diff --git "a/0dFLT4oBgHgl3EQfoy-j/content/tmp_files/load_file.txt" "b/0dFLT4oBgHgl3EQfoy-j/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/0dFLT4oBgHgl3EQfoy-j/content/tmp_files/load_file.txt" @@ -0,0 +1,572 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf,len=571 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='12133v1 [gr-qc] 28 Jan 2023 The first variation of the matter energy-momentum tensor with respect to the metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' and its implications on modified gravity theories Zahra Haghani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' ∗ Tiberiu Harko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' † and Shahab Shahidi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' ‡ 1School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Damghan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Damghan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 41167-36716,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Iran 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Babes-Bolyai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 1 Kogalniceanu Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 400084 Cluj-Napoca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Romania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 3Department of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' National Institute of Physics and Nuclear Engineering (IFIN-HH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Bucharest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 077125 Romania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 4Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 19 Ciresilor Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 400487 Cluj-Napoca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Romania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (Dated: January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 2023) The first order variation of the matter energy-momentum tensor Tµν with respect to the metric tensor gαβ plays an important role in modified gravity theories with geometry-matter coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' and in particular in the f(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' T ) modified gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' We obtain the expression of the variation δTµν/δgαβ for the baryonic matter described by an equation given in a parametric form, with the basic thermodynamic variables represented by the particle number density, and by the specific entropy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The first variation of the matter energy-momentum tensor turns out to be independent on the matter Lagrangian, and can be expressed in terms of the pressure, the energy- momentum tensor itself, and the matter fluid four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' We apply the obtained results for the case of the f(R, T ) gravity theory, where R is the Ricci scalar, and T is the trace of the matter energy-momentum tensor, which thus becomes a unique theory, also independent on the choice of the matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' A simple cosmological model, in which the Hilbert-Einstein Lagrangian is generalized through the addition of a term proportional to T n is considered in detail, and it is shown that it gives a very good description of the observational values of the Hubble parameter up to a redshift of z ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='+h,04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='Cv, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='+d I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' INTRODUCTION There are at least three theoretical perspectives [1] that could be used to explain the large amount of re- cent observations, which strongly suggest a faster and faster expanding Universe [2, 3], with a composition in which ordinary matter represents only 5% of its com- position, the rest being represented by the dark energy, and the dark matter [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The first point of view is represented by the dark constituents theory, which adds two more components to the total energy mo- mentum tensor of the Universe, representing dark mat- ter and dark energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Therefore the cos- mological dynamics is described by the field equation Gµν = κ2T bar µν + κ2T DM µν (φ, ψµ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=') + κ2T DE µν (φ, ψµ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='), where T bar µν , T DM µν (φ, ψµ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='), and T DE µν (φ, ψµ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=') represent the energy-momentum tensors of baryonic matter, dark matter, and dark energy, respectively, with φ and ψµ rep- resenting scalar or vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' A well studied dark con- stituent model is represented by the quintessence (scalar field) description of dark energy [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In the dark geometry approach, an exclusively ge- ometric attitude on the gravitational phenomena is adopted, by explaining the cosmological dynamics through the modification of the geometry underly- ∗ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='haghani@du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='ir † tiberiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='harko@aira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='ro ‡ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='shahidi@du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='ir ing the Einstein field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Hence, the ex- tended Einstein equations become in this approach Gµν = κ2T bar µν + κ2T (geom) µν (gµν, R, □R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='), where Tµν is the energy-momentum tensor of ordinary matter, and T (geom) µν (gµν, R, □R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=') is a purely geometric term, ob- tained from the metric, torsion τ, nonmetricity Q, exten- sions of Riemann geometry etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=', and which can effectively mimic dark energy, dark matter, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Some typical example of dark geometric theories are the f(R) [7], f(Q) [8], hybrid metric-Palatini gravity [9] theories, or gravi- tational theories based on the Weyl-Cartan-Weitzenb¨ock [10], or Weyl [11, 12], and Finsler geometries [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The third avenue for the understanding of the gravitational and cosmological phenomena is rep- resented by the dark coupling approach, in which the standard Einstein gravitational equations are generalized to take the mathematical form Gµν = κ2Tµν + κ2T (coup) µν (R, Lm, T, □R, □T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='), where the effective energy-momentum tensor T (coup) µν (gµν, R, Lm, T, □R, □T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=') of the theory is built up by considering the maximal extension of the Hilbert-Einstein Lagrangian, by abandoning its additive structure in matter and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In the dark coupling approach, matter is represented either by the trace T of the matter energy-momentum tensor, by the matter Lagrangian Lm or by some scalar made by Tµν such as TµνT µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The dark coupling approach is also a theoretical an- swer to the problem of the maximal extension of the additive Hilbert-Einstein Lagrangian, which automati- 2 cally implies a non-additive structure of the action in the geometric and matter variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In a general form the requirement of the maximal extension of the grav- itational action can be implemented by assuming that the Lagrangian of the gravitational field is an arbitrary function of the curvature scalar R, and of the matter Lagrangian Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' One of the interesting features of the dark coupling models is that they imply the presence of a nonminimal geometry-matter coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Dark cou- plings are not restricted to Riemannian geometry, but they can be considered in the framework of the exten- sions of Riemann geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Typical examples of dark coupling theories are the f (R, Lm) [15, 16], f(R, T ) [17], f (R, T, RµνT µν) [18], f(τ, T ) [19], f(Q, T ) [20], or the f (R, T, Q, Tm) [21] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Other gravitational theories implying geometry-matter coupling have been considered in [22–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' One of the interesting consequences of the dark cou- pling theories is the reconsideration of the role of the or- dinary (baryonic) matter in the cosmological dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Through its coupling to gravity, matter becomes a key element in the explanation of cosmic dynamics, and re- covers its central role gravity, which is minimized or even neglected in the dark constituents and dark geometric type theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' An important implication of the geometry- matter coupling is that the matter energy-momentum tensor is generally not conserved, and thus an extra- force is generated, acting on massive particles moving in a gravitational field, with the particles following non- geodesic paths [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The possibility of the existence of such couplings between matter and geometry have opened interesting, and novel pathways for the study of gravitational phenomena [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' However, the dependence of the gravitational action in the dark coupling theories on Lm gives a new rele- vance to the old problem of the degeneracy of the matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Two, physically inequivalent expressions of the matter Lagrangian, Lm = −ρ, and Lm = P, lead to the same energy-momentum tensor for matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' This re- sult has important implications for dark coupling gravity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' For example, in the framework of the f (R, Lm) theory, it was shown in [29] that adopting for the La- grangian density the expression Lm = p, where p is the pressure, in the case of dust the extra force vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' However, for the form Lm = ρ of the matter Lagrangian, the extra-force does not vanish [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In [31] it was shown, by using the variational formulation for the derivation of the equations of motion, that both the matter La- grangian, and the energy-momentum tensor, are uniquely and completely determined by the form of the geometry- matter coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Therefore, the extra-force never van- ishes as a consequence of the thermodynamic properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In [32] it was shown that if the particle number is conserved, the Lagrangian of a barotropic per- fect fluid with P = P(ρ) is Lm = −ρ � c2 + � P(ρ)/ρ2dρ � , where ρ is the rest mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' This result can be used successfully in the study of the modified theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The result is based on the assumption that the Lagrangian does not depend on the derivatives of the metric, and that the particle number of the fluid is a con- served quantity, ∇µ (ρuµ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The matter Lagrangian also plays an important role in the f(R, T ) theory of gravity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In theories with geometry-matter coupling another im- portant quantity, the variation of the energy-momentum tensor with respect to the metric does appear, and plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The corresponding second order ten- sor is denoted as Tµν, and it is introduced via the defini- tion [17] Tµν ≡ gρσ δTρσ δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' If the matter Lagrangian does not depend on the deriva- tives of the metric, one can obtain for Tµν a mathemat- ical expression that also contains the second variation of the matter Lagrangian with respect to the metric, δ2Lm/δgµνδgαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The Lagrangian of the electromagnetic field is quadratic in the components of the metric tensor, and hence its second variation gives a non-zero contri- bution to Tµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' However, the case of ordinary baryonic matter is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' At first sight, by taking into account the explicit forms of the matter Lagrangians, Lm = −ρ, or Lm = p, no explicit dependence on the metric does appear, as opposed, for example, to the case of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' This would suggest that the second variation of the matter Lagrangian always iden- tically vanishes, no matter what its functional form is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' This conclusion may be valid indeed for some special forms of the equation of state, but it is not correct if one adopts a general thermodynamic description of the baryonic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' It is the goal of the present Letter to investigate the problem of the second variation of the perfect fluid mat- ter Lagrangian with respect to the metric tensor com- ponents, and to analyze its impact on modified gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' As a first step in our analysis, we obtain, from general thermodynamic considerations, the expressions of the variations with respect to the metric and of the baryonic matter energy density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Once these expressions are known, a straightforward calculation, in- volving the computation of the second variation of the energy density and pressure, gives the first variation of the matter energy-momentum tensor with respect to the metric, which also allows to obtain the tensor Tµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The basic result of our investigation is that the tensor Tµν is independent of the choice of the matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The effect of the second order correction is estimated in a cosmological background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' As a specific example we will concentrate on the f(R, T ) gravity theory, in which the tensor Tµν plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The present Letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The general thermodynamic formalism used for the calculation of the second variation of the matter Lagrangian is discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The general expression for the second vari- ation of the matter Lagrangian, and of the variation of the energy-momentum tensor is presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 3 Some cosmological applications of the obtained results are presented in Section III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' We then briefly review the basics of the f(R, T ) gravity theory in Section IV and outline its cosmological implications for a simple choice f(R, T ) = α|T |n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Finally, we discuss and conclude our results in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' THERMODYNAMICS AND GEOMETRY In order to obtain the second variation of the baryonic matter Lagrangian, it is necessary to review the deriva- tion of its first variation using thermodynamics consid- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The first law of the thermodynamic is given by dU = T dS − PdV + µdN, (1) where U is the total energy, µ is the chemical potential, related to the change in the number of particles in the system, N is the particle number and V is the volume en- closing the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' An important thermodynamic relation is the Gibbs-Duhem equation, U = T S − PV + µN, (2) which follows from the extensivity of the energy, U(λX) = λU(X), where λ is a constant, and from Euler’s theorem of the homogeneous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Let us define the particle number density n = N/V and entropy per particle s = S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The first law of ther- modynamics (1) and the Gibbs-Duhem relation (2) can be simplified to [33, 34] dρ = T nds + µ′dn, (3) ρ = µ′n − P, (4) where µ′ = µ+T s and we have defined the energy density as ρ = U/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Also, by taking the differential of the Gibbs- Duhem relation (2) we obtain dU = T dS + SdT − PdV − V dP + Ndµ + µdN, and using the first law of thermodynamics (1), one can obtain dP = sdT + ndµ = ndµ′ − nT ds, (5) implying that ρ = ρ(s, n) and P = P(µ′, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Now, we define the particle number flux Jµ = √−gnuµ, (6) and the Taub current [34] Vµ = µ′uµ, (7) where uµ is the fluid 4-velocity, and n, the particle num- ber density, can be obtained according to the relation, n = � gµνJµJν g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (8) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' With the above definition, one obtains J ≡ � −JµJµ = √−gn, Jµ = Juµ, (9) V ≡ � −VµV µ = µ′, V µ = V uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (10) In the context of general relativity, it is well-known that there are two equivalent baryonic matter La- grangians corresponding to Lm = −ρ, Lm = p, (11) It should be noted that from the definition of the energy-momentum tensor as Tµν = − 2 √−g δ(√−gLm) δgµν , (12) both Lagrangians in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (11) give the same result, Tµν = (ρ + P)uµuν + Pgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (13) As a next step in our study, we introduce the basic assumptions that the variations of the entropy density s and of the ordinary matter number flux vector density Jµ = nuµ√−g, satisfy the two independent constraints [35], δs = 0, (14) and δJµ = 0, (15) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Hence, in the following we impose the re- striction that the entropy and particle production rates re- main unchanged during the dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' There- fore, the entropy and particle number currents satisfy the conservation equations δ (Jµ∂µs) = 0 and ∇µ (nuµ) = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The first of these relations is obtained by taking the divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (14), contracting the ob- tained expression with Jµ, and by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' By taking the variation of the particle number n, with the use of the assumptions previously introduced, we find [35], δn = n 2 (−g) uµuν �δgµν g − gµν g2 δg � = n 2 (uµuν + gµν) δgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (16) In order to obtain the variation of the energy- momentum tensor, we need to find the variations of the energy density and pressure with respect to the metric, namely, δρ/δgµν and δP/δgµν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In the case of isentropic processes, we have δρ = ρ + P n δn, (17) δP = n dµ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (18) 4 Let the equation of state for matter be given as ρ = ρ (n, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Then, since δs = 0, from the thermodynamic relation (∂ρ/∂n)s = w = (ρ + P) /n, we obtain δρ = wδn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The variation of n is given by Eq (16), while the vari- ation of µ′ from equation (10) can be obtained as, δµ′ = δV = −VµVν 2V δgµν = −1 2µ′uµuνδgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (19) These relations give the thermodynamic variations of the energy density and pressure with respect to the met- ric as, δρ δgµν = 1 2(ρ + P)(gµν + uµuν), (20) δP δgµν = −1 2(ρ + P)uµuν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (21) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (19) and (20) can be obtained in a direct way by starting from the definition of the matter energy- momentum tensor, as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' If the matter Lagrangian does not depend on the derivatives of the metric tensor, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (12) we obtain Tµν = Lmgµν − 2 δLm δgµν , (22) giving δLm δgµν = 1 2Lmgµν − 1 2Tµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (23) If we take now Lm = −ρ, from the above equation we find δ(−ρ) δgµν = −1 2ρgµν − 1 2Tµν = −1 2(ρ + P) (gµν + uµuν) , (24) where we have used the expression (13) for the energy- momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' For Lm = P, we obtain δP δgµν = 1 2Pgµν − 1 2Tµν = −1 2(ρ + P)uµuν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (25) Hence, we have recovered the expressions of the varia- tions with respect to the metric of the energy and pres- sure variations, previously obtained from first principle thermodynamic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' THE FIRST VARIATION OF THE MATTER ENERGY-MOMENTUM TENSOR Now, we have all the necessary tools for computing the second variation of the energy density and of the pressure of a perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Taking into account that δgµν = −gµαgνβδgαβ, (26) and δuµ δgαβ = uν δgµν δgαβ , (27) respectively, one immediately obtains δ2P δgαβδgµν ≡ δ δgαβ � δp δgµν � = 1 4(ρ + P) � gµβuαuν + gµαuβuν + gνβuαuµ + gναuβuµ − 1 2gαβuµuν − 1 2gµνuαuβ � , (28) and δ2(−ρ) δgαβδgµν = δ2P δgαβδgµν − 1 4(ρ + P)(gαβgµν − gµαgνβ − gµβgνα), (29) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Here, since the energy density and pressure are scalars, we expect that the second variation is sym- metric with respect to the change (αβ) ⇄ (µν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Hence, we have implemented this symmetry to the above expres- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' After a little algebra one can obtain from its definition (12), and by assuming that the matter Lagrangian does not depend on the derivatives of the metric tensor, the variation of the energy-momentum tensor as δTµν δgαβ = 1 2Lm(gαβgµν − gµαgνβ − gµβgνα) − 1 2Tαβgµν − 2 δ2Lm δgαβδgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (30) Therefore, after substituting the expressions of the sec- ond variations of the matter Lagrangians, we find the im- portant result that for both baryonic matter Lagrangians in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (11), we obtain, δTµν δgαβ = 1 2P(gαβgµν − gµαgνβ − gµβgνα) − 1 2Tαβgµν − 2 δ2P δgαβδgµν , (31) implying that the expression of δTµν/δgαβ is indepen- dent on the choice of the matter Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' This is not 5 the case for the approximate result obtained by neglect- ing the second variation of the matter Lagrangian with respect to the metric, δTµν δgαβ ≈ 1 2Lm(gαβgµν − gµαgνβ − gµβgνα) − 1 2Tαβgµν, (32) which obviously depends on the choice of Lagrangian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' It should be noted at this moment that the energy- momentum tensor, and its variation, should be indepen- dent to the choice of the baryonic matter Lagrangian, as we have summarized in the previous Section on thermo- dynamics grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (31) can also be written in the form, δTµν δgαβ = 1 2P(gνβgαµ + gναgβµ) − 1 2 � Tανgµβ + Tβνgµα + Tαµgνβ + Tβµgνα − 1 2Tµνgαβ + 1 2Tαβgµν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (33) Also, by defining a modified energy-momentum tensor ¯Tµν = (ρ + P)uµuν + 1 2Pgµν, (34) one can write the first variation of the energy-momentum tensor as δTµν δgαβ = −1 2 � ¯Tβνgµα + ¯Tανgµβ + ¯Tαµgνβ + ¯Tβµgνα − 1 2 ¯Tµνgαβ + 1 2 ¯Tαβgµν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (35) \u200cIn the well-known f(R, T ) gravity theories [17], on en- counters with the expression gµνδTµν/δgαβ, which enters into the modified field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' With the result given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (33), we define Tαβ ≡ gµν δTµν δgαβ = −1 4(12 ¯Tαβ − ¯Tgαβ), (36) where ¯T = −ρ + P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Alternatively, we also have, δT δgαβ = Tαβ + Tαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (37) In the comoving frame one can then obtain, Tµ ν = 1 4diag (11ρ + 7P, −ρ − 5P, −ρ − 5P, −ρ − 5P) δµ ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (38) Taking the trace of the above expression, one finds T ≡ gµνTµν = 2(ρ − P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (39) The approximate results, obtained by neglecting the second variation of the matter Lagrangian, are, Tµ ν ≈ −1 2(ρ + 3P)δµ ν , (40) for Lm = −ρ, and Tµ ν ≈ 1 2(ρ − P)δµ ν , (41) for Lm = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' For the approximate result with Lm = −ρ we obtain T ≈ −2(ρ + 3P), while for Lm = P we obtain T ≈ 2(ρ − P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' We thus arrive to the interesting conclusion that the approximate result with Lm = P still gives the correct answer for the trace of the tensor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Cosmological implications In order to determine the effect of the new term in the variation of the energy-momentum tensor, let us find its behavior for a conserved matter source in a flat FLRW Universe, with the line element ds2 = −dt2 + a2(t) � dx2 + dy2 + dz2� , (42) where a is the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In this case, one has for the baryonic matter density ρm, assumed to be in the form of dust, the expression ρm = Ωm0 a3 , (43) 6 where Ω0m is the present time density abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' For the variation of the density of the radiation we have ρr = Ωr0 a4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (44) Assume that the Universe is filled with dust and radi- ation, with ρ = ρm + ρr = Ωm0 a3 + Ωr0 a4 , P = 1 3ρr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (45) In this case, one obtains T = 2Ωm0(1 + z)3 + 4 3Ωr0(1 + z)4, (46) where we have introduced the redshift z, defined as 1 + z = 1 a, (47) and Ωm,0 and Ωr,0 are the current values of the dust and radiation abundances, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='305, and Ωr0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='3 × 10−5, respectively [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 1 we have depicted the evolution of the new term T as a function of the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' As a result, we expect that the new term changes the behavior of the cosmological models in theories in which the first order variation of the energy-momentum tensor with respect to the metric is present in the gravitational field equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' There are major differences as compared with the approximate relation for Lm = −ρ, but the two relations coincide for Lm = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' f(R, T ) GRAVITY Now let us consider a typical gravitational theory in which the above results can have an important influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Consider the action [17], S = � d4x√−g(κ2R + f(R, T ) + Lm), (48) where f(R, T ) is an arbitrary function of the Ricci scalar R, and of the trace of the energy-momentum tensor T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' We suppose that the Universe is filled with a perfect fluid with the matter energy-momentum having the form (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The field equations can be obtained as κ2Gµν − 1 2fgµν + fRRµν + (gµν□ − ∇µ∇ν)fR = 1 2Tµν − fT Tµν − fT Tµν, (49) where the last term is computed as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' It should be noted that using the correct result Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (36), the choice of the matter Lagrangian is irrelevant, both cases with Lm = −ρ and Lm = P giving the same field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' With the use of the mathematical identity (□∇ν − ∇ν□) fR = Rµν∇µfR, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The behavior of the extra term T as a function of the redshift z for the new correct expression (solid curve), and for the previously considered approximate relation for Lm = −ρ (dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The approximate relation with Lm = P for T exactly coincides with the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' after taking the divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (49) we obtain the conservation equation in the f (R, T ) gravity theory in the form �1 2 − fT � ∇µTµν = (Tµν + Tµν) ∇µfT + fT � ∇µTµν + 1 2∇νT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (50) As one can see from the field equations (49), the dy- namical behavior in f(R, T ) gravity essentially depends on the tensor Tµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In this Letter, we will consider a sim- ple case that indicates the importance of the new term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Let us assume that f(R, T ) = α|T |n, and P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In this case, the field equations reduce to κ2Gµν = 1 2Tµν + 1 2α|T |ngµν − nαǫ|T |n−1(Tµν + Tµν), (51) where ǫ = sign(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Here we have T = −ρ and then ǫ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The Friedmann and Raychaudhuri equations are then h2 = ¯ρm − 1 2β(7n + 2)¯ρn m, (52) h′ = −3 2 (¯ρm − 4βn¯ρn m) , (53) where we have used the following set of dimensionless variables, τ = H0t, H = H0h, ¯ρ = ρ 6κ2H2 0 , β = (6κ2H2 0)n−1α, (54) and we have denoted by H0 the current value of the Hub- ble parameter, and by a prime the derivative with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' As an indicator of the decelerating/accelerating 15 10 5 correct approximate(Lm=-p) 10E 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 :N7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 z 50 100 150 200 250 300 350 400 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 z −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='4 q FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The behavior of the Hubble parameter H and of the deceleration parameter q as a function of the redshift for the best fit values of the parameters as given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The dashed line represents the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='2 Ω m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The behavior of the matter density parameter Ωm as a function of redshift for the best fit values of the parameters as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The dashed line represents the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' evolution we introduce the deceleration parameter, de- fined as q = d dτ 1 h − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (55) Note that from the normalized Friedmann equation (52), and by taking into account that at the present time we have h(present) = 1, we can obtain the coupling β as β = − 2(1 − Ωm0) (2 + 7n)Ωn m0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (56) In order to find the best fit value of the parameter n, H0 and Ωm0, we use the Likelihood analysis using the ob- servational data on the Hubble parameter in the redshift range z ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='07, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='36) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In the case of independent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='3 Ω m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='025 n 64 66 68 70 72 H 0 64 66 68 70 72 H 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='025 n FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The corner plot for the values of the parameters H0, Ωm0 and n with their 1σ and 2σ confidence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' data points, the likelihood function can be defined as L = L0e−χ2/2, (57) where L0 is the normalization constant and the quantity χ2 is defined as χ2 = � i �Oi − Ti σi �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (58) Here i counts the data points, Oi are the observational value, Ti are the theoretical values, and σi are the errors associated with the ith data obtained from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' By maximizing the likelihood function, the best fit val- ues of the parameters n, Ωm0 and H0 at 1σ confidence 8 level, can be obtained as Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='224+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='023, H0 = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='352+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='391 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='418, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='020+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (59) Also, with the use of equation (56) we obtain β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='747+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content='026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (60) The redshift evolution of the Hubble function, of the deceleration parameter q, and of the matter density pa- rameter Ωm = ¯ρm/h2 are represented, for this model, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Also, the corner plot for the values of the parameters H0, Ωm0 and n with their 1σ and 2σ confidence levels is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' DISCUSSIONS AND FINAL REMARKS In the present Letter we have obtained the complete expression of the first variation of the matter energy- momentum tensor with respect to the metric gµν, and of its associated tensor Tµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The full estimation of this term requires the calculation of the second variations of the matter Lagrangian with respect to the metric, a term which was generally ignored in the previous investiga- tions of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The expression of δ2Lm/δgµνδgαβ can be calculated straightforwardly from the first varia- tion δLm/δgµν, which can be obtained for the two possi- ble choices of the matter Lagrangian either from thermo- dynamic considerations, or in a direct way by using the definition of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The main re- sult of this Letter is that the first variation of the matter energy-momentum tensor, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (31), is indepen- dent of the choice of the matter Lagrangian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' both possible choices lead to the same expression (31), depending only on the thermodynamic pressure, and its second variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The variation of the energy-momentum tensor can also be expressed in terms of the pressure, and the energy- momentum tensor itself, or in a compact form in terms of a generalized energy-momentum tensor, formally de- fined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The new form of the variation of the matter energy- momentum tensor may have some important implications on modified gravity theories with geometry-matter cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' As an important example we have considered the particular case of the f(R, T ) gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' We have investigated the cosmological implications of a particular representation of the f(R, T ) gravity, with action given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' (48), in which the standard Hilbert-Einstein La- grangian is corrected by a general term f(R, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' As a simple case we have taken f(R, T ) = α|T |n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The gener- alized Friedmann equations take a simple form, and they allow a complete analysis of the cosmological features of this simple model, and a full fitting of the observational cosmological data, which permits the determination of the optimal values of the free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The model gives an excellent description of the observational data for the Hubble function, up to a redshift of z ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' In this redshift range the model basically coincides with the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The transition from acceleration to decel- eration takes place a redshift that again coincides with the ΛCDM value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Moreover, the deceleration parameter q basically coincides with the ΛCDM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' How- ever, significant differences in the behavior of the matter density do appear at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The search for the “true” physical quantities from which the matter energy-momentum tensor can be ob- tained (−ρ or P) in a variational formulation is still go- ing on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Interestingly enough, the two possible matter Lagrangians are not equivalent in any sense (physical or mathematical), but their functional variation coincides, leading to the same energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' However, as shown in the present Letter, the first variation of the matter energy-momentum tensor is independent on the adopted form of the matter Lagrangian, making the mod- ified gravity theories containing this term unique, and well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Hence, the study of the various orders of variations of the matter Lagrangians and of the energy- momentum tensor turns out to be an important field of research, which could lead to a new understanding of the mathematical formalism, and of the astrophysical and cosmological implications of the modified gravitational theories, and in particular of the f(R, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Nihan Katirci for useful discussions, and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' The work of TH is sup- ported by a grant of the Romanian Ministry of Educa- tion and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-2255 (PNCDI III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Harko and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Lobo, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' D 29, 2030008 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Weinberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Mortonson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Eisenstein, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Hirata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Riess, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Rozo, Physics Reports 530, 87 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' [3] D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' C 70, :373 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' Harko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFLT4oBgHgl3EQfoy-j/content/2301.12133v1.pdf'} +page_content=' N.' 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