Diego Saez-Gomez

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tudy bations of the are kept quasi-static [13, 15]. approximation Let us point out for II. that this Θ f(R, F(R,T) µν kind this = T −2T ) approximation of GRAVITY models. gravity µν − pg µν InTHEORIES Section . may VI we apply our results to two particu (7 tion odelsof and thenumerical first-order results perturbed are obtained fields [16, and23] compared and therefore with the requires ΛCDM model. Finally in Section VII we conclu us iththe Let theequations us main start conclusions bymotion writingbecome of the this general investigation. action for f(R, T ) gravities [4], on II, we briefly review the state-of-the-art of f(R, T S = II. S G + f(R, S m T = 1 ) gravity. ) GRAVITY 2κ 2 d 4 x √ Section osmological equations f R R for µν f(R, − 1 2 fg T )=f µν − (g 1 (R) µν +f −∇ 2 (T µ ∇) ν models ) f R = − as THEORIES −g κwell 2 −(f(R, fas T Tµν T )+L + f Tm pg ) µν , . (8 omentum conservation for such models. Then, Section IV addressed I. FORMULAE ons ere we where, for have f(R, Let us Action, κstart 2 dropped =8πG, T )=f 1 by writing R (R)+f the explicit is the 2 (T general Ricci ) models dependences scalar while of actionand forT Section f in R f(R, represents Vand deals T . T ) gravities the with trace the [4], of the energy-momentum tensor, i.e., T = tis is while kind straightforward L of models. m is the matter Into Section see that Lagrangian. VI we theapply usual As usual our continuity the results energy-momentum toequation two particular is not satisfied for the field equations (8), an S = S c 1 (R/R HS ) n f(R) =R − R HS c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n G + S m = 1 tensor is defined as, dsequently comparedthe with covariant the ΛCDM derivative model. ofFinally the energy-momentum in Section VII we − b) 2κ 2 d 4 x √ tensor conclude is not null in general ∇ µ T µν = 0. In order t n. ain the modified continuity equation, let us take the covariant−g (f(R, T )+L m ) , T 1+cR n . µν = 2 δS derivative m √ −g δg µν . of the equation (8), here, κ Field 2 =8πG, R∇ is µ the f Ricci scalar and T represents the trace of the energy-momentum tensor, i.e., T = T equations, R R µν − 1 (R, S EH = d 4 x √ −g hile Then, T ) GRAVITY L m byis varying the matter the THEORIES action Lagrangian. with2respect fg µν − (g µν −∇ µ ∇ ν ) f R = − κ 2 + f T Tµν − f T Θ µν As usual to the energy-momentum metric field g µν , the tensor R +2κ 2 field is equations defined as, are obtained, L m → S = d 4 x √ −g f(R)+2κ 2 L m or f(R, T ) fgravities T µν = √ 2 R (R, T )R[4], µν − 1 δS m 2 f(R, T )g µν − (g µν −∇ µ ∇ ν ) f R (R, T )=− −g δg µν . κ 2 + f T (R, T ) T µν − f T (R, T )Θ µν , where the energy-momentum tensor is defined as: T µν (m) = −√ 2 δ ( √ where = 1 → the subscripts on the function f(R, T ) mean differentiation −gL m ) hen, by varying the action with respect to the metric field g µν with respect to R or T , and the tensor Θ 2κ , the field equations −g δg µν are obtained, defined 2 d 4 x √ f R ∇ µ R µν + R µν ∇ µ f R − 1 2 g µν(f R ∇ µ R + f T ∇ µ T ) − (g µν ∇ µ −∇ µ ∇ µ ∇ ν ) f R = −g (f(R, T )+L m ) , (1) as, f R (R, T )R µν − 1 ∇ µ − κ 2 + f T represents the trace of Divergence of the field 2 f(R, the energy-momentum T )g µν − (g µν −∇ µ ∇ ν ) f R (R, )=− κ 2 + f T (R, T ) T µν − f T (R, T )Θ µν , equations Θ µν ≡ g αβ δT tensor, T Tµν − f i.e., T T Θ = µν . (9 T µ µ, αβ δg µν = −2T µν g µν L m +2g αβ δL m the energy-momentum tensor is defined as, us, using the identities ∇ µ R . µν − 1 2 Rg µν = 0, and (∇ν − ∇ ν ) f R (R, T )=R δg µν g µν αβ ∇ µ f R , the covariant derivativ the here energy-momentum the subscripts on the function f(R, T ) mean differentiation with respect to R or T , and the tensor Θ µν efined Note T µν that as, = √for 2 δS m tensor needs to satisfy, a regular . f(R, T ) function, in absence of any kind of matter, (2) the corresponding f(R) gravity equa −g δg µν are recovered, and consequently ∇ µ T the corresponding properties and the well-known solutions for f(R) gravity are µν = f T 1 he satisfied metric by fieldf(R, g µν T , the ) theories field equations inκclassical Θ 2 + f µν ≡ gare αβ δT T 2vacuum g µν∇ µ T − (T µν + Θ µν )∇ µ ln f T −∇ µ Θ obtained, αβ δg µν = −2T (for a review on f(R) µν − g µν L m +2g αβ theories, δL µν . (10 m see [1]). Moreover, here w . interested to study the behavior of this kind of theories for spatially flatδg FLRW µν g αβ spacetimes, which are express nce, comoving for coordinates by the line element, −∇ In a ote that µ ∇ general, perfect ν ) f for R (R, the fluid T )=− divergence with an κ a regular f(R, 2 + of equation f the T ) function, T (R, equations T ) of state T in µν − is not p f absence T (R, null, = wρ, T which beingw )Θ of any kind µν of matter, the corresponding f(R) gravity equati re recovered, and consequently the corresponding ds 2 properties = a 2 (η) , may give a constant, (3) rise to an the anomalous 0−component behavior of of the the covarian ivative matter (10) turns content. out to become, dη and 2 −the dxwell-known 2 solutions for f(R) gravity are a atisfied ) mean by differentiation f(R, T ) theories with respect in classical to Rvacuum or T , and (for the a review tensoron Θ µν f(R) is κ 2 + w − 3 f theories, see [1]). Moreover, here we terested where a(η) to study is the the scale behavior factor T − (1 + w)Tf 2 in of conformal TT T ˙ + 3(1 + w) H(κ 2 − f this kind of time theories η. Then, for spatially the main T ) − 2f flatissue TR (4HḢ FLRW arises spacetimes, on + the Ḧ) T =0. (11 T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, content #Phys. which Rev. D84, are of 024020 the expressed (2011) Unive
ere right lethand us remind side (r.h.s.). that T Thus, = T it µ = may ρ −lead 3p. toThe violations last equation of the usual differs evolution from theofusual different continuity species equation in theon th ,-null T Nevertheless, ) mean rightdifferentiation hand in side the next (r.h.s.). with section Thus, respect F(R,T) we it focus may to Rour lead orattention T to , and violations gravity the on atensor model of theΘ that usual keeps evolution the usual of the continuity differentequation species in th µν is iverse. d. Nevertheless, in the next section we focus our attention on a model that keeps the usual continuity equatio hanged. δT αβ δg µν = −2T µν − g µν L m +2g αβ δL m . (4) III. δg µν f 1 g(R)+f αβ 2 (T ) TYPE THEORIES III. f 1 (R)+f 2 (T ) TYPE THEORIES section, absencewe of any choose kind the ofalgebraic matter, the function corresponding f(R, T ) to f(R) be agravity sum ofequations two independent functions nonding thisA section, particular properties wecase, choose and the thewell-known algebraic function solutionsf(R, for f(R) T ) togravity be a sum areof also two independent functions acuum (for a review on f(R) theories, f(R, T see )=f [1]). 1 (R)+f Moreover, 2 (T ) here we are d of theories for spatially flat FLRW spacetimes, f(R, T )=f which 1 (R)+f are expressed 2 (T ) in (12) (12 (R) and f 2 (T ), respectively depend on the curvature R and the trace T . The generalized Einstein equations ere rgy yield fdensity, Friedmann 1 (R) andwith f equations, 2 (T ), respectively depend on the curvature R and the trace T . The generalized Einstein equation (8) ds 2 yield = a 2 (η) f 10 ≡ f 1 (R 0 ), f 1R0 ≡ df 1 (R 0 )/dR 0 , f 20 ≡ f 2 (T 0 ), f 2T0 ≡ df 2 (T 0 )/dT 0 and c 2 dη 2 − dx 2 s = p 0 /ρ 0 . Th tinuity equation (11) for Lagrangians given by (12) yields (5) − 3Hf 1R +3H f 1R0 − a2 2 f 10 = −κ 2 a 2 ρ 0 +(1+c 2 s)ρ 0 a 2 f 2T0 + a2 l time η. Then, the main ∇ 2 f 20 , (13) − 3Hf 1R 0 +3H f 1R0 − a2 f 1R 0 + Hf 1R 0 − (H +2H 2 )f 1R0 + a2 2 f 2 f 10 = −κ 2 a 2 ρ 0 +(1+c 2 s)ρ 0 a 2 f 2T0 + a2 µ T µ issue arises on the content of the Universe is 0 ν = 1 1 10 = −κ 2 2 c 2 sρ 0 − a2 2 f 2 f 20 , (13 ensor, defined in (2). Since we are κ 2 δ ν µ ∂ µ −interested f 2T0 on flat 2 f 20 + c 2 sρ 0 f 2T0 + T µ 0 ν ∂ µf 2T0 , (15 FLRW cosmologies, less matter, radiation,... ) can be well described 20 . (14) f 1R 0 + Hf 1R 0 − (H +2H 2 )f 1R0 + a2 by 2 f perfect fluids, whose wing explicitly that the energy-momentum tensor is not 10 = a priori −κ 2 a 2 covariantly c 2 sρ 0 − a2 2 f conserved 20 . in f(R, T ) theories. Thus (14 these theories, the test particles moving in a gravitational e prime holds for the derivative with respect to η, H ≡ a field do not follow geodesic lines. By exploring th ation /a and the subscript 0 holds for unperturbed ere ndT (15) quantities: the prime Rholds 0 denotes for the thederivative scalar curvature with respect corresponding to η, Hto ≡the a µν =(ρ for + p)u ν = µ u 0 component, ν − pg µν . one gets Continuity equation, /a unperturbed and (6) the subscript metric, ρ0 holds the unperturbed for unperturbe kground quantities: R 0 denotes the scalar curvature corresponding to the unperturbed metric, ρ 0 the unperturbe ρ 0 +3Hρ 0 (1 + c 2 1 s)= κ 2 (1 + c 2 − f s)ρ 0 f2T 0 + c 2 sρ 0f 2T0 + 1 2T0 2 f 20 . (16 e that The whether usual continuity f 2 vanishes equation (i.e., is f(R) recovered theories) when, or characterizes a non-running cosmological constant, both f 2T vanish, and then the continuity equation in these scenarios becomes f 2T0 f ρ 0 +3H 2 (T )=α √ 1+c 2 T + β s ρ0 =0 . (17 n order to Lagrangians such as (12) consistent with the standard conservation equation, the r.h.s. of (15) has t ish leading to the differential equation 1 F. Alvarenga, A. Cruz-Dombriz, S. Houndjo, M. Rodrigues, DSG, to appear in PRD, ArXiv:1302.1866
Page 1 and 2: Cosmological constraints on modifie
Page 3 and 4: Brief Article The Author April 6, 2
Page 5 and 6: FLRW cosmologies 12 3 No Big Bang (
Page 7 and 8: FLRW cosmologies Future singulariti
Page 9 and 10: Viable f(r) gravity Viability condi
Page 11 and 12: 3 Cosmological evolution in viable
Page 13 and 14: Cosmological evolution in viable f(
Page 15 and 16: Cosmological evolution in viable f(
Page 17 and 18: 8 − 3H 2 − 2Ḣ = 1 κ 2 p m +
Page 19 and 20: (z = 0) = H 0 = 100 hkms −1 Mpc
Page 21 and 22: (z = 0) = H 0 = 100 hkms −1 Mpc
Page 23 and 24: effective repulsive force 12h R ∼
Page 25 and 26: Testing F(R) gravity with Sne Ia da
Page 27: Testing F(R) gravity with Sne Ia da
Page 31 and 32: e previous section, we study two pa
Page 33 and 34: en at redshift z = 1000 where δ is

Diego Saez-Gomez

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