Growth factor in f(T,T) gravity
GGrowth factor in f ( T, T ) gravity Gabriel Farrugia ∗
1, 2 and Jackson Levi Said †
1, 2 Department of Physics, University of Malta, Msida, MSD 2080, Malta Institute of Space Sciences and Astronomy,University of Malta, Msida, MSD 2080, Malta (Dated: 2 April 2018)We investigate the growth factor for sub-horizon modes during late times in f ( T, T ) gravity,where T is the torsion scalar and T is the trace of the stress-energy tensor. This is achieved byobtaining the modified M´esz´aros equation, which describes the evolution of the perturbations ofthe matter energy density, and obtaining numerical results. Such results are obtained by solvingthe modified continuity equation and analysing the behaviour of the solutions of the latter usingvarious constraints on the integration constants. Furthermore, the role of the anisotropic term π S is investigated. PACS numbers: 04.50.Kd, 95.30.Sf, 98.80.Jk
I. INTRODUCTION
The theory of general relativity (GR) has beenstudied for over 100 years, but yet it fails to fullyexplain what the cause of the accelerated expan-sion of the universe is [1–5]. One of the promi-nent models used is the dark energy model whichattempts to explain the late-time acceleration asa result of a kind of energy related to the cos-mological constant. Alternative models and the-ories have tried to explain this phenomenon inthe context of curvature, such as f ( R ) gravity,modified Gauss-Bonnet gravity f ( G ) and witha general coupling between the Ricci scalar andthe Gauss-Bonnet terms in f ( R, G ) gravity (seean extensive review on the cosmological implica-tions in Refs. [6, 7] and references therein), andalso via a coupling between matter and curvaturethrough f ( R, T ) gravity, where T is the trace ofthe stress-energy tensor [8–12], among other ap-proaches [13].Recently, there has been an increase of inter-est in a different type of alternative theory ofgravity, one which does not use curvature butuses torsion instead, which is called teleparallelgravity [14, 15]. This makes use of a differentconnection to the Christoffel symbol, called theWeitzenb¨ock connection, which is a curvature-free quantity (which contrasts the Levi-Civitaconnection, used in curvature based models,which is a torsion-free quantity) and vierbeinfields instead of a metric field. From this, torsionbased quantities can be derived, most notablythe torsion scalar T (not to be confused with the ∗ [email protected] † [email protected] trace of the stress-energy tensor), which can beused to explain gravity in terms of torsion. Thus,this torsion scalar replaces the idea of the Ricciscalar R .It was shown that the torsion formalism isequivalent to that of GR, called TeleparallelEquivalent General Relativity (TEGR), up toa boundary term difference [16–18]. However,there are some subtle differences, such as hav-ing force-field equations analogues to those fromelectromagnetism instead of a geodesic equa-tion, which allows the weak equivalence principle(WEP) to be violated, something which in GR isnot possible [19, 20].As GR was generalised to f ( R ) gravity, telepar-allel gravity was generalised to f ( T ) gravity.Some interesting implications were discovered,such as allowing violations in the local Lorentztransformations (see [21] and references therein).However, recently it has been shown that localLorentz invariance can be restored making thetheory to be covariant as those in curvature mod-els [22]. Furthermore, in contrast to f ( R ) gravity,the resulting field equations remain second order(in contrast to fourth order theories), making thefield equations simpler to work with. Thus, al-though GR and TEGR are equivalent ways indescribing gravity, f ( R ) and f ( T ) gravity mod-els are fundamentally different, resulting in a newway to investigate the cosmological implicationsof such a theory [23]. Nonetheless, various in-vestigations in f ( T ) gravity have been appliedwithin the realm of cosmology, including ther-modynamics Ref. [24], reconstruction Ref. [25],cosmological solutions Ref. [26] and late-time ac-celeration Refs. [27, 28] (detailed discussions onthe topic of late-time acceleration can be foundin Ref. [29]; for a detailed review on f ( T ) gravity,see Ref. [21] and references therein). a r X i v : . [ g r- q c ] D ec In a similar way to what was done to f ( R, T )gravity (where here T is the trace of the stress-energy tensor), f ( T ) gravity can then be gener-alised into f ( T, T ) gravity, where T is the traceof the stress energy tensor. The study of cosmo-logical solutions in this theory has been investi-gated in Ref. [30], as well as the aspect of recon-struction, thermodynamics and stability in Ref.[31]. It is also possible to investigate the intro-duction of the trace of the stress-energy tensor fornon-linear couplings in curvature based gravity,say f ( G, T ) gravity, which has been recently pro-posed in Ref. [32]. However, given the simplicityof the resulting field equations in the torsionalperspective (of that being second order), it ismuch more reasonable to investigate the couplingbetween gravitation and matter through torsionrather than curvature (since the resulting fieldequations would be fourth order). This torsionand matter coupling further opens possibilitiesto describe what the nature of dark energy is,or more precisely what is causing the observedacceleration.In this paper, we investigate the implicationsof such a theory in the realm of the growthevolution of the inhomogeneities of the universeduring late times. This was originally consid-ered by M´esz´aros for GR in 1974 [33], and theperturbed equation is sometimes referred to asM´esz´aros equation or M´esz´aros effect [34]. Thiseffect was investigated in various alternative andmodified theories, including f ( R ) Refs. [35–37], f ( T ) gravity Refs. [38, 39], and in holographic f ( T ) gravity [40], yielding possible physical re-sults when compared with ΛCDM models. Giventhe recent volume of work in the torsional de-scription of gravity, we are interested in findingout what happens to the M´esz´aros effect in the f ( T, T ) theory of gravity. In the case where T is introduced in f ( R, T ) gravity, this was inves-tigated by Alvarenga et. al [41], and the growthfactor was found to be dependent on the sub-horizon mode, which contrasts with what one ob-tains in GR and observational data. This puts inquestion the possibility of having such a theoryas a possible candidate to explain late-time ac-celeration. Thus, we investigate whether such aresult is also observed for f ( T, T ) gravity.The paper is divided as follows. A briefoverview of teleparallelism and f ( T, T ) gravityis given in Sec. II, followed by the derivation ofthe modified M´esz´aros equation for the inhomo-geneous universes in Sec. III. Afterwards, somesolutions and also potential ansatz functions ofthe continuity equation are discussed in Sec. IV.Using these solutions, in Sec. V, numerical re- sults for the growth factor are analysed. Finally,a conclusion about the results is given in Sec. VI. II. AN OVERVIEW OF f ( T, T ) GRAVITYA. Connections, action and field equations
In order to obtain a torsion based theory, onerequires a new connection, the Weitzenb¨ock con-nection (cid:98) Γ αµν , which is defined as (cid:98) Γ µρν ≡ e ρa ∂ µ e aν + e ρa ω abµ e bµ , (1)where e aρ and e aµ are referred to as vierbeins(or tetrads) along with their respective inverses,and ω abµ is called the purely inertial spin con-nection which is related to the inertial effects ofthe system under consideration [15, 22]. Here,the Latin indices transform like a flat space co-ordinate, while the Greek indices transform likeglobal coordinates. In this way, these vierbeinscan be used to relate to the metric tensor g µν de-pending on the local position x on the spacetimemanifold by g µν ( x ) ≡ e aµ ( x ) e bν ( x ) η ab , (2)where η ab is the Minkowski metric tensordiag(1 , − , − , − x will be suppressed. Using this connection, the torsion tensor can be defined as T µρν ≡ (cid:98) Γ µνρ − (cid:98) Γ µρν . (3)The difference between the Weitzenb¨ock andLevi-Civita connections is expressed by the con-torsion tensor K αµν K αµν ≡ (cid:98) Γ αµν − Γ αµν , (4)where Γ αµν is the Levi-Civita connection. Thecontorsion tensor can also be expressed in termsof the torsion tensor as K λµν = 12 (cid:16) T λµ ν + T λν µ − T λµν (cid:17) . (5)In this way, the superpotential tensor can be de-fined as S µνρ ≡ (cid:0) K µνρ + δ µρ T ανα − δ νρ T αµα (cid:1) . (6)Using Eq. (3) and (6) leads to the torsion scalar T ≡ S µνρ T ρµν = 14 T ρµν T ρµν + 12 T ρµν T νµρ − T ρρµ T νµν , (7)which defines the action for teleparallel gravityto be S = 116 πG (cid:90) d x e T + (cid:90) d x e L m , (8)where e = det (cid:0) e Aµ (cid:1) = √− g and L m is the mat-ter Lagrangian. As is done in GR, the torsionscalar in the action can be generalised to be-come a general function of both the torsion scalarand the trace of the energy momentum tensor T ,which results in S = 116 πG (cid:90) d x e [ T + f ( T, T )] + (cid:90) d x e L m , (9) where f ( T, T ) represents the generalised func-tion. This is the analogue of f ( R, T ) gravitywhere T is the trace of energy of the momen-tum tensor. By varying the action with respectto the inverse vierbein field δe Aρ (analogous totaking variations with the inverse metric tensor δg µν in the metric formalism in f ( R ) gravity Ref.[42]), the following field equations field equationsare obtained(1 + f T ) (cid:104) e − ∂ σ ( eS ρσa ) − T bνa S νρb + ω baν S bνρ (cid:105) + ( f T T ∂ σ T + f T T ∂ σ T ) S ρσa + e ρa (cid:18) T + f (cid:19) + f T (cid:18) em T ρa + pe ρa (cid:19) = 4 πG em T ρa . (10)where em T ρα is the stress-energy tensor, which interms of the matter Lagrangian is given by em T ρβ = − e − ∂ ( e L m ) ∂e aρ (the full details of the derivation isgiven in Appendix I). In the case where the spinconnection is zero, the field equations reduce tothose found in Refs. [23, 30].It should be mentioned that the field equa-tions listed in Refs. [23, 30] are derived withinthe pure vierbein formalism (the only dynamicvariable is the vierbein) where the purely iner-tial spin connection, w cab , is assumed to vanishin all frames. This formulation results in hav-ing a breaking of the local Lorentz symmetry for f ( T ) (cid:54) = T [42–44] due to previous assumption.This led to formulations of what are called goodand bad tetrads (see Ref. [45] for more details).Recently, Krˇsˇs´ak and Saridakis show that thislocal Lorentz invariance problem can be solvedby allowing a non-zero purely inertial spin con-nection, i.e. by taking the covariant formulationof the theory [46]. Nonetheless, one can choosevierbeins which make the purely inertial spin con-nection vanish and still allow for local Lorentz in-variance in this theory (such vierbeins are calledproper vierbeins), whilst reducing the field equa-tions to the standard pure vierbein ones. Thevierbein considered in this paper is such typeof vierbein and hence the field equations reduce to those in Refs. [23, 30] and will still be lo-cal Lorentz invariant. However, if a non-propervierbein is chosen, one first requires to find thespin connection before deriving the field equa-tions, which due to covariance, will be identical. B. Cosmologies in f ( T, T ) gravity One can analyse some basic properties of thefield equations by considering a spatially flatFriedmann-Lemaitre-Robertson-Walker (FLRW)metric ds = dt − a ( t ) (cid:0) dx + dy + dz (cid:1) , (11)where a ( t ) is the scale factor in terms of cosmictime. For such a metric, a diagonal vierbein fieldof the form e Aµ = diag (1 , a, a, a ) , (12)is considered. In this case, T = − H . Using thefield equations in Eq. (10), this gives rise to thetwo GR modified equations(1 + f T ) 3 H + f + T f T ρ + p ) = 4 πGρ, (13)(1 + f T ) (cid:16) H + ˙ H (cid:17) + f + T − H ˙ Hf T T + H ( ˙ ρ − p ) f T T = − πGp. (14)These equations can be rearranged into a morefamiliar form H = 8 πG ρ − H f T − f − f T ρ + p ) , (15)˙ H = − πG ( p + ρ ) + f T ρ + p ) − H ( ˙ ρ − p ) f T T + 12 H ˙ Hf T T − ˙ Hf T . (16)Thus, by analysing the equations, one can definean effective dark energy (DE) pressure p DE andenergy density ρ DE as follows8 πG ρ DE ≡ − H f T − f − f T ρ + p ) , (17) − πGp DE ≡ πGρ DE + f T ρ + p ) − H ( ˙ ρ − p ) f T T + 12 H ˙ Hf T T − ˙ Hf T , (18)which in turn can be used to define an ef-fective equation of state parameter w DE to be w DE ≡ p DE ρ DE = − − f T ρ + p ) − H ( ˙ ρ − p ) f T T + 12 H ˙ Hf TT − ˙ Hf T − H f T − f − f T ρ + p ) . (19) Note that this expression only makes sense giventhe provision that the denominator is non-zero.In this way, the effective equations become morefamiliar to the GR counterpart H = 8 πG ρ + ρ DE ) , (20)˙ H = − πG ( p + ρ + p DE + ρ DE ) . (21)Together, they give rise to a modified continuityequation˙ ρ + ˙ ρ DE = − H ( ρ + p + ρ DE + p DE ) . (22)Assuming that an equation of state for the mat-ter pressure and density with equation of stateparameter w , i.e. p = wρ , the continuity equa-tion reduces to˙ ρ + ˙ ρ DE = − H [ ρ (1 + w ) + ρ DE (1 + w DE )] . (23)At this point, one finds a coupling relation be-tween the matter energy density and the effectiveDE energy density. In other words, this impliesthat the stress-energy tensor is not divergencefree [23]. This occurs due to the matter and tor-sion coupling in the gravitational Lagrangian. Infact, removing such coupling restores the diver-gence free property in f ( T ) gravity [38]. Othertheories also result in such lack of divergence freeproperty, for example f ( R, T ) gravity Ref. [8] and f ( R, L m ) theories [47, 48], where the cou-pling of matter and curvature is the cause ofsuch divergencelessness. Having this not diver-gence free means that the standard GR continu-ity equation does not hold, and the matter evo-lution is influenced by this coupling. This in-fluences standard fluid evolutions (e.g. photonsenergy density would necessarily evolve as a − ),which are well defined from Maxwell-Boltzmannstatistics [34]. Thus, this seemingly results intocontradictions. However, this shortcoming canbe resolved by choosing the right f functionwhich results into the stress-energy tensor to betruly divergenceless . This concept, formulatedand used in Ref. [23], allows to determine someof such possible functions. In this way, the La-grangian is restricted. However, the full detailsof this approach are given in Section IV, wheresuch solutions are extracted. In the subsequentsections however, this condition is not assumedto allow for generality.For the time being, let us consider the particu-lar case in which this effective dark energy fluidbecomes an effective cosmological constant, i.e.one which requires the condition that w DE = − f T ρ + p ) − H ( ˙ ρ − p ) f T T + 12 H ˙ Hf T T − ˙ Hf T . (24)By rearranging Eq. (16), the following expressionis obtained˙ H (cid:0) − H f T T + f T (cid:1) = − πG ( p + ρ )+ f T ρ + p ) − H ( ˙ ρ − p ) f T T , (25)which when combined with Eq. (24), the follow-ing relation is obtained − f T ρ (1 + w ) + H ˙ ρ (1 − w ) f T T = 4 πGρ (1 + w ) (2 T f
T T + f T ) , (26)where the equation of state and the torsion scalar T = − H have been used. One can note the de-pendence on ˙ ρ and H in the equations, the for-mer being dependent on the continuity equationwhilst the latter can be expressed in terms of T .Since ˙ ρ depends also on ˙ ρ DE and ρ DE , one hasto use the definition of this effective dark energyand combine with Eq. (16) to create a differentialequation in terms of T and T only [the ρ termscan be expressed in terms of T by T = ρ (1 − w ),except for the case w = 1 / f which effectively have the sameeffect as a cosmological constant. However, onecan note one clear solution which is when f is aconstant, which results in the standard ΛCDMmodel. III. INHOMOGENEOUS EVOLUTIONA. Metric and Field Equations
In order to analyse the evolution of the inho-mogeneities of the universe, we shall consider thestandard scalar perturbed FLRW metric (up tofirst order), which is of the form [49, 50] ds = (1 + 2 φ ) dt − a ( t ) (1 − ψ ) δ ij dx i dx j , (27)for some scalar functions φ and ψ . Since themetric is generated by a vierbein field, one hasto choose such a field which generates the abovemetric. One trivial choice would be e Aµ = (cid:18) φ a (1 − ψ ) δ mi (cid:19) . (28)As it is argued in Zheng and Huang’s paper Ref.[38], in the case of f ( T ) gravity, this results incompatibility issues with the integrated Sachs-Wolfe effect. For this reason, they proposed anon-diagonal vierbein field of the form e Aµ = (cid:0) δ AB + χ AB (cid:1) ¯ e Bµ , (29) where ¯ e A = δ A , ¯ e Ai = aδ Ai and χ AB = (cid:18) φ ∂ i w∂ i ¯ w δ ij ψ + ∂ i ∂ j h + (cid:15) ijk ∂ k ˜ h (cid:19) . (30)where w and ˜ w are two degrees of freedom ofmass dimension and h and ˜ h are parity-violatingterms. Using this vierbein, the following metrictensor is obtained g µν = (cid:18) φ a∂ i ( w + ˜ w ) a∂ i ( w + ˜ w ) − a [(1 − ψ ) δ ij − ∂ i ∂ j h ] (cid:19) . (31) To obtain the FLRW metric, a Newtoniangauge is considered, being ˜ w = − w and h = 0.Having the vierbein field set, one can obtain thefield equations from Eq. (10). For more de-tails about the quantities being considered, seeAppendix II, where the veirbein field, superpo-tential and torsion tensors, torsion scalar andstress-energy tensor are all defined. It shouldbe mentioned that in this paper the effect of theanisotropic term π S is considered. The equationsare as follows: (a) Zero order equations: These are the onesfound in Eqs. (13) and (14). (b)
First order equations: In the following,the “tensor” E ρA corresponds to the free indicesof the field equations, and hence the equationbeing considered are given by E : (1 + f T ) (cid:104) a − ∂ ψ − H ˙ ψ − H φ (cid:105) + 3 H (cid:20) f T T (cid:16) H ˙ ψ + 12 H φ − a − H∂ w (cid:17) + f T T (cid:0) δρ − δp − ∂ π S (cid:1) (cid:21) + f T (cid:0) δρ − δp − ∂ π S (cid:1) + f T δρ + δp ρ + p (cid:20) f T T (cid:16) H ˙ ψ + 12 H φ − a − H∂ w (cid:17) + f T T (cid:0) δρ − δp − ∂ π S (cid:1) (cid:21) = 4 πGδρ (32) E i : − a − ∂ i (cid:16) ˙ ψ + Hφ (cid:17) (1 + f T ) − a − ∂ i ψ (cid:16) − H ˙ Hf T T + ( ˙ ρ − p ) f T T (cid:17) + f T ρ + p ) ∂ i v = 4 πG ( ρ + p ) ∂ i v, (33) E i : (1 + f T ) (cid:104) a − H∂ i φ + a − ∂ i ˙ ψ (cid:105) − a − H (cid:20) f T T (cid:16) H∂ i (cid:16) ˙ ψ + Hφ (cid:17) − a − H∂ i ∂ w (cid:17) + f T T ∂ i (cid:0) δρ − δp − ∂ π S (cid:1) (cid:21) − a f T ( ρ + p ) ∂ i v = − πGa ( ρ + p ) ∂ i v, (34) T r (cid:0) E ij (cid:1) : (1 + f T ) (cid:20) H ˙ φ + 3 H φ + 3 H ˙ ψ + 2 ˙ Hφ + ¨ ψ − a − ∂ ( ψ − φ ) (cid:21) + f T T (cid:16) − H ˙ ψ − H φ − H ˙ H ˙ ψ − H ˙ Hφ − H ¨ ψ − H ˙ φ + 8 a − H ∂ w + 12 H ˙ Ha − ∂ w + 4 a − H ∂ ˙ w (cid:17) − f T (cid:18) δρ − δp − ∂ π S (cid:19) + f T T (cid:20) (cid:16) − H − ˙ H (cid:17) (cid:0) δρ − δp − ∂ π S (cid:1) − H (cid:0) δ ˙ ρ − δ ˙ p − ∂ ˙ π S (cid:1) + ( ˙ ρ − p ) (cid:18) ˙ ψ + 2 Hφ − a − ∂ w (cid:19) (cid:21) + 12 H ˙ H (cid:104) H (cid:16) ˙ ψ + Hφ (cid:17) − a − H∂ w (cid:105) f T T T + f T T T (cid:110) H ˙ H (cid:0) δρ − δp − ∂ π S (cid:1) − H ( ˙ ρ − p ) (cid:104) H (cid:16) ˙ ψ + Hφ (cid:17) − a − H∂ w (cid:105)(cid:111) − H ( ˙ ρ − p ) (cid:0) δρ − δp − ∂ π S (cid:1) f T T T = 4 πG (cid:18) δp + ∂ π S (cid:19) , (35) E ij , i (cid:54) = j : 12 a − (1 + f T ) ∂ j ∂ i ( φ − ψ ) − a − ∂ j ∂ i w (cid:16) − H ˙ Hf T T + ( ˙ ρ − p ) f T T (cid:17) − f T ∂ j ∂ i π S = − πG∂ j ∂ i π S . (36)Similar to what was obtained by Zheng andHuang, the Parity-violating term ˜ h vanisheswhilst the w term survives [38]. Similar to Harko et. al , the existence of the anisotropic term π S and the fractional energy density and pressureare retained in the equations [30]. These are al-ways coupled with derivatives of T except for π S in the last equation which is independent of thefunction f . In this case, the GR equations areobtained for f ( T, T ) = 0 and π S = 0. B. Conservation Equations - Continuity andVelocity
The following conservation equations were de-rived using the field equations. Nonetheless, these can be obtained by taking the divergenceof the stress-energy tensor. (a)
Continuity equation: The zero-order formof the continuity equation is the one given in theprevious section, Eq. (22). Substituting for theeffective dark energy fluid yields4 πG [ ˙ ρ + 3 H ( ρ + p )] = 3 H f T ( ρ + p ) + f T (cid:0) ρ − ˙ p (cid:1) − H ˙ H ( ρ + p ) f T T + ρ + p ρ − p ) f T T . (37)On the other hand, the first order equation is4 πG (cid:20) δ ˙ ρ + 3 H (cid:18) δρ + δp + ∂ π S (cid:19) − ρ + p ) ˙ ψ + ( ρ + p ) ∂ v (cid:21) = f T (cid:20) ρ + p (cid:16) ∂ v − ψ (cid:17) + H (cid:0) δρ + 3 δp + ∂ π S (cid:1) + 14 (cid:16) δ ˙ ρ − δ ˙ p − ∂ ˙ π S (cid:17) (cid:21) + f T T (cid:20) − a − ∂ ψ ( ˙ ρ − p ) + 2 ( ρ + p ) (cid:0) H ˙ ψ + 9 H φ − a − H ∂ w + 3 ˙ H ˙ ψ + 3 H ¨ ψ + 6 H ˙ Hφ + 3 H ˙ φ − a − H∂ ˙ w − a − ˙ H∂ w (cid:1) − H ˙ H ( δρ + δp )+ 3 H (cid:16) H + ˙ ψ (cid:17) ( ˙ ρ − p ) + ( ˙ ρ + ˙ p ) (cid:16) H ˙ ψ + 6 H φ − a − H∂ w (cid:17) (cid:21) + f T T (cid:20) H ρ + p ) (cid:0) δρ − δp − ∂ π S (cid:1) + 14 (cid:0) δρ − δp − ∂ π S (cid:1) ( ˙ ρ − p ) + 12 ( ρ + p ) (cid:16) δ ˙ ρ − δ ˙ p − ∂ ˙ π S (cid:17) + 12 ( ˙ ρ + ˙ p ) (cid:0) δρ − δp − ∂ π S (cid:1)(cid:21) − H ˙ Hf T T T ( ρ + p ) (cid:16) H ˙ ψ + 12 H φ − a − H∂ w (cid:17) + f T T T ρ + p ) ( ˙ ρ − p ) (cid:0) δρ − δp − ∂ π S (cid:1) + f T T T ρ + p ) (cid:104) ( ˙ ρ − p ) (cid:16) H ˙ ψ + 12 H φ − a − H∂ w (cid:17) − H ˙ H (cid:0) δρ − δp − ∂ π S (cid:1)(cid:105) (38) (b) Velocity equation: Since the velocity is a first order quantity, there is not a zeroth orderequation, but instead only a first order one, which is given to be − πG (cid:2) a ( ρ + p ) ∂ i ˙ v + a ˙ p∂ i v + 2 a H ( ρ + p ) ∂ i v + ( ρ + p ) ∂ i φ + ∂ i δp + ∂ i ∂ π S (cid:3) = − a f T (cid:0) ρ + p (cid:1) ∂ i ˙ v + 14 a f T ( ˙ ρ − p ) ∂ i v − a Hf T ( ρ + p ) ∂ i v − f T ( ρ + p ) ∂ i φ + 14 f T ∂ i (cid:0) δρ − δp − ∂ π S (cid:1) . (39) C. Deriving the f ( T, T ) M´esz´aros equation
In what follows, we investigate how the inho-mogeneous structure grows during matter dom-inated eras, i.e. our interest lies in what hap-pens during the matter dominated era ( p = δp =0). This is achieved by investigating sub-horizonmodes ( k >> aH ) and obtaining the f ( T, T )equivalent of the M´esz´aros equation. To makethe calculations simpler, the equations will beFourier transformed but no new symbols shall beapplied to avoid confusion. I. Sub-horizon approximations
We start off by defining the gauge invariantfractional matter perturbation δ m to be given by δ m ≡ δρρ − Ha v. (40) From Eq. (33) and (34) f T T (cid:16) H ˙ ψ + 12 H φ − H ˙ Hψ − a − H k w (cid:17) = f T T (cid:2) − H (cid:0) δρ − δp − k π S (cid:1) − ψ ( ˙ ρ − p ) (cid:3) . (41)To eliminate φ from the equation, we use Eq.(36) φ (1 + f T ) = ψ (1 + f T ) + aw (cid:20) − H ˙ Hf T T + (cid:0) ˙ ρ − p (cid:1) f T T (cid:21) + a π S ( f T − πG ) , (42)which when combined yields(1 + f T ) f T T (cid:16) H ˙ ψ − H ˙ Hψ − a − H k w (cid:17) + 12 H f T T (cid:26) ψ (1 + f T ) + aw (cid:20) − H ˙ Hf T T + (cid:0) ˙ ρ − p (cid:1) f T T (cid:21) + a π S ( f T − πG ) (cid:27) = (1 + f T ) f T T (cid:2) − H (cid:0) δρ − δp − k π S (cid:1) − ψ ( ˙ ρ − p ) (cid:3) . (43)In the case of sub-horizon modes, the equationreduces to H ψ + δρ ∼ k (cid:0) a − Hw + π S (cid:1) . (44)However, from the definition of δ m Eq. (40), onefinds that the its order is δ m ∼ δρH + Ha v. (45)Thus, we get H ψ + H δ m + H a v ∼ k (cid:0) a − Hw + π S (cid:1) . (46)Note that this condition is only true if f is alsodependent on T . For f = f ( T ), the δρ term isnot present in Eq. (43), which results into H ψ ∼ k a − Hw. (47)However, since we are mostly interested in whathappens when the function f is also dependenton T , this detail shall be ignored (the details for f ( T ) gravity can be found in Zheng and Huang’spaper [38]). Using the first order trace equation Eq. (35),its order is φ ∼ ψ + aHw + a π S . (48)On the other hand, for a matter dominated uni-verse, the velocity equation Eq. (39) reduces to˙ v + 2 Hv + φa + k π S a ρ = f T πGa ρ (cid:20) a ρ ˙ v − a ˙ ρv + 12 ρφ − ρδ m + 14 a ρvH + 34 k π S (cid:21) , (49)whose order is given by Hv + φa + k π S a H ∼ δ m a . (50)For the f = f ( T ) case, the right hand side (RHS)of Eq. (49) is zero, and hence the order becomes Hv + φa ∼ k π S a H . (51)Now, combining with Eq. (44) φa + ψa ∼ k (cid:18) wa H + π S a H (cid:19) , (52)which when combined with Eq. (48) yields ψa ∼ Hwa + aπ S + k (cid:18) wa H + π S a H (cid:19) . (53)Therefore, in the sub-horizon limit, one concludesthat ψa >> Hwa + π S . (54)By Eq. (48), this implies φ (cid:39) ψ . Note thatfrom the previous relation, ψ >> a π S . Lastly,by Eq. (33) and taking its order, one concludes Hφ + Hψ ∼ a H v = ⇒ Hv ∼ ψa >> Hwa + π S . (55) II. The M´esz´aros equation
Using Eq. (32), (34) and the definition of the δ m , one finds4 πGδ m = (1 + f T ) k ψa ρ − a f T Hv + 12 (cid:26) f T T (cid:104) H (cid:16) ˙ ψ + Hφ (cid:17) − a − Hk w (cid:105) + f T T (cid:0) ρδ m + 3 Ha ρv − k π S (cid:1) (cid:27) + f T ρ (cid:0) ρδ m − Ha ρv − k π S (cid:1) . (56)Using the subhorizon relationships, this reducesto (cid:18) πG − f T − f T T ρ (cid:19) δ m = (1 + f T ) k ψa ρ . (57)For simplicity, we shall define the following quan-tity A ≡ πG − f T − f T T ρ. (58) Now, from Eq. (33) and Eq. (35), one finds4 πGk v − H ˙ Hf T T k waρ = − (1 + f T ) k ˙ ψa ρ + 12 H ˙ Hf T T k ψa ρ − ˙ ρf T T k ψa ρ + f T k v + (cid:0) πG − f T (cid:1) k π S Hρ − (1 + f T ) k ψa ρ H − ˙ ρf T T k waρ H, (59)which for sub-horizon modes reduces to4 πGk v = − (1 + f T ) k ˙ ψa ρ + 12 H ˙ Hf T T k ψa ρ − ˙ ρf T T k ψa ρ + f T k v − (1 + f T ) k ψa ρ H. (60)By differentiating with respect to time Eq.(57), and combining with the latter Eq. (60) andusing again Eq. (57) gives A ˙ δ m + (cid:20) ˙ A + A (cid:18) ˙ ρρ + 3 H (cid:19)(cid:21) δ m = (cid:18) − πG + f T (cid:19) k v. (61)Now, the sub-horizon limit of the velocity equa-tion Eq. (60) is (cid:18) − f T πG (cid:19) ˙ v + 2 Hv + (cid:18) − f T πG (cid:19) φa + (cid:18) − f T πG (cid:19) k π S a ρ = − f T πG δ m a . (62)By differentiating with respect to time Eq. (61),combined with the previous equation Eq. (62)and the fact that φ (cid:39) ψ followed by Eq. (57) andEq. (61) yields the modified M´esz´aros equation A ¨ δ m + (cid:40) A + A (cid:18) ˙ ρρ + 3 H (cid:19) + 2 AH (cid:18) − f T πG (cid:19) − + A ddt (cid:20) ln (cid:18) − πG + f T (cid:19)(cid:21)(cid:41) ˙ δ m + (cid:26) ddt (cid:20) ˙ A + A (cid:18) ˙ ρρ + 3 H (cid:19) (cid:21) + 2 H (cid:18) − f T πG (cid:19) − (cid:20) ˙ A + A (cid:18) ˙ ρρ + 3 H (cid:19)(cid:21) − πG (cid:18) − f T πG (cid:19) ρA f T − k f T a + ddt (cid:20) ln (cid:18) − πG + f T (cid:19)(cid:21) (cid:20) ˙ A + A (cid:18) ˙ ρρ + 3 H (cid:19)(cid:21) (cid:27) δ m − πG (cid:18) − f T πG (cid:19) k π S a ρ = 0 . (63)Let us analyse some of the properties of this evolution equation. One immediately notes thatthe equation is now dependent on the sub-horizonmode k , present in the two terms k f T a , πG (cid:18) − f T πG (cid:19) k π S a ρ . (64)This contrasts from the GR M´esz´aros equation,but agrees with the f ( R, T ) model [41]. For thisto make sense, we either require each term tovanish or their sum to vanish. In the first case,the following conditions need to be met π S = 0 , f T = 0 . (65)Having π S = 0 means that no fluid anisotropyexists while f T = 0 implies that f = f ( T ), inother words, the theory reduces reduces to stan-dard f ( T ) gravity models.In the second case, this leads to the followingrelation π S = − k ρf T δ m πG − f T . (66)Recall that this π S is not the same anisotropyterm defined in the stress-energy tensor, but itsFourier transform. This couples the effect ofthe trace of the stress-energy tensor with theanisotropic term. Furthermore, if π S = 0, thisleads to f T = 0, and hence to the standard f ( T ) model as before. One also notes the de-pendence of π S to be inversely proportional to k , and hence decreases for larger values of k , inwhich for sub-horizon modes leads to the effectof anisotropy to be small. This makes sense sincethe effect of anisotropy should be small. Further-more, one can also note the order of this expres-sion to be π S ∼ H δ m k , (67)which is in line with Eq. (46). However,one should carefully interpret this result. Themodified M´esz´aros equation was obtained underthe approximation of sub-horizon modes, so thisequality only holds for such approximations. Ifone solves the evolution exactly, the form of π S will most likely change, but it should reduce toEq. (66) for sub-horizon modes. Nonetheless,one can use this relation in two different ways.The first is by letting the parameters of the func-tion f set the value of π S , while the second is bysetting a form of π S , which leads to a constrainton the parameters of f . These are investigatedin Section V.Lastly, in the case of f ( T ) gravity, we have A =4 πG and ˙ ρ +3 Hρ = 0 (by continuity equation Eq.(37)). This reduces the equation to¨ δ m + 2 H ˙ δ m − πG f T ρδ m = k π S a ρ . (68) This agrees with Zheng and Huang’s result ex-cept for the π S term, which survives even forTEGR (i.e. f = 0) [38]. This shows that theevolution is dependent on the sub-horizon modeeven in TEGR provided that the universe hasnon-zero anisotropy. Unless this term is inverselydependent on k by at least k , this will leadto, eventually large, deviations for larger sub-horizon modes. IV. CONTINUITY EQUATIONSOLUTIONS
In this section, we investigate some possiblesolutions of the continuity equation Eq. (37).Considering matter dominated times, the matterenergy density should evolve as ρ ∝ a − (i.e. vol-umetric). For this reason, as discussed in SectionII, the case in which the standard GR continuityequation is satisfied was considered. This simpli-fies the continuity equation into0 = f T + 8 ˙ Hf T T + 2 T f T T , (69)where we have used the fact that for matter domi-nated universes, T = ρ . Clearly, for any function f ( T, T ) = g ( T ), this is a solution of the differ-ential equation. However, our main interest liesin the non-zero T solutions. Two such solutionshave been found to satisfy the differential equa-tion, which were also obtained by Diego et. al [23]. A. f ( T, T ) = g ( T ) For such functions, Eq. (69) reduces to0 = g (cid:48) + 2 T g (cid:48)(cid:48) , (70)where primes are derivatives with respect to T .This leads to the solution, g ( T ) = c √T + c ,where c and c are integration constants ( c rep-resents a cosmological constant). The evolutionof the universe for such a model can be seen bysubstituting in Eq. (13), which gives H = 8 πG ρ − c ρ / − c . (71)Since c and c are constants, by evaluating theequation for matter dominated times gives c = 3 (cid:115) πG H Ω m (cid:104) H (Ω m − − c (cid:105) , (72)0where the relations ρ = ρ a − with ρ being aconstant of proportionality, and the matter den-sity parameter Ω m beingΩ m ≡ ρ ρ c = 8 πGρ H , (73)with ρ c being the critical density have been used.Substituting back into Eq. (71) yields H = H Ω m a − c − a / (cid:104) H (Ω m − − c (cid:105) . (74)Thus, the evolution is only dependent on the pa-rameter c . Note that c = 0 gives the condition c = 6 H (Ω m − c as c ≡ H (Ω m − (cid:15) ) , (75)where (cid:15) is an arbitrary constant. This reducesEq. (74) to H = H (cid:18) Ω m a − Ω m + 1 − (cid:15) + (cid:15)a / (cid:19) . (76) B. f ( T, T ) = T g ( T ) In this case, we consider a rescaling of the tor-sion scalar through some function of the trace ofthe stress-energy tensor. In this case, the Eq.(69) becomes0 = g (cid:48) (cid:32) − H H (cid:33) + 2 T g (cid:48)(cid:48) . (77)From Eq. (13) and (14), one finds˙ HH = 31 + g (cid:18) T g (cid:48) − g (cid:19) , (78)which when combined with the previous equationyields 0 = g (cid:48) (cid:18) − T g (cid:48) g (cid:19) + 2 T g (cid:48)(cid:48) . (79)This results in the following solution g = − − c √T + c , (80)where c and c are integration constants. Notethat in this case, the action for f ( T, T ) gravityEq. (9) becomes S = − πG (cid:90) d x e T c √T + c + (cid:90) d x e L m . (81) where the negative sign can be countered by as-signing negative values to both the c c c = 0, c serves the role of arescaling constant without a cosmological con-stant, with c = − H = − πG c (2 c + c √ ρ ) . (82)Evaluating at today’s time gives a way to definethe constants c and c to the Hubble constantand today’s energy density by H = − πG c (2 c + c √ ρ ) . (83)Hence, the expression for the Hubble parametercan be rewritten as HH = 2 c + c √ ρ c + c √ ρ . (84)Note that the constants relation can be rewrit-ten as c (cid:112) T + c (cid:18) c (cid:112) T + 3 H πG (cid:19) + 4 c = 0 , (85)which is simply a quadratic in c , whose solutionsare given by c = − c √T − m ± (cid:115) c Ω m √T + 1Ω m . (86)This means that we do not have two degrees offreedom but essentially one, c . Note that al-though a quadratic in c was arrived at, thiscan be treated vice-versa to be a quadratic in c where c becomes to degree of freedom. Thissolution for c only produces for real roots, andhence we require4 c Ω m √T + 1Ω m ≥ ⇒ c ≥ − √T m . (87)By defining c to be c ≡ − (cid:15) √T m , (88)for some constant (cid:15) which due to the conditionfor c forces the condition (cid:15) ≤
1, reduces thesolution for c to c = 12Ω m (cid:16) (cid:15) − ± √ − (cid:15) (cid:17) . (89)1This in turn reduces Eq. (83) to HH = − (cid:15) + a − / (cid:0) (cid:15) − ± √ − (cid:15) (cid:1) − (cid:15) − ± √ − (cid:15) . (90)Thus, the evolution is only dependent on themagnitude of (cid:15) . In the extremal case when (cid:15) = 1,the evolution reduces to HH = 2 + a − / . (91)For the case when (cid:15) = 0 (i.e. c = 0), c has twosolutions, 0 or − / Ω m . The first case becomesnon-physical because Eq. (83) leads to H = 0.On the other hand, the second case reduces theequation to HH = a − / , (92)which is precisely the evolution of a matter dom-inated universe. However, this only makes sensefor Ω m = 1, since the action in this case is repre-sented by S = (cid:90) Ω m T + S m . (93)This is a rescaling scenario, and such rescalingshould deviate slightly from TEGR (i.e. from T ). Since only matter dominated universes areconsidered, this forces the matter density to bethe only present component in the universe, andhence Ω m = 1 (this also follows for only matteruniverses in GR). Thus, since we are consideringmatter dominated universes, but closer to the ob-servational value of Ω m ≈ .
3, such case wouldbe non-physical.
C. Potential ansatz functions
Other functions have been considered, howeverthey prove to be inconsistent with the field equa-tions. In particular, the following two have beenconsidered. I. f ( T, T ) = T g ( T ) Since a rescaling of T was considered, the con-verse is now assumed, i.e. a rescaling of T . Inthis case, the continuity equation becomes0 = g + 8 ˙ Hg (cid:48) , (94)where prime denotes a derivative with respect to T . Again, using Eq. (13) and (14) gives the following relationship˙ H (1 + T g (cid:48) + 2 T T g (cid:48)(cid:48) ) = T − T g , (95)which when substituted in the previous equationyields 0 = g + g (cid:48) (2 T − T g ) + 2 T T gg (cid:48)(cid:48) . (96)To convert this equation into a differential equa-tion of T only, Eq. (13) was used to form a rela-tion between T and T which is T (cid:18) πG − g T g (cid:48) (cid:19) = − T . (97)The result of this is that Eq. (96) is expressed as0 = 16 πGg − g − T gg (cid:48) + 32 πGT g (cid:48) + 4 T g (cid:48) − T gg (cid:48)(cid:48) . (98)Solving this differential equation analytically isextremely difficult. However, one can note that g being a constant is a possible solution. Assuming g = c , where c is the constant, reduces thedifferential equation to0 = 16 πGc − c . (99)which gives two solutions, c = 0 or c =16 πG/
3. In the first case, this means g = 0 = ⇒ f = 0, which thus boils down to TEGR (with-out cosmological constant). In the second case,substituting in the energy density equation gives T = 0 = ⇒ H = 0, which is not a physicalsolution. Since we are looking for a non-trivialsolution, the only possible solution in this casewould be the non-constant solution, which can-not be found analytically.However, one can analyse the differential equa-tion using perturbation techniques by treating T as a ‘first order’ quantity. Assuming a solutionof the form g ≈ g + g + g + . . . leads to thefollowing system of equations0 = 16 πGg − g , (100)0 = 16 πGg − T g g (cid:48) + 32 πGT g (cid:48) , (101)0 = 16 πGg − g − T ( g g (cid:48) + g g (cid:48) )+ 32 πGT g (cid:48) + 4 T g (cid:48) − T g g (cid:48)(cid:48) (102)...The first equation leads to g = 0 or g = 16 πG/ g equation leads to g = 0.Similarly, this leads to g = 0, and so forth.Thus, the solution becomes g ≈ g , which boils2down to the two cases discussed previously. Thismight indicate that other solutions might notexist or such solution cannot be expanded as apower series solution, ultimately leading to nosolutions for a possible rescaling of T . II. f ( T, T ) = αT n T m In this case, we consider the possibility of hav-ing a product solution, where α, n, m are con-stants. The continuity equation for this case re-duces to αmT n T m − (cid:16) HnT − + 2 T ( m − T − (cid:17) . (103) This leads to the following possibilities, α = 0, m = 0, T = 0, T = 0 or the bracketed term tobe zero. The first case reduces to TEGR (with-out cosmological constant), the second reduces to f ( T ) gravity, while the third and the fourth givenon-physical results. Thus, we consider the last,non-trivial case. This can be re-expressed as2 m − − H H n = 0 . (104)However, since both n, m are constants, this re-quires ˙ H/H to be constant. This simply leadsto H = H a − / , which reduces the condition to2 m + 2 n = 1 . (105)Note that this can only occur when n (cid:54) = 0 sincewhen n = 0, no constraint on the evolution of H would need to be set, and this sets m = 1 /
2. Thissimply reduces to the first solution encounteredin Section IV. Thus, we shall now consider the case in which n (cid:54) = 0. In order for this conditionto be consistent, the solution for H has to be con-sistent with the field equations. Let us considerthe energy density equation Eq. (13), in whichcase, reduces to H = 8 πG ρ − αT n T m m − n + 1) . (106)Recall that ρ = ρ a − . Since α is a constantindependent of time, substituting for both H and ρ should make the equation independent of time.This leads to the following H a − = 8 πG ρ a − − α − n H n ρ m a − n − m (2 m − n + 1) . (107)Since the scale factor is the only function whichdepends on time, all powers of a must cancel.This sets another condition for n, m , being − n − m = − ⇒ n + m = 1 . (108)However, this contradicts the previous conditionEq. (105). Thus, the only solutions are α √T andthe constant solution (obtained by taking both n, m equal to 0). V. NUMERICAL RESULTS
In this section, we shall consider various dif-ferent scenarios concerning the evolution of thegrowth factor using the solutions found in theprevious section. Since these solutions are basedon the GR condition set on the continuity condi-tion, the evolution differential equation Eq. (63)becomes A ¨ δ m + (cid:40) A + 2 AH (cid:18) − f T πG (cid:19) − + A ddt (cid:20) ln (cid:18) f T − πG (cid:19)(cid:21)(cid:41) ˙ δ m + (cid:26) ¨ A + 2 ˙ AH (cid:18) − f T πG (cid:19) − − πG (cid:18) − f T πG (cid:19) − ρA f T + ˙ A ddt (cid:20) ln (cid:18) f T − πG (cid:19)(cid:21) − k f T a (cid:27) δ m − πG (cid:18) − f T πG (cid:19) k π S a ρ = 0 . (109)For simplicity, we define D ≡ δ m ( a ) /δ m ( a i ),where a i is some initial scale factor, in which it isconsidered to be 0.1. In the following, we analysethe evolution of D with a . A. Numerical Results for π S = 0 In the case where π S = 0, the differential equa-tion still remains dependent on k due to the pres-3ence of the following term k f T a δ m . (110)Thus, for any function considered here, unless f is a function of torsion (or a cosmological con-stant), the evolution will be dependent on k . Forthis reason, one finds that for the functions con-sidered in this paper, the evolution of δ m willeventually cause either oscillations or acceleratedgrowth as the value of k changes, which is non-physical due to the fact that the evolution of δ m should not change with the value of k (or at leastdoes not deviate from a ΛCDM solution by muchfor every sub-horizon k value). I. f = c √T + c From the previous section, it was found that c and c are dependent on each other through Eq.(72). Ultimately, the evolution Eq. (76) can beexpressed by a single parameter (cid:15) defined in Eq.(75). Thus, the evolution of δ m can be analysedby varying the values of (cid:15) .The first case considered is where (cid:15) = 1 − Ω m ( c = 0), which basically neglects the effect ofthe cosmological constant. As shown in Fig. 1,the effect of k is already dominant, even for suf-ficiently small sub-horizon modes. Furthermore,the solution is oscillatory, with increasing periodsfor larger sub-horizon modes, and is far from theΛCDM solution. a ( t ) D ( a ) Λ CDM k = H k = H k = H FIG. 1: Evolution for the f = c √T + c modelwith c and c defined by Eqs. (72) and (75) having (cid:15) = 1 − Ω m (i.e. c = 0). Note that the solutionis very far from the ΛCDM solution and oscillatory.With increasing k , the number of periods increase. Afterwards, the non-zero cosmological constantcase is considered, for two different scenarios, apositive and a negative (cid:15) (see Fig. 2 and 3).In the former, one notes that initially, the ef-fect of k causes a slight deviation from ΛCDM( k = 50 H and k = 100 H ), being a slowergrowth rate. As the sub-horizon modes increase, this leads to much larger deviations, eventuallyleading to an oscillatory motion for much largervalues of k (compare k = 200 H with k = 500 H and k = 1000 H ). On the other hand, for nega-tive (cid:15) , a similar scenario happens for small sub-horizon modes but having a faster growth ratethan ΛCDM. As k increases, the growth fac-tors increases extremely rapidly and shadows theΛCDM evolution. a ( t ) D ( a ) Λ CDM k = H k = H k = H k = H k = H FIG. 2: Evolution for the f = c √T + c model with c and c defined by Eqs. (72) and (75) having (cid:15) =10 − . Note that the solution is close to the ΛCDMsolution for smaller modes but starts to deviate as k is increased. a ( t ) D ( a ) Λ CDM k = H k = H k = H k = H k = H FIG. 3: Evolution for the f = c √T + c model with c and c defined by Eqs. (72) and (75) having (cid:15) = − − . Note that the solution is close to the ΛCDMsolution for smaller modes but starts to deviate as k is increased. II. f = − T − T c √T + c For this case, it is found that c and c are de-pendent on each other through Eq. (89), whichexpresses the evolution Eq. (90) by a single pa-rameter (cid:15) defined in Eq. (88). Hence, the evolu-tion of δ m can again be analysed by varying thevalues of (cid:15) .The first case considered is the extremal case (cid:15) = 1, which gives only one type of evolution.As shown in Fig. 6, the effect of k is alreadydominant, even for sufficiently small sub-horizonmodes. In fact, the solution is also oscillatory,4with increasing periods for larger sub-horizonmodes, similar to the one found in Fig. 1. a ( t ) D ( a ) Λ CDM k = H k = H k = H FIG. 4: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) having (cid:15) = 1. The solution deviates completely from ΛCDMand becomes oscillatory. By increasing k , the numberof periods increase. Subsequently, a positive and a negative (cid:15) wasconsidered. In each case, this gives rise to twopossible evolutions due to the presence of theplus/minus sign in Eq. (90). Let us first con-sider the positive (cid:15) case.In this scenario, the effect of k on the positivesolution is immediately evident (Fig. 5). Forincreasing k , the number of periods increase ex-tremely rapidly, making it deviate greatly fromΛCDM. On the other hand, the negative solu-tion is somewhat close to what happens in Fig.2, with the difference that the smaller modes( k = 50 H and k = 100 H ) have a larger growthprofile than ΛCDM (Fig. 6). a ( t ) D ( a ) Λ CDM k = H FIG. 5: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) forthe positive solution and (cid:15) = 10 − . The solutionis oscillatory with increasing periods as k increases.Note that the solution deviates greatly from ΛCDM. Lastly, for negative (cid:15) , a very similar behaviourto Fig. 5 is observed for the positive solution(Fig. 7). On the other hand, the negative solu-tion is again somewhat close to what happens inFig. 3, with the only difference being that thesmaller modes ( k = 50 H and k = 100 H ) have a ( t ) D ( a ) Λ CDM k = H k = H k = H k = H k = H FIG. 6: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) for thenegative solution and (cid:15) = 10 − . The solution is closeto the ΛCDM solution for smaller modes but deviatesfor increasing modes. a larger growth than the ones observed in theformer (Fig. 8). a ( t ) D ( a ) Λ CDM k = H FIG. 7: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) forthe positive solution and (cid:15) = − − . The solutionis oscillatory with increasing periods as k increases.Note that the solution deviates greatly from ΛCDM. a ( t ) D ( a ) Λ CDM k = H k = H k = H k = H k = H FIG. 8: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) for thenegative solution and (cid:15) = − − . The solution isclose to the ΛCDM solution for smaller modes butdeviates for increasing modes. B. Numerical Results for π S (cid:54) = 0 As discussed in Section III and from the resultsof the previous section, the k dependence causesdeviations from the ΛCDM solution. The onlyway that this can be avoided is by either neglect-ing both the effects of π S and f T , or having theirsum be equal to zero. Since having both of themzero leads to standard f ( T )-ΛCDM models, thesecond case is considered, which results in the re-lation Eq. (66). In this section, the case where π S (cid:54) = 0 was considered, where constraints on π S were set to obtain different growth evolutions. I. f = c √T + c For the first model, using Eq. (72), the expres-sion for the anisotropic term becomes π S = − δ m k H Ω m a − πG × H (Ω m − − c H Ω m a − / − H (Ω m −
1) + c . (111)Note that by setting π S sets c , i.e. the cosmo-logical constant (and ultimately c ). Thus, weonly have one free parameter, which is π S . Letus consider some cases. (a) π S = constant: For this to occur, theright hand side (RHS) must become independentof δ m , k and a . This means that c must be de-pendent on these quantities. However, since c isa constant, this becomes a contradiction. Hence, π S cannot be constant (except for 0 which re-duces to f ( T )-ΛCDM models). (b) π S ∝ δ m /k : Suppose that π S = − γ H Ω m πG δ m k , (112)where γ is a function of time. Substituting andrearranging leads to c = (cid:0) γa (cid:1) − (cid:26) H (Ω m − − γ (cid:104) H Ω m a / − H (Ω m − a (cid:105) (cid:27) . (113)However, c is a constant, and hence the RHSmust become independent of time. By differenti-ating the expression and solving the differentialequation in γ gives a solution of the form γ = 1 − a + ηa / , (114) where η is a non-zero constant (since for η = 0, γ = − a − leading the numerator of the RHS tostill depend on time). By substituting back, wefind that c = 6 H (cid:20) Ω m − − m η (cid:21) . (115)One can note that this is a modification from theΛCDM solution, provided by the last term whichis only dependent on the value of η . As | η | → ∞ ,the value of c becomes the ΛCDM value [be-ing 6 H (Ω m − γ → π S → η , we can constrainthe value of the cosmological constant (and hence c ). Conversely, one can choose a specific formfor c (like Eq. (75)) to constrain the value for η (and hence c ). Since by choosing the rightparameters would yield the same plots, only theconstraints of η were considered.In Fig. 9, some values of η are considered. Onecan note that positive η yields faster growthswhilst negative η yields slower growths whencompared to ΛCDM. Furthermore, as the mag-nitude of η increases, the closer to the ΛCDMsolution gets. This is expected given the rela-tionship between c and η . a ( t ) D ( a ) Λ CDM η = (- ) η = (- ) η = η = FIG. 9: Evolution for the f = c √T + c model with c and c defined by Eqs. (72) and (115) for various η . Note that the solution can be made sufficientlyclose to the ΛCDM solution. On the other hand, for values of η which arecloser to zero start to deviate from the ΛCDMsolution, as can be seen in Fig. 10. This isagain due to the relationship between c and η ;for | η | <
1, the modification term starts to be-come large, effectively becoming large comparedto the ΛCDM model value. However, the samebehaviour as the previous case is retained, wherepositive η yields faster growths whilst negative η yields slower growths when compared to ΛCDM.Thus, from the discussions about the value of η , one can easily see that | η | has to be suffi-ciently large for it to be closely mimic the ΛCDMgrowth evolution. This imposes a constraint on6 a ( t ) D ( a ) Λ CDM η = (- ) η = (- ) η = η = FIG. 10: Evolution for the f = c √T + c model with c and c defined by Eqs. (72) and (115) for various η . One notes that the evolutions start to deviate forvalues of | η | <
1. This is due to the singularity at η = 0. the cosmological constant c , which simply statesthat the value should not deviate much from theΛCDM value; otherwise the growth factor willstart to either grow much faster (or much slowerdepending on the sign of η ) than the standardΛCDM evolution. II. f = − T − T c √T + c In the second model, Eq. (66) becomes π S = δ m k c T T πG T / (cid:16) c + c √T (cid:17) + 3 c T . (116)Using Eq. (84), the anisotropic term can be sim-plified into π S = δ m k − H T a − c πG √T a − / (cid:0) c + c √T (cid:1) − c H (117)Similar to the previous case, by setting π S sets c and c . Let us consider some cases. (a) π S = constant: For this to occur, theright hand side (RHS) must become independentof δ m , k and a . This means that both c and c must be dependent on these quantities. However,since both of them are constant, this becomesa contradiction. Hence, π S cannot be constant(except for 0 which becomes the re-scaling f ( T )model, which is essentially a contradiction as dis-cussed in Section IV). (b) π S ∝ δ m /k : Suppose that π S = − γ H T δ m k , (118) where γ is a function of time. Substituting andrearranging leads to c = a γ (cid:20) πG (cid:112) T a − / (cid:16) c + c (cid:112) T (cid:17) − c H (cid:21) . (119)However, c is a constant, and hence the RHSmust become independent of time. By differenti-ating the expression and solving the differentialequation in γ gives a solution of the form γ = ηa − / a / c H − πG √T (cid:0) c + c √T (cid:1) , (120)where η is a constant. By substituting back, wefind that c = − η. (121)By determining c , c can be determined usingEq. (89), which in terms of η gives η ≤ √T m . (122)This limits the possible choices of η . One canalso constrain η using c , however the plots turnout to be equivalent as long as the right constantsare chosen. As was done in Section IV, one candefine η to be η ≡ (cid:15) √T m , (123)where (cid:15) is a rescaling constant. Thus, the condi-tion reduces to having (cid:15) ≤ (cid:15) are considered.Recall that in this case, c has two solutions andhence there can be two types of evolution. Forthe positive solution, one finds that no value for (cid:15) can describe a ΛCDM like evolution (exceptfor (cid:15) = 1, but with a much faster growth rate).There also seems to be a value for which thegrowth factors transition from being completelygrowing faster than ΛCDM to decreasing theirgrowth rate and eventually fall below the latter.Furthermore, for smaller values, the solutions be-comes unstable.For Fig. 12, the negative solution for various (cid:15) are considered. In this case, an evolution be-haviour was noticed. For positive (cid:15) , the growthfactors are growing faster than the ΛCDM solu-tion, while the negative values grow slower. Onealso notes that (cid:15) = − . (cid:15) valuesbeing closer to latter.7 a ( t ) D ( a ) Λ CDM ϵ = ϵ = ϵ = ϵ = ϵ = (- ) FIG. 11: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) for thepositive solution and various (cid:15) . The solutions in thiscase deviate from the ΛCDM solution for every (cid:15) , andbecomes unstable for negative values of (cid:15) . a ( t ) D ( a ) Λ CDM ϵ = ϵ = ϵ = ϵ = (- ) ϵ = (- ) FIG. 12: Evolution for the f = − T − T c √T + c modelwith c and c defined by Eqs. (88) and (89) for thenegative solution and various (cid:15) . The solution can bemade close to the ΛCDM solution by choosing theright (cid:15) . VI. CONCLUSION
The main result of the paper is the modifiedM´esz´aros equation for f ( T, T ) gravity Eq. (63).It was found that the equation depends on thesub-horizon mode k , which contrasts from whatoccurs in GR and ΛCDM models. However, thiseffect was also found in f ( R, T ) models, andgiven the similarity between teleparallel gravityand general relativity, it is not surprising to findyet another similarity [41].In this case however, it was found that theGR limit of this modified M´esz´aros equation isstill dependent on k due to the presence of theanisotropic term π S . Thus, this proposes two op-tions: either this term vanishes (i.e. π S = 0) orthe anisotropy is a function of the wave numbersuch that its effect does not cause non-physicalresults (for example π S ∝ k n with n ≤ − f ( T, T ) models due to the presence of k f T a δ m . Since the func-tion f here is of zero order, this cannot be de-pendent on k , leading to problems. In fact, for π S = 0 models, the growths found for the twofunctions considered provided non-physical re-sults (either oscillatory or with varying growthswhich eventually deviate greatly from ΛCDM).Thus, two possible solutions were considered, thefirst being a constant f . This case would lead tostandard f ( T )-ΛCDM models (provided that theeffect of π S is negligible), which is not of interesthere. Thus, the second non-trivial case is consid-ered, where the sum of this term and anisotropicterm becomes zero.For this scenario, a coupling between theanisotropic term and the integration constants ofthe functions considered were found. This led todifferent evolutions, independent of k , in whichsome of them being close to the ΛCDM growth.Even though this might seem as a possible so-lution, one has to keep in mind that this setsa very specific form of how π S behaves, whichcan be unrealistic. The terms in the modifiedM´esz´aros equation have to cancel exactly, oth-erwise this would leave a k dependence, whichfurther strengthens this unrealistic possibility.Nonetheless, this leaves an avenue for further in-vestigation.This leaves f ( T, T ) models being unable toexplain growth evolution (except for cosmolog-ical constant models), unless the nature of theanisotropic term can be given very specific val-ues. Acknowledgements
The authors would like to thank Diego S´aez-G´omez for his comments and suggestions on anearlier version of this manuscript. The researchwork disclosed in this paper is partially fundedby the ENDEAVOUR Scholarships Scheme. [1] A. G. Riess and Others, Astron. J. , 1009(1998), astro-ph/9805201.[2] S. Perlmutter and Others, Astrophys. J. ,565 (1999), astro-ph/9812133.[3] G. Hinshaw and Others, Astrophys. J. Suppl. , 19 (2013), 1212.5226.[4] V. de Sabbata and C. Sivaram, in
In *Erice1990, Proceedings, Gravitation and modern cos-mology* 19-36. (1990).[5] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. , 559 (2003), astro-ph/0207347. [6] S. Nojiri and S. D. Odintsov, Phys. Rept. ,59 (2011), 1011.0544.[7] A. De Felice and S. Tsujikawa, Living Rev. Rel. , 3 (2010), 1002.4928.[8] T. Harko, F. S. N. Lobo, S. Nojiri, andS. D. Odintsov, Phys. Rev. D84 , 24020 (2011),1104.2669.[9] S. Chakraborty, Gen. Rel. Grav. , 2039(2013), 1212.3050.[10] G. Sun and Y.-C. Huang, Int. J. Mod. Phys. D25 , 1650038 (2016), 1510.01061.[11] G. P. Singh, B. K. Bishi, and P. K. Sahoo, Int.J. Geom. Meth. Mod. Phys. , 1650058 (2016).[12] V. Fayaz, H. Hossienkhani, Z. Zarei, and N. Az-imi, Eur. Phys. J. Plus , 22 (2016).[13] T. Clifton, P. G. Ferreira, A. Padilla, and C. Sko-rdis, Phys. Rept. , 1 (2012), 1106.2476.[14] V. C. De Andrade, L. C. T. Guillen, and J. G.Pereira, in Recent developments in theoreticaland experimental general relativity, gravitationand relativistic field theories. Proceedings, 9thMarcel Grossmann Meeting, MG’9, Rome, Italy,July 2-8, 2000. Pts. A-C (2000), gr-qc/0011087.[15] R. Aldrovandi and J. G. Pereira,
TeleparallelGravity: An Introduction , Fundamental Theo-ries of Physics (Springer Netherlands, 2012).[16] J. Garecki, in
Hypercomplex Seminar 2010: (Hy-per)Complex and Randers-Ingarden Structuresin Mathematics and Physics Bedlewo, Poland,July 17-24, 2010 (2010), 1010.2654.[17] J. W. Maluf, Annalen Phys. , 339 (2013),1303.3897.[18] H. I. Arcos and J. G. Pereira, Int. J. Mod. Phys.
D13 , 2193 (2004), gr-qc/0501017.[19] R. Aldrovandi, J. G. Pereira, and K. H. Vu,Braz. J. Phys. , 1374 (2004), gr-qc/0312008.[20] R. Aldrovandi, J. G. Pereira, and K. H. Vu, Gen.Rel. Grav. , 101 (2004), gr-qc/0304106.[21] Y.-F. Cai, S. Capozziello, M. De Laurentis, andE. N. Saridakis (2015), 1511.07586.[22] M. Krˇsˇs´ak (2015), 1510.06676.[23] D. S´aez-G´omez, C. S. Carvalho, F. S. N. Lobo,and I. Tereno (2016), 1603.09670.[24] I. G. Salako, M. E. Rodrigues, A. V. Kpadonou,M. J. S. Houndjo, and J. Tossa, JCAP ,060 (2013), 1307.0730.[25] K. Bamba, R. Myrzakulov, S. Nojiri, and S. D.Odintsov, Phys. Rev. D85 , 104036 (2012),1202.4057.[26] A. Paliathanasis, J. D. Barrow, and P. G. L.Leach, Phys. Rev.
D94 , 023525 (2016),1606.00659.[27] G. R. Bengochea and R. Ferraro, Phys. Rev.
D79 , 124019 (2009), 0812.1205. [28] E. V. Linder, Phys. Rev.
D81 , 127301 (2010),[Erratum: Phys. Rev.D82,109902(2010)],1005.3039.[29] S. Nesseris, S. Basilakos, E. N. Saridakis, andL. Perivolaropoulos, Phys. Rev.
D88 , 103010(2013), 1308.6142.[30] T. Harko, F. S. N. Lobo, G. Otalora, and E. N.Saridakis, JCAP , 21 (2014), 1405.0519.[31] E. L. B. Junior, M. E. Rodrigues, I. G. Salako,and M. J. S. Houndjo, Class. Quant. Grav. ,125006 (2016), 1501.00621.[32] M. Sharif and A. Ikram (2016), 1608.01182.[33] P. Meszaros, Astron. Astrophys. , 225 (1974).[34] S. Dodelson, Modern Cosmology , AcademicPress (Academic Press, 2003).[35] X. Fu, P. Wu, and H. W. Yu, Eur. Phys. J.
C68 ,271 (2010), 1012.2249.[36] K. Bamba, A. Lopez-Revelles, R. Myrzakulov,S. D. Odintsov, and L. Sebastiani, Class. Quant.Grav. , 015008 (2013), 1207.1009.[37] I. de Martino, M. De Laurentis, andS. Capozziello, Universe , 123 (2015),1507.06123.[38] R. Zheng and Q.-G. Huang, JCAP , 2(2011), 1010.3512.[39] S. Basilakos, Phys. Rev. D93 , 083007 (2016),1604.00264.[40] K. Karami, A. Abdolmaleki, S. Asadzadeh,and Z. Safari, Phys. Rev.
D88 , 084034 (2013),1111.7269.[41] F. G. Alvarenga, A. de la Cruz-Dombriz, M. J. S.Houndjo, M. E. Rodrigues, and D. S´aez-G´omez,Phys. Rev.
D87 , 103526 (2013), 1302.1866.[42] T. P. Sotiriou, B. Li, and J. D. Barrow, Phys.Rev.
D83 , 104030 (2011), 1012.4039.[43] R. Ferraro and F. Fiorini, Phys. Rev.
D75 ,084031 (2007), gr-qc/0610067.[44] B. Li, T. P. Sotiriou, and J. D. Barrow, Phys.Rev.
D83 , 064035 (2011), 1010.1041.[45] N. Tamanini and C. G. Boehmer, Phys. Rev.
D86 , 044009 (2012), 1204.4593.[46] M. Krˇsˇs´ak and E. N. Saridakis, Class. Quant.Grav. , 115009 (2016), 1510.08432.[47] T. Harko, Phys. Lett. B669 , 376 (2008),0810.0742.[48] J. Wang and K. Liao, Class. Quant. Grav. ,215016 (2012), 1212.4656.[49] T. S. Pereira, G. A. M. Marug´an, andS. Carneiro, JCAP , 29 (2015), 1505.00794.[50] G. F. R. Ellis, R. Maartens, and M. A. H.MacCallum, Relativistic Cosmology (CambridgeUniversity Press, 2012).
Appendix I
The action is composed of two Lagrangians to form a single Lagrangian of the form, L = L grav + L M , (124)where L grav ≡ e [ T + f ( T, T )] / πG and L M ≡ e L m , which denote the gravitational and matterLagrangians respectively. The field equations are obtained by taking small variations of the action9with respect to the inverse vierbein, and are found through the Euler-Lagrange equations [15] ∂ L ∂e aρ − ∂ σ ∂ L ∂ (cid:0) ∂ σ e aρ (cid:1) = 0 . (125)The gravitational component of the first term is expanded as follows,16 πG ∂ L grav ∂e aρ = e (cid:20) (1 + f T ) ∂T∂e aρ + f T ∂ T ∂e aρ (cid:21) + [ T + f ( T, T )] ∂e∂e aρ = e (1 + f T ) ∂T∂e aρ + f T ∂ (cid:18) g αβ em T αβ (cid:19) ∂e aρ + [ T + f ( T, T )] ∂e∂e aρ = e (1 + f T ) ∂T∂e aρ + f T g αβ ∂ em T αβ ∂e aρ + em T αβ ∂g αβ ∂e aρ + ( T + f ) ∂e∂e aρ , (126)whilst the gravitational component of the second term is expanded to16 πG∂ σ ∂ L grav ∂ (cid:0) ∂ σ e aρ (cid:1) = ∂ σ (cid:34) e (1 + f T ) ∂T∂ (cid:0) ∂ σ e aρ (cid:1) + ef T ∂ T ∂ (cid:0) ∂ σ e aρ (cid:1) (cid:35) = (1 + f T ) ∂ σ (cid:34) e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) (cid:35) + e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) f T T ∂ σ T + f T ∂ σ (cid:34) e ∂ T ∂ (cid:0) ∂ σ e aρ (cid:1) (cid:35) + e ∂ T ∂ (cid:0) ∂ σ e aρ (cid:1) f T T ∂ σ T = (1 + f T ) ∂ σ (cid:34) e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) (cid:35) + e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) f T T ∂ σ T + f T ∂ σ (cid:34) e ∂ T ∂ (cid:0) ∂ σ e aρ (cid:1) (cid:35) + e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) f T T ∂ σ T . (127)On the other hand, under the assumption that the matter Lagrangian does not depend on thederivatives of the inverse vierbein field, the matter components of the Euler-Lagrange terms become, ∂ L M ∂e aρ = ∂ ( e L m ) ∂e aρ , (128) ∂ σ ∂ L M ∂ (cid:0) ∂ σ e aρ (cid:1) = 0 . (129)This condition simplifies the second gravitational term to16 πG∂ σ ∂ L grav ∂ (cid:0) ∂ σ e aρ (cid:1) = (1 + f T ) ∂ σ (cid:34) e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) (cid:35) + e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) f T T ∂ σ T + e ∂T∂ (cid:0) ∂ σ e aρ (cid:1) f T T ∂ σ T . (130)As given in Refs. [15, 46], the following relations for the derivatives are given, ∂T∂ (cid:0) ∂ σ e aρ (cid:1) = − S ρσa , (131) ∂T∂e aρ = − T bνa S νρb + 4 ω baν S bνρ , (132) ∂e∂e aρ = ee ρa , (133) ∂g αβ ∂e aρ = − g ρβ e αa − g ρα e βa . (134)0Therefore, the field equations for f ( T, T ) gravity become(1 + f T ) (cid:104) e − ∂ σ ( eS ρσa ) − T bνa S νρb + ω baν S bνρ (cid:105) + ( f T T ∂ σ T + f T T ∂ σ T ) S ρσa + e ρa (cid:18) T + f (cid:19) + f T g αβ ∂ em T αβ ∂e aρ + em T αβ (cid:0) − g ρβ e αa − g ρα e βa (cid:1) = − πGe − ∂ ( e L m ) ∂e aρ . (135)By defining the stress-energy tensor to be, em T ρa ≡ − e − ∂ ( e L m ) ∂e aρ , (136)and considering a perfect fluid representation, we have [8] g αβ ∂ em T αβ ∂e aρ = 4 em T ρa +2 pe ρa . (137)Therefore, the final field equations are given to be,(1 + f T ) (cid:104) e − ∂ σ ( eS ρσa ) − T bνa S νρb + ω baν S bνρ (cid:105) + ( f T T ∂ σ T + f T T ∂ σ T ) S ρσa + e ρa (cid:18) T + f (cid:19) + f T (cid:18) em T ρa + pe ρa (cid:19) = 4 πG em T ρa . (138) Appendix II