaluate three aspects of the calibration
+using Tau A: i) the global calibration scale in intensity, ii) the
+polarization angle calibration, and iii) the polarization efficiency.
+We discuss them in detail here.
+2.6.1
+Global recalibration in intensity
+Tau A and Cas A are the two main primary calibrators of QUI-
+JOTE MFI (Génova-Santos et al. 2023). Daily observations of these
+sources in raster scan mode are used to obtain the overall gain scale
+in intensity for each MFI channel in every observing period. How-
+ever, as daily calibrator observations might suffer from 1/ 𝑓 noise
+and other uncertainties, we recalibrate the MFI wide survey maps
+in the post-processing stage. For this recalibration, we use Tau A as
+the reference source, because it is located on a cleaner background
+than Cas A.
+For this, we first generate wide survey maps for each individual
+period (four maps in total for each horn and frequency). These four
+maps per period are degraded to one degree angular resolution,
+and then we apply beam ��tting photometry (hereafter BF1d) on
+Tau A. The derived flux densities are compared, accounting for
+colour corrections, with a spectral emission model that we have
+specifically obtained for Tau A, using WMAP and Planck data
+together with some ancillary measurements, and applying the same
+BF1d methodology. The new model will be presented and discussed
+in detail in a separate paper (Génova-Santos & Rubiño-Martín, in
+preparation), and builds on that presented in Weiland et al. (2011),
+but including several improvements: i) improved treatment of the
+colour-corrections and beam effects on WMAP data, ii) inclusion
+of Planck data, iii) improved variability model. The adopted Tau A
+model for the recalibration of MFI maps has the shape
+𝑆𝜈(Tau A) = 358.3
+�
+𝜈
+22.8 GHz
+�−0.297
+Jy.
+(9)
+This model is evaluated at epoch 2016.3, which corresponds to
+the effective central epoch of the wide survey, and we use a secular
+decrease of −0.218 % yr−1 (Weiland et al. 2011). From this compar-
+ison, we derive global recalibration factors for each MFI frequency
+map and for each individual period, accounting for the secular de-
+crease of Tau A and the effective epochs in each period (see values
+in Column 4 of Table 2). The mean value of these recalibration fac-
+tors results in an overall 4 per cent recalibration of the wide survey
+maps. The accuracy of the MFI wide survey intensity calibration is
+discussed in Sect. 5.2.
+2.6.2
+Polar angle recalibration
+The reference angle for each MFI polarimeter (i.e. the reference for
+𝜃pm in equations 3 and 4) changes across the spectral band, and
+thus from band to band. For this reason, the reference angle for
+each frequency map is calibrated separately, despite of the fact that
+the two frequency bands of the same horn share the same polar
+modulator. This procedure is based on daily Tau A observations,
+and it is described in Génova-Santos et al. (2023). In particular, the
+adopted model for the Tau A angle in Galactic coordinates is given
+by
+𝛾Tau A = 𝛾0 + 𝑅𝑀𝜆2,
+(10)
+where 𝑅𝑀 = −1406 ± 12 deg m−2 and 𝛾0 = −88.31◦ ± 0.25◦. Our
+daily calibration provides a reference polar angle for Tau A with a
+statistical error of approximately 1◦ within a period. But similarly
+to the intensity calibration, daily observations of Tau A might suffer
+from 1/ 𝑓 noise or other effects, so the polar angles of the final wide
+survey maps are recalibrated in each period with Tau A again. As
+for the global recalibration in intensity, we also use BF1d in Tau A
+to extract the fluxes in Stokes Q and U parameters in the maps per
+period. From there, recalibration offsets in the reference angles are
+computed for each channel and each period, and applied in order to
+generate the final maps. The accuracy of the angle calibration in the
+MFI wide survey is discussed in Sect. 5.5.
+MNRAS 000, 1–58 (2022)
+
+Transfer function. 11 GHz
+1.3
+EE
+ = 1/(1 - C, res/Ce,map)
+BB
+1.2
+FDEC TT
+FDEC EE
+FDEC BB
+1.1
+1.0
+101
+10216
+Rubiño-Martín et al.
+Table 6. Polarization efficiency for horns 2, 3 and 4 in period 6. Error bars
+for all measurements are 2 per cent. See text for details.
+Channel
+𝜌corr
+𝜌uncorr
+217
+0.84
+0.98
+219
+0.86
+0.96
+311
+0.89
+0.98
+313
+0.83
+0.97
+417
+1.00
+0.93
+419
+0.99
+0.91
+Table 7. Change in the polarization efficiency for horns 2, 3 and 4 in period
+6 due to errors in the 𝑟-factor. See text for details.
+Channel
+Horn 2
+Horn 3
+Horn 4
+Low freq, corr
+−0.075
+0.021
+−0.006
+High freq, corr
+−0.113
+0.029
+0.016
+Low freq, uncorr
+0.028
+0.004
+0.005
+High freq, uncorr
+−0.020
+−0.002
+0.011
+2.6.3
+Polar efficiency
+Detailed measurements of the polar efficiency of the MFI polarime-
+ters in horns 2, 3 and 4 were obtained in period 6, once the MFI
+observations concluded. The description of the instrumental setup
+and the final measurements are presented in Génova-Santos et al.
+(2023), and summarized in Table 6.
+In order to transfer this polar efficiency information to the other
+observing periods where we do not have laboratory measurements,
+we use again BF1d photometry on Tau A, using the MFI wide
+survey maps per period. The polar efficiency in each period 𝑝 is
+transferred from period 6 according to the relative value of the Tau
+A polarized intensity 𝑃TauA(𝑝) in that period and in period 6, i.e.
+using the ratio 𝑃TauA(𝑝)/𝑃TauA(6). On average, this photometry
+method introduces errors of approximately 1 % for horn 3, and 2 %
+for horns 2 and 4.
+Finally, we also account for possible errors in the determination
+of the 𝑟 factors in equations 3 and 4, using wide survey data as
+follows. As shown in Appendix D, an error 𝜖 in the determination
+of the 𝑟 factors translates into a modification of the polar efficiency,
+and the appearance of a small leakage term in the TOD polarization
+timeline which is proportional to the intensity map. We use the
+PICASSO map-making code to fit for an intensity-to-polarization
+leakage global component in period 6 data, in a two step process.
+First, we solve for the intensity map 𝐼 for each case (i.e. horn,
+frequency and channel), and then we use it to fit for an additional
+term 𝛼𝐼 when solving for the polarization map in equations 3 and
+4. These values are used to correct for the polar efficiency of each
+channel in period 6, using the equations derived in Appendix D.
+Table 7 shows the effective correction terms 𝛼 ≡ 𝜖/(2𝑟). We can
+see that in the case of horn 3, this correction introduces a change
+of 2–3 per cent in correlated channels, and below 1 per cent for
+uncorrelated channels. Horn 4 is almost unaffected, while the largest
+correction factor appears for the correlated channels in horn 2. The
+accuracy of the polar efficiency calibration in the MFI wide survey
+is discussed again in Sect. 5.1.
+3
+MFI WIDE SURVEY MAPS: INTENSITY AND
+POLARIZATION
+Following the methodology described in the previous section, we
+produced intensity and polarization maps for each MFI horn and fre-
+quency. Images of these individual maps (per horn and frequency)
+are shown in Appendix C, at their original resolution (i.e. the angular
+resolution listed in Table 3). The resulting maps cover a sky fraction
+of 𝑓sky = 0.75, 0.71 and 0.73 (equivalent to sky areas of 30 900,
+29 300 and 30 100 deg2) for horns 2, 3 and 4, respectively. All MFI
+maps are produced in CMB thermodynamic units (mKCMB). For
+simplicity, throughout this paper we drop the subindex CMB and
+use the notation mK. Nevertheless, we recall that the correction
+to Rayleigh-Jeans units is very small at MFI frequencies (at most
+1 per cent at 19 GHz). Smoothed maps at 1◦ resolution are gen-
+erated by convolving those original maps with the corresponding
+transfer function 𝑇ℓ ≡ 𝑊1 deg
+ℓ
+/𝑊MFI
+ℓ
+, which converts the spherical
+harmonic window function for each horn (𝑊MFI
+ℓ
+) into a gaussian
+beam with FWHM= 1◦ (𝑊1 deg
+ℓ
+). All maps are displayed in Galactic
+coordinates. We recall that QUIJOTE-MFI Stokes Q and U param-
+eter maps and data follow the COSMO convention for polarization
+angles from HEALPix. Grey regions correspond to the sky areas
+not observed by QUIJOTE MFI: the southern sky (approximately
+below 𝛿 = −34◦); a small area around the North Celestial Pole
+(NCP) for some of the horns (depending on their location in the
+MFI focal plane); and the band of geostationary satellites close to
+declination zero degrees, which mainly emit at 11 and 13 GHz.
+Appendix C also contains the associated number of hits (𝑁hit)
+and weight maps. Both set of maps are outputs of the PICASSO
+map-making code. The hit maps (𝑁hit) correspond to the total num-
+ber of 40 ms samples in each HEALPix pixel of 𝑁side = 512 reso-
+lution. The weight maps correspond to the propagation through the
+map-making process of the errors (weights) associated with each
+individual 40 ms sample. Both sets of maps clearly show the im-
+print of the scanning strategy of the QUIJOTE MFI wide survey.
+The ring structures around the North Celestial Pole correspond to
+the boundaries of the different elevations considered in the survey.
+Due to projection effects, the number of hits is significantly larger
+in those borders (and thus, the noise levels are smaller). In the low
+declination band of the maps (below the masked area due to geosta-
+tionary satellites), the number of hits is significantly lower due to
+the combined effect of a lower number of observations at these low
+elevations (mainly 30◦, 35◦ and 40◦), and projection effects. We
+recall that the number of hits in the intensity maps is larger than in
+polarization due to the fact that some intensity data are not used in
+polarization (period 1 data are not used for any polarization maps;
+data from period 2 are not used in polarization for horn 4; and data
+from period 5 are not used in polarization for horn 2; see summary
+information in Table 8).
+The final QUIJOTE MFI wide survey maps at 11 and 13 GHz
+presented in Fig. 1 and 2 are directly the maps from horn 3, smoothed
+to 1◦ resolution. The final maps at 17 and 19 GHz in Fig. 3 and 4
+have been produced as a linear combination of those for horns 2 and
+4. For simplicity in the computation of effective beams, frequencies
+and colour corrections, we adopted constant weights for this com-
+bination. We have checked that the resulting maps have comparable
+noise levels to the maps obtained using spatially-varying weights
+based on the actual weight maps for each individual map in the
+combination. Thus, the combined maps at 17 GHz can be obtained
+as
+𝑚17 = 𝑤2,17𝑚2,17 + 𝑤4,17𝑚4,17
+(11)
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+17
+Table 8. List of periods contributing to each final MFI map per horn.
+Column 1 indicates the map per horn with the usual notation: the first
+number indicates the horn/pixel (column 2), and second and third numbers
+indicate the nominal frequency (column 3). Column 4 shows the list of
+periods contributing to the map based on the correlated channels 𝑉x+y and
+𝑉x−y. Column 5 shows the list of periods used for the map based on the
+uncorrelated channels 𝑉x and 𝑉y. The final map is the combination of both
+correlated and uncorrelated maps.
+Map
+Horn/Pixel
+Nominal Freq. (GHz)
+Corr
+Uncorr
+Intensity
+311
+3
+11
+1,2,5,6
+1,2,5,6
+313
+3
+13
+1,2,5,6
+1,2,5,6
+217
+2
+17
+1,2,5,6
+1,2,5,6
+219
+2
+19
+1,2,5,6
+1,2,5,6
+417
+4
+17
+1,2,5,6
+1,2,5,6
+419
+4
+19
+1,2,5,6
+1,2,5,6
+Polarization
+311
+3
+11
+2,5,6
+5,6
+313
+3
+13
+2,5,6
+5,6
+217
+2
+17
+2,6
+6
+219
+2
+19
+2,6
+6
+417
+4
+17
+5,6
+5,6
+419
+4
+19
+5,6
+5,6
+Table 9. Constant weight factors used to produce the combined 17 and
+19 GHz MFI wide survey maps. We include only the weight factors for
+horn 4, as those for horn 2 can be obtained as 𝑤2,17 = 1 − 𝑤4,17 and
+𝑤2,19 = 1 − 𝑤4,19.
+I
+Q
+U
+𝑤4,17
+0.362
+0.732
+0.732
+𝑤4,19
+0.419
+0.788
+0.788
+for 𝑚 = 𝐼, 𝑄,𝑈, and similarly for 19 GHz, we have
+𝑚19 = 𝑤2,19𝑚2,19 + 𝑤4,19𝑚4,19.
+(12)
+Table 9 contains the final weights used for this linear combination.
+These values have been derived from the white noise level of the
+individual frequency maps for each horn, using optimal (inverse
+variance) weights. We note that horn 2 dominates the linear com-
+bination in intensity, while horn 4 contributes with a higher weight
+to the polarization maps. The actual values of noise levels for these
+maps are discussed in Sect. 4.3.
+The final maps in polarization (Figs. 1–4) are dominated by the
+Galactic synchrotron emission (the spectral index of the observed
+signal is discussed below in Sect. 7 and 8). Large scale features such
+as the Fan region or the North Polar Spur are clearly seen in the four
+frequency maps. The MFI instrument is not optimized to measure
+the intensity signal, and thus the intensity maps present worse noise
+properties. In particular, the two highest frequency channels show
+clear large scale 1/ 𝑓 residuals, particularly at negative declinations,
+due to the fact that they are observed only with the lower elevations
+(higher air masses).
+3.1
+Analysis masks
+Figure 11 shows the footprint of the different analysis masks which
+are specific for the QUIJOTE wide survey. There are three distinct
+regions that are considered when building these masks:
+• Satellite band ("sat"). The masked region around declination
+zero is used to block the RFI contamination of geostationary satel-
+lites mainly affecting 11 and 13 GHz maps. In the MFI pipeline,
+the emission from each geostationary satellite is flagged at the TOD
+level using a mask of 5◦ radius around each satellite. Other satellites
+or RFI signals are flagged as described in Section 2. After this pro-
+cess, the resulting masked area (with zero number of hits) is located
+approximately between declinations −10◦ to −2◦ (note that geo-
+stationary satellites are seen at slightly negative declinations from
+the Teide Observatory). The proposed mask to remove the satellite
+band (−12◦ < 𝛿 < 6◦) is a conservative choice based on a close
+inspection of the final maps, extending the unobserved area by two
+degrees in the negative declination direction, and by eight degrees
+in the positive direction. This choice accounts for low-level RFI
+residuals in the intensity maps (some of the residual RFI signals
+corrected during the post-processing stage are located in that area),
+while keeping a relatively high number of hits per pixel.
+• North Celestial Pole ("NCP") region. Given the latitude of the
+Teide Observatory (28.30◦N) and the minimum elevation observed
+with QUIJOTE MFI (EL= 30◦), some of the maps present a small
+area of unobserved pixels around the NCP, depending on the loca-
+tion of the MFI horns in the focal plane. The maximum observed
+declination is approximately 86◦ for horn 3, and 87.5◦ for horn 2.
+Horn 4 covers up to 90◦ in declination. In any case, the pixels sur-
+rounding this NCP area are only accesible with the lowest elevation
+bands, which usually present the largest levels of atmospheric con-
+tamination in the intensity maps, particularly at 19 GHz. For this
+reason, for some of the analysis we mask the region above 𝛿 = 70◦,
+in order to keep a sky area that is observed practically by all the
+elevations considered in the survey.
+• Low (negative) declinations ("lowdec"). Similarly to the NCP
+area, this region is only observed when using low elevations (below
+40◦), and thus the corresponding intensity maps, specially at the
+two highest frequencies, are more affected by 1/ 𝑓 residuals from
+atmospheric emission (see Fig. 3 and 4, and also the individual maps
+for horns 2 and 4 in Appendix C). The proposed mask to exclude
+this area covers all declinations below 𝛿 = −12◦.
+All different combinations of those three masked regions produce
+the reference set of specific masks for the MFI wide survey used
+in this and all accompanying papers. In particular, unless other-
+wise stated, the default analysis mask used in most of the scientific
+analyses in this paper, and in particular, in all power spectrum
+computations, corresponds to the superposition of the three re-
+gions (sat+NCP+lowdec). This mask preserves a sky fraction of
+𝑓sky = 0.418, equivalent to approximately 17 200 deg2.
+4
+DATA VALIDATION
+In order to characterize the properties of the wide survey maps,
+we carry out a number of tests and studies in this section. Most
+of them rely on different types of null tests, which can be used to
+detect possible remaining systematic effects in the data, including
+residual RFI signals, calibration issues, changes in the operational
+or instrumental conditions, or even unknown effects.
+MNRAS 000, 1–58 (2022)
+
+18
+Rubiño-Martín et al.
+Figure 11. Footprint of the wide survey in Galactic coordinates, and pro-
+posed analysis masks. The background image corresponds to the 9-yr
+WMAP-K band polarized intensity map (Bennett et al. 2013). Light colours
+indicate the observed MFI wide survey regions. The band excluded due to
+satellite contamination corresponds to −12◦ < 𝛿 < 6◦. The default mask
+adopted for the analyses in this paper preserves the band 6◦ < 𝛿 < 70◦,
+which is marked as the brightest region in the image. This mask is labelled
+as sat+NCP+lowdec (see text for details).
+4.1
+Null tests
+A “null test” is defined as the difference between the maps produced
+from two independent sub-sets of files from the full data base, which
+are expected to give the same signal under the assumption of a
+perfect calibration and no systematic effects. Null tests have been
+shown to be a powerful mean to assess the contribution of residual
+systematic effects in CMB analyses (e.g. Planck Collaboration et al.
+2014c, 2016d). For the characterization of the QUIJOTE MFI wide
+survey data, we produced the following set of null tests:
+(i) Half mission. The full database is divided in two halves. The
+separation is done according to the calendar date inside each period
+and each elevation, producing maps labelled as “half1” and “half2”.
+In this way, both null test maps contain data from all periods, and
+have a similar sky coverage. This is the reference null test used to
+characterize the overall noise properties.
+(ii) Rings. The MFI wide survey maps are produced using the so-
+called nominal observing mode, in which the QUIJOTE telescope
+scans the sky using a circular scanning strategy with a continuous
+movement in azimuth direction while maintaining a constant ele-
+vation. Each azimuth scan is called a "ring". For this null test, the
+full database is divided in odd (“rings1”) and even (“rings2”) rings.
+With the nominal azimuth scan speed of 12 deg s−1, each ring is
+completed in 30 s, so this null test can be used to test for instrumen-
+tal variations in these short time scales. As the instrument gain is
+stable in time scales much longer than one minute, this null test is
+not expected to reflect gain variations, and will essentially contain
+white noise plus a 1/ 𝑓 -noise component in scales of 30 s.
+(iii) Daynight. In order to evaluate possible residual system-
+atic effects due to day-night variations of the system gain or cal-
+ibration factors, this null test is produced by dividing the full
+database into day observations (“daynight1”) and night observa-
+tions (“daynight2”). For simplicity, we define here “day” as all
+observations from 8 AM to 8 PM (UT).
+(iv) PWV. Using the information from GPS measurements at the
+Teide Observatory of the precipitable water vapour (PWV) content
+of the atmosphere during each individual observation5, we divide
+5 The GNSS antenna that provides these PWV measurements is located at
+the full data base in two sets of low (“pwv1”) and high (“pwv2”) pwv
+values. As in the case of the half mission null test, the separation is
+done inside each period and elevation, to guarantee that both splits
+contain a similar sky coverage. As a reference, the resulting median
+pwv in these two data splits is 2 mm and 5.2 mm, for "pwv1" and
+"pwv2", respectively.
+(v) Halfrings. This null test separates the data by dividing each
+ring in two halves. Data taken with telescope azimuth values 0◦ ≤
+𝐴𝑍 ≤ 180◦ correspond to "halfring1", while data with 𝐴𝑍 > 180◦
+are part of "halfring2". Although these maps are expected to be
+noisier than the other null tests due to 1/ 𝑓 contributions (note that
+in this case we are basically decreasing by a factor of two the number
+of independent crossings in each pixel when solving the conjugate
+gradient inside the map-making algorithm), they are still extremely
+useful to detect residual RFI signals arising from local structures,
+which usually appear at fixed AZ values. Moreover, these maps can
+be also used to test residual pointing errors.
+(vi) 𝑇BEM. As explained in Génova-Santos et al. (2023), the
+overall gain of the instrument is strongly correlated with the physical
+temperature in the electronic boxes containing the Back-End Module
+(BEM) of the MFI. As a further test to explore possible residual
+variations after our gain model correction, we use the values of one
+of the temperature sensors 𝑇BEM, which is monitored every second
+as part of the house-keeping data, to separate the data in two halves,
+according to low ("tbem1") and high ("tbem2") values of the BEM
+temperature. As a reference, the median temperature for these two
+data splits is 8.1◦C and 16.1◦C, respectively. As for the half mission
+and PWV null tests, we do the division in two halves for each period
+and elevation configuration separately, and then we combine the
+sub-lists. For simplicity, we refer to this case as "tbem null test" in
+the text.
+Two separated lists of calibrated TOD files are produced for
+each one of those six null tests cases, and the corresponding maps
+ℎ1 and ℎ2 are produced with fully independent runs of the map-
+making code. The post-processing of each null test is identical to
+the procedure applied to the full maps. From this point, a "null-test
+difference map" can be produced for each case, as
+𝑛 = ℎ1 − ℎ2
+𝑤
+,
+(13)
+where the normalizing weight is computed as
+𝑤 =
+√︃
+(𝑤1 + 𝑤2)(𝑤−1
+1 + 𝑤−1
+2 ).
+(14)
+Here 𝑤1 and 𝑤2 are the individual weight maps of the null tests
+ℎ1 and ℎ2, respectively. They are computed as 𝑤𝑖 = 1/𝜎2
+𝑖 , with
+𝑖 = 1, 2. Defined in this way, equation 13 provides a map with
+similar noise levels as the residual noise for the weighted-sum of
+the two halves (see e.g. Planck Collaboration et al. 2014b, 2016f).
+4.1.1
+Null tests with a common baseline solution
+For those six cases listed above we have also produced a different
+set of null test maps, named as "null test with common baselines",
+as follows. First, we run the map-making code for the complete
+database, and record the baseline solutions. Then, each pair of null
+test maps is generated using that recorded solution, instead of solv-
+ing for the baselines with half of the data only, as it was the case
+before. By construction, this procedure cancels out an important
+the Izaña Atmospheric Observatory (IZO) just 1.4 km away from QUIJOTE,
+and virtually at the same altitude (≈ 10 m below).
+MNRAS 000, 1–58 (2022)
+
+QUljOTE MFI masks
+6 =.70
+6
+12QUIJOTE MFI wide survey
+19
+Figure 12. Half-mission null test difference maps for horn 3 11 GHz. Top row shows the Stokes I (left), Q (centre) and U (right) difference maps for the case
+of "independent baselines". Bottom row corresponds to the case of "common baselines" (see text for details). For display purposes, all maps are smoothed to 1
+degree resolution. The colour scale corresponds to ±1 mK for the intensity maps, and ±0.3 mK for polarization.
+part of the 1/ 𝑓 noise contribution associated with long time-scale
+variations, partly due to the fact that the baseline solution is better
+constrained when using the full database. Differences between the
+two halves ℎ1 and ℎ2 now will be entirely due to the fact that each
+half uses different input data, and not to the possible uncertainties
+in the determination of the baseline solution. For this reason, these
+null test maps are found to be particularly useful to study those
+variations in the data which can be (mainly) ascribed to calibration
+uncertainties, instrument changes or to variability of the sky signal.
+Thus, these maps will be used specifically in Section 5.2 to assess
+the internal calibration of the wide survey. For all the remaining
+analyses, and in particular, for assessing the noise levels in the wide
+survey maps, we will always use the default set of null tests maps
+("with independent baselines").
+As illustration, Figs. 12, 13 and 14 present few examples of
+null test difference maps for horn 3 11 GHz, after smoothing to one
+degree resolution. Fig. 12 shows the half mission difference map
+both for the "independent baselines" and the "common baselines"
+cases. Fig. 13 contains the ring, halfring and tbem null tests for the
+case of independent baselines, while Fig. 14 shows the same three
+cases for the "common baselines" solution.
+4.1.2
+Other data splits
+In addition to the null tests described above, other data splits have
+been considered and generated for the MFI wide survey. In particu-
+lar, we generated the four "maps per period", in correspondence to
+periods 1, 2, 5 and 6, both for the case of "independent baselines",
+and also with "common baselines". Although these four maps per
+period do not have exactly the same sky coverage (e.g. elevation 30
+is only used in period 5) or the same format (e.g. polarization maps
+are not generated in period 1), they are still very useful for valida-
+tion purposes (RFI residuals, gain model, calibration), as shown in
+the following sections. Moreover, these maps are also used for the
+study of transients and in particular, to characterise the potential
+variability of some bright point sources (see e.g. Herranz et al.
+2023).
+4.2
+Assessing systematic effects with null tests in power
+spectra and maps
+4.2.1
+Power spectra
+Fig. 15 presents the binned raw power spectra (i.e. uncorrected for
+the beam and pixel window functions) of the six null-test difference
+maps described in the previous section and computed using eq. 13,
+compared to the raw power spectra of the final maps for each horn
+and frequency. For simplicity, we show only two cases, for horn
+3 (11 GHz) and horn 4 (17 GHz). The equivalent figures for other
+horns and frequencies provide qualitatively similar results. In this
+section, the 𝐶ℓ’s are computed with the publicly available code
+Xpol6, which is based on a pseudo-𝐶ℓ estimator, and accounts for
+incomplete sky coverage (Tristram et al. 2005). The mask adopted
+for this computation is the default one described in Section 3.1
+(NCP+sat+lowdec), using a 5◦ apodization with a cosine function,
+as implemented in the NaMaster library (Alonso et al. 2019). In
+all panels, we show as a reference the angular power spectrum of
+the final map in black, and the spectra of the different "null test
+difference maps" (eq. 13) in various colours. For completeness,
+these figures also include the power spectra (as dotted lines) of the
+null-test difference maps for the case of "common baselines". We
+also include the ideal white noise level for each map, computed
+from the normalized weights (see Sect. 4.3.2 for details).
+All six null test difference maps present a similar behaviour,
+being asymptotically flat at high multipoles when reaching the white
+noise level, and increasing at low multipoles (large angular scales) as
+expected for residual 1/ 𝑓 noise. A comparison of these six null test
+power spectra provides a useful tool to identify and isolate different
+sources of systematic effects or calibration errors. In polarization,
+6 https://gitlab.in2p3.fr/tristram/Xpol
+MNRAS 000, 1–58 (2022)
+
+half noise map (311, I)
+-1.0
+1.0 mKhalf noise map (311, Q
+-0.30
+0.30 mKhalf noise map (3l1, U)
+-0.30
+0.30 mKhalf noise map (311, I)
+-1.0
+1.0 mKhalf noise map (311, Q
+-0.30
+0.30 mKhalf noise map (3l1, U)
+-0.30
+0.30 mK20
+Rubiño-Martín et al.
+Figure 13. Three examples of null test difference maps for horn 3 11 GHz, for the case of "independent baselines": ring (top), halfring (centre) and tbem
+(bottom). From left to right, each row shows the Stokes I (left), Q (centre) and U (right) difference maps. For display purposes, all maps are smoothed to 1
+degree resolution. The colour scale corresponds to ±1 mK for the intensity maps, and ±0.3 mK for polarization.
+all null test spectra are basically consistent among them, except the
+ring case, which presents a slightly lower level of 1/ 𝑓 residuals at
+low multipoles. This behaviour is expected because the ring null
+test maps probe noise variations in scales of one minute, while the
+others cases (half, daynight, tbem, pwv) probe longer time scales.
+We also note that the halfring null test tends to be slightly above
+the other noise estimates, but again this is expected as this null
+test uses basically half of the possible crossings for each pixel,
+and thus the baseline solution is less constrained. However, this
+is not the case of halfring null test with common baselines, as
+in this case the baseline solution was obtained with the complete
+dataset. For the intensity maps, the qualitative behaviour is similar
+to polarization, although the scatter among the null tests in the 1/ 𝑓
+residuals at low multipolesislarger, particularlyat11 GHz where the
+RFI contamination due to geostationary satellites was higher. In this
+case, the largest 1/ 𝑓 residuals at low multipoles correspond to the
+tbem, daynight and halfring cases, as expected. By construction, the
+halfring case amplifies the presence of residual RFI signals. In the
+case of tbem and daynight, this might indicate some low-level RFI
+residual which becomes visible when splitting the data according
+to the daily gain variations. We have confirmed that this is indeed
+the case, by constructing a new set of maps excluding period 1
+in intensity, which was the period most affected by RFI due to the
+absence of the extended shielding in the telescope. When generating
+the halfring null test for the case of no period 1, that small excess
+disappears. Finally, we note that the power spectra for the null test
+difference maps with "common baselines" present a significantly
+lower level of 1/ 𝑓 residuals, as anticipated.
+4.2.2
+Maps
+Visual inspection of the null test difference maps provides comple-
+mentary information to the one obtained from the power spectra
+analysis, in terms of identifying localised features due to systematic
+effects. For example, the halfring null test maps (see the example
+for horn 3 at 11 GHz in Fig. 13 and 14) can be used to assess
+the residual systematic effects due to uncertainties in the pointing
+model. As described in Génova-Santos et al. (2023), the pointing
+model solution for each MFI horn provides a reconstruction of the
+pointing with an overall 1 arcmin accuracy. Any residual pointing
+error will produce a characteristic feature in the halfring null-test
+map, as each one of the two sub-maps (halfring1 and halfring2)
+uses totally different ranges of local coordinates of the telescope.
+Indeed, the morphology and amplitude of the features appearing in
+the intensity map along the Galactic plane, both around the Galactic
+centre and the Cygnus area, match the expected residual signals for
+a shift of 1 arcmin between the halfring1 and halfring2 sub-maps.
+Null test difference maps can also be used for assessing the
+level of residuals in real space. For example, a cross correlation
+analysis of each null test difference map (𝑛) with the corresponding
+signal map (𝑚) can be used to trace the presence of both errors in
+the overall gain model or time-dependent RFI residuals. As usual,
+MNRAS 000, 1–58 (2022)
+
+ring noise map (311, I)
+-1.0
+1.0 mKring noise map (311, Q)
+-0.30
+0.30 mKring noise map (311, U)
+-0.30
+0.30 mKhalfring noise map (311, I)
+-1.0
+1.0 mKhalfring noise map (311, Q
+-0.30
+0.30 mKhalfring noise map (311, U)
+-0.30
+0.30 mKtbem noise map (311, I)
+-1.0
+1.0 mKtbem noise map (311, Q)
+-0.30
+0.30 mKtbem noise map (311, U)
+-0.30
+0.30 mKQUIJOTE MFI wide survey
+21
+Figure 14. Same as Fig. 13, but for the case of "common baselines" difference maps. The colour scale corresponds also to ±1 mK for the intensity maps, and
+±0.3 mK for polarization.
+Table 10. Cross-correlation in real space between the half mission difference
+maps and the final signal maps. Columns 2–4 correspond to the case of half
+mission maps with common baselines, while columns 5–7 show the results
+for the case of independent baselines. Error bars are of the order of 0.1 in
+all cases.
+Channel
+𝛼T
+𝛼Q
+𝛼U
+𝛼T
+𝛼Q
+𝛼U
+[%]
+[%]
+[%]
+[%]
+[%]
+[%]
+Common baselines
+Indep. baselines
+217
+0.2
+0.5
+0.9
+−4.8
+0.4
+0.8
+219
+0.3
+1.4
+1.3
+−1.8
+1.4
+1.3
+311
+−0.1
+0.0
+0.7
+0.1
+−0.6
+0.8
+313
+−0.2
+0.3
+0.9
+−0.1
+0.5
+1.0
+417
+0.3
+−0.2
+−0.6
+−3.0
+−0.5
+−0.6
+419
+1.2
+−0.1
+0.0
+−0.4
+−0.1
+0.2
+a cross-correlation coefficient 𝛼 can be obtained as the minimum
+variance estimator that minimizes 𝑛 − 𝛼𝑚 (see e.g. Hernández-
+Monteagudo & Rubiño-Martín 2004). Table 10 presents the corre-
+lation coefficients 𝛼, in percent units, for the case of the half mission
+null tests both for common and independent baselines. The analysis
+is carried out using the standard mask NCP+sat+lowdec defined
+in Sect. 3.1. These numbers are consistent with the power spectra
+analyses described in the previous subsection, and lie below the
+calibration uncertainty of the wide survey (see details in Sect. 5
+and Table 16). In particular, for horn 3, these values are within one
+per cent, both in intensity (𝐼) and polarization (𝑄, 𝑈). Moreover, in
+polarization all values are below 1.4 per cent.
+4.3
+Noise characterization: 1/ 𝑓 noise and correlations
+Noise parameters for the MFI instrument have been described in
+(Génova-Santos et al. 2023), and are summarized in Table 3. Those
+values determine some of the noise properties of the final wide sur-
+vey maps. Here, we use the half-mission difference maps (hereafter
+HMDM), constructed as in equation 13 and for the case of "inde-
+pendent baselines", to assess the overall noise properties of the MFI
+wide survey, including white noise levels, 1/ 𝑓 -type components
+and correlation properties. The analyses are done both in harmonic
+(Sect. 4.3.1) and real (Sect. 4.3.2) space, using the standard mask
+defined as NCP+sat+lowdec in Sect. 3.1, which contains the region
+in the declination range 6◦ < 𝛿 < 70◦. In addition, and due to
+the MFI receiver design, there are well-known noise correlations
+at the TOD level (also called "common mode 1/ 𝑓 noise") between
+channels of the same horn, which are inherited by the final maps.
+We use the cross-spectra of different HMDM to characterize these
+noise correlations at the map level, both between the two frequen-
+cies of the same horn (Sect. 4.3.3) and between the correlated and
+uncorrelated channels contributing to a given map (Sect. 4.3.4).
+MNRAS 000, 1–58 (2022)
+
+ring noise map (311, I)
+-1.0
+1.0 mKring noise map (311, Q)
+-0.30
+0.30 mKring noise map (311, U)
+-0.30
+0.30 mKhalfring noise map (311, I)
+-1.0
+1.0 mKhalfring noise map (311, Q
+-0.30
+0.30 mKhalfring noise map (311, U)
+-0.30
+0.30 mKtbem noise map (311, I)
+-1.0
+1.0 mKtbem noise map (311, Q
+-0.30
+0.30 mKtbem noise map (311, U)
+-0.30
+0.30 mK22
+Rubiño-Martín et al.
+Figure 15. Binned raw power spectra (Δℓ = 11) of the six null test difference maps discussed in the text, for horn 3 at 11 GHz (left) and horn 4 at 17 GHz
+(right). For comparison, we also include as dashed lines the spectra of the null test difference maps for the case of "common baselines". Black solid lines depict
+the spectra of the signal maps, while the horizontal dashed lines indicate the ideal white noise level for each map (see text for details).
+4.3.1
+Noise properties in harmonic space
+Our analysis of the noise properties in harmonic space is shown in
+Fig. 16 and Table 11. The power spectra for the HMDM are com-
+puted using NaMaster and then fitted with the following empirical
+model:
+𝐶ℓ = 𝐶w
+�
+1 +
+�ℓk
+ℓ
+� 𝛼�
+,
+(15)
+which accounts for a 1/ 𝑓 noise component projected on sky. We
+fit for the three parameters in this equation in two steps. First, we
+obtain the white noise level 𝐶w as the average level of the angu-
+lar power spectrum at high multipoles (ℓ ∈ [700, 800] for TT, and
+ℓ ∈ [600, 800] for EE and BB). Then, the knee-multipole ℓk and the
+slope 𝛼 are obtained analytically after fitting for a linear relation in
+log10(𝐶ℓ − 𝐶w) 𝑣𝑠 log10(ℓ), in the multipole range ℓ ∈ [20, 100]
+for both intensity and polarization. To have a better fit in the high
+multipole range for the EE and BB case of horn 2, we use here
+the range ℓ ∈ [80, 300]. The parameter 𝐶w, which represents the
+white noise level of the full maps, can be translated into the com-
+monly used quantity 𝜎1-deg, the equivalent noise level (rms) of the
+map for a 1-degree beam, with the relation 𝜎1-deg = √︁𝐶w/Ω1-deg,
+where Ω1-deg is the solid angle of a Gaussian beam with a FWHM
+of 1-degree, which corresponds to 0.345 msr= 1.133 deg2. These
+numbers (third column in Table 11) can be directly compared to
+those obtained with real space statistics in the next subsection7.
+In summary, for the intensity spectra, horn 3 presents the low-
+est noise levels both for the 1/ 𝑓 and the white noise components,
+while horn 4 is the most noisy one. However, in polarization, horn
+4 has a much better performance, yielding the lowest noise levels,
+while horn 2 is the noisiest in this case. Although the noise levels
+for horn 3 in polarization are slightly higher than those for horn 4,
+7 Note that if we want to quote the map sensitivity in the usual units of
+𝜇K.arcmin (or 𝜇K.deg), we can not use directly 𝜎1-deg, as we have to
+account for the √︁Ω1-deg factor. For instance, the white noise level of the
+MFI 311 map in polarization is 42.2 𝜇K per 1-degree beam, or equivalently,
+44.9 𝜇K.deg = 2695.1 𝜇K.arcmin, consistently with the reported 𝐶w value.
+MNRAS 000, 1–58 (2022)
+
+Horn 3 11 GHz △lbin=11.0
+half
+halfring
+pwv
+Map
+ring
+daynight
+tbem
+10
+10-1
+[mk?]
+10-2
+10
+10-4
+10-5
+10-6
+10-3
+10-4
+10
+10-6
+10-7
+10-3
+10-4
+[mk2]
+10-
+5
+10-
+6
+101
+102Horn417GHzAlbin=11.0
+half
+halfring
+pwv
+Map
+ring
+daynight
+tbem
+100
+10-1
+[mk2]
+10-2
+10
+10-4
+10-5
+10-6
+10-3
+10-4
+[mk²]
+10
+5
+10-
+10-7
+10-3
+10-4
+[mk?]
+10
+10-
+10-
+101
+102QUIJOTE MFI wide survey
+23
+Figure 16. Best-fit solutions to the power spectra of the half-mission differ-
+ence maps (HMDM). Using eq. 15, we obtain the best-fit models depicted
+here as dotted lines. The corresponding coefficients are listed in Table 11.
+given that the sky signal is significantly brighter at lower frequencies
+(see Fig. 15), the wide survey polarization maps of horn 3 (11 and
+13 GHz) have the better signal-to-noise ratios. Regarding the cor-
+related noise component, we find that the noise spectra in intensity
+are dominated by the 1/ℓ component down to scales of 1 degree,
+as a consequence of the large 1/ 𝑓 noise in the intensity TODs.
+In polarization, we find typical knee-multipoles of ℓk = 54–86 for
+horns 3 and 4, as expected for the significantly lower correlated
+noise component.
+4.3.2
+Noise properties in real space
+First, we normalize the HMDM by dividing each individual pixel by
+the square root of its covariance as computed from the map weights
+(i.e., 𝜎𝑖 = 𝑤−1/2
+𝑖
+). We recall that those weights are propagated
+through the pipeline and the map-making code, and were computed
+Table 11. Noise levels from the fit to the noise power spectra based on
+the parametric equation 15, computed from the half-mission null tests with
+independent baselines. In polarization, we show the results of the fit to the
+EE spectra. Results for BB are fully consistent.
+Channel
+𝐶w
+𝜎1-deg
+𝛼
+ℓk
+[mK2 sr]
+[𝜇K]
+Intensity (TT)
+217
+6.13 × 10−6
+133.5
+1.50
+228.8
+219
+1.05 × 10−5
+174.5
+1.82
+229.3
+311
+2.56 × 10−6
+86.3
+1.27
+221.4
+313
+1.29 × 10−6
+61.3
+1.60
+192.5
+417
+1.07 × 10−5
+176.4
+1.45
+230.4
+419
+1.40 × 10−5
+201.7
+1.82
+243.6
+Polarization (EE)
+217
+1.21 × 10−6
+59.4
+1.20
+145.0
+219
+1.87 × 10−6
+73.7
+1.30
+173.7
+311
+6.13 × 10−7
+42.2
+1.24
+86.0
+313
+4.95 × 10−7
+37.9
+1.35
+75.3
+417
+4.42 × 10−7
+35.8
+1.06
+53.5
+419
+5.02 × 10−7
+38.2
+1.24
+73.2
+Table 12. Recalibration factor of the noise standard deviation included in
+the weight maps, based on null test maps.
+Map
+H2,17
+H2,19
+H3,11
+H3,13
+H4,17
+H4,19
+Half mission null test
+I
+4.974
+5.596
+3.424
+3.016
+4.695
+5.108
+Q
+1.723
+2.001
+1.471
+1.372
+1.285
+1.292
+U
+1.723
+1.999
+1.473
+1.373
+1.285
+1.292
+Ring null test
+I
+4.896
+5.449
+3.410
+2.993
+4.641
+4.978
+Q
+1.717
+1.994
+1.471
+1.370
+1.286
+1.289
+U
+1.716
+1.991
+1.473
+1.370
+1.285
+1.291
+from the variance of each individual 40 ms sample in the TOD. For
+this normalized map, we fit for the standard deviation within the
+reference mask. The results are shown in Table 12. As expected,
+these values are reasonably close to unity for the case of the polar-
+ization maps, while in intensity these factors are greater than 3 in
+all cases. These deviations from unity are generally consistent with
+the level of 1/ 𝑓 noise in each case (see e.g. Table 11). This set of
+values could be used to renormalize the weight maps, so they would
+be representative of the actual noise levels, while preserving the
+underlying spatial distribution of the hit maps. Indeed, these factors
+are used to estimate the ideal white noise of each map at the power
+spectrum level. For example, the dashed lines in Fig. 15 are com-
+puted with these rescaled weight maps. Moreover, these rescaled
+weight maps can be used to produce signal-to-noise maps for each
+frequency (see Appendix C1).
+As a second analysis, we repeat the same procedure but now we
+normalize each difference map according to the square root of the
+number of hits. Taking into account that hits correspond to 40 ms
+samples, we can obtain from here representative normalization val-
+ues to describe the noise standard deviation as
+𝜎 =
+𝜎0
+√𝑁hit
+.
+(16)
+MNRAS 000, 1–58 (2022)
+
+FitnoiseClAl=10.0
+10-2
+h2 17GHz
+h2 19GHz
+h3 11GHz
+10-3
+h3 13GHz
+[mk2]
+h4 17GHz
+h4 19GHz
+10-5
+10-B
+10-4
+[mk²]
+10-
+10-6
+10-3
+10-4
+[mk2]
+l adb
+10-
+-5
+10-6
+101
+10224
+Rubiño-Martín et al.
+Table 13. Characteristic value of the sensitivity for each channel, 𝜎0, in
+units of mK s1/2. Based on the half-mission null test maps.
+Map
+H2,17
+H2,19
+H3,11
+H3,13
+H4,17
+H4,19
+Half mission null test
+I
+5.896
+7.445
+3.481
+2.422
+7.939
+8.427
+Q
+1.878
+2.280
+1.371
+1.188
+1.101
+1.059
+U
+1.875
+2.273
+1.372
+1.188
+1.100
+1.064
+Table 14. Mean noise figures in the final MFI maps, in units of 𝜎1-deg (𝜇K
+per 1-degree beam), using real-space statistics. A variance map is estimated
+based on the half-mission nulltest maps, computing the variance within a
+circle of 1 degree radius. Those values are then converted into 𝜎1-deg.
+Map
+H2,17
+H2,19
+H3,11
+H3,13
+H4,17
+H4,19
+Half mission null test
+I
+136.6
+184.8
+88.3
+65.0
+184.2
+214.8
+Q
+59.4
+76.4
+40.5
+35.9
+34.2
+32.7
+U
+59.4
+76.1
+40.6
+35.9
+34.1
+32.9
+Our results are shown in Table 13. The values obtained for the
+MFI wide survey in polarization are comparable to those obtained
+for raster scan observations with the MFI in smaller regions (see
+e.g. last column in Table 1 from Génova-Santos et al. 2017), and
+represent the actual sensitivity of the instrument.
+Finally, we can also estimate the noise variance directly from
+the HMDM, using apertures of 1-degree radius across the same
+mask. The average values obtained from this analysis are given in Ta-
+ble 14. To facilitate the comparison with the numbers in the previous
+subsection, these values are re-scaled by the factor
+√︃
+Ωpix/Ω1-deg,
+so they represent 𝜎1-deg. In summary, the final combined maps of
+the MFI wide survey in polarization present sensitivities within the
+range 35–40 𝜇K per 1-degree beam for the four frequencies.
+4.3.3
+Noise correlations between frequencies of the same horn
+Two MFI frequency channels from the same horn have a corre-
+lated ("common mode") 1/ 𝑓 noise component, due to the fact that
+they share the same LNA. This fact is particularly relevant for the
+intensity maps, which are strongly dominated by correlated noise.
+Because of this reason, our final wide survey maps at 11 and 13 GHz
+have correlated noise between them, as is the case for the maps at
+17 and 19 GHz.
+In order to characterize the actual degree of correlation be-
+tween two wide-survey maps obtained from the same horn, we use
+the normalized cross-spectra between the corresponding null-test
+difference maps. As in the previous section, we use as a reference
+the HMDM for the case of independent baselines. Following the
+notation in Sect. 4.1, here 𝑛ℎ, 𝑓 represents the half-mission differ-
+ence map for horn ℎ and frequency 𝑓 (see eq. 13). Then, for a given
+horn ℎ(= 2, 3, 4), the normalized correlation between the lowest
+frequency band 𝑓1 and the highest frequency band 𝑓2, is given by
+𝜌ℓ ≡
+𝐶
+𝑛ℎ, 𝑓1×𝑛ℎ, 𝑓2
+ℓ
+√︃
+𝐶
+𝑛ℎ, 𝑓1
+ℓ
+𝐶
+𝑛ℎ, 𝑓2
+ℓ
+,
+(17)
+where 𝐶
+𝑛ℎ, 𝑓1×𝑛ℎ, 𝑓2
+ℓ
+is the cross-spectrum between the two difference
+maps, and 𝐶
+𝑛ℎ, 𝑓𝑖
+ℓ
+for 𝑖 = 1, 2 represents the auto-spectra.
+Figure 17 shows this normalized cross-spectrum 𝜌ℓ in the
+final MFI wide survey maps for horns 2, 3 and 4, both in intensity
+and polarization. In intensity, the resulting noise correlation is of
+the order of 75–85 per cent for the three horns, being relatively
+flat in the multipole range 20 ≲ ℓ ≲ 300. In polarization, the
+correlation is found to be ∼ 20–60 per cent depending on the horn,
+with a moderate dependence on the multipole, being slightly lower at
+higher multipoles (smaller scales). In order to obtain a representative
+value for this correlation, we compute the average (and standard
+deviation) of 𝜌ℓ in the multipole range [20, 200]. For TT, we obtain
+85.0 ± 0.3 %, 76.5 ± 0.4 % and 84.1 ± 0.3 % for horns 2, 3 and
+4, respectively. In polarization, for EE we obtain 60.7 ± 1.0 %,
+32.8 ± 1.4 % and 20.9 ± 1.2 %, and for BB we have 60.8 ± 0.9 %,
+36.2 ± 1.1 % and 21.7 ± 1.2 %, again for horns 2, 3 and 4. This
+high degree of correlation has to be taken into account when doing
+combined analyses of the two frequency maps of the same horn.
+As a consistency check, and in order to test that these inter-
+frequency correlations are entirely due to instrumental (common
+mode) 1/ 𝑓 noise, and not to external correlated signals produced
+either by the atmosphere or by RFI, we performed the same analysis
+but now comparing two frequencies coming from two different
+horns. In particular, we evaluated the cross-correlation of horn 2 at
+17 GHz with horn 4 at 19 GHz, obtaining −0.64 ± 2.47 %, 0.47 ±
+0.96 % and −0.49 ± 0.96 % for TT, EE, and BB, respectively. In
+addition, the cross-correlation of horn 4 at 17 GHz with horn 2 at
+19 GHz gives −0.32 ± 2.13 %, 0.99 ± 0.83 % and 0.72 ± 0.76 %,
+again for TT, EE and BB. In both cases, the results are consistent
+with zero within the error bar.
+4.3.4
+Noise correlations between channels
+As described above, for any given horn and frequency sub-band
+of MFI, we produce two versions of the intensity and polarization
+maps, the so-called correlated (𝑥c) and uncorrelated (𝑥u) maps.
+Due to the MFI design, we expect a high degree of correlation
+between the noise affecting those two versions of the intensity maps,
+due to the fact that they all share the same LNAs and there is
+no cancellation of the 1/ 𝑓 noise in any of the sums of channels
+contributing to 𝑥c and 𝑥u. We can use the same methodology applied
+in the previous sub-section to characterize this correlation level of
+the noise between correlated and uncorrelated channels maps for
+a given horn and frequency. We also use the half-mission null test
+maps as a reference for this analysis. But now, in the post-processing
+stage, we generate two independent versions for each individual
+map, using either the correlated or the uncorrelated information
+only. With these maps, and using again eq. 13, for a given horn and
+frequency we can produce 𝑛c and 𝑛u, the half-mission difference
+maps of the correlated and uncorrelated channels, respectively. In
+analogy to equation 17, we now compute
+𝜌ℓ ≡
+𝐶𝑛c×𝑛u
+ℓ
+√︃
+𝐶𝑛c
+ℓ 𝐶𝑛u
+ℓ
+,
+(18)
+where 𝐶𝑛c×𝑛u
+ℓ
+is the cross-spectrum between the two difference
+maps, and 𝐶𝑛c
+ℓ and 𝐶𝑛u
+ℓ
+are the auto-spectra.
+Fig. 18 shows the resulting correlation level between correlated
+and uncorrelated channels. As expected, we find a very high degree
+of correlation (of the order of 90 per cent) in intensity, and a signal
+consistent with zero in polarization (both for EE and BB spectra).
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+25
+Figure 17. Cross-correlation spectra of the half-mission difference maps between the two frequencies of the same horn, for TT (left), EE (centre) and BB
+(right).
+Figure 18. Cross-correlation spectra of the half-mission difference maps between the correlated and uncorrelated channels from the same horn and frequency,
+for TT (left), EE (centre) and BB (right).
+Table 15. Average inter-channel correlations < 𝜌ℓ > of the half-mission
+difference maps between the correlated and uncorrelated channels for a
+given horn and frequency. The values correspond to the mean and standard
+deviation of the 𝜌ℓ displayed in Figure 18, computed in the multipole range
+20–200.
+Channel
+TT (%)
+EE (%)
+BB (%)
+217
+97.14 ± 0.08
+−1.36 ± 1.10
+−0.91 ± 1.37
+219
+87.91 ± 0.34
+2.40 ± 1.19
+−0.60 ± 1.04
+311
+85.82 ± 0.26
+3.81 ± 0.75
+2.38 ± 1.18
+313
+79.65 ± 0.42
+−0.32 ± 0.89
+−2.01 ± 0.91
+417
+97.95 ± 0.04
+−3.26 ± 1.07
+−1.22 ± 1.12
+419
+91.56 ± 0.22
+−2.81 ± 0.80
+−0.61 ± 1.15
+Again, as a representative value for this correlation, we compute
+the average and standard deviation of 𝜌ℓ in the multipole range
+[20, 200]. The results are shown in Table 15. These average corre-
+lation values in intensity are used in the pipeline in order to produce
+the final combinations of correlated and uncorrelated channels, as
+described in Section 2.4.1.
+4.4
+Impact of residuals on the power spectra: atmospheric
+and RFI corrections
+As described in Sect. 2, the MFI wide-survey pipeline incorporates
+several steps tailored to correct for the contribution of atmospheric
+and RFI signals in the final maps. Atmospheric corrections are
+applied at the TOD level (see Sect. 2.2.4), and for intensity maps
+only. When projected on maps, they appear as large scale patterns
+with an increasing amplitude in frequency (see Fig. 7). RFI signals
+are corrected both in the intensity and polarization maps, in two
+stages. First, RFI signals at the TOD level are corrected using spatial
+templates as described in Sect. 2.2.3. When projected on sky, they
+also appear as large scale patterns with a moderate amplitude (≲
+0.5 mK) and presenting a higher amplitude in intensity (see Fig. 6).
+Later, in the post-processing stage (Sect. 2.4), any residual RFI
+signals emerging after co-adding all data in the map-making process
+are corrected using a function of the declination. In terms of relative
+amplitude, this is by far the largest correction applied to the MFI
+wide-survey polarization data, with its amplitude being higher in
+the 11 and 13 GHz channels due to the emission of geo-stationary
+satellites entering through the far sidelobes. Indeed, the effective
+transfer function of the MFI wide survey in polarization is mainly
+determined by this effect (see Sect. 2.5).
+In order to quantify the relative importance of these three
+MNRAS 000, 1–58 (2022)
+
+Correlationlow/highfreguencies
+TT
+EE
+BB
+100
+80
+60
+%
+40
+20
+h2 17x19
+0
+h3 11x13
+-20
+h4 17x19
+101
+102
+103101
+102
+103101
+102
+103
+l
+l
+lCorrelation corr/uncorr
+TT
+EE
+BB
+100
+80
+60
+%
+40
+20
+217
+313
+0
+219
+417
+-20
+311
+419
+101
+102
+103101
+102
+103101
+102
+103
+l
+l
+l26
+Rubiño-Martín et al.
+Figure 19. Raw angular power spectra of the ATMOS (red), RFI (green) and FDEC (blue) patterns removed from the MFI wide survey maps, for horn 3 at
+11 GHz (top row) and horn 4 at 19 GHz (bottom row). For each case, we represent TT (left), EE (centre) and BB (right) spectra. Solid black lines correspond to
+the angular power spectra of the corresponding wide survey maps, while dashed lines correspond to the half-mission difference maps. All spectra are computed
+using the default analysis mask (NCP+sat+lowdec).
+corrections, and to evaluate the possible impact of any residual
+systematic effects due to uncorrected contamination in the final
+wide survey maps, we have computed the angular power spectra
+of those patterns that are removed from the maps, and we have
+compared them with the spectra of the final maps and the half-
+mission noise levels. Figure 19 shows the resulting power spectra
+for the two extreme frequency values (11 and 19 GHz) taken here as
+representative cases, with 11 GHz being the one with highest RFI
+contamination, and 19 GHz the one with the highest atmospheric
+contamination. In this plot, we use the notation of ATMOS, RFI
+and FDEC for "atmospheric", "RFI at TOD level using a function
+of azimuth", and "RFI at the map level using function of declination"
+corrections, respectively.
+Regarding the atmospheric contribution (ATMOS) to the in-
+tensity power spectra, the removed pattern is subdominant at all an-
+gular scales in the 11 GHz case when compared to the noise level.
+At 19 GHz, we have a similar behaviour at small angular scales
+(ℓ >∼ 20). However, the atmospheric residuals become comparable
+to the noise levels for multipoles ℓ ≲ 20, as can be anticipated from
+the visual inspection of Fig. 7.
+For the RFI contribution at the TOD level, the removed patterns
+both in intensity and polarization are always below the noise levels
+at all frequencies, although they become comparable to the noise at
+large angular scales ℓ ≲ 20. Thus, in this RFI case, as well as for
+ATMOS, any residual systematic effect with an amplitude being a
+fraction of the applied correction will have negligible impact at the
+power spectrum level.
+Finally, for the removed FDEC patterns, the largest amplitude
+is found at 11 GHz, as anticipated from Fig. 8. At this frequency, the
+removed pattern in intensity is above the correlated noise level for
+multipoles ℓ ≲ 20. Moreover, in polarization, the applied correction
+is found to be critical, in the sense that its amplitude is above the sky
+signal for multipoles ℓ ≲ 30. When looking at the 19 GHz FDEC
+patterns, in intensity the corrected amplitude is always below the
+noise levels for all multipoles, while in polarization again becomes
+comparable to the sky signal for ℓ ≲ 20. In this case, although the
+underlying assumption for modelling residual RFI signals using a
+function of the declination is very robust and well tested, it is im-
+portant to keep in mind that residual contributions might have an
+impact on the polarization maps of the MFI wide survey on large
+angular scales. In addition, as explained in Sect. 2.5, the FDEC pro-
+cedure also affects the same multipole range by introducing a signal
+error in the reconstructed sky. For these reasons, in the following
+sections involving scientific analyses based on power spectra of the
+polarization signals in the wide survey, we adopt the conservative
+choice of restricting the study to multipoles ℓ ≥ 30.
+4.5
+Inter-frequency comparison of the MFI maps
+As an additional validation test, here we present an inter-frequency
+comparison of the MFI wide survey maps, together with a com-
+parison with external data. For this test, we rely on the assump-
+tion that the average spectral index of the polarized synchrotron
+emission in the QUIJOTE maps is 𝛽 = −3.1 (see discussion be-
+low in Sect. 8). We then rescale the MFI wide survey maps at 1
+degree resolution to the central frequency of the WMAP K-band
+map, 𝜈 = 22.8 GHz, accounting for colour correction factors both
+for MFI and WMAP maps. Figure 20 shows the rescaled MFI po-
+larization maps at 11 and 13 GHz compared to WMAP-K, while
+Figure 21 shows the differences for pairs of those maps (313−311,
+311−WMAP, 313−WMAP). A visual inspection shows that there
+is obvious polarized emission in the Galactic plane which is not
+consistent with the 𝛽 = −3.1 spectral index, mainly towards the
+Galactic centre or the Fan region (𝑙 ≈ 135◦). In the maps we can
+also identify some residual intensity to polarization leakage in the
+Cygnus area (around 𝑙 ≈ 80◦). However, the large scale emission
+MNRAS 000, 1–58 (2022)
+
+TT H3, 11GHZ
+TT
+RFI
+100
+noise
+FDEC
+ATMOS
+10-2
+C,T [mk?]
+10-4
+10-6
+10-8
+10-10
+101
+102EE H3. 11GHZ
+10-2
+EE
+RFI
+noise
+FDEC
+10-4
+CEE [mK?2]
+10-6
+10-8.
+10-10
+101
+102BB H3, 11GHZ
+10-2
+BB
+RFI
+noise
+FDEC
+10-4
+10-6
+10-8.
+10-10
+101
+102TT H4,. 19GHZ
+TT
+RFI
+100
+noise
+FDEC
+ATMOS
+10-2
+C}T [mk?]
+10-4,
+10-6
+10-8,
+10-10
+101
+102EE H4. 19GHZ
+10-2
+EE
+RFI
+noise
+FDEC
+10-4
+CEE [mK?2]
+10-6.
+10-8.
+10-10
+101
+102BB H4, 19GHZ
+10-2
+BB
+RFI
+noise
+FDEC
+10-4
+CBB[mK2]
+10-6
+10-8.
+10-10
+101
+102QUIJOTE MFI wide survey
+27
+Figure 20. Comparison of the rescaled polarization MFI maps at 11 and 13 GHz with the 9-yr WMAP-K band map (Bennett et al. 2013). MFI maps are
+rescaled to 23 GHz using an average spectral of 𝛽 = −3.1, and accounting for colour corrections. All maps use the same colour scale, saturated at ±0.1 mK.
+From left to right, we show MFI 11 GHz (rescaled), MFI 13 GHz (rescaled) and WMAP-K. Top row: Stokes Q maps. Bottom row: Stokes U maps. For display
+purposes, to facilitate the comparison of the different structures near the mask edges, we applied here the QUIJOTE MFI sky mask to the WMAP map.
+Figure 21. Inter-frequency comparison of the rescaled maps shown in Fig. 20. Top (bottom) row shows differences of Stokes Q (U) maps. First column shows
+the difference between the rescaled 11 and 13 GHz MFI maps. Second and third column show the MFI 11 GHz minus WMAP-K, and MFI 13 GHz minus
+WMAP-K maps, respectively. All maps use the same colour scale as in Fig. 20, saturated at ±0.1 mK.
+far from the Galactic plane is largely suppressed in this difference,
+showing a good consistency of the MFI and WMAP-K maps. The
+residual emission in the difference map 313-311 is basically consis-
+tent with the expected noise level for the difference of both maps, as
+shown in Fig. 22. In this comparison, we use the EE power spectra
+for the rescaled maps using the default QUIJOTE mask with the
+Galactic cut |𝑏| > 10◦, and restricting the comparison to multipoles
+ℓ ≥ 30.
+5
+ACCURACY OF THE WIDE SURVEY CALIBRATION
+In this section we assess the overall calibration uncertainty of the
+QUIJOTE MFI wide survey maps in intensity and polarization,
+using the information described in the pipeline paper (Génova-
+Santos et al. 2023) to account for known systematics, and also
+presenting a set of consistency checks based on the null test maps,
+in order to evaluate the impact of unknown systematics. Table 16
+shows the summary of all types of uncertainties considered in this
+MNRAS 000, 1–58 (2022)
+
+QUOTE Q H3 11GHz (1deg, rescaled)
+mK
+-0.1
+0.1QUOTE Q H3 13GHz (1deg, rescaled)
+mK
+-0.1
+0.1WMAP 23GHz Q (1deg)
+mK
+-0.1
+0.1QUlOTE U H3 11GHz (1deg, rescaled)
+mK
+-0.1
+0.1QUlOTE U H3 13GHz (1deg, rescaled)
+mK
+-0.1
+0.1WMAP 23GHz U (1deg)
+mK
+-0.1
+0.1MFl H3 13GHz - MFI311 (Q, 1deg,rescaled)
+mK
+-0.1
+0.1MFl H3 11GHz - WMAP (Q, 1deg, rescaled
+mK
+-0.1
+0.1MFl H3 13GHz - WMAP (Q, 1deg, rescaled
+mK
+-0.1
+0.1MFl H3 13GHz - MFI311 (U, 1deg, rescaled)
+mK
+-0.1
+0.1MFl H3 11GHz - WMAP (U, ldeg, rescaled)
+mK
+-0.1
+0.1MFl H3 13GHz - WMAP (U, 1deg, rescaled)
+mK
+-0.1
+0.128
+Rubiño-Martín et al.
+Figure 22. EE power spectra of the inter-frequency comparison of the MFI
+rescaled maps 313−311, shown in the first column of Fig. 21. Black and red
+solid lines show the EE power spectra of the rescaled MFI 11 and 13 GHz
+maps, respectively. Blue solid line is the power spectrum of the difference
+map 313−311, while the yellow dashed line shows the expected noise level
+for that difference map, assuming an average inter-frequency correlation of
+32.8 % (see Sect. 4.3.3).
+work, as well as the impact of each of them in the overall calibration
+error budget.
+5.1
+Statistical uncertainty and known systematics
+5.1.1
+Calibration model
+An important contribution to the global systematic uncertainty bud-
+get comes from calibration uncertainties, and in particular, the cali-
+brator model. As discussed in Génova-Santos et al. (2023), the two
+main amplitude calibrators of QUIJOTE MFI are Tau A and Cas
+A, which are amongst the brightest sources on the sky in this fre-
+quency range. As explained in subsection 2.6, the wide survey maps
+have been recalibrated using flux densities extracted on these maps
+at the position of Tau A. These flux densities are measured with
+sensitivities better than 0.3 % in all frequencies (see also Table 24
+and Sect. 9) while the internal calibration accuracy of QUIJOTE
+is better than 1 % as shown below in subsection 5.2. Therefore in
+our case the dominant error component is associated with the cal-
+ibration models that are used as reference. As will be discussed in
+detail in Génova-Santos & Rubiño-Martín in prep. (see also sub-
+section 2.6), using different tests we estimate that the Tau A model
+has an uncertainty of ≈ 4 % in our frequency range. We believe
+this value is dominated by calibration errors of the different data
+that are used to model this spectrum. In the case of Tau A there is
+also an important contribution due to the modelling of its secular
+decrease, which leads to errors when data taken at different epochs
+are combined to model its spectrum. We decide to set a conservative
+overall calibration uncertainty of 5 %. The reliability of this number
+is supported by the tests on radiosources and planets presented in
+Section 9, as well as other calibration tests based on the detection
+of primary CMB anisotropies shown in Sect. 5.3.
+5.1.2
+Colour corrections
+The overall 5 % calibration uncertainty would strictly apply to any
+analysis performed in our maps on sources or regions with a power-
+law spectrum with index 𝛼 = −0.3, as that of Tau A (our primary
+calibrator). For a different spectrum, uncertainties in the colour
+corrections must be factored in. These are mainly associated with
+errors in the measurement or characterisation of the instrument
+bandpasses. MFI bandpasses were last measured in 2020, for the
+instrumental configuration corresponding to period 6. The statisti-
+cal uncertainties of these measurements are very low, such that they
+lead to errors in the global calibrated antenna temperature below
+0.01% for a range of spectral indices 𝛼 ∈ [−3, +3] and for all horns
+and frequencies. On the other hand, MFI suffered various modifi-
+cations over its lifetime (see Table 2), which may have introduced
+modifications in the actual bandpass shapes of periods 1, 2 and 5
+with respect to period 6.
+For the MFI wide survey, we conservatively assign errors to
+the colour corrections by comparing the last bandpass measurement
+from period 6 with a previous one performed in 2013 during period
+1. Through comparing the colour correction coefficients obtained in
+both cases we find that channel 219 presents the largest differences,
+and in this case the error scales approximately as 𝜖×|𝛼+0.3| %, with
+𝜖 = 1.03. Note that the error increases as the spectral index of the
+observation, 𝛼, departs from that of the primary calibrator. For 311
+and 313, we obtain 𝜖 ≈ 0.01 and 𝜖 ≈ 0.53 respectively. For 217 we
+have 𝜖 ≈ 0.51, while for horn 4 we have values between 𝜖 = 0.2–0.4.
+We must note that these uncertainties are somewhat conservative,
+as differences between the two measured bandpasses may not be
+entirely real, but could also be due to shortcomings in the 2013
+measurements, which are deemed much less reliable than those of
+2020 due to measurement techniques (see details in Génova-Santos
+et al. (2023)). Taking this into account, and the fact that errors in the
+other channels are smaller, as a conservative choice for this paper,
+in Table 16 we have assigned an overall 0.5 × |𝛼 + 0.3| % error to
+colour corrections for 11 and 13 GHz, and 1 × |𝛼 + 0.3| % for 17
+and 19 GHz.
+Note that these errors in the colour correction coefficients
+should impact the consistency checks presented in Section 9, where
+we compare with models flux densities of sources with spectral in-
+dices ranging between −0.3 and −1.2, and of planets with 𝛼 ≈ 2, or
+those presented below in this section where we correlate our maps
+with templates tracing the CMB anisotropies or the CMB dipole
+that also have 𝛼 ≈ 2. In the former case, we find differences of
+∼ 5% which we are confident are due to uncertainties in the source
+calibration models. For the CMB anisotropies and CMB dipole, the
+differences are ∼ 3 % and ∼ 10 % respectively, and are driven by
+statistical noise (see Sect. 5.3).
+5.1.3
+Beams
+One of the instrumental aspects that are most carefully charac-
+terised in CMB experiments are the beams and derived window
+functions, as they have a direct impact on the amplitude of the
+derived power spectrum and thence on cosmological parameters.
+In QUIJOTE MFI this is even more important as its calibration is
+tied to unresolved point sources. Comparison between beam radial
+profiles derived from observations of bright point sources and the
+numerical optical simulation based on CST software8 described in
+Génova-Santos et al. (2023) demonstrates an accuracy in the deter-
+mination of the intensity beam typically below the 2 % level (with
+respect to the centre of the main beam). Given that the MFI maps
+are (re)calibrated using a beam-fitting photometry on point sources,
+errors in the beams will directly impact the global map temperature
+8 https://www.3ds.com/products-services/simulia/products/
+cst-studio-suite/
+MNRAS 000, 1–58 (2022)
+
+Mask: default + Ibl > 10°
+10-6
+311 rescaled
+Difference 313-311
+313 rescaled
+Expected noise 313-311
+10-7
+10-8
+102QUIJOTE MFI wide survey
+29
+Table 16. Accuracy of the calibration in the QUIJOTE MFI wide survey data. Second column indicates if the type of uncertainty is applicable to intensity (I)
+and/or to polarization (P) maps.
+Type of uncertainty
+Applies to
+11 GHz
+13 GHz
+17 GHz
+19 GHz
+Method
+Reference
+Calibration model
+I,P
+5 %
+5 %
+5 %
+5 %
+Model for calibrators
+Sect. 5.1.1
+Colour corrections𝑎
+I,P
+0.5 %
+0.5 %
+1 %
+1 %
+Bandpass measurements
+Sect. 5.1.2
+Beam uncertainty
+I,P
+2 %
+2 %
+2 %
+2 %
+CST beam model, Tau A
+Sect. 5.1.3
+Zero level [mK]
+I
+−0.74 ± 0.20
+−0.59 ± 0.22
+0
+0
+Plane-parallel model
+Sect. 5.4
+I→P leakage
+P
+0.65 %
+0.4 %
+0.8 %
+0.9 %
+Cygnus area
+Sect. 5.1.4
+Polarization efficiency
+P
+3 %
+3 %
+4 %
+4 %
+Lab measurements, Tau A
+Sect. 5.1.5
+Polarization angle (deg)
+P
+0.6
+0.9
+1.0
+3.2
+Tau A, WMAP/Planck
+Sect. 5.5
+Unknown systematics:
+Real space (𝜇K/beam)
+I
+< 53
+< 49
+< 118
+< 224
+Null tests at 𝑁side = 64
+Sect. 5.2.1
+Real space (𝜇K/beam)
+P
+< 12
+< 15
+< 10
+< 13
+Null tests at 𝑁side = 64
+Sect. 5.2.1
+Harmonic space (30 < ℓ < 200)
+I
+0.2 %
+0.3 %
+0.5 %
+0.7 %
+Null tests
+Sect. 5.2.2
+Harmonic (30 < ℓ < 200)
+P
+3 %
+4 %
+6 %
+6 %
+Null tests
+Sect. 5.2.2
+Overall calibration error𝑏
+I
+5 %
+5 %
+5 %
+5 %
+Overall calibration error𝑏
+P
+5 %
+5 %
+6 %
+6 %
+𝑎 These numbers should be multiplied by |𝛼 + 0.3|, being 𝛼 the spectral index of the source.
+𝑏 Obtained as the maximum value of the following errors: for intensity, calibration, beam uncertainty and unknown systematics in harmonic space;
+and for polarization, we add also I→P leakage and polar efficiency.
+scale. We confidently estimate the error in this temperature scale
+to be below 2 %. Note though that in extracting flux densities of
+point sources using the same beam-fitting photometry that is used
+for the main calibration, these errors would be largely suppressed.
+In Table 16, we adopt a conservative value of 2 per cent, which cor-
+responds to the maximum error associated with the determination
+of the brightness of a beam-filling emission.
+In polarization, a detailed description of the MFI beams can be
+found in Génova-Santos et al. (2023), where we use the CST optical
+simulations and the Mueller matrix formalism. Due to the MFI
+optical design, the cross-polar terms are significantly smaller than
+the copolar terms. For example, for horn 3, the cross-polar terms are
+less than 0.05 % of the copolar beams across the band. This implies
+that the diagonal components of the Mueller matrix (𝑀II and 𝑀QQ)
+can be considered nearly identical (with that accuracy). Moreover,
+the leakage terms 𝑀IQ and 𝑀QI are also identical in this limit, and
+are given by one half of the difference of the copolar beams at 0◦
+and 90◦. As shown in Génova-Santos et al. (2023), these terms have
+a quadrupolar structure with two positive and two negative lobes,
+with typical peak amplitudes (relative to the copolar peak) of ≲ 1 %.
+As shown below in Sect. 9, when studying bright compact sources
+in the MFI wide survey, these patterns are clearly visible around Tau
+A (in Stokes 𝑈 parameter, because most of the signal appears in 𝑄)
+and Cas A (in this case, as the source is essentially unpolarized, they
+are seen both in 𝑄 and 𝑈 maps, rotated by 45◦). When integrated on
+scales larger than the beam, these patterns average to zero, and thus
+have minimum impact on the photometry analyses (see also Leahy
+et al. 2010, for the case of Planck beams). For example, for the MFI
+311 map, the impact on a photometry measurement using either
+aperture photometry in 1 deg, or beam fitting, is well below 0.05 %
+across the full frequency band. Thus, we neglect this contribution
+to the overall calibration error due to beam uncertainties, and in
+Table 16 we adopt the same calibration uncertainty in polarization
+as for intensity beams.
+5.1.4
+Intensity-to-polarization leakage
+Despite of the fact that the MFI is a true polarimeter, in the sense
+that the polarization signal is produced directly for each individual
+horn and frequency band, there are several known systematic effects
+that may lead to spurious polarization signals, particularly in bright
+regions in intensity. In the previous subsection we have already dis-
+cussed, for bright point sources, the intensity-to-polarization leak-
+age (hereafter IPL) terms due to beam non-idealities. Here, we
+discuss the IPL terms arising from the bandpass mismatch between
+the two pairs of channels that contribute to a given polarization
+timeline. For the MFI instrument, the 𝑟-factors in equations 3 and
+4 are determined using Tau A observations (see details in Génova-
+Santos et al. 2023). When observing a sky region with a bright
+intensity emission, the effective 𝑟-factor might change depending
+on the spectral index of the sky emission, particularly if it differs
+from that of Tau A (𝛼 = −0.3). Using the detailed measurements of
+the bandpasses, we have estimated that for spectral indices typical
+of Galactic emission (𝛼 ∈ [−1.5, 0]), the amount of signal leaked
+into Stokes 𝑄 or 𝑈 due to this effect is typically below 0.2 % of
+the intensity signal. For a CMB spectrum (𝛼 ≈ 2), it is still below
+0.5 %.
+Here, we provide an independent confirmation of the order
+of magnitude of the IPL in the MFI wide survey maps using the
+sky emission in the Cygnus region, located at Galactic coordinates
+(𝑙, 𝑏) = (80◦, 0◦). Figure 23 shows this area in more detail. As
+the intensity emission in this region is dominated by free-free, it
+is expected to be almost unpolarized. We use a cross-correlation
+analysis (similar to the one used in Sect. 4.2.2) to obtain the corre-
+lation coefficient 𝛼 that minimizes 𝑄 − 𝛼𝐼 within a region centred
+at (𝑙, 𝑏) = (80◦, 0◦) with a radius of 5◦. The values are always
+below 1 per cent for all cases, as expected. For 311, we find 0.10 %
+and 0.65 % for Stokes Q and U, respectively. The largest values are
+found for 419, where we obtain 0.91% and −0.41% for Stokes Q
+and U. This effect in the Cygnus area is clearly seen in the maps
+of Fig. 21 and 23. The values reported in Table 16 correspond to
+the most conservative case (Stokes Q or U) at each frequency, in
+absolute value.
+MNRAS 000, 1–58 (2022)
+
+30
+Rubiño-Martín et al.
+Figure 23. Minimaps of 15◦ × 15◦ around the Cygnus region, located at
+Galactic coordinates (𝑙, 𝑏) = (80◦, 0◦). We show the horn 3 11 GHz (top)
+and horn 4 19 GHz maps (bottom) at their original resolution. The circle
+indicates the region where the IPL is computed (see text for details). The two
+bright compact objects in the polarization maps located outside the circle,
+W63 and Cygnus A, are discussed in Sect. 9.
+5.1.5
+Polarization efficiency
+As discussed in Sect. 2.6, the calibration of the polarization ef-
+ficiency of the MFI wide survey data is done in two steps. First,
+we use laboratory measurements taken at the end of period 6 to
+calibrate the polar efficiency of each individual MFI channel. In
+addition, we use the wide survey data in period 6 to add also the
+correction factors to these polar efficiencies associated with a pos-
+sible error in the determination of the 𝑟-factors. These procedures
+provide a determination of the polarization efficiency in period 6
+with a relative accuracy of 2 %. Then, in a second step these values
+from period 6 are transferred to the other two periods that are used
+in the construction of the MFI wide survey polarization maps (i.e. 2
+and 5), using beam fitting photometry (BF1d) measurements on Tau
+A. The error budget for these factors is given by the accuracy of the
+flux density extraction, which is found to be of the order of 1 % for
+horn 3, and 2 % for horns 2 and 4. As they correspond to systematic
+errors, we adopt the conservative approach of adding them linearly,
+and we quote an overall 3 % error in the polar efficiency for horn 3,
+and 4 % for horns 2 and 4. In the following subsections we evaluate
+unknown systematic effects in the polarization maps, noting that in
+those cases, the global errors include the polar efficiency error. In
+addition, in Sect. 9 we also discuss the polarization fraction of Tau
+A and Cyg A, and the polarized flux in W63, as further consistency
+tests for this polar efficiency calibration.
+5.2
+Internal calibration of the wide survey and consistency
+checks: evaluating unknown systematics
+Following the methodologies outlined in Planck Collaboration et al.
+(2014c) and Planck Collaboration et al. (2014d), we use internal
+consistency checks based on null test maps and other data splits of
+the wide survey in order to estimate the impact of systematic effects
+Table 17. Systematic effects in the MFI wide survey maps, evaluated in the
+maps degraded to 𝑁side = 64. The excess signal (last column) is computed
+as the quadratic difference between the values for half and ring null test
+difference maps. See text for details.
+Channel
+T,Q,U
+p-p (half)
+rms (half)
+rms (ring)
+Excess rms
+[𝜇K]
+[𝜇K]
+[𝜇K]
+[𝜇K]
+217
+T
+1177.8
+249.8
+224.6
+109.3
+217
+Q
+410.8
+88.8
+86.9
+18.5
+217
+U
+417.0
+87.7
+86.7
+12.8
+219
+T
+1736.3
+363.0
+297.8
+207.6
+219
+Q
+552.6
+116.0
+113.2
+25.5
+219
+U
+539.7
+115.1
+113.4
+19.2
+311
+T
+736.7
+153.8
+144.3
+53.3
+311
+Q
+283.4
+59.3
+58.5
+10.1
+311
+U
+282.5
+59.5
+58.3
+12.0
+313
+T
+538.5
+113.0
+101.7
+49.3
+313
+Q
+241.9
+51.2
+49.2
+14.3
+313
+U
+239.1
+51.0
+48.7
+15.1
+417
+T
+1586.5
+332.8
+304.5
+134.3
+417
+Q
+210.4
+45.0
+44.5
+6.8
+417
+U
+209.8
+44.8
+44.6
+4.1
+419
+T
+2053.8
+429.9
+352.1
+246.6
+419
+Q
+232.9
+48.8
+48.2
+7.3
+419
+U
+233.2
+49.5
+48.2
+11.3
+in the overall calibration. This is particularly useful for assessing
+the impact of "unknown systematics", i.e. those for which we do
+not have specific measurements or numerical simulations. For the
+MFI wide survey, and given that we want to focus on the relative
+calibration of the instrument, we use as a reference the set of null
+test maps and data splits labelled as "with common baselines" in
+Sect. 4.1.
+5.2.1
+Unknown systematics in real space
+Uncertainties due to (unknown) calibration or systematics effects at
+the pixel scale have been calculated using the HMDM for common
+baselines, degraded to 𝑁side = 64. At this resolution, each pixel
+roughly corresponds to the beam size. The reference mask for the
+analysis is the default one (sat+NCP+lowdec) as defined in Sect. 3.1.
+Table 17 lists the rms values and peak-to-peak (p-p) variation
+for the HMDM. Following Planck Collaboration et al. (2014c), the
+p-p values are computed as the difference between the 99 % and
+the 1 % quantiles in the pixel value distribution, in order to neglect
+possible outliers9. A comparison between these numbers for the
+half-mission null tests and those for the ring null tests is useful
+for checking residual calibration and/or systematic effects on large
+angular scales. Given that the ring null test maps cancel out possible
+variations in scales longer than 30 s (i.e. the duration of one azimuth
+scan), they can be used as our best estimate of the noise level, which
+includes white noise and 1/ 𝑓 on degree scales. Any variation on
+scales longer than one minute, due either to calibration uncertainties
+in the gain model or systematic effects, will appear as a signal excess
+in the HMDM. As illustration, the top panel in Fig. 14 shows the ring
+null-test difference maps for the 311 (horn 3 at 11 GHz) case. The
+results of this comparison are shown in Table 17. Column 5 presents
+the rms value for the ring difference maps, and column 6 shows the
+9 Note that for a Gaussian distribution, we should have p-p=4.65𝜎.
+MNRAS 000, 1–58 (2022)
+
+0
+50
+100150200
+-1
+0
+1
+2
+3
+4
+-6
+-4
+-2
+0
+2
+4
+6
+4
+00
+2
+0
+b
+-2
+-4
+-6
+ 11 GHz
+Q 11 GHz
+U 11 GHz
+86848280787674
+86 84 82 80 78 76 74
+86848280787674
+1 (deg)
+1 (deg)
+1 (deg)0
+20
+40
+60
+80
+100
+-1.0-0.50.00.51.01.5
+-1.0-0.50.00.51.0
+mK
+CMB
+6
+4
+0.0
+2
+0
+b
+-2
+-4
+-6
+I 19 GHz
+ 19. GHz
+U.19. GHz
+86848280787674
+86848280787674
+86848280787674
+1 (deg)
+1 (deg)
+1 (deg)QUIJOTE MFI wide survey
+31
+signal excess in the half-mission difference maps. Comparing these
+values with those in Tables 11 and 14 for the noise levels for the wide
+survey, we find that in polarization, the rms excess due to unknown
+systematics is well below the white noise levels, with typical values
+in the range 5–20𝜇K. In intensity, we find a similar situation for horn
+3 and the 17 GHz frequency maps of horns 2 and 4. For the two maps
+at 19 GHz (horns 2 and 4), the residuals are slightly larger than the
+white noise levels, but still well below the total noise contribution in
+those channels (column 5). As a reference, for horn 3, the residuals
+at beam scales are of the order of ∼ 50𝜇K. These numbers are used
+to complete the main table 16, appearing as "unknown systematics"
+in real space. As a conservative choice, the values for horns 2 and 4
+are combined linearly instead of using a quadratic combination.
+5.2.2
+Unknown systematics in harmonic space
+We use the ratio of cross-power spectra of the null test maps with
+some external maps, as the reference tool to validate the calibration
+in harmonic space. The use of cross-spectra to external maps min-
+imises the effects of noise bias on the power spectrum estimation.
+In practice, given two maps 1 and 2 that we want to compare, we
+compute
+𝐴1,2 =
+��
+𝐶1,X
+ℓ
+𝐶2,X
+ℓ
+�
+ℓ
+�
+X
+,
+(19)
+where 𝐶𝑖,X
+ℓ
+is the cross-spectrum of map 𝑖 (=1, 2) with some other
+external map X, with X running over all possible uncorrelated ex-
+ternal maps, and the brackets represent the (unweighted) average
+in a given multipole range (< ... >ℓ) or over all external maps
+(< ... >X), respectively. For completeness, we also evaluate the
+uncertainty on this parameter (𝜎𝐴1,2) as the standard deviation of
+those ratios over the external maps,
+𝜎𝐴1,2 =
+1
+√𝑛X
+𝑠𝑡𝑑𝑋
+��
+𝐶1,X
+ℓ
+𝐶2,X
+ℓ
+�
+ℓ
+�
+,
+(20)
+where 𝑛X is the number of external maps involved in the analysis.
+In this section, all cross-spectra are obtained using Xpol. The
+reference mask adopted for this computation is the default one
+(sat+NCP+lowdec), which preserves the declination range 6◦ ≤
+𝛿 ≤ 70◦. This mask is apodized using a 5◦ cosine function, as
+implemented in the NaMaster library (Alonso et al. 2019). All
+maps have been smoothed to a common resolution of one degree.
+For MFI, the ratios are evaluated and averaged within the multipole
+range ℓ = 30 to ℓ = 200. The lower value of ℓ = 30 guarantees
+that the pseudo-𝐶ℓ estimation is not affected by mode coupling due
+to incomplete sky coverage, and constitutes a conservative choice
+regarding possible large scale residuals due to RFI and atmosphere,
+as discussed in the previous section. As external maps, we decided
+to use low frequency maps (≤ 70 GHz) from satellites, in order to
+have similar foreground components to the signal in the QUIJOTE
+maps. In particular, we use the 9-year WMAP maps (Bennett et al.
+2013) for bands K, Ka, Q and V, and the PR2 Planck-LFI maps
+at 30, 44 and 70 GHz corrected from bandpass leakage (Planck
+Collaboration et al. 2016b).
+5.2.2.1
+Intra-nulltest calibration. We first evaluate the relative
+calibration of the wide survey, using the six null test maps described
+in Sect. 4.1, namely half (mission), rings, halfring, daynight, pwv
+and tbem. For each case, we compare the relative calibration of the
+Figure 24. Intra-nulltest calibration of the MFI widey survey. We show the
+consistency of the null test maps, for intensity (TT, top) and polarization
+(average of EE and BB, bottom).
+two maps in each pair ℎ1 and ℎ2, as in equation 19, and we evaluate
+the error bar using equation 20.
+Fig. 24 shows the result both for intensity (TT) and polarization
+(average of EE and BB) data. In intensity, we find a good consistency
+of all the different data splits well within one per cent. At 11 and
+13 GHz, the maximum discrepancy is found to be 0.3 %. The average
+of the six null test cases is consistent with one (perfect relative
+calibration) within 0.2 %. At 17 and 19 GHz, the maps from horn
+4 present a maximum discrepancy of 0.7 %, and the scatter of the
+six measurements stays within 0.5 %. Horn 2, which is known to be
+the noisiest one, presents the larger discrepancy of −1.6 % for the
+half-mission null test, and the average of the six values is consistent
+with one within 1 %.
+In polarization, we find larger values of the scatter, as expected
+due to the lower signal-to-noise ratios of these maps, although we
+remind that in this case our analysis also probes possible time vari-
+ations of the polarization efficiency values on top of the global cal-
+ibration. For horn 3, the maximum discrepancy is associated with
+the halfring null test, which presents deviations of +7 % for 311,
+and -8 % for 313. However, we note that this null test is expected to
+be noisier than the others, due to the lower number of independent
+crossings in each half. The average of the six measurements is fully
+consistent with one, and has a scatter of 2.9 % and 3.8 % for 11 and
+13 GHz, respectively. For horn 4, we find a maximum discrepancy
+of 6.4 %. The average of the six measurements is again consistent
+with one, and the scatter is 3.8 % and 2.8 % for 17 and 19 GHz.
+Finally, for horn 2, as in intensity, we find the largest scatter of the
+MNRAS 000, 1–58 (2022)
+
+Intra-nulltest calibration TT
+1.03
+half
+halfrings
+pwv
+rings
+daynight
+tbem
+1.02
+1.01
+A
+1.00
+0.99
+0.98
+217
+219
+311
+313
+417
+419
+Channelntra-nultestcalibrationEE+BB
+half
+halfrings
+pwv
+1.3
+rings
+daynight
+tbem
+1.2
+A
+1.1
+1.0
+0.9
+217
+219
+311
+313
+417
+419
+Channel32
+Rubiño-Martín et al.
+measurements. The largest discrepancy is found to be 17 % but with
+a large error bar. The average of the six measurements is slightly
+biased towards positive values of 𝐴 for 219, but not significantly
+(two sigmas). The scatter of the measurements is 5.9 % and 5.2 %
+for 217 and 219, respectively.
+In summary, the internal calibration scale of the MFI wide
+survey seems to be consistent within 0.7 per cent in intensity for all
+horns, reaching 0.2 % for horn 3. In polarization, we find consistency
+within 3–4 per cent in for horns 3 and 4, and within 10 % for horn 2.
+To put in context these values, it is useful to compare them with the
+expected scatter in the 𝐴 values in the case of a perfectly calibrated
+instrument with the realistic noise levels of the MFI wide survey.
+For this purpose, we have repeated this analysis using simulations
+including realistic 1/ 𝑓 noise levels as in Sect. 5 of Guidi et al.
+(2021). According to these simulations, the expected scatter of the
+six null tests in intensity is within 0.1–0.2 %, while in polarization
+we expect 2 % for horn 3 and horn 4 at 17 GHz, and we could have
+up to 5–6 % for horn 2 and horn 4 at 19 GHz. We stress that these
+numbers are driven by the 1/ 𝑓 noise in the maps, and therefore they
+represent the actual sensitivity of this method to detect calibration
+errors. Any calibration uncertainty due to systematic effects in the
+real data will add to these values.
+When comparing these values from simulations with those
+found for real sky measurements, we find that they are consistent in
+intensity, but the real data produce slightly larger scatter in polar-
+ization. This small excess of uncertainty in the polarization values
+from the real maps can be ascribed to polarization efficiency sys-
+tematic errors. As a conservative approach, we decided to quote as
+calibration uncertainty in Table 16 the final numbers obtained from
+this test, thus including also the 1/ 𝑓 noise contribution.
+5.2.2.2
+Inter-period calibration. We now evaluate the time sta-
+bility of the wide survey calibration, using the four maps per period
+described in Sect. 4.1.2, again for the case of "common baselines".
+We also note that period 1 only has observations at high elevations,
+so in order to have a common sky coverage for this comparison in
+the four maps, we restrict the analysis in this particular case to a
+sky mask covering the declination range 8◦ ≤ 𝛿 ≤ 50◦. As usual,
+this extended mask is apodized using a 5◦ cosine function, as im-
+plemented in the NaMaster library (Alonso et al. 2019). Fig. 25
+shows the comparison of the 𝐴 factors for the four maps by period
+used for the wide survey (periods 1, 2, 5 and 6), when compared to
+the total final map for each horn and frequency.
+In intensity, the internal consistency is found to be again better
+than 1 %. The largest discrepancy in absolute value is found for the
+map 419 in period 1, at the level of -1.5 %. The standard deviation
+of the four 𝐴 values for each horn and frequency is found to be
+∼ 0.5 % for channels in horns 2 and 3, and 0.7–1 % for horn 4. In
+polarization, we recall that some periods are not used for the final
+maps. In particular, period 1 is not used in polarization, period 2
+is not used for horn 4, and period 5 is not used for horn 2. The
+maximum discrepancy with respect to the final map is found in 313
+for period 5, at the level of −3.7 %. Taken as a whole, these values
+suggest that the calibration scale is stable within 1 per percent in
+intensity, and within 2 per cent in polarization, during the six years
+of observations covered by the wide survey.
+5.2.2.3
+Inter-horn calibration for horns 2 and 4.
+Given that
+the frequencies of 17 and 19 GHz are observed with horns 2 and 4,
+we also carry out an inter-horn comparison of the final wide survey
+maps at these frequencies using the same methodology as above,
+and where the 𝐴 factor in equation 19 now compares the ratio of the
+Figure 25. Inter-period consistency checks, in intensity (TT, top) and polar-
+ization (average of EE and BB, bottom). We show the 𝐴 factor computed as
+in equation 19, when comparing the map per period (i.e. using the data of
+that given period only) to the total final map, for each horn and frequency.
+two maps of a given frequency from the two horns. In this case, we
+obtain two values, 𝐴217,417 and 𝐴219,419. The results are displayed
+in Fig. 26 both for intensity (TT) and polarization (EE and BB, here
+plotted separately). We find that the relative calibration of the wide
+survey between horns 2 and 4 is consistent within 0.2 per cent in
+intensity. In polarization, this test is not providing very restrictive
+results due to the high noise levels of horn 2 in comparison to horn
+4. Nevertheless, we can conclude that the relative calibration of the
+two 17 GHz maps is found to be consistent within 2 per cent, while
+for 19 GHz we find consistency within 4 per cent if we average the
+values for EE and BB. In this later case, our simulations show that
+the separated values for EE or BB alone might differ by more than
+4 per cent in the ideal case of a perfect calibration, due to the (white
+plus 1/ 𝑓 ) noise levels.
+5.2.3
+Summary of the internal calibration tests
+The overall calibration uncertainty quoted for the QUIJOTE MFI
+wide survey maps is 5 % in intensity for all frequency maps, 5 %
+in polarization for 11 and 13 GHz, and 6 % in polarization for the
+combined 17 and 19 GHz maps (see last two rows in Table 16).
+These values are mainly limited by the physical modelling of the
+point-sources (Tau A, Cas A) used to calibrate the experiment. In
+intensity, all the tests in this section show that the internal consis-
+tency of the calibration and gain model, which spans 6 years of
+measurements, is within the one per cent level. In polarization, the
+MNRAS 000, 1–58 (2022)
+
+Inter-period calibration TT
+1.020
+p1
+p2
+p5
+p6
+1.015
+1.010
+1.005
+A 1.000
+0.995
+0.990
+0.985
+0.980
+217
+219
+311
+313
+417
+419
+ChannelInter-periodcalibrationEE+BB
+1.03
+p2
+p5
+p6
+1.02
+1.01
+1.00
+0.99
+0.98
+0.97
+0.96
+217
+219
+311
+313
+417
+419
+ChannelQUIJOTE MFI wide survey
+33
+Figure 26. Inter-horn consistency check between horns 2 and 4, in intensity
+(top) and polarization (bottom).
+internal consistency tests show that the calibration is controlled at
+the 2–3 per cent level for frequencies 11, 13 and 17 GHz, while for
+19 GHz, and particularly for horn 2, this uncertainty could be up to
+6 %. However, we note that in this later case, the quoted uncertainty
+includes calibration errors, polarization efficiency uncertainties and
+1/ 𝑓 noise contributions.
+5.3
+Other calibration tests
+5.3.1
+CMB anisotropies
+CMB anisotropies in intensity can be measured in the QU��OTE
+MFI wide survey maps using a cross-correlation with an external
+CMB template. We follow the methodology described and validated
+in Section 6.5 of Guidi et al. (2021), and use a template fitting
+method with two templates: a reference CMB map (mCMB), and
+a "foreground" map to account for chance alignments between the
+CMB and the Galactic foregrounds (f). The basic assumption is that
+the QUIJOTE map (mMFI) can be written as a linear combination
+of these two maps as
+mMFI = 𝐴mCMB + 𝐵f + n,
+(21)
+where 𝐴 and 𝐵 are the parameters of the linear combination, and n
+represents a noise component. Using the cross spectra of the QUI-
+JOTE maps with both external templates, 𝐶MFI,CMB
+ℓ
+and 𝐶MFI,f
+ℓ
+,
+we can extract both 𝐴 and 𝐵 parameters. As shown in Guidi et al.
+(2021), this method produces unbiased results for the CMB recon-
+struction (𝐴 = 1), provided that there is a perfect consistency with
+Figure 27. Relative amplitude of the CMB signal in the QUIJOTE MFI
+maps, using cross-correlations with the Planck SMICA map. Error bars are
+obtained using rotations of the CMB map. For consistency, we show that the
+average signal of the cross-correlation with rotated CMB maps is consistent
+with zero, as expected.
+the calibration of the CMB map. Thus, the method can be used as
+an additional calibration test.
+Here, we use as a reference the SMICA 2018 map (Planck Col-
+laboration et al. 2020d), but we have checked that consistent values
+are obtained using other versions of the Planck CMB map (NILC,
+COMMANDER, SEVEM). As foreground template, we use the
+WMAP 9-year K-band map (Bennett et al. 2013), after subtracting
+the CMB component. The analysis mask is the same as in Guidi
+et al. (2021), which combines the default QUIJOTE analysis mask
+(NCP+sat+lowdec) with the Planck common confidence mask for
+temperature analyses (Planck Collaboration et al. 2020d), apodized
+with a simple 2-degree smoothing. All cross-spectra in this section
+are computed using Xpol. Error bars are obtained using rotations
+of the CMB map in steps of Δ𝑙 = 18◦, as in Guidi et al. (2021).
+The analysis is carried out in the multipole range [100, 200], but
+consistent results are obtained in other ranges (e.g. we also tested
+[30, 200], although the overall significance is lower in this case due
+to the larger 1/ 𝑓 contribution of lower multipoles). The final results
+are shown in Figure 27 and Table 18. The CMB signal is detected
+in all channels, with a significance larger than 10-sigma in all cases.
+These error bars are consistent with the level of 1/ 𝑓 noise in the
+QUIJOTE maps (see Table 4 in Guidi et al. (2021)). We note that,
+due to the strongly correlated noise in the MFI intensity maps, es-
+timates from the same horn tend to deviate in the same direction.
+All values are consistent with 𝐴 = 1, providing an independent
+confirmation of the calibration scale of the maps. Finally, we also
+provide a combined measurement of the CMB signal present in the
+QUIJOTE MFI maps, using a weighted average combination of all
+channels and accounting for the noise correlation between frequen-
+cies of the same horn. The overall result (1.02 ± 0.03) provides
+a 35-sigma detection of the CMB anisotropies in the QUIJOTE
+MFI intensity maps, and shows a consistent calibration with Planck
+within three per cent.
+5.3.2
+CMB dipole
+As an additional calibration test, we present here the detection of the
+CMB dipole in the MFI wide survey maps, using a cross-correlation
+technique similar to the one used in the previous subsection for the
+CMB anisotropies. For this analysis, specific MFI wide survey maps
+are generated excluding the dipole removal and the atmospheric cor-
+MNRAS 000, 1–58 (2022)
+
+nter-hornscalibrationTT
+1.002
+1.001
+A
+1.000
+0.999
+17
+19
+Frequency[GHz]Inter-horns calibration EE,BB
+EE
+1.02
+BB
+1.00
+0.98
+0.96
+0.94
+0.92
+17
+19
+Frequency[GHz]CMB Cross-correlations l E[100, 20o]
+1.2
+1.0
+0.8
+A
+0.6
+<rot(A)>
+0.4
+0.2
+0.0
+217
+219
+3i1
+3i3
+417
+419
+Channels34
+Rubiño-Martín et al.
+Table 18. Relative amplitude (𝐴) of the CMB component in the QUIJOTE-
+MFI wide survey maps with respect to the SMICA Planck map, obtained with
+cross-correlations in the multipole range 100–200. Error bars are obtained
+using rotations of the CMB map.
+Channel
+A
+Uncertainty
+217
+1.080
+0.068
+219
+1.086
+0.086
+311
+1.010
+0.037
+313
+1.005
+0.033
+417
+1.030
+0.086
+419
+0.974
+0.097
+Combined
+1.019
+0.029
+Figure 28. MFI wide survey 311 (horn 3 at 11 GHz) map, with the dipole
+component not removed from the map. For display purposes, the map has
+been downgraded to resolution 𝑁side = 256.
+rection steps in the post-processing stage of the pipeline. Figure 28
+shows one example of these maps, for the case of horn 3 at 11 GHz.
+We use a template fitting method in real space with three tem-
+plates: a reference CMB dipole template map (mdip), a "foreground"
+map to account for the Galactic component (f), and a constant map
+accounting for a residual monopole term (𝐶). As in the previous
+section, we assume that the MFI wide survey maps (mMFI) can be
+written as a linear combination of those three templates as
+mMFI = 𝐴mdip + 𝐵f + 𝐶 + n,
+(22)
+where 𝐴, 𝐵 and 𝐶 are the three coefficients to be obtained and
+n represents the noise component. The dipole template map mdip
+is prepared following the methodology outlined in Sect. 4.4.2 of
+Guidi et al. (2021), including both the solar and orbital CMB dipole
+terms with the measured amplitudes by the Planck collaboration.
+The dipole prediction is generated at the TOD level, and then this
+is projected into a sky map using the PICASSO map-making algo-
+rithm. For the Galactic template, we use again the WMAP 9-year
+K-band map after subtracting the CMB component. For this analy-
+sis, all maps are degraded to a common resolution of one degree.
+The analysis mask combines the default QUIJOTE analysis mask
+(NCP+sat+lowdec), the Planck confidence CMB mask for temper-
+ature analyses (Planck Collaboration et al. 2020d), and a Galactic
+mask |𝑏| < 30◦, in order to avoid a possible bias in the dipole
+determination due to the Galactic emission.
+We first validate the methodology using end-to-end simula-
+tions of the MFI wide survey including the dipole component and
+realistic 1/ 𝑓 noise levels as in Guidi et al. (2021). We find that
+our approach provides unbiased estimates of the dipole amplitude
+Table 19. Fitting for the CMB dipole in the MFI wide survey maps. We
+present the relative amplitude with respect to the expected CMB dipole, and
+the associated uncertainty. See text for details.
+Channel
+Relative amplitude
+Uncertainty
+217
+1.04
+0.22
+219
+0.97
+0.47
+311
+0.88
+0.09
+313
+0.92
+0.12
+417
+0.99
+0.30
+419
+1.23
+0.67
+Combined
+0.92
+0.09
+(i.e. 𝐴 = 1) for all MFI frequency maps, with typical errors of
+few percent. We have also tested the impact of the three different
+corrections that are applied to the maps (RFI, FDEC and ATMOS)
+on the reconstructed dipole amplitude 𝐴. In summary, we find that
+including or not the RFI and FDEC corrections does not bias the
+recovered 𝐴 value. However, the ATMOS correction significantly
+affects the recovered amplitude, especially in the high frequency
+MFI bands. This is expected because the atmospheric templates are
+built on approximately one hour timescales, and on those scales the
+CMB dipole component is a very stable signal in the azimuth scans
+(rings). Because of this, the ATMOS correction is not applied for
+this analysis.
+The measured values in real data are presented in Table 19, for
+each one of the MFI wide survey maps separately. Error bars have
+been estimated using the following methodology. We rely on the
+null test maps for independent baselines as the most representative
+method to capture large angular scale noise in the maps. Thus, we
+repeat the analysis and detect the CMB dipole in the half1/2, pwv1/2,
+tbem1/2 and daynight1/2 maps. The reported values correspond to
+the average dipole of the 8 cases, and the error bar is the scatter of the
+8 measurements, taken to be a representative error of the method.
+We have tested that we obtain almost identical results if we carry
+out the analysis on maps with no FDEC and/or RFI corrections.
+Finally, we also present the weighted average combination of
+all channels, accounting for the correlation between frequencies of
+the same horn. The value is 𝐴 = 0.92 ± 0.09, which corresponds to
+a 10-sigma detection of the CMB dipole, and it is consistent with
+the Planck calibration within nine per cent.
+5.3.3
+Bright point sources and planets
+Bright radio sources and planets have been used extensively as a
+basic calibration test for MFI wide survey maps in several stages
+of the pipeline. Indeed, the maps in each period are recalibrated in
+order to match the Tau A model in intensity (Sect. 2.6). Below in
+Sect. 9 we present a detailed study of few bright objects (Tau A,
+Cas A, Cyg A, 3C274, W63, Jupiter and Venus), which could be
+seen as a further validation test of the overall calibration scale of
+the experiment.
+5.4
+Setting the zero levels
+The QUIJOTE MFI wide survey intensity maps produced by our
+default pipeline are insensitive to the true absolute zero level
+(monopole) of the sky emission. A monopole signal is essentially
+unconstrained for QUIJOTE MFI, as a global constant added to
+MNRAS 000, 1–58 (2022)
+
+H3, 11GHz (with dipole)
+-10
+mK
+10QUIJOTE MFI wide survey
+35
+the full TOD database is not changing the map-making solution af-
+ter the basic TOD processing. Indeed, in the post-processing stage
+maps are corrected of any residual monopole and dipole signals.
+In order to estimate the zero levels of these maps in intensity,
+we follow a methodology similar to the one adopted by WMAP
+(Bennett et al. 2003), and we assume a plane-parallel model for the
+Galactic emission. In that case, the zero level of the maps can be
+estimated by fitting a cosecant model of the form:
+Δ𝑇 = 𝐴 csc(|𝑏|) + 𝐵.
+(23)
+For this analysis, we use the smoothed maps at 1◦ angular reso-
+lution, and degrade them to 𝑁side = 64 in order to have approx-
+imately independent pixels. We carry out the fit independently in
+both hemispheres, using the Galactic latitude ranges 15◦ < 𝑏 < 90◦
+and −90◦ < 𝑏 < −15◦ for the northern and southern hemispheres,
+respectively. We mask the satellite band, and in the case of the
+northern sky, our analysis also excludes the region in Galactic lon-
+gitude corresponding to the North Polar Spur (0◦ ≤ 𝑙 ≤ 35◦). Error
+bars are computed using the scatter of the results around the mean
+value, when adding realistic noise simulations. For this analysis,
+we use 100 of the simulations described in Sect. 6.2. The reference
+results adopted here correspond to the northern hemisphere, due to
+the larger sky fraction covered by the QUIJOTE MFI footprint. For
+QUIJOTE MFI 11 GHz (horn 3), we have 𝐵 = −0.74 ± 0.20 mK,
+where the error bar includes both the effect of varying sky emission
+and the noise variance contained in the simulations. Similarly, for
+QUIJOTE MFI 13 GHz (horn3) we have 𝐵 = −0.59 ± 0.22 mK.
+The results for the southern hemisphere are consistent with those
+(−0.59 ± 0.27 mK and −0.42 ± 0.26 mK for 11 and 13 GHz, re-
+spectively), although they have larger error bars. For the other two
+frequency bands (17 and 19 GHz), and both for horns 2 and 4, the
+zero levels are statistically consistent with zero in both hemispheres
+(with typical error bars of 1.2–1.3 mK). These values are inserted
+in Table 16. Finally, we note that there are other methods in the
+literature for deriving the zero levels of radio maps (see e.g. Wehus
+et al. 2017), which could be applied here. However, we emphasize
+that those analyses should be done carefully, due to the special filter-
+ing of large angular scales (FDEC) applied to the MFI wide survey
+maps.
+5.5
+Polarization angle
+As described in Génova-Santos et al. (2023), the reference angle for
+each MFI observation is calibrated using daily Tau A observations.
+Our calibration scheme provides a reference angle for each period
+and channel, as this value changes across the spectral band, from
+horn to horn, and also with the instrument configuration. As this
+daily calibration might suffer from 1/ 𝑓 noise uncertainties, the final
+QUIJOTE MFI wide survey maps are recalibrated again using Tau
+A in each period (see Sect. 2.6). Here, we can evaluate the error
+budget associated with the polarization angle in the wide survey
+maps using Tau A. As a reference method, we use aperture pho-
+tometry in the polarization maps smoothed to 1 degree. We adopt
+an integration radius of 𝑟1 = 1.5◦ for the primary aperture, and
+an outer annulus between 𝑟1 and 𝑟2 =
+√
+2𝑟1 to correct for the lo-
+cal background contribution. The photometry results are described
+in Table 24 and Sect. 9. Table 20 presents the error budget in the
+polarization angle obtained using two methodologies. First, col-
+umn 2 presents the scatter (standard deviation) of the Tau A angle
+measurements obtained from the null test maps with independent
+baselines (half1/2, pwv1/2, ring1/2, daynight1/2 and halfring1/2).
+On the other hand, column 3 presents the statistical error obtained
+Table 20. Error budget for the polarization angle in the wide survey, based
+on Tau A photometry. We include the error budget from the scatter of the
+measurements in the different null tests (column 2) and the statistical error
+obtained from the photometry method (column 3).
+Channel
+Error (null tests)
+Error (stat.)
+(deg)
+(deg)
+217
+0.71
+0.91
+219
+0.98
+0.96
+311
+0.44
+0.50
+313
+0.34
+0.67
+417
+1.18
+0.64
+419
+1.70
+0.59
+Comb. 17 GHz
+0.96
+0.53
+Comb. 19 GHz
+1.43
+0.51
+Table 21. Comparison of the reconstructed angles in the QUIJOTE MFI
+wide survey data to WMAP-K (column 2), LFI30 (column 3) and MFI 311
+(column 4). See text for details.
+Channel
+WMAP-K
+LFI30
+MFI-311
+(deg)
+(deg)
+(deg)
+217
+−2.8 ± 1.5
+−3.3 ± 1.5
+−4.2 ± 1.5
+219
+0.8 ± 3.0
+0.4 ± 3.0
+−0.4 ± 2.9
+311
+0.6 ± 0.6
+−0.5 ± 0.6
+–
+313
+−1.2 ± 0.6
+−2.0 ± 0.6
+−2.2 ± 0.6
+417
+−1.2 ± 1.0
+−1.6 ± 1.0
+−2.3 ± 1.0
+419
+0.5 ± 3.6
+0.0 ± 3.6
+−0.9 ± 3.5
+Comb. 17 GHz
+−1.6 ± 0.9
+−2.2 ± 0.9
+−2.8 ± 0.9
+Comb. 19 GHz
+0.9 ± 3.2
+0.3 ± 3.2
+−0.5 ± 3.1
+from the propagation of the errors from the photometry measure-
+ment in the final maps. As a conservative approach, we keep the
+highest value of each pair as representative of the error budget in
+the angle determination from Tau A. We see that the uncertainty
+changes from 0.5◦ for 311, to 1.7◦ for 419.
+As a further consistency check for the polarization angle cal-
+ibration, we compare the measured MFI wide survey polarization
+angle maps with those from WMAP 9-year K-band map (Bennett
+et al. 2013) and Planck PR4 LFI30 data (Planck Collaboration et al.
+2020f). Table 21 presents the results of this comparison, including
+also an internal comparison to the MFI 311 map. The analysis is car-
+ried out smoothing all maps to 1 degree resolution, and degrading
+them to 𝑁side = 64, in order to match approximately the beam scale
+in one pixel. We use the standard analysis mask (NCP+sat+lowdec),
+but in addition, we keep only those high signal-to-noise pixels with
+a nominal uncertainty in the MFI 311 angle 𝜎𝜙311 ≤ 2◦. In order
+to avoid bright regions that might bias the comparison, pixels that
+have an absolute value in 𝑄 or 𝑈 that is greater than 2 mK in the
+WMAP K-band after being rescaled to 11.1 GHz using a spectral
+index of −3.0 are also flagged. Finally, we also exclude the bright
+Cygnus area removing all pixels within 5 degrees around the loca-
+tion (𝑙, 𝑏) = (80◦, 0◦). The resulting analysis area has 𝑓sky = 0.124.
+In order to correct for residual zero level differences between
+the MFI and the WMAP/Planck maps (e.g. due to unresolved point
+sources), we use a TT plot technique between WMAP-K and each
+MFI Stokes Q and U map within the analysis mask, and we remove
+the fitted zero levels from the MFI maps. We note that the resulting
+values are basically consistent with zero (within the error), but of
+MNRAS 000, 1–58 (2022)
+
+36
+Rubiño-Martín et al.
+the order of 20 𝜇K for 311 and 313. Although small, they might
+introduce measurable differences (at the level of a degree) in our
+analysis. For each MFI map, we compute the weighted mean of the
+difference between the two angles (e.g. 𝜙MFI − 𝜙WMAP for the first
+case), using as weights the inverse variance of the angle, which in
+turn is derived from the 𝑄 and𝑈 weight maps. Error bars in Table 21
+are generated with a Monte Carlo method using 100 of the noise
+simulations described in Sect. 6.2. We add each noise simulation to
+the corresponding MFI map, and repeat the same procedure. The
+error bar corresponds to the standard deviation of the 100 values.
+In general, all the measured differences are statistically consistent
+with zero given the noise uncertainty. MFI 311 (horn 3 at 11 GHz)
+is consistent with both WMAP-K and LFI30 within the quoted
+uncertainty of 0.6◦. The situation is similar for the 19 GHz maps
+(both horns 2 and 4). However, we note that there is a moderate
+tension with the MFI 313, which deviates in the case of LFI30 up
+to 3.3 sigmas, and the 17 GHz cases, which deviates 2.4 sigmas for
+the combined map of horn 2 and horn 4. In order to investigate this
+possible discrepancy, we have repeated the analysis but using all the
+different null test maps with independent baselines (half1/2, pwv1/2,
+ring1/2, daynight1/2, halfring1/2). The error is now computed as
+the standard deviation of all those values. The result for the MFI
+313 comparison with LFI30 now gives −2.0 ± 0.9, showing that
+maybe the error in this case is slightly underestimated. While we
+are still finding a discrepancy, the significance is now reduced to 2.2
+sigmas. Another point that we have studied is the possible impact
+of Faraday rotation in this comparison. Using the Galactic Faraday
+depth maps from Hutschenreuter & Enßlin (2020), we estimate that
+in our analysis region the mean rotation measure is −11.9 rad m−2.
+This would introduce differences of the order of approximately
+−0.4◦ between MFI311/MFI313 and LFI30. Although this value is
+not enough to explain the discrepancy, it helps to further decrease
+the tension below the 2 sigma level.
+The final results in Table 16 contain the worst case value based
+on the three values reported in this section (two values for Tau
+A in Table 20, and the standard deviation of the comparison with
+WMAP/Planck in Table 21).
+6
+SIMULATIONS
+6.1
+Sky signal
+Some of the analyses in this paper make use of sky simulations.
+Our reference sky simulations were developed within the context
+of the RADIOFOREGROUNDS project10, and are described in
+detail in Sect. 5.2 of Guidi et al. (2021). They contain different
+foreground components from the Planck FFP10 sky model (Planck
+Collaboration et al. 2020b,c), a CMB realization, and the CMB
+dipole contribution. For some applications, these sky simulations
+are projected into the MFI wide survey TODs, and the PICASSO
+map-making code is used to generate synthetic maps with the same
+flagging and number of hits as in the real wide survey data. These
+simulated data can also include a noise contribution, injected at
+the TOD level. As explained in this paper, this approach has been
+extensively used to validate some aspects of the pipeline (map-
+making, transfer function, null tests, determination of the CMB
+dipole, etc.). These sky simulations are also used below to evaluate
+the statistical errors associated with the power spectra (see Sect. 7).
+10 www.radioforegrounds.eu
+6.2
+Simulated noise maps
+In addition to the end-to-end noise simulations that have been pro-
+duced as explained in the previous subsection, we also construct
+noise simulations for the different channels (i.e., pair frequency-
+horn) maps, starting from the HMDM of totally independent splits.
+The simulations aim to account for the measured anisotropic be-
+haviour, spatial correlations and the correlations between the two
+frequency channels of the same horn in the wide survey maps (∼ 60–
+80% in intensity, and ∼ 20% in polarization, as seen in Sect. 4.3.3).
+The anisotropic behaviour follows the properties of the cor-
+responding Local Variance (LV) maps per channel and per Stokes
+parameter. These LV maps are estimated from the HMDM maps, by
+assigning, at each pixel at resolution 𝑁side = 512, the variance com-
+puted from the surrounding pixels at a given distance (39 arcmin;
+this value has been chosen as a compromise to have enough pixels to
+provide an accurate estimation and, at the same time, to preserve as
+much information at small scale as possible). The estimation of the
+variance takes into account only those pixels which are within the
+observed sky at each channel. Each one of the HMDM are normal-
+ized by dividing them by the square root of the corresponding LV
+maps. The non-observed pixels of the normalized HMDM are filled
+with a Gaussian random realization with unit dispersion, building
+in this manner extended-normalized HMDM maps.
+We now compute the noise spatial correlation by computing
+the TT, TE and BB angular power spectra (APS) of the extended-
+normalized HMDM maps, and from there, we derive a model of
+these APS. This is done by estimating a smoothed version of the
+observed APS of the extended-normalized HMDM maps for each
+channel, using a polynomial fit (of order 4), and by defining the max-
+imum multipole that provides a variance (at the map level) as close
+as possible to the one of the corresponding extended-normalized
+HMDM maps. Following this process, we end-up with a model for
+the noise correlations that provides the right power level.
+A noise simulation is now generated by drawing a Gaussian
+random map in harmonic space, following the corresponding mod-
+els of the noise APS for each frequency map. The maps (T, Q and
+U) are further multiplied by the square root of the corresponding LV
+map. We use the correlation coefficient between frequency maps of
+the same horn to further modify the simulated map of the second
+member of the pair (e.g., the 13 GHz frequency channel in the case
+of horn 3). In particular, we construct the final version of the simu-
+lated map of the second member of the pair as a linear combination
+of the first member of the pair and the initial version of the second
+map, taking into account the correlation coefficient. In this way,
+all the pixel-based statistics are maintained for the two members of
+the pair, as well as the correlation. The APS of the first member
+are also maintained, but, eventually, we modify the APS properties
+of the second member and the cross-correlation. The correlation at
+pixel level is imposed for T, Q and U. Notice that for the Stokes
+parameters this is done as if they were scalars. Nevertheless, the
+properties of the polarization intensity are preserved, although we
+are not able to reproduce the observed cross-correlation in 𝑃. We
+find that this approximation is the most adequate for our further
+analyses, since most of them are addressed in the pixel domain. As
+illustration, Figure 29 shows the power spectra for a subset of 100
+noise simulations for horn 3 at 11 GHz.
+7
+POWER SPECTRA OF THE WIDE SURVEY MAPS
+In this section we study the main properties of the auto- and cross-
+spectra of the MFI wide survey maps. We consider three masks,
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+37
+Figure 29. Power spectra (TT, EE, and BB) for 100 noise simulations of the
+311 map (horn 3 at 11 GHz). The red line shows the reference noise power
+spectrum for the half mission difference maps (labelled as HD) which was
+used to generate the simulations. The individual power spectrum for each
+simulation is shown in light grey, and the average of those 100 simulations
+in light blue.
+corresponding to different Galactic latitude cuts (|𝑏| > 5◦, 10◦
+and 20◦), which are always combined with the default QUIJOTE
+analysis mask (NCP+sat+lowdec). As usual, each of these three
+masks is apodized with a five degree apodization kernel and the
+cosine function implemented in Alonso et al. (2019). All spectra
+have been computed with the NaMaster code, enabling for the
+option of "purification" of E and B modes, which allows a better
+reconstruction of the E and B mixing matrix for cut-sky spectra. In
+Appendix E we discuss the validity of the use of this pseudo-Cℓ
+approach for the wide survey maps.
+Throughout this section, all power spectra have been corrected
+by the MFI beam window functions, as well as the pixel window
+function (which in this case corresponds to a HEALPix map with
+𝑁side = 512). Noise levels (𝑁ℓ) are estimated from the half-mission
+difference maps (with independent baselines), and then subtracted
+from the corresponding power spectra of the maps, in order to
+obtain the spectrum of the sky signal, 𝐶sky
+ℓ
+= 𝐶map
+ℓ
+− 𝑁ℓ. We have
+tested that using another estimate of the noise power spectra (e.g.
+the average of several null test difference maps) produces consistent
+results to those presented in this section. All spectra are binned
+using Δℓ = 10. In all figures in this section, we represent band
+power values 𝐷ℓ = ℓ(ℓ + 1)𝐶𝑋𝑌
+ℓ
+/2𝜋, where 𝑋,𝑌 ∈ {𝑇, 𝐸, 𝐵}.
+Uncertainties in the power spectra of the maps 𝜎(𝐶map
+ℓ
+) are
+estimated using 100 simulations including sky signal (Sect. 6.1) and
+realistic noise simulations (Sect. 6.2). The same noise simulations
+Figure 30. TT, EE and BB spectra for |𝑏| > 5◦, and for all frequencies
+(11, 13, 17, 19 GHz), represented as solid circles with their corresponding
+uncertainties. As a reference, dashed lines depict the noise spectra 𝑁ℓ for
+each case, using the same colour scheme.
+are also used to estimate the uncertainties in the noise level, 𝜎(𝑁ℓ).
+The quoted uncertainties in 𝜎(𝐶sky
+ℓ
+) are obtained as the quadratic
+sum of both 𝜎(𝐶map
+ℓ
+) and 𝜎(𝑁ℓ).
+Figure 30 shows the (auto) power spectra (TT, EE, BB) of
+the wide survey maps, for the particular case of the Galactic mask
+with |𝑏| > 5◦, combined with the default QUIJOTE analysis mask
+(NCP+sat+lowdec). We have a high significance detection of TT,
+particularly for the two lowest frequencies. At these frequencies,
+MNRAS 000, 1–58 (2022)
+
+Noise simulations 11GHz, H3
+Sims
+10-3
+[mk2]
+<sims>
+HD
+10-4
+10-5
+10-4
+[mk?]
+10-5
+10-6
+CBB [mK2]
+10-5
+10-6
+101
+102TT, Ibl>5°
+1.000
+0.100
+[mk?]
+D
+0.010
+11 GHz 
+13 GHz
+17 GHz
+19GHz
+0.001
+50
+100
+150
+200
+Multipole lEE, lbl>5°
+11
+GHz
+13 GHz
+17 GHz
+19 GHz
+mk²]
+10
+D
+10
+10-
+50
+100
+150
+200
+Multipole lBB,
+Ibl>5°
+11(
+GHz
+13 GHz
+17 GHz
+19 GHz
+D
+10
+50
+100
+150
+200
+Multipole l38
+Rubiño-Martín et al.
+Table 22. Best fit results obtained after fitting the model in equation 24 to
+the wide survey EE and BB power spectra at 11 GHz, in the multipole range
+30 < ℓ < 200. No colour corrections were applied when fitting the spectra.
+Mask
+|𝑏| > 5◦
+|𝑏| > 10◦
+|𝑏| > 20◦
+𝑓sky
+0.38
+0.34
+0.27
+EE and BB fitted separately
+𝐴EE [𝜇K2]
+1.52 ± 0.15
+1.05 ± 0.18
+0.81 ± 0.19
+𝐴BB [𝜇K2]
+0.52 ± 0.15
+0.20 ± 0.12
+0.18 ± 0.13
+𝛼EE
+−3.00 ± 0.16
+−2.72 ± 0.26
+−2.96 ± 0.36
+𝛼BB
+−3.08 ± 0.42
+−3.13 ± 0.87
+−3.12 ± 1.03
+𝑐EE [𝜇K2]
+0.07 ± 0.09
+−0.13 ± 0.11
+−0.09 ± 0.12
+𝑐BB [𝜇K2]
+0.10 ± 0.09
+−0.06 ± 0.09
+−0.09 ± 0.09
+𝐴BB/𝐴EE
+0.34 ± 0.10
+0.19 ± 0.12
+0.22 ± 0.18
+Joint EE and BB analysis
+𝐴EE [𝜇K2]
+1.49 ± 0.12
+0.97 ± 0.13
+0.78 ± 0.14
+𝛼EE (= 𝛼BB)
+−3.04 ± 0.13
+−2.83 ± 0.21
+−3.03 ± 0.29
+𝑐EE (= 𝑐BB) [𝜇K2]
+0.09 ± 0.06
+−0.08 ± 0.06
+−0.08 ± 0.07
+𝐴BB/𝐴EE
+0.36 ± 0.04
+0.26 ± 0.07
+0.26 ± 0.08
+the polarized emission is dominated by Galactic synchrotron. The
+EE synchrotron signal is clearly detected at large angular scales
+(ℓ ≲ 100) for 11 and 13 GHz, and the BB signal is also significantly
+detected in that range for 11 GHz. In the next three subsections
+we discuss the angular and frequency dependence of these spectra.
+A multi-frequency analysis of the power spectra of the MFI wide
+survey maps, in combination with WMAP and Planck data, will be
+presented in a separate paper (Vansyngel et al. 2023).
+7.1
+Fitting the EE and BB auto-spectra at 11 GHz
+Figure 31 shows the TT, EE and BB auto-spectra at 11 GHz, for
+three masks with Galactic latitude cuts |𝑏| > 5◦, 10◦ and 20◦.
+We focus here on the polarization spectra, EE and BB. Following
+Krachmalnicoff et al. (2018), we fit for these spectra in the multipole
+range 30 < ℓ < 300, using the following parameterization
+𝐶XX
+ℓ
+= 𝐴XX
+�
+ℓ
+80
+� 𝛼XX
++ 𝑐XX,
+(24)
+where 𝑋 ∈ {𝐸, 𝐵}, 𝐴XX is the amplitude of the spectrum at the pivot
+multipole ℓ = 80, 𝛼XX is the slope of the multipole dependence,
+and 𝑐XX is a global constant which represents the contribution of
+unresolved (Poisson distributed) radio sources.
+The power spectra are fitted using the EMCEE ensemble sam-
+pler (Foreman-Mackey et al. 2013), and using a standard Gaus-
+sian likelihood function. Our best-fit results, obtained from the
+marginalised posterior distributions for each parameter, are given
+in Table 22. First, we fit for the EE and BB power spectra sep-
+arately. In all three cases, the global constants 𝑐EE and 𝑐BB are
+statistically consistent with zero, as expected given the noise lev-
+els of the wide survey maps, and the expected contribution from
+radio sources at these frequencies, estimated to be ≲ 30 𝜇K.deg at
+11 GHz (Puglisi et al. 2018; Herranz et al. 2023). Both the EE and
+BB spectra present similar values of the slope, and no dependence
+on the Galactic latitude cut is observed. When combining the ratios
+of the EE and BB signals, we find that 𝐴BB/𝐴EE is of the order
+of 0.2 for the two higher Galactic cuts (|𝑏| > 10◦ and |𝑏| > 20◦),
+and we obtain 0.34 �� 0.10 for the lowest cut (|𝑏| > 5◦). In order to
+increase the significance of this measurement, and based on these
+Figure 31. TT, EE and BB spectra for QUIJOTE MFI 11GHz, as a function
+of the Galactic cut. Dashed lines represent the corresponding noise spectra
+𝑁ℓ for each case, using the same colour scheme.
+results, we repeat the analysis now assuming that both EE and BB
+spectra have the same slope (𝛼EE = 𝛼BB) and Poissonian terms
+contributions (𝑐𝐸𝐸 = 𝑐𝐵𝐵). In this case, we can fit simultaneously
+for the EE and BB spectra using four parameters (𝐴EE, 𝛼EE, 𝑐EE
+and 𝐴BB/𝐴EE). The results for the amplitudes and slopes are con-
+sistent with the values obtained in the previous case. Regarding the
+ratio of the amplitudes, we have now a higher significance, with
+𝐴BB/𝐴EE = 0.26 ± 0.08 for the |𝑏| > 20◦ case. In summary, the
+MFI wide survey data at 11 GHz show more power in the EE spectra
+MNRAS 000, 1–58 (2022)
+
+TT
+ 11GHZ
+1.00
+[mk²]
+0.10
+D
+Ibl> 5°
+Ibl>10°
+Ib1>20°
+0.01
+50
+100
+150
+200
+250
+Multipole lEE 1 1GHz
+Ibl> 
+lbl>10°
+lb/>20°
+mk²]
+10
+D
+10-
+50
+100
+150
+200
+Multipole lBB 
+3 11GHz
+Ibl> 5°
+lb/>10°
+lb/>20°
+mk²]
+10°
+D
+10
+50
+100
+150
+200
+Multipole lQUIJOTE MFI wide survey
+39
+Table 23. Best fit results obtained after fitting a constant model to the
+wide survey EB and TB power spectra at 11 GHz, in the multipole range
+30 < ℓ < 150. No colour corrections are applied.
+Mask
+|𝑏| > 5◦
+|𝑏| > 10◦
+|𝑏| > 20◦
+𝐴EB [𝜇K2]
+−0.014 ± 0.037
+0.002 ± 0.038
+0.043 ± 0.041
+𝐴EB/𝐴EE (ℓ = 80)
+−0.010 ± 0.025
+0.002 ± 0.038
+0.057 ± 0.059
+𝐴TB [𝜇K2]
+−0.17 ± 0.24
+−0.15 ± 0.20
+−0.21 ± 0.19
+𝐴TB/𝐴EE (ℓ = 80)
+−0.11 ± 0.16
+−0.15 ± 0.20
+−0.28 ± 0.28
+than BB, with a typical BB/EE ratio of a factor of 0.26. This value
+is approximately half of the equivalent BB/EE ratio for thermal dust
+emission, as derived from Planck observations at 353 GHz (Planck
+Collaboration et al. 2016a, 2020e).
+Our numbers for the synchrotron emission at 11 GHz can be
+compared with others in the literature. Planck Collaboration et al.
+(2020d) found 𝛼EE = −2.84 ± 0.05, 𝛼BB = −2.76 ± 0.09 and
+𝐴BB/𝐴EE = 0.34 for the synchrotron map at 30 GHz obtained with
+Commander (Eriksen et al. 2008), and analysing a sky area of
+𝑓sky = 0.78 and a multipole range ℓ = 4-140. Following a similar
+methodology to the one used here, Martire et al. (2022) carried
+out a combined analysis of WMAP-K band and Planck LFI30 data,
+finding very stable values for the slopes and BB/EE ratios as a
+function of the sky mask. For the case of a mask preserving 50 % of
+the sky, they obtain 𝛼EE = −2.79 ± 0.05, 𝛼BB = −2.77 ± 0.15, and
+𝐴BB/𝐴EE = 0.22 ± 0.02. In both cases, the values are consistent
+with our results at 11 GHz. On the other hand, using S-PASS data
+at 2.3 GHz, Krachmalnicoff et al. (2018) find significantly larger
+values of the BB/EE ratio for similar Galactic cuts in the southern
+sky, with values of 0.87 ± 0.02 for |𝑏| > 20◦, and 0.64 ± 0.03 for
+|𝑏| > 30◦.
+7.2
+TE, TB and EB spectra at 11 GHz
+Figure 32 shows the TE, EB and TB power spectra for the 11 GHz
+map, evaluated in the same sky masks as in the previous subsection
+(see also Fig. 31). Given that the power spectra of the HMDM is
+statistically consistent with zero in all three cases (TE, EB and TB),
+we do not apply the 𝑁ℓ correction in this subsection. Error bars are
+computed using the same methodology described above. However,
+for the EB spectra, we also add in quadrature the uncertainty on
+the power spectrum due to the polarization angle (Table 16), using
+equation 5 in Minami et al. (2019), and assuming that the underlying
+EB power spectrum is zero.
+We detect a positive cross-correlation between the total inten-
+sity T and the E-mode polarization (TE> 0) at large angular scales
+for the three considered Galactic cuts (up to ℓ ≲ 80 for |𝑏| > 5◦,
+and ℓ ≲ 50 for |𝑏| > 10◦ and |𝑏| > 20◦). Beyond ℓ >∼ 150, this
+TE cross spectrum becomes very noisy. We also find a null corre-
+lation in TB and EB in the range 30 ≲ ℓ ≲ 150, as expected for
+a parity-invariant emission process and an accurate calibration of
+the polarization angle. Beyond this multipole range, the error bars
+increase significantly, in particular for the TB case.
+We provide a quantitative measurement of the TB/EE and
+EB/EE ratios by fitting these spectra to a constant value (i.e. 𝐶TB
+ℓ
+=
+𝐴TB, and 𝐶EB
+ℓ
+= 𝐴EB), in the range 30 ≲ ℓ ≲ 150. The results are
+presented in Table 23, where we have used the EE fits from Table 22.
+For the synchrotron emission, the MFI 311 maps provide upper
+limits on the EB signal at the level of 4 per cent of the EE component
+at ℓ = 80 for the |𝑏| > 10◦ cut. These results are consistent with
+Figure 32. TE, EB and TB spectra for QUIJOTE MFI 11GHz, as a function
+of the Galactic cut.
+those found in Martire et al. (2022) for WMAP/Planck. Similarly,
+for the TB component we provide upper limits at the level of 20 % of
+the EE component. We recall that for the thermal dust emission, the
+Planck satellite found a positive TE signal at large scales, a weakly
+positive TB, and a EB statistically consistent with zero (Planck
+Collaboration et al. 2016a, 2020e).
+7.3
+Frequency dependence of the EE and BB signal
+We carry out a simultaneous fit of all the power spectra shown in
+Figure 30, using the parameterization from eq. 24, but assuming that
+the amplitudes are related via a power law dependence in frequency
+with a temperature spectral index 𝛽s,EE. In practice, the amplitude
+MNRAS 000, 1–58 (2022)
+
+TB
+3 11GHZ
+0.010
+0.005
+[mk²]
+0.000
+D
+-0.005
+ lbl> 5°
+/b/>10°
+Ibl>20°
+-0.010
+50
+100
+150
+200
+Multipole lTE 11GHz
+0.010
+0.005
+D. [mk"]
+0.000
+-0.005
+ lbl> 5°
+/b/>10°
+Ibl>20°
+-0.010
+50
+100
+150
+200
+Multipole lEB 11GHz
+0.006
+Ibl> 5°
+Ibl>10°
+Ibl>20°
+0.004
+0.002
+[mk?]
+0.000
+D
+-0.002
+-0.004
+-0.006
+50
+100
+150
+200
+Multipole l40
+Rubiño-Martín et al.
+at a given frequency channel 𝜈 is computed as:
+𝐴EE(𝜈) = 𝐴EE
+�
+𝜈
+11.1 GHz
+�2𝛽s,EE
+(25)
+where 𝐴EE represents the EE amplitude in the MFI 311 map. There-
+fore, for this fit, we have seven parameters, namely 𝐴EE and 𝛼EE for
+the amplitude and angular dependence of the synchrotron signal at
+11 GHz; the spectral index 𝛽s,EE describing the frequency depen-
+dence, and four constant coefficients 𝑐11
+EE, 𝑐13
+EE, 𝑐17
+EE and 𝑐19
+EE, ac-
+counting for the unresolved source contributions at each frequency.
+For this analysis, we also introduce the colour correction term based
+on the fitted spectral index, using values reported in Table 4. For
+the |𝑏| > 5◦ mask, we obtain 𝐴EE = 1.48 ± 0.13 𝜇K2, 𝛼EE =
+−2.97 ± 0.13 and 𝛽s,EE = −2.99 ± 0.14. Similarly, we repeat the
+analysis for the BB power spectra, finding 𝐴BB = 0.47 ± 0.12 𝜇K2,
+𝛼BB = −3.14 ± 0.33, and 𝛽s,BB = −2.79 ± 0.35.
+In both cases, the first two parameters are in agreement with
+the values reported in Table 22, taking into account that the colour
+correction term for the 11 GHz map and for a spectral index of
+𝛽 ≈ −3 is 0.967. The power spectrum of the synchrotron emission
+detected in the MFI wide survey maps scales with an average index
+of −2.99 for EE. The BB analysis is consistent with this value. The
+weighted average of the two values is −2.96 ± 0.13, consistent with
+the result of −2.96 ± 0.09 for the 50 per cent mask obtained in
+Martire et al. (2022) for the combination of WMAP-K and LFI30
+data. Our value also agrees with the study carried out in the next
+section for a real space analysis. A more detailed analysis on the
+reconstruction of the synchrotron spectral index with QUIJOTE
+MFI wide survey data is presented in two accompanying papers (de
+la Hoz et al. 2023a; Vansyngel et al. 2023).
+8
+BASIC PROPERTIES OF THE WIDE SURVEY MAPS
+8.1
+Spectral index of the MFI sky emission
+8.1.1
+Intensity
+We first investigate the spectral dependence of the intensity emission
+in the MFI wide survey maps. We use as a reference the MFI 11 GHz
+map, which presents the largest signal-to-noise, and we evaluate the
+spectral index of the sky emission when comparing it to the Haslam
+408 MHz (Haslam et al. 1982) and WMAP-K 9-year maps (Bennett
+et al. 2013). The version of the Haslam map used here corresponds
+to the destriped map from Remazeilles et al. (2015). For this spectral
+analysis in real space, all external maps are filtered using the FDEC
+procedure, degraded to 2◦ angular resolution, and then downgraded
+to 𝑁side = 64 resolution. Zero levels of all maps are corrected as
+in Sect. 5.4. Colour corrections for MFI-311 and WMAP-K are
+taken into account. The analysis region is restricted to the sky area
+covered by MFI 11 GHz, but excluding the satellite band (satband)
+as described in Sect. 3.1. For each 𝑁side = 64 pixel 𝑝 within the
+allowed mask, we solve for the spectral index 𝛽(𝑝) using a standard
+gaussian likelihood function L, which for the case of Haslam and
+MFI 11 GHz reads
+−2 ln L(𝑝) =
+�
+𝐼408(𝑝)
+�
+11.1
+0.408
+�𝛽( 𝑝)
+− 𝑐𝑐11(𝛽(𝑝))𝐼11(𝑝)
+�2
+𝜎(𝑝)2
+,
+(26)
+where 𝑐𝑐11 is the colour correction for MFI 11 GHz, and the noise
+term 𝜎 is evaluated using 1000 noise simulations for MFI (see
+Sect. 6.2), and accounting for a 10 per cent calibration error in the
+Figure 33. Spectral index of the intensity emission in the QUIJOTE 11 GHz
+map. Top: Spectral index of 𝛽408MHz−11GHz. The average index is 𝛽 = −2.9.
+As expected, the Galactic plane regions have a flatter index, while the regions
+off the plane have steeper values. Bottom: Spectral index of 𝛽11GHz−23GHz.
+The average spectral index in this case is 𝛽 ≈ −2.6. In this colour scale,
+dark red corresponds to AME dominated regions.
+Haslam map. Similarly, for the spectral index in intensity between
+MFI 11 GHz and WMAP-K, we use the same approach, accounting
+for the WMAP noise levels in the evaluation of the noise term.
+Figure 33 shows the results for the case of 𝛽408MHz−11GHz
+(top panel) and 𝛽11GHz−23GHz (bottom panel), both for the inten-
+sity emission. Figure 35 shows a histogram with the distribution of
+spectral indices in both maps. The median intensity spectral index
+𝛽408MHz−11GHz in the full analysis mask is −2.90, with a standard
+deviation of the values across the map of 0.20. This value is con-
+sistent with the expectation for the average synchrotron emission at
+these frequencies (see e.g. Platania et al. 1998; de Oliveira-Costa
+et al. 1999; Fernández-Cerezo et al. 2006). Moreover, the spatial de-
+pendence confirms the well-known steepening of the spectral index
+at high Galactic latitudes (see e.g. the 408 MHz–23 GHz spectral
+index map in Bennett et al. 2003).
+The 𝛽11GHz−23GHz intensity spectral index presents a much
+broader distribution of values, due to the presence of multiple spec-
+tral components (AME, free-free and synchrotron). In order to avoid
+extreme values for low signal-to-noise (high Galactic latitude) pix-
+els, in this case we also add a broad gaussian prior 𝛽 = −3.1±0.5 to
+the likelihood in equation 26. We have checked that this has a mini-
+mal impact in the final histogram. The median spectral index in this
+case is −2.59, and the standard deviation of the values is 0.43. Some
+of the bright AME dominated regions (Perseus, Lambda Orionis and
+rho Ophiucus) are clearly visible in dark red colour, while free-free
+dominated regions (e.g. Cygnus area) appear as light red. A more
+detailed study of the spectral properties of the sky emission in in-
+tensity along the Galactic plane (|𝑏| ≤ 10◦) in the MFI wide survey
+MNRAS 000, 1–58 (2022)
+
+Spectral index in intensity (Haslam to MFll1)
+-3.4
+β408MHz - 11GHz
+-2.2Spectral index in intensity (MFIl1 to WMAP-K)
+-3.4
+β11GHz - 23GHzQUIJOTE MFI wide survey
+41
+Figure 34. Top: Spectral index map of the polarized emission between
+QUIJOTE 11 GHz and WMAP 23 GHz. Bottom: the associated error map.
+maps is carried out in an accompanying paper (Fernandez-Torreiro
+et al. 2023). We also present a component separation analysis of the
+full MFI maps in de la Hoz et al. (2023b).
+8.1.2
+Polarization
+In polarization, the 𝛽11GHz−23GHz spectral index presents a cleaner
+interpretation in this case, as we are dominated by synchrotron
+emission only. Figure 34 presents the recovered polarization spec-
+tral index map, following the same methodology as for the intensity.
+The fit is carried out simultaneously in Stokes Q and U parameters,
+and in order to obtain a stable solution for high Galactic latitude
+pixels, we add a Gaussian prior 𝛽 = −3.1 ± 0.3 to the likelihood
+in equation 26. The bottom panel in that figure shows the asso-
+ciated error map, derived from the posterior distribution. Fig. 35
+includes also the histogram of these polarization 𝛽11GHz−23GHz
+values, showing that the median value is −3.09, and the standard
+deviation is 0.14. For comparison, we also include in this figure
+the histogram of spectral index values for the PySM synchrotron
+model 1 (Thorne et al. 2017), which in turn corresponds to "Model
+4" of Miville-Deschênes et al. (2008) calculated from a combi-
+nation of Haslam and WMAP 23 GHz polarization data using a
+model of the Galactic magnetic field. We find that in the same sky
+mask, the PySM spectral index map peaks at a higher value and
+presents a much narrower distribution (−2.99 ± 0.06). As a further
+consistency check, Appendix F presents the results for the same
+analysis carried out in this section, but using the MFI 13 GHz map
+as reference. We can see that both the mean values and widths of
+the distributions discussed here are consistently reproduced in this
+case. These values for 𝛽11GHz−23GHz in polarization are consistent
+with those measured in the range 22.8–100 GHz (𝛽s ∼ −3.1) by
+Figure 35. Histogram of spectral index values obtained from Figures 33 and
+34. We show in dashed lines the mean of the prior adopted in the determina-
+tion of the spectral index in polarization. For comparison, we also include
+the histogram of spectral index values from the PySM synchrotron model 1
+(Thorne et al. 2017). We recall that in the intensity case for 𝛽11GHz−23GHz
+(blue line), the 11 GHz map contains free-free and AME in addition to syn-
+chrotron, and thus the histogram presents a different shape with a broader
+distribution (see text for details).
+other authors (Dunkley et al. 2009; Fuskeland et al. 2014, 2021;
+Harper et al. 2022). A more detailed study of the spectral properties
+of the sky emission in polarization using the MFI wide survey maps
+in combination with WMAP and Planck, including a discussion on
+synchrotron spectral curvature, is carried out in an accompanying
+paper (de la Hoz et al. 2023a).
+8.2
+E- and B-mode maps
+As a complementary view of the relative power distribution in the E-
+and B-mode components for the synchrotron emission traced by the
+QUIJOTE MFI wide survey map, we have obtained in this section
+E- and B-mode maps. We use the full QUIJOTE observed area, but
+we mask the satellite band (satband) as described in Sect. 3.1. In
+order to minimize the impact of E/B mixing (Lewis et al. 2001),
+we apodize this analysis mask using a Gaussian kernel of 2◦. E-
+and B-mode maps are then generated using the standard HEALPix
+routines anafast and synfast, as
+𝐸( ˆ𝑛) =
+∞
+∑︁
+ℓ=2
+ℓ
+∑︁
+𝑚=−ℓ
+𝑎E
+ℓ,𝑚𝑌ℓ,𝑚( ˆ𝑛)
+𝐵( ˆ𝑛) =
+∞
+∑︁
+ℓ=2
+ℓ
+∑︁
+𝑚=−ℓ
+𝑎B
+ℓ,𝑚𝑌ℓ,𝑚( ˆ𝑛),
+(27)
+where 𝑎E
+ℓ,𝑚 and 𝑎B
+ℓ,𝑚 are the corresponding harmonic coefficients.
+Figure 36 shows the derived maps for MFI 11 GHz. As ex-
+pected from the power spectrum analysis in Sect. 7, there is sig-
+nificantly more power in the E-mode than in the B-mode map.
+Moreover, most of the brightest synchrotron features in the polar-
+ized intensity map (North Polar Spur, Fan region, Galactic centre)
+appear mostly in the E-mode, as expected due to the underlying
+magnetic field structure. Strongly polarized radio sources (Tau A,
+Cyg A) appear in these E- and B-mode maps with the characteristic
+quadrupole patterns with two positive and two negative lobes, and
+with the B-mode profile rotated by 45◦ with respect to the E-mode
+map (see e.g. Diego-Palazuelos et al. 2021).
+MNRAS 000, 1–58 (2022)
+
+-3.4
+β11GHz - 23GHz
+-2.7Error in the spectral index in polarization (MFll1 to WMAP-K)
+0
+(β11GHz - 23GHz)
+0.41.2
+Intensity（408MHz-11GHz）
+Intensity
+(11GHz-23GHz
+1.0
+Polarization
+（11GHz-23GHz
+Prior β=-3.1±0.3
+(normalized)
+PySM
+0.8
+0.6
+count
+0.4
+ixel
+0.2
+0.0
+-4
+-3
+-2
+Spectral index β42
+Rubiño-Martín et al.
+Figure 36. E and B-mode maps at 11 GHz. Most of the brightest features in
+the QUIJOTE map (North Polar Spur, Fan region, Galactic plane) appear in
+the E-mode map.
+8.3
+Bright structures in the polarized intensity maps
+The MFI wide survey polarized intensity maps are dominated by
+several bright and extended structures (see Figure 5). We discuss
+some of them in four accompanying papers: the Fan region (Ruiz-
+Granados et al. 2023), the Haze and Galactic center (Guidi et al.
+2023), the North Polar Spur (Watson et al. 2023), and other syn-
+chrotron loops and spurs (Peel et al. 2023).
+8.4
+AME in the MFI wide survey maps
+The MFI wide survey maps can be used to characterize the spec-
+tral properties of the AME, both in intensity and polarization. In
+particular, in Fernandez-Torreiro et al. (2023) we present a study
+of the diffuse AME emission in intensity along the Galactic plane
+(|𝑏| ≤ 10◦), while Poidevin et al. (2023) characterizes the SED in
+intensity for 52 compact sources with AME. Finally, two additional
+papers update the constraints in intensity and polarization of the
+AME in several Galactic regions (Tramonte et al. 2023; Lopez-
+Caraballo et al. 2023).
+9
+BRIGHT COMPACT SOURCES AND PLANETS IN THE
+WIDE SURVEY
+Despite its coarse angular resolution a high number of point sources
+are detected to high significance in the QUIJOTE MFI wide survey
+data. In a companion paper, where we discuss radio source de-
+tectability in these maps and derived statistical properties (Herranz
+et al. 2023), we show that we detect 235 point sources at S/N> 3
+at 11 GHz, while 85 are detected at S/N> 5. As a further consis-
+tency check of the global amplitude calibration, in this section we
+compare with models the recovered flux densities on four of the
+brightest sources having well characterised spectra (Tau A, Cas A,
+Cyg A and 3C274), and in two planets (Jupiter and Venus). We
+also calculate polarization flux densities in three bright polarized
+sources (Tau A, Cyg A and W63) to assess the accuracy of the
+polarization calibration.
+9.1
+Compact sources in intensity
+Tau A (also known as the Crab nebula), Cas A and Cyg A are
+amongst the brightest compact sources in the microwave range, and
+hence they have traditionally been used to calibrate experiments op-
+erating in this frequency range, including CMB experiments (Baars
+et al. 1977). Using WMAP data, Weiland et al. (2011) presented
+updated spectrum models in the range ∼ 1–300 GHz of these three
+sources and of 3C274 (also known as Virgo A or M87) and 3C58.
+Here we will focus on Tau A, Cas A, Cyg A and 3C274, while 3C58
+will be discussed in detail in Ruiz-Granados et al. (2023).
+Figure 37 shows the MFI wide survey maps on the positions of
+these four sources (Tau A, Cas A, Cyg A and 3C274) at 11 GHz and
+19 GHz, smoothed to a common angular resolution of 1◦. We note
+that Tau A and Cas A are the two main calibrators of QUIJOTE MFI,
+and thus we have much more sensitive data on these two sources
+obtained in raster mode. However we focus here on the wide survey
+maps only, in order to provide another consistency check for the
+calibration scheme.
+We extracted total-intensity flux densities on these maps us-
+ing a beam-fitting photometry (BF1d), consisting in fitting a 1◦-
+FWHM Gaussian beam superimposed on a flat background. We
+applied colour corrections following the methodology described in
+Génova-Santos et al. (2023), and using for each source a spectral
+index derived from the model. We compare these flux densities with
+spectral emission models that we have specifically derived for these
+sources, and which will be presented in a separate paper (Génova-
+Santos & Rubiño-Martín, in preparation). While in that paper we
+discuss models extracted with different photometry techniques, here
+we compare with models derived from WMAP and Planck maps
+convolved to a common resolution of 1◦ and using the same BF1d
+technique that we applied to QUIJOTE MFI. In particular, the Tau A
+model was used in Sect. 2.6 to recalibrate the wide survey maps. As
+it will be discussed in depth in Génova-Santos & Rubiño-Martín (in
+prep.), the uncertainties of these models are of the order of 3–5 %,
+and are driven not by the statistical noise of the individual obser-
+vations which is well below this value, but by systematic effects
+and calibration uncertainties of the fitted data, which lead to higher
+model-fitting residuals than would be expected in the presence of
+just statistical errors. In the cases of Tau A and Cas A, modelling of
+their secular decrease also introduces significant uncertainty.
+Final QUIJOTE MFI flux densities, for each horn and fre-
+quency, and relative deviation with respect to the fitted intensity
+models, are quoted in Table 24. All values are referred to date
+2016.3 (1 April, 2016), which roughly corresponds to the middle
+of the wide survey observations. It can be seen that in most cases
+the measured flux densities deviate less than 3–5 % with respect
+to the models, while in the case of Tau A, which is the main am-
+plitude calibrator, the deviations are within 1 % (the difference is
+not exactly zero due to the way the different periods are calibrated
+and combined; see section 2.6). The level of these deviations is
+expected given the typical model uncertainties, and therefore these
+results give full confidence to our global calibration strategy and the
+MNRAS 000, 1–58 (2022)
+
+MFI 11GHz - E modes
+mKMFI 11GHz - B modes
+mKQUIJOTE MFI wide survey
+43
+Figure 37. Minimaps of 5◦ × 5◦ size around four bright radiosources: Tau A (first row), Cas-A (second row), Cygnus-A (third row), and 3C274 (bottom row)
+at 11 GHz (first three columns are I, Q, and U), and 19 GHz (columns 4 to 6 are I, Q, U, respectively). For display purposes, we use the MFI maps degraded to
+a common angular resolution of one degree.
+quoted uncertainty (see Table 16). A detailed discussion on the vari-
+ability of these four sources (and others in the wide survey maps)
+can be found in Herranz et al. (2023).
+9.2
+Planets in intensity
+Venus and Jupiter are also detected to high significance in the QUI-
+JOTE MFI wide survey data. Owing to its orbital motion Venus
+declination varies roughly between ±27◦. Given Tenerife’s latitude
+(28.3◦ N), when its declination is close to 27◦ it is always visible in
+any of the elevations considered in the wide-survey. On the contrary,
+when it reaches its minimum declination of −27◦ it culminates at
+elevation 35.5◦, and therefore it is only picked up in observations at
+elevations 30 or 35◦. The distance between this planet and the Earth
+changes between 0.27 and 1.74 A.U., meaning that there is a factor
+≈ 42 variation between its minimum and maximum brightness. At
+19 GHz its flux density is expected to vary between 10.9 and 445 Jy.
+Then, during its inferior conjunction it is amongst the brightest
+sources on the sky at the QUIJOTE MFI frequencies. In the case
+of Jupiter, being an external planet, this variation is much smaller.
+Its distance to Earth varies between 4.1 and 6.4 A.U., producing a
+variation of its flux density at 19 GHz between 26.1 and 61.1 Jy.
+Between 2012 and 2016 its declination was always positive, reach-
+ing 23◦, meaning that it was picked up in most of the wide survey
+data. Between 2016 and 2018 its declination dropped below zero,
+reaching −22◦, and therefore during this period it was only visible
+on the wide survey observations performed at low elevations.
+While further details will be given in a future paper where
+we will discuss planets and other bright astronomical sources, here
+we briefly describe the procedure we have developed to estimate
+planets’ brightness temperatures. We implemented a specific map-
+making in which we rotate the coordinates of QUIJOTE MFI wide
+survey data to planet-centred coordinates, to produce planet-centred
+maps. We use the same final calibrated data that were used to pro-
+MNRAS 000, 1–58 (2022)
+
+50100150200250300
+-20
+-15
+-10
+-5
+0
+0.5
+0.0
+0.5
+1.0
+mKcMB
+mK
+CMB
+-4
+(deg)
+-5
+-6
+b
+-7
+-8
+Tau A I 11 GHz
+Tau A Q 11 GHz
+Tau A U 11 GHz
+187186185184183
+187186185184183
+187186185184183
+1 (deg)
+1 (deg)
+1 (deg)0
+20
+40
+60
+80
+100
+-6
+-4
+-2
+0
+0.0
+0.1
+0.2
+0.3
+mKcMB
+-4
+(deg)
+-5
+-6
+b
+-7
+-8
+Tau A I 19 GHz
+Tau A Q 19 GHz
+Tau A U 19 GHz
+187186185184183
+187186185184183
+187186 185 184183
+1 (deg)
+1 (deg)
+1 (deg)50
+100
+150
+200
+250
+0.5
+0.0
+0.5
+1.0
+0.5
+0.0
+0.5
+1.0
+mK
+CMB
+mK
+CMB
+0
+-1
+(deg)
+2-
+b
+-3
+-4
+Cas A I 11 GHz
+Cas A Q 11 GHz
+Cas A U 11 GHz
+114 113 112 111110
+114 113 112 111 110
+114 113 112 111 110
+1 (deg)
+1 (deg)
+1 (deg)10
+20
+30
+40
+50
+60
+-0.2
+-0.1
+0.0
+0.1
+0.050.000.050.100.15
+mK,
+CMB
+mK,
+CMB
+0
+-1
+(deg
+2-
+b
+-3
+-4
+Cas A I 19 GHz
+Cas A Q 19 GHz
+Cas A U 19
+GHz
+114 113 112 111 110
+114113112111110
+114 113 112111110
+1 (deg)
+1 (deg)
+1 (deg)20
+40
+60
+80
+100
+0
+1
+2
+3
+-4
+-3
+-2
+-1
+0
+mKcMB
+8
+7
+00
+(deg
+6
+b
+5
+4
+Cyg A I 11 GHz
+Cyg A Q 11 
+ GHz
+Cyg A U. 11 GHz
+78
+76
+75
+74
+78
+76
+75
+74
+78
+76
+75
+74
+1 (deg)
+1 (deg)
+1 (deg)0
+5
+10
+15
+20
+25
+30
+0.0
+0.10
+0.20
+0.3
+-0.2
+0.1
+0.0
+8
+7
+80
+(deg
+6
+b
+5
+4
+Cyg A I 19 GHz
+Cyg. A..
+Q
+19
+GHz
+Cyg A U 19 GHz
+78
+76
+75
+74
+78
+76
+75
+74
+78
+76
+75
+74
+1 (deg)
+1 (deg)
+1 (deg)0
+5
+10
+15
+20
+25-0.3-0.2-0.10.00.10.2
+0.4
+-0.2
+0.0
+0.2
+mKcMB
+76
+80 75
+(deg
+b
+74
+73
+3C274 I 11 GHz
+3C274
+11 GH:
+C274 U 11 GHz
+72
+286 285 284 283 282
+286 285 284 283 282
+286 285 284 283 282
+1 (deg)
+1 (deg)
+1 (deg)0
+1
+2
+3
+4
+5 -0.10
+-0.05
+0.00
+0.05
+0.10
+0.0
+0.10
+mK
+CMB
+mK,
+CMB
+76
+a0 75
+(deg
+b
+74
+73
+3C274 I 19 GHz
+3C274 Q 19 GHz
+3C274 U 19 GHz
+72
+286285284283282
+286285284283282
+286 285 284 283 282
+1 (deg)
+1 (deg)
+1 (deg)44
+Rubiño-Martín et al.
+Table 24. Flux densities (Jy), in intensity and in polarization, extracted from the QUIJOTE MFI wide survey maps at one degree resolution on Tau A, Cas A,
+Cyg A and 3C274. Intensity measurements are based on BF1d photometry, while the polarization measurements used AP1d. For the intensity measurements,
+inside parentheses we quote the percent deviation of flux densities with respect to predictions from spectral models. Tau A and Cas A values are referred to an
+effective date corresponding to 1 April 2016. All flux densities include colour corrections.
+Source
+Stokes
+311 (11.1 GHz)
+313 (12.9 GHz)
+217 (16.7 GHz)
+417 (17.0 GHz)
+219 (18.7 GHz)
+419 (19.0 GHz)
+Tau A
+I
+440.0 ± 0.9 (-0.8)
+427.4 ± 0.8 (+0.7)
+391.2 ± 0.8 (-0.5)
+393.4 ± 0.8 (+0.6)
+377.9 ± 0.7 (-0.6)
+378.8 ± 0.8 (+0.2)
+Q
+−29.27 ± 0.51
+−31.20 ± 0.51
+−28.00 ± 0.83
+−28.12 ± 0.43
+−26.36 ± 1.52
+−28.42 ± 0.66
+U
+0.63 ± 0.51
+0.90 ± 0.73
+1.43 ± 0.89
+1.05 ± 0.63
+0.34 ± 0.89
+1.87 ± 0.59
+Cas A
+I
+340.9 ± 1.8 (-1.1)
+309.7 ± 1.8 (-0.5)
+255.8 ± 1.9 (-2.4)
+256.3 ± 1.9 (-1.0)
+236.2 ± 2.1 (-2.8)
+235.7 ± 1.9 (-2.0)
+Q
+−1.18 ± 0.62
+−0.01 ± 0.53
+0.32 ± 0.57
+−0.93 ± 0.32
+−0.34 ± 0.65
+−1.25 ± 0.64
+U
+0.15 ± 0.34
+−0.90 ± 0.39
+0.26 ± 0.47
+−0.28 ± 0.45
+1.18 ± 0.72
+0.29 ± 0.51
+Cyg A
+I
+129.3 ± 1.0 (-4.1)
+108.7 ± 1.0 (-3.5)
+79.2 ± 1.0 (-4.3)
+78.1 ± 1.0 (-3.5)
+69.5 ± 0.9 (-3.8)
+67.5 ± 1.0 (-4.8)
+Q
+3.93 ± 0.61
+1.69 ± 0.64
+−0.55 ± 0.54
+0.41 ± 0.45
+−1.24 ± 0.66
+0.59 ± 0.38
+U
+−5.95 ± 0.44
+−4.64 ± 0.39
+−2.23 ± 0.59
+−1.60 ± 0.45
+−1.52 ± 0.98
+−1.26 ± 0.44
+3C274
+I
+34.2 ± 0.1 (-5.3)
+30.9 ± 0.1 (-3.8)
+25.6 ± 0.2 (-3.1)
+25.9 ± 0.2 (-0.5)
+22.3 ± 0.3 (-8.1)
+24.0 ± 0.3 (+0.2)
+Q
+−0.26 ± 0.48
+0.39 ± 0.52
+−0.54 ± 0.77
+−0.19 ± 0.38
+0.45 ± 1.10
+−0.24 ± 0.48
+U
+−0.74 ± 0.44
+−0.81 ± 0.56
+−2.14 ± 0.72
+−0.97 ± 0.45
+−2.63 ± 1.19
+−1.35 ± 0.47
+duce the final maps that are presented in this paper. In order to ac-
+count for the 1/𝑑2 effect we define distance bins (3 bins for Jupiter
+and 6 for Venus), and produce individual maps for each bin. We
+have verified in the final map that the (symmetrized) beam shape
+is well preserved, this being a health check both for the tailored
+map-making that we use here as well as for the pointing model. On
+these maps we apply a beam-fitting photometry to derive flux den-
+sities for each distance bin and for each horn/frequency. These flux
+densities are then colour-corrected using a Rayleigh-Jeans spectrum
+(spectral index 𝛼 = 2). In addition to data maps for each redshift bin
+we produce maps of 1/𝑑2 using the same noise weights and flags
+that are applied to the data. These maps are later used to calculate
+an effective distance at the position of the planet. Using this infor-
+mation we fit the flux densities measured in each bin to a 1/𝑑2 law
+in order to derive the final brightness temperatures.
+Our Venus and Jupiter brightness temperatures derived from
+QUIJOTE MFI are listed in Table 25 and plotted in Figure 38, in
+comparison with other data at similar frequencies, as well as with
+various models giving the spectral dependency of the brightness
+temperatures of these planets. In both cases we have corrected for the
+planet absorption of the CMB monopole, and therefore the quoted
+values represent the intrinsic brightness temperature of the planets.
+In the case of Venus, it is seen that the ancillary measurements seem
+a bit high with respect to the Bellotti (2015) and Fahd (1992) models
+and therefore we performed a power-law fit to the data in the range
+7–100 GHz (dashed line in the figure) and use this fit as a reference
+to compare with the QUIJOTE MFI values. In the case of Jupiter we
+use as reference the model of Karim et al. (2018), which seems to
+trace better the ancillary data, and in particular the VLA data from
+de Pater et al. (2019) below the ammonia absorption at 23 GHz. As
+can be seen in Table 25, both for Venus and Jupiter the QUIJOTE
+MFI measurements deviate always less than 5 % from the models
+(note that in some cases the statistical error bar is larger than this
+value), which bestows confidence to our calibration strategy.
+9.3
+Polarized sources
+Figure 37 also shows wide-survey polarization maps of Tau A,
+Cas A, Cyg A and 3C274, projected in Galactic coordinates and
+convolved to an angular resolution of one degree. Clear polarized
+emission is seen in Tau A, mainly concentrated in the 𝑄 map,
+as expected due to its polarization angle (see e.g. Weiland et al.
+2011). The 𝑈 map shows the typical cloverleaf pattern (with the
+expected peak-to-peak amplitude of ∼ 1 % with respect to the total
+intensity) arising from the differences between the two co-polar
+beams (Génova-Santos et al. 2023). This pattern is also visible in the
+𝑄 and 𝑈 maps of Cas A, more notably at 11 GHz. Due to it being a
+very young shell-type supernova remnant (SNR), the magnetic field
+of Cas A is expected to be radial (Anderson et al. 1995). Being ∼ 5′
+across, this source is unresolved by the QUIJOTE MFI beam and
+therefore we expect zero integrated polarization. Clear polarized
+emission is also seen in Cyg A. A rotation of the polarization angle
+is apparent between 11 and 19 GHz, which is due to the two jets of
+this radio galaxy having different rotation measures, the so-called
+Laing-Garrington effect (Laing 1988). In the case of 3C274, we
+only have a marginal polarization detection in the 𝑈 maps. This is
+expected given our noise levels (between 0.5–1 Jy), and the fact that
+the measured polarization fraction at 23 GHz is approximately 4 %
+(Weiland et al. 2011).
+In order to minimize systematic effects introduced by differ-
+ences between the two co-polar beams, we extract flux densities
+in polarization through an aperture photometry technique on maps
+smoothed to one-degree angular resolution (AP1d). The circular
+aperture radius (𝑟1) is taken to be 𝑟1 = 1.5◦ for Tau A and 3C274,
+and 𝑟1 = 1.3◦ for Cas A and Cyg A, due to the larger foreground
+contamination in the surroundings of the latter two sources. In the
+case of Cas A, we also mask the region centred at Galactic co-
+ordinates (𝑙, 𝑏) = (111.11◦, −0.53◦), using an exclusion radius of
+0.7◦. The background emission in all four cases is corrected using
+the mean of the signal in the annulus between 𝑟1 and 𝑟2 =
+√
+2𝑟1.
+Table 24 shows the Stokes 𝑄 and 𝑈 flux densities measured on Tau
+A, Cas A, Cyg A and 3C274.
+We now discuss the first three cases in detail, as well as the
+bright polarized emission in W63. For this discussion, we also ap-
+ply the same methodology (i.e. AP1d for polarization and BF1d for
+intensity) to derive the photometry values for these sources using
+WMAP 9-year data (Bennett et al. 2013) and Planck 2018 maps
+(Planck Collaboration et al. 2020a) at the common one-degree res-
+olution. Specially for the cases of Tau A and Cas A, and for Planck
+LFI, we correct for the intensity-to-polarization leakage due to band-
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+45
+Table 25. Brightness temperatures (in Kelvin) of Jupiter and Venus extracted from the QUIJOTE MFI wide survey data. Inside parentheses we quote the
+percent deviation with respect to predictions from spectral models.
+Planet
+311 (11.1 GHz)
+313 (12.9 GHz)
+217 (16.7 GHz)
+417 (17.0 GHz)
+219 (18.7 GHz)
+419 (19.0 GHz)
+Jupiter
+176.3 ± 0.4 (+0.8)
+170.3 ± 1.7 (+2.6)
+153.9 ± 8.7 (+1.0)
+148.7 ± 2.2 (-1.9)
+145.7 ± 14.5 (-0.9)
+141.0 ± 8.0 (-3.6)
+Venus
+578.3 ± 14.4 (-4.6)
+568.2 ± 4.7 (-2.5)
+546.6 ± 5.7 (+0.4)
+533.1 ± 9.5 (-1.6)
+526.7 ± 10.9 (-0.4)
+518.0 ± 11.5 (-1.6)
+Figure 38. Venus (top) and Jupiter (bottom) brightness temperatures derived
+from the QUIJOTE MFI wide survey data (red and yellow) in comparison
+with ancillary data, and with various models. Venus data have been obtained
+from Bellotti (2015); Hafez et al. (2008); Dahal et al. (2021), while the
+plotted models are from Bellotti (2015); Fahd (1992). We also show a
+power-law fit to the observed data in the range 7–100 GHz that we use to
+compare with the QUIJOTE MFI measurements. Jupiter data come from
+Hafez et al. (2008); Gibson et al. (2005); de Pater et al. (2019); Karim et al.
+(2018); Weiland et al. (2011); Planck Collaboration et al. (2016e, 2017b).
+We plot the model by Karim et al. (2018) and the ESA1 model.
+pass mismatch following the methodology described in Appendix C
+of Planck Collaboration et al. (2016h), using the maps of projection
+factors described in Planck Collaboration et al. (2016c), and the
+spectral index of each source derived from the intensity SED.
+9.3.1
+Tau A and Cyg A
+Figure 39 shows our results for the polarization fractions in Tau A
+and Cyg A at one degree resolution. We include also our WMAP (for
+both sources) and Planck (only for Tau A) measurements, as well
+as ancillary measurements both for Tau A (Kuz’min & Udal’Tsov
+1959; Mayer & Sloanaker 1959; Mayer et al. 1962; Davies & Ver-
+schuur 1963; Hollinger et al. 1964; Mayer et al. 1964; Morris &
+Berge 1964; Boland et al. 1966; Gardner & Whiteoak 1966; Hobbs
+& Haddock 1967a; Sastry et al. 1967; Satoh et al. 1967; Hobbs
+1968; Hollinger & Hobbs 1968; Mayer & Hollinger 1968; Seielstad
+& Weiler 1968; Johnston & Hobbs 1969; Dmitrenko et al. 1970;
+Wright 1970; Green et al. 1975; Hafez et al. 2008; Aumont et al.
+2010) and for Cyg A (Mayer et al. 1962; Hollinger et al. 1964; Sobol-
+eva 1966; Boland et al. 1966; Mezger & Schraml 1966; Hobbs &
+Haddock 1967b).
+For Tau A, we show in solid grey lines Monte Carlo realiza-
+tions of a simple model for the spectral dependency of its polar-
+ization fraction that accounts for the Faraday depolarization (Burn
+1966), and that is consistent with both the ancillary measurements
+at low frequencies (𝜈 ≲ 10 GHz) and existing measurements of
+the Faraday dispersion in the region (e.g. Bietenholz & Kronberg
+1991). This model will be described in detail in a separate paper
+(Génova-Santos & Rubiño-Martín, in preparation). We note that in
+our recalibration strategy of the MFI wide survey maps, we use
+Tau A for fixing the intensity calibration scale and the polarization
+angle, while the polarization amplitude is essentially given by inde-
+pendent polarization efficiency measurements. Thus, this analysis
+provides a consistency test on the MFI polarization calibration.
+Cyg A data points clearly show the effect of the Faraday de-
+polarization produced by the Laing-Garrington effect. It is evident
+from this plot that the maximum alignment between the polarization
+directions of the two jets occurs at ≈ 10 GHz, and then the measured
+polarization fractions decrease in both sides of the spectrum. Mod-
+elling this effect is complicated and beyond the scope of this paper.
+The QUIJOTE MFI measurements are in good agreement with the
+other measurements, again providing confidence in our calibration
+strategy.
+9.3.2
+Cas A
+Figure 40 shows the polarization fraction measured in Cas A with
+QUIJOTE MFI. We also include our photometry results for WMAP
+and Planck, and ancillary data from the literature (Mayer et al. 1962;
+Hollinger et al. 1964; Hobbs & Haddock 1967a; Sastry et al. 1967;
+Seielstad & Weiler 1968; Vinyaikin 2014). All values are noise-
+debiased using the PMAS estimator (Plaszczynski et al. 2014). The
+intensity-to-polarization leakage due to the co-polar beam asym-
+metry is almost cancelled in the integrated flux densities thanks to
+the positive and negative structure of this pattern, leading to inte-
+grated polarization fractions in QUIJOTE of around ∼ 0.3 %. Note
+that similar levels are detected in WMAP, and could also be due to
+beam effects as discussed in Weiland et al. (2011). At face value,
+these numbers can be considered as a conservative upper limit on
+the overall intensity-to-polarization leakage in the MFI wide survey
+maps.
+MNRAS 000, 1–58 (2022)
+
+46
+Rubiño-Martín et al.
+Figure 39. Consistency checks on polarized sources detected on the QUI-
+JOTE MFI the wide survey. We show polarization fractions measured in
+Tau A and in Cyg A, in comparison with our WMAP and Planck results
+obtained using the same methodology, and with ancillary measurements
+(see the complete list of references in the main text). In the case of Tau
+A we overplot in grey models for the polarization fraction that account for
+Faraday rotation. The Cyg A data show the Laing-Garrington effect arising
+from different rotation measures in the two lobes of this galaxy.
+9.3.3
+W63 region
+As an additional test of the polarization calibration of the MFI,
+we also investigate the polarized intensities of W63, another SNR
+which appears as a very bright extended structure in the polarization
+maps at these frequencies. The top panel in Fig. 41 shows the MFI
+11 GHz Stokes I, Q and U maps for this object. The total-intensity
+emission of W63 is practically embedded inside the emission of
+the Cygnus X star-forming complex, so it is difficult to extract re-
+liable total-intensity flux density estimates in this case. However,
+the polarization signal is reasonably isolated. Thus, we only discuss
+its polarized flux density here. In order to capture all the flux in
+the region, we use an aperture radius of 𝑟1 = 2◦. As in the pre-
+vious cases, we carry out this analysis in the smoothed maps at
+one degree resolution (i.e. AP1d photometry). The bottom panel in
+Fig. 41 shows the SED in polarized intensity 𝑃 =
+√︁
+𝑄2 + 𝑈2 derived
+from our photometry measurements, including also our results for
+WMAP and Planck applying the same methodology. All values are
+noise-debiased using the PMAS estimator. Error bars account for
+the photometry error plus the corresponding calibration uncertain-
+ties added in quadrature. For MFI, we use the values reported in
+Figure 40. Polarization fractions measured on Cas A in QUIJOTE MFI wide
+survey data, in comparison with other measurements. WMAP and Planck
+results are obtained using the same methodology as for the MFI maps values.
+The complete list of ancillary measurements is given in the main text. Upper
+limits are represented with arrows. At degree-beam scales, the polarized
+emission of Cas A is expected to be zero, so these measurements serve as
+a consistency check for the overall intensity-to-polarization leakage of the
+MFI wide survey maps.
+Table 16, while for WMAP and Planck data we adopt the conser-
+vative value of 3 %, as done for similar analyses (see e.g. Planck
+Collaboration et al. 2014a; Poidevin et al. 2019; Cepeda-Arroita
+et al. 2021). The WMAP and Planck polarized intensity flux in
+W63 can be fitted to a power-law 𝑃 = 6.97(𝜈/22.8GHz)−0.68 Jy,
+that is depicted by the dashed line. The QUIJOTE MFI data are con-
+sistent within 1-sigma with the fitted model, which gives additional
+confidence to our calibration strategy in polarization.
+10
+DATA RELEASE AND DESCRIPTION OF THE DATA
+PRODUCTS
+Together with this paper, there is a series of further publications
+containing scientific results derived from the QUIJOTE-MFI wide
+survey maps presented here. The titles of all the papers in the series
+begin with "QUIJOTE scientific results", and comprise:
+IV. A northern sky survey at 10–20 GHz with the Multi-
+Frequency Instrument (this paper).
+V. The microwave intensity and polarization spectra of the
+Galactic regions W49, W51 and IC443 (Tramonte et al. 2023).
+VI. The Haze as seen by QUIJOTE (Guidi et al. 2023).
+VII. Galactic AME sources in the QUIJOTE-MFI North Hemi-
+sphere Wide-Survey (Poidevin et al. 2023).
+VIII. Diffuse polarized foregrounds from component separation
+with QUIJOTE-MFI (de la Hoz et al. 2023a).
+IX. Radio sources in the QUIJOTE-MFI wide survey maps (Her-
+ranz et al. 2023).
+X. AME variability along the Galactic Plane in the QUIJOTE-
+MFI wide survey (Fernandez-Torreiro et al. 2023, in prep.).
+XI. Polarized synchrotron loops and spurs in the QUIJOTE-MFI
+wide survey (Peel et al. 2023, in prep.).
+XII. Analysis of the polarized synchrotron emission at the power
+spectrum level in the MFI wide survey (Vansyngel et al. 2023, in
+prep.).
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+47
+Figure 41. Top: Minimaps of 6◦ × 6◦ size around W63 for the MFI 11 GHz
+I, Q, and U maps at one degree angular resolution. A circle with radius of 2◦
+indicates the integration area for the photometry analysis. Bottom: Polarized
+intensity measurements on W63 with the QUIJOTE MFI wide survey data,
+in comparison with WMAP and Planck measurements. We overplot with
+a dashed line a power-law fit representing the spectrum of the synchrotron
+emission fitted to the WMAP and Planck data.
+XIII. SNRs in the QUIJOTE-MFI wide survey (Lopez-Caraballo
+et al. 2023, in prep.).
+XIV. The FAN region as seen by QUIJOTE-MFI (Ruiz-
+Granados et al. 2023, in prep.).
+XV. The North Polar Spur as seen by QUIJOTE-MFI (Watson
+et al. 2023, in prep.).
+XVI. Diffuse intensity foregrounds from component separation
+with QUIJOTE-MFI (de la Hoz et al. 2023b, in prep.).
+In addition, we have a dedicated paper describing the MFI data
+processing pipeline (Génova-Santos et al. 2023). The distribution
+of released data products associated with the QUIJOTE-MFI wide
+survey papers contain the following items:
+• Four frequency maps (11, 13, 17, 19 GHz) in intensity and
+polarization, both at native and one degree resolution. Maps at 11
+and 13 GHz correspond to those produced from MFI horn 3. Maps
+at 17 and 19 GHz correspond to the weighted average of horns 2
+and 4, as described in Sec. 3.
+• The associated weight and hit maps for each frequency map at
+native resolution.
+• One set of null tests maps (half1/2 for independent baselines).
+• Instrument Model (IMO), containing central frequencies,
+beams properties, beam profiles and window functions for each
+MFI horn, bandpasses and colour corrections.
+• The default analysis mask (sat+NCP+lowdec), as well as the
+satellite mask (sat). The later is applied to all the released maps.
+11
+CONCLUSIONS
+This paper presents and characterizes the properties of the QUI-
+JOTE wide survey maps of the northern sky carried out with the
+MFI instrument. They result from approximately 9 000 h of obser-
+vations spread over six years between 2013 and 2018, and include
+four frequency maps at 11.1, 12.9, 16.8 and 18.8 GHz, with angu-
+lar resolutions between 55 and 39 arcmin. The maps cover around
+29 000 deg2 with sensitivities in linear polarization (Stokes Q and
+U parameters) within 35–40 𝜇K per 1-degree beam. Although the
+MFI instrument is not optimized for intensity measurements, we
+also present the corresponding intensity maps at those four fre-
+quencies, with sensitivities in the range 65–200 𝜇K per 1-degree
+beam.
+Together with the description of the specific aspects of the MFI
+pipeline related to the production of the wide survey maps, we have
+presented a detailed validation of the maps, a characterization of
+residual systematic effects (Sect. 4), and an extensive study of their
+calibration accuracy (Sect. 5 and Table 16). The overall calibration
+uncertainty of the polarization maps is 5 % for the two lowest fre-
+quency channels, and 6 % for the highest ones. These final maps
+and other derived data products are part of a public data release
+associated with this paper.
+Although a full description of the science results obtained
+from these maps are given in the accompanying papers listed in
+Sect. 10, this paper presents some global properties of the Galactic
+foregrounds at these frequencies, and in particular, the polarized
+synchrotron emission. The average synchrotron spectral index in
+polarization between 11 GHz and the WMAP 23 GHz is found to
+be 𝛽 = −3.07 ± 0.16, showing a much broader distribution (by
+a factor ∼ 2.7) than the one adopted in current synchrotron sky
+models (e.g. Miville-Deschênes et al. 2008). Most of the large-
+scale polarized synchrotron features in the MFI maps appear in
+the E-mode map, which shows significantly more power than the
+B-mode at these frequencies. Based on the analysis of the angular
+power spectra of the measured polarized signal, we find that the
+BB/EE ratio at multipole scales of ℓ = 80 is 0.26 ± 0.07 for a
+Galactic cut |𝑏| > 10◦. This value is consistent with that found
+for WMAP/Planck low frequency maps (Martire et al. 2022), but
+it is significantly different from the values obtained for the S-PASS
+polarized signal at 2.3 GHz (Krachmalnicoff et al. 2018), suggesting
+that probably there is some contribution of Faraday rotation and/or
+depolarization at lower frequencies than those probed by QUIJOTE
+MFI. We also find a positive correlation in the TE spectrum for
+11 GHz at large angular scales (ℓ ≲ 80), while the EB and TB signals
+are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150, as
+expected for the synchrotron emission, as its polarization orientation
+is dictated by the Galactic magnetic field lines.
+The MFI instrument was decommissioned in 2018. At this
+moment, QT2 is operating with a combination of the TGI and FGI
+instruments in a single cryostat. In addition to QUIJOTE, there
+are two other CMB polarization experiments at the Teide Observa-
+tory and providing a similar sky coverage: GroundBird and LSPE-
+STRIP. GroundBird (Honda et al. 2020) is a MKIDs array with two
+bands centered at 145 and 220 GHz, installed back in 2019. STRIP
+(Addamo et al. 2021) is part of LSPE, a combined programme of
+ground-based and balloon-borne polarization observations. STRIP
+will operate in the 42 and 90 GHz bands, and will be installed at the
+Teide Observatory in 2023. The QUIJOTE collaboration is devel-
+oping a new instrument at these frequencies, called MFI2, with an
+expected sensitivity three times better than the former MFI (Hoyland
+et al. 2022). The new MFI2 is now in the final integration phase, and
+MNRAS 000, 1–58 (2022)
+
+10
+20
+30
+40
+0.0
+0.51.0
+1.5
+2.0
+2.5
+0
+1
+2
+3
+mK,
+CMB
+mK
+CMB
+8
+7
+00
+6
+(deg
+5
+b
+4
+3
+V63 I 11 GHz
+W63Q
+Q11GHz
+W63 U 11 GHz
+85
+84
+83
+82
+81
+80
+85
+84
+83
+82
+81
+80
+85
+84
+83
+82
+81
+80
+1 (deg)
+1 (deg)
+1 (deg)48
+Rubiño-Martín et al.
+it is using a digital back-end based on Field Programmable Gate Ar-
+rays (FPGAs), that will allow us to identify and filter the RFI signals
+from geostationary satellites directly in the data processing stage. A
+new wide survey at these frequencies (10–20 GHz) will be carried
+out with MFI2 at the first QUIJOTE telescope (QT1) starting 2023.
+DATA AVAILABILITY
+All data products described in Sect. 10 can be freely downloaded
+from the QUIJOTE web page11, as well as from the RADIOFORE-
+GROUNDS platform12. They include also an Explanatory Supple-
+ment describing the data formats. Maps will be submitted also to
+the Planck Legacy Archive (PLA) interface and the LAMBDA site.
+Any other derived data products described in this paper (null test
+maps, simulations, etc) are available upon request to the QUIJOTE
+collaboration.
+ACKNOWLEDGEMENTS
+We thank the staff of the Teide Observatory for invaluable
+assistance in the commissioning and operation of QUIJOTE.
+The QUIJOTE experiment is being developed by the Instituto
+de Astrofisica de Canarias (IAC), the Instituto de Fisica de
+Cantabria (IFCA), and the Universities of Cantabria, Manch-
+ester and Cambridge. Partial financial support was provided
+by the Spanish Ministry of Science and Innovation under
+the projects AYA2007-68058-C03-01, AYA2007-68058-C03-02,
+AYA2010-21766-C03-01, AYA2010-21766-C03-02, AYA2014-
+60438-P, ESP2015-70646-C2-1-R, AYA2017-84185-P, ESP2017-
+83921-C2-1-R,
+AYA2017-90675-REDC
+(co-funded
+with
+EU
+FEDER funds), PGC2018-101814-B-I00, PID2019-110610RB-
+C21, PID2020-120514GB-I00, IACA13-3E-2336, IACA15-BE-
+3707, EQC2018-004918-P, the Severo Ochoa Programs SEV-
+2015-0548 and CEX2019-000920-S, the Maria de Maeztu Pro-
+gram MDM-2017-0765, and by the Consolider-Ingenio project
+CSD2010-00064 (EPI: Exploring the Physics of Inflation). We
+acknowledge support from the ACIISI, Consejeria de Economia,
+Conocimiento y Empleo del Gobierno de Canarias and the European
+Regional Development Fund (ERDF) under grant with reference
+ProID2020010108. This project has received funding from the Eu-
+ropean Union’s Horizon 2020 research and innovation program un-
+der grant agreement number 687312 (RADIOFOREGROUNDS).
+This research made use of computing time available on the
+high-performance computing systems at the IAC. We thankfully
+acknowledge the technical expertise and assistance provided by the
+Spanish Supercomputing Network (Red Española de Supercom-
+putación), as well as the computer resources used: the Deimos/Diva
+Supercomputer, located at the IAC. This research used resources of
+the National Energy Research Scientific Computing Center, which is
+supported by the Office of Science of the U.S. Department of Energy
+under Contract No. DE-AC02-05CH11231. The PWV data used in
+the tests presented in Section 4 comes from the Izaña Atmospheric
+Observatory (IZO), and have been made available to us by the Izaña
+Atmospheric Research Center (AEMET). SEH and CD acknowl-
+edge support from the STFC Consolidated Grant (ST/P000649/1).
+FP acknowledges support from the Spanish State Research Agency
+11 http://research.iac.es/proyecto/quijote
+12 http://www.radioforegrounds.eu/
+(AEI) under grant number PID2019-105552RB-C43. DT acknowl-
+edges the support from the Chinese Academy of Sciences (CAS)
+President’s International Fellowship Initiative (PIFI) with Grant N.
+2020PM0042. Some of the presented results are based on observa-
+tions obtained with Planck (http://www.esa.int/Planck), an
+ESA science mission with instruments and contributions directly
+funded by ESA Member States, NASA, and Canada. We acknowl-
+edge the use of the Legacy Archive for Microwave Background
+Data Analysis (LAMBDA). Support for LAMBDA is provided by
+the NASA Office of Space Science. Some of the results in this pa-
+per have been derived using the HEALPix (Górski et al. 2005) and
+healpy (Zonca et al. 2019) packages. We also use Numpy (Harris
+et al. 2020), Matplotlib (Hunter 2007) and the sklearn module
+(Pedregosa et al. 2011).
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+APPENDIX A: DATA FLAGGING IN THE MFI WIDE
+SURVEY
+Tables A1, A2, A3 and A4 show the percentage of data used (and
+flagged) for each period, elevation and horn in the MFI wide survey.
+APPENDIX B: IMPACT OF FDEC FILTERING ON
+POLARIZATION MAPS
+In this appendix, we investigate the impact of the FDEC filtering
+on some of the scientific analyses carried out in this paper and
+in other papers of the associated release (Sect. 10). In particular,
+we consider here a photometry method (aperture photometry) and
+correlation method (the so called TT plot).
+For this study, we use the sky signal simulations presented in
+Sect. 6.1. Figure B1 shows the simulated (noiseless) sky maps in
+polarization at 11 GHz used as reference. We apply the FDEC filter
+to these maps, and show in the same figure the resulting filtered
+maps, as well as the residual maps (i.e. difference between the
+original and the filtered map). As shown in Sect. 2.5, the FDEC
+filtering effectively corresponds to a high-pass filter, which removes
+the zero mode for any line of constant declination on a map in
+local (equatorial) coordinates. The image illustrates again that the
+effective transfer function of the FDEC filter leaves unaltered all
+scales with ℓ >∼ 30, because the residual maps only contain large
+scale features.
+B1
+Impact of FDEC on photometry methods: aperture
+photometry
+From Fig. B1, one would expect that all analyses in real space
+involving "local" analyses (e.g. the photometry extraction of a com-
+pact source with a local determination of the background), should
+be unaffected by the FDEC filtering. To test this hypothesis, we
+take as a reference one of the photometry methods used in this pa-
+per: the aperture photometry method (AP1d) described in Sect. 9.
+Then, we apply AP1d to all possible pixels in the simulated maps
+within the MFI wide survey sky mask, both to the original and to
+the filtered maps. For this analysis, we use a reference aperture of
+𝑟1 = 1◦, and the background is estimated in the annulus between
+𝑟1 and 𝑟2 =
+√
+2𝑟1. We find that the maximum difference between
+the photometry on both Stokes Q and U parameters obtained in the
+original map and the filtered one is 0.06 Jy, while the standard de-
+viation of the difference of the two photometry methods is 0.007 Jy.
+Both values are significantly smaller than the typical error in the
+photometry (see e.g. the results presented in Table 24, in Sect. 9),
+thus confirming that we can safely neglect any impact on the pho-
+tometry due to the FDEC filtering. For completeness, we repeat the
+analysis for a larger aperture of 𝑟1 = 2◦, and find that in this case the
+maximum difference is 0.4 Jy, with a standard deviation of 0.037 Jy.
+B2
+Impact of FDEC on correlation analyses: recovery of the
+spectral index
+We now evaluate the impact of the FDEC filtering on the recovery
+of spectral index of the sky emission. To this end, we use the same
+simulation set described above, taking as a reference the simulated
+maps at 11 and 23 GHz. We now apply two different methods to
+reconstruct the spectral index 𝛽 of the sky emission between 11 and
+23 GHz.
+First, we carry out a direct evaluation of the spectral index at
+the pixel level (𝑁side = 512) in the original (unfiltered) maps, and
+also in the filtered ones. This methodology is similar to the one used
+in Sect 8. It is important to emphasize that both maps (the simulated
+MFI 11 GHz and the simulated WMAP 23 GHz) have to be filtered
+with the FDEC, in order to have consistent scales between the two.
+The results are shown in Fig. B2. The reconstructed spectral index is
+fully consistent with the input one. If we restrict the comparison to
+pixels with high emission (polarized intensity at 11 GHz greater than
+0.1 mK), and we compare the reconstructed maps after degrading to
+2◦ (to be consistent with Sect 8), we find that the median difference
+Δ𝛽 between the reconstructed and original spectral index is 0.0005,
+while the standard deviation of the difference is 0.02.
+Second, we use a correlation analysis method (also called TT
+plot) to recover the spectral index of the emission between 11 and
+23 GHz. For this analysis, we degrade the simulated maps at 1
+degree resolution to 𝑁side = 64, in order to have approximately
+independent pixels. Then, we divide the observed sky in patches of
+∼ 7.3◦, using as a reference the pixels of a 𝑁side = 8 HEALPix
+map. Within each patch, we carry out a TT plot analysis assuming
+a typical error in each map corresponding to 3 per cent of the sky
+signal, and accounting for errors in both axes (see e.g. Fuskeland
+et al. 2014). Fig. B3 shows the obtained results from the original
+maps (top panel), and the FDEC filtered maps (bottom panel). As
+expected, the spatial distribution of the reconstructed index has a
+good correspondence with the maps shown in Fig. B2. A numerical
+comparison of both maps in Fig. B3 gives that median difference
+Δ𝛽 between the two maps is -0.0007, and the standard deviation of
+the difference is 0.02.
+Summarising, we can obtain an unbiased reconstruction of the
+spectral index of the sky signal, provided that both maps are filtered
+in the same way using FDEC. In practice, this means that when
+doing these type of analyses using QUIJOTE MFI wide survey
+maps and external ancillary data, we must filter first the external
+maps using the same procedure as for the MFI maps. If the FDEC
+filtering is not applied to the external ancillary data, we find that
+for these simulations the standard deviation of the reconstructed
+spectral index can be as large as 0.3. This issue is further discussed
+in other papers in the series (see e.g. Appendix C in de la Hoz et al.
+2023a).
+APPENDIX C: QUIJOTE MFI WIDE SURVEY MAPS PER
+HORN AT ORIGINAL RESOLUTION
+Figures C1, C2 and C3 show the final MFI wide survey maps at their
+original resolution (quoted as beam FWHM in Table 3), obtained
+for horns 2, 3 and 4 respectively. The intensity maps of horns 2
+and 4 show some large angular-scale residual patterns, particularly
+visible in the highest frequency map (19 GHz). These are due to a
+combination of residual instrumental and atmospheric 1/ 𝑓 noise.
+Figures C4, C5 and C6 show the corresponding weight maps at
+the original resolution. Figures C7, C8 and C9 show the maps with
+the number of individual TOD samples in each pixel (the so called
+"hit maps", 𝑁hit). They correspond to the total number of 40 ms
+samples in each HEALPix pixel of 𝑁side = 512 resolution. The ring
+structures correspond to lines of constant declination, and indicate
+the edges of the declination limits of observations performed at
+different elevations. Due to projection effects, the number of hits
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+51
+Table A1. Fraction of data used in period 1 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Column 1 indicates
+the elevation, columns 2 and 3 show the horn and frequency (0 for low and 1 for high). Columns 4 and 5 show the percentage of used data in correlated and
+uncorrelated channels, respectively. Columns 6 and 7 show the percentage of flagged data during the post-processing stage, and columns 8 and 9 show the
+percentage of flagged data due to Sun, Moon and planets (Mars, Venus, Jupiter). Last column indicates the range of dates when each elevation was observed.
+Elevation
+Horn
+Freq
+Used c
+Used u
+Flag1 c
+Flag1 u
+Flag2 c
+Flag2 u
+Range of Dates
+(deg)
+(%)
+(%)
+(%)
+(%)
+(%)
+(%)
+60
+2
+0
+71.7
+73.9
+16.8
+14.1
+1.4
+1.4
+5/2013–3/2014
+60
+2
+1
+60.7
+49.3
+29.5
+42.6
+1.4
+1.4
+5/2013–3/2014
+60
+3
+0
+33.1
+58.7
+52.8
+16.7
+6.6
+6.6
+5/2013–3/2014
+60
+3
+1
+32.8
+62.7
+54.6
+13.8
+6.6
+6.6
+5/2013–3/2014
+60
+4
+0
+73.4
+74.2
+6.7
+5.7
+1.5
+1.5
+5/2013–3/2014
+60
+4
+1
+56.9
+63.1
+27.8
+19.9
+1.5
+1.5
+5/2013–3/2014
+65
+2
+0
+81.6
+84.5
+13.8
+10.7
+2.0
+2.0
+5/2013–3/2014
+65
+2
+1
+73.8
+55.8
+22.1
+40.9
+2.0
+2.0
+5/2013–3/2014
+65
+3
+0
+39.3
+70.8
+54.7
+18.5
+1.8
+1.8
+5/2013–3/2014
+65
+3
+1
+30.9
+72.8
+62.2
+11.5
+1.8
+1.8
+5/2013–3/2014
+65
+4
+0
+86.3
+87.1
+4.5
+3.6
+1.9
+1.9
+5/2013–3/2014
+65
+4
+1
+70.6
+77.7
+22.0
+14.1
+1.9
+1.9
+5/2013–3/2014
+Table A2. Fraction of data used in period 2 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Same format as in
+Table A1.
+Elevation
+Horn
+Freq
+Used c
+Used u
+Flag1 c
+Flag1 u
+Flag2 c
+Flag2 u
+Range of Dates
+(deg)
+(%)
+(%)
+(%)
+(%)
+(%)
+(%)
+30
+2
+0
+57.1
+56.9
+37.3
+37.5
+2.0
+2.0
+8/2014–3/2015
+30
+2
+1
+53.1
+46.1
+41.7
+49.4
+2.0
+2.0
+8/2014–3/2015
+30
+3
+0
+31.0
+37.9
+64.5
+56.4
+1.7
+1.7
+8/2014–3/2015
+30
+3
+1
+29.4
+41.6
+66.8
+53.0
+1.7
+1.7
+8/2014–3/2015
+30
+4
+0
+57.4
+58.2
+37.4
+36.5
+1.8
+1.8
+8/2014–3/2015
+30
+4
+1
+51.2
+53.0
+44.1
+42.1
+1.8
+1.8
+8/2014–3/2015
+40
+2
+0
+49.2
+51.8
+43.2
+40.1
+1.9
+1.9
+8/2014–1/2015
+40
+2
+1
+46.7
+38.5
+46.1
+55.5
+1.9
+1.9
+8/2014–1/2015
+40
+3
+0
+28.6
+51.5
+65.4
+38.0
+5.2
+5.2
+8/2014–1/2015
+40
+3
+1
+22.0
+41.1
+73.5
+50.6
+5.2
+5.2
+8/2014–1/2015
+40
+4
+0
+55.1
+56.1
+36.9
+35.8
+2.0
+2.0
+8/2014–1/2015
+40
+4
+1
+52.4
+51.7
+40.0
+40.8
+2.0
+2.0
+8/2014–1/2015
+50
+2
+0
+64.7
+65.7
+22.7
+21.4
+2.0
+2.0
+8/2014–10/2015
+50
+2
+1
+61.2
+60.9
+27.0
+27.4
+2.0
+2.0
+8/2014–10/2015
+50
+3
+0
+41.9
+55.4
+47.3
+30.2
+7.0
+7.0
+8/2014–10/2015
+50
+3
+1
+35.8
+55.5
+54.3
+29.2
+7.0
+7.0
+8/2014–10/2015
+50
+4
+0
+62.4
+62.8
+20.2
+19.7
+2.1
+2.1
+8/2014–10/2015
+50
+4
+1
+53.1
+53.1
+31.4
+31.3
+2.1
+2.1
+8/2014–10/2015
+60
+2
+0
+58.3
+59.4
+31.7
+30.3
+2.0
+2.0
+6/2014–9/2014
+60
+2
+1
+58.6
+30.5
+31.4
+64.3
+2.0
+2.0
+6/2014–9/2014
+60
+3
+0
+8.8
+50.4
+87.6
+29.0
+7.0
+7.0
+6/2014–9/2014
+60
+3
+1
+0.0
+50.8
+100.0
+29.8
+7.0
+7.0
+6/2014–9/2014
+60
+4
+0
+55.5
+54.5
+28.9
+30.1
+2.0
+2.0
+6/2014–9/2014
+60
+4
+1
+52.8
+53.0
+32.4
+32.0
+2.0
+2.0
+6/2014–9/2014
+65
+2
+0
+73.8
+79.3
+21.4
+15.5
+3.3
+3.3
+8/2014–10/2014
+65
+2
+1
+76.0
+44.8
+19.0
+52.5
+3.3
+3.3
+8/2014–10/2014
+65
+3
+0
+36.9
+66.4
+56.9
+21.6
+3.2
+3.2
+8/2014–10/2014
+65
+3
+1
+36.1
+67.2
+56.3
+17.6
+3.2
+3.2
+8/2014–10/2014
+65
+4
+0
+71.6
+76.9
+20.0
+14.2
+3.2
+3.2
+8/2014–10/2014
+65
+4
+1
+65.3
+68.4
+27.2
+23.7
+3.2
+3.2
+8/2014–10/2014
+is significantly larger in those boundaries. In the low declination
+band of the maps, particularly for negative declinations, the number
+of hits is significantly lower due to the combined effect of smaller
+number of observations at low elevations (mainly 30◦, 35◦ and 40◦)
+and projection effects. We recall that the number of hits in intensity
+is larger than in polarization, due to the fact that some intensity
+data are not used in polarization, as shown in Table 1 (period 1
+is not used for any polarization maps, data from period 2 are not
+used in polarization for horn 4, and data from period 5 are not
+used in polarization for horn 2). Finally, Fig. C10 shows the 𝑟cond
+maps in polarization, and Fig. C11 shows the normalized covariance
+𝑐𝑜𝑣(𝑄,𝑈), both at original resolution.
+MNRAS 000, 1–58 (2022)
+
+52
+Rubiño-Martín et al.
+Table A3. Fraction of data used in period 5 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Same format as in
+Table A1.
+Elevation
+Horn
+Freq
+Used c
+Used u
+Flag1 c
+Flag1 u
+Flag2 c
+Flag2 u
+Range of Dates
+(deg)
+(%)
+(%)
+(%)
+(%)
+(%)
+(%)
+40
+2
+0
+57.6
+57.7
+33.9
+33.7
+2.0
+2.0
+8/2016–10/2016
+40
+2
+1
+50.9
+49.6
+41.6
+43.1
+2.0
+2.0
+8/2016–10/2016
+40
+3
+0
+60.8
+61.4
+27.6
+26.9
+5.1
+5.1
+8/2016–10/2016
+40
+3
+1
+46.1
+45.0
+45.2
+46.5
+5.1
+5.1
+8/2016–10/2016
+40
+4
+0
+59.5
+59.2
+32.4
+32.7
+2.0
+2.0
+8/2016–10/2016
+40
+4
+1
+43.4
+49.5
+50.7
+43.8
+2.0
+2.0
+8/2016–10/2016
+50
+2
+0
+65.6
+66.1
+21.6
+21.1
+2.3
+2.3
+8/2016–10/2016
+50
+2
+1
+62.0
+61.7
+26.1
+26.5
+2.3
+2.3
+8/2016–10/2016
+50
+3
+0
+57.3
+56.6
+28.7
+29.5
+6.9
+6.9
+8/2016–10/2016
+50
+3
+1
+57.8
+56.3
+26.8
+28.6
+6.9
+6.9
+8/2016–10/2016
+50
+4
+0
+62.5
+62.6
+20.6
+20.5
+2.1
+2.1
+8/2016–10/2016
+50
+4
+1
+45.5
+54.0
+41.4
+30.6
+2.1
+2.1
+8/2016–10/2016
+60
+2
+0
+79.4
+79.5
+7.1
+6.9
+2.0
+2.0
+8/2016–9/2016
+60
+2
+1
+77.0
+76.6
+9.8
+10.3
+2.0
+2.0
+8/2016–9/2016
+60
+3
+0
+61.9
+62.3
+13.2
+12.6
+7.0
+7.0
+8/2016–9/2016
+60
+3
+1
+67.2
+64.5
+7.3
+11.1
+7.0
+7.0
+8/2016–9/2016
+60
+4
+0
+72.7
+73.4
+7.2
+6.3
+1.9
+1.9
+8/2016–9/2016
+60
+4
+1
+60.3
+67.9
+23.0
+13.2
+1.9
+1.9
+8/2016–9/2016
+Table A4. Fraction of data used in period 6 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Same format as in
+Table A1.
+Elevation
+Horn
+Freq
+Used c
+Used u
+Flag1 c
+Flag1 u
+Flag2 c
+Flag2 u
+Range of Dates
+(deg)
+(%)
+(%)
+(%)
+(%)
+(%)
+(%)
+35
+2
+0
+58.0
+56.6
+33.7
+35.4
+1.9
+1.9
+12/2017–6/2018
+35
+2
+1
+47.7
+47.2
+45.5
+46.1
+1.9
+1.9
+12/2017–6/2018
+35
+3
+0
+63.5
+62.0
+26.1
+27.8
+4.2
+4.2
+12/2017–6/2018
+35
+3
+1
+51.6
+51.4
+39.9
+40.1
+4.2
+4.2
+12/2017–6/2018
+35
+4
+0
+61.9
+62.6
+33.0
+32.3
+1.8
+1.8
+12/2017–6/2018
+35
+4
+1
+42.2
+24.5
+54.5
+73.5
+1.8
+1.8
+12/2017–6/2018
+50
+2
+0
+68.2
+68.3
+18.3
+18.1
+2.8
+2.8
+3/2017–4/2017
+50
+2
+1
+60.3
+59.9
+28.0
+28.4
+2.8
+2.8
+3/2017–4/2017
+50
+3
+0
+61.9
+61.6
+22.5
+23.0
+7.3
+7.3
+3/2017–4/2017
+50
+3
+1
+59.6
+56.6
+24.1
+27.9
+7.3
+7.3
+3/2017–4/2017
+50
+4
+0
+67.6
+67.6
+13.7
+13.8
+2.6
+2.6
+3/2017–4/2017
+50
+4
+1
+55.4
+52.7
+28.6
+32.1
+2.6
+2.6
+3/2017–4/2017
+60
+2
+0
+73.9
+73.9
+14.3
+14.3
+0.7
+0.7
+12/2016–2/2017
+60
+2
+1
+72.1
+72.0
+16.4
+16.5
+0.7
+0.7
+12/2016–2/2017
+60
+3
+0
+55.6
+55.6
+22.3
+22.3
+5.8
+5.8
+12/2016–2/2017
+60
+3
+1
+61.8
+61.6
+15.4
+15.6
+5.8
+5.8
+12/2016–2/2017
+60
+4
+0
+68.4
+68.5
+13.4
+13.2
+0.7
+0.7
+12/2016–2/2017
+60
+4
+1
+66.6
+65.8
+15.6
+16.6
+0.7
+0.7
+12/2016–2/2017
+65
+2
+0
+85.3
+85.2
+8.8
+8.9
+3.0
+3.0
+3/2017–4/2017
+65
+2
+1
+83.5
+83.2
+10.8
+11.1
+3.0
+3.0
+3/2017–4/2017
+65
+3
+0
+59.5
+59.5
+29.1
+29.0
+3.2
+3.2
+3/2017–4/2017
+65
+3
+1
+71.3
+69.3
+12.4
+14.9
+3.2
+3.2
+3/2017–4/2017
+65
+4
+0
+81.9
+81.9
+8.2
+8.2
+3.1
+3.1
+3/2017–4/2017
+65
+4
+1
+80.2
+76.9
+10.1
+13.8
+3.1
+3.1
+3/2017–4/2017
+70
+2
+0
+85.1
+85.1
+9.4
+9.4
+2.2
+2.2
+2/2017–4/2017
+70
+2
+1
+84.3
+84.3
+10.4
+10.4
+2.2
+2.2
+2/2017–4/2017
+70
+3
+0
+67.4
+67.5
+28.8
+28.7
+2.5
+2.5
+2/2017–4/2017
+70
+3
+1
+83.8
+82.6
+10.7
+12.0
+2.5
+2.5
+2/2017–4/2017
+70
+4
+0
+86.5
+86.5
+7.8
+7.8
+2.4
+2.4
+2/2017–4/2017
+70
+4
+1
+85.6
+84.8
+8.9
+9.7
+2.4
+2.4
+2/2017–4/2017
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE MFI wide survey
+53
+Figure B1. Example of application of FDEC filtering in simulations of the polarized MFI signal. Top (bottom) row corresponds to Stokes Q (U) parameters.
+Left column shows the simulated MFI 11 GHz map at 1 deg resolution; middle column corresponds to the same map, after applying the FDEC filtering; and
+last column shows the difference of the previous two maps. All maps use the same colour scale, saturated at ±1 mK.
+Figure B2. Impact of the application of FDEC filtering in the reconstruction
+of the spectral index in real space. We use simulations of the polarized sky
+signal in MFI 11 GHz and WMAP 23 GHz. Top panel shows the (true)
+underlying spectral index of the simulated signal between 11 and 23 GHz,
+within the MFI observing mask. Bottom panel shows the reconstructed
+spectral index after applying the FDEC filtering to both simulated maps (11
+and 23 GHz).
+Figure B3. Impact of the application of FDEC filtering in the reconstruction
+of the spectral index using correlation analysis (TTplot). We carry out the
+correlation analysis in regions defined by HEALPix pixels of 𝑁side = 8, and
+extract the spectral index of the polarized sky signal between MFI 11 GHz
+and WMAP 23 GHz. Top panel shows the (true) underlying spectral index
+of the simulated signal within the MFI observing mask. Bottom panel shows
+the reconstructed spectral index after applying the FDEC filtering to both
+simulated maps (11 and 23 GHz).
+MNRAS 000, 1–58 (2022)
+
+Simulated MFl 11GHz Q (1deg) - original
+mkSimulated MFI 11GHz Q (1deg) - no FDEC
+mkSimulated MFl 11GHz Q (1deg) - FDEC
+mk
+一Simulated MFl 11GHz U (ldeg) - original
+mkSimulated MFI 11GHz U (ldeg) - no FDEC
+mkSimulated MFl 11GHz U (1deg) - FDEC
+mkSimulated spectral index in polarization (MFil1 to WMAP-K)
+-3.4
+β11GHz - 23GHz
+-2.7Recovered spectral index in polarization (MFll1 to WMAP-K) - after FDEC
+-3.4
+β11GHz - 23GHz
+-2.7Simulated spectral index in polarization (MFil1 to WMAP-K)
+-3.4
+β11GHz - 23GHz
+-2.7Recovered spectral index in polarization (MFll1 to WMAP-K) - after FDEC
+-3.4
+β11GHz - 23GHz
+2.754
+Rubiño-Martín et al.
+Figure C1. Original resolution QUIJOTE MFI wide survey maps for horn 2. Maps are shown in Galactic coordinates. All figures use the same linear colour
+scale, saturated at 20 mKCMB for intensity (first column) and 2 mKCMB in polarization for Stokes Q (second column) and Stokes U (third column) parameters.
+For display purposes, maps are downgraded to 𝑁side = 256.
+Figure C2. Same as Fig. C1, but for QUIJOTE MFI wide survey maps for horn 3.
+C1
+Signal-to-noise of the QUIJOTE MFI maps
+From the maps at original resolution shown in Figs. C1–C3, and
+the noise variance maps estimated from the inverse of the weights
+presented in Figs. C4–C6 and rescaled by the factors reported in
+Table 12, we can produce signal-to-noise maps for the MFI wide
+survey. To this end, we downgrade these maps to a HEALPix reso-
+lution of 𝑁side = 64, which roughly corresponds to the beam size of
+the maps. Table C1 presents some basic statistics about the fraction
+of 𝑁side = 64 pixels in the maps observed about a certain signal-to-
+noise significance. As a reference, the 11 GHz polarized intensity
+map has 52 % of its pixels with a signal-to-noise ratio larger than 3.
+APPENDIX D: IMPACT IN THE POLARIZATION TOD OF
+AN ERROR IN THE DETERMINATION OF THE
+𝑟-FACTOR
+We illustrate this effect using the particular case of uncorrelated
+channels in the first MFI configuration, but the result is equivalent
+for correlated channels and for all MFI configurations. We follow
+the notation introduced in Génova-Santos et al. (2023), and used in
+equation 1. Following the notation of Jarosik et al. (2003), the MFI
+response for the two uncorrelated channels, 𝑥 and 𝑦, in the first MFI
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE 1 H2 17GHZ
+5
+mK
+20QUIJOTE Q H2 17GHz
+mK
+2QUIJOTE U H2 17GHZ
+2
+mKQUIJOTE 1 H2 19GHZ
+5
+mK
+20QUIJOTE Q H2 19GHz
+mK
+2QUIJOTE U H2 19GHzZ
+mK
+2QUIJOTE I H3 11GHZ
+5
+mK
+20QUIJOTE Q H3 11GHz
+mK
+2QUIJOTE U H3 11GHzZ
+2
+mKQUIJOTE I H3 13GHZ
+5
+mK
+20QUIJOTE Q H3 13GHZ
+2
+mKQUIJOTE U H3 13GHZ
+2
+mK
+2QUIJOTE MFI wide survey
+55
+Figure C3. Same as Fig. C1, but for QUIJOTE MFI wide survey maps for horn 4.
+Figure C4. QUIJOTE MFI wide survey weight maps for horn 2. Top row is 17 GHz, and bottom row is 19 GHz. Each row shows, from left to right, the weight
+maps for Stokes I, Q and U.
+configuration is given by
+𝑉x =
+𝑠x𝑔2
+1
+2
+�
+𝐼 + 𝜌x(𝑄 cos 𝜃 − 𝑈 sin 𝜃)
+�
+(D1)
+𝑉y =
+𝑠y𝑔2
+2
+2
+�
+𝐼 + 𝜌y(−𝑄 cos 𝜃 + 𝑈 sin 𝜃)
+�
+(D2)
+where 𝜃 stands for the argument of the cosine and sine in MFI
+receivers (i.e. 𝜃 = 4𝜃pm + 2𝛾p, as in equations 3 and 4), 𝑠x and 𝑠y
+represent the responsivities of the detectors in the two branches, 𝑔1
+and 𝑔2 represent the voltage gains of the two amplifiers in each MFI
+polarimeter, and 𝜌x and 𝜌y are the polar efficiencies in each branch.
+The 𝑟-factor is defined as
+𝑟u ≡
+𝑠x𝑔2
+1
+𝑠y𝑔2
+2
+.
+(D3)
+If we are using an incorrect 𝑟-factor 𝑟′u = 𝑟u + 𝜖, where 𝑟u is
+the correct underlying value, then we have
+𝑉x − 𝑟′
+u𝑉y = 𝑠x𝑔2
+1
+�� 𝜌x + 𝜌y
+2
++ 𝜖
+2𝑟u
+𝜌y
+�
+(𝑄 cos 𝜃 − 𝑈 sin 𝜃) − 𝜖
+2𝑟u
+𝐼
+�
+.
+(D4)
+We find that this error on the 𝑟-factor translates into an effective
+modification of the polar efficiency, and the appearance of a constant
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE 1 H4 17GHZ
+5
+mK
+20QUIJOTE Q H4 17GHZ
+mK
+2QUIJOTE U H4 17GHzZ
+2
+mKQUIJOTE 1H4 19GHZ
+5
+mK
+20QUIJOTE Q H4 19GHz
+2
+mKQUIJOTE U H4 19GHzZ
+2
+mK
+2QUIJOTE WEIGHTS 1 H2 17GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS Q H2 17GHz
+0
+mk-2
+30QUIJOTE WEIGHTS U H2 17GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS 1 H2 19GHz
+0
+mk-2
+30QUIJOTE WEIGHTS Q H2 19GHz
+0
+mk-2
+30QUIJOTE WEIGHTS U H2 19GHZ
+0
+mk-2
+3056
+Rubiño-Martín et al.
+Figure C5. QUIJOTE MFI wide survey weight maps for horn 3.
+Figure C6. QUIJOTE MFI wide survey weight maps for horn 4.
+offset factor in the polarization timeline. For the particular case of
+𝜌x = 𝜌y, then the effective polar efficiency is rescaled by the factor
+𝜌x → 𝜌x
+�
+1 +
+𝜖
+2𝑟u
+�
+.
+(D5)
+Finally, we note that for the intensity timeline, the same effect gener-
+ates an overall calibration shift, and a small polarization-to-intensity
+leakage term. The first term is absorbed once we carry our a re-
+calibration of the instrument, while the second one can be safely
+ignored, as the polarization fraction of the sky emission is already
+small (typically well below 10 per cent).
+APPENDIX E: POWER SPECTRUM ESTIMATORS FOR
+MFI WIDE SURVEY MAPS
+Throughout this paper, we have been using two power spectrum esti-
+mation codes, both based on a pseudo-Cℓ approach: Xpol (Tristram
+et al. 2005) and NaMaster (Alonso et al. 2019). In this appendix,
+we show that both methods produce consistent results for the typical
+sky masks adopted in this paper. For this comparison, we take as
+a reference case the MFI 11 GHz wide survey map and the default
+QUIJOTE mask (NCP+sat+lowdec) combined with the Galactic
+cut |𝑏| > 10◦. In addition, we justify the use of the pseudo-spectra
+approach by comparing these results with those from an optimal
+estimator based on a fast implementation of a quadratic maximum-
+likelihood (QML) estimator (ECLIPSE, Bilbao-Ahedo et al. 2021).
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE WEIGHTS I H3 11GHz
+0
+mk-2
+30QUIJOTE WEIGHTS Q H3 11GHz
+0
+mk-2
+30QUIJOTE WEIGHTS U H3 11GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS 1 H3 13GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS Q H3 13GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS U H3 13GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS I H4 17GHz
+0
+mk-2
+30QUIJOTE WEIGHTS Q H4 17GHz
+0
+mk-2
+30QUIJOTE WEIGHTS U H4 17GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS 1 H4 19GHZ
+0
+mk-2
+30QUIJOTE WEIGHTS Q H4 19GHz
+0
+mk-2
+30QUIJOTE WEIGHTS U H4 19GHZ
+0
+mk-2
+30QUIJOTE MFI wide survey
+57
+Figure C7. QUIJOTE MFI wide survey hit maps for horn 2. They show the total number of 40 ms samples in each HEALPix pixel of 𝑁side = 512 resolution.
+Figure C8. QUIJOTE MFI wide survey hit maps for horn 3.
+Running this QML code is computationally very expensive, so the
+comparison is limited to this case only.
+Figure E1 shows the (binned) low multipoles points of the an-
+gular power spectra and cross-spectra (30 ≤ ℓ ≤ 80) computed with
+those three codes using the same mask. For the case of NaMaster
+we use the "purification" option. The conclusion is that, within the
+multipole range used in this paper (ℓ ≥ 30), all methods provide
+consistent results, so it is justified to use the pseudo-Cℓ approach for
+our computations. In this work, we use equally Xpol or NaMaster
+for TT, EE and BB. For the cross-spectrum analysis in Sect. 7, we
+use the NaMaster code, as it provides slightly closer results to the
+(optimum) QML solution.
+APPENDIX F: SPECTRAL INDEX OF THE MFI 13 GHZ
+SKY EMISSION
+In this appendix we repeat the same analysis carried out in Sect. 8.1,
+but using now as a reference the MFI 13 GHz map. Figure F1 shows
+the result for the intensity spectral index in 𝛽408MHz−13GHz (top
+panel) and 𝛽13GHz−23GHz (bottom panel), while Fig. F2 presents the
+polarization spectral index map 𝛽13GHz−23GHz. Again, in Fig. F3
+we show the histogram with the distribution of spectral indices in
+both maps.
+In general, all results are consistent with those obtained using
+MFI 11 GHz as the reference map. In intensity, the median spectral
+index 𝛽408MHz−13GHz in the full analysis mask is −2.83, with a
+standard deviation of the values across the map of 0.19; and the
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE NHITS I H2 17GHZ
+50
+nhits
+1000QUIJOTE NHITS Q H2 17GHZ
+50
+nhits
+500QUIJOTE NHITS U H2 17GHZ
+50
+nhits
+500QUIJOTE NHITS I H2 19GHz
+50
+nhits
+1000QUIJOTE NHITS Q H2 19GHZ
+50
+nhits
+500QUIJOTE NHITS U H2 19GHZ
+50
+nhits
+500QUIJOTE NHITS I H3 11GHZ
+50
+nhits
+1000QUIJOTE NHITS Q H3 11GHZ
+50
+nhits
+500QUIJOTE NHITS U H3 11GHZ
+50
+nhits
+500QUIJOTE NHITS I H3 13GHZ
+50
+nhits
+1000QUIJOTE NHITS Q H3 13GHZ
+50
+nhits
+500QUIJOTE NHITS U H3 13GHZ
+50
+nhits
+50058
+Rubiño-Martín et al.
+Figure C9. QUIJOTE MFI wide survey hit maps for horn 4.
+Figure C10. QUIJOTE MFI wide survey 𝑟cond maps for all four horns. During the post-processing stage, all pixels with 𝑟cond > 3 are removed from the final
+wide survey polarization maps.
+𝛽13GHz−23GHz spectral index has a median of −2.83 and a standard
+deviation of 0.46. We note that in this latter case, there is a peak
+around −3.1, which is due to the adopted prior. In polarization, the
+𝛽13GHz−23GHz spectral index presents a median value −3.09, and
+the standard deviation is 0.13.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE NHITS I H4 17GHZ
+50
+nhits
+1000QUIJOTE NHITS Q H4 17GHZ
+50
+nhits
+500QUIJOTE NHITS U H4 17GHZ
+50
+nhits
+500QUIJOTE NHITS I H4 19GHZ
+50
+nhits
+1000QUIJOTE NHITS Q H4 19GHZ
+50
+nhits
+500QUIJOTE NHITS U H4 19GHZ
+50
+nhits
+500QUIJOTE RCOND H2 17GHZ
+0
+rcond
+3QUIJOTE RCOND H2 19GHZ
+0
+rcond
+3QUIJOTE RCOND H3 11GHZ
+0
+rcond
+3QUIJOTE RCOND H3 13GHZ
+0
+rcond
+3QUIJOTE RCOND H4 17GHZ
+0
+rcond
+3QUIJOTE RCOND H4 19GHZ
+0
+rcond
+3QUIJOTE MFI wide survey
+59
+Figure C11. QUIJOTE MFI wide survey normalized covariance (𝑐𝑜𝑣 (𝑄, 𝑈)) maps for all four horns.
+Figure E1. Comparison of the angular power spectrum estimators used in this work. Using the default QUIJOTE analysis mask together with the Galactic cut
+|𝑏| > 10◦, we evaluate the TT, EE, BB auto-spectra and the TE, EB and TB cross-spectra of the MFI 11 GHz map, using Xpol and NaMaster (both based
+on pseudo-Cℓ formalism), and ECLIPSE (based on a QML approach). As a reference, in the first three panels we also show the corresponding noise power
+spectrum. For display purposes, the different data points have been shifted by Δℓ = 1. See text for details.
+MNRAS 000, 1–58 (2022)
+
+QUIJOTE COV(Q,U) H2 17GHz
+-0.0001
+cov(Q,U)/(QQ Qu)
+0.0001QUIJOTE COV(Q,U) H2 19GHz
+-0.0001
+cov(Q,U)/(QQ Qu)
+0.0001QUIJOTE COV(Q,U) H3 11GHz
+-0.0001
+cov(Q,U)/(Qo Qu)
+0.0001QUIJOTE COV(Q,U) H3 13GHz
+-0.0001
+cov(Q,U)/(QQ Qu)
+0.0001QUIJOTE COV(Q,U) H4 17GHz
+-0.0001
+cov(Q,U)/(QQ Qu)
+0.0001QUIJOTE COV(Q,U) H4 19GHz
+-0.0001
+cov(Q,U)/(QQ Qu)
+0.0001TT. Ibl>10°
+EE.Ibl>10°
+BB.Ibl>10°
+100.000
+10-
+10
+TT Nmt pure ·
+EE Nmt pure ·
+BB Nmt pure ·
+TT QML O
+EE QML O
+BB QML
+10.000
+TT Xpol ·
+EE Xpol ·
+BB Xpol ●
+10-2E
+10-2
+1.000
+[mk']
+[mk']
+a
+a
+0.100
+10~4
+10-*
+0.010E
+0.001L
+10-5
+10-5
+30
+40
+50
+60
+70
+80
+30
+40
+50
+60
+70
+80
+30
+40
+50
+60
+70
+80
+Multipole t
+Multipole t
+Multipole l
+EB
+TE
+TB
+0.0010
+0.010
+0.010
+EB Nmt pure ·
+TE Nmt pure ·
+TB Nmt pure ·
+EB QML O
+TE QML 
+TB QML
+EB Xpol
+TE Xpol ·
+TB Xpol
+0.0005
+0.005
+0.005
+[mk']
+[mk']
+[eyw]
+0.0000
+0.000
+0.000
+6
+0.0005
+0.005
+0.005
+0.0010LI
+0.010L
+30
+40
+50
+60
+70
+80
+30
+40
+50
+60
+70
+80
+30
+40
+50
+60
+70
+80
+Multipole t
+Multipole t
+Multipole t60
+Rubiño-Martín et al.
+Table C1. Fraction of 𝑁side = 64 pixels with signal-to-noise ratio (SNR)
+above a certain threshold in the four QUIJOTE-MFI frequency maps (horns
+2 and 4 have been combined). We report the SNR both for the intensity (I)
+and the (noise debiased) polarized intensity (𝑃 =
+√︁
+𝑄2 + 𝑈2) maps.
+11 GHz
+13 GHz
+17 GHz
+19 GHz
+Intensity (I)
+SNR> 1
+0.88
+0.90
+0.86
+0.82
+SNR> 2
+0.78
+0.81
+0.72
+0.64
+SNR> 3
+0.70
+0.73
+0.59
+0.49
+SNR> 4
+0.64
+0.66
+0.48
+0.36
+SNR> 5
+0.58
+0.60
+0.39
+0.26
+Polarized intensity (P)
+SNR> 1
+0.87
+0.82
+0.69
+0.70
+SNR> 2
+0.70
+0.60
+0.38
+0.38
+SNR> 3
+0.52
+0.42
+0.19
+0.16
+SNR> 4
+0.38
+0.29
+0.10
+0.06
+SNR> 5
+0.29
+0.21
+0.06
+0.02
+Figure F1. Spectral index of the intensity emission in the QUIJOTE 13 GHz
+map. Top: Spectral index of 𝛽408MHz−13GHz. The average index is approxi-
+mately −2.8. Bottom: Spectral index of 𝛽13GHz−23GHz. The average spectral
+index is also 𝛽 = −2.8. In this colour scale, dark red corresponds to AME
+dominated regions.
+Figure F2. Top: Spectral index map of the polarized emission between
+QUIJOTE 13 GHz and WMAP 23 GHz. Bottom: error map.
+Figure F3. Histogram of spectral index values obtained from Figures F1
+and F2. We show in dashed lines the mean of the prior adopted in the
+determination of the spectral index in polarization. For comparison, we also
+include the histogram of spectral index values from the PySM synchrotron
+model 1 (Thorne et al. 2017).
+MNRAS 000, 1–58 (2022)
+
+Spectral index in intensity (Haslam to MFl13)
+-3.4
+β408MHz - 13GHz
+-2.2Spectral index in intensity (MFI13 to WMAP-K)
+-3.4
+β13GHz - 23GHz-3.4
+β13GHz - 23GHz
+-2.7Error in the spectral index in polarization (MFl13 to WMAP-K)
+0
+(β13GHz - 23GHz)
+0.41.2
+Intensity（408MHz-13GHz
+Intensity（13GHz-23GHz
+1.0
+Polarization（13GHz-23GHz）
+Prior β=-3.1±0.3
+(normalized)
+PySM
+0.8
+0.6
+count
+Pixel 
+0.4
+0.2
+0.0
+-4
+-3
+-2
+Spectral index β
\ No newline at end of file