J Inflation and dark energy from the Brans-Dicke theory

We consider the Brans-Dicke theory motivated by the f(R) = R+ αRn − βR2−n model to obtain a stable minimum of the Einstein frame scalar potential of the Brans-Dicke field. As a result we have obtained an inflationary scalar potential with non-zero value of residual vacuum energy, which may be a source of dark energy. In addition we discuss the probability of quantum tunnelling from the minimum of the potential. Our results can be easily consistent with PLANCK or BICEP2 data for appropriate choices of the value of n and ω.


Introduction
Inflationary models [1][2][3][4] are widely accepted as a solution to problems of classical cosmology, such as flatness, horizon and monopole problem. They may also be responsible for the generation of primordial inhomogeneities and their predictions on that are consistent with astronomical observations of cosmic microwave background (CMB) [5][6][7][8]. During the inflationary era one obtains the accelerated growth of the FRW scale factor, which is ended by the so-called reheating characterized by the generation of relativistic particles. Inflation may be generated by e.g. scalar fields minimally or non-minimally coupled to gravity. The latter case will be analysed in this paper.
On the other hand the series of experiments [7][8][9] 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) theory [10][11][12][13][14][15] or the Brans-Dicke field.
The Brans-Dicke theory is widely discussed in the context of cosmic inflation [16], dark energy (DE) [17], stability of stars [18], gravitational collapse and quantum gravity corrections [19,20]. It may be interpreted as a generalisation of f (R) theory with an auxiliary field ϕ := F (R) = f (R), where prime denotes the derivative with respect to the Ricci scalar R. The Brans-Dicke theory may be expressed in the Jordan frame (where the non-minimal coupling of ϕ to the gravity is explicit) or in the Einstein frame (where the conformal transformation modifies the action into its canonical, minimally coupled form). In the classical theory (i.e. without quantum gravity corrections) those two approaches are equivalent to each other.
In [21] we have analysed the R+αR n theory, which appeared to be a realistic inflationary model. Under some assumptions it has a stable minimum which provides the graceful exit and reheating of the universe. Depending on the value of the n parameter its primordial inhomogeneities may be consistent with PLANCK [8] or BICEP2 [5,6] data. Similar analysis was also performed in ref. [22]. By extending this model into f (R) = R + αR n − βR 2−n we -1 -

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have obtained an improved model, which provides a minimum of an Einstein frame scalar potential for all values of the n parameter as well as the non-zero vacuum energy of order of (nδ) 1/(n−1) , which could be responsible for the present acceleration of the Universe.
In this paper we want to extend this analysis into the Brans-Dicke theory by taking non-zero value of the ω BD . We keep scalar potentials in the same form they appeared in [21] and we look for analytical and numerical solutions for the evolution of the background as well as for the primordial perturbations. The motivation to extend the analysis into the Brans-Dicke theory with non-zero ω is the following: first of all this is the most natural extension of the f (R) model. Second of all the non-zero value of ω may have influence on the Einstein frame potential, on the primordial power spectrum of the primordial inhomogeneities, and on the issue of eternal inflation.
The structure of this paper is as follows. In section 2 we analyse the U ∝ (ϕ−1) n/(n−1) model: we discuss analytical inflationary solutions and primordial inhomogeneities. In section 3 we generalise this model to obtain the non-zero vacuum energy of the Einstein frame potential in its minimum. In section 4 we discuss numerical study of the evolution of the model with dust modelling the contribution of matter fields to energy density. In section 5 we investigate the stability of the DE solution and the possibility of quantum tunnelling of ϕ from the minimum of the Einstein frame scalar potential. Finally, we conclude in section 6.

The modified Starobinsky inflation
Let us consider a Brans-Dicke theory in the flat FRW space-time with the metric tensor of the form ds 2 = −dt 2 + a(t) 2 (d x) 2 . Then the Jordan frame action is of the form where ω = const and S m is the action of matter fields. Then, for the homogeneous field ϕ the field's equation of motion and the first Friedmann equation become [10] ϕ + 3Hφ where β = 2ω + 3, U ϕ := dU dϕ and ρ M and P M are energy density and pressure of matter fields respectively. 1 Let us note that U may be interpreted as a energy density, but U ϕ is not an effective force in eq. (2.2). One can define the effective potential and its derivative -the effective force, by where C is an unknown constant of integration. The effective potential shall be interpreted as a source of an effective force, but not as energy density.

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Let us consider a generalisation of the Starobinsky inflation motivated by the f (R) theory from ref. [23], namely (2.6) In the n = 2 case eq. (2.6) recovers the Starobinsky potential. As mentioned in ref. [21] the considered potential has a minimum in the Einstein frame only for certain values of n parameter. For other values of n it may obtain negative or complex values, which requires additional terms in the potential. Thus, in order to obtain the graceful exit and reheating for all n one needs to extend this potential into more general form, which shall be done in further parts of this paper. Different modifications of the Starobinsky model were also discussed in refs. [24,25].

Slow-roll solutions
During the cosmic inflation the field is in its slow-roll regime, which means thatφ 3Hφ andφ Hϕ. In such a case, for the potential from eq. (2.6) the number of e-folds until the end of inflation is equal to The initial conditions were chosen to satisfy N (ϕ = 1) = 0, since ϕ = 1 is a typical value of the field at the end of inflation. This result is approximately the same for the Einstein frame analysis in the slow-roll regime. Let us assume that during the slow-roll era we are in the ϕ 1 regime and that at the moment of 50-60 e-folds before the end of inflation ϕ satisfies ϕ β (2 − n) (1 − n) 2 . Then the Jordan frame potential satisfies U ∝ ϕ n/(n−1) and eqs. , In such a case one obtains a power-law inflation with a constant slow-roll parameter and a scale factor proportional to t 1/ . Hubble parameter needs to be positive in order to obtain the expansion of the universe. Thus, from eq. (2.8) one requires n < 2. Inflationary evolution appears for < 1, which gives the lower limit for n. Finally one obtains the following allowed range for n n ∈ 1 + 1 2β

Einstein frame analysis
The gravitational part of the action may obtain its canonical (minimally coupled to ϕ) form after transformation to the Einstein frame. Let us assume that ϕ > 0. Then for the Einstein frame metric tensorg

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one obtains the action of the form of where∇ is the derivative with respect to the Einstein frame coordinates. Matter fields are now explicitly coupled to ϕ due to the fact that In order to obtain the canonical kinetic term for ϕ let us use the Einstein frame scalar field φ The action in terms ofg µν and φ looks as follows is the Einstein frame scalar potential. Let us note that (ϕU ϕ − 2U ) from eq. (2.2) 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, most 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 paper we will often 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 paper 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 → 0 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 frame analysis, neither in U nor in U eff . The infinite barrier of the Einstein frame potential comes from the singularity ofg µν at ϕ = 0 and therefore is non-physical. Nevertheless the Einstein frame singularity has an analogue in the Jordan frame. In ref. [26] it has been proven that in the similar theory with non-minimal coupling one obtains the curvature singularity when the function of non-minimal coupling is going to zero. This singularity comes from small inhomogeneities or anisotropies of spacetime, which are always present due to quantum fluctuations of the scalaron and matter fields. There are several Brans-Dicke or f (R) models of dark energy (see e.g. ref. [27]), where ϕ(in the perfectly homogeneous and isotropic case) may pass ϕ = 0 and obtain negative values. In the perfectly homogeneous and isotropic case, as well as in the more realistic scenario in which small initial inhomogeneities are included, the ϕ → 0 limit causes serious problems, namely repulsive gravity or singularity respectively. Thus, one shall avoid this limit, which is the case of the model described in the next section. The comparison between Jordan and Einstein frame potentials is shown in figure 1.
The Einstein frame scalar potential for f (R) = R + αR n as a function of ϕ has the following form  Figure 1. Upper panels show the Einstein frame scalar potential from the eq. (3.1) for different values of n. Inflation takes place for ϕ 1 and it ends when ϕ reaches the local minimum of V (ϕ) at ϕ = ϕ min 1. The ϕ < 0 regime is separated from the minimum by the infinite wall of the potential at ϕ = 0. Lower left panel presents the Jordan frame scalar potential U (ϕ) for different n. The U (ϕ) always decreases with ϕ and it has no minimum. The lower right panel shows the effective potential U eff (ϕ) (which is the source of the effective force in the EOM) for different values of ρ. The minimum is deeper and it is getting closer to ϕ = 1 as the ρ M starts to dominate over U (ϕ). Besides U eff these plots are β independent, since we are using the Jordan frame field as a variable.
where the last term parametrizes the deviation from the Starobinsky potential. Let us note that the slow-roll solutions for the Brans-Dicke field, Hubble parameter and the number of e-folds are the same in Einstein and Jordan frames. In the φ/ √ β 1 limit one finds ϕ 1 + φ 2/β and the Einstein frame potential takes the following form The φ field is minimally coupled to gravity, so the model evolves like in the V (φ) = λφ n n−1 in the GR frame.
For ϕ < 0 one needs to define the Einstein frame according to the following procedure: a) Define the Einstein frame metric tensorg µν = −ϕg µν . Then dt = √ −ϕdt,ã = √ −ϕa, where √ −ϕ remains real. b) Rewrite the action from eq. (2.1) usingg µν which gives the canonical for of the GR action with a negative sign and the BD field part with a ghost-like kinetic term.
The problem of initial conditions. Let us note that due to stronger limits on the tensorto-scalar ratio r the PLANCK data favour the plateau-like potentials (like in Starobinsky inflation) and disfavour the power-law potentials (like m 2 φ 2 ). Nevertheless, the plateau-like models of inflation struggle with some difficulties mentioned in ref. [28]. One of the problems, which appears within the analysis of the Starobinsky model, is the fact that the Einstein frame potential is limited from above by the scale of the order of M 2 M 2 p M 4 p . Thus, if one would set initial conditions at the Planck scale the potential term would be always subdominant and the (∂ i φ) 2 term may dominate the universe, which would lead to strong inhomogeneities. On the other hand potentials without the upper bound shall generate the Planck scale initial conditions to be (∂ i φ) 2 ∼φ 2 ∼ V (φ), 2 so the potential term can dominate the evolution of the field and homogenize the universe.
The modification of the Starobinsky potential that we consider in this paper solves the problem of initial conditions. For n = 2 the Einstein frame potential is of the form This potential is not limited from above, so V can have large contribution to the energy density at the Planck scale. Another advantage of this model is that inflation can start for any sufficiently big ϕ and therefore to obtain successful inflation without significant inhomogeneities one needs only one bubble at the Planck scale in which V (φ) dominates.
Another issue related to initial conditions at the Planck scale is the eternal inflation. In the Starobinsky model for sufficiently big φ one obtains δφ q > δφ c , where δφ q and δφ c are changes of the vev of the scalaron according to quantum fluctuations and classical evolution respectively. If δφ q > δφ c then the evolution of the field is dominated by quantum fluctuations, which may lead to the so-called eternal inflation. For the slow-roll approximation one finds δφ q V /3/2π and δφ c V φ /V . Let us define ϕ q , such as δφ q = δφ c for ϕ = ϕ q and ϕ p , for which V (ϕ p ) = M 4 p . Then ϕ q > ϕ p for In such a case the evolution of the scalaron below the Planck scale is always classical and one shall never obtain the eternal inflation. In the Brans-Dicke theory one can always chose β, which satisfies eq. (2.18), which is an advantage over the f (R) theory.

The generation of primordial inhomogeneities
The simplest way to calculate the power spectrum of primordial inhomogeneities is to quantize curvature perturbations in the Einstein frame. As shown in ref. [10] this procedure performed in the slow-roll regime gives the same results as the Jordan frame quantization. Then one obtains (2.21) The spectral index n s and the tensor-to-scalar ratio r are approximately equal to The lower left panel of figure 2 shows the power spectrum on the (n s , r) plane for the inflationary potential (2.6) for different n and β. Namely, every point on that plot corresponds to the unique combination of (n, β). We compare the results with PLANCK and BICEP2 experimental data (orange and blue areas respectively). The upper panels of figure 2 show r and n s as a function of n for different β. Note that the n-dependence of n s gets weakened for bigger β. Figure 3 shows areas in the parameter space (n, β), which give results consistent with PLANCK and BICEP2. Note that for β 70 one can obtain result which fit the data from both experiments. 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 spectrum to obtain α as a function of n and β. The result obtained from eqs. (2.6), (2.7), (2.21) in the slow-roll regime is plotted in figure 2.
The issue of primordial non-Gaussianities in the Brans-Dicke theory was widely analysed in ref. [30]. The authors argue that |f local NL | 1, since in the slow-roll regime of the single field inflation the f local NL is proportional to slow-roll parameters.

The dark energy model
Deficiencies of the model discussed in the previous section can be bypassed by considering modification of the scalar potential motivated by the f (R) = R + αR n − δR 2−n theory [21], namely where (3. 2) The R function could be interpreted as the Ricci scalar in the f (R) theory. However in the considered model (for β = 3) it has no connection with the curvature. Let us require α 1, δ 1 and αδ 1. Then in the δ → 0 limit one restores the inflationary potential from section 2. Both models give the same results under the slow-roll approximation and they generate the same power spectrum of primordial inhomogeneities during inflation. However, the potential from eq. (3.1) has several advantages: it has a minimum at ϕ ∼ 1 for all values of n in which it has a non-zero vacuum energy. In order to satisfy experimental data one needs δ ≪ 1. Let us note that the DE solution lies in the stable minimum of the Einstein frame potential. This prevents the field from rolling down to the negative values of ϕ at small energies, which is the case in e.g. R − δR 2−n model described in ref. [27], in which the scalar potential obtains real values only for ϕ < 1. This is another advantage of the model considered in this section, for which all values of ϕ ale allowed. Different model which combines Inflation and DE was described in ref. [32].
The Einstein frame potential has a minimum at ϕ min 2 n (n − 1)(1 + 2nαδ). The minimum is slightly shifted with respect to ϕ = 1, which is the GR vacuum case. The value Hence, this model predicts some amount of vacuum energy. Let us clarify that ϕ min could be considered as a local minimum of V . The R function (and therefore the potential U ) has no global minimum, its minimal value is equal to 0 (at the ϕ → −∞ limit) and it continuously grows with ϕ. An example of Jordan and Einstein frame potentials and minima of V (ϕ) are plotted in figure 1. The existence of a stable minimum is one of the main differences between this model and DE models motivated by f (R) = R − δR 2−n theory, in which the auxiliary field may roll down towards negative ϕ [27]. As we will show the minimum of the Einstein frame potential (visible in figure 1) prevents the ϕ from obtaining negative values for any initial conditions with ϕ(0) > 0 and real values of the Hubble parameter.

Viability of the dark energy model
Let us denote the present time as t 0 . In order to obtain successful model of dark energy one needs to satisfy several conditions: 1) The kinetic term of the ϕ is proportional to ϕ −1 . Thus, to avoid the ghost state for β > 0 one needs ϕ 0 := ϕ(t 0 ) > 0. This condition is satisfied as long as the field lies in the minimum of its potential after inflation. The possibility of passing ϕ = 0 via the classical evolution or quantum tunnelling is described in section 5.
2) To avoid the negative mass square for a scalar field degree of freedom one needs which means that R 0 > 0. This condition is satisfied for any values of ϕ.

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3) The model needs to obtain the correct low-energy limit and to satisfy consistency with local gravity constraints. This means that the low-energy action shall be of the form of After the ϕ field is stabilised in its minimum (which in the dust domination era is exactly GR minimum) it produces the vacuum energy, which is a source of Λ. The S m comes from radiation and dust produced during the reheating of the universe. l satisfies conditions for a viable DE model.

4)
The model needs to have sufficiently long period of dust domination, during which the energy density of DE and radiation are negligible. In ref. [33] it was proven that DE models which base on R − δR 2−n Lagrangian density (including Brans-Dicke models with the Jordan frame scalar potential motivated by such a theory) may suffer from lack of the dust-dominated era. After the radiation-domination epoch one obtains domination of dust and DE with the Einstein frame energy fractionΩ φ = 1 3β . Problems with the correct form of the dust-domination era exclude models with 2 − n < 0 and with 2 − n > 1. Luckily in our model 0 < 2 − n < 1 and one can obtain the correct form of the dust-dominated era, which is presented in figure 5. The pure R − δR 2−n model suffers from two issues: (a) the ϕ → 0 limit, (b) their scalar potential is also not well defined for all ϕ. None of these are the case for the model considered in this paper.

Numerical analysis of the dark energy model
where the index " N " denotes the derivative with respect to N . Since in the Jordan frame eq. (2.4) 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 contribute to the U eff , so the field is not shifted from the minimum of the potential V . However, during the dust domination era the U eff is modified and ϕ oscillates around ϕ = 1. The evolution of ϕ and ϕ N during radiation, dust and DE domination eras is presented in figure 4. The evolution of the Hubble parameter and ρ M /3 is plotted in figure 5. We have assumed that the Universe is initially dominated by radiation and that the field starts from the ϕ = ϕ min . When the dust starts to dominate the field rolls up to ϕ = 1, which is the GR limit of the theory. When the dust becomes subdominant the ϕ rolls to ϕ min and one obtains the dark energy with the barotropic parameter ω = −1. The evolution of ϕ, ϕ N , H and energy densities of ϕ and dust in the f (R) case (i.e. for β = 3) have been presented in figure 8 and 9 in ref. [21]. As shown in figure 5 the energy density of ϕ obtains the constant value when ϕ = ϕ min . During that period one finds H, ϕ = const, which implies that

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where Ω DE is a density parameter of DE. This equation is valid when DE completely dominates over the dust. From eq. (4.3) one finds the connection between theoretical predictions of this model and astronomical observations. Another issue worth detailed analysis is the reheating after inflation. This has been analysed in the context of more sophisticated f (R) theories, which also combine inflationary paradigm and DE solutions (see refs. [14,15] for details). These models are based on the Starobinsky inflation limit, which is different from our approach. In our paper problems of eternal inflation and initial conditions at the Planck scale are solved by the modification of the Starobinsky model, which is not the case in refs. [14,15]. The other difference is the existence of local maximum of the Einstein frame potential, which may lead to e.g. topological inflation.

Stability of the vacuum
As discussed in previous sections, classical evolution of the field ϕ brings it to rest at the minimum of its effective potential near ϕ ≈ 1. However the Jordan frame potential U (ϕ) (shown in the middle row of figure 1) which gives the energy density, has no minimum and decreases towards smaller values of field ϕ. This indicates that the vacuum at ϕ ≈ 1 may actually be a metastable minimum and the true vacuum may be located in the region of negative values of ϕ. In this section we shall check if one can find initial conditions that -11 - could result in classical evolution ending at negative values of ϕ. We will also discuss whether quantum tunneling can allow our false vacuum to decay into a deeper global minimum.
First we will show that classically the field can cross ϕ = 0 only if it starts from unphysical initial conditions. In our model the field ϕ can evolve towards negative values, instead of rolling to the minimum of the effective force, only if the Hubble parameter (4.2) is imaginary. This happens because the imaginary Hubble parameter changes the sign of the kinetic part of scalar field equation of motion (4.1), and the effective force (shown in lower right panel of figure 1) pushes the field away from ϕ = 1. In our model the numerator of the squared Hubble parameter (4.2) is always positive. Thus during evolution the sign of H 2 can only be changed by its denominator which would correspond to a point of infinite friction for the field. Hence, obtaining a ghost field is possible only if we set it as such from the beginning of its evolution.
To discuss quantum tunneling we will use the standard formalism [34,35], assuming that vacuum decay proceeds through nucleation of true vacuum bubbles within our false vacuum. Such bubble is an O(4) symmetric scalar field configuration ϕ = ϕ(τ ), with the metric given by ds 2 = dτ 2 + r(τ ) 2 (dΩ) 2 . Here dΩ is an infinitesimal element of the 3D sphere, and r(τ ) is the radius of that sphere. The resulting metric tensor is of the form of the F RW metric with the curvature parameter k = +1. Euclidean action in Jordan Frame takes the form where R = 6 rττ r + rτ r 2 − 1 r 2 . The scalar field equation of motion reads The first Friedmann equation reads

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and differs from (2.2), (2.3) only by the term 1/r 2 corresponding to the curvature of the sphere, and by the different sign of the effective force. The appropriate boundary condition at the true vacuum (negative ϕ) is [36]: When the field evolves from such initial conditions in inverted effective potential from figure 1, it is clear the field will simply roll toward smaller values. And so no tunneling towards positive ϕ is possible.

Conclusions
In this paper we have analysed the Brans-Dicke theory motivated by f (R) = R+αR n −δR 2−n model (with α 1 and δ 1) and we have obtained two crucial features of the observable Universe in a single framework: we show how to obtain successful inflation and the nonzero residual value of the Ricci scalar in an extension of the BD Starobinsky-like model. In section 2 we have shown that the high energy limit of our theory, namely the Brans-Dicke theory with a Jordan frame scalar potential proportional to (ϕ − 1) n/(n−1) can be easily consistent with PLANCK or BICEP2 data for appropriate choices of the value of n and β. We have shown that the inflationary model considered in this paper does not suffer from the problem of initial conditions and that for certain values of n and β parameters the eternal inflation would not appear even for energies close to the Planck scale.
In section 3 we have considered a full theory, with a potential from eq. (3.1). In this case 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 δ (as a function of n and β) can be read from the measured value of the present DE energy density.
In section 4 we have performed numerical analysis of the late-time evolution of the model with dust employed as a matter field. During the radiation domination era the ρ M −3p M = 0, so the effective potential in the Jordan frame obtains its vacuum form. Thus the field stays at ϕ = ϕ min . During the dust domination era one finds ϕ = 1, which corresponds to the GR limit of the theory. When matter starts to be subdominant the ϕ rolls to its minimum in ϕ = ϕ min 1. Even before that moment the energy density of the scalar field becomes constant and the energy density of the ϕ evolves like DE with barotropic parameter ω ϕ = −1.
Finally, we have considered the possibility of quantum tunnelling of the Brans-Dicke field to the region of its negative values. Inspection of the equations of motion in the presence of gravitational background leads to the conclusion, that there are no bounce solutions interpolating between positive and negative values of the BD field. On the other hand there exist rolling solutions connecting the two regions. However, they correspond to unphysical initial conditions. Hence one is led to the conclusion, that the vacuum near ϕ = 1 is stable.