Inflationary cosmology in modified gravity theories

We review inflationary cosmology in modified gravity such as $R^2$ gravity with its extensions in order to generalize the Starobinsky inflation model. In particular, we explore inflation realized by three kinds of effects: modification of gravity, the quantum anomaly, and the $R^2$ term in loop quantum cosmology. It is explicitly demonstrated that in these inflationary models, the spectral index of scalar modes of the density perturbations and the tensor-to-scalar ratio can be consistent with the Planck results. Bounce cosmology in $F(R)$ gravity is also explained.


I. INTRODUCTION
Inflation in the early universe has recently been studied much more extensively because of the BICEP2 experiment [1] in terms of the primordial gravitational waves, in addition to the Wilkinson Microwave anisotropy probe (WMAP) [2][3][4][5][6] and the Planck satellite [7,8] on the unisotropy of the cosmic microwave background (CMB) radiation. For a standard inflationary scenario like chaotic inflation [9], the existence of the inflaton field is assumed, whose potential contributes to inflation.
Indeed, the WMAP and Planck data [2][3][4][5][6][7][8] support a kind of the trace-anomaly driven inflation with the R 2 term. Such a theory can be regarded as modified gravity because the R 2 term or its higher derivative term of the trace-anomaly term R, which leads to the long enough inflation and graceful exit from it [21], is the effective action of gravity, where is the covariant d'Alembertian for a scalar quantity 1 .
In this paper, we review the main results in Refs. [26][27][28]. The main purpose of this paper is to explain the recent developments on inflationary models to realize the Planck results in the so-called R 2 gravity (namely, the action consist of the Einstein-Hilbert term plus R 2 term) with further extensions, which can be regarded as a kind of F (R) gravity.
Particularly, we consider inflation (i) derived by modification terms of gravity [26], (ii) through the quantum anomaly [27], and (iii) in R 2 gravity in the framework of the so-called loop quantum cosmology (LQC) [29][30][31][32][33] to include quantum effects [28] (for reviews on LQC, see, for example, [34][35][36][37][38][39][40]). In addition, we state the recent progress of the bounce cosmology in F (R) gravity by presenting the important consequences in Refs. [41,42]. We use units The organization of the paper is the following. In Sec. II, we study inflation induced by 1 In Refs. [23][24][25], the features of inflation in non-local gravity including such a non-local term as R has been analyzed in detail. modification of gravity. In Sec. III, we explore the trace-anomaly driven inflation in modified gravity. In Sec. IV, we investigate R 2 gravity in the context of LQC. Furthermore, in Sec. V, we reconstruct F (R) gravity to realize the cosmological bounce in LQC. Finally, conclusions are described in Sec. VI.

II. INFLATION INDUCED BY MODIFICATION OF GRAVITY
In this section, we review inflation in modified gravity, particularly F (R) gravity, based on Ref. [26]. The deviation of F (R) gravity from general relativity may be interpreted as a kind of quantum corrections in the early universe, or such a modification of gravity could be motivated by the so-called ultraviolet (UV) completion of quantum gravity. In fact, the Starobinsky inflation [21] can be regarded as inflation induced by the modification term of R 2 from general relativity. We here attempt to examine inflation by the other forms of modification of gravity.

A. Conformal transformation
We first explain the conformal transformation from F (R) gravity in the Jordan frame to the corresponding scalar field theory in the Einstein frame [19,43]. The action of F (R) gravity is represented as where g is the determinant of the metric tensor g µν . We use a conformal transformationĝ µν = Ω −2 g µν with Ω 2 ≡ F R , where the hat denotes quantities in the Einstein frame, and the subscription of F R denotes the derivative with respect to R as F R (R) ≡ dF (R)/dR. Here, we introduce a scalar field ϕ ≡ − 3/2 (1/κ) ln F R . Through the conformal transformation, the action in the Einstein frame reads [44,45] with F R = exp − 2/3κϕ . This is the action for a canonical scalar field ϕ with its potential V (ϕ). For the Starobinsky inflation model [21] with
For the slow-roll regime, we impose the slow-roll approximations ofφ 2 /2 ≪ V (ϕ) on the Friedmann equation and |φ| ≪ |3Hφ| on the EoM for ϕ, so that we can find 3H 2 /κ 2 ≈ V (ϕ) ≈ constant and 3Hφ + dV (ϕ)/dϕ ≈ 0. Furthermore, we define the slow-roll During inflation, these parameters should be much smaller than unity. In addition, the number of e-folds is Here, a i (ϕ i ) and a f (ϕ f ) are the values of the scale factor a (the scalar field ϕ) at the initial time t i and end of time t f of inflation, respectively. Moreover, in deriving the second approximate equality, we have used the second gravitational field equation with the slow-roll approximation. The amplitude of the power spectrum for the curvature perturbations is expressed as ∆ 2 R = κ 2 H 2 / (8π 2 ǫ) ≈ κ 4 V / (24π 2 ǫ), where in the second approximate equality follows from the Friedmann equation operated the slow-roll approximation, and n s and r are written as n s −1 = −6ǫ+2η and r = 16ǫ [46,47]. The detailed explanations on the reconstruction of potential of inflationary models has been executed in Ref. [48].

C. Reconstruction of F (R) gravity
There are two possible ways to reconstruct F (R) gravity models to realize inflation. One is to start from the action in the Einstein frame. The other is to reconstruct the action in the Jordan frame. In this subsection, we consider the former way. We also explain the latter way in the next subsection.
The main purpose of our investigations is that we study the generalization of the Starobin-sky inflation model. To execute it, we take an appropriate form of V (ϕ) in the Jordan frame, which is an extended form from that in the Starobinsky inflation model. By taking the derivative of Eq. (II.2) with respect to R, we find Through this equation, i.e., Eq. (II.2), we reconstruct the form of F (R) in the Jordan frame from the potential V (ϕ) in the Einstein frame.

Extension of the Starobinsky inflation model
As the simplest model, we explore the following potential where c 1 ( = 0), c 2 , and c 3 are constants. In this case, from Eq. (II.3) we have 2c 1 F 2 R + c 2 F R − RF R = 0. By solving this equation with F R = 0, we eventually find that the corresponding form of F (R) can be expressed as the small curvature regime [49][50][51][52].
In the following, we set c 3 = 0 for simplicity and introduce a positive γ 1 (> 0) to express c 1 as c 1 = γ 1 / (4κ 2 ). Here, γ 1 has the mass dimension 2 and the dimensionless quantity γ 1 /M Pl ≪ 1, so that in the higher-curvature regime, the correction to the Einstein gravity can appear. We explore the inflationary dynamics in this extended model with the potential . We consider the case that the inflaton slowly rolls from the initial value with its large negative amplitude down to the minimum of the potential as V (ϕ = 0) = −γ 1 / (4κ 2 ) (< 0). In this case, from the gravitational field equations we find that the exponential inflation can be realized as a(t) = a i exp (H inf t) with the Hubble parameter H inf ≡ (1/2) γ 1 /3 during inflation and a i a constant. Furthermore, the solution |ϕ| ≫ 1 and the slow-roll parameters are ǫ = (4/3) These slow-roll parameters become of order of unity when ϕ approaches ϕ f ≈ −0.17 3/2/κ. For |ϕ i | ≫ |ϕ f |, the number of e-folds is given we have ϕ i ≈ 1.07M Pl . The slow-roll parameters are also represented as ǫ ≈ 3/ (4N 2 e ) and |η| ≈ 1/N. As a consequence, we obtain Here, we remark that ∆ 2 The observations obtained from the Planck satellite suggest n s = 0.9603±0.0073 (68% CL) and r < 0.11 (95% CL) [7]. In this model, for n s < 1 and r < 0.11, we see that n s > 1 − 0.11/3 = 0.809. Accordingly, for N e = 60, we acquire n s = 0.967 and r = 3.00 × 10 −3 .
Thus, in this model, the spectral index n s of the curvature perturbations and the tensorto-scalar ratio r consistent with the Planck result can be realized. Various descriptions of inflationary models in terms of scalar field models [53] and perfect fluid as well as F (R) gravity [54] have been examined. Moreover, the effects of quantum corrections on inflation have been explored in Refs. [55][56][57]. We note that the BICEP2 experiment has recently detected the B-mode polarization of the cosmic microwave background (CMB) radiation with the tensor to scalar ratio r = 0.20 +0.07 −0.05 (68% CL) [1]. There have been proposed several discussions on the method to obtain this result regarding the subtraction of the foreground data, e.g., Refs. [58][59][60][61][62]. A study to support the BICEP2 results has also been reported in Ref. [63]. Very recently, the collaboration between BICEP2/Keck and Planck has released the result of r < 0.12 for the wave number k = 0.05 Mpc −1 of tensor mode of the density perturbations [64].

Power-law corrections to general relativity
Next, we examine the following potential.
For this potential in the Einstein frame, a model in which a generic power-law correction term is added to the Einstein-Hilbert term is reconstructed as Here, β (> 0) is a dimensionful positive constant, R c and Λ p are constant, and q > 1 (q = 2). Inflationary models in such a power-law type gravity with q 2 has also been examined in Ref. [65]. Through the same procedures as those executed for the previous case in Eqs. (II.4) and (II.5), namely, by deriving the Hubble parameter at the inflationary stage, the scale factor, the solution of the inflaton ϕ, the slow-roll parameters, and the number of e-folds, if n is close to 2, we find Indeed, for q = 1.99, we find n s = 0.962 and r = 1.08 × 10 −3 . Therefore, this inflationary model can be compatible with the Planck analysis.

D. Reconstruction method of F (R) gravity in the Jordan frame
In this subsection, we reconstruct the form of F (R) in the Jordan frame. We note that cosmology in the Einstein frame may differ from that in the Jordan frame due to their physical non-equivalence. Hence, it is more convenient to consider these theories in the Einstein and Jordan frames as different cosmological theories. Here, we discuss inflation in F (R) gravity without its transformation to scalar-tensor theory. Eventually, the results may be different. Nevertheless, we demonstrate that F (R) inflation is also consistent with the observations by the Planck satellite. From the other point of view, for the fairness, it should also be remarked that there are the debates on the issue of the equivalence between the (Jordan and Einstein) conformal frames in Refs. [66][67][68]. Especially, the investigations in Ref. [68] seems to support equivalence of conformal frames for inflationary scenarios. It is an issue of presentation rather than substance, but in the interest of fairness other points of view should be mentioned.
The reconstruction method of F (R) gravity proposed in Ref. [69] is as follows (for another reconstruction method of F (R) gravity, see Refs. [70][71][72][73]). We consider the action of F (R) gravity with matter action S matter as We define the number of e-folds asN ≡ ln (a * /a) with a * the scale factor at the fiducial time t * . We By solving this equation inversely, we getN =N(R). In the flat FLRW space-time, the Friedmann equation can represented as the second order differential equation of F (R) with respect to R, given by )/dN 2 , and ρ matter is the energy density of matter.
As an example found in Ref. [54], we study an exponential formḠN = H 2 (N) =Ḡ 1 e τN + G 2 , where G 1 (< 0), G 2 (> 0), and τ (> 0) are constants. For this expression, we have . When the matter contribution is negligible, namely, ρ matter = 0, the solution of Eq. (II.10) is derived as (II.11) Here, ω ± and ϑ are defined as In this section, we review inflation by the quantum anomaly in the framework of F (R) gravity by following Ref. [27]. The effect of the trace anomaly on inflation in F (T ) gravity with T the torsion scalar in teleparallelism has also been studied in Ref. [74] (the explanations of teleparallelism exist, e.g., in Refs. [15,19]).

A. Quantum anomaly
It is known that the quantum anomaly appears via the procedure of the renormalization.
For four-dimensional space-time, the trace of the energy momentum tensor T (QA) µν originating from the quantum anomaly becomes [75][76][77][78][79] T (QA)µ Here, the brackets denotes the vacuum expectation value. Moreover, R µνρσ is the Riemann tensor, R µν is the Ricci tensor, R is the scalar curvature, C µνρσ is the Weyl tensor, to whose square W corresponds, G is the Gauss-Bonnet invariant, and = g µν ∇ µ ∇ ν with ∇ µ the covariant derivative associated with the metric tensor g µν is the covariant d'Alembertian.
In addition, the coefficients are defined as Accordingly, for the Yang-Mills theory in the curved spacetime, the R 2 term plays a role of correction of the higher curvature to the Einstein gravity or it contributes to the energy-momentum tensor as matter.

B. F (R) gravity with the quantum anomaly
The action describing F (R) gravity is given by In the FLRW background, the gravitational field equations are represented as Here, ρ eff and P eff are the effective energy density and pressure and they obey the equation of the conservation lowρ eff + 3H (ρ eff + P eff ) = 0. Their expressions are given by with the contributions from the quantum anomaly to the effective energy density and pressure whereρ is an integration constant, and in deriving these expressions, we have used the conservation equation for ρ eff and P eff . Theρ term corresponds to the energy density of radiation of the quantum state [80]. In what follows, we takeρ = 0 because at the inflationary stage around the Planck scale, the contribution of radiation can be neglected in comparison with that of the quantum anomaly as well as that of deviation of modified gravity from general relativity.

C. de Sitter solutions by the trace anomaly
We consider the case that f (R) is given by an exponential form [49,50,81] f where Λ c (> 0) and R c (> 0) are positive constants. For R/R c ≪ 1, i.e., in the late-time (e.g., present) universe, f (R) approaches zero, and therefore our model becomes equivalent to R 2 gravity with the quantum anomaly. While for R/R c ≫ 1, namely, in the early universe such as the inflationary stage, the term of Λ c plays a role of the cosmological constant. When we expand the exponential term as exp where we impose the condition Λ c < 3/ (8α 2 κ 2 ) with α 2 > 0 so that the solution can be real.
We examine the instability condition of the de Sitter solution. If the de Sitter solution describes inflation, it should be unstable because inflation has to end.
We represent the perturbation as H = H de Sitter + δH(t), where |δH(t)| ≪ 1. By combining it with the Friedmann equation, we obtain δḦ(t) where we have neglected the terms proportional to exp (−R/R c ) in Eq. (III.12) because the stability of the solution is only related to Λ c . The solution for δH(t) is written as δH(t) =H exp (λ ± t), whereH is a constant and λ ± ≡ −3H de Sitter ± √ D /2 (the subscriptions ± of λ ± correspond to the signs of ± in the r.h.s.). The de Sitter solutions are unstable (and adopted to describe the inflation) only if the value of λ + is a real and positive number, namely, where α 2 > 0 and α 4 > 0 have been used.

D. Trace-anomaly driven inflation
We investigate the observable quantities of the spectral index n s of the power spectrum for the the scalar mode of the density perturbations and the tensor-to-scalar ratio r in the trace-anomaly driven inflation in exponential gravity, namely, inflation is described by the de Sitter solutions in Eq. (III.13). In the slow-roll inflation, for the exponential form of f (R) in (III.12), we have As a result, we acquire For u = 3, Λ c α 2 κ 2 = 0.125, and N e = 76, we acquire n s = 0.960 and r = 1.20 × 10 −3 .
Consequently, the trace-anomaly driven inflation in exponential gravity can explain the Planck results.

IV. R 2 GRAVITY IN LOOP QUANTUM COSMOLOGY (LQC)
In this section, we review R 2 gravity and its cosmological dynamics in LQC with the holonomy corrections along the investigations in Ref. [28] 3 .

A. F (R) gravity in LQC
We explain F (R) gravity in the framework of LQC [87][88][89]. We consider the Einstein frame as in usual LQC only for the FLRW background with its spatially flatness [90]. In this case, the relation {β LQC ,V volume } = γ BI /2 is satisfied [91]. Here, { } denotes the Poisson bracket in terms of classical variablesβ LQC ≡ γ BIĤ with γ BI the Barbero-Immirzi parameter and the volumeV volume ≡â 3 , whereâ = √ F R a. Moreover,β LQC andV volume are canonically conjugated quantities with each other, and these variables are the unique combination for a loop quantization [92]. We note that the hat shows the quantities in the Einstein frame.
It is necessary to take the Hilbert space, in which the quantum states are described by (almost) periodic functions, so that the property of the discrete space can be included.
For this purpose, we use the Hamiltonian with the general holonomy corrections [35,93].

B. R 2 gravity in LQC
For R 2 gravity, whose action is given by S = d 4 x √ −g [F (R)/ (2κ 2 )] with F (R) = R + α S κ 2 R 2 , there appears curvature singularities in the early universe. In what follows, when we consider R 2 gravity, we analyze this action. On the other hand, for R 2 gravity in the context of LQC, it is possible that no singularity happens. We show this point below.

C. Loop quantum R 2 gravity in the Einstein frame
We further analyze the dynamics of R 2 gravity (i.e, F (R) = R + α S κ 2 R 2 ) in the Einstein frame, where equations become simpler than those in the Jordan frame. The equation of motion forφ is expressed by d 2φ /dt 2 + 3Ĥ dφ/dt + dV (φ)/dφ = 0 with We introduce a variableΨ ≡ exp 2/3κφ .
From the equation of motion forφ, we obtain In addition, the energy density is expressed aŝ This implies thatĤ = 0 at (Ψ(t), dΨ(t)/dt) = (1, 0). Forρ =ρ critical , we find critical . This depicts an ellipse for A > 0, a parabola for A = 0, and a hyperbola for A < 0. There exists only the critical point at (Ψ(t), dΨ(t)/dt) = (1, 0), whereρ = 0. All of the trajectories start from this point and come back to it. Therefore, it corresponds to both the beginning and end points of the universe.
As a consequence, thanks to the holonomy corrections, in the Einstein frame, the bounces can occur whenρ =ρ critical . The universe evolves from the contraction phase (Ĥ < 0). Its trajectory oscillates near the critical point (Ψ(t), dΨ(t)/dt) = (1, 0) and the oscillatory amplitude becomes large. Eventually, the trajectory approaches the lineρ =ρ critical and the bounce happens. After that, the expansion phase (Ĥ > 0) begins and the trajectory goes back to the critical point (Ψ(t), dΨ(t)/dt) = (1, 0) with its oscillating behavior.
On the other hand, the case of − sign is forΨ > 1. In this case, it shows an ellipse. If the trajectory intersects this curve in the Einstein frame, the bounce happens in the Jordan frame. At the bouncing point, the relationĤ = (1/2) dΨ/dt /Ψ has to be met.
In the Einstein frame, the universe first contracts and finally expands at the critical point (Ψ(t), dΨ(t)/dt) = (1, 0). With the relation between H andĤ and its time derivative it is seen that in the Jordan frame, the universe begins and ends at the point (H,Ḣ) = (0, 0).
It should be emphasized that the holonomy corrections yield the bounce in the Jordan frame, and hence, if they are absent, a singularity appears at the early stage of the universe.

V. BOUNCING COSMOLOGY IN F (R) GRAVITY
In this section, we review the cosmological bounce from F (R) gravity. Especially, we present the consequences found in Refs. [41,42]. The bouncing behaviors in various modified gravity theories have also been investigated in Refs. [97][98][99][100][101][102]. We show that it is possible to reconstruct an F (R) gravity theory in which the matter bounce can happen in the framework of LQC. where I 1 and I 2 are constants. We can set I 1 = 1, so that the Einstein-Hilbert term should be included.
In Ref. [42], the reconstruction of various modified gravity theories including F (R), F (G), and F (T ) gravity theories has been performed, where F (G) is an arbitrary function of the Gauss-Bonnet invariant G, to describe the two-times bouncing phenomena, called superbounce [103,104], and the ekpyrotic scenario [105] in the context of LQC.

VI. CONCLUSIONS
In the present paper, we have reviewed inflationary models in modified gravity theories such as F (R) gravity including R 2 gravity with extended terms so that we can generalize the Starobinsky inflation in R 2 gravity and derive its important properties to be useful clues to obtain the information on physics in the early universe.
First, we have studied inflationary cosmology by modification terms of gravity, especially, inflation in F (R) gravity, by following Ref. [26]. The Starobinsky inflation in R 2 gravity is considered to be the seminal and significant idea of inflationary models in modified gravity theories. We have made the coformal transformation from the Jordan frame (namely, F (R) gravity) to the Einstein frame (i.e., general gravity plus the scalar field theory), and given slow-roll dynamics of inflation in the Einstein frame. In addition, we have reconstructed F (R) gravity models, which are an extended version of the Starobinsky inflation model in R 2 gravity and general relativity with power-law correction terms.
Second, we have explored the trace-anomaly driven inflation in F (R) gravity along the discussions in Ref. [27]. We have first explained the quantum anomaly appearing through the process of the renormalization in four-dimensional space-time. We have further discussed F (R) gravity with the quantum anomaly and the de Sitter solutions for inflation due to the trace anomaly.
Third, based on Ref. [28], we have examined inflation in R 2 gravity and the cosmological evolutions for LQC with the holonomy corrections. We have analyzed R 2 gravity for LQC in both the Einstein and Jordan frames. We have found that in the Jordan frame, owing to the holonomy corrections, the bounce can happen, and accordingly the cosmic singularities can be removed, although such singularities appear in ordinary R 2 gravity.
In these three inflationary models, we have shown that the spectral index of scalar modes of the density perturbations and the tensor-to-scalar ratio can be compatible with the Planck analysis.
Furthermore, we have presented the recent developments of the bounce cosmology in F (R) gravity obtained in Refs. [41,42]. It has been performed that an F (R) gravity theory can be reconstructed, where the matter bounce occurs in the context of LQC, thanks to the reconstruction method of F (R) gravity. Recently, the reconstruction of F (R), F (G), and F (T ) gravity theories have also been executed, in which the super-bounce (i.e., two-times bounce behaviors) and the ekpyrotic scenario for LQC can be realized.
In this work, we have concentrated on the accelerating universe from F (R) gravity. Note, however, that it is possible to extend this study for more complicated versions of effective gravity, which comes from quantum gravity. Particularly, it has recently been demonstrated that successful inflation consistent with the Planck data may emerge from multiplicativelyrenormalizable higher derivative quantum gravity in Ref. [106].