Trace-Anomaly Driven Inflation in F(T)

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Physics Letters B 731 (2014) 257–264

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Physics Letters B

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Trace-anomaly driven inﬂation in f (T ) gravity and in minimal massive bigravity ∗ Kazuharu Bamba a, , Shin’ichi Nojiri a,b, Sergei D. Odintsov b,c,d,e,f

a Kobayashi–Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan b Department of Physics, Nagoya University, Nagoya 464-8602, Japan c Instituciò Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain d Institut de Ciencies de l’Espai (CSIC-IEEC), Campus UAB, Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain e Tomsk State Pedagogical University, Kievskaya Avenue, 60, 634061 Tomsk, Russia f National Research Tomsk State University, 634050, Tomsk, Russia

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Article history: We explore trace-anomaly driven inflation in modified gravity. It is explicitly shown that in T 2 teleparallel Received 29 January 2014 gravity, the de Sitter inflation can occur, although quasi de Sitter inflation happens in R2 gravity. Accepted 20 February 2014 Furthermore, we investigate the influence of the trace anomaly on inflation. It is found that in f (T ) Available online 26 February 2014 gravity, the de Sitter inflation can end because it becomes unstable due to the trace anomaly, whereas Editor: J. Hisano also in higher derivative gravity, the de Sitter inflation can be realized and it will be over owing to the trace anomaly for smaller parameter regions in comparison with those in teleparallelism. The instability of the de Sitter inflation in T 2 gravity and R2 gravity (both with taking account of the trace anomaly) is examined. In addition, we study trace-anomaly driven inflation in minimal massive bigravity, where the contribution from the massive graviton acts as negative cosmological constant. It is demonstrated that the de Sitter inflation can occur and continue for long enough duration. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction [7] where new dilaton-dependent terms appear in the conformal anomaly and brane-world gravity [8,9] where it was called Brane For a long time, it is expected that vacuum quantum effects New World. and/or quantum gravity effects should play an important role in Recent cosmological data [10–13] indicate that trace-anomaly the early-time cosmological evolution [1,2]. It was realized in [3] driven inflation or its closely related R2-version may be one of the that quantum effects due to conformally invariant fields (confor- most realistic candidates of inflationary theory. This has attracted mal or trace-anomaly induced effective action) which is expected a number of researchers to study different aspects of R2 inflation to be most relevant at the early universe may lead to the early- (for a very incomplete list of related papers, see [14] and refer- time de Sitter universe. This may be considered to be an exam- ences therein). ple of an eternal inflation model. Nevertheless, making deeper In this Letter, we investigate trace-anomaly driven inflation in analysis and adjusting the coefficient of higher (total) derivative modified gravity of teleparallel type [15] and in massive bigrav- counter-term in the trace-anomaly, the very successful model of ity. In particular, we study the analog of an R2 inflation model [4, trace-anomaly driven inflation (also called Starobinsky inflation 14] in extended teleparallelism, namely, so-called f (T ) gravity. The [4]) has been constructed (for its extended discussion, see [5]). main reason why f (T ) gravity has vigorously been explored in the The 2R term in the trace anomaly permits the long-lived inflation literature is that current cosmic acceleration [16–19] as well as in- with the possibility of the exit [4]. Note that in the anomaly- flation [20] can occur in f (T ) gravity. Hence, various accelerating induced action such a term is a higher-derivative R2 term. Note cosmology features of f (T ) gravity itself have also been explored, also that the trace-anomaly induced action is just an effective e.g., in Refs. [21–29]. We find that for T 2 gravity, there can exist gravitational action with R2-term and non-local term, so it may the de Sitter solution. This is different from the fact that in usual be considered as natural contribution to modified gravity. Fur- R2 gravity a quasi de Sitter inflation can happen. Furthermore, we thermore, trace-anomaly driven inflation has been extended for examine the influence of the trace anomaly on T 2 inflation and gravity with anti-symmetric torsion [6], dilaton-coupled gravity demonstrate that since the de Sitter solution is not stable, inflation can be over. In other words, by taking the trace anomaly into consideration, the so-called graceful exit problem from inflation * Corresponding author. can be solved as in seminal paper [4]. We also observe that in

http://dx.doi.org/10.1016/j.physletb.2014.02.041 0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 258 K. Bamba et al. / Physics Letters B 731 (2014) 257–264 convenient higher derivative gravity, the de Sitter inflation can be Lagrangian of matters. If f (T ) = T , this theory is pure teleparal- realized and finally it can end owing to the existence of the trace lelism. anomaly, although for much smaller parameter regions than those It follows from the action in Eq. (2.3) that we have [16] in teleparallelism. Moreover, we explore trace-anomaly driven in- 1 μν − λ ρ νμ + μν + 1 ν flation in other modified gravity, i.e., in minimal massive bigravity. ∂μ eSa f ea T μλ Sρ f Sa ∂μ(T ) f ea f It has recently been presented the non-linear ghost-free massive e 4 2 gravity (de Rham, Gabadadze, and Tolley (dRGT)) model [30,31]). = κ ρ ν ea Tmatter ρ , (2.5) The dRGT model has non-dynamical background, but models with 2 dynamical metric have been found in Ref. [32] (for a recent very ν with Tmatter ρ the energy–momentum tensor of matters and the detailed review of massive gravity, see [33]). It is shown that the prime denoting the derivative with respect to T .Also,therelation de Sitter inflation with long enough duration can occur, and that of R to T is given by [15] its instability can lead to the end of the de Sitter inflationary stage. = = = μ ρ We use units of kB c h¯ 1, where c is the speed of light, and R =− T + 2∇ T μρ , (2.6) 2 ≡ 2 denote the gravitational constant 8π GN by κ 8π/MPl with the μ −1/2 where ∇ is the covariant derivative in terms of the metric ten- Planck mass of M = G = 1.2 × 1019 GeV. Pl N sor gμν . It shows that General Relativity is equivalent to T -gravity, The Letter is organized as follows. In Section 2, we briefly de- while f (T ) gravity is not a covariant theory because only a piece scribe the formulation of teleparallelism. In Section 3, we explore of curvature is kept in T . (quasi de Sitter) inflation in R2 gravity in the Einstein frame. The 2 comparison with T gravity is made, where only eternal de Sitter 3. R2 inflationary model inflation is possible. Also, in Section 4, we investigate the trace- anomaly driven inflation in f (T ) gravity. We show that in f (T ) 3.1. R2 gravity gravity, taking account of the trace anomaly makes the exit possi- 2 ble like in usual R gravity. Furthermore, we study trace-anomaly The action of an R2 theory is given by1 driven inflation in minimal massive bigravity in Section 5, where it √ 1 1 is demonstrated that the massive term acts as negative cosmolog- S = d4x −g R + R2 , (3.1) ical constant. Finally, conclusions are given in Section 6. 2 2 2 κ 6MS where M is a mass scale. Substituting the relation in Eq. (2.6) into 2. Teleparallelism S the action in Eq. (3.1),wefind We present the essence of teleparallelism [15].Weintroduce √ 1 = 4 − − + ∇μ ρ orthonormal tetrad components, i.e., the so-called viervein fields S d x g T 2 T μρ 2κ2 μ = ea(x ), where a 0,...,3withtheromanindexa denoting μ 1 2 the tangent space of a manifold at a point x .Wedescribe + T + 2∇μT ρ a b 2 μρ the metric with the tetrad components as gμν = ηabe e , where 6M μ ν S the Greek indices μ, ν (= 0,...,3) show coordinates on the √ μ 4 1 1 2 manifold, and hence ea can be regarded as its tangent vec- = d x −g −T + T 2κ2 6M2 tor. Moreover, the inverse of the viervein fields can be derived S from the equation ea eν = δν . By using the Weitzenböck connec- μ a μ 2 μ ρ μ ρ 2 (W)ρ ≡ ρ a ρ ≡ + T ∇ T μρ + ∇ T μρ , (3.2) tion Γ νμ ea ∂μeν , we define the torsion tensor as T μν 3M2 (W)ρ − (W)ρ = ρ a − a S Γ νμ Γ μν ea (∂μeν ∂νeμ). In this case, curvature vanishes and only non-zero torsion exists. where in deriving Eq. (3.2),wehavedoneapartialintegration.By To construct the torsion scalar, we further make the superpo- comparing the action in Eq. (3.1) with that in Eq. (3.2),weseethe μν μν difference of the action of R2 gravity from that of T 2 gravity. tential Sρ and the contorsion tensor K ρ ,givenby In general, the action of F (R) gravity is represented as μν 1 μν μ αν ν αμ √ Sρ ≡ K ρ + δρ T α − δ T α , 4 F (R) 2 ρ S = d x −g . (3.3) 2κ2 μν 1 μν νμ μν K ρ ≡− T ρ − T ρ − Tρ . (2.1) = +[ 2 ] 2 2 For the action in Eq. (3.1),wehaveF (R) R 1/(6MS ) R .With an auxiliary field ψ, the action in Eq. (3.3) is rewritten to Eventually, we can build the torsion scalar as √ 4 1 dF(ψ) T ≡ S μν T ρ S = d x −g F (ψ) + (R − ψ) . (3.4) ρ μν 2κ2 dψ 1 ρμν ρμν ρ νμ = T Tρμν + 2T Tνμρ − 4Tρμ T ν . (2.2) By varying this action with respect to ψ,wefind(R − ψ)d2 F (ψ) 4 /dψ 2 = 0. Provided that d2 F (ψ)/dψ 2 = 0, we obtain R = ψ.Com- The action of extended teleparallelism, namely, f (T ) gravity, is bining this relation with the action in Eq. (3.4) leads to the origi- written as nal action of F (R) gravity with matters in Eq. (3.3).Moreover,by defining ϑ ≡ dF(ψ)/dψ, we express the action in Eq. (3.4) as 4 f (T ) S = d x |e| + Smatter, (2.3) 2κ2 √ 1 2 = 4 − 2 with the matter action For the action representing an R term as√S d x gR ,thetraceofthe energy–momentum tensor is defined by gμν (2/ −g )(δS/δgμν ),andthisgives2R. 4 Then, effectively the R2 term in Eq. (3.1) comes from 2R in the trace anomaly. Smatter = d x Lmatter(gμν,Φi). (2.4) Furthermore, the coefficient of the 2R term is not fixed in the trace anomaly and | |= the trace anomaly produces one more non-local term in the effective gravitational Here, we√ express the determinant of the metric gμν as e action. Nevertheless, thanks to the above relation between R2 and 2R, R2 inflation a = − L det(eμ) g, and matter(gμν,Φi ) with Φi matter fields is the is qualitatively very similar to trace-anomaly driven inflation. K. Bamba et al. / Physics Letters B 731 (2014) 257–264 259 √ ϑ R 1 1 S = d4x −g − Υ(ϑ) , (3.5) S = d4x |e| T + T 2 2κ2 2κ2 6M2 S 1 Υ(ϑ)≡ ϑ(ψ)ψ − F ϑ(ψ) . (3.6) 4 1 μ ρ 1 2 2 = d x |e| T + 2∇ T μρ + T , (3.13) 2κ 2 2 2 κ 6MS Through the conformal transformation g → gˆ ≡ Ω2 g with μν μν μν where in deriving the second equality, we have taken into account 2 = ,weacquire[34] Ω ϑ the fact that ∇μT ρ is a total derivative. μρ ˆ In comparison of the action in Eq. (3.2) for R2 gravity with that 4 R 1 μν S = d x −gˆ − gˆ ∂μφ∂νφ − V (φ) , (3.7) in Eq. (3.13) for T 2 gravity, we understand that the form of these 2κ2 2 actions are different from each other. Hence, it is expected that the ˆ ϑ R − F cosmological evolutions in these two theories would be different. V (φ) ≡ , (3.8) 2 2κ2ϑ 2 Indeed, for R gravity, the de Sitter inflation cannot occur, whereas √ for T 2 gravity, the de Sitter inflation can be realized, as is investi- where we have introduced a scalar field φ ≡ ( 3/2/κ) ln ϑ, and gated below. the hat means quantities in the Einstein frame. For R2 inflation It is known that there does not exist the conformal transfor- model with the action (3.3),wefind[34] mation from an f (T ) gravity theory to a scalar field theory in 2 pure teleparallelism, different from the case in F (R) gravity [25]. 3MS 2 V (φ) = 1 − exp(− 2/3κφ) . (3.9) Thus, in f (T ) gravity, through the conformal transformation, it is 4κ2 impossible to move to the Einstein frame, i.e., the corresponding We suppose the four-dimensional flat Friedmann–Lemaître– action consists of pure teleparallel term T /(2κ2) plus the kinetic Robertson–Walker (FLRW) space–time whose metric is given by and potential terms of a scalar field, and examine the properties of inflation in the Einstein frame, as explained in the preceding 2 = 2 − 2 i 2 ds dt a(t) dx . (3.10) subsection. i=1,2,3 To compare cosmological consequences for the action in a = Eq. (3.13) in teleparallelism with those for an R2 inflation model In this background, we choose the tetrad components as eμ in convenient gravity, we examine solutions for the action in (1,a(t), a(t), a(t)).Thus,wefindT =−6H2 with H = a˙/a, where Eq. (3.13) at the inflationary stage. In the flat FLRW background, the dot denotes the time derivative. the gravitational field equations can be written in the equivalent The values of quantities in terms of the curvature perturbations 2 forms of those in general relativity: in R inflation model are summarized as follows [11].Forφ φf = ≡ with φ(t tf) φf, where tf is the end of inflation, slow-roll pa- 3 2 1 ˙ 2 H = ρeff, − 2H + 3H = peff, (3.14) rameters and η and the number of e-folds Na from the end of κ2 κ2 inflation at t = t to the era when the curvature perturbation with f where ρ and p are the effective energy density and pressure the comoving wave number k = k crossed the horizon are de- eff eff WMAP of the universe, respectively, given by scribed by [11,35] 2 1 1 1 1 dV(φ) 3 ρeff ≡ I, peff ≡− (4J + I), (3.15) ≡ = , 2κ2 2κ2 2 V (φ) dφ 4N2 a with 1 d2 V (φ) 1 η ≡ =− , 2 2 2 I ≡−T − f + 2Tf = 6H − f − 12H f , (3.16) V (φ) dφ Na ˙ 2 ˙ tf J ≡ 1 − f − 2Tf H = 1 − f + 12H f H. (3.17) 3 Na ≡ Hdt= exp( 2/3κφ). (3.11) We should note that ρ and p in Eq. (3.15) satisfy the standard 4 eff eff t continuity equation identically Furthermore, for the super-horizon modes k aH, the scalar spec- ρ˙eff + 3H(ρeff + peff) = 0. (3.18) tral index of primordial curvature perturbations ns and the tensor- 2 2 to-scalar ratio r are expressed as [36,37] For the action in Eq. (3.13),wehave f (T ) = T +[1/(6M )]T .It S follows from Eq. (3.14) that we have 0 = f − 2Tf , whose solutions 2 d ln R(k) 2 are given by ns − 1 ≡ =−6 + 2η − , d lnk k=k Na WMAP = √MS = 12 Hinf 0, ( constant). (3.19) r = 16 = , (3.12) 3 2 Na Hence, we have a de Sitter solution at the inflationary stage, and 2 where R is the amplitude of scalar modes of the primordial eventually from the first equation in (3.14) and Eq. (3.19) we obtain − curvature perturbations at k = 0.002 Mpc 1.ThePLANCKresults = = √MS are ns = 0.9603 ± 0.0073 (68% errors) and r < 0.11 (95% up- a(t) ac exp(Hinft) ac exp t , (3.20) 3 per limit) [12]. For instance, if Na = 50, we find ns = 0.96 and −3 r = 4.8 × 10 ( 0.11). where ac is a constant. It is not known whether the de Sitter solution could be attractor or not, and therefore it is not clear if 3.2. Inflation in T 2 gravity the inflation could be eternal or not is not. However, by including quantum effects such as the trace anomaly, the de Sitter solution From the action in Eq. (3.1),weseethereasonwhyT 2 infla- is unstable against even small perturbations around this de Sitter tion does not work in f (T ) gravity. Our starting action with only solution, so that the de Sitter inflation can be over, as is seen in gravity part and without matter is given by Section 4.2. 260 K. Bamba et al. / Physics Letters B 731 (2014) 257–264

2 On the other hand, it is known that for F (R) gravity, at the TT =−ρT + 3pT =− (I + 3J ) de Sitter point, which is a solution for the case without matter, κ2 2 2 we have F (R)R − 2F (R) = 0. If F (R) = R +[1/(6M )]R ,wefind 2 S =− 6H2 + 3H˙ − f − 3Hf˙ R = 0. Moreover, we mention the case of an R2 inflation model κ2 in the ordinary curvature gravity. In this case, the pure de Sit- − 12H2 f + 36H2 Hf˙ (4.5) ter inflation does not occur unlike in teleparallelism demonstrated , above. Qualitatively, for |H¨ /(M2 H)| 1 and |−H˙ 2/(M2 H2)| 1, S S where ρ and p are the energy density and pressure coming from = − 2 − = [ − − 2 − T T we have H Hi (MS /6)(t ti) and a ai Hi(t ti) (MS /12)(t the torsion scalar, respectively. Then by including the contribution 2] ti) , where Hi and ai are the values of the Hubble parameter and from the trace anomaly, the equations in (3.14) are modified and = scale factor at the initial time t ti of inflation, respectively. As we obtain a result, what is observed here is that the qualitative behavior of 2 eternal inflation in T gravity for teleparallelism is different from 2 0 =− − f − 3Hf˙ − 12H2 f + 36H2 Hf˙ inflation in R2 gravity. κ2 2 2 d d 4. Trace anomaly − b˜ + b˜ + 3H 12H2 + 6H˙ 3 dt2 dt ˜ 4 2 In this section, we examine the cosmological influences of the + 24b H + H H˙ . (4.6) trace anomaly on the energy density of the universe in modified gravity theories. Especially if we consider the de Sitter space, where H is a constant, H = Hc,Eq.(4.6) reduces to an algebraic equation: 4.1. Effects of the trace anomaly 2 2 ˜ 4 0 =− − f − 12H f + 24b H . (4.7) First of all, we should note that quantum effects of conformally- κ2 c c invariant fields on classical gravitational background are just the same as in T -gravity (i.e., pure teleparallelism) due to its equiva- When we do not include the contribution from the trace anomaly, ˜ lence to general relativity (GR). Furthermore, conformal invariance that is, b = 0in(4.7), in teleparallelism is proved to be the same as in usual gravity. This requires non-minimal coupling with curvature [26]. =− + 2 fEH 2T 12Hc , (4.8) We now include the massless quantum effects by taking into =− 2 account the trace anomaly TA, which has the following well- satisfies Eq. (4.7) (note that T 6Hc ), which is equivalent to the known form [38]: Einstein theory with a cosmological term. Particularly when ˜ 2 ˜ ˜ = + n TA = b F + 2R + b G + b 2R, (4.1) f (T ) T βT , (4.9) 3 where β and n are constants, from Eq. (4.7) we find that the fol- ˜ ˜ ˜ 2 F with b, b , and b constants , where is the square of the 4D lowing relations are satisfied: Weyl tensor, and G is the Gauss–Bonnet invariant, which are given by − 2 = + − − 2 n 1 + 2 ˜ 2 = Hc 0, and/or 1 (2n 1)β 6Hc 2Hinfb κ 0. 1 2 μν μνρσ F = R − 2Rμν R + Rμνρσ R , (4.10) 3 2 μν μνρσ G = R − 4Rμν R + Rμνρσ R . (4.2) For instance, when n = 2, if the following relation is met:

When we consider N scalars, N1/2 spinors, N1 vector fields, N2 9β − ˜ (= 0 or 1) gravitons and NHD higher-derivative conformal scalars, b > 0, (4.11) κ2 b˜ and b˜ are given by + + + − there appears a solution corresponding to the de Sitter space, and ˜ N 6N1/2 12N1 611N2 8NHD b = , the de Sitter (i.e., exponential) inflation can be realized. The condi- 2 ˜ 120(4π) tion (4.11) is always satisfied as long as β>0, because b < 0for + + + − ˜ N 11N1/2 62N1 1411N2 28NHD the ordinary matter. b =− . (4.3) 360(4π)2 On the other hand, in curvature gravity, the trace TC of the energy–momentum tensor in the scalar curvature becomes ˜ ˜ 2 ˙ 2 ¨ We should note that b > 0 and b < 0 for the usual matter, ex- TC = (1/κ )[−2( f (R)R − f ) + 9 f (R)H + 12 f (R)H + 3 f (R) + cept the higher-derivative conformal scalars. Notice that b˜ can be 6 f (R)H˙ − R] with R = 6(2H2 + H˙ ), where the prime shows the 2 shifted by the finite renormalization of the local counterterm R , derivative with respect to R, and the trace anomaly is expressed... as ˜ ˜ ˙ 2 ˜ ˙ 2 2 ˙ 4 ˜ ˜ ¨ so b canbeanarbitrarycoefficient.IntheFLRWuniverse,wefind TA =−12bH + 24b (−H + H H + H ) − 2(2b + 3b )(H + 7H H + ˙ 2 2 ˙ ˜ n˜ ˜ 4H + 12H H) [39].WetakeF (R) = R + β R with β and n˜ con- R = 12H2 + 6H˙ , F = 0, G = 24 H4 + H2 H˙ . (4.4) stants for the action in Eq. (3.3). In this case, the de Sitter solution ˜ ˜ H = Hc, where Hc is a constant, can be obtained if the following Eqs. (3.15), (3.16), and (3.17) tell that the trace TT of the = −˜ + ˜ ˜ 2 n˜ + ˜ 2 + 2 ˜ ˜ 4 relation is satisfied: 0 ( n 2)β(12Hc ) 12Hc 24κ b Hc . energy–momentum tensor in the torsion scalar is given by ˜ = ˜ 2 = ˜ = For n 2, this relation can be met only if Hc 0 and/or b − 2 ˜ 2 1/(2κ Hc ). In comparison with the case of teleparallelism, for 2 Note that the prime in b˜ and b˜ is not the derivative with respect to T but just curvature gravity, there exist only smaller regions of the model a superscription as a notation. parameters to realize the de Sitter inflation. K. Bamba et al. / Physics Letters B 731 (2014) 257–264 261

4.2. Instability of the de Sitter solution becomes positive. Consequently, when χ > 0, δ = exp(χt) in- creases in time, and thus it follows from Eq. (4.13) that H de- By following the procedure proposed in Ref. [5], we study the creases, that is, the de Sitter solution becomes unstable. instability of the de Sitter solution leading to the end of inflation. For F (R) = R + β˜R2 (i.e., n˜ = 2), we examine the small pertur- = +[ 2 ] 2 For f (T ) T 1/(6MS ) T , the gravitational field equation (4.6) bation around the de Sitter solution with Eq. (4.13). Substituting including the contribution of the trace anomaly reads this into the gravitational field equations and only remaining the linear term in δ,weobtain 6 1 1 0 = 2H2 − 36 H4 + H˙ − 36 H2 H˙ 2 2 2 ˜ ˜ 2 ... κ (6M ) (6M ) 2 −18β˜ + 2b + 3b κ H δ S S inf 2 ... ˜ ˜ ˜ 2 2 ¨ − ˜ + ˜ 2 + ¨ + 2 ˙ + ˙ 2 + 2 −72β + 7 2b + 3b κ H δ b b κ H 7H H 12H H 4H inf 3 2 ˜ ˜ ˜ 2 2 − 6 1 + 72β˜H − 4 2b − b + 3b H H δ˙ inf κ inf inf ˜ 2 4 2 ˙ + 4b κ H + H H . (4.12) − + ˜ 2 2 2 + + ˜ 2 2 2 = 24 1 4b κ Hinf Hinfδ 12 1 2b κ Hinf Hinf 0.

We explore the small perturbation from the de Sitter solution (4.19) as We consider the form δ = exp(τt) with τ a constant. If τ > 0, thedeSittersolutionbecomesunstable,becausetheamplitude H = H 1 − δ(t) ,δ 1, (4.13) inf of δ increasing in time without the upper bound. It follows with Hinf the constant Hubble parameter at the inflationary stage from the investigations in the last part of Section 4.1 that for t = 0). Here, ˜ = 2 = ˜ = for the case that the small perturbation vanishes (δ( ) n 2 and Hinf 0, if the de Sitter solution exists, we have b we only study the case that δ>0, i.e., H < Hinf because for δ<0, − 2 2 1/(2κ Hinf). With this relation, we see that the first term in the H becomes larger in time and eventually diverges. By substitut- 2 third line of Eq. (4.19) is represented as 24Hinfδ, and the last term ing this expression into Eq. (4.12) and taking only the leading (i.e., in Eq. (4.19) is zero. By plugging δ = exp(τt) with Eq. (4.19),we linear) order terms of δ,weacquire find 2 ... 2 ˜ + ˜ 2 + ˜ + ˜ 2 2 ¨ ˜ ˜ ˜ 2 3 b b κ Hinf δ 7 b b κ H δ 2 −18β + 2b + 3b κ Hinfτ 3 3 inf + − ˜ + ˜ + ˜ 2 2 2 1 2 2 72β 7 2b 3b κ Hinfτ − − 2 − ˜ + ˜ 2 2 1 36 Hinf 12 b b κ Hinf 2 ˜ ˜ 2 2 6M2 3 − 6 72β˜H − 4 2b + 3b κ H − 1 H τ S inf inf inf + 2 = + ˜ 2 2 ˙ 24Hinf 0. (4.20) 4b κ Hinf Hinfδ With the procedure used in teleparallelism shown above, we con- 3 2 1 2 ˜ 2 2 sider the case that the change of H is slow enough for the τ term + 4H 1 − 36 H + 4b κ H = 0. (4.14) inf 2 inf inf to be negligible in compared with the τ 2 term. As a result, we ac- 6MS quire the approximate solution We take the form of δ as δ = exp(χt), where χ is a constant. If χ > 0, since the amplitude of δ grows as the cosmic time passes, 3Z ± D˜ the de Sitter solution becomes unstable, so that the universe can τ = , (4.21) W exit from the inflationary stage. By plugging the expression δ = 2 exp(χt) with Eq. (4.14), we have the cubic equation in terms of χ with 3 2 ˜ ˜ ˜ 2 χ + 7Hinfχ W ≡−72β + 7 2b + 3b κ , (4.22) − [ 2 ] 2 − [ ˜ + ˜ ] 2 2 + ˜ 2 2 ˜ ˜ 2 2 1 36 1/(6MS ) Hinf 12 (2/3)b b κ Hinf 4b κ Hinf Z ≡ −W + 3 2b + 3b H − 1 (4.23) − χ κ inf , [ ˜ + ˜] 2 (2/3)b b κ D˜ ≡ Z2 − W 2 9 48 Hinf. (4.24) 2 2 ˜ 2 2 4Hinf{1 − 36[1/(6M )]H + 4b κ H } + S inf inf = 0. (4.15) If we choose the ‘−’signinEq.(4.21),forW < 0 and Z > 0, we ˜ ˜ [(2/3)b + b ]κ2 find τ > 0. When τ > 0, the amplitude of the perturbation diverges in time, and thus the de Sitter solution becomes unstable. This Provided... that at the initial stage of inflation, H varies slowly so ¨ 3 that H H H. In this case, by neglecting the χ term in Eq. (4.15) means the de Sitter inflation can end. Thus, we demonstrated that 2 2 and solving it approximately, we find trace-anomaly driven inflation in T or R gravity has different behavior towards its exit. 1 7Q − 12H2 √ χ = inf ± D , (4.16) 2 7Hinf 5. Trace-anomaly driven inflation in minimal massive bigravity where To check the universality of trace-anomaly driven inflation, we ˜ 1 − 36[1/(6M2)]H2 + 4b κ2 H2 consider it in another version of modified gravity, so-called mas- Q ≡ S inf inf , (4.17) ˜ ˜ sive ghost-free bigravity, which has been very popular recently 7[(2/3)b + b ]κ2 [33]. 7Q − 12H2 2 For the bimetric gravity, some cosmological solutions describing D ≡ inf − Q 16 . (4.18) the accelerating universe have been investigated in several papers 7Hinf [40–46]. Particularly, together with Ref. [42], the ghost-free, bigrav- ˜ ˜ ˜ Since b > 0, b < 0, and b is an arbitrary constant, Q can be nega- ity theory with cosmological solutions describing the accelerating tive. If Q < 0, we find D > 0, and χ with the ‘+’signinEq.(4.16) universe has been examined in Ref. [43]. In the work, there has 262 K. Bamba et al. / Physics Letters B 731 (2014) 257–264 3 3 been presented an extensive and more detailed cosmological anal- 2 = μ ν = 2 − 2 + i 2 ysis of the model at the background level showing the existence of dsg gμν dx dx a(t) dt dx , a late-time accelerated expansion without resorting to a cosmolog- μ,ν=0 i=1 ical constant. In this section, we include the trace anomaly to the 3 3 bigravity models and show the existence of the solution describ- 2 = μ ν =− 2 2 + 2 i 2 ds f fμν dx dx c(t) dt b(t) dx . (5.7) ing de Sitter space–time, which may correspond to inflation in the μ,ν=0 i=1 early universe, so-called anomaly-driven inflation. We now investigate the minimal ghost-free bigravity model We assume a, b, and c are positive. In this case, from the (t,t) whose action is given by [30,32] component of (5.3) we find =− 2 2 − 2 2 2 − + 2 = 2 4 − (g) + 2 4 − ( f ) 0 3M g H 3m Meff a ab a ρmatter, (5.8) Sbi M g d x det gR M f d x det fR and (i, j) components yield + 2m2 M2 d4x − det g 3 − tr g−1 f + det g−1 f eff = 2 ˙ + 2 + 2 2 2 − − + 2 0 M g 2H H m Meff 3a 2ab ac a pmatter. (5.9) + 4 L d x matter(gμν,Φi). (5.1) Here ρmatter and pmatter are the energy density and pressure of matter fields, respectively. On the other hand, the (t,t) component Here Meff is defined by of (5.4) leads to 1 1 1 3 = + 2 2 2 2 2 a . (5.2) 0 =−3M K + m M c 1 − , (5.10) 2 2 2 f eff 3 Meff M g M f b (g) ( f ) and from (i, j) components we find In (5.1), R and R are the scalar curvatures for gμν and fμν , − respectively. The tensor g 1f isdefinedbythesquarerootof 3 2 ˙ 2 2 2 a c 2 gμρ f , namely, ( g−1 f )μ ( g−1 f )ρ = gμρ f . 0 = M 2K + 3K − 2LK + m M − c , (5.11) ρν ρ ν ρν f eff b2 Then by the variation over gμν ,weget ˙ with K = b/b and L = c˙/c. Both Eqs. (5.5) and (5.6) yield the iden- 1 2 (g) (g) 2 2 −1 0 = M gμν R − Rμν + m M gμν 3 − tr g f tical equation: g 2 eff ˙ ca ˙ 1 − −1ρ 1 − −1ρ cH = bK or = b. (5.12) + fμρ g 1 f + fνρ g 1 f a 2 ν 2 μ ˙ If a˙ = 0, we obtain c = ab/a˙. On the other hand, if a˙ = 0, we find + Tmatter μν. (5.3) b˙ = 0, that is, a and b are constant and c can be arbitrary. On the other hand, through the variation over fμν ,weobtain WesupposethedeSitterspace–timeforthemetricgμν or a and provided that b and c have the following forms: 2 1 ( f ) ( f ) 0 = M fμν R − Rμν f 2 = 1 = 1 = 1 a , b , c , (5.13) 1 H0t K0t L0t + 2 2 −1 − −1 ρ m Meff det f g fμρ g f 2 ν with constants H0, K0, L0.Then(5.12) gives

1 − ρ − = − fνρ g 1 f + det g 1 f fμν . (5.4) K0 L0, (5.14) 2 μ and therefore Eqs. (5.8) and (5.9) have the following forms: ∇μ By multiplying the covariant derivative g with respect to the = L metric g by Eq. (5.3) and using the Bianchi identity 0 =− 2 2 − 2 2 − 0 + μ (g) μ 0 3M H 3m M 1 ρmatter, (5.15) ∇ ( 1 g R(g) − R ) and the conservation law ∇ T = 0, g 0 eff H g 2 μν μν g matter μν 0 we have L = 2 2 + 2 2 − 0 + 0 3M g H0 3m Meff 1 pmatter. (5.16) =− ∇μ −1 H0 0 gμν g tr g f 1 − − Eqs. (5.10) and (5.11) give an identical equation: + ∇μ −1 1ρ + −1 1ρ g fμρ g f fνρ g f . (5.5) 2 ν μ 3 2 2 2 2 L0 μ 0 =−3M L + m M 1 − . (5.17) Similarly by using the covariant derivative ∇ with respect to the f 0 eff 3 f H0 metric f ,from(5.4),weget = = Without matter, that is, when ρmatter pmatter 0, Eq. (5.15) or μ − 1 − −1ν σμ Eq. (5.16) conflicts with (5.17).Infact,ifH0 does not vanish, 0 =∇ det f 1 g − g 1 f g = = f 2 σ Eq. (5.15) or Eq. (5.16) with ρmatter pmatter 0 tells L0 > H0, which conflicts with (5.17). 1 −1μ − g−1 f gσν + det g−1 f f μν . (5.6) We include the contribution from the trace anomaly in 2 σ Eq. (4.1).3 In the FLRW universe, we find The identities (5.5) and (5.6) impose constraints on the solutions. We take the FLRW universes for the metric g and use the μν 3 Note that as the trace anomaly gives non-local and R2-term contributions to conformal time t. Furthermore, we suppose the form of the metric the effective gravitational action, the ghost-free feature of massive bigravity may be fμν as follows lost. K. Bamba et al. / Physics Letters B 731 (2014) 257–264 263 2 ˙ 6 24H H 0 = 6M2 H2 + H˙ + 12m2 M2 12a2 − 9ab − 3ac R = H2 + H˙ , F = 0, G = . (5.18) g eff 2 4 a a ˜ 2 ˙ ˜ 24b H H 2 b d 2 d 6 2 ˙ Then Eqs. (5.15) and Eq. (5.16) give + − a H + H . (5.25) a2 3 a2 dt dt a2 L = 2 2 + 2 2 − 0 + ˜ 4 We now investigate if the de Sitter solution can be stable or not. 0 12M g H0 12m Meff 1 24b H0, (5.19) H0 Eq. (5.25) might be compared with Newton’s equation of motion, 2 0 = F − m d x in the classical mechanics. As a consequence, we which can be solved with respect to L0: dt2 ˜ can find the stability depends on the sign of b and the strength M2 H3 ˜ 5 ˜ g 0 2b H0 of the stability on the magnitude of b [4,9].EvenifthedeSit- L0 = H0 + + . (5.20) 2 2 2 2 ter solution is unstable, if we choose b˜ large enough [4,5], which m Meff m Meff may correspond to the large mass in the classical mechanics, the Then substituting of L0 in the above expression into (5.17),weob- de Sitter solution might be long-lived. Then, it represents realistic tain trace-anomaly driven inflation in massive bigravity. = 2 0 F H0 6. Conclusions M2 H2 ˜ 4 2 2 2 g 0 2b H0 ≡−3M H 1 + + In the present Letter, we have studied trace-anomaly driven in- f 0 2 2 2 2 m M m M 2 eff eff flation in T teleparallel gravity. In particular, it has clearly been M2 H2 ˜ 4 3 demonstrated that the de Sitter inflation can be realized in T 2 2 2 g 0 2b H0 + m M 1 − 1 + + . (5.21) gravity, while realistic quasi de Sitter inflation occurs in R2 gravity. eff m2 M2 m2 M2 eff eff In addition, we have examined the effects of the trace anomaly 2 If Eq. (5.21) has a real and positive solution with respect to H2 and on inflation in T gravity. As a result, we have shown that the 0 de Sitter inflation can occur and finally the universe can exit from L0 in (5.20) is positive for the obtained H0, the de Sitter space– 2 → 2 it due to the de Sitter space instability solution coming from the time can be generated by the trace anomaly. When H0 0, F (H0) 2 ∼− 2 + 2 2 trace anomaly. On the other hand, in convenient gravity, if the behaves as F (H0) 3(M f M g )H0 < 0. On the other hand, 2 ˜ trace anomaly is taken into account, in R gravity the realistic de 8(b )3 H12 2 →+∞ 2 ∼− 0 Sitter inflation can happen. However, compared with the case in when H0 ,wefindF (H0) 4 4 > 0. We should note m Meff ˜ teleparallelism, the parameter regions leading to the de Sitter so- b < 0 in general. Hence we find Eq. (5.21) has surely a real and lution are more narrow. In this case, there are two sources for the 2 2 positive solution with respect to H0. We also see that the value of exit from inflation: from the classical R -term and the similar term 2 2 H0 gives a positive value of L0 for (5.20). The function F (H0) can coming from the trace anomaly. ˜ be rewritten as a function F (L0/H0) of L0/H0 by using (5.20), Moreover, trace-anomaly driven inflation has been explored in minimal massive ghost-free bigravity. It has been observed that the ˜ = 2 F (L0/H0) F H0 de Sitter inflation can occur and continue for long enough time, and that eventually its instability induced by the trace anomaly can L 2 L 3 =− 2 2 0 + 2 2 − 0 lead to the end of it. Thus, in this massive bigravity, trace-anomaly 3M f H0 m Meff 1 . (5.22) H0 H0 driven inflation with long enough duration can be realized. Fi- nally, we remark that for massive bigravity, the contribution of Thus we find F˜ (0) = m2 M2 > 0 and when L /H →+∞, eff 0 0 the graviton mass plays a role of negative cosmological constant. ˜ 2 2 L0 3 ˜ F (L0/H0) ∼−m M ( ) < 0. Therefore F (L0/H0) vanishes for eff H0 It is interesting that one can use the universality of trace-anomaly 2 2 finite and positive L0/H0, which corresponds to H0 satisfying driven inflation or its R -cousin for the unified description of infla- 2 = 2 2 = tion and late-time acceleration as it has been proposed in Ref. [47]. F (H0) 0. Accordingly for H0 which obeys F (H0) 0, L0 is surely positive, and therefore the de Sitter space–time can be generated Such a scenario may be realized in F (R) bigravity [45]. by the trace anomaly. We should note that the space–time described by fμν is also the de Sitter space whose length parameter Acknowledgements is given by 1/L0.In(5.22), the first term is positive and hence the second term must be positive. This means L0 < H0, that is, the S.D.O. appreciates the Japan Society for the Promotion of Sci- expansion in the space–time described by fμν is slow, compared ence (JSPS) Short Term Visitor Program S-13131 and is grateful to with the expansion in the space–time described by gμν . the very warm hospitality at Nagoya University, where the work In case of the standard Einstein gravity, instead of (5.19),we was initiated. The work is supported in part by the JSPS Grant-in- obtain Aid for Young Scientists (B) # 25800136 (K.B.); that for Scientific Research (S) # 22224003 and (C) # 23540296 (S.N.); and MINECO = 2 2 + ˜ 4 0 12M g H0 24b H0, (5.23) (Spain), FIS2010-15640 and AGAUR (Generalitat de Catalunya), con- tract 2009SGR-345, and Ministry of Education and Science project whose solution is given by (Russia) (S.D.O.). M2 2 =− g References H0 . (5.24) 2b˜ [1] I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective Action in Quantum Gravity, In (5.19), the contribution from the mass term 12m2 M2 (1 − L0 ) eff H0 IOP, Bristol, UK, 1992, 413 p. acts as a negative cosmological constant but the contribution [2] V. Mukhanov, S. Winitzki, Introduction to Quantum Effects in Gravity, Cam- makes H2 or curvature larger than that in the Einstein gravity in bridge Univ. Press, Cambridge, UK, 2007, 273 p. 0 [3] J.S. Dowker, R. Critchley, Phys. Rev. D 13 (1976) 3224; (5.24). M.V. Fischetti, J.B. Hartle, B.L. Hu, Phys. 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