Cosmological applications of F(T, T G ) gravity
aa r X i v : . [ g r- q c ] O c t Cosmological applications of F ( T, T G ) gravity Georgios Kofinas ∗ and Emmanuel N. Saridakis
2, 3, † Research Group of Geometry, Dynamical Systems and Cosmology,Department of Information and Communication Systems EngineeringUniversity of the Aegean, Karlovassi 83200, Samos, Greece Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece Instituto de F´ısica, Pontificia Universidad de Cat´olica de Valpara´ıso, Casilla 4950, Valpara´ıso, Chile
We investigate the cosmological applications of F ( T, T G ) gravity, which is a novel modified gravi-tational theory based on the torsion invariant T and the teleparallel equivalent of the Gauss-Bonnetterm T G . F ( T, T G ) gravity differs from both F ( T ) theories as well as from F ( R, G ) class of cur-vature modified gravity, and thus its corresponding cosmology proves to be very interesting. Inparticular, it provides a unified description of the cosmological history from early-times inflationto late-times self-acceleration, without the inclusion of a cosmological constant. Moreover, thedark energy equation-of-state parameter can be quintessence or phantom-like, or experience thephantom-divide crossing, depending on the parameters of the model.
PACS numbers: 04.50.Kd, 98.80.-k, 95.36.+x
I. INTRODUCTION
Since theoretical arguments and observational datasuggest that the universe passed through an early-timesinflationary stage and resulted in a late-times acceleratedphase, a large amount of research was devoted to explainthis behavior. In general, one can follow two ways toachieve it. The first direction is to alter the universecontent by introducing additional fields, canonical scalar,phantom scalar, both scalars, vector fields etc, that is in-troducing the concepts of the inflaton and/or the darkenergy, which can be extended in a huge class of mod-els (see [1–3] and references therein). The second wayis to modify the gravitational sector instead (see [4] andreferences therein). Note however that one can in prin-ciple transform from one approach to the other, sincethe important point is the number of degrees of freedombeyond standard model particles and General Relativity(GR) [5].In modified gravitational theories one usually extendsthe curvature-based Einstein-Hilbert action. However, adifferent and interesting class of gravitational modifica-tion arises when one extends the action of the equivalenttorsional formulation of General Relativity. In partic-ular, since Einstein’s years it was known that one canconstruct the so-called “Teleparallel Equivalent of Gen-eral Relativity” (TEGR) [6–10], that is attributing grav-ity to torsion instead of curvature, by using instead ofthe torsion-less Levi-Civita connection the curvature-lessWeitzenb¨ock one. In such a formulation the gravitationalLagrangian results from contractions of the torsion tensorand is called the “torsion scalar” T , similarly to the Gen-eral Relativity Lagrangian, i.e. the “curvature scalar” R ,which is constructed by contractions of the curvature ten- ∗ Electronic address: gkofi[email protected] † Electronic address: Emmanuel˙[email protected] sor. Hence, similarly to the f ( R ) extensions of GeneralRelativity [11, 12], one can construct f ( T ) extensions ofTEGR [13, 14]. The interesting feature in this exten-sion is that f ( T ) does not coincide with f ( R ) gravity,despite the fact that TEGR coincides with General Rel-ativity. Since it is a new gravitational modification class,its corresponding cosmological behavior and black holesolutions have been studied in detail [13–19].However, apart from the simple modifications of cur-vature gravity, one can construct more complicated ac-tions introducing higher-curvature corrections such asthe Gauss-Bonnet combination G [20, 21] or arbitraryfunctions f ( G ) [22–24], Weyl combinations [25], Love-lock combinations [26, 27] etc. Hence, one can follow thesame direction starting from the teleparallel formulationof gravity, and construct actions involving higher-torsioncorrections. Indeed, in our recent work [28] we first con-structed the teleparallel equivalent of the Gauss-Bonnetterm T G (which is a new quartic torsional scalar whichreduces to a topological invariant in four dimensions),and then, using also the torsion scalar T , we constructed F ( T, T G ) gravity (see [29–31] for different constructionsof torsional actions). This is a new class of gravitationalmodification, since it is different from both f ( T ) gravityas well as from f ( R, G ) gravity.Since F ( T, T G ) gravity is a novel modified gravity the-ory, in the present work we are interested in investigat-ing its cosmological applications. In particular, after ex-tracting the Friedmann equations, we define the effectivedark energy sector and the various observables such asthe density parameters and the dark energy equation-of-state parameter. Then, considering specific F ( T, T G )ansatzes we investigate the inflation realization and thelate-times acceleration. The plan of the work is as fol-lows: In section II we review F ( T, T G ) gravity. In sectionIII we apply it in a cosmological framework, extractingthe corresponding equations and defining the various ob-servables, while in section IV we analyze some specificcases. Finally, section V is devoted to the conclusions. II. F ( T, T G ) GRAVITY
Let us give a brief review of F ( T, T G ) gravity [28]. Al-though in this manuscript we are interested in its cos-mological application, we will present the formulation in D -dimensions where it is non-trivial, and then discuss F ( T, T G ) gravity in D = 4. In the teleparallel formula-tion of gravity theories, the dynamical variables are thevielbein field e a ( x µ ) and the connection 1-forms ω ab ( x µ )which defines the parallel transportation . We can ex-press them in components in terms of coordinates as e a = e µa ∂ µ and ω ab = ω abµ dx µ = ω abc e c , while we definethe dual vielbein as e a = e aµ dx µ . The vielbein commu-tation relations read[ e a , e b ] = C cab e c , (1)where the structure coefficients functions C cab are writ-ten as C cab = e µa e νb ( e cµ,ν − e cν,µ ) , (2)with a comma denoting differentiation. Thus, we can de-fine the torsion tensor, expressed in tangent componentsas T abc = ω acb − ω abc − C abc , (3)while the curvature tensor is R abcd = ω abd,c − ω abc,d + ω ebd ω aec − ω ebc ω aed − C ecd ω abe . (4)Furthermore, we use the metric tensor g to make thevielbein orthonormal g ( e a , e b ) = η ab , where η ab =diag( − , , ... g µν = η ab e aµ e bν , (5)and indices a, b, ... are raised/lowered with the Minkowskimetric η ab . Finally, it proves convenient to define thecontorsion tensor as K abc = 12 ( T cab − T bca − T abc ) = −K bac . (6)We now impose the condition of teleparallelism,namely R abcd = 0, which holds in all frames. One wayto realize this condition is by assuming the Weitzenb¨ockconnection ˜ ω λµν defined in terms of the vielbein in all co-ordinate frames as ˜ ω λµν = e λa e aµ,ν , or expressed in thepreferred tangent-space components as ˜ ω abc = 0. TheRicci scalar ¯ R corresponding to the usual Levi-Civitaconnection can be expressed as [8, 9] e ¯ R = − eT + 2( eT νµν ) ,µ , (7) In this manuscript the notation is as follows: Greek indices µ, ν, ...run over all space-time coordinates, while Latin indices a, b, ...run over the tangent space. where we have defined the “torsion scalar” T as T = 14 T µνλ T µνλ + 12 T µνλ T λνµ − T νµν T λλµ , (8)and e = det ( e aµ ) = p | g | .One can now clearly see that the Lagrangian den-sity e ¯ R of General Relativity, that is the one calculatedwith the Levi-Civita connection, and the torsional den-sity − eT differ only by a total derivative. Hence, theEinstein-Hilbert action S EH = 12 κ D Z M d D x e ¯ R, (9)up to boundary terms is equivalent to the action S (1) tel = − κ D Z M d D x e T (10)in the sense that varying (9) with respect to the met-ric and varying (10) with respect to the vielbein givesrise to the same equations of motion ( κ D is the D -dimensional gravitational constant) [10]. That is whythe above theory, where one uses torsion to describe thegravitational field and imposes the teleparallelism condi-tion, was dubbed by Einstein as Teleparallel Equivalentof General Relativity (TEGR).The recipe of the construction of TEGR was to ex-press the Ricci scalar R corresponding to a general con-nection as the Ricci scalar ¯ R calculated with the Levi-Civita connection, plus terms arising from the torsiontensor. Then, by imposing the teleparallelism condi-tion R abcd = 0, we obtained that ¯ R is equal to a tor-sion scalar plus a total derivative. Hence, we can followthe same steps, but using the Gauss-Bonnet combination G = R − R µν R µν + R µνκλ R µνκλ instead of the Ricciscalar. In [28] we have derived the teleparallel equiva-lent of Gauss-Bonnet gravity characterized by the newtorsion scalar T G , as well as the equations of motion ofthe modified gravity defined by the function F ( T, T G ).In the following, we give the corresponding expressionswhen restricted to the Weitzenb¨ock connection ω abc = 0(the tildes are withdrawn for notational simplicity). It is e ¯ G = eT G +total diverg. , (11)where ¯ G is the Gauss-Bonnet term calculated by theLevi-Civita connection, and T G = ( K a ea K ea b K a fc K fa d − K a a a K a eb K efc K fa d +2 K a a a K a eb K ea f K fcd +2 K a a a K a eb K ea c,d ) δ a b c da a a a , (12)with the generalized δ being the determinant of the Kro-necker deltas. Thus, T G is the teleparallel equivalent of¯ G , in the sense that the action S (2) tel = 12 κ D Z M d D x e T G , (13)varied in terms of the vielbein gives exactly the sameequations with the action S GB = 12 κ D Z M d D x e ¯ G , (14)varied in terms of the metric.Having constructed the teleparallel equivalent of cur-vature invariants, one can be based on them in order tobuild modified gravitational theories. Thus, one can startfrom an action where T is generalized to F ( T ), resultingto the so-called F ( T ) gravity [13–19]. Similarly, one canextend T G to F ( T G ) in the action, and since T G is quar-tic in torsion then F ( T G ) cannot arise from any F ( T ).Hence, in [28] we combined both possible extensions andwe constructed the F ( T, T G ) modified gravity S = 12 κ D Z d D x e F ( T, T G ) , (15)which is clearly different from both F ( T ) theory as wellas from F ( R, G ) gravity [22–24], and therefore it is novelgravitational modification. We mention that TEGR(and therefore GR) is obtained for F ( T, T G ) = − T ,while the usual Einstein-Gauss-Bonnet theory arises for F ( T, T G ) = − T + αT G , with α the Gauss-Bonnet cou-pling.Let us now give the equations of motion in D = 4which is our main interest in the present paper. Varyingthe action (15) in terms of the vierbein, after varioussteps, we finally obtain [28]2( H [ ac ] b + H [ ba ] c − H [ cb ] a ) ,c +2( H [ ac ] b + H [ ba ] c − H [ cb ] a ) C ddc +(2 H [ ac ] d + H dca ) C bcd +4 H [ db ] c C a ( dc ) + T acd H cdb − h ab +( F − T F T − T G F T G ) η ab = 0 , (16)where H abc = F T ( η ac K bdd − K bca ) + F T G (cid:2) ǫ cprt (cid:0) ǫ adkf K bkp K dqr + ǫ qdkf K akp K bdr + ǫ abkf K kdp K dqr (cid:1) K qft + ǫ cprt ǫ abkd K fdp (cid:0) K kfr,t − K kfq C qtr (cid:1) + ǫ cprt ǫ akdf K dfp (cid:0) K bkr,t − K bkq C qtr (cid:1)(cid:3) + ǫ cprt ǫ akdf h(cid:0) F T G K bkp K dfr (cid:1) ,t + F T G C qpt K bk [ q K dfr ] i (17)and h ab = F T ǫ akcd ǫ bpqd K kfp K fcq . (18)We have used the notation F T = ∂F/∂T , F T G = ∂F/∂T G , the (anti)symmetrization symbol contains thefactor 1 /
2, while the antisymmetric symbol ǫ abcd has ǫ = 1, ǫ = − F ( T, T G ) modified gravity itself, before proceeding to itscosmological investigation. The first has to do with the Lorentz violation. In particular, as we discussed also in[28], under the use of the Weitzenb¨ock connection thetorsion scalar T remains diffeomorphism invariant, how-ever the Lorentz invariance has been lost since we havechosen a specific class of frames, namely the autoparal-lel orthonormal frames. Nevertheless, the equations ofmotion of the Lagrangian eT , being the Einstein equa-tions, are still Lorentz covariant. On the contrary, whenwe replace T by a general function f ( T ) in the action,the new equations of motion will not be covariant underLorentz rotations of the vielbein, although they will in-deed be form-invariant, and the same features appear inthe F ( T, T G ) extension. However, this is not a deficit (itis a sort of analogue of gauge fixing in gauge theories),and the theory, although not Lorentz covariant, is mean-ingful. Definitely, not all vielbeins will be solutions of theequations of motion, but those which solve the equationswill determine the metric uniquely.The second comment is related to possible acausali-ties and problems with the Cauchy development of aconstant-time hypersurface. Indeed, there are worksclaiming that a departure from TEGR, as for instance in f ( T ) gravity, with the subsequent local Lorentz violation,will lead to the above problems [32]. In order to exam-ine whether one also has these problems in the presentscenario of F ( T, T G ) gravity, he would need to perform avery complicated analysis, extending the characteristicsmethod of [32] for this case, although at first sight onedoes expect to indeed find them. Nevertheless, even ifthis proves to be the case, it does not mean that the the-ory has to be ruled out, since one could still handle f ( T )gravity (and similarly F ( T, T G ) one) as an effective the-ory, in the regime of validity of which the extra degreesof freedom can be removed or be excited in a healthy way(alternatively one could reformulate the theory using La-grange multipliers) [32]. However, there is a possibilitythat these problems might be related to the restrictinguse of the Weitzenb¨ock connection, since the formulationof TEGR and its modifications using other connections(still in the “teleparallel class”) does not seem to be prob-lematic, and thus, the general formulation of F ( T, T G )gravity that was presented in [28] might be free of theabove disadvantages. These issues definitely need fur-ther investigation, and the discussion is still open in theliterature. III. F ( T, T G ) COSMOLOGY
In this section we apply F ( T, T G ) gravity in a cos-mological framework. Firstly, we add the matter sectoralong the gravitational one, that is we start by the totalaction S tot = 12 κ Z d x e F ( T, T G ) + S m , (19)where S m corresponds to a matter energy-momentumtensor Θ µν and κ = 8 πG is the four-dimensional New-ton’s constant. Secondly, in order to investigate the cos-mological implications of the above action, we consider aspatially flat cosmological ansatz ds = − N ( t ) dt + a ( t ) δ ˆ i ˆ j dx ˆ i dx ˆ j , (20)where a ( t ) is the scale factor and N ( t ) is the lapse func-tion (the hat indices run in the three spatial coordinates).This metric arises from the diagonal vierbein e aµ = diag( N ( t ) , a ( t ) , a ( t ) , a ( t )) (21)through (5), while the dual vierbein is e µa =diag( N − ( t ) , a − ( t ) , a − ( t ) , a − ( t )), and its determinant e = N ( t ) a ( t ) .Considering as usual N ( t ) = 1 and inserting the vier-bein (21) into relations (8) and (12), we find T = 6 ˙ a a = 6 H (22) T G = 24 ˙ a a ¨ aa = 24 H (cid:0) ˙ H + H (cid:1) , (23)where H = ˙ aa is the Hubble parameter and dots denotedifferentiation with respect to t . Additionally, inserting(21) into the general equations of motion (16), after somealgebra we obtain the Friedmann equations F − H F T − T G F T G + 24 H ˙ F T G = 2 κ ρ (24) F −
4( ˙ H + 3 H ) F T − H ˙ F T − T G F T G + 23 H T G ˙ F T G + 8 H ¨ F T G = − κ p , (25)where the right hand sides arise from the independentvariation of the matter action, considering it to corre-spond to a perfect fluid with energy density ρ and pres-sure p (it is Θ tt = ρ , Θ ˆ i ˆ j = pa δ ˆ i ˆ j , Θ ˆ i ˆ i = 3 p ). In theabove expressions it is ˙ F T = F T T ˙ T + F T T G ˙ T G , ˙ F T G = F T T G ˙ T + F T G T G ˙ T G , ¨ F T G = F T T T G ˙ T + 2 F T T G T G ˙ T ˙ T G + F T G T G T G ˙ T G + F T T G ¨ T + F T G T G ¨ T G , with F T T , F T T G , ...denoting multiple partial differentiations of F with re-spect to T , T G . Finally, ˙ T , ¨ T and ˙ T G , ¨ T G are obtainedby differentiating (22) and (23) respectively with respectto time.Let us make a comment here on the derivation of theabove Friedmann equations. We mention that we fol-lowed the robust way, that is we first performed the gen-eral variation of the action resulting to the general equa-tions of motion (16), and then we inserted the cosmo-logical ansatz (21), obtaining (24) and (25). This pro-cedure in principle does not give the same results withthe shortcut procedure where one first inserts the cos-mological ansatz (21) in the action and then performsvariation with respect to N and a , since variation andansatz-insertion do not commute in general, especiallyin theories with higher-order derivatives [33, 34]. Thisshortcut method is a sort of minisuperspace procedure since the (potential) additional degrees of freedom otherthan those contained in the scale factor are frozen. How-ever, as we show in the Appendix, following the shortcutprocedure in the cosmological application of the scenarioat hand leads exactly to the two Friedmann equations(24) and (25) of the robust procedure.Setting F ( T, T G ) = − T in equations (24), (25), weget the standard cosmological equations of General Rel-ativity without a cosmological constant. For F ( T, T G ) = F ( T ) we get F − H F T = 2 κ ρ (26) F −
4( ˙ H + 3 H ) F T − H ˙ F T = − κ p , (27)which are recognized as the standard equations of F ( T )gravity. However, note that due to the various conven-tions adopted in the literature in the definitions of theRiemann tensor, the torsion tensor and the Minkowskimetric, which may differ by a total sign, our function F ( T ) may correspond to F ( − T ) in some other works (forinstance [14]).In order to parametrize the deviation of the theory F ( T, T G ) from GR, we write F ( T, T G ) = − T + f ( T, T G ).Thus, the modification of GR (for instance the effectivedark energy component) is included in the function f .Equations (24), (25) are then written as6 H + f − H f T − T G f T G + 24 H ˙ f T G = 2 κ ρ (28)2(2 ˙ H + 3 H ) + f −
4( ˙ H + 3 H ) f T − H ˙ f T − T G f T G + 23 H T G ˙ f T G + 8 H ¨ f T G = − κ p . (29)The Friedmann equations (28), (29) can be rewritten inthe usual form H = κ ρ + ρ DE ) (30)˙ H = − κ ρ + p + ρ DE + p DE ) , (31)where the energy density and pressure of the effectivedark energy sector are defined as ρ DE = − κ ( f − H f T − T G f T G + 24 H ˙ f T G ) (32) p DE = 12 κ h f −
4( ˙ H + 3 H ) f T − H ˙ f T − T G f T G + 23 H T G ˙ f T G + 8 H ¨ f T G i . (33)In terms of the initial function F , we can write ρ DE , p DE as ρ DE = 12 κ (6 H − F +12 H F T + T G F T G − H ˙ F T G ) (34) p DE = 12 κ h − H +3 H )+ F −
4( ˙ H +3 H ) F T − H ˙ F T − T G F T G + 23 H T G ˙ F T G +8 H ¨ F T G i . (35)Since the standard matter is conserved independently,˙ ρ + 3 H ( ρ + p ) = 0, we obtain from (32), (33) that thedark energy density and pressure also satisfy the usualevolution equation˙ ρ DE + 3 H ( ρ DE + p DE ) = 0 . (36)Finally, we can define the dark energy equation-of-stateparameter as w DE = p DE ρ DE . (37) IV. SPECIFIC CASES
In the previous section we extracted the Friedmannequations of general F ( T, T G ) cosmology, and we definedthe effective dark energy sector. Thus, in this sectionwe proceed to the investigation of some specific F ( T, T G )cases, focusing on the evolution of observables such asthe various density parameters Ω i = 8 πGρ i / (3 H ) andthe dark energy equation-of-state parameter w DE . A. F ( T, T G ) = − T + β p T + β T G + α T + α T p | T G | Since T G contains quartic torsion terms, it will in gen-eral and approximately be of the same order with T .Therefore, T and p T + β T G are of the same order,and thus, if one of them contributes during the evolu-tion the other will contribute too. Therefore, it wouldbe very interesting to consider modifications of the form F ( T, T G ) = − T + β p T + β T G , which are expectedto play an important role at late times. Note that thecouplings β , β are dimensionless, and so, no new massscale enters at late times. Nevertheless, in order to de-scribe the early-times cosmology, one should additionallyinclude higher order corrections like T . Since the scalar T G is of the same order with T , it should be also in-cluded. However, since T G is topological in four dimen-sions it cannot be included as it is, and therefore we usethe term T p | T G | which is also of the same order with T and non-trivial. Thus, the total function F is takento be F ( T, T G ) = − T + β p T + β T G + α T + α T p | T G | . (38)In summary, when the above function is used as an ac-tion, it gives rise to a gravitational theory that can de-scribe both inflation and late-times acceleration in a uni-fied way.In order to examine the cosmological evolution of auniverse governed by the above unified action, we per-form a numerical elaboration of the Friedman equations(30), (31), with ρ DE , p DE given by equations (34), (35),under the ansatz (38). In Fig. 1 we present the early-times, inflationary solutions for four parameter choices.As we observe, inflationary, de-Sitter exponential expan- -30 -27 -24 a ( t ) t FIG. 1:
Four inflationary solutions for the ansatz F ( T, T G ) = α T + α T p | T G | − T + β p T + β T G , cor-responding to a) α = − . , α = 8 , β = 0 . , β = 1 (black-solid), b) α = − , α = 8 , β = 0 . , β = 1 (red-dashed), c) α = 8 , α = 8 , β = 0 . , β = 1 (blue-dotted),d) α = 20 , α = 5 , β = 0 . , β = 1 (green-dashed-dotted).All parameters are in Planck units. sions can be easily obtained (with the exponent of the ex-pansion determined by the model parameters), althoughthere is not an explicit cosmological constant term in theaction, which is an advantage of the scenario. This wasexpected, since one can easily extract analytical solutionsof the Friedmann equations (30), (31) with H ≈ const (inwhich case T and T G as also constants).Let us now focus on the late-times evolution. In Fig. 2we depict the evolution of the matter and effective darkenergy density parameters, as well as the behavior of thedark energy equation-of-state parameter, for a specificchoice of the model parameters.As we see, we can obtain the observed behavior, whereΩ m decreases, resulting to its current value of Ω m ≈ . DE = 1 − Ω m increases. Concerning w DE , we cansee that in this example it lies in the quintessence regime.However, as it is usual in modified gravity [35], themodel at hand can describe the phantom regime too, fora region of the parameter space, which is an additionaladvantage. In particular, in Fig. 3 we depict the cosmo-logical evolution for a parameter choice that leads w DE tothe phantom regime, while the density parameters main-tain their observed behavior. Similarly, note that thescenario can also exhibit the phantom-divide crossing [3]too. Finally, note that one can use dynamical-systemsmethods in order to examine in a systematic way the late-times cosmological behavior of the scenario at hand, in-dependently of the initial conditions of the universe [36]. w D E z DEm
FIG. 2:
Upper graph: The evolution of the dark energy den-sity parameter Ω DE (black-solid) and the matter density pa-rameter Ω m (red-dashed), as a function of the redshift z , forthe ansatz F ( T, T G ) = α T + α T p | T G |− T + β p T + β T G with α = 0 . , α = 0 . , β = 2 . , β = 1 . .Lower graph: The evolution of the corresponding dark energyequation-of-state parameter w DE . All parameters are in unitswhere the present Hubble parameter is H = 1 , and we haveimposed Ω m ≈ . , Ω DE ≈ . at present. DEm w D E z FIG. 3:
Upper graph: The evolution of the dark energydensity parameter Ω DE (black-solid) and the matter den-sity parameter Ω m (red-dashed), as a function of the red-shift z , for the ansatz F ( T, T G ) = α T + α T p | T G | − T + β p T + β T G , with α = 0 . , α = 0 . , β = 2 . , β = 2 . Lower graph: The evolution of the corresponding darkenergy equation-of-state parameter w DE . All parameters arein units where the present Hubble parameter is H = 1 , andwe have imposed Ω m ≈ . , Ω DE ≈ . at present. B. F ( T, T G ) = − T + f ( T + β T G ) One can go beyond the simple model of the previousparagraph. In particular, since as we already mentioned T G contains quartic torsion terms, it will in general andapproximately be of the same order with T . Therefore,it would be interesting to consider modifications of theform F ( T, T G ) = − T + f ( T + β T G ). The involved build-ing block is an extension of the simple T , and thus, it cansignificantly improve the detailed cosmological behaviorof a suitable reconstructed F ( T ). The general equations(28), (29) in this case become6 H + f − (24 H T + β T G ) f ′ + 24 β H (2 T ˙ T + β ˙ T G ) f ′′ = 2 κ ρ (39)2(2 ˙ H + 3 H ) + f − [8( ˙ H +3 H ) T +8 H ˙ T + β T G ] f ′ + nh β T G H − HT i (2 T ˙ T + β ˙ T G )+8 β H (2 T ˙ T + β ˙ T G ) · o f ′′ +8 β H (2 T ˙ T + β ˙ T G ) f ′′′ = − κ p , (40)where f ′ , f ′′ , f ′′′ denote the derivatives of the function f and are evaluated at T + β T G .As a representative example we choose the case F ( T, T G ) = − T + β ( T + β T G )+ β ( T + β T G ) , that iskeeping up to fourth-order torsion terms (one could eas-ily proceed to the investigation of other ansatzes in thisclass and to the detailed description of a unified picture ofinflation and late-times acceleration). As expected, thehigher-order torsion terms are significant at early times,and thus they can easily drive inflation. In Fig. 4 weshow the early-times, inflationary solutions for five pa-rameter choices, changing in particular the value of β inorder to see the effect of T G on the evolution (the value of β is irrelevant since the linear T G term does not have anyeffect, since T G , similarly to G , is topological invariant).As we observe, inflationary evolution, that is de-Sitterexponential expansions can be easily obtained, and theexpansion-exponent is determined by the model param-eters. The significant advantage is that the exponentialexpansion is obtained without an explicit cosmologicalconstant term in the action. Again, this was expectedsince we can easily extract analytical solutions of theFriedmann equations (39), (40) with H ≈ const (in whichcase T and T G as also constants). Additionally, note thatin this case the inflation realization is more efficient com-paring to the model of the previous subsection, since itleads to more e-foldings in less time, as expected sincenow higher-order torsion terms are considered.In the above analysis we showed that f ( T, T G ) cos-mology can be very efficient in describing the evolutionof the universe at the background level. However, beforeconsidering any cosmological model as a candidate for thedescription of nature it is necessary to perform a detailedinvestigation of its perturbations, namely to examinewhether the obtained solutions are stable. Furthermore, -30 -27 -24 a ( t ) t FIG. 4:
Five inflationary solutions for the ansatz F ( T, T G ) = − T + β ( T + β T G ) + β ( T + β T G ) , corresponding to a) β = − . , β = 1 , β = − , β = − (black-solid), b) β = − . , β = 1 , β = − , β = − (red-dashed), c) β = − . , β = 1 , β = − , β = − (blue-dotted), d) β = − . , β = 1 , β = − , β = − (green-dashed-dotted),e) β = − . , β = 1 , β = − , β = − (yellow-dashed-dotted-dotted). All parameters are in Planck units. especially in theories with local Lorentz invariance viola-tion, new degrees of freedom are introduced, the behaviorof which is not guaranteed that is stable (for instance thisis the case in the initial version of Hoˇrava-Lifshitz gravity[37], in the initial version of de Rham-Gabadadze-Tolleymassive gravity [38], etc), and this makes the pertur-bation analysis of such theories even more imperative.Although such a detailed and complete analysis of thecosmological perturbations of f ( T, T G ) gravity is neces-sary, its various complications and lengthy calculationsmake it more convenient to be examined in a separateproject [39]. However, for the moment we would like tomention that in the case of simple f ( T ) gravity, the per-turbations of which have been examined in detail [15, 17],one does obtain instabilities, but there are many classesof f ( T ) ansantzes and/or parameter-space regions, wherethe perturbations are well-behaved. This is a good indi-cation that we could expect to find a similar behavior in f ( T, T G ) gravity too, although we need to indeed verifythis under the detailed perturbation analysis. V. CONCLUSIONS
In this work we investigated the cosmological appli-cations of F ( T, T G ) gravity, which is a modified gravitybased on the torsion scalar T and the teleparallel equiv-alent of the Gauss-Bonnet combination T G . F ( T, T G )gravity is different from both the simple F ( T ) theory,as well as from the curvature modification F ( R, G ), and thus it is a novel class of gravitational modification. First,we extracted the general Friedmann equations and wedefined the effective dark energy sector consisting of tor-sional combinations. Then, choosing specific F ( T, T G )ansatzes we performed a detailed study of various ob-servables, such as the matter and dark energy densityparameters and the dark energy equation-of-state param-eters.Amongst the huge number of possible ansatzes, an in-teresting option is the construction of terms of the sameorder as T or T using T G , for instance new combina-tions of the form T + α p | T G | or T + βT G can partic-ipate in the F ( T, T G ) function. The resulting cosmol-ogy leads to interesting behaviors. Firstly, the scenariocan describe the inflationary regime, without an infla-ton field. Secondly, at late times it provides an effectivedark energy sector which can drive the acceleration ofthe universe, along with the correct evolution of the den-sity parameters, without the need of a cosmological con-stant. Furthermore, the dark energy equation-of-stateparameter can be quintessence or phantom-like, or ex-periences the phantom-divide crossing, depending on theparameters of the model. Another possible ansatz is the F ( T, T G ) = − T + f ( T + β T G ), the simplest applica-tion of which can also easily lead to inflationary behav-ior. These features make the proposed modified gravitya good candidate for the description of Nature. Acknowledgments
The research of ENS is implemented within the frame-work of the Operational Program “Education and Life-long Learning” (Actions Beneficiary: General Secretariatfor Research and Technology), and is co-financed by theEuropean Social Fund (ESF) and the Greek State.
Appendix A: Shortcut procedure for the extractionof the field equations of F ( T, T G ) cosmology In this Appendix we follow the shortcut procedure inorder to derive the field equations of F ( T, T G ) cosmology.In particular, instead of performing the variation of theaction (19) in terms of the general vierbein, obtaining thegeneral field equations (16), and then insert into themthe cosmological vierbein ansatz (21), we will first insertthe cosmological vierbein ansatz into the action and thenperform the variation in terms of the scale factor a ( t )and the lapse function N ( t ). Although in principle theshortcut procedure is not guaranteed that it will give thesame results as the first robust method [33, 34], especiallywhen the action involves higher-order derivatives, in thisspecific example it proves that we do obtain the sameresults indeed.Under the cosmological ansatz (21), namely e aµ = diag( N ( t ) , a ( t ) , a ( t ) , a ( t )) , (A1)the scalars T and T G become T = 6 ˙ a N a = 6 H (A2) T G = 24 ˙ a N a (cid:16) ¨ aN a − ˙ N ˙ aN a (cid:17) = 24 H (cid:16) ˙ HN + H (cid:17) . (A3)Therefore, insertion into the total action (19) gives S tot = 12 κ Z dt N a F (cid:16) H , H (cid:16) ˙ HN + H (cid:17)(cid:17) + S m . (A4)Let us now perform the variation of (A4) with respectto N and a . Since δ N H = − H δNN , we obtain δ N T = − H δNNδ N T G = − H h(cid:16) ˙ HN + H − H ˙ N N (cid:17) δNN + H N ( δN ) (cid:5) i . Similarly, since δ a H = ( δa ) (cid:5) Na − H δaa , we acquire δ a T = 12 HN a h ( δa ) (cid:5) − N Hδa i δ a T G = 24 HN a " HN ( δa ) (cid:5)(cid:5) + (cid:16) HN + 2 H − H ˙ NN (cid:17) ( δa ) (cid:5) − N H (cid:16) ˙ HN + H (cid:17) δa . Therefore, variation of the gravitational part of the action(A4) with respect to N and a gives δ N S = 12 κ Z dt a " F − H F T + 24 a (cid:16) a H N F T G (cid:17) (cid:5) − H (cid:16) ˙ HN + H − H ˙ N N (cid:17) F T G δN (A5) δ a S = 32 κ Z dt N a ( F − H F T − H (cid:16) ˙ HN + H (cid:17) F T G − a N " a HF T + 2 a H (cid:16) HN + 2 H − H ˙ NN (cid:17) F T G (cid:5) + 8 a N (cid:16) a H N F T G (cid:17) (cid:5)(cid:5) ) δa, (A6)where F T ≡ ∂F/∂T and F T G ≡ ∂F/∂T G . Additionally, variation of S m gives δS m = 12 Z d x e Θ µν δg µν , and its time-dependent part is δS m = − Z dt N a Θ tt δN + Z dt N a Θ ˆ i ˆ i δa , (A7)where hat indices run in the three spatial coordinates.In summary, taking into account the total action varia-tion, and setting as usual N = 1 in the end, the obtainedfield equations, that is the Friedmann equations, take theform F − H F T − H (cid:0) ˙ H + H (cid:1) F T G + 24 a (cid:0) a H F T G (cid:1) (cid:5) = 2 κ Θ tt (A8) F − H F T − H (cid:0) ˙ H + H (cid:1) F T G − a h a HF T + 4 a H (cid:0) ˙ H + H (cid:1) F T G i (cid:5) + 8 a (cid:0) a H F T G (cid:1) (cid:5)(cid:5) = − κ Θ ˆ i ˆ i . (A9)Additionally, if we consider the matter energy-momentum tensor to correspond to a perfect fluid of en-ergy density ρ and pressure p , we insert in the above fieldequations Θ tt = ρ , Θ ˆ i ˆ j = pa δ ˆ i ˆ j , Θ ˆ i ˆ i = 3 p .Lastly, we can re-organize the terms, performing theinvolved time derivatives, resulting in the end to F − H F T − T G F T G + 24 H ˙ F T G = 2 κ ρ (A10) F −
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