The Starobinsky model within the f(R,T) formalism as a cosmological model
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
The Starobinsky model within the f ( R, T ) formalism as acosmological model P.H.R.S. Moraes ,a , R.A.C. Correa , ,b , G. Ribeiro ,c Address(es) of author(s) should be givenReceived: date / Accepted: date
Abstract
In this paper we derive a cosmological modelfrom the f ( R, T ) theory of gravity, for which R is theRicci scalar and T is the trace of the energy-momentumtensor. We consider f ( R, T ) = f ( R ) + f ( T ), with f ( R )being the Starobinksy model R + αR and f ( T ) = γT ,with α and γ constants. We find that from such a func-tional form, it is possible to describe the cosmologicalscenario of a radiation-dominated universe, which hasshown to be a non-trivial feature within the f ( R, T )formalism.
Keywords f ( R, T ) gravity · Starobinsky model · radiation era · cosmological models The f ( R ) theories of gravity [1,2,3] are an optimistic al-ternative to the shortcomings General Relativity (GR)faces as the underlying gravitational theory for a cosmo-logical model [4,5,6]. They can account for the cosmicacceleration [7,8], providing a great match between the-ory and cosmological observations [9,10,11], and alsofor inflation [12,13,14,15,16,17,18] and dark matter is-sues [19,20,21].One of the crucial troubles surrounding GR is thatapparently it cannot be quantized, although attemptsto do so have been proposed, as String Theory [22,23,24] (check also [25,26] for reviews on the topic), andcan, in future, provide us a robust and trustworthymodel of gravity - quantum mechanics unification.Meanwhile it is worthwhile to attempt to considerthe presence of quantum effects in gravitational the-ories. Those effects can rise from the consideration ofterms proportional to the trace of the energy-momentum tensor T in the gravitational part of the f ( R ) action,yielding the f ( R, T ) gravity theories [27]. Those theo-ries were also motivated by the fact that although f ( R )gravity is well behaved in cosmological scales, the So-lar System regime seems to rule out most of the f ( R )models proposed so far [28,29,30,31].Despite its recent elaboration, f ( R, T ) gravity hasalready been applied to a number of areas, such as Cos-mology [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]and Astrophysics [47,48,49,50,51,52].By deeply investigating the outcomes and featuresof an f ( R, T ) or f ( R ) model, one realizes the strong re-lation they have with the functional form of the chosenfunctions and free parameter values. In fact, a reliablemethod to constraint those “free” parameters to valuesthat yield realistic models can bee seen in [53] and [54]for f ( R, T ) and f ( R ) models, respectively.In f ( R ) gravity a reliable and reputed functionalform was proposed by A.A. Starobinsky as [55] f ( R ) = R + αR , (1)which is known as Starobinsky Model (SM), with α aconstant. It predicts quadratic corrections of the Ricciscalar to be inserted in the gravitational part of theEinstein-Hilbert action.An analysis of matter density perturbations in SMwas presented in [56]. Black hole studies were made for R gravity in [57]. The consideration of wormholes insuch theories can be appreciated in [58,59].Our proposal in this paper is to construct a cosmo-logical scenario from an f ( R, T ) functional form whose R − dependence is the same as in the SM, i.e., with aquadratic extra contribution of R , as in Eq.(1). The a r X i v : . [ g r- q c ] D ec T − dependence will be considered to be linear, as γT ,with γ a constant. Therefore, we will take f ( R, T ) = R + αR + γT. (2)Despite the high number of considerations of the SMin f ( R ) cosmology (check also [60,61,62]), it has notbeen considered for the R − dependence in f ( R, T ) mod-els for cosmological purposes so far, only in the studyof astrophysical compact objects [50,51,52]. We believethis is due to the expected high nonlinearity of the re-sulting differential equation for the scale factor. Any-how, the consideration of quantum corrections togetherwith quadratic geometrical terms can imply interest-ing outcomes in a cosmological perspective as it did inthe astrophysical level (check [50,51,52]). Therefore wepresent here a reliable and well referenced method toobtain solutions for such a cosmological scenario.Here let us stress that the f ( R, T ) formalism ex-hibits a sort of shortcoming for a specific era of the Uni-verse evolution. One could ask what are the predictionsof f ( R, T ) gravity in the regime T = 0. It is natural tothink that for different functional forms formulated tothe f ( R, T ) function, the regime T = 0 makes f ( R, T )gravity to recover f ( R ) theories. The regime T = 0 isachieved for p = ρ/
3, with p and ρ being the pressureand density of the Universe, respectively, which is theequation of state (EoS) of radiation. Therefore, from acosmological perspective it becomes intuitive to thinkthat f ( R, T ) gravity itself is not able to describe the erain which the Universe was dominated by radiation . Itwould only recover the f ( R ) outcomes.The T = 0 issue surrounding the f ( R, T ) formalismwas already investigated in [33,36,37,63,64]. In [33], inorder to be able to describe the radiation era of theUniverse, a scalar field was invoked in f ( R, T ) gravity,namely the f ( R, T φ ) gravity. In [36] such a descriptionbecame possible only in a five-dimensional space-time,while in [37] the speed of light was considered a variableand an alternative scenario to inflation was obtained.Here, instead, one of our goals is to check if restric-tively the choice of the SM for the R dependence in the f ( R, T ) function is able to make f ( R, T ) formalism todescribe a radiation-dominated universe.The SM in f ( R ) formalism is known to successfullydescribe the accelerated periods of the Universe evolu-tion, namely the inflationary and dark energy eras [55,59,65,66,67,68]. Would it also be a powerful tool tohelp f ( R, T ) gravity to be able to describe the radia- In [33] it was deeply discussed that this non-contributionregime of f ( R, T ) gravity can also be expected in vacuum; forinstance, in the study of gravitational waves propagation. tion era of the Universe? Let us address this questionin the next sections. f ( R, T ) formalism
Originally proposed as a generalization of the f ( R ) the-ories, the f ( R, T ) gravity considers the gravitationalpart of the model action to be dependent not only ona general function of the Ricci scalar R , but also on ageneral function of the trace of the energy-momentumtensor T , as S grav = 116 π Z d x √− gf ( R, T ) , (3)with g being the determinant of the metric and f ( R, T )the function of R and T . Moreover, throughout thisarticle we will consider natural units.By varying action (3) with respect to the metric g µν ,one obtains the following field equations: f R ( R, T ) R µν − f ( R, T ) g µν + ( g µν (cid:3) − ∇ µ ∇ ν ) , (4) f R ( R, T ) = 8 πT µν − f T ( R, T ) T µν − f T ( R, T ) Θ µν . (5)In (4), R µν is the Ricci tensor, f R ( R, T ) = ∂f ( R, T ) /∂R , f T ( R, T ) = ∂f ( R, T ) /∂T , (cid:3) is the D’Alambert opera-tor, ∇ µ is the covariant derivative and Θ µν = − T µν − pg µν , with the energy-momentum tensor T µν being con-sidered the one of a perfect fluid.Moreover, the covariant divergence of the energy-momentum tensor in f ( R, T ) gravity reads [69,70] ∇ µ T µν = f T ( R, T )8 π − f T ( R, T ) [( T µν + Θ µν ) ∇ µ ln f T ( R, T )+ ∇ µ Θ µν − (1 / g µν ∇ µ T ] . (6) f ( R, T ) = R + αR + γT model αR +1) G µν − αR g µν = (8 π + γ ) T µν + γ ρ − p ) g µν . (7)In Eq.(7), G µν is the usual Einstein tensor and we havealready taken the trace of the energy-momentum tensorof a perfect fluid to be ρ − p . The elegant form in whichEq.(7) is presented makes straightforward to recoverGR when α, γ → Φ = Φ ( t ) ≡ (cid:18) ˙ aa (cid:19) + ¨ aa , (8)with a = a ( t ) being the scale factor and dots represent-ing time derivatives, the non-null components of Eq.(7)for a flat Friedmann-Robertson-Walker metric are: Φ − ¨ aa − αΦ (cid:18) Φ − ¨ aa + Φ (cid:19) = 16 [(16 π + 3 γ ) ρ + γp ] , (9) Φ + ¨ aa − αΦ (cid:18) Φ + ¨ aa − Φ (cid:19) = −
12 [(16 π − γ ) p + γρ ] . (10)It is worthwhile reinforcing that, as required, thelimits α, γ → ρ + 3 ˙ aa ( ρ + p ) = ˜ γ ( ˙ ρ − ˙ p ) , (11)where ˜ γ ≡ ¯ γ/ [2(1 − γ )] with ¯ γ ≡ γ/ ( γ − π ) . It is worthmentioning that by making γ →
0, GR is once againrecovered.
As we can see in the previous section, the equations (9)-(10) are nonlinear second-order differential equations. Itis worth pointing out that nowadays the nonlinearity isfound in many areas of Physics, including CondensedMatter [72,73,74], Field Theory [75,76,77,78] and alsoCosmology [79,80,81]. In a cosmological context, thenonlinear effects can play an important role to under-stand the dynamics of the Universe. For instance, in arecent work it has been shown that in a cosmologicalscenario with Lorentz symmetry breaking, the so-calledoscillons [82] in the early Universe have passed througha phase transition that changed their internal structure[83].Unfortunately, as a consequence of the nonlinearity,in general we lose the capability of getting the completesolutions. However, in this section we will show that In accordance with recent cosmic microwave background tem-perature fluctuations observations [71].
Equations (9)-(10) can be solved analytically in orderto get the general solutions of the system.In order to eliminate the explicit dependence on theterm ¨ a/a , we add the Eqs.(9) and (10) to conclude that12 αΦ ( Φ −
1) + Φ = 8 π ρ + (cid:18) γ − π (cid:19) p. (12)It is important to remark that there is no restrictionin adding these equations and that such a mathematicalapproach was shown to be very useful [84]. Also, in [77],it was used in order to find a class of traveling solitonsin Lorentz and CPT breaking systems.Now we will focus on getting analytical solutionsfor Eq.(12). Looking at it, it is natural to think thatthe functions ρ and p can be represented by polynomialfunctions of third degree in Φ . In fact, such a represen-tation is constantly used in studies concerning oscillontheories [85,86,87,88,89,90]. In those cases, this math-ematical procedure allows to obtain the fundamentalcharacteristics of the oscillons, such as their field con-figuration, lifetime, amplitude and rate of decaying. Byusing this approach we will have a specific class of so-lutions, but with the great advantage of its analyticalform.Therefore, with the above motivation, we assumethat ρ and p are related by a general polytropic equationof state [91]: p ( t ) = K [ ρ ( t )] γ . (13)In (13), K and γ are constants.By substituting the above form of p in Eq.(11), weobtain the following constraints ρ ( t ) = A [ a ( t )] − /Γ , p ( t ) = A K [ a ( t )] − /Γ , (14)where A is an arbitrary constant of integration. More-over, we are using the following definition Γ ≡ (1 − ˜ γ ) [1 − ˜ γK/ (1 − ˜ γ )]1 + K . (15)It is important to remark that, in order to avoid singu-larities, we must impose that Γ < Φ + Φ − (1 / α ) Φ = 0 , (16)where we are using the indentification K ≡ π π − γ . (17) Now, in order to solve Eq.(16) and consequently finda class of analytical solutions for the scale factor, weimpose that α = 1 /
3. Thus, we can see from Eq.(16)that there are two different roots for Φ , which are givenby Φ = 0 , (18) Φ = 12 . (19)Thus, after some mathematical manipulations, wecan obtain the following analytical solutions for thescale factor a ( t ) = p A t + B, (20) a ( t ) = A e − t/ p B e t + C, (21)where A i , B i and C are arbitrary constants of integra-tion, with i = 1 , H = ˙ a/a , shows us theexpansion rate of the Universe in time, whereas the de-celeration parameter, expressed by q = − ¨ aa/ ˙ a , is suchthat negative values stand for an accelerated expansionwhile positive values, for a decelerated expansion.Let us start by analysing solution (20). Such a scalefactor evolves in time according to Fig.1 below. t a Fig. 1
Time evolution of the scale factor from Equation (20).The solid (red) line stands for A = 0 .
3, while the dotted (blue)and dot-dashed (green) lines, for A = 0 . A = 0 .
1, re-spectively. For all curves, B = 0 . The referred Hubble parameter reads H = A A t + B ) , (22)which is depicted in Fig.2.Moreover, independently of the values of the con-stants A and B , Eq.(20) yields q = 1. t H Fig. 2
Time evolution of the Hubble parameter from Equation(22). The solid (red) line stands for A = 0 .
3, while the dotted(blue) and dot-dashed (green) lines, for A = 0 . A = 0 . B = 0 . The behaviour of the cosmological parameters a , H and q obtained above are in agreement with a universedominated by radiation. In order to verify this, let usrecall that the standard Friedmann equations are ob-tained in the present model by making α, γ = 0 in (9)-(10) and read3 (cid:18) ˙ aa (cid:19) = 8 πρ, (23)2 ¨ aa + (cid:18) ˙ aa (cid:19) = − πp. (24)In order to make standard Friedmann equations aboveto describe a radiation-dominated universe, one usu-ally assumes p = ρ/ a ( t ) ∼ t ,exactly as in Eq.(20), obtained from the f ( R, T ) for-malism.Furthermore, Fig.1 shows that a = 0 as t →
0. Infact, a null value for a would indicate the origin of theUniverse. However, since we are treating the radiationdominated universe, t = 0 does not describe the Big-Bang. Rather, it describes the time in which radiationstarts dominating the Universe dynamics. In this way,the fact that a = 0 for low values of time is in agreementwith a radiation dominated universe. We can also seethat a increases with time, corroborating an expandinguniverse.The Hubble parameter behaviour of Figure 2 alsostrengthens our argument. Firstly, we can see that it de-creases with time, as it should happen in an expandinguniverse. Secondly, since H ∼ t − H , with t H being theHubble time, at the end of the stage in which the Uni-verse dynamics was dominated by radiation, H must be = 0. High values of time in Fig.2 (and also in Fig.1)indicate the end of the radiation era rather than thepresent or future epochs of the Universe, in which H asymptotically tends to 0. Such an asymptotically be-haviour for H can be seen, for instance, in [33,36], forwhich high values of time stand for present and futureepochs of the Universe evolution.Moreover, the value which we obtained for the de-celeration parameter, i.e., q = 1, also is in accordancewith a radiation dominated universe. The fact that it ispositive means that during this stage, the Universe ex-pansion was decelerating (in fact, the expansion startedto accelerate some few billion years ago [71]). Also, fromthe time proportionality obtained for a from the stan-dard Friedmann equations above, i.e., a ∼ t , the de-celeration parameter definition − ¨ aa/ ˙ a yields exactly1, i.e., our model has the same features of a standardcosmology radiation-dominated universe.Now, using Eq.(21), we find the following results forthe cosmological parameters H = A e − t ( B e t + C )2 √ C − B e t t , (25) q = C (6 B e t −
1) + B e t ( C + B e t ) . (26)The evolution of these quantities in time can be ap-preciated in Figs.3-4 below. H (cid:72) t (cid:76) Fig. 3
Time evolution of the Hubble parameter from Equation(25). The (blue) dotted line stands for A = 2 and B = 1 .
5, the(green) dot-dashed stands for A = 3 and B = 1 . A = B = 1. In all curves, C = − . The cosmological model constructed from Eq.(21) isquite more general than the one presented earlier. Thisfeature can easily be checked by investigating Fig.4. Fordifferent values of the constants involved, q departsfrom 1, which stands for a radiation-dominated era. As (cid:45) (cid:45) q (cid:72) t (cid:76) Fig. 4
Time evolution of the deceleration parameter fromEquation (26). The (blue) dotted line stands for A = 2 and B = 1 .
5, the (green) dot-dashed stands for A = 3 and B = 1 . A = B = 1. Inall curves, C = − . time passes by, q assumes the value 0 .
5, which is thedeceleration parameter of a matter-dominated universe.This can be checked by taking p = 0 in Eq.(24). Fig.4also shows that the model predicts a transition from adecelerated to an accelerated phase of expansion of theUniverse, since the deceleration parameter eventuallyassumes negative values. These values are in agreementwith observations, as one can check, for instance, the192 ESSENCE SNe Ia data [92]. It is known that for a small but non-negligible periodof time the dynamics of the early universe was domi-nated by radiation. During this epoch, the density andtemperature of photons were high enough to preventatoms, (and consequently) stars and galaxies to form.In such a stage, the EoS of the Universe is writ-ten as p = ρ/
3. For a perfect fluid, such an EoS yields anull trace of the energy-momentum tensor and thereforeone expects, in this regime, f ( R, T ) gravity to simplyretrieve f ( R ) gravity. Indeed, no contributions from theformer are expected since the dependence on T disap-pears.Such an f ( R, T ) formalism shortcoming has gener-ated some important discussions. In [33], in order tosurpass such an unpleasant feature, the authors haveformulated a cosmological scenario for the f ( R, T φ ) grav-ity, with φ being a scalar field. They have showed thateven in the regime T = 0, the field equations of themodel present extra contributions, when compared tothose from f ( R ) gravity, coming from the trace of theenergy-momentum tensor of the scalar field. Such aformalism originated the possibility of studying grav- itational waves in f ( R, T ) gravity [49] (recall that the T = 0 regime is also obtained in vacuum).Here, instead, we have proposed a quadratic cor-rection for the R -dependence of the f ( R, T ) function.Motivated by the application of the SM in f ( R ) cos-mology [55,60,61,62] and f ( R, T ) astrophysics [50,51,52], we intended here to check if from the f ( R, T ) = R + αR + γT theory, one could derive a healthy cos-mological scenario.In constructing our model, we have obtained a highlynonlinear set of differential equations for the scale factor a , from which important and informative cosmologicalparameters are obtained.Remarkably, for small values of time, the values ofour scale factor solution presented in Fig.1 are not closeto 0. The restriction of this model to the radiation eraof the Universe can be checked also in Fig.2, in whichwe can see that for high values of time (end of radiationera) the Hubble parameter does not tends asymptoti-cally to 0, which is expected in a recent universe (check,for instance, [33]).We have presented from solution (20) a formalismwhich makes f ( R, T ) gravity able to generate a cosmo-logical scenario in which radiation dominates the dy-namics of the Universe. The relevance of such a con-struction lies on the fact that one does not expect f ( R, T )gravity to be capable of describing such a stage of theUniverse without simply recovering f ( R ) gravity. Here,instead, we have shown that besides predicting a varietyof well behaved cosmological and astrophysical scenar-ios in f ( R ) gravity, the SM within the f ( R, T ) gravitysolves the T = 0 issue of f ( R, T ) theories.On the other hand, solution (21) is related to amore complete cosmological scenario. It predicts, fromthe analysis of the referred deceleration parameter, theradiation, matter and dark energy-dominated eras, aswell as the transition among these stages, in a contin-uous form, which is certainly a milestone in theoreticalcosmology.
Acknowledgements
PHRSM would like to thank S˜ao PauloResearch Foundation (FAPESP), grant 2015/08476-0, for finan-cial support. RACC thanks to CAPES for financial support.
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