Prepared for submission to JCAP Inflation and dark energy from f ( R ) gravity

Abstract. The standard Starobinsky inflation has been extended to the R + αRn − βR2−n model to obtain a stable minimum of the Einstein frame scalar potential of the auxiliary field. As a result we have obtained obtain a scalar potential with non-zero value of residual vacuum energy, which may be a source of Dark Energy. Our results can be easily consistent with PLANCK or BICEP2 data for appropriate choices of the value of n.


data for appr
priate choices of the value of n.

Introduction

Inflationary scenario for the Early Universe has become one of the standard assumptions in theoretical model building.It can account naturally for the wealth of present observational data.The existence of rapid expansion stage in the history of the Universe can give us a natural explanation of the homogeneity, flatness and horizon problems, and provides an explanation of the process of seeding large-scale structure and temperature anisotropies of the cosmic microwave backgroun radiation (see refs.[1][2][3] for reviews).

One of the first inflationary models was the Starobinsky inflation [4], based on the f (R) = R + R 2 /6M 2 Lagrangian density.In this theory during inflation the R 2 term dominates, which provides a stage of the de-Sitter-like evolution of space-time.The inflationary potential has a stable minimum, which allows for the graceful exit, reheating and good low-energy limit of the theory.However, the recent observational data from the BICEP2 experiment [5,6] suggest significant amount of primordial gravitational waves, which disfavours the Starobinsky inflation.As shown in [7], a small modification of the f (R) function, namely f (R) = R + αR n , can generate inflation which fits the BICEP2 data.This model has already been partially discussed in [8].Different modifications of the Starobinsky mo el were also discussed in Ref. [9,10].

On the other hand the series of experiments [11][12][13] convincingly suggests the existence of the so-called Dark Energy (DE) with barotropic parameter close to −1.One of the possible sources of DE may be a non-zero vacuum energy of a scalar field, which in principle can be the f (R) theo y [8,14,15] or the Brans-Dicke field.

In this paper we demonstrate how to extend the f (R) = R + αR n model to combine successfully both features of the observable Universe in a single framework: we show how to obtain successful inflation and the non-zero residual value of the Ricci scalar in an extension of the Starobinsky model.Namely, we use the f (R) = αR n − βR 2−n Lagrangian density, which provides inflation from the αR n and a stable minimum of the scalar potential of an auxiliary field.The potential has a non-zero vacuum energy density, which is the source of the DE in the late-times evolution.Let us note that some inflationary potentials, which generate DE (motivated by modified theories of gravity) were already introduced in the Ref. [16] In the following draft we use the convention 8πG = M −2 pl = 1, where M pl ∼ 2×10 18 GeV is the reduced Planck mass.

The structure of this paper is as follows.In the Sec. 2 we analyse the f (R) = R + αR n model: we discuss analytic solutions, parameters of inflation and primordial inhomogeneities.In the Sec. 3 we generalise this model to f (R) = R + αR n − βR 2−n , which leads to the nonzero vacuum energy of the Einstein frame potential.In the Sec. 4 we discuss numerical study of the evolution of the model with dust modelling the contribution of matter fields to energy density.Fina

y, we conclude in t
e Sec. 5.


The R + αR n model

Let us consider an f (R) theory in the flat FRW space-time with the metric tensor of the form of ds 2 = −dt 2 + a(t) 2 (d x) 2 .Then, = F − H Ḟ + (ρ M + P M ) ,(2.2)
where F (R) = df dR and ρ M and P M are energy density and pressur of matter fields respectively.

The action of , U (ϕ) = 1 2 (RF − f ) ,(2.4)
which means that the f (R) gravity can be expressed as a Brans-Dicke theory in Jordan frame with ω BD = 0.In the Jordan frame the auxiliary field ϕ is non-minimally coupled to gravity, which creates a deviation from the General Relativity (GR) frame.The first Friedmann equation and continuity equ M + 3H(ρ M + P M ) = 0 . (2.7)
where U ϕ = dU dϕ .Let us note that U may be interpreted as a energy density, but U ϕ is not an effective force in the Eq.(2.5).One can define the effective potential and its derivati ef f := dU ef f dϕ ,(2.8)
where C is unknown constant of integration.The effective potential shall be interpreted as a source of an effective force, but not as a energy density.

The gravitational part of the action may obtain its canonical (minimally coupled to ϕ) form after transfo t = √ ϕdt , ã = √ ϕa (2.9)
which gives 2 − U (ϕ) ϕ 2   ,(2.10)
where ∇ is the derivative with respect to xµ .In order to obtain the canonical kinetic term for ϕ let us use the Einstein frame scala field φ = 3 2 log ϕ .(2.11)
The action in terms of gµν 2 = U (ϕ) ϕ 2 ϕ=ϕ(φ)
.

(2.13)

Let us note that (ϕU ϕ − 2U ) from the Eq.(2.5) can be expressed as ϕ 3 V ϕ , so the minimum of V shall also be the minimum of the effective potential in the Jordan frame.In fact, all important features of the potential, like existence of minima and barriers between them, which determine the evolution of the field in the Einstein frame are reflected in the evolution of the field in the Jordan frame.In further parts of this draft we will refer to the Einstein frame potential, even though we consider the Jordan frame as the primordial (or defining) one.Since all of the analysis performed in this draft is classical, descriptions in both frames give the same physical results.However, the description in the Einstein frame is more intuitive, due to the canonical form of the scalar field's kinetic term and the minimal coupling between the field and the gravity.The only exception is the ϕ → ∞ limit, which usually leads to V → ∞ due to the ϕ −2 term in the potential.This infinity comes from the singularity of the Einstein frame metric tensor and it does not appear in the Jordan f ame analysis.

For the vac n > 0) ,(2.1 n−2 Ṙ . (2.15)
In the regime F 1, F > 1 (n ∝ t 1/ . (2.16)
The same results would be obtained for the power-low inflation in GR frame (for minimally coupled scalar field) with the potential V = V 0 e λφ , where = λ 2 /2.However, such a model would generate different spectrum of primordial inhomogeneities.Let us note that the can be interpreted as a slow-roll parameter only for F 1 and > 1/F .The second condition comes from the fact, that the correction to the equation of motion coming from the GR low-energy limit shall be of the order of 1/F ∼ R/f (R).The slow-roll parameter will be denoted as H end 1 F .
(2.17)

The inflation ends when H ∼ 1, which happens in the F < 1 regime.Thus at the moment of the end of inflation one finds ϕ 1 for all realistic values of n.Besides H let us define slow-roll parameters F F H Ḟ .(2.18)
To obtain inflation one must satisfy H , F , η F 1. To satisfy this condition in the F > 1 2 (1 + √ 3) .
To satisfy Ḣ < 0 and H > 0 one needs n < 2. Thus the allowed range for n is gi 3), 2 . (2.19)
Let us check whether the assumption F 1 is valid during the last 50-60 e-folding of inflation, during which the primordial inhomogeneities were generated.The evolution of a slow-roll parameters (and their F > 1 limit) as a function of time for different values of n has been presented at fig. 1, 2, 3. Values of the α parameter have been chosen to satisfy normalisation of primordial curvature perturbations at 50−60 e-folds before the end f inflation.

To obtain the number of e-folds let us note that for the slow-roll approximation of Eq. (2.5,2.6,2. ) n ,(2.20)
where N and Ñ are the number of e-folds in Jordan and Einstein frames respectively.Usually the log(ϕ) term is subdominant and one can write  This result has been obtained using the slow-roll approximation and its accuracy is of order of few F n 1.8 Α 3.
The Einstein frame scalar potential for f (R) = R + αR n as a function of ϕ has a fo s the deviation from the Starobinsky potential.Let us discuss case by case spe .

a) To obtain the real values of the potential for all ϕ together with the stable minimum at ) cannot be used to describe the evolution of space-time during that period.
ϕ = 1 one needs 2 − n n − 1 = 2l 2k + 1 ,(2.23)
where k, l ∈ N and l ≤ k.To satisfy these conditions one need to assum that
n = 2(1 − 2 a 10 −b ) ,(2.24)
where a, b ∈ N, a ≥ b and a < b log 2 (10)
+log 2 (3− √ 3) to obtain accelerated expansion of the space-time.This case is presented at the left panel of the Fig. 6.For this class of potentials the reheating of the Universe takes place during the oscillations period, after the scalar field reach its minimum.

b) The re l values of the potential for all ϕ and a potential without any minimum is obtained for
2 − n n − 1 = 2l + 1 2k + 1 ,(2.25)
where k, l ∈ N and l < k.Potential obtains negative values for ϕ < 1 and it heads to −∞ for ϕ → 0. This case is presented at the middle panel of the Fig. 6.Such a potential is non-physical and without additional terms it cannot be used to generate inflation.

c) In al

other cases the potential V becomes imaginary
for any ϕ < 1 and it has no minimum.This case is presented at the right panel of the Fig. 6.The inflation ends with the reheating ince there is no oscillation phase such a potential requires strong couplings between the auxiliary field and matter fields.


The generation of primordial inhomogeneities

The power spectrum of the Jordan frame primordial curvature perturbations at the superhorizon scales has the following form [8]
P R 2H 2 F 3 Ḟ 2 H 2π 2 .
(2.26)

Let us note that the Eq.(2.2) can be expressed as H = − F (1 − η F ), so for the slowroll approximation one finds H − F .This approximation is valid for 2 − n 1, since F = (n − 1) H for F > 1.Thus, the spectral index n s and tensor to scalar ratio r are of the form of [8]
n s 1 − 4 H + 2 F − 2η F 1 − 6 H − 2η F , r 48 2 F 48 2 H .
(2.27)

The normalisation of primordial inhomogeneities requires that P 1/2 R ∼ 5×10 −5 at the moment of 50 to 60 e-folds before the end of inflation.One can use the normalisation of the power sp lues of α.From Eq. (2.5,2.6,2.14,2.26) in the slow-roll regime one finds the α = α(n), which (for realistic values of n) is plotted at the Fig. 4 The (n s , r) plane (which describes the shape of primordial curvatur on of the R + αR n model we present in the Appendix A the analysis o nimum, in cases when the minimum does exist.3 The R + αR n − βR 2−n model Deficiencies of the model discussed in the previou Section can be bypassed by considering further modification of t ar potential and (as we will show) to generate dark e y
f (R) = R + αR n − βR 2−n ,(3.1)
where α and β are positive constants and n satisfies the Eq.(2.19).Let us require α 1, β 1 and αβ 1.This means that the αR n R βR 2−n during inflation, so the results obtained in the sec. 2 are still v ne finds
R(ϕ) = 4(2 − n)nαβ + (ϕ − 1) 2 + ϕ − 1 2nα 1 n−1 (3.3)
Let us note that R > 0 for any value of ϕ.For ϕ − 1 αβ one obtains
R(ϕ) ϕ − 1 nα 1 n−1 ⇒ αR n βR 2−n 1 αβ ϕ − 1 n 2 1 . (3.4)
Thus, the βR 2−n term does not have any influence on inflation and generation of the large scale structure of the universe.

The Jordan and Einstein frames scalar potentials look as follows
U (ϕ) = 1 2 (n − 1) αR n (ϕ) + βR 2−n (ϕ) , V = 1 ϕ 2 U (2 − n)nαβ − 1 2(2 − n)α 1 n−1 (nβ) 1 n−1 1 − (2 − n)nαβ n − 1 , (3.6)
where the last term is the Taylor expansion with respect to beta.The minimum is slightly shifted with respect to R = 0, which is the GR vacuum case.Hence, this model predicts some amount of vacuum energy.Around the minimum one finds
αR n−1 min ∼ nαβ 1 , βR 1−n min ∼ 1 n ∼ O(1) ,(3.7)
which means that the inflationary term is negligible and R ∼ βR 2−n .The existence of the αR n term is still important, since it provides the minimum and real values of a scalar potential for all ϕ.Let us clarify that R min is not the minimal value of R. The Ricci scalar has no minimum, its minimal value is equal to 0 (a

the ϕ → −∞ limit) and it continuously
grows with ϕ.The value of V at the minimum for small values of β reads
V (ϕ min ) n 8(n − 1) 2 (nβ) 1 n−1 n − 1 − n 2 αβ ∼ 1 2 β 1 n−1 ,(3.8)
where
ϕ min := F (R min ) 2 n value of ϕ at the density of the DE shall be of order of β 1 n−1 and that we need β ≪ 1 to fit the dark energy data.An example of a potential and its minimum are plotted at the Fig. 7.The existence of a stable minimum is one of the main differences between this model and a R βR 2−n dark energy model, in which the auxiliary field rolls down towards negative The minimum of the Einstein frame potential (visible at both panels of the Fig. 7) prevents the ϕ from obtaining negative values for ny solution with inflationary initial conditions.


Conditions for whether this model satisfies conditions pointed out in the Ref. [8].The R 0 denoted the value of the Ricci scalar today 1) To avoid the ghost state one needs
F > 0 for R ≥ R 0 ,(3.9)
which means that
R 0 > R(ϕ = 0) = 1 + 4nαβ(2 − n) − 1 2αn 1 n−1 .(3.10)
This conditio is satisfied for any ϕ with initial condition ϕ > 0, due to the existence of the ential.This is an advan n which F < 0 in late times.

2) To avoid the negative mass square for a scalar field degree of freedom one needs
F > 0 for R ≥ R 0 ,(3.11)
which means that R 0 > 0. This condition is obviously satisfied, since the universe at the present ti

onstraints one needs
f (R) → R − 2Λ for R ≥
0 ,(3.12)
After the ϕ field is stabilised in its minimum it produces the vacuum energy, which is a source of Λ.The non-zero values of R comes from radiation and dust produced during the reheating of the universe, as well as from the non-zero minimal value of the Ricci scalar.

4) For the stability and the presence of a 1 + 4n (2 − n) αβ − 1 2(2 − n)α 1 n−1 (nβ) 1 n−1 . (3.14) At R = R r the RF /F takes the form of RF F (r = −2) = 1 2 + 8αβ 1 + 4 n 2 − 2n + 2 αβ − (n − 1) 1 + 4n (2 − n) αβ ,(3.15

Numerical analysis of the dark energy model

The non-zero value of the Einstein frame potential at the minimum rises a possibility of obtaining a realistic solution to the dark energy problem.To analyse low energy solutions of the Jordan frame equations of motion let us use the number of e-folds (defined by N := log(a)) as a time variable.Then Eq. (2.5,2.6)read
H 2 (ϕ N N + 3ϕ N ) + H N Hϕ N + 2 3 (ϕU ϕ − 2U ) = 1 3 (ρ M − 3p M ) ,(4.1)H 2 = ρ M + U 3 (ϕ + ϕ N ) ,(4.2)
where the index " N " denotes the derivative with respect to N .Since in the Jordan frame the Eq.(2.7) is satisfied one finds ρ M = ρ I e −3(1+w)N , where w = p M /ρ M is a barotropic parameter.After the inflation ϕ oscillates around ϕ min and reheats the universe by the particle production.Thus, after oscillations one obtains the radiation domination era, for which w = 1/3 and ρ M − 3p M = 0.The radiation increases the cosmic friction term but does not contributes to the U ef f , so the field is not shifted from the minimum.However, during the dust domination era the U ef f is modified and ϕ oscillates around ϕ = 1.The evolution of ϕ and ϕ N during the dust/DE domination era is presented at the Fig. 8.The evolution of the Hubble parameter and ρ M /3 is plotted at the Fig. 9.We have assumed that the field starts from the ϕ 1 (which i the GR limit of the theory), but numerical analysis shows that the late-time results do not depend on initial conditions.For instance the initial value of the field in the Einstein frame minimum the only difference is slightly longer period of oscillations around ϕ = 1.As lo ns-Dicke field the field oscillates and stabilises above the ϕ min .During that period |ϕ − 1| 1 and the R term in f (

dominates.T
us one recovers the GR limit of the theory.When the dust becomes subdominant the ϕ rolls to its minimum and one obtains the Dark Energy with the barotropic parameter ω = −1.

As shown in the Fig. 9 the Hubble parameter obtains the constant value when the auxiliary field is in its minimum.The H = const implies that R = 12H 2 = 4ρ Λ , where ρ Λ is the DE energy density.While ϕ is in its minimum one finds R = R min , so at the late times one finds
ρ Λ = R min /4 1 4 (nβ) 1 n−1 ⇒ β 1 n (4ρ Λ ) n−1 .(4.3)
The ρ Λ becomes constant even before the DE domination era.Thus, the Eq.4.3 shall be satisfied at the present time.


Conclusions

In this note we have demonstrated how to extend the f (R) = R + αR n model to combine successfully both features of the observable Universe in a single framework: we show how to obtain successful inflation nd the non-zero residual value of the Ricci scalar in an extension of the Starobinsky model.Our results can be easily consistent with PLANCK or BICEP2 data for appropriate choices of the value of n.In this case the Einstein frame potential of the auxiliary field has a minimum only when (2 − n)/(n − 1) is of the form of a fraction with even and odd numbers in the numerator and the denominator respectively.The potential has a minimum at ϕ = 1 and V (ϕ = 1) = 0, which means that there is no vacuum energy.For all other forms of (2 − n)/(n − 1) the minimum does not exist and the potential goes to −∞ or becomes complex for ϕ < 1.

In the section 3 we have generalised this model into R + αR n − βR 2−n , with α 1 and β 1.In this case the Ricci scalar is always bigger than zero.The Einstein frame scalar potential is real for all ϕ and it has a minimum for all n.The potential has non-zero value at the minimum, which may become a source of DE.The value of the parameter α is set by the normalisation of primordial inhomogeneities, while the value of the parameter β can be read from the measured value of the present DE energy density.

In the section 4 we have performed numerical analysis of the late-time evolution of the R + αR n − βR 2−n model with dust employed as a matte

field.During the
adiation domination era the ρ M − 3p M = 0, so the effective potential in the Jordan frame obtains

ts vacuum form.Thus the field hold
ϕ = ϕ min .During the dust domination era one finds

FFigure 2 .
2
Figure 2. Analytical results versus numerical simulation of the slow-roll parameters and their F > 1 limit.Dots correspond to N = 60 and N = 50 (left and right dots respectively).At the moment of the horizon crossing the condition F > 1 is satisfied and one can use the analytical solution from the Eq.(2.16) to describe the evolution of space-time during that period.


FFigure 3 .
3
Figure 3. Ana ytical results versus numerical simulatio s.Dots correspond to N = 60 and N = 50 (left and right dots respectively).At the moment of the horizon crossing one obtains F1, so the analytical solution from th g that period.


Figure 4 .
4
Figure 4. Left panel: The analytical so of n at the moment of = 50 or N = 60 (solid green and dashed red lines respectively).Right panel: The U ef f in the vacuum case (solid blue line) and for the dust with ρ M = 2 × 10 10 (dotted red line), ρ M = 4 × 10 10 (thick dotted brown line),