Minimal inflationary cosmologies and constraints on reheating
MMinimal inflationary cosmologies and constraints on reheating
Debaprasad Maity ∗ and Pankaj Saha † Department of Physics, Indian Institute of Technology Guwahati.Guwahati, Assam, India (Dated: February 8, 2019)
Abstract
With the growing consensus on simple power law inflation models not being favored by the PLANCKobservation, dynamics for the non-standard form of the inflaton potential gain significant interest in therecent past. In this paper, we analyze in great detail classes of phenomenologically motivated inflationarymodels with non-polynomial potential which are the generalization of the potential introduced in [1]. Afterthe end of inflation, inflaton field will coherently oscillate around its minimum. Depending upon the initialamplitude of the oscillation and coupling parameters standard parametric resonance phenomena will occur.Therefore, we will study how the inflationary model parameters play an important role in understanding theresonant structure of our model under study. Subsequently, the universe will go through the perturbativereheating phase. However, without any specific model consideration, we further study the constraints onour models based on model independent reheating constraint analysis.
Keywords: ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] F e b ontents I. Introduction II. The Model n s , r, dn ks ) 6C. End of inflation and general equation of state 11 III. Pre-heating: parametric resonance and their dependence on inflation scale φ ∗ IV. Model independent constraints from reheating predictions V. Summary and Conclusion VI. Acknowledgement A. Towards derivation of our model potential B. Validity of /N expansion for φ ∗ < M p References I. INTRODUCTION
The inflation [2–4] is a model independent mechanism proposed to solve some of the outstandingproblems in standard Big-Bang cosmology. It is an early exponential expansion phase of ouruniverse, which sets the required initial condition for the standard Big-Bang evolution. Over theyears large number of models have been introduced to realize this mechanism [5], and explain thecosmological observations [6]. Out of the large number of models, a particularly interesting class ofmodels that have recently been studied is called α -attractor[7]. It has gained significant attentionsbecause it unifies a large number of existing inflationary models. In this paper, we will introducenew classes of inflationary models generalizing the model proposed in [1]. In order to explain theobservation, we phenomenologically consider classes of non-polynomial potentials, which could bederived from a general scalar-tensor theory in certain limit (shown in appendix-A). At this pointlet us motivate the reader mentioning the important points of our study. It is well known thatthe general power law canonical potentials of the form V ( φ ) ∼ | φ | n are not cosmologically viablebecause of its prediction of large tensor to scalar ratio. In addition, because of super-Planckian2alue of the field excursion, the effective field theory description may be invalid. One of our goals inthis paper is to circumvent the above mentioned problems in the framework of canonical scalar fieldmodel. Therefore, we generalize the power law form of the potential to non-polynomial form sothat it can fit well with the observation, and also the inflaton assumes sub-Plankian field excursion.After the inflation, the inflaton will go through the oscillatory phase. Initially because of largeoscillation amplitude, the inflaton can decays through parametric resonance depending upon theinflaton coupling with the reheating fields. Considering a specific model ( n = 2), we figure outthe parameter region where broad parametric resonance happens. Our analysis shows that as wedecrease the inflationary energy scale, the instability bands evolve into wider band, thereby, enhancethe strength of the resonance. However, the number of stability/instability region decreases withdecreasing the scale we introduced in the model. Detail analysis of this phenomena will be done inour future work. After few initial oscillations, resonant decay will naturally reduce the amplitudeof the inflaton oscillation significantly. Therefore, the perturbative reheating starts to play its roletill the radiation domination begins. In this paper we will not discuss about the usual perturbativereheating. However, we should mention that to the best of our knowledge detailed analysis of thisperturbative phase for arbitrary power law inflaton potential has not been done. We defer thisstudies for our future publication. However, what we have done instead is the model independentreheating constraint analysis based on the works [8, 10], and understand the possible constraintson the model for the successful reheating to be realized.We structured our paper as follows: In section-II, we generalize the model introduced in [1],and study in detail the cosmological dynamics of inflaton starting from inflation to reheating. Wecompute important cosmological parameters such as scalar spectral index ( n s ), the tensor to scalarratio ( r ), and the spectral running ( dn ks ) and fit with the experimental observations. From thosecosmological observations, we constrain the parameters of our model. After the end of inflation,the inflaton starts to have coherent oscillation around the minimum of the potential, during whichthe universe will undergo reheating phase. We also compute the effective equation of state of theoscillating inflaton for our subsequent studies. In section-III, we will discuss about how the newscale φ ∗ controls the resonance structure during during the first few oscillation of the inflationfield. This is very important for pre-heating phenomena. For this we will only consider one singlemodel with n = 2. Further detail of this pre-heating phase will be discussed elsewhere. In section-IV, we have done the model independent reheating constraint analysis considering the importantconnection between the end of reheating and the cosmic microwave background (CMB) anisotropy.Finally we concluded and discussed about our future work.3 - ϕ ( ϕ ) ϕ * FIG. 1:
An illustration of the dependence of the shape of the potential with the scale φ ∗ for n = 2 , λ = 1 when the two classof the model considered became identical. As we decrease φ ∗ , the CMB normalization changes the parameter m such that theheight as well as the width of the potential decreases. This fact will result in decrease in the field excursion as well as the scalarto tensor ratio and also have significant effect on the post-inflationary dynamics. II. THE MODEL
As we have discussed in the introduction and also tried to construct in the appendix, our startingpoint in this section is the non-polynomial potential which has dominant power law behavior aroundits minimum. Therefore, we will start by considering the following phenomenological forms of thepotential, V ( φ ) = λ m − n φ n (cid:16) φφ ∗ (cid:17) n λ m − n φ n (cid:18) (cid:16) φφ ∗ (cid:17) (cid:19) n , (1)In the above form of the potentials, we have introduced two free parameters ( m or λ, n ). Where,the parameter λ is defined for n = 4, which has been studied as minimal Higgs inflation in [1].For other value of n , we can set λ = 1. For large value of φ ∗ the potential is plateau like andthe associated inflationary scale is Λ = λm − n φ n ∗ . A simple illustration of the above form of thepotentials is shown in the fig.1. Through out the paper, we will refer type-I for the first form andtype-II for the second form of the potential. One can further generalize our model by consideringthe potential to be dependent only upon the modulus of the inflaton field. Therefore, all the oddvalues of n can be included. For the sake of simplicity we will stick to only even values of n .Another simple generalization of our model can be done by defining V ( φ ) q as a new potential.Where, q would a new parameter. 4 . Background Equations In this section we will study in detail the background dynamics using the above form of thepotentials. We will start with the following action, S = (cid:90) d x √− g (cid:34) M p R − ∂ µ φ∂ µ φ − V ( φ ) (cid:35) (2)Where M p = √ πG is the reduced Planck mass. Assuming the usual Friedmann-Robertson-Walker(FRW) background ansatz for the space-time ds = dt − a ( t ) ( dx + dy + dz ) , (3)the system of equations governing the dynamics of inflaton and scale factor are3 M p H = 12 ˙ φ + V ( φ ) (4)2 M p ˙ H = − ˙ φ (5)¨ φ + 3 H ˙ φ + V (cid:48) ( φ ) . (6)Where, the usual definition of Hubble constant is H = ˙ a/a . As we have seen, our potential isasymptotically flat for large field value compared to φ ∗ . This is condition which is required for theinflationary dynamics is automatically satisfied. The flatness conditions for the potential duringinflation are written in terms of the slow-roll parameters, which are defied as (cid:15) ≡ M p (cid:18) V (cid:48) V (cid:19) = n M p φ n ∗ φ ( φ n ∗ + φ n ) φ ∗ n M p φ ( φ ∗ + φ ) η ≡ M p (cid:18) V (cid:48)(cid:48) V (cid:19) = nM p φ n ∗ (( n − φ n ∗ − ( n +1) φ n ) φ ( φ n ∗ + φ n ) φ ∗ nM p ( φ ∗ ( n − − φ ) φ ( φ ∗ + φ ) . (7)During inflation (cid:15) (cid:28) | η | (cid:28)
1. Therefore, the end of inflation is usually set by the condition (cid:15) = 1. Let us also define a higher order slow-roll parameter related to the third derivative of thepotential for spectral running. The expression for the higher order slow-roll parameter is as follows: ξ ≡ M p (cid:18) V (cid:48) V (cid:48)(cid:48)(cid:48) V (cid:19) = n M p φ n ∗ (( n − n +2 ) φ n ∗ − ( n − ) φ n ∗ φ n + ( n +3 n +2 ) φ n ) φ ( φ n ∗ + φ n ) φ ∗ n M p ( φ ∗ ( n − n +2 ) +3 φ ∗ (2 − n ) φ +12 φ ) φ ( φ ∗ + φ ) . (8)In addition to provide the successful inflation, all the aforementioned slow-roll parameters playvery important role in controlling the dynamics of cosmological perturbation during inflation.An important cosmological parameter which quantifies the amount of inflation is called e-folding5umber ( N ), which plays crucial role in solving the horizon and flatness problem of standardBig-Bang. The e-folding number is expressed as as N = ln (cid:18) a end a in (cid:19) = a end (cid:90) a in d lna = t end (cid:90) t in Hdt (cid:39) φ end (cid:90) φ in √ (cid:15) | dφ | M p . (9)As we have mentioned the inflation ends when (cid:15) = 1, and one can use the eq.(9) to find the valueof the inflaton at the beginning of the inflation. By solving the aforementioned condition, we canexpress the e-folding number N into the following form, N = φ ∗ nM p (cid:104) n +2) ( ˜ φ ( n +2) − ˜ φ ( n +2) end ) + ( ˜ φ − ˜ φ end ) (cid:105) (cid:39) φ ∗ nM p n +2) ˜ φ ( n +2) φ ∗ nM p (cid:104) ( ˜ φ − ˜ φ end ) + ( ˜ φ − ˜ φ end ) (cid:105) (cid:39) φ ∗ nM p ˜ φ . (10)Where we have defined, ˜ φ = φ/φ ∗ . In the above expressions for N , we have ignored the contributioncoming from φ end , and also the squared term. We have numerically checked the validity of thoseexpressions for a wide range of value of φ ∗ ≤ O ( M p ). From cosmological observations one needs N (cid:39) −
60, such that the scales of our interest in CMB were in causal contact before theinflation. By using the above mentioned boundary conditions for the inflaton we have solved forthe homogeneous part of inflaton φ ( t ) and the scale factor a ( t ). One particular solution has beengiven in fig.6, with a specific value of the efolding number. Next we study the perturbation aroundinflationary background and derive the relevant cosmological parameters associated the variouscorrelation functions of fluctuation. B. Computation of ( n s , r, dn ks ) As we described in the introduction, the very idea of inflation was introduced to solve someoutstanding problems of standard Big-Bang cosmology. Soon it was realized that inflation alsoprovides seed for the large-scale structure of our universe through quantum fluctuation. All thecosmologically relevant inflationary observables are identified with various correlation functionsof those primordial fluctuations calculated in the framework of quantum field theory. We havecurvature and tensor perturbation. The two and higher point correlation functions of those fluctu-ation are parametrized by power spectrum(see, [11–13] for a comprehensive review of CosmologicalPerturbation Theory). The scalar curvature power spectrum is given by P R = 18 π (cid:15) H M p (cid:12)(cid:12)(cid:12)(cid:12) k = aH = 112 π V M p ( V (cid:48) ) . (11)Once, we know the power spectrum, cosmological quantity of our interests are the spectral tilt andits running. During inflation a particular inflaton field value corresponds to a particular momentum6ode exiting the horizon. Hence by using the following relation to the leading order in slow-rollparameters, dd lnk = ˙ φH ddφ , one obtains the following inflationary observables, n s − ≡ d ln P R d lnk = − (cid:15) + 2 η (12) dn ks ≡ dnd lnk = − ξ + 16 (cid:15)η − (cid:15) . (13)Similarly we can compute the tensor power spectrum P T for the gauge invariant tenor perturbation h ij . To quantify this, standard practice is to define tensor-to-scalar ratio r = P T P R = 16 (cid:15). (14)Once we have all the expression for cosmological quantities in terms of slow roll parameters, byusing eqs.(7,10), and considering φ ∗ ≤ O (1) in unit of M p , we express ( n s , r, dn ks ) in terms of n , N and φ ∗ , as 1 − n s = n +1)( n +2) 1 N N ; dn ks = − (2+3 n + n )( n +2) N − N (15) r = n (cid:16) φ ∗ M p (cid:17) n ( n +2) n ( n +2)] n +1)( n +2) N n +1)( n +2) φ ∗ M p n N At this point, we want to emphasize the fact that the above expansions for all the spectralquantities in large- N limit may not always be valid for all inflaton field values as has been pointedout recently in [14]. This fact is indeed true if we look at the figs.(4,5), where, values of ( n s , r ) aredeviating from the analytic expressions eq.15 for large φ ∗ > M p . Therefore, for our model, aboveexpansion in large- N for ( n s , r, dn ks ) are valid only in the regime of small-field inflation. Along theline of argument provided in [14], we have analytically shown our claim for n = 2 in appendix-B.From the above analytic expressions for ( n s , r, dn ks ), some important observations are as follows:For type-II class of models, we see that the value of ( n s , dn ks ) are insensitive to the value of n . Thisfact can also bee seen from the fig.(5), where, no approximation has been made. In particular oneobserves that for φ ∗ (cid:38) M p , value of ( n s , r ) start to deviate form each other for different valuesof n . However, in general tensor to scalar ratio behaves as r ∝ n (1 / to the leading order in N .Therefore, form the PLANCK observation considering the upper bound r < .
07, we can put theupper limit on n for a fixed value of φ ∗ . From the current observations, it turns out to be verydifficult to uniquely fix the form of the potential. Therefore, we need more theoretical inputs tofigure out the full form of the potential. For type-I models, all quantities are dependent on n ,7hich can also be seen from fig.(4). However, an important fact emerges in the limit n → ∞ forType-II potentials. If we take n → ∞ limit, the expressions of ( n s , r, dn ks ) reduce to1 − n s → N ; r → (cid:18) φ ∗ M p (cid:19) N ; dn ks = − N , (16)which can be identified as a particular model within the class of recently proposed ‘ α -attractor’[7].We have numerically checked the aforementioned asymptotic limit of n s in terms of n , as can beseen in fig.(2). As one observes, from eq.(16), for N = 50, the scalar spectral index n s → . n → ∞ , which is the central value of PLANCK observation. Therefore, for a wide rangeof parameter values, we have infinite possible models corresponding to n = 2 , , , ... , which cansuccessfully explain cosmological observations made by PLANCK [6]. From the field theory point ofview, UV completion of our model is an important issue. Specifically the supergravity formulationof those form of the potential could be an important direction to study. We defer this for ourfuture studies. ϕ * = ϕ * = ϕ * =
10 10 20 30 40 500.9610.9620.9630.9640.9650.9660.967 n n s (a)Asymptotic behavoir of n s for model I ϕ * = ϕ * = ϕ * = - n r (b)Asymptotic Behavior of r for Model II FIG. 2:
Variation of n s with n for Model-I with n = 2 , , , ... , as we increase n , n s → .
96 and variation of r with n forModel-II, it is evident that the value of r will satisfy the PLANCK bound for φ ∗ (cid:28) So far all we have discussed is directly related to the cosmological observation made byPLANCK. Another important quantity of theoretical interest we would like to compute is Lythbound [15] ∆ φ . This quantity measures the difference of field values which is traversed by theinflaton field during inflation. This is so calculated that for a particular model ∆ φ is the maxi-mum possible value for a particular efolding number. Inflation is a semi-classical phenomena. It isbelieved that natural cut off scale for any theory minimally or non-minimally coupled with gravityis Planck scale M p . Therefore, amount of inflaton field value can naturally be a good measure totell us the effective validity of a model under study in the effective field theory language. Hence8 .960 0.965 0.970 0.975 0.9800.0010.010 n s Log r N = = Planck TT + lowP + lensing + ext + BK14 n = = = = (a)Model I n s Log r N = = Planck TT + lowP + lensing + ext + BK14 n = = = = (b)Model II FIG. 3:
Plot of n s vs r when φ ∗ = 0 . M p for the two potential plotted on Planck 2015 background, as we have seen in ourcalculation that, for the second type of potential, the calculated quantities are largely independent of n . While for the firstpotential type the change of n has significant effect on the spectral quantities. - - - - ϕ * n s n = = = = (a) n s vs log φ ∗ for Model-I - - - - ϕ * r n = = = = (b) r vs log φ ∗ for Model-I FIG. 4:
The dependence of n s and r on the scale φ ∗ for fixed number of efolding ( N = 50) for Potential of type-I. - - - - ϕ * n s n = = = = (a) n s vs log φ ∗ for Model-II - - - - ϕ * r n = = = = (b) r vs log φ ∗ for Model-II FIG. 5:
The dependence of n s and r on the scale φ ∗ for fixed number of efolding ( N = 50) for Potential of type-II the calculated expression for the field excursion in terms of N and φ ∗ are:∆ φ (cid:38) M p N (cid:114) r M p (cid:16) nn +2 (cid:17) n ( n +2)] ( n +1 n +2 ) N n +2) M p (cid:16) φ ∗ M P (cid:17) N (17)All the quantities we have discussed so far is independent of m or λ . (At this point let us again9 ∗ M p n Model I Model II n s r dn ks ∆ φ n s r dn ks ∆ φ × − -0.00066 0.39 0.969 4 × − -0.00066 0.394 0.966 2 × − -0.00066 0.12 0.969 5 × − -0.00060 0.476 0.965 3 × − -0.00069 0.06 0.969 7 × − -0.00060 0.518 0.964 1 × − -0.00070 0.04 0.969 8 × − -0.00060 0.551 2 0.969 4 × − -0.0006 3.53 0.969 4 × − -0.0006 3.534 0.966 9 . × − -0.0007 2.13 0.969 6 × − -0.0006 4.06 0.964 3 . × − -0.0007 1.47 0.969 7 × − -0.0006 4.38 0.964 1 . × − -0.0007 1.1 0.969 8 × − -0.0006 4.7TABLE I: The spectral quantities for different values of n for 50 efolding. The two values of φ ∗ are chosen to illustrate thatwe can have both small field and large field inflation depending on the value of φ ∗ . The general trend for the variation of thesequantities with φ ∗ is illustrated in the figures (4-5) remind the reader that for n (cid:54) = 4, λ is a dimensionless quartic coupling parameter. While for n (cid:54) = 4, m is dimensionful parameter, and we set λ = 1). However, comparing the inflationarypower spectrum with the PLANCK normalization we will determine the value of m or λ and thencalculate all the other quantities of our interest. The expression for the power spectrum of thecurvature perturbation is P R = λ π n (cid:16) mM p (cid:17) − n (cid:16) φ ∗ M p (cid:17) n n +2 [ n ( n + 2) N ] n +1)( n +2) π λ √ n (cid:16) mM p (cid:17) (4 − n ) (cid:16) φ ∗ M p (cid:17) ( n − N = 2 . × − . (18)As mentioned we considered the PLANCK normalization: P R at the pivot scale k/a = 0 . M pc − ,and corresponding estimated scalar spectral index is n s = 0 . ± . n s , r ) space and compared with the experimental values n s = 0 . ± . r < .
11 in fig.(3). In the table-(III), we have given some sample values ofall the cosmologically relevant quantities for different values of theoretical parameters. As wehave mentioned already, we found infinitely many model potentials with a universal shape. Mostinteresting case would probably be for n = 4. In the recent paper [1], it has been identified as aminimal Higgs inflation. Of course this identification is not straight forward. However for smallfield value, we can certainly Taylor expand the potential, and identify the coupling λ as Higgsquartic coupling which can be set to its electroweak value. However, renormalization analysisneeds to be done in order to do this identification.At this point let us re-emphasize the fact that the observation made by PLANCK strongly10 w = n − n +2 p = 3(1 + w ) p from fittingModel I Model II2 0 3 3.12 3.124 disfavors the usual power law inflation with n ≥
2. In this paper we showed that problems of thosepower law inflationary models can be cured with a non-polynomial generalization of the potential.We plotted the dependence of the ( n s , r ) on the inflationary energy scale φ ∗ in figs.(4-5). For bothtype of models, it is clearly matching with our approximate analytic expression eq.(16). In thesubsequent section we will see how the reheating prediction will constrain the value of N dependingupon the reheating temperature consistent with PLANCK. C. End of inflation and general equation of state
In this section we will be interested in the dynamics of the inflaton field after the inflation.During this phase the inflaton field oscillates coherently around the minimum of the potential.At the beginning the oscillation dynamics will naturally be dependent upon the inflation scale φ ∗ because of the large amplitude. This is the stage during which non-perturbative particle productionwill be effective. Therefore, resonant particle production will take place and conversion of energyfrom the inflaton to matter particles will be highly efficient. This phenomena is usually known aspre-heating of the universe. In this section we will discuss about the late time behaviour of theinflaton, specifically focusing on the dynamics of the energy density of the inflaton field. After themany oscillations, when the amplitude of the inflaton decreases much below the φ ∗ , the dynamicswill be controlled by usual power law potential. As we have emphasized the coherent oscillation isvery important in standard treatment of reheating. For any models of inflation this is thought to bean important criteria to have successful reheating. In this section, we will first discuss the evolutionof inflaton and its energy density in full generality for all classes of potentials. As mentioned before,at late time the potential can be approximated as V ( φ ) = λm − n φ n . (19)In cosmology for any dynamical field such as inflaton, one usually defines the equation of stateparameter w . For the oscillating inflaton, when the time scale of oscillation about the minimum11
06 107 108 109 110 111 112 t0.050.100.15 ϕ ( t ) Model - I, n = λ = α =
10 20 30 40 50 ln ( aa ini ) - - - - ln ( ρρ ini ) FIG. 6:
The evolution of the scalar field with time (in arbitrary unit)and with scale factor for Potential of type I with n = 2,and φ ∗ = 0 .
01. time is measure in the unit of ( m φ ∗ ) of a potential is small enough compared to the background expansion time scale, by using virialtheorem effective equation of state for a potential of the form V ( φ ) ∝ φ n can be expressed as[16] w ≡ P φ ρ φ (cid:39) (cid:104) φV (cid:48) ( φ ) (cid:105) − (cid:104) V (cid:105)(cid:104) φV (cid:48) ( φ ) (cid:105) + (cid:104) V (cid:105) = n − n + 2 . (20)Therefore, in an expanding background, the evolution of energy density ρ φ of the inflaton averagedover many oscillation will follow, ˙ ρ φ + 3 H (1 + w ) ρ φ = 0 . (21)At late time we relate the energy density( ρ φ ) of the universe (assuming that the universe is domi-nated by a single component) and the scale factor ( a ) as ρ φ ∝ a − w ) = a − p . (22)In the table-II, we provide some theoretical as well as numerically fitting values corresponding tothe equation of state parameter w of the inflaton and the power law evolution of the energy densitynamely the value of p .In the following sections will be considering those equation of state parameters and studytheir role in the subsequent cosmological evolution. We will first discuss about the constraint onreheating phenomena by taking the model independent approach, where explicit dynamics duringreheating phase will not be considered. III. PRE-HEATING: PARAMETRIC RESONANCE AND THEIR DEPENDENCE ON IN-FLATION SCALE φ ∗ Reheating is an important phase of the early universe, when all the matter field is assumedto be produced from the decay of inflaton. Initial study on this mechanism was based on the12erturbative quantum field theory [17–20]. However, it was soon realized that this approach maynot be efficient enough for successful reheating. In general reheating phenomena is a complicatednon-linear dynamics of inflaton coupled with matter fields at finite temperature and the process oftheir thermalization. In the seminal work by Kofman, Linde and Starobinsky[21, 22] (see also[23]),the idea of non-perturbative resonant production of particles has been introduced[24]. Though thefull non-linear theory of preheating is still not well understood but a significant advancement inthis field has been made and lot of works are going on [25–27]. It is generally believed that thereheating phase usually happens in two stages. In the first stage, particle production is due toparametric resonance known as ‘preheating’ followed by the perturbative reheating.Therefore, in this section we discuss about non-perturbative particle production via parametricresonance phenomena for a specific model ( n = 2). It is evident that initial few oscillations after theend of inflation play important role at this stage. Therefore, we will see how the inflationary scale φ ∗ which controls the shape of the potential, effects the structure of the resonance for reheatingfields. Hence, through resonance structure, we may be able to further restrict the parameter spaceof our model. We defer the detailed of this resonance phenomena for our future study.The inflaton field oscillating coherently after inflation acts as a classical external force leadingto the production and growth of quantum boson fields via the bose condensation. We write theLagrangian for the daughter scalar field χ as. L χ = 12 ∂ µ χ∂ µ χ − m χ χ − g φ χ . (23)Where, m χ is the mass of the χ particle. The matter field χ satisfies the following equation:¨ χ + 3 H ˙ χ − a ∇ χ + ( m χ + g φ ) χ = 0 . (24)Decomposing the scalar field operator into Fourier modes, χ ( t, x ) = (cid:90) d k (2 π ) / [ a k χ k ( t ) e i k · x + a k † χ k ( t ) e − i k · x ] , (25)the mode equation for χ k ( t ) takes the following from¨ χ k + 3 H ˙ χ k + (cid:18) k a + m χ + g φ (cid:19) χ k = 0 . (26)Where, a k , a k † are the creation and annihilation operators respectively. The parametric resonancephenomena with periodic background force can be best explained though the stability/instabilitydiagram arising from the above generalized Mathieu equation[28, 29]. To study this in the expand-ing cosmological background we rescale the field variable χ k and the inflaton field φ as follows χ k → a − X k ; φ → a − Φ . X (cid:48)(cid:48) k + ω k X k = 0 (27)where ω k ≡ k a m + g φ m a Φ ( t ) + ∆ , ∆ ≡ −
34 (3 H + 2 ˙ H ) , and “prime” is taken with respect to rescaled dimensionless time variable z = mt . As we are onlyconsidering n = 2, soon after the inflation ends, the background evolution approximately satisfies H ≡ ˙ H (cid:28) m . Therefore, we set ∆ (cid:39) φ as the initial amplitude of the coherent inflaton oscillation.If we ignore the expansion of the universe, eq.(27) can be identified as a Hill’s differentialequation with parameters κ = k m a and q = g φ m a . The solution of this equation is known to exhibitparametric resonance depending on the value of the parameters ( q, κ ). If the value of the parameters( q, κ ) is within certain ‘instability bands’ the solution of the eq.(27) grows exponentially as X k ∝ exp ( µ k z ). Where, Floquet exponent, µ k , parametrizes the strength of the resonance. Therefore,depending upon the value of µ k , corresponding χ -particle of momentum k will grow exponentially.This indefinite growth of any mode is just the consequence of neglecting the expansion of theuniverse as well as the back-reaction of the produced χ particles. When the expansion is includedthe parameters in the Hill’s equation becomes time dependent. However, it can be seen that therelative change in q during oscillation is 1 m ˙ qq = − Hm . (28)During reheating period, H (cid:28) m , hence, q parameter can be taken as constant.The structure of resonance for chaotic inflationary model has been well studied. As has beenmentioned, the scale φ ∗ plays very important role during inflation. Therefore, our main goal wouldbe to understand the role of φ ∗ on the structure of resonance. To compare the effect for different φ ∗ we consider n = 2, and measure time in unit of m which is the value of m corresponding to φ ∗ = 10 M p . We present the contour plots for the Floquet exponent in ( q, κ ) space for differentvalues of φ ∗ as shown in fig.7. We also show how the zero mode function ( X k =0 ) grows for φ ∗ = 10in fig.8 for two values of q , one taken from inside and another just outside the unstable region. Thestability/instability chart has been computed for the first oscillation taking a ( t ) = 1. Therefore,with time the magnitude of ( q, κ ) parameters decrease which measures duration of preheatingperiod. 14 * = 10 ( ) * = 1 ( ) * = 0.1 M p ( ) * = 0.01 M p ( ) FIG. 7:
Stability/instability charts for the non-perturbative production of reheating field χ (Eq.(26) in ( k m , q ) space. - - - t X k κ =
0, q = ϕ * = (a) - - t X k κ =
0, q = ϕ * = (b) FIG. 8:
The zero mode of the produced field ( X k = a χ ) for values of slightly inside (a) and outside (b) of the instabilityband. Now we will examine the effects of φ ∗ on the structure of resonance. The very first point thatwe would like to point out is the dependence of the Floquet exponent on inflation scale φ ∗ . Wecan clearly see that as we decrease φ ∗ from (10 , , . , .
01) in unit of Planck, the maximum valueof Floquet exponent increases as (0 . , . , . ,
5) respectively. Therefore, with decreasing value of φ ∗ , the resonance becomes stronger and simultaneously the band width also increases. This can15lso be seen from the approximate analytic expression for the band width∆ k (cid:39) (cid:18) mm (cid:19) (0 . g ˙ φ ) . (29)Where, velocity of φ field is measured in unit of m − . However, in order to compare the resultsfor different φ ∗ , we set a fixed time scale m corresponding to φ ∗ = 10. Hence, as we decrease φ ∗ , m decreases from CMB normalization, which enhance the frequency of the inflaton oscillation andconsequently the resonance band width becomes wider. Moreover, for large φ ∗ , long wavelengthmodes of χ field needs stronger coupling to get excited. From the effective field theory pointof view, small scale inflation suggests the parameter should be φ ∗ < g (cid:38) (10 − , − ) for φ ∗ (cid:39) (0 . , . M p . This is in sharp contrast with the usual chaotic inflation, where inflationaryobservables do not have much effect on the reheating coupling parameter g , and consequently thereheating temperature if we consider the shifted minimum of the inflaton potential. We will dodetail lattice study on this issue in the subsequent publication.To this end, let us mentioned an important point which we will defer for our future studies.As we decrease the value of φ ∗ below 1 M p , the effective mass m eff = V (cid:48)(cid:48) ( φ ) of the inflation fieldbecomes negative in certain range of inflaton field values after the end of inflation and remains so forfirst few oscillations. This will lead to tachyonic preheating and will have important consequencesspecifically with regard to the gravitational production. We will differ detailed study on this issuefor our future work. IV. MODEL INDEPENDENT CONSTRAINTS FROM REHEATING PREDICTIONS
After inflation, reheating is the most important phase, where, all the visible matter energywill be pumped in. In this section, we will try to constrain our model parameters without anyspecific mechanism of reheating. The background evolution of cosmological scales from inflationto the present day and the conservation of entropy density provide us important constraints onreheating as well as our model parameters. Reheating is the supposed to be the integral part ofthe inflationary paradigm. However, because of the single observable universe, it is very difficult tounderstand this process by the present day cosmological observation. Thermalization process erasesall the information about the initial conditions which is the most important part of this phase.To understand this phase an indirect attempt has been made in the recent past [8, 9, 30] throughthe evolution of cosmological scales and the entropy density by parametrizing it by reheatingtemperature ( T re ), equation of state ( w re ), and efolding number ( N re ). In this section we follow16he reference [10] by taking into account the two stage reheating phase generalizing the formalismof [30]. Our main goal is to understand the possible constraint on our minimal inflationary models.As we have seen from previous analysis, all the cosmological quantities during inflation can beexpressed in terms of two main parameters ( m or λ, φ ∗ ) for a particular model. Because of two stagereheating process, the suitable reheating parameters are as follows, ( N re = N re + N re , T re , w re , w re ).Where, N re , N re are efolding number during the first and second stage of the reheating phase withthe equation of state w re , w re respectively. At the initial stage the oscillating inflaton will be thedominant component, and at the end radiation must be the dominant component. Therefore,instead of taking the equation of state as free parameters, we will be considering only the followingparticular case w re = n − n + 2 ; w re = 13 . (30)We also assumed the change of reheating phase from the first to the second stage as instantaneous.A particular scale k going out of the horizon during inflation will re-enter the horizon duringusual cosmological evolution. This fact will provide us an important relation among different phasesof expansion parametrizing by enfolding number as followsln (cid:18) ka H (cid:19) = ln (cid:18) a k H k a H (cid:19) = − N k − (cid:88) i =1 N ire − ln (cid:18) a re H k a H (cid:19) , (31)In the above expressions, use has been made of k = a H = a k H k . Where, ( a re , a ) are thecosmological scale factor at the end of the reheating phase and at the present time respectively.( N k , H k ) are the efolding number and the Hubble parameter respectively for a particular scale k which exits the horizon during inflation. Therefore, following mathematical expressions will beused in the final numerical calculation, H k = (cid:115) V ( φ k )3 M p = (cid:16) λφ n ∗ M p (cid:17) m − n ˜ φ n k (
1+ ˜ φ nk ) (cid:16) λφ n ∗ M p (cid:17) m − n ˜ φ n k (
1+ ˜ φ k ) n , (32) N k = 1 M p (cid:90) φ k φ end √ (cid:15) dφ (cid:39) φ ∗ nM p (cid:104) n +2) ˜ φ ( n +2) k + ˜ φ k (cid:105) φ ∗ nM p (cid:104) ˜ φ k + ˜ φ k (cid:105) (33) φ k and φ end are the inflaton field values corresponding to a particular scale k crossing theinflationary horizon, and at the end of inflation respectively. In the above expressions, we haveignored the contribution coming from the inflaton field value φ end . It is important to note that, inprinciple we can write the field value at a particular scale k in terms of n s , r , by inverting thoserelations. Because of non-linear form, we will numerically solve those. The above unknown efolding17 .95 0.96 0.97 0.98 0.99020406080100 n s N r e , N k - n s T r e [ G e V ] FIG. 9:
Variation of ( N re (solid) , N k (dotted) , T re ) as a function of n s have been plotted for φ ∗ = 0 . M p . This is the plot forModel-I. (Blue, red, magenta, brown, green) curves correspond to n = (2 , , , , w re , w re ) = (( n − / ( n + 2) , /
3) during reheating. We also consider N re = N re . The lightblue shaded region corresponds to the 1 σ bounds on n s from Planck. The brown shaded region corresponds to the 1 σ boundsof a further CMB experiment with sensitivity ± − [31, 32], using the same central n s value as Planck. Temperatures belowthe horizontal red line is ruled out by BBN. The deep green shaded region is below the electroweak scale, assumed 100 GeVfor reference. numbers during reheating will certainly be dependent upon the energy densities ( ρ end , ρ re ), at theend of inflaton (beginning of reheating phase) and at the end of the reheating phase( beginning ofthe standard radiation dominated phase);ln (cid:18) ρ end ρ re (cid:19) = 3(1 + w re ) N re + 3(1 + w re ) N re = 3 (cid:88) i =1 (1 + w ire ) N ire . (34)Above two eqs.(31,34), can be easily generalized for multi-stage inflation with different equationof state parameters. As has been mentioned, after the end of reheating standard evolution ofour universe is precisely known in terms of energy density and the equilibrium temperature ofthe relativistic degrees of freedom such as photon and the neutrinos. Therefore, the equilibriumtemperature after the end of reheating phase, T re , is related to temperature ( T , T ν ) of the CMBphoton and neutrino background at the present day respectively, as follows g re T re = (cid:18) a a re (cid:19) (cid:18) T + 6 78 T ν (cid:19) . (35)The basic underlying assumption of the above equation is the conservation of reheating entropyduring the the evolution from the radiation dominated phase to the current phase. g re is thenumber of relativistic degrees of freedom after the end of reheating phase. We also use the followingrelation between the two temperatures, T ν = (4 / / T . For further calculation, we define aquantity, γ = N re /N re . If we identify the scale of cosmological importance k as the pivot scaleof PLANCK, so that k/a = 0 . M pc − , and the corresponding estimated scalar spectral index n s = 0 . ± . .95 0.96 0.97 0.98 0.99020406080100 n s N r e , N k - n s T r e [ G e V ] FIG. 10:
Variation of ( N re (solid) , N k (dotted) , T re ) as a function of n s have been plotted for φ ∗ = 0 . M p . This is the plotfor Model-II. (Blue, red, magenta, brown, green) curves correspond to n = (2 , , , , w re , w re ) = (( n − / ( n + 2) , /
3) during reheating. We also consider N re = N re . Fromthe left figure, one clearly sees that the behavior of N k is independent of n N re = 4(1 + γ )(1 − w re ) + γ (1 − w re ) . − ln V end H k − N k (36) T re = (cid:34)(cid:18) g re (cid:19) a T k H k e − N k (cid:35) wre γ (1+ wre wre − γ (3 wre − (cid:20) . V end π g re (cid:21) γ (1 − wre γ (1 − wre . (37)In the above derivation, we have used g re = 100. Before discussing any further, let us providethe general descriptions of the figures we have drawn in this section. As has been mentioned before,we have considered specific values of equation of state parameter ( w re , w re ) = (( n − / ( n + 2) , / / , /
3) which is realized for n = 4. Analyticallyone can check that at this special point both ( T re , N re ) become indeterministic seen in eq.(37). Thisfact corresponds to all the vertical solid red lines in ( n s vs T re ) and ( n s vs N re ) plots. We haveconsidered γ = 1 as our arbitrary choice. Each curve corresponds to different values of n . Onthe same plot of ( n s vs N re ), we also plotted ( n s vs N k ) corresponding to the dotted curves fordifferent models. One particularly notices that for the second type model in fig.(10), behavior of( n s vs N k ) is same for all different value of n . This universality is inherited from the fact that n s does not really depend upon n . Therefore, background dynamics because of the second type modelfor different values of n are almost universal. However, prediction of ( T re , N re ) are dependentupon the value of n through the equation of state parameter eq.(30). At this stage, we would liketo remind the reader again that for a wide range of φ ∗ , all the models predict very small valueof tensor to scalar ratio r . Therefore, we will be discussing all the constraints without explicitlymentioning r . Given the overall description of all the plots, we now set to discuss the predictionand constraints for two different models. In the table-(III) we provide the important numbers for19 .95 0.96 0.97 0.98 0.99020406080100 n s N r e , N k - n s T r e [ G e V ] FIG. 11:
Variation of ( N re (solid) , N k (dotted) , T re ) as a function of n s have been plotted for for three different values of φ ∗ .(Blue, magenta, purple) curves are for φ ∗ = (0 . , . , M p respectively. We consider only n = 2 for Model-I. All the otherparameters remain the same as for the previous plots. reheating temperature and the efolding number. We provided only the limiting values of T re whichare still allowed from the cosmological observation. n Model-I Model-II n s T re (GeV) N re N k n s T re (GeV) N re N k × × × × × × × × × × × × × × × × Some sample values of ( n s , T re , N re , N k ) are give for two different models for n = (2 , , , n = 4, ( T re , N re ) become indeterministic. All these prediction are for φ ∗ = 0 . M p . From the figure we see that for a very small change in n s , the variation of reheating temperatureis very high. Therefore, reheating temperature provides tight constraints on the possible values ofefolding number N re during reheating phase. Except for n = 4, if we restrict the value of T re (cid:38) GeV, the efolding number turned out to be N re (cid:46)
35 during reheating. As an example, for n = 2,we find spectral index lies within 0 . (cid:46) n s (cid:46) . × (cid:38) T re (cid:38) × in unit of GeV. This restrictionin turn fixed the possible range of efolding number within a very narrow range 50 < N <
54 for n = 2. For other value of n , the ranges are provided in the table-III. What we can infer from ouranalysis in this section is that reheating constraint does not allow n to be vary large specificallyfor type-I model. Whereas for type-II mode, the prediction of n s is almost independent of n for20 .95 0.96 0.97 0.98 0.99020406080100 n s N r e , N k n s T r e [ G e V ] FIG. 12:
Variation of ( N re (solid) , N k (dotted) , T re ) as a function of n s have been plotted for for three different values of φ ∗ .(Blue, magenta, purple) curves are for φ ∗ = (0 . , . , M p respectively. We consider only n = 6 for Model-I. All the otherparameters remain the same as for the previous plots. φ ∗ < M p However in the figs.11,12,we have plotted the dependence of various reheating parametersfor different values of φ ∗ . We have plotted only for type-I model and n = 2 ,
6. For all the othermodels qualitative behaviors of those plots will be same, except n = 4.With increasing value of the equation of state, efolding number during reheating N re , decreasesfor a fixed value of reheating temperature. This essentially means that as one increases the valueof inflationary equation of state w , faster will be the thermalization process, therefore, earlier willbe the radiation dominated phase. In our subsequent full numerical solutions, we have observedthis fact considering the evolution of all the important components during reheating. We alsonoted that as we decrease the value of γ , the prediction of ( T re , N re ) will be controlled by inflationequation of state parameter w re . V. SUMMARY AND CONCLUSION
Before we conclude, let us summarize the main results of our study. As emphasized, we tried toconstrain specific classes of inflationary models based on the inflation and dark matter abundance.Effective field theory consideration constraints φ ∗ to be less than unity in Planck unit. As a resultwe have sub-Planckian field excursion during inflation. In this regime of φ ∗ the value of inflationaryobservables ( n s , r, dn ks ) saturate to a constant value depending on the e-folding number N andthe power law index n . Furthermore, requirement of broad parametric resonance constraints thecoupling parameter g (cid:38) − for n = 2 , φ ∗ = 0 .
1. For fixed n , if we further reduce the value of φ ∗ , the lower limit on g increases, however, resonance becomes stronger and broader. Therefore,instant transfer of energy from inflaton to reheating field is possible within few oscillations of theinflaton field. Detailed analysis of this issue will be done in our subsequent publication.In the first part of this paper, we have constructed two new classes of inflationary model with21on-polynomial modification of the inflaton potential. In the appendix we have tried to constructsuch potential from a most general non-minimal scalar tensor theory. In certain region of theparameter space, our models coincide with the aforementioned scalar-tensor theory. It would beinteresting to construct such potential from more fundamental approach. Interesting property ofthese classes of potentials is that they have infinitely large flat plateau. Therefore, the inflation canbe naturally realized because of this shift symmetry. As a result, the predictions of the models forinflationary observables are not very much sensitive to the detail form specifically near the minimumof the potential. Importantly our model fits extremely well with latest cosmological observationmade by PLANCK. All the necessary scales assume below Planck scale value, which may implythat our model predictions are robust against quantum correction. Detailed computation on theultra-violate effect on our model could be important and we left it for our future work. Dependingupon the choice of scale, in our model we realize both large field as well as small field inflation.However, for both the cases, the prediction of tensor to scalar ratio ( r ) turned out to be significantlysmall. In the end we have studied model independent reheating constraint analysis and discussabout further constraint coming from the reheating when connecting with the CMB anisotropy.Another important aspect of our model is that we can have significantly low inflationary Hubblescale H ∗ unlike the usual power law inflationary model. The value could be as low as ∼ GeVfor different values of n = 4 , , φ ∗ (cid:39) . M p . It is well known that low value of H ∗ could beinteresting in the context of Higgs vacuum instability. As has been pointed out in [33, 34], during aswell as after the inflation the quantum fluctuation of Higgs field can destabilize the standard modelmetastable Higgs vacuum at around Λ I = 10 GeV. However, this instability crucially dependsupon the value of H ∗ , and also the height of the Higgs potential. Therefore, comparing the naivescale dependence between Λ I and H ∗ , our model have potential to save the Higgs vacuum fromdecaying into the global vacuum. We leave the detailed study on this issue for our future work. VI. ACKNOWLEDGEMENT
We would like to thank our HEP and Gravity group members for their valuable comments anddiscussions.
Appendix A: Towards derivation of our model potential
In this section starting from non-minimal scalar-tensor theory, we will try to construct ourmodel potentials which were a priori ad hoc in nature. As is well known, inflationary models based22n power law potential V ( φ ) ∼ φ n are simple but have been ruled out in general because of theirlarge prediction of tensor to scalar ratio. Moreover, the models with large plateaus (Starobinskyor α -attractors) are found to be most favored form the PLANCK observation. While most of theseplateau models can be cast into exponential potential, plateau potentials with power-law formhave also been discussed in super gravity[35, 36] and non-minimal coupling to gravity[37, 38]. Inthis section we will try to construct our model based on this non-minimally coupled scalar-tensortheory. We will see, how simple power-law potentials in the Jordan frame can give rise to the plateaupotentials of desired form in the Einstein frame. However, this transformed models coincide withour minimal models only in a limiting regime (weak conformal coupling). At this point let us pointout that equivalence between the Einstein frame and Jordon frame is an important question toask. This issue has been discussed [39–45], from theoretical as well as cosmological point of views.Nevertheless, our motivation in this section is to construct our desired form of the potentialswhich we have shown to be in different class of models rather tan α attractor model. We startwith the following non-minimally coupled scalar-tensor theory, S J = (cid:90) d x √− g (cid:20) Ω( ϕ )2 M p R − ω ( ϕ )2 g µν ∂ µ ϕ∂ ν ϕ − V ( ϕ ) (cid:21) , (A1)where, Ω( ϕ ) , ω ( ϕ ) are arbitrary function of a scalar field ϕ . We will chose a specific form of thosefunction for our later purpose. To get the action in the Einstein frame, one performs the followingconformal transformation as, ˜ g µν = Ω( ϕ ) g µν , (A2)The action in the Einstein frame can be written as[46] S E = (cid:90) d x (cid:112) − ˜ g (cid:34) M p R − F ( ϕ )˜ g µν ∂ µ ϕ∂ ν ϕ − ˜ V ( ϕ ) (cid:35) (A3)Where, we have assumed that ω ( ϕ ) = Ω( ϕ ) and F and the new potential can be found to be, F ( ϕ ) = 3 M p (cid:48) ( ϕ )Ω ( ϕ ) + 1 ; ˜ V ( ϕ ) = V ( ϕ )Ω ( ϕ ) (A4)Now, we choose the following non-minimal coupling function [47, 48], for Ω ( ϕ ),Ω ( ϕ ) = ξ ( ϕM p ) n (cid:104) ξ ( ϕM p ) (cid:105) n . (A5)Therefore, applying (A5), we find F and ˜ V as, F ( ϕ ) = n ξ (cid:16) ϕMp (cid:17) n − (cid:104) ξ (cid:16) ϕMp (cid:17) n (cid:105) + 1 n ξ (cid:16) ϕMp (cid:17) (cid:20) ξ (cid:16) ϕMp (cid:17) (cid:21) + 1 ; ˜ V ( ϕ ) = V ( ϕ )1+ ξ (cid:16) ϕMp (cid:17) n V ( ϕ ) (cid:104) ξ ( ϕMp ) (cid:105) n (A6)23e use the following field redefinition dφdϕ = F ( ϕ ) (A7)to transform the non-minimal into the action of a minimally coupled scalar field with canonicalkinetic term, S E = (cid:90) d x (cid:112) − ˜ g (cid:34) M p R − ˜ g µν ∂ µ φ∂ ν φ − ˜ V ( φ ) (cid:35) (A8)At this point we can integrate eq.(A7), to find the new field in terms of the old field, and constructthe modified potential as a function of new field. It is clear from the above set of transformationsthat for entire range of parameter ξ , it is very difficult to reproduce our model. However, in theregime of weak coupling ξ << F ∼
1, hence we can approximately write, using eq(A7); ϕ ∼ φ φ ( φ is some integration constant). Considering Jordan frame potential as power-law: V ( ϕ ) ≈ ϕ n ,one gets plateau potential as ˜ V ( φ ) = λ m − n φ n (cid:16) φφ ∗ (cid:17) n λ m − n φ n (cid:104) φφ ∗ ) (cid:105) n , (A9)where, we identify φ ∗ as M p /ξ n for Type-I potential and M p /ξ for type-II potential. Therefore,in the weak coupling regime, ξ (cid:28) φ ∗ >
1, the non-minimal scalar tensor theory can give riseto a large class of minimal cosmologies such as ours which do not belong the α -attractor model. Appendix B: Validity of /N expansion for φ ∗ < M p In this section we consider n = 2 case, as we can analytically compute the expression for thepower spectrum. Let us start by defining x ≡ φ/φ ∗ , and using eq.(9), we define the number ofefolding by the following integral expression from field value x k for which scales exit the horizonto the end of inflation with field value x e , N k = x e (cid:90) x k (cid:18) φ ∗ M p (cid:19) V ( x ) V (cid:48) ( x ) dx. (B1)The x k corresponds to a scale k which exits the horizon, and x e is the reduced field value at theend of inflation( (cid:15) ( x e ) = 1 ). Therefore, the field value at the horizon crossing turned out to be, x k = (cid:104) − (cid:112) f k (cid:105) , (B2)24here f k is given by f k = 8 M p φ ∗ N k + x e (2 + x e ) . (B3)The expression for x e is given by x e = 2 / (cid:16) φ ∗ M p (cid:17) / (cid:34) − (cid:114) (cid:16) φ ∗ M p (cid:17) (cid:35) / − (cid:34) − (cid:114) (cid:16) φ ∗ M p (cid:17) (cid:35) / / (cid:16) φ ∗ M p (cid:17) / . (B4)In the limit φ ∗ < M p , which is necessary for 1 /N k expansion we see from the last expression that x e simply reduces to x e ∼ / (cid:18) M p φ ∗ (cid:19) / ; x k ∼ f / k . Now the slow-roll parameters in terms of x k reduces to (cid:15) ∼ M p φ ∗ f / k = 1 (cid:20) N k (cid:16) M p φ ∗ (cid:17) N k + 2 / (cid:16) M p φ ∗ (cid:17) / (cid:21) / ∼ / φ ∗ M p N / k . The final term in the above expression is the leading order in N k for φ ∗ < M p . Similarly, thesecond slow-roll parameter of our interest reduces to | η | ∼
34 1 N k (B5)All the above leading order expressions for the slow roll parameters match exactly with our generalexpression for n s and r given in terms of (cid:15) and η in Eqs.(15-16) and it is clear that for φ ∗ (cid:28) M p /N k − expansion and slow-roll parameters are consistent.Another interesting limit arises for φ ∗ (cid:29) M p , in this case as φ ∗ increases, x e →
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