Beyond-one-loop quantum gravity action yielding both inflation and late-time acceleration

# Beyond-one-loop quantum gravity action yielding both inflation and late-time acceleration

E. Elizalde, S. D. Odintsov, L. Sebastiani, and R. Myrzakulov
Consejo Superior de Investigaciones Científicas, ICE/CSIC-IEEC, Campus UAB, Carrer de Can Magrans s/n, 08193 Bellaterra (Barcelona) Spain
International Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, and Tomsk State Pedagogical University, Tomsk, Russia
Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
Department of General & Theoretical Physics and Eurasian Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan
Dipartimento di Fisica, Università di Trento, Italy
###### Abstract

A unified description of early-time inflation with the current cosmic acceleration is achieved by means of a new theory that uses a quadratic model of gravity, with the inclusion of an exponential -gravity contribution for dark energy. High-curvature corrections of the theory come from higher-derivative quantum gravity and yield an effective action that goes beyond the one-loop approximation. It is shown that, in this theory, viable inflation emerges in a natural way, leading to a spectral index and tensor-to-scalar ratio that are in perfect agreement with the most reliable Planck results. At low energy, late-time accelerated expansion takes place. As exponential gravity, for dark energy, must be stabilized during the matter and radiation eras, we introduce a curing term in order to avoid nonphysical singularities in the effective equation of state parameter. The results of our analysis are confirmed by accurate numerical simulations, which show that our model does fit the most recent cosmological data for dark energy very precisely.

###### pacs:
04.50.Kd, 04.60.Bc, 95.36.+x, 98.80.Cq

## I Introduction

It is well accepted nowadays that the Universe underwent a period of strong and extremely quick accelerated expansion, namely the inflation stage, immediately after its origin (usually termed as the Big Bang singularity). From the very first proposal of the inflationary paradigm in 1981, by Guth Guth () and Sato Sato (), several attempts to describe this early-time acceleration have been carried out (see Refs. infrev (), for some reviews).

Moreover, cosmological data Planck () clearly show that the Universe is experiencing now a new phase of accelerated expansion, which can be explained either in terms of the existence of a dark energy fluid Li:2011sd (); Kunz:2012aw (); Bamba:2012cp () or by modifying Einstein’s gravity. In this respect, one of the most popular classes of modified gravity theories is -gravity. Here, the gravitational action is given by a function of the Ricci scalar only (for a review, see Refs. reviewFr ()). Many authors have investigated -gravity as an alternative for dark energy and its properties, showing that theories of this class are able to fulfill the constrains imposed by local and cosmological tests reviewFr (); oldFR (). In particular, exponential models of modified gravity as sound alternatives to dark energy have become quite popular in the last decade, since they represent a simple and natural way to mimic the cosmological constant term of the standard CDM model at large curvature nostriexp (); altriexp (); SuperBamba (); oscillation1 (); oscillation2 (); oscillation3 ().

As first suggested in Ref. inlation+DE (), it may be interesting and natural—e.g., as the first step towards the construction of a more fundamental theory—to try to unify the early-time and late-time cosmological accelerations in one single model. In this respect, it is worth noting that, at high curvature, when early-time inflation occurs, quantum gravity effects have to be incorporated to the theory. Starting from this crucial observation, we would like to present here high-curvature corrections to General Relativity (GR) under the form of a higher-derivative quantum gravity model B (). We thus enter the domain of quantum field theory, where basic renormalization group (RG) considerations lead to a RG improvement of the effective action. Indeed, this technique has been successfully developed in quantum field theory in curved space-time El () and permits to construct an effective action which goes beyond the one-loop approximation, because it renders it possible to sum over all leading log-terms of the theory. Since we are here interested in the Friedmann-Robertson-Walker (FRW) space-time solutions, we will be finally led to work with a higher-derivative multiplicatively renormalizable quantum gravity theory Stelle (); Fradkin () through the use of RG-improved techniques. A model of this kind was first discussed in Ref. qg1 () and subsequently extended to the case of with quantum electrodynamics in Ref. qg2 (). Here, we further extend the formulation in order to go beyond these results and obtain a unified description of a viable inflationary scenario with the Friedmann and dark energy Universe stages. To reproduce the dark energy sector we make use of an exponential model of -modified gravity, including an additional term to stabilize the theory during the radiation and the matter eras. Our approach is phenomenological and we add the dark energy -function by hands. Of course, in that case we follow several conditions. First of all, we choose the exponential form of so that it would be qualitatively similar to RG improved effective action under discussion. Second, its choice is done in such a way that inflationary universe scenario which follows from our RG improved effective action is not modified by dark energy -term which gives non-essential impact to inflation. Third, our purpose is to formulate the unified description of the early-time inflation with dark energy. For that reason, our exponential -term is chosen so that to cancel some singularities (past-time oscillations) of the whole theory. Of course, the addition of such term to quadratic action makes the theory to be non-renormalizable. In this sense, our complete action is classical modified gravity theory where some terms relevant for early-time inflation are inspired by quantum gravity considerations while the exponential -term is chosen so that it is negligible at the inflationary epoch.

The paper is organized as follows. In Section II we discuss our model of a RG-improved effective action for higher-derivative renormalizable quantum gravity. The equations for the running coupling constants in front of the gravitational invariants are obtained, and explicit forms for these running coupling constants are derived. Section III is devoted to the application of high-derivative quantum gravity to inflation. The field equations for FRW space-time are presented and the quasi-de Sitter (dS) solution describing the early-time acceleration is found. Furthermore, we show that this solution is unstable and that the model has a graceful exit from inflation, leading moreover to an amount of inflation (number of -folds) that is large enough in order to get the necessary thermalization of the observable Universe. In Section IV we derive the spectral index and the tensor-to-scalar ratio for cosmological perturbations. We show that these parameters are in agreement with the most recent analysis of the Planck satellite data. At the end of inflation, the model can be recast under the form of an -correction to General Relativity, with a reheating mechanism able to convert the energy of inflation into the one corresponding to standard matter and radiation. In Section V we introduce exponentially modified gravity for dark energy. We recover the de Sitter solution for the current cosmic acceleration, and show that this solution is a final attractor of the system. Modified gravity for dark energy needs a mechanism to avoid singularities during the radiation and matter eras, that is why in Section VI we device a suitable logarithmic correction which stabilizes the theory at large curvature. In Section VII, we provide a numerical simulation of the late-time acceleration occurring in our model. We should remark that the whole gravitational Lagrangian of the theory is here considered, which shows that the high-curvature corrections for inflation do not affect the dynamics of our model at late times. Actually, this model proves to be stable and to fit remarkably well the dark energy parameters coming from the latest analysis of Planck’s data. Conclusions and final remarks are given in Section VIII.

## Ii RG-improved effective action for higher-derivative quantum gravity

It is well-known that the Hilbert-Einstein action which classically describes the gravitational field gives rise to a non-renormalizable theory since, at high energy, the strong interaction is affected by divergences that cannot be canceled by a finite number of counter terms. This means that some ultraviolet completion is necessary at high energy scales. In this context, the development of a renormalizable quantum gravity theory is extremely interesting and can be also very useful in other areas of field theory. To wit, gravitation functions may act as a cut-off for interactions in elementary particle theory and it should be mentioned that a number of speculations are open in that sense.

For the pure gravitational action, renormalizability can be achieved by using quadratic terms on the Ricci scalar and Ricci tensor B (); Stelle (); Fradkin (); Hooft (), but the resulting theory is not ghost free and in the end some extension (eventually, non-local) is required Stelleplus (). The ghost problem in higher-derivative gravity is not solved so far but there are some hopes that it may be solved in non-perturbative approach. We do not go to the discussion of this problem because, as we explained in the introduction, we will consider the effective gravity theory which is not renormalizable due to presence of the dark energy -term added by hands, so this is just classical modified gravity. Furthermore, effectively Friedmann equations of our complete theory are just the same as for corresponding ghost-free -gravity obtained from our theory by dropping the Weyl-squared term which does not give any contribution to the equations of motion.

In this paper we will use the approach of Ref. B () (see also the references therein), where, by starting from quadratic higher-derivative gravity, it is possible to obtain a renormalizable model by computing the one-loop divergences of the theory.

Quadratic higher-derivative gravity models are quite interesting, since they represent a modification of Einstein’s gravity at high curvature, where the phenomenology of early-time inflation can be strictly connected with them. The very general action of this theory is given by,

 I=∫Md4x√−g(Rκ20−~Λ+aR2+bRμνRμν+cRμνξσRμνξσ+d□R), (II.1)

where is the determinant of the metric tensor , is the space-time manifold, and the covariant d’Alembertian, being the covariant derivative operator associated with the metric . The Hilbert-Einstein action is given by the Ricci scalar , while and are the higher curvature corrections to GR, and being the Ricci and the Riemann tensors, respectively. Here, encodes the mass scale of the theory and are constant parameters. Finally, is a cosmological constant term, which should not be confused with the cosmological constant for dark energy. If we here introduce the Gauss-Bonnet four-dimensional topological invariant, , and the square of the Weyl tensor,  ,

 G=R2−4RμνRμν+RμνξσRμνσξ,C2=13R2−2RμνRμν+RξσμνRξσμν, (II.2)

we can write

 RμνRμν=C22−G2+R23,RμνξσRμνξσ=2C2−G+R23. (II.3)

As the Gauss-Bonnet and the surface term do not contribute to the dynamical field equations of the model, we can drop them down from the action, which will result in terms of , , and only.

Using the results of the one-loop calculatios in the above theory, we can now proceed with it RG improvement, in analogy with RG-improved calculations carried out in quantum field theory in curved space-time El (); rginfl (). In this way (see, also, Ref. qg1 ()), we obtain a RG-improved action for higher-derivative quantum gravity, which reads

 I=∫Md4√−g[Rκ2(t′)−~Λ(t′)−ω(t′)3λ(t′)R2+1λ(t′)C2+fDE(R)+Lm]. (II.4)

Note that here, to the RG-improved action we have added , which is the Lagrangian corresponding to standard matter, and the function of the Ricci scalar , which has been introduced by hand to support the late-time cosmic acceleration. The precise role of this term will be discussed in Section V. In this way we unify in a single model the higher curvature corrections, which account for quantum gravity effects for inflation, with a phenomenological -gravity for the dark energy era. Notice also that QG corrections in can be safely neglected, since inflation is assumed to occur owing to the presence of purely gravitational terms.

The running coupling constants , , and are obtained from one-loop quantum gravity corrections and, thus, we deal with a renormalization-group-improved effective action from multiplicatively-renormalizable quantum gravity. This kind of Lagrangian has been investigated in several papers El (); rginfl (). In its simplest formulation  El (), the RG-improved effective action follows from the solution of the RG equation applied to the complete effective action of the multiplicatively renormalizable theory. As a final result, the one-loop coupling constants are expressed in terms of the log term of a characteristic mass scale in the theory, namely

 t′=t′02log[RR0]2,0

where is a positive number and is the curvature at which the quantum gravity effects disappear. The running coupling constants for the gravitational action (II.4) obey the one-loop RG equations B (); Fradkin (),

 dλdt′ = −β2λ2,dωdt′=−λ(ωβ2+β3),dκ2dt′=κ2γ,d~Λdt′=β4(κ2)2−2γ~Λ, (II.6)

and are closely connected with the -functions and , namely

 β2=13310,β3=103ω2+5ω+512,β4=λ22(5+14ω2)+λ3κ4~Λ(20ω+15−12ω),γ=λ(103ω−136−14ω). (II.7)

We observe that the renormalization procedure does not allow us to set . We also should note that, when the coefficients in the action (II.1) are not constant, the Gauss-Bonnet and the -term contribute to the dynamics of the model, resulting into additional RG equations. As stated before, in this work we will use the simplest renormalized theory of Ref. B () with and Weyl corrections only, but in the next chapter we will offer a detailed comparison with the extended model in FRW space-time.

The important remark is in order. Despite to the fact that standard scalar radiation/matter fields quickly are shifted away during the early-time accelerated expansion, their presence may slightly affect the behavior of the running coupling constants in (II.6)–(II.7). In Ref. qg2 () the scalar electrodynamics model non-minimally coupled with higher-derivative gravity has been analyzed in the framework of quadratic gravity by taking into account the whole form of quantum corrections including the matter. The coupling with gravity leads to an effective field potential in the Lagrangian together with additional terms in the beta functions (II.7). However, if one considers simple models where the matter fields are decoupled from gravity, the corrections to the RG equations above can be ignored as curvature terms are dominant if compare with scalar potential similar to Starobinsky inflation with matter. The matter Lagrangian may contain some log-terms of the field whose contribution appears in the perturbed equations, but leads to negligible corrections in the spectral index and in the tensor-to-scalar ratio of the model. For these reasons, assuming Standard Model for the matter Lagrangian, we can use the results in (II.6)–(II.7). We explicitly checked that account of matter contribution (scalar theory and/or scalar electrodynamics) in RG improved effective action (II.4) does not lead to any qualitative changes in the results of this paper.

From the first equation in (II.6), we immediately get:

 λ(t′)=λ(0)1+λ(0)β2t′,0<λ(0), (II.8)

where is a positive integration constant, corresponding to the value of at , namely in (II.5).

The equation for exhibits two fixed points, namely and , but only the first of them is stable, being an attractor of the system for large values of , namely when  Fradkin (). We can easily demonstrate this extreme by considering the behavior of , , around the fixed points, namely

 dω(t′)dt′ ≃ (II.9) = −λ(t′)(203ω+1585)|ω1,2δω(t′),

 ω(t′)=ω1,2+c0(1+λ(0)β2t′)q,q=1β2(203ω+1585)|ω1,2, (II.10)

where is a constant and we have used (II.8). Thus, the solution is stable at for with , while for with it diverges. This means that, for large values of , the function tends to the attractor at . By taking into account that , when and , the function grows up with and approaches , as

 ω(t′)=ω1+c0(1+λ(0)β2t′)p,p=(103)(ω1−ω2)β2≃1.36, (II.11)

where we have taken the average value of between and . In this paper we will set , so that

 ω(t′)=ω1=−0.02. (II.12)

Note that the term gives a positive contribution inside the gravitational action and helps to avoid finite-time future singularities at large curvature.

Now, it is possible to get the form of from the third equation in (II.6),

 κ2(t′)=κ20(1+λ(0)β2t′)Z/β2,Z=(103ω1−136−14ω1)=10.27, (II.13)

where corresponds to the mass scale of the theory at small curvature, namely the Planck mass ,

 κ20=16πM2Pl,M2Pl=1.2×1028eV. (II.14)

Finally, we derive the form of from the last equation in (II.6), which reads

 d(κ4~Λ)dt′=β4≡λ(t′)22(5+14ω(t′)2)+λ(t′)(κ4(t′)~Λ(t′))(203ω(t′)+5−16ω(t′)). (II.15)

When is constant, this equation can be solved with respect to the dimensionless function , and leads to

 κ4~Λ=−3λ(0)(1+20ω21)4ω1(1+λ(0)β2t′)(−1+30ω1+6β2ω1+40ω21)+κ40~Λ0(1+λ(0)β2t′)W/β2,W=203ω1+5−16ω1=13.2, (II.16)

where we have used (II.8). Since we would like to completely avoid the quantum induced effects at small curvature, we require that when , by fixing the integration constant as

 ~Λ0=3λ(0)κ40((1+20ω21)4ω1(−1+30ω1+6β2ω1+40ω21))≃−11×λ(0)κ40. (II.17)

On the other hand, due to the fact that , at large curvature

 ~Λ≃~Λ0(1+λ(0)β2t′)X/β2≃~Λ0(1+λ(0)β2t′)0.55,X=(2Z−W)≃7.34, (II.18)

and the cosmological constant from the RG improved effective action tends to decrease.

A general remark is here in order. Quantum corrections must disappear towards the end of inflation, when . For this reason the boundary term must be chosen according to the condition

 λ(0)≪|1β2t′0log[4Λ/R0]|, (II.19)

where corresponds to the de Sitter (final) curvature attractor of the dark energy epoch of our Universe today, being the cosmological constant. Only in this way can we be sure that quantum corrections are negligible in the whole curvature range . Thus, the parametrization in (II.18) guarantees that the cosmological constant term does not play any significative role, neither at high curvature nor at small curvature, and thus we can neglect its contribution.

In the next chapter, the possibility to get an early-time inflation from our gravitational model will be discussed. Contributions from the modified function and the matter Lagrangian inside the action turn out to be negligible, and we will consider inflation as a manifestation of high-curvature corrections to GR, which take into account quantum effects.

## Iii Early-time inflation in higher-derivative gravity

Let us consider the general form of flat FRW space-time

 ds2=−N(t)2dt2+a(t)2(dx2+dy2+dz2), (III.20)

where is the scale factor of the Universe and is an einbein function of the cosmological time. If we take the variation of the Weyl term, as

 (III.21)

we immediately note that on the FRW metric (III.20),

 C2=0,1λ(t′)δ(√−gC2)=0, (III.22)

so that the square of the Weyl tensor does not enter into the Friedmann field equations of the theory. In what follows, we will fix the usual gauge as .

The evolution of the model in the vacuum at high curvature is governed by the first Friedmann-like equation. If we neglect the contribution of and in (II.4), the first Friedmann-like equation reads qg1 ()

 0 = (III.23) +6(H2+˙H)Δ(t′)t′0R−6H⎡⎣dΔ(t′)dt′(t′0R)2−Δ(t′)t′0R2⎤⎦˙R−~Λ(t′),

with

 Δ(t′)=[R(κ2(t′))2dκ2(t′)dt′+R2ddt′(ω(t′)3λ(t′))+d~Λ(t′)dt′]. (III.24)

In the above expressions, the dot denotes time derivative, and the Ricci scalar is given by,

 R=12H2+6˙H,H=˙aa, (III.25)

with the Hubble parameter.

Inflation is described by a (quasi) de Sitter solution, where the Hubble parameter is almost a constant, namely . On the de Sitter solution , the system leads to

 6H2dSκ2(t′)+Δ(t′)t′02−~Λ(t′)=0,Δ(t′)=⎡⎣12H2dS(κ2(t′))2dκ2(t′)dt′+144H4dSddt′(ω(t′)3λ(t′))+d~Λ(t′)dt′⎤⎦. (III.26)

It is interesting to compare these expressions with the ones from higher-derivative quantum gravity including the Gauss-Bonnet and -terms. As we observed in the preceding section, when one deals with running coupling constants, the most general form of a quadratic higher-derivative theory is given by (II.4) plus the following gravitational part

 IG,□R=−∫Md4x√−g[γ(t′)G−ζ(t′)□R], (III.27)

with and functions of the scale parameter . In this case, the first Friedmann-like equation of the model reads qg1 ()

 0 = (III.28) +6(H2+˙H)Δ(t′)t′0R−6H⎡⎣dΔ(t′)dt′(t′0R)2−Δ(t′)t′0R2⎤⎦˙R−24H3dγ(t′)dt′t′0˙RR−6H[dγ(t′)dt′t′0˙GR] −3A˙R2−2B˙R2R+6ddt[2A(4H2+3˙H)˙R+BH˙R2]+18H[2A(4H2+3˙H)˙R+BH˙R2] −36(3H2+˙H)AH˙R−72Hddt(AH˙R)−12d2dt2(AH˙R)−~Λ(t′),

where is still given by (III.24), the Gauss-Bonnet on FRW space-time corresponds to

 G=24H2(H2+˙H), (III.29)

and

 A=(dζ(t′)dt′t′0R),B=⎡⎣d2ζ(t′)dt′2(t′0R)2−dζ(t′)dt′t′0R2⎤⎦. (III.30)

We see that, when are constant, we recover (III.23). For the de Sitter solution we have now

 6H2dSκ2(t′)+Δ(t′)t′02−~Λ(t′)+12H4dSdγdt′t′0=0, (III.31)

and (of course) only the Gauss-Bonnet gives an additional contribution with respect to the case in (III.26).

Generally speaking, for large values of the Hubble parameter, when we neglect , Eq. (III.26) assumes the form

 −α(t′)H4dS+β(t′)H2dS≃0, (III.32)

where and are dimensionfull (positive) functions of the scale parameter . Thus, the de Sitter solution reads,

 H2dS≃β(t′)α(t′),t′=t′02log[RdSR0]2. (III.33)

If we add to the model the Gauss-Bonnet contribution, the first Friedmann-like equation for constant Hubble parameter (III.31) takes the (asymptotic) form:

 (−α(t′)+12dγdt′t′0)H4dS+β(t′)H2dS≃0, (III.34)

 H2dS≃β(t′)[α(t′)−12(dγ/dt′)t′0]. (III.35)

We conclude that a contribution of the Gauss-Bonnet kind in the gravitational action as increases the curvature of inflation if , and vice-versa, it decreases the curvature of the inflationary Universe provided that .

Let us return to the simplified action (II.4) with (III.23)–(III.26). By using the set of equations (II.6)–(II.7), one derives, from (III.26),

 0 = 6H2κ2−t′048(κ2)2ω2(480H4(κ2)2ω2(4ω(2ω+3)+1)+24κ2λωH2(−40ω2+26ω+3) (III.36) −3λ2(20ω2+1)),

where the functions are assumed to be constant with respect to time.

By taking into account the expressions of and derived in the preceding section (note that in (III.32) turns out to be constant), by taking (see (II.17)–(II.18) with ), we derive, in the limit when the quantum corrections are relevant,

 H2dSκ20≃0.107t′0(λ(0)t′)0.77. (III.37)

In fact, our de Sitter solution emerges from the one-loop corrections encoded in (III.24). In Starobinsky’s inflationary scenario, where the coefficients of the action are constant and come from the trace-anomaly Staro (), the -term alone supports the de Sitter solution of inflation, while the Hilbert-Einstein term permits to slowly exit from the accelerated phase (this role can be played also by different power functions with  mioStaro ()). However even in the Starobinsky model, when one takes the asymptotic limit in the Friedmann equation (), the mass scale of the theory is implicitly considered to be smaller than the Planck mass, in order to avoid super-Planckian curvatures, implying a sort of “running” mechanism (for some recent works on -gravity in this respect, see Refs. Rinaldi ()).

We will now proceed with the investigation of the graceful exit from inflation. Let us consider a small perturbation around the quasi de Sitter solution described by (III.37), namely

 H=HdS+δH(t),|δH(t)/HdS|≪1. (III.38)

In the limit , if we neglect the contribution of , Eq. (III.23) reads

 0=(κ0˙δH)⎡⎣t′0⎛⎝(HdSκ0)2(34.344−0.913t′0t′)+0.001t′+0.003t′0t′3(HdSκ0)2(λ(0)t′)1.54+0.346t′0−0.086t′t′2(λ(0)t′)0.77⎞⎠+19.152t′(HdSκ0)2⎤⎦ −0.043t′2t′0(HdSκ0)2(λ(0)t′)0.77+0.001t′20(λ(0)t′)1.54+0.087t′t′20(HdSκ0)2(λ(0)t′)0.77+2×10−4t′t′0(λ(0)t′)1.5400⎤⎦ +(HdSκ0)δH[0.223(λ(0)t′)0.77+0.172λ(0)t′0(λ(0)t′)1.77−30.528t′0(HdSκ0)2], (III.39)

and, for , it leads to qg1 ()

 D0δH(t)+t′[19.152(HdSκ0)(κ0˙δH(t))+6.384(κ20¨δH(t))]≃0, (III.40)

with

 D0=[0.223(λ(0)t′)0.77−30.528t′0(H2dSκ20)]≃−28.444t′0(H2dSκ20). (III.41)

The solution of Eq. (III.40) is

 δH(t)=h±exp[A±t],A±=⎡⎣HdS2⎛⎝−3± ⎷9−0.627D0(H2dSκ20)t′⎞⎠⎤⎦,|h±/HdS|≪1, (III.42)

where are integration constants corresponding to the plus and minus signs inside , respectively. Since is negative, the solution turns out to to be unstable, namely, for ,

 δH(t)≃h+eA+t,A+≃1.486(HdSt′0t′), (III.43)

with to make the Hubble parameter decreasing.

If we introduce and as the initial and the final time of the early-time acceleration, respectively, we can set , and thus obtain

 H=HdS(1−eA+(t−te)),ti≪te, (III.44)

so that the Hubble parameter tends to vanish at the end of inflation.

The number of -folds is a valid parametrization frequently used in the study of early-time inflation. It is defined as

 N=ln(a(te)a(t)). (III.45)

By taking into account that , we get

 t=te−NHdS. (III.46)

Thus, the Hubble parameter during inflation behaves as

 H≃HdS⎛⎝1−e−A+NHdS⎞⎠. (III.47)

When the Hubble parameter is given by the de Sitter solution , while, when , the Hubble parameter decreases and goes to zero (i.e. ), allowing the model to exit from the inflationary phase.

The total -fold number of the corresponding inflation

 N=ln(a(te)a(ti))≃HdS(te−ti), (III.48)

must be large enough in order to appropriately lead to the thermalization of our observable Universe and to solve the problem of the initial conditions of the Friedmann Universe, too. In general it is required that . Moreover, given that when it must be , we can identify as the minimal value of -folds at which inflation can start, namely

 60≤Nmin=HdSA+≤N. (III.49)

It turns out that (III.47) can be rewritten as

 H≃HdS(1−e−NNmin). (III.50)

We observe that encodes the curvature expansion rate of inflation as

 Nmin≃(1.486)−1log[RdSR0], (III.51)

where we have used (III.43) with (II.5). Once the curvature expansion rate is fixed, by taking into account Eq. (III.37), we obtain a relation between the curvature at the time of inflation and the boundary parameter ,

 RdS=1.284(t′0κ20)(λ(0)t′0log[RdS/R0])0.77. (III.52)

In the next section we will investigate the cosmological perturbations left at the end of inflation, and we will derive the spectral index and the tensor-to-scalar ratio of the model. Accurate comparison with astronomical data will establish the value of the curvature expansion rate .

## Iv Cosmological perturbations during inflation

During the inflationary stage the Hubble parameter slowly decreases in the so called “slow-roll approximation” regime, provided the following conditions are met

 |˙HH2|≪1,|¨HH˙H|≪1. (IV.53)

In particular, the slow-roll parameter ,

 ϵ=−˙HH2≡1HdHdN, (IV.54)

must be small and positive during inflation, while it tends towards one when the early-time acceleration ends. Using (III.50), we obtain

 ϵ≃e−NNminNmin. (IV.55)

One of the most important prediction of inflation consists in the description of the anisotropies of our Universe at galactic scale. In this respect, perturbation theory is the key mechanism to calculate the inhomogeneities left at the end of the primordial accelerated expansion, and leads to the derivation of the spectral index and of the tensor-to-scalar ratio for scalar and tensorial perturbations, respectively. Therefore, only if these indexes fit the inferred values in our observable Universe will the theory be considered to be viable and to lead to a realistic description of inflation.

An important remark is in order. If one considers the full form of the renormalizable action (II.4), the Weyl term appears. As we recalled before, it actually does not contribute to the Friedmann-like equations of the theory: neither inflation itself, nor the graceful exit phase, -fold number (III.48), nor the slow-roll parameter (IV.55) depend on it. However, when one introduces the cosmological perturbations around the FRW metric, the Weyl term plays a role in the perturbed equations Derue (); corea1 (); corea2 (). In Refs. corea1 () it has been shown that in pure Weyl conformal gravity the scalar perturbations do not propagate in the de Sitter background, while the vector and the tensor power spectra are constant, as a consequence of the invariance of the Weyl tensor with respect to conformal transformations. This behavior seems to be confirmed in Weyl invariant scalar tensor theories corea2 (). In our case, however, the Weyl term may change the evolution of the cosmological perturbations of the model. For example, if one starts with scalar perturbations in the Newton’s gauge,

 ds2=−(1+2Φ(t,x))dt2+a(t)2(1−2Ψ(t,x))(dx2+dy2+dz2), (IV.56)

with and scalar functions of the space-time coordinates, we derive . In the background of Einstein’s gravity one has , while in a -theory of gravity , and we see that the square of the Weyl tensor is here different from zero.

It lies beyond the scope of our work to investigate these implications of the Weyl tensor in the perturbative theory of -gravity, namely because in our action the square of the Weyl tensor is coupled with the Ricci scalar at the inflationary scale, rendering the system very involved. Cosmological perturbation theory in the presence of the Weyl term is still a debated subject (see e.g. Ref. Man ()). In what follows, we will neglect its contribution in dealing with a -gravity model. Thus, the spectral index and the tensor-to-scalar ratio read corea (),

 (1−ns)≃−2ϵdϵdN,r≃48ϵ2, (IV.57)

where for the tensor-to-scalar ratio we have used second order corrections (the first order ones simply vanish). By using (IV.55), we immediately get

 (1−ns)=2Nmin,r=48N2mine−2NNmin, (IV.58)

where we set during inflation. Recent analysis of Planck data Planck () constraint these quantities as and . Therefore, the general condition to realize a realistic inflationary scenario is

 Nmin=HdSA+≃60,60≃Nmin≤N. (IV.59)

The second condition is trivially satisfied by realistic models of inflation. On the other side, the first condition fix the curvature expansion rate since, by using (III.51), we obtain

 RdS≃R0e89. (IV.60)

Now we must involve (III.52) in order to fix the boundary term

 RdS=R0e89≃0.040(t′0κ20)(λ(0)t′0)0.77. (IV.61)

Without loss of generality, we can set

 t′0=1, (IV.62)

and introduce the Planck mass in (II.14), namely

 κ20=3.49×10−55eV−2. (IV.63)

By taking into account the value of the cosmological constant (see also Ref. Barrow ()),

 Λ=11.895×10−67eV2≃10−122M2Pl, (IV.64)

we obtain a realistic ratio by setting

 λ(0)=8×10−16. (IV.65)

In this case the right hand side of (II.19) yields and the condition is well satisfied. Here, we must stress that the model predicts inflation with curvature at least 130 orders of magnitude larger than the curvature of the Universe today. Indeed, if the curvature of inflation is smaller, condition (II.19) is not fulfilled and the matter/radiation eras with the following late-time acceleration may be not well reproduced, so that a unified description fails.

At the end of the early-time acceleration stage some reheating mechanism is involved with the purpose to convert the inflation energy into standard matter and radiation (e.g., the quark gluon plasma). Since quantum gravity effects then disappear, the gravitational Lagrangian in (II.4) reads (we still neglect dark energy in )

 L≃√−g[Rκ20+0.02λ(0)R2+1λ(0)C2], (IV.66)

where we have taken into account (II.12) and (II.16) with (II.17). Once again, the square of the Weyl tensor does not contribute to the field equations in FRW space-time and the reheating mechanism coincides with the one for -inflation. The first Friedmann-like equation leads to

 ¨H−˙H22H+λ(0)0.24κ20H=−3H˙H, (IV.67)

with the oscillating solution

 H≃43(t−tr)cos2⎡⎣ ⎷λ(0)0.12κ20(t−tdS)2⎤⎦, (IV.68)

where is the time at reheating, and, since , we get a matter-like cosmological evolution. By taking into account that the Hubble parameter tends to vanish at the end of inflation111In the second reference of SuperBamba () the critical value of the curvature during reheating for a combined model of -inflation with exponential -gravity for dark energy has been accurately calculated as . (), and , we derive for the Ricci scalar

 R≃−4(t−tr) ⎷λ(0)0.12κ20sin⎡⎣ ⎷λ(0)0.12κ20(t−tr)⎤⎦. (IV.69)

And using the Lagrangian of a scalar bosonic field222The production of bosons is favored with respect to that of fermions during reheating infrev (). with mass and non-minimally coupled with gravity,

 Lχ=−gμν∂μχ∂νχ2−m2χχ22−ξRχ22, (IV.70)

being a coupling constant, one obtains the field equation

 □χ−m2χχ−ξRχ=0. (IV.71)

By decomposing the field into Fourier modes with momentum , on FRW space-time, we get

 ¨χk+3H˙χk+(m2χ+ξR)χk=0. (IV.72)

Now we can introduce a conformal time , such that

 d2dη2uk+m2effa(t)2uk=0,uk=a(t)χk, (IV.73)

where the effective mass is given by

 m2eff=[m2χ+(ξ−16)R]. (IV.74)

Since the Ricci scalar oscillates as in Eq. (IV.69), the number of massive particles changes with time and the reheating mechanism takes place. We should note that, even in the case of minimal coupling with gravity , the effective mass still depends on and we do get reheating. After particle production, when the energy density of radiation and ultrarelativistic matter becomes dominant and the -term in the Lagrangian vanishes, due to the condition (II.19), the usual radiation/matter era can start.

The second part of our work is thus devoted to the study of Friedmann cosmology; in the next section we will introduce modified gravity for the dark energy sector through the function in (II.4). We will see that a complete picture of the matter era and of dark energy de Sitter expansion can be recovered in our model, thus confirming that high-curvature corrections of the model disappear at small curvature, as expected.

## V Dark energy from exponential gravity

In Refs. nostriexp (); altriexp (); HS () several versions of viable modified gravity for the dark energy epoch have been proposed and investigated. They belong to a class of so-called “one-step models”, which produce reasonable description of the dark energy evolution of our Universe today. They incorporate a vanishing cosmological constant in the flat limit ()