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# Minimal inflationary cosmologies and constraints on reheating

Debaprasad Maity<EMAIL_ADDRESS>Pankaj Saha<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Guwahati.
Guwahati, Assam, India

###### Abstract

With the growing consensus on simple power law inflation models not being
favored by the PLANCK observation, dynamics for the non-standard form of the
inflaton potential gain significant interest in the recent past. In this
paper, we analyze in great detail classes of phenomenologically motivated
inflationary models with non-polynomial potential which are the generalization
of the potential introduced in mhiggs . After the end of inflation, inflaton
field will coherently oscillate around its minimum. Depending upon the initial
amplitude 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 the resonant
structure of our model under study. Subsequently, the universe will go through
the perturbative reheating phase. However, without any specific model
consideration, we further study the constraints on our models based on model
independent reheating constraint analysis.

###### Contents

  1. I Introduction
  2. II The Model
    1. II.1 Background Equations
    2. II.2  Computation of $(n_{s},r,dn_{s}^{k})$
    3. II.3 End of inflation and general equation of state
  3. III Pre-heating: parametric resonance and their dependence on inflation scale $\phi_{*}$
  4. IV  Model independent constraints from reheating predictions
  5. V Summary and Conclusion
  6. VI Acknowledgement
  7. A Towards derivation of our model potential
  8. B Validity of $1/N$ expansion for $\phi_{*}<M_{p}$

##  I Introduction

The inflation guth ; linde1 ; steinhardt  is a model independent mechanism
proposed to solve some of the outstanding problems in standard Big-Bang
cosmology. It is an early exponential expansion phase of our universe, which
sets the required initial condition for the standard Big-Bang evolution. Over
the years large number of models have been introduced to realize this
mechanism encyclopedia , and explain the cosmological observations PLANCK .
Out of the large number of models, a particularly interesting class of models
that have recently been studied is called $\alpha$-attractorlinde . It has
gained significant attentions because it unifies a large number of existing
inflationary models. In this paper, we will introduce new classes of
inflationary models generalizing the model proposed in mhiggs . In order to
explain the observation, we phenomenologically consider classes of non-
polynomial potentials, which could be derived from a general scalar-tensor
theory in certain limit (shown in appendix-A). At this point let us motivate
the reader mentioning the important points of our study. It is well known that
the general power law canonical potentials of the form $V(\phi)\sim|\phi|^{n}$
are not cosmologically viable because of its prediction of large tensor to
scalar ratio. In addition, because of super-Planckian value of the field
excursion, the effective field theory description may be invalid. One of our
goals in this paper is to circumvent the above mentioned problems in the
framework of canonical scalar field model. Therefore, we generalize the power
law form of the potential to non-polynomial form so that 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 large oscillation amplitude, the inflaton can decays
through parametric resonance depending upon the inflaton coupling with the
reheating fields. Considering a specific model ($n=2$), we figure out the
parameter region where broad parametric resonance happens. Our analysis shows
that as we decrease the inflationary energy scale, the instability bands
evolve into wider band, thereby, enhance the strength of the resonance.
However, the number of stability/instability region decreases with decreasing
the scale we introduced in the model. Detail analysis of this phenomena will
be done in our future work. After few initial oscillations, resonant decay
will naturally reduce the amplitude of the inflaton oscillation significantly.
Therefore, the perturbative reheating starts to play its role till the
radiation domination begins. In this paper we will not discuss about the usual
perturbative reheating. However, we should mention that to the best of our
knowledge detailed analysis of this perturbative phase for arbitrary power law
inflaton potential has not been done. We defer this studies for our future
publication. However, what we have done instead is the model independent
reheating constraint analysis based on the works kamionkowski ; debuGB , and
understand the possible constraints on the model for the successful reheating
to be realized.

We structured our paper as follows: In section-II, we generalize the model
introduced in mhiggs , and study in detail the cosmological dynamics of
inflaton starting from inflation to reheating. We compute important
cosmological parameters such as scalar spectral index $(n_{s})$, the tensor to
scalar ratio $(r)$, and the spectral running $(dn_{s}^{k})$ and fit with the
experimental observations. From those cosmological 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 which
the universe will undergo reheating phase. We also compute the effective
equation of state of the oscillating inflaton for our subsequent studies. In
section-III, we will discuss about how the new scale $\phi_{*}$ controls the
resonance structure during during the first few oscillation of the inflation
field. This is very important for pre-heating phenomena. For this we will only
consider one single model 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 important connection between the
end of reheating and the cosmic microwave background (CMB) anisotropy. Finally
we concluded and discussed about our future work.

##  II The Model

As we have discussed in the introduction and also tried to construct in the
appendix, our starting point in this section is the non-polynomial potential
which has dominant power law behavior around its minimum. Therefore, we will
start by considering the following phenomenological forms of the potential,

$\displaystyle
V(\phi)=\begin{cases}\lambda\frac{m^{4-n}\phi^{n}}{1+\left(\frac{\phi}{\phi_{*}}\right)^{n}}\\\
\lambda\frac{m^{4-n}\phi^{n}}{\left(1+\left(\frac{\phi}{\phi_{*}}\right)^{2}\right)^{\frac{n}{2}}},\end{cases}$
(1)

In the above form of the potentials, we have introduced two free parameters
$(m~{}or~{}\lambda,n)$. Where, the parameter $\lambda$ is defined for $n=4$,
which has been studied as minimal Higgs inflation in mhiggs . For other value
of $n$, we can set $\lambda=1$. For large value of $\phi_{*}$ the potential is
plateau like and the associated inflationary scale is $\Lambda=\lambda
m^{4-n}\phi_{*}^{n}$. A simple illustration of the above form of the
potentials is shown in the fig.1. Through out the paper, we will refer type-I
for the first form and type-II for the second form of the potential. One can
further generalize our model by considering the potential to be dependent only
upon the modulus of the inflaton field. Therefore, all the odd values 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
$V(\phi)^{q}$ as a new potential. Where, $q$ would a new parameter.

Figure 1: An illustration of the dependence of the shape of the potential with
the scale $\phi_{*}$ for $n=2,\lambda=1$ when the two class of the model
considered became identical. As we decrease $\phi_{*}$, the CMB normalization
changes the parameter $m$ such that the height as well as the width of the
potential decreases. This fact will result in decrease in the field excursion
as well as the scalar to tensor ratio and also have significant effect on the
post-inflationary dynamics.

###  II.1 Background Equations

In this section we will study in detail the background dynamics using the
above form of the potentials. We will start with the following action,

$\displaystyle S~{}=~{}\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]$
(2)

Where $M_{p}=\frac{1}{\sqrt{8\pi G}}$ is the reduced Planck mass. Assuming the
usual Friedmann-Robertson-Walker(FRW) background ansatz for the space-time

$ds^{2}=dt^{2}-a(t)^{2}(dx^{2}+dy^{2}+dz^{2}),$ (3)

the system of equations governing the dynamics of inflaton and scale factor
are

$\displaystyle 3M_{p}^{2}H^{2}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)$ (4)
$\displaystyle 2M_{p}^{2}\dot{H}=-\dot{\phi}^{2}$ (5)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi).$ (6)

Where, the usual definition of Hubble constant is $H={\dot{a}}/a$. As we have
seen, our potential is asymptotically flat for large field value compared to
$\phi_{*}$. This is condition which is required for the inflationary dynamics
is automatically satisfied. The flatness conditions for the potential during
inflation are written in terms of the slow-roll parameters, which are defied
as

$\displaystyle\epsilon\equiv\frac{M_{p}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{n^{2}M_{p}^{2}\phi_{*}^{2n}}{2\phi^{2}\left(\phi_{*}^{n}+\phi^{n}\right)^{2}}\\\
\frac{\phi_{*}^{4}n^{2}M_{p}^{2}}{2\phi^{2}\left(\phi_{*}^{2}+\phi^{2}\right)^{2}}\end{cases}$
$\displaystyle\eta\equiv M_{p}^{2}\left(\frac{V^{\prime\prime}}{V}\right)$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{nM_{p}^{2}\phi_{*}^{n}\left((n-1)\phi_{*}^{n}-(n+1)\phi^{n}\right)}{\phi^{2}\left(\phi_{*}^{n}+\phi^{n}\right)^{2}}\\\
\frac{\phi_{*}^{2}nM_{p}^{2}\left(\phi_{*}^{2}(n-1)-3\phi^{2}\right)}{\phi^{2}\left(\phi_{*}^{2}+\phi^{2}\right)^{2}}.\end{cases}$
(7)

During inflation $\epsilon\ll 1$ and $|\eta|\ll 1$. Therefore, the end of
inflation is usually set by the condition $\epsilon=1$. Let us also define a
higher order slow-roll parameter related to the third derivative of the
potential for spectral running. The expression for the higher order slow-roll
parameter is as follows:

$\xi\equiv
M_{p}^{4}\left(\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}}\right)=\begin{cases}\frac{n^{2}M_{p}^{4}\phi_{*}^{2n}\left(\left(n^{2}-3n+2\right)\phi_{*}^{2n}-4\left(n^{2}-1\right)\phi_{*}^{n}\phi^{n}+\left(n^{2}+3n+2\right)\phi^{2n}\right)}{\phi^{4}\left(\phi_{*}^{n}+\phi^{n}\right)^{4}}\\\
\frac{\phi_{*}^{4}n^{2}M_{p}^{4}\left(\phi_{*}^{4}\left(n^{2}-3n+2\right)+3\phi_{*}^{2}(2-3n)\phi^{2}+12\phi^{4}\right)}{\phi^{4}\left(\phi_{*}^{2}+\phi^{2}\right)^{4}}.\end{cases}$
(8)

In addition to provide the successful inflation, all the aforementioned slow-
roll parameters play very important role in controlling the dynamics of
cosmological perturbation during inflation. An important cosmological
parameter which quantifies the amount of inflation is called e-folding number
$(N)$, which plays crucial role in solving the horizon and flatness problem of
standard Big-Bang. The e-folding number is expressed as as

$\displaystyle
N=ln\left(\frac{a_{end}}{a_{in}}\right)=\int\limits_{a_{in}}^{a_{end}}d\hskip
2.84544ptlna=\int\limits_{t_{in}}^{t_{end}}Hdt\simeq\int\limits_{\phi_{in}}^{\phi_{end}}\frac{1}{\sqrt{2\epsilon}}\frac{|d\phi|}{M_{p}}.$
(9)

As we have mentioned the inflation ends when $\epsilon=1$, and one can use the
eq.(9) to find the value of the inflaton at the beginning of the inflation. By
solving the aforementioned condition, we can express the e-folding number $N$
into the following form,

$N=\begin{cases}\frac{\phi_{*}^{2}}{nM_{p}^{2}}\left[\frac{1}{(n+2)}(\tilde{\phi}^{(n+2)}-\tilde{\phi}_{end}^{(n+2)})+\frac{1}{2}(\tilde{\phi}^{2}-\tilde{\phi}_{end}^{2})\right]&\simeq\frac{\phi_{*}^{2}}{nM_{p}^{2}}\frac{1}{(n+2)}\tilde{\phi}^{(n+2)}\\\
\frac{\phi_{*}^{2}}{nM_{p}^{2}}\left[\frac{1}{4}(\tilde{\phi}^{4}-\tilde{\phi}_{end}^{4})+\frac{1}{2}(\tilde{\phi}^{2}-\tilde{\phi}_{end}^{2})\right]&\simeq\frac{\phi_{*}^{2}}{4nM_{p}^{2}}\tilde{\phi}^{4}.\end{cases}$
(10)

Where we have defined, $\tilde{\phi}={\phi}/{\phi_{*}}$. In the above
expressions for $N$, we have ignored the contribution coming from
$\phi_{end}$, and also the squared term. We have numerically checked the
validity of those expressions for a wide range of value of $\phi*\leq{\cal
O}(M_{p})$. From cosmological observations one needs $N\simeq 50-60$, such
that the scales of our interest in CMB were in causal contact before the
inflation. By using the above mentioned boundary conditions for the inflaton
we have solved for the homogeneous part of inflaton $\phi(t)$ and the scale
factor $a(t)$. One particular solution has been given in fig.6, with a
specific value of the efolding number. Next we study the perturbation around
inflationary background and derive the relevant cosmological parameters
associated the various correlation functions of fluctuation.

###  II.2  Computation of $(n_{s},r,dn_{s}^{k})$

As we described in the introduction, the very idea of inflation was introduced
to solve some outstanding problems of standard Big-Bang cosmology. Soon it was
realized that inflation also provides seed for the large-scale structure of
our universe through quantum fluctuation. All the cosmologically relevant
inflationary observables are identified with various correlation functions of
those primordial fluctuations calculated in the framework of quantum field
theory. We have curvature and tensor perturbation. The two and higher point
correlation functions of those fluctuation are parametrized by power
spectrum(see, cpt1 ; cpt2 ; baumann-tasi  for a comprehensive review of
Cosmological Perturbation Theory). The scalar curvature power spectrum is
given by

$\mathcal{P_{\mathcal{R}}}=\frac{1}{8\pi^{2}}\frac{1}{\epsilon}\frac{H^{2}}{M_{p}^{2}}\bigg{|}_{k=aH}=\frac{1}{12\pi^{2}}\frac{V^{3}}{M_{p}^{6}(V^{\prime})^{2}}.$
(11)

Once, we know the power spectrum, cosmological quantity of our interests are
the spectral tilt and its running. During inflation a particular inflaton
field value corresponds to a particular momentum mode exiting the horizon.
Hence by using the following relation to the leading order in slow-roll
parameters, $\frac{d}{d\hskip
2.84544ptlnk}=\frac{\dot{\phi}}{H}\frac{d}{d\phi}$, one obtains the following
inflationary observables,

$\displaystyle n_{s}-1\equiv\frac{d\hskip
2.84544ptln\mathcal{P_{\mathcal{R}}}}{d\hskip 2.84544ptlnk}=-6\epsilon+2\eta$
(12) $\displaystyle dn_{s}^{k}\equiv\frac{dn}{d\hskip
2.84544ptlnk}=-2\xi+16\epsilon\eta-24\epsilon^{2}.$ (13)

Similarly we can compute the tensor power spectrum $\mathcal{P}_{T}$ for the
gauge invariant tenor perturbation $h_{ij}$. To quantify this, standard
practice is to define tensor-to-scalar ratio

$\displaystyle r=\frac{\mathcal{P}_{T}}{\mathcal{P}_{R}}=16\epsilon.$ (14)

Once we have all the expression for cosmological quantities in terms of slow
roll parameters, by using eqs.(7,10), and considering
$\phi_{*}\leq{\cal{O}}(1)$ in unit of $M_{p}$, we express
$(n_{s},r,dn_{s}^{k})$ in terms of $n$,$N$ and $\phi_{*}$, as

$\displaystyle 1-n_{s}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{2(n+1)}{(n+2)}\frac{1}{N}\\\
\frac{3}{2N}\end{cases}~{}~{};~{}~{}dn_{s}^{k}=\begin{cases}-\frac{(2+3n+n^{2})}{(n+2)^{2}}\frac{1}{N^{2}}\\\
-\frac{3}{4N^{2}}\end{cases}$ (15) $\displaystyle r$ $\displaystyle=$
$\displaystyle\begin{cases}8n^{2}\left(\frac{\phi_{*}}{M_{p}}\right)^{\frac{2n}{(n+2)}}\frac{1}{[n(n+2)]^{\frac{2(n+1)}{(n+2)}}N^{\frac{2(n+1)}{(n+2)}}}\\\
\frac{\phi_{*}}{M_{p}}\frac{n^{\frac{1}{2}}}{N^{\frac{3}{2}}}\end{cases}$

At this point, we want to emphasize the fact that the above expansions for all
the spectral quantities in large-$N$ limit may not always be valid for all
inflaton field values as has been pointed out recently in Martin:2016iqo .
This fact is indeed true if we look at the figs.(4,5), where, values of
$(n_{s},r)$ are deviating from the analytic expressions eq.15 for large
$\phi_{*}>1~{}M_{p}$. Therefore, for our model, above expansion in large-$N$
for $(n_{s},r,dn^{k}_{s})$ are valid only in the regime of small-field
inflation. Along the line of argument provided in Martin:2016iqo , we have
analytically shown our claim for $n=2$ in appendix-B.

From the above analytic expressions for $(n_{s},r,dn_{s}^{k})$, some important
observations are as follows: For type-II class of models, we see that the
value of $(n_{s},dn_{s}^{k})$ are insensitive to the value of $n$. This fact
can also bee seen from the fig.(5), where, no approximation has been made. In
particular one observes that for $\phi_{*}\gtrsim 10M_{p}$, value of
$(n_{s},r)$ start to deviate form each other for different values of $n$.
However, in general tensor to scalar ratio behaves as $r\propto n^{(1/2)}$ to
the leading order in $N$. Therefore, form the PLANCK observation considering
the upper bound $r<0.07$, we can put the upper limit on $n$ for a fixed value
of $\phi*$. From the current observations, it turns out to be very difficult
to uniquely fix the form of the potential. Therefore, we need more theoretical
inputs to figure out the full form of the potential. For type-I models, all
quantities are dependent on $n$, which can also be seen from fig.(4). However,
an important fact emerges in the limit $n\to\infty$ for Type-II potentials. If
we take $n\to\infty$ limit, the expressions of $(n_{s},r,dn_{s}^{k})$ reduce
to

$\displaystyle 1-n_{s}\to\frac{2}{N};~{}~{}~{}~{}r\to
8\left(\frac{\phi_{*}}{M_{p}}\right)^{2}\frac{1}{N^{2}}~{}~{};~{}~{}dn_{s}^{k}=-\frac{1}{N^{2}},$
(16)

which can be identified as a particular model within the class of recently
proposed ‘$\alpha$-attractor’linde . We have numerically checked the
aforementioned asymptotic limit of $n_{s}$ in terms of $n$, as can be seen in
fig.(2). As one observes, from eq.(II.2), for $N=50$, the scalar spectral
index $n_{s}\to 0.96$ as $n\to\infty$, which is the central value of PLANCK
observation. Therefore, for a wide range of parameter values, we have infinite
possible models corresponding to $n=2,3,4,5...$, which can successfully
explain cosmological observations made by PLANCK PLANCK . From the field
theory point of view, UV completion of our model is an important issue.
Specifically the supergravity formulation of those form of the potential could
be an important direction to study. We defer this for our future studies.

(a) Asymptotic behavoir of $n_{s}$ for model I (b) Asymptotic Behavior of r
for Model II

Figure 2: Variation of $n_{s}$ with $n$ for Model-I with $n=2,4,6,...$, as we
increase $n$, $n_{s}\to 0.96$ and variation of $r$ with $n$ for Model-II, it
is evident that the value of $r$ will satisfy the PLANCK bound for
$\phi_{*}\ll 1$

(a) Model I (b) Model II

Figure 3: Plot of $n_{s}$ vs $r$ when $\phi_{*}=0.1M_{p}$ for the two
potential plotted on Planck 2015 background, as we have seen in our
calculation that, for the second type of potential, the calculated quantities
are largely independent of $n$. While for the first potential type the change
of $n$ has significant effect on the spectral quantities.

(a) $n_{s}$ vs $log_{10}\phi_{*}$ for Model-I (b) $r$ vs $log_{10}\phi_{*}$
for Model-I

Figure 4: The dependence of $n_{s}$ and $r$ on the scale $\phi_{*}$ for fixed
number of efolding $(N=50)$ for Potential of type-I.

(a) $n_{s}$ vs $log_{10}\phi_{*}$ for Model-II (b) $r$ vs $log_{10}\phi_{*}$
for Model-II

Figure 5: The dependence of $n_{s}$ and $r$ on the scale $\phi_{*}$ for fixed number of efolding $(N=50)$ for Potential of type-II $\frac{\phi_{*}}{M_{p}}$ |  $n$ | Model I | Model II
---|---|---|---
|  |  $n_{s}$ |  $r$ |  $dn_{s}^{k}$ |  $\Delta\phi$ |  $n_{s}$ |  $r$ |  $dn_{s}^{k}$ |  $\Delta\phi$
0.01 |  2 |  0.969 |  $4\times 10^{-5}$ |  -0.00066 |  0.39 |  0.969 |  $4\times 10^{-5}$ |  -0.00066 |  0.39
|  4 |  0.966 |  $2\times 10^{-6}$ |  -0.00066 |  0.12 |  0.969 |  $5\times 10^{-5}$ |  -0.00060 |  0.47
|  6 |  0.965 |  $3\times 10^{-7}$ |  -0.00069 |  0.06 |  0.969 |  $7\times 10^{-5}$ |  -0.00060 |  0.51
|  8 |  0.964 |  $1\times 10^{-7}$ |  -0.00070 |  0.04 |  0.969 |  $8\times 10^{-5}$ |  -0.00060 |  0.55
1 |  2 |  0.969 |  $4\times 10^{-3}$ |  -0.0006 |  3.53 |  0.969 |  $4\times 10^{-3}$ |  -0.0006 |  3.53
|  4 |  0.966 |  $9.6\times 10^{-4}$ |  -0.0007 |  2.13 |  0.969 |  $6\times 10^{-3}$ |  -0.0006 |  4.0
|  6 |  0.964 |  $3.5\times 10^{-4}$ |  -0.0007 |  1.47 |  0.969 |  $7\times 10^{-3}$ |  -0.0006 |  4.3
|  8 |  0.964 |  $1.7\times 10^{-4}$ |  -0.0007 |  1.1 |  0.969 |  $8\times 10^{-3}$ |  -0.0006 |  4.7
Table 1: The spectral quantities for different values of $n$ for 50 efolding.
The two values of $\phi_{*}$ are chosen to illustrate that we can have both
small field and large field inflation depending on the value of $\phi_{*}$.
The general trend for the variation of these quantities with $\phi_{*}$ is
illustrated in the figures (4-5)

So far all we have discussed is directly related to the cosmological
observation made by PLANCK. Another important quantity of theoretical interest
we would like to compute is Lyth bound lythbound  $\Delta\phi$. This quantity
measures the difference of field values which is traversed by the inflaton
field during inflation. This is so calculated that for a particular model
$\Delta\phi$ is the maximum possible value for a particular efolding number.
Inflation is a semi-classical phenomena. It is believed that natural cut off
scale for any theory minimally or non-minimally coupled with gravity is Planck
scale $M_{p}$. Therefore, amount of inflaton field value can naturally be a
good measure to tell us the effective validity of a model under study in the
effective field theory language. Hence the calculated expression for the field
excursion in terms of $N$ and $\phi_{*}$ are:

$\displaystyle\Delta\phi\gtrsim
M_{p}N\sqrt{\frac{r}{8}}=\begin{cases}M_{p}\left(\frac{n}{n+2}\right)\frac{1}{\left[n(n+2)\right]^{\left(\frac{n+1}{n+2}\right)}}N^{\frac{1}{(n+2)}}\\\
\frac{M_{p}}{2}\left(\frac{\phi_{*}}{M_{P}}\right)^{\frac{1}{2}}N^{\frac{1}{4}}\end{cases}$
(17)

All the quantities we have discussed so far is independent of $m$ or
$\lambda$. (At this point let us again remind the reader that for $n\neq 4$,
$\lambda$ is a dimensionless quartic coupling parameter. While for $n\neq 4$,
$m$ is dimensionful parameter, and we set $\lambda=1$). However, comparing the
inflationary power spectrum with the PLANCK normalization we will determine
the value of $m$ or $\lambda$ and then calculate all the other quantities of
our interest. The expression for the power spectrum of the curvature
perturbation is

$\displaystyle\mathcal{P_{R}}=\begin{cases}\frac{\lambda}{12\pi^{2}n^{2}}\left(\frac{m}{M_{p}}\right)^{4-n}\left(\frac{\phi_{*}}{M_{p}}\right)^{\frac{n^{2}}{n+2}}\left[n(n+2)N\right]^{\frac{2(n+1)}{(n+2)}}\\\
\frac{2}{3\pi^{2}}\frac{\lambda}{\sqrt{n}}\left(\frac{m}{M_{p}}\right)^{(4-n)}\left(\frac{\phi_{*}}{M_{p}}\right)^{(n-1)}N^{\frac{3}{2}}\end{cases}=2.4\times
10^{-9}.$ (18)

As mentioned we considered the PLANCK normalization: $\mathcal{P}_{R}$ at the
pivot scale $k/a_{0}=0.05Mpc^{-1}$, and corresponding estimated scalar
spectral index is $n_{s}=0.9682\pm 0.0062$.

After having all our necessary expressions for all the cosmological
quantities, we have plotted our main results in $(n_{s},r)$ space and compared
with the experimental values $n_{s}=0.968\pm 0.006$ and upper limit on
$r<0.11$ in fig.(3). In the table-(3), we have given some sample values of all
the cosmologically relevant quantities for different values of theoretical
parameters. As we have mentioned already, we found infinitely many model
potentials with a universal shape. Most interesting case would probably be for
$n=4$. In the recent paper mhiggs , it has been identified as a minimal Higgs
inflation. Of course this identification is not straight forward. However for
small field value, we can certainly Taylor expand the potential, and identify
the coupling $\lambda$ as Higgs quartic coupling which can be set to its
electroweak value. However, renormalization analysis needs to be done in order
to do this identification.

At this point let us re-emphasize the fact that the observation made by PLANCK
strongly disfavors the usual power law inflation with $n\geq 2$. In this paper
we showed that problems of those power 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 $\phi*$ in
figs.(4-5). For both type of models, it is clearly matching with our
approximate analytic expression eq.(II.2). In the subsequent section we will
see how the reheating prediction will constrain the value of $N$ depending
upon the reheating temperature consistent with PLANCK.

###  II.3 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
$\phi_{*}$ because of the large amplitude. This is the stage during which non-
perturbative particle production will be effective. Therefore, resonant
particle production will take place and conversion of energy from the inflaton
to matter particles will be highly efficient. This phenomena is usually known
as pre-heating of the universe. In this section we will discuss about the late
time behaviour of the inflaton, specifically focusing on the dynamics of the
energy density of the inflaton field. After the many oscillations, when the
amplitude of the inflaton decreases much below the $\phi_{*}$, the dynamics
will be controlled by usual power law potential. As we have emphasized the
coherent oscillation is very important in standard treatment of reheating. For
any models of inflation this is thought to be an important criteria to have
successful reheating. In this section, we will first discuss the evolution of
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

$V(\phi)=\lambda m^{4-n}\phi^{n}.$ (19)

$n$ |  $\textit{w}=\frac{n-2}{n+2}$ |  $p=3(1+\textit{w})$ | p from fitting
---|---|---|---
|  |  |  Model I |  Model II
$2$ |  0 |  3 |  3.12 |  3.12
4 |  $\frac{1}{3}$ |  4 |  3.99 |  3.93
6 |  $\frac{1}{2}$ |  4.5 |  4.56 |  4.45
8 |  $\frac{3}{5}$ |  4.8 |  4.83 |  4.74
Table 2: The variation of inflation energy density with scale factor for
various potential

In cosmology for any dynamical field such as inflaton, one usually defines the
equation of state parameter w. For the oscillating inflaton, when the time
scale of oscillation about the minimum of a potential is small enough compared
to the background expansion time scale, by using virial theorem effective
equation of state for a potential of the form $V(\phi)\propto\phi^{n}$ can be
expressed asmukhanov.book

$\textit{w}\equiv\frac{P_{\phi}}{\rho_{\phi}}\simeq\frac{\langle\phi
V^{\prime}(\phi)\rangle-\langle 2V\rangle}{\langle\phi
V^{\prime}(\phi)\rangle+\langle 2V\rangle}=\frac{n-2}{n+2}.$ (20)

Therefore, in an expanding background, the evolution of energy density
$\rho_{\phi}$ of the inflaton averaged over many oscillation will follow,

$\displaystyle\dot{\rho}_{\phi}+3H(1+\textit{w})\rho_{\phi}=0.$ (21)

At late time we relate the energy density$(\rho_{\phi})$ of the universe
(assuming that the universe is dominated by a single component) and the scale
factor $(a)$ as

$\rho_{\phi}\propto a^{-3(1+\textit{w})}=a^{-p}.$ (22)

In the table-2, we provide some theoretical as well as numerically fitting
values corresponding to the equation of state parameter w of the inflaton and
the power law evolution of the energy density namely the value of $p$.

Figure 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 $\phi_{*}=0.01$.
time is measure in the unit of $(\frac{m^{2}}{\phi_{*}})$

In the following sections will be considering those equation of state
parameters and study their role in the subsequent cosmological evolution. We
will first discuss about the constraint on reheating phenomena by taking the
model independent approach, where explicit dynamics during reheating phase
will not be considered.

##  III Pre-heating: parametric resonance and their dependence on inflation
scale $\phi_{*}$

Reheating is an important phase of the early universe, when all the matter
field is assumed to be produced from the decay of inflaton. Initial study on
this mechanism was based on the perturbative quantum field theory
Albrecht:1982mp ; Dolgov:1982th ; Abbott:1982hn ; Traschen:1990sw . However,
it was soon realized that this approach may not be efficient enough for
successful reheating. In general reheating phenomena is a complicated non-
linear dynamics of inflaton coupled with matter fields at finite temperature
and the process of their thermalization. In the seminal work by Kofman, Linde
and StarobinskyKofman:1994rk ; Kofman:1997yn  (see alsoShtanov:1994ce ), the
idea of non-perturbative resonant production of particles has been
introducedLandau:mec . Though the full non-linear theory of preheating is
still not well understood but a significant advancement in this field has been
made and lot of works are going on Bassett:2005xm ; Allahverdi:2010xz ;
Amin:2014eta . It is generally believed that the reheating phase usually
happens in two stages. In the first stage, particle production is due to
parametric resonance known as ‘preheating’ followed by the perturbative
reheating.

Therefore, in this section we discuss about non-perturbative particle
production via parametric resonance phenomena for a specific model ($n=2$). It
is evident that initial few oscillations after the end of inflation play
important role at this stage. Therefore, we will see how the inflationary
scale $\phi_{*}$ which controls the shape of the potential, effects the
structure of the resonance for reheating fields. Hence, through resonance
structure, we may be able to further restrict the parameter space of 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 leading to the production and growth of quantum boson fields
via the bose condensation. We write the Lagrangian for the daughter scalar
field $\chi$ as.

$\mathcal{L}_{\chi}=\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi-\frac{1}{2}m_{\chi}^{2}\chi^{2}-\frac{1}{2}g^{2}\phi^{2}\chi^{2}.$
(23)

Where, $m_{\chi}$ is the mass of the $\chi$ particle. The matter field $\chi$
satisfies the following equation:

$\displaystyle{\ddot{\chi}}+3H\dot{\chi}-\frac{1}{a^{2}}\nabla^{2}\chi+(m_{\chi}^{2}+g^{2}\phi^{2})\chi=0.$
(24)

Decomposing the scalar field operator into Fourier modes,

$\displaystyle\chi(t,x)=\int\frac{d^{3}{\bf k}}{(2\pi)^{2/3}}[a_{\bf
k}~{}\chi_{k}(t)e^{i{\bf k}\cdot{\bf x}}+{a}_{\bf
k}\dagger~{}\chi_{k}(t)e^{-i{\bf k}\cdot{\bf x}}],$ (25)

the mode equation for $\chi_{k}(t)$ takes the following from

$\displaystyle\ddot{\chi}_{k}+3H\dot{\chi}_{k}+\left(\frac{k^{2}}{a^{2}}+m_{\chi}^{2}+g^{2}\phi^{2}\right)\chi_{k}=0.$
(26)

Where, $a_{\bf k},{a}_{\bf k}\dagger$ are the creation and annihilation
operators respectively. The parametric resonance phenomena with periodic
background force can be best explained though the stability/instability
diagram arising from the above generalized Mathieu equationMcLachlan ; MW . To
study this in the expanding cosmological background we rescale the field
variable $\chi_{k}$ and the inflaton field $\phi$ as follows

$\displaystyle\chi_{k}\to a^{-\frac{3}{2}}X_{k}~{}~{};~{}~{}\phi\to
a^{-\frac{3}{2}}\Phi.$

Where we have set the initial value of the scale factor to unity at the point
from where we started our numerical computation. With the above rescaling, the
mode Eq.(26) now turns out to be

$X_{k}^{\prime\prime}+\omega^{2}_{k}X_{k}=0$ (27)

where

$\omega^{2}_{k}\equiv\frac{k^{2}}{a^{2}m^{2}}+\frac{g^{2}\phi^{2}_{0}}{m^{2}a^{3}}\Phi^{2}(t)+\Delta~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Delta\equiv-\frac{3}{4}(3H^{2}+2\dot{H}),$

and “prime” is taken with respect to rescaled dimensionless time variable
$z=mt$. As we are only considering $n=2$, soon after the inflation ends, the
background evolution approximately satisfies $H^{2}\equiv\dot{H}\ll m^{2}$.
Therefore, we set $\Delta\simeq 0$ in the above eq.26 to simplify our
computation. Due to rescaling, the rescaled background inflaton field
oscillates with almost constant amplitude. We set $\phi_{0}$ 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 differential equation with parameters $\kappa=\frac{k^{2}}{m^{2}a^{2}}$
and $q=\frac{g^{2}\phi_{0}^{2}}{m^{2}a^{3}}$. The solution of this equation is
known to exhibit parametric resonance depending on the value of the parameters
$(q,\kappa)$. If the value of the parameters $(q,\kappa)$ is within certain
‘instability bands’ the solution of the eq.(27) grows exponentially as
$X_{k}\propto exp(\mu_{k}z)$. Where, Floquet exponent, $\mu_{k}$, parametrizes
the strength of the resonance. Therefore, depending upon the value of
$\mu_{k}$, corresponding $\chi$-particle of momentum $k$ will grow
exponentially. This indefinite growth of any mode is just the consequence of
neglecting the expansion of the universe as well as the back-reaction of the
produced $\chi$ particles. When the expansion is included the parameters in
the Hill’s equation becomes time dependent. However, it can be seen that the
relative change in $q$ during oscillation is

$\frac{1}{m}\frac{\dot{q}}{q}=-\frac{3H}{m}.$ (28)

During reheating period, $H\ll m$, hence, $q$ parameter can be taken as
constant.

The structure of resonance for chaotic inflationary model has been well
studied. As has been mentioned, the scale $\phi_{*}$ plays very important role
during inflation. Therefore, our main goal would be to understand the role of
$\phi_{*}$ on the structure of resonance. To compare the effect for different
$\phi_{*}$ we consider $n=2$, and measure time in unit of $m_{10}$ which is
the value of $m$ corresponding to $\phi_{*}=10M_{p}$. We present the contour
plots for the Floquet exponent in $(q,\kappa)$ space for different values of
$\phi_{*}$ as shown in fig.7.

Figure 7: Stability/instability charts for the non-perturbative production of
reheating field $\chi$ (Eq.(26) in $(\frac{k^{2}}{m^{2}},q)$ space.

We also show how the zero mode function $(X_{k=0})$ grows for $\phi_{*}=10$ in
fig.8 for two values of $q$, one taken from inside and another just outside
the unstable region.

Figure 8: The zero mode of the produced field ( $X_{k}=a^{\frac{3}{2}}\chi$)
for values of slightly inside (a) and outside (b) of the instability band.

The stability/instability chart has been computed for the first oscillation
taking $a(t_{0})=1$. Therefore, with time the magnitude of $(q,\kappa)$
parameters decrease which measures duration of preheating period.

Now we will examine the effects of $\phi_{*}$ on the structure of resonance.
The very first point that we would like to point out is the dependence of the
Floquet exponent on inflation scale $\phi_{*}$. We can clearly see that as we
decrease $\phi_{*}$ from $(10,1,0.1,0.01)$ in unit of Planck, the maximum
value of Floquet exponent increases as $(0.4,0.8,2.25,5)$ respectively.
Therefore, with decreasing value of $\phi_{*}$, the resonance becomes stronger
and simultaneously the band width also increases. This can also be seen from
the approximate analytic expression for the band width

$\displaystyle\Delta
k\simeq\left(\frac{m}{m_{10}}\right)^{\frac{1}{2}}(0.4g\dot{\phi})^{\frac{1}{2}}.$
(29)

Where, velocity of $\phi$ field is measured in unit of $m^{-1}$. However, in
order to compare the results for different $\phi_{*}$, we set a fixed time
scale $m_{10}$ corresponding to $\phi_{*}=10$. Hence, as we decrease
$\phi_{*}$, $m$ decreases from CMB normalization, which enhance the frequency
of the inflaton oscillation and consequently the resonance band width becomes
wider. Moreover, for large $\phi_{*}$, long wavelength modes of $\chi$ field
needs stronger coupling to get excited. From the effective field theory point
of view, small scale inflation suggests the parameter should be $\phi_{*}<1$
in unit of Planck mass. Further, the broad parametric resonance constrains the
coupling parameter $g\gtrsim(10^{-3},10^{-2})$ for
$\phi_{*}\simeq(0.1,0.01)M_{p}$. This is in sharp contrast with the usual
chaotic inflation, where inflationary observables do not have much effect on
the reheating coupling parameter $g$, and consequently the reheating
temperature if we consider the shifted minimum of the inflaton potential. We
will do detail 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 $\phi_{*}$ below $1M_{p}$, the
effective mass $m_{eff}^{2}=V^{\prime\prime}(\phi)$ of the inflation field
becomes negative in certain range of inflaton field values after the end of
inflation and remains so for first few oscillations. This will lead to
tachyonic preheating and will have important consequences specifically with
regard to the gravitational production. We will differ detailed study on this
issue for our future work.

##  IV  Model independent constraints from reheating predictions

After inflation, reheating is the most important phase, where, all the visible
matter energy will be pumped in. In this section, we will try to constrain our
model parameters without any specific mechanism of reheating. The background
evolution of cosmological scales from inflation to the present day and the
conservation of entropy density provide us important constraints on reheating
as well as our model parameters. Reheating is the supposed to be the integral
part of the inflationary paradigm. However, because of the single observable
universe, it is very difficult to understand this process by the present day
cosmological observation. Thermalization process erases all 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
liddle ; kamionkowski ; cook  through the evolution of cosmological scales and
the entropy density by parametrizing it by reheating temperature $(T_{re})$,
equation of state $(w_{re})$, and efolding number $(N_{re})$. In this section
we follow the reference debuGB  by taking into account the two stage reheating
phase generalizing the formalism of liddle . 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
be expressed in terms of two main parameters $(m~{}or~{}\lambda,\phi_{*})$ for
a particular model. Because of two stage reheating process, the suitable
reheating parameters are as follows,
$(N_{re}=N^{1}_{re}+N^{2}_{re},T_{re},w^{1}_{re},w^{2}_{re})$. Where,
$N^{1}_{re},N^{2}_{re}$ are efolding number during the first and second stage
of the reheating phase with the equation of state $w^{1}_{re},w^{2}_{re}$
respectively. At the initial stage the oscillating inflaton will be the
dominant 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 following particular case

$\displaystyle
w^{1}_{re}=\frac{n-2}{n+2}~{}~{};~{}~{}~{}w^{2}_{re}=\frac{1}{3}.$ (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 during usual cosmological evolution. This fact will provide us an
important relation among different phases of expansion parametrizing by
enfolding number as follows

$\displaystyle\ln{\left(\frac{k}{a_{0}H_{0}}\right)}=\ln{\left(\frac{a_{k}H_{k}}{a_{0}H_{0}}\right)}=-N_{k}-\sum_{i=1}^{2}N^{i}_{re}-\ln{\left(\frac{a_{re}H_{k}}{a_{0}H_{0}}\right)},$
(31)

In the above expressions, use has been made of $k=a_{0}H_{0}=a_{k}H_{k}$.
Where, $(a_{re},a_{0})$ are the cosmological 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 be used in the final numerical calculation,

$\displaystyle H_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{V(\phi_{k})}{3M_{p}^{2}}}=\begin{cases}\left(\frac{\lambda\phi_{*}^{n}}{3M_{p}^{2}}\right)^{\frac{1}{2}}\frac{m^{\frac{4-n}{2}}\tilde{\phi}_{k}^{\frac{n}{2}}}{\left(1+\tilde{\phi}_{k}^{n}\right)^{\frac{1}{2}}}\\\
\left(\frac{\lambda\phi_{*}^{n}}{3M_{p}^{2}}\right)^{\frac{1}{2}}\frac{m^{\frac{4-n}{2}}\tilde{\phi}_{k}^{\frac{n}{2}}}{\left(1+\tilde{\phi}_{k}^{2}\right)^{\frac{n}{4}}},\end{cases}$
(32) $\displaystyle N_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{M_{p}}\int_{\phi_{end}}^{\phi_{k}}\frac{1}{\sqrt{2\epsilon}}d\phi\simeq\begin{cases}\frac{\phi_{*}^{2}}{nM_{p}^{2}}\left[\frac{1}{(n+2)}\tilde{\phi}_{k}^{(n+2)}+\frac{1}{2}\tilde{\phi}_{k}^{2}\right]\\\
\frac{\phi_{*}^{2}}{nM_{p}^{2}}\left[\frac{1}{4}\tilde{\phi}_{k}^{4}+\frac{1}{2}\tilde{\phi}_{k}^{2}\right]\end{cases}$
(33)

Figure 9: Variation of $(N_{re}(\mbox{solid}),N_{k}(\mbox{dotted}),T_{re})$ as
a function of $n_{s}$ have been plotted for $\phi_{*}=0.01M_{p}$. This is the
plot for Model-I. (Blue, red, magenta, brown, green) curves correspond to
$n=(2,4,6,8,30)$. Each curve corresponds to a specific set of equation of
state parameters $(w^{1}_{re},w^{2}_{re})=((n-2)/(n+2),1/3)$ during reheating.
We also consider $N^{1}_{re}=N^{2}_{re}$. The light blue shaded region
corresponds to the $1\sigma$ bounds on $n_{s}$ from Planck. The brown shaded
region corresponds to the $1\sigma$ bounds of a further CMB experiment with
sensitivity $\pm 10^{-3}$ limit1 ; limit2 , using the same central $n_{s}$
value as Planck. Temperatures below the horizontal red line is ruled out by
BBN. The deep green shaded region is below the electroweak scale, assumed 100
GeV for reference.

$\phi_{k}$ and $\phi_{end}$ are the inflaton field values corresponding to a
particular scale $k$ crossing the inflationary horizon, and at the end of
inflation respectively. In the above expressions, we have ignored the
contribution coming from the inflaton field value $\phi_{end}$. It is
important to note that, in principle we can write the field value at a
particular scale $k$ in terms of $n_{s},r$, by inverting those relations.
Because of non-linear form, we will numerically solve those. The above unknown
efolding numbers during reheating will certainly be dependent upon the energy
densities $(\rho_{end},\rho_{re})$, at the end of inflaton (beginning of
reheating phase) and at the end of the reheating phase( beginning of the
standard radiation dominated phase);

$\displaystyle\ln\left(\frac{\rho_{end}}{\rho_{re}}\right)=3(1+w^{1}_{re})N^{1}_{re}+3(1+w^{2}_{re})N^{2}_{re}=3\sum_{i=1}^{2}(1+w^{i}_{re})N^{i}_{re}.$
(34)

Above two eqs.(31,34), can be easily generalized for multi-stage inflation
with different equation of state parameters. As has been mentioned, after the
end of reheating standard evolution of our universe is precisely known in
terms of energy density and the equilibrium temperature of the relativistic
degrees of freedom such as photon and the neutrinos. Therefore, the
equilibrium temperature after the end of reheating phase, $T_{re}$, is related
to temperature $(T_{0},T_{\nu 0})$ of the CMB photon and neutrino background
at the present day respectively, as follows

$\displaystyle
g_{re}T_{re}^{3}=\left(\frac{a_{0}}{a_{re}}\right)^{3}\left(2T_{0}^{3}+6\frac{7}{8}T_{\nu
0}^{3}\right).$ (35)

The basic underlying assumption of the above equation is the conservation of
reheating entropy during the the evolution from the radiation dominated phase
to the current phase. $g_{re}$ is the number of relativistic degrees of
freedom after the end of reheating phase. We also use the following relation
between the two temperatures, $T_{\nu 0}=(4/11)^{1/3}T_{0}$. For further
calculation, we define a quantity, $\gamma=N^{2}_{re}/N^{1}_{re}$. If we
identify the scale of cosmological importance $k$ as the pivot scale of
PLANCK, so that $k/a_{0}=0.05Mpc^{-1}$, and the corresponding estimated scalar
spectral index $n_{s}=0.9682\pm 0.0062$, we arrive at the following equation
for the efolding number during reheating period, and the reheating
temperature,

Figure 10: Variation of $(N_{re}(\mbox{solid}),N_{k}(\mbox{dotted}),T_{re})$
as a function of $n_{s}$ have been plotted for $\phi_{*}=0.01M_{p}$. This is
the plot for Model-II. (Blue, red, magenta, brown, green) curves correspond to
$n=(2,4,6,8,30)$. Each curve corresponds to a specific set of equation of
state parameters $(w^{1}_{re},w^{2}_{re})=((n-2)/(n+2),1/3)$ during reheating.
We also consider $N^{1}_{re}=N^{2}_{re}$. From the left figure, one clearly
sees that the behavior of $N_{k}$ is independent of $n$

$\displaystyle
N_{re}=\frac{4(1+\gamma)}{(1-3w_{re1})+\gamma(1-3w_{re2})}\left[61.6-\ln\left(\frac{V_{end}^{\frac{1}{4}}}{H_{k}}\right)-N_{k}\right]$
(36) $\displaystyle
T_{re}=\left[\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\frac{a_{0}T_{0}}{k}H_{k}e^{-N_{k}}\right]^{\frac{3[(1+w_{re1})+\gamma(1+w_{re2})]}{(3w_{re1}-1)+\gamma(3w_{re2}-1)}}\left[\frac{3^{2}.5V_{end}}{\pi^{2}g_{re}}\right]^{\frac{1+\gamma}{(1-3w_{re1})+\gamma(1-3w_{re2})}}.$
(37)

In the above derivation, we have used $g_{re}=100$. Before discussing any
further, let us provide the 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
$(w^{1}_{re},w^{2}_{re})=((n-2)/(n+2),1/3)$ in compatible with our model
discussed in the previous section. Important to mention regarding a special
point in the aforementioned state space is $(1/3,1/3)$ which is realized for
$n=4$. Analytically one can check that at this special point both
$(T_{re},N_{re})$ become indeterministic seen in eq.(37). This fact
corresponds to all the vertical solid red lines in $(n_{s}~{}vs~{}T_{re})$ and
$(n_{s}~{}vs~{}N_{re})$ plots. We have considered $\gamma=1$ as our arbitrary
choice. Each curve corresponds to different values of $n$. On the same plot of
$(n_{s}~{}vs~{}N_{re})$, we also plotted $(n_{s}~{}vs~{}N_{k})$ corresponding
to the dotted curves for different 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 model for different values of $n$ are almost
universal. However, prediction of $(T_{re},N_{re})$ are dependent upon the
value of $n$ through the equation of state parameter eq.(30). At this stage,
we would like to remind the reader again that for a wide range of $\phi_{*}$,
all the models predict very small value of tensor to scalar ratio $r$.
Therefore, we will be discussing all the constraints without explicitly
mentioning $r$. Given the overall description of all the plots, we now set to
discuss the prediction and constraints for two different models. In the
table-(3) we provide the important numbers for reheating temperature and the
efolding number. We provided only the limiting values of $T_{re}$ which are
still allowed from the cosmological observation.

Figure 11: Variation of $(N_{re}(\mbox{solid}),N_{k}(\mbox{dotted}),T_{re})$ as a function of $n_{s}$ have been plotted for for three different values of $\phi_{*}$. (Blue, magenta, purple) curves are for $\phi_{*}=(0.01,0.1,10)M_{p}$ respectively. We consider only $n=2$ for Model-I. All the other parameters remain the same as for the previous plots. $n$ | Model-I | Model-II
---|---|---
|  $n_{s}$ |  $T_{re}$(GeV) |  $N_{re}$ |  $N_{k}$ |  $n_{s}$ |  $T_{re}$(GeV) |  $N_{re}$ |  $N_{k}$
2 |  0.9723 0.9702 |  $1\times 10^{15}$ $1\times 10^{3}$ |  0.4 32 |  54 50 |  0.9723 0.9702 |  $2\times 10^{13}$ $1\times 10^{3}$ |  0.4 32 |  54 53
6 |  0.9670 0.9679 |  $1\times 10^{14}$ $2\times 10^{3}$ |  00 23 |  53 55 |  0.9724 0.9728 |  $4\times 10^{14}$ $4\times 10^{3}$ |  00 24 |  54 56
8 |  0.9659 0.9673 |  $7\times 10^{13}$ $1\times 10^{3}$ |  0.6 23 |  53 55 |  0.9725 0.9730 |  $3\times 10^{14}$ $2\times 10^{3}$ |  0.4 24 |  55 57
30 |  0.9625 0.9653 |  $6\times 10^{13}$ $1\times 10^{3}$ |  00 21 |  52 56 |  0.9726 0.9736 |  $6\times 10^{14}$ $1\times 10^{3}$ |  0.5 23 |  55 59
Table 3: Some sample values of $(n_{s},T_{re},N_{re},N_{k})$ are give for two
different models for $n=(2,6,8,30)$. As we have mentioned, for $n=4$,
$(T_{re},N_{re})$ become indeterministic. All these prediction are for
$\phi_{*}=0.01M_{p}$.

From the figure we see that for a very small change in $n_{s}$, the variation
of reheating temperature is very high. Therefore, reheating temperature
provides tight constraints on the possible values of efolding number $N_{re}$
during reheating phase. Except for $n=4$, if we restrict the value of
$T_{re}\gtrsim 10^{3}$ GeV, the efolding number turned out to be
$N_{re}\lesssim 35$ during reheating. As an example, for $n=2$, we find
spectral index lies within $0.9723\lesssim n_{s}\lesssim 0.9702$. Within this
range of spectral index, the reheating temperature has to be within $1\times
10^{15}\gtrsim T_{re}\gtrsim 1\times 10^{3}$ in unit of GeV. This restriction
in 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-3. What we can infer from our analysis in this section is that reheating
constraint does not allow $n$ to be vary large specifically for type-I model.
Whereas for type-II mode, the prediction of $n_{s}$ is almost independent of
$n$ for $\phi_{*}<M_{p}$ However in the figs.11,12,we have plotted the
dependence of various reheating parameters for different values of $\phi_{*}$.
We have plotted only for type-I model and $n=2,6$. For all the other models
qualitative behaviors of those plots will be same, except $n=4$.

With increasing value of the equation of state, efolding number during
reheating $N_{re}$, decreases for a fixed value of reheating temperature. This
essentially means that as one increases the value of inflationary equation of
state $w$, faster will be the thermalization process, therefore, earlier will
be the radiation dominated phase. In our subsequent full numerical solutions,
we have observed this fact considering the evolution of all the important
components during reheating. We also noted that as we decrease the value of
$\gamma$, the prediction of $(T_{re},N_{re})$ will be controlled by inflation
equation of state parameter $w^{1}_{re}$.

Figure 12: Variation of $(N_{re}(\mbox{solid}),N_{k}(\mbox{dotted}),T_{re})$
as a function of $n_{s}$ have been plotted for for three different values of
$\phi_{*}$. (Blue, magenta, purple) curves are for
$\phi_{*}=(0.01,0.1,10)M_{p}$ respectively. We consider only $n=6$ for
Model-I. All the other parameters remain the same as for the previous plots.

##  V Summary and Conclusion

Before we conclude, let us summarize the main results of our study. As
emphasized, we tried to constrain specific classes of inflationary models
based on the inflation and dark matter abundance. Effective field theory
consideration constraints $\phi_{*}$ to be less than unity in Planck unit. As
a result we have sub-Planckian field excursion during inflation. In this
regime of $\phi_{*}$ the value of inflationary observables
$(n_{s},r,dn_{s}^{k})$ saturate to a constant value depending on the e-folding
number $N$ and the power law index $n$. Furthermore, requirement of broad
parametric resonance constraints the coupling parameter $g\gtrsim 10^{-3}$ for
$n=2,\phi_{*}=0.1$. For fixed $n$, if we further reduce the value of
$\phi_{*}$, 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 the inflaton 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 with non-polynomial modification of the inflaton potential.
In the appendix we have tried to construct such potential from a most general
non-minimal scalar tensor theory. In certain region of the parameter space,
our models coincide with the aforementioned scalar-tensor theory. It would be
interesting to construct such potential from more fundamental approach.
Interesting property of these classes of potentials is that they have
infinitely large flat plateau. Therefore, the inflation can be naturally
realized because of this shift symmetry. As a result, the predictions of the
models for inflationary observables are not very much sensitive to the detail
form specifically near the minimum of the potential. Importantly our model
fits extremely well with latest cosmological observation made by PLANCK. All
the necessary scales assume below Planck scale value, which may imply that our
model predictions are robust against quantum correction. Detailed computation
on the ultra-violate effect on our model could be important and we left it for
our future work. Depending upon 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
significantly small. In the end we have studied model independent reheating
constraint analysis and discuss about 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 Hubble scale $H_{*}$ unlike the usual power law inflationary
model. The value could be as low as $\sim 10^{9}$ GeV for different values of
$n=4,6,8$ for $\phi_{*}\simeq 0.0001M_{p}$. It is well known that low value of
$H_{*}$ could be interesting in the context of Higgs vacuum instability. As
has been pointed out in 1407.3141 ; 1506.04065 , during as well as after the
inflation the quantum fluctuation of Higgs field can destabilize the standard
model metastable Higgs vacuum at around $\Lambda_{I}=10^{11}$ GeV. However,
this instability crucially depends upon the value of $H_{*}$, and also the
height of the Higgs potential. Therefore, comparing the naive scale dependence
between $\Lambda_{I}$ and $H^{*}$, our model have potential to save the Higgs
vacuum from decaying 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 and discussions.

##  Appendix A Towards derivation of our model potential

In this section starting from non-minimal scalar-tensor theory, we will try to
construct our model potentials which were a priori ad hoc in nature. As is
well known, inflationary models based on power law potential
$V(\phi)\sim\phi^{n}$ are simple but have been ruled out in general because of
their large prediction of tensor to scalar ratio. Moreover, the models with
large plateaus (Starobinsky or $\alpha$-attractors) are found to be most
favored form the PLANCK observation. While most of these plateau models can be
cast into exponential potential, plateau potentials with power-law form have
also been discussed in super gravityDimopoulos:2014boa ; Dimopoulos:2016zhy
and non-minimal coupling to gravityEshaghi:2015rta ; Broy:2016rfg . In this
section we will try to construct our model based on this non-minimally coupled
scalar-tensor theory. We will see, how simple power-law potentials in the
Jordan frame can give rise to the plateau potentials of desired form in the
Einstein frame. However, this transformed models coincide with our minimal
models only in a limiting regime (weak conformal coupling). At this point let
us point out that equivalence between the Einstein frame and Jordon frame is
an important question to ask. This issue has been discussed Futamase:1987ua ;
Kaiser:1994vs ; Hwang:1996np ; Deruelle:2010ht ; Postma:2014vaa ;
Banerjee:2016 ; Bhattacharya:2017pqc , from theoretical as well as
cosmological point of views.

Nevertheless, our motivation in this section is to construct our desired form
of the potentials which we have shown to be in different class of models
rather tan $\alpha$ attractor model. We start with the following non-minimally
coupled scalar-tensor theory,

$S_{J}=\int
d^{4}x\sqrt{-g}\left[\frac{\Omega(\varphi)}{2}M_{p}^{2}R-\frac{\omega(\varphi)}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V(\varphi)\right],$
(38)

where, $\Omega(\varphi),\omega(\varphi)$ are arbitrary function of a scalar
field $\varphi$. We will chose a specific form of those function for our later
purpose. To get the action in the Einstein frame, one performs the following
conformal transformation as,

$\tilde{g}_{\mu\nu}=\Omega(\varphi)g_{\mu\nu},$ (39)

The action in the Einstein frame can be written asFujii:2003pa

$S_{E}=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{p}^{2}}{2}\tilde{R}-\frac{1}{2}F^{2}(\varphi)\tilde{g}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\tilde{V}(\varphi)\right]$
(40)

Where, we have assumed that $\omega(\varphi)=\Omega(\varphi)$ and $F$ and the
new potential can be found to be,

$F^{2}(\varphi)=\frac{3M_{p}^{2}}{2}\frac{\Omega^{\prime
2}(\varphi)}{\Omega^{2}(\varphi)}+1~{}~{};~{}~{}\tilde{V}(\varphi)=\frac{V(\varphi)}{\Omega^{2}(\varphi)}$
(41)

Now, we choose the following non-minimal coupling function Kallosh:2013tua ;
Galante:2014ifa , for $\Omega^{2}(\varphi)$,

$\displaystyle\Omega^{2}(\varphi)=\begin{cases}1+\xi(\frac{\varphi}{M_{p}})^{n}\\\
\left[1+\xi(\frac{\varphi}{M_{p}})^{2}\right]^{\frac{n}{2}}.\end{cases}$ (42)

Therefore, applying (42), we find F and $\tilde{V}$ as,

$\displaystyle
F^{2}(\varphi)=\begin{cases}\frac{3n^{2}\xi^{2}\left(\frac{\varphi}{M_{p}}\right)^{2(n-1)}}{8\left[1+\xi\left(\frac{\varphi}{M_{p}}\right)^{n}\right]^{2}}+1\\\
\frac{3n^{2}\xi^{2}\left(\frac{\varphi}{M_{p}}\right)^{2}}{8\left[1+\xi\left(\frac{\varphi}{M_{p}}\right)^{2}\right]^{2}}+1\end{cases}~{}~{};~{}~{}\tilde{V}(\varphi)=\begin{cases}\frac{V(\varphi)}{1+\xi\left(\frac{\varphi}{M_{p}}\right)^{n}}\\\
\frac{V(\varphi)}{\left[1+\xi(\frac{\varphi}{M_{p}})^{2}\right]^{\frac{n}{2}}}\end{cases}$
(43)

We use the following field redefinition

$\frac{d\phi}{d\varphi}=F(\varphi)$ (44)

to transform the non-minimal into the action of a minimally coupled scalar
field with canonical kinetic term,

$S_{E}=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{p}^{2}}{2}\tilde{R}-\tilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\tilde{V}(\phi)\right]$
(45)

At this point we can integrate eq.(44), to find the new field in terms of the
old field, and construct the modified potential as a function of new field. It
is clear from the above set of transformations that for entire range of
parameter $\xi$, it is very difficult to reproduce our model. However, in the
regime of weak coupling $\xi<<1$, $F\sim 1$, hence we can approximately write,
using eq(44); $\varphi\sim\phi_{0}\phi$ ($\phi_{0}$ is some integration
constant). Considering Jordan frame potential as power-law:
$V(\varphi)\approx\varphi^{n}$, one gets plateau potential as

$\displaystyle\tilde{V}(\phi)=\begin{cases}\frac{\lambda~{}m^{4-n}\phi^{n}}{1+\left(\frac{\phi}{\phi_{*}}\right)^{n}}\\\
\frac{\lambda~{}m^{4-n}\phi^{n}}{\left[1+(\frac{\phi}{\phi_{*}})^{2}\right]^{\frac{n}{2}}},\end{cases}$
(46)

where, we identify $\phi_{*}$ as $M_{p}/\xi^{\frac{1}{n}}$ for Type-I
potential and $M_{p}/\xi^{\frac{1}{2}}$ for type-II potential. Therefore, in
the weak coupling regime, $\xi\ll 1$ or $\phi_{*}>1$, the non-minimal scalar
tensor theory can give rise to a large class of minimal cosmologies such as
ours which do not belong the $\alpha$-attractor model.

##  Appendix B Validity of $1/N$ expansion for $\phi_{*}<M_{p}$

In this section we consider $n=2$ case, as we can analytically compute the
expression for the power spectrum. Let us start by defining
$x\equiv\phi/\phi_{*}$, and using eq.(9), we define the number of efolding by
the following integral expression from field value $x_{k}$ for which scales
exit the horizon to the end of inflation with field value $x_{e}$,

$\displaystyle
N_{k}=\int\limits_{x_{k}}^{x_{e}}\left(\frac{\phi_{*}}{M_{p}}\right)^{2}\frac{V(x)}{V^{\prime}(x)}~{}dx.$
(47)

The $x_{k}$ corresponds to a scale $k$ which exits the horizon, and $x_{e}$ is
the reduced field value at the end of inflation( $\epsilon(x_{e})=1$ ).
Therefore, the field value at the horizon crossing turned out to be,

$x_{k}=\left[-1+\sqrt{1+f_{k}}\right]^{\frac{1}{2}},$ (48)

Where $f_{k}$ is given by

$f_{k}=8\frac{M_{p}^{2}}{\phi^{2}_{*}}N_{k}+x_{e}^{2}(2+x_{e}^{2}).$ (49)

The expression for $x_{e}$ is given by

$x_{e}=\frac{2^{1/6}\left(\frac{\phi_{*}}{M_{p}}\right)^{1/3}}{3\left[-1+\sqrt{1+\frac{2}{27}\left(\frac{\phi_{*}}{M_{p}}\right)^{2}}\right]^{1/3}}-\frac{\left[-1+\sqrt{1+\frac{2}{27}\left(\frac{\phi_{*}}{M_{p}}\right)^{2}}\right]^{1/3}}{2^{1/6}\left(\frac{\phi_{*}}{M_{p}}\right)^{1/3}}.$
(50)

In the limit $\phi_{*}<M_{p}$, which is necessary for $1/N_{k}$ expansion we
see from the last expression that $x_{e}$ simply reduces to

$x_{e}\sim
2^{1/6}\left(\frac{M_{p}}{\phi_{*}}\right)^{1/3}~{}~{};~{}~{}x_{k}\sim
f_{k}^{1/4}.$

Now the slow-roll parameters in terms of $x_{k}$ reduces to

$\displaystyle\epsilon\sim\frac{2M_{p}^{2}}{\phi_{*}^{2}}\frac{1}{f_{k}^{3/2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\left[8N_{k}\left(\frac{M_{p}}{\phi_{*}}\right)^{2}N_{k}+2^{2/3}\left(\frac{M_{p}}{\phi_{*}}\right)^{4/3}\right]^{3/2}}$
$\displaystyle\sim$
$\displaystyle\frac{1}{2^{7/2}}\frac{\phi_{*}}{M_{p}}\frac{1}{N_{k}^{3/2}}.$

The final term in the above expression is the leading order in $N_{k}$ for
$\phi_{*}<M_{p}$. Similarly, the second slow-roll parameter of our interest
reduces to

$\displaystyle|\eta|\sim\frac{3}{4}\frac{1}{N_{k}}$ (51)

All the above leading order expressions for the slow roll parameters match
exactly with our general expression for $n_{s}$ and $r$ given in terms of
$\epsilon$ and $\eta$ in Eqs.(15-II.2) and it is clear that for $\phi_{*}\ll
M_{p}$ $1/N_{k}-$expansion and slow-roll parameters are consistent.

Another interesting limit arises for $\phi_{*}\gg M_{p}$, in this case as
$\phi_{*}$ increases, $x_{e}\to 0$. Expanding eq.(48) for small $f_{k}(<1)$,
(which occurs when $\phi_{*}>2M_{p}\sqrt{2N_{k}}$) one arrives,

$x_{k}\sim\left(\frac{f_{k}}{2}\right)^{1/2},$ (52)

where $f_{k}=8N_{k}\frac{M^{2}_{p}}{\phi^{2}_{*}}$. Therefore, the slow roll
parameters turn into the following simple expressions to the leading order

$\displaystyle\epsilon\sim|\eta|\sim
2\frac{M_{p}^{2}}{\phi_{*}^{2}}\frac{1}{x_{k}^{2}}\sim\frac{1}{2N_{k}}$ (53)

This can be identified with the chaotic inflation limit. Indeed as we have
seen before for large value of $\phi_{*}$, the potentials can be represented
as $\phi^{n}$ during inflation. The results in this case are consistent with
large filed models producing large value of scalar-to-tensor ratio which
however, is in tension with Planck data. Finally we also check the validity of
our expansion numerically for other values of $n$.

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