#  On the running of the spectral index to all orders: a new model dependent
approach to constrain inflationary models

Moslem Zarei2,3 m.zarei[AT]cc.iut.ac.ir 2Department of Physics, Isfahan
University of Technology, Isfahan 84156-83111, Iran  3 School of Astronomy,
Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531,
Tehran, Iran

###### Abstract

In conventional model independent approaches, the power spectrum of primordial
perturbations is characterized by such free parameters as spectral index, its
running, the running of running, and the tensor-to-scalar ratio. In this work
we, show that, at least for simple inflationary potentials, one can find the
primordial scalar and tensor power spectra exactly by resumming over all the
running terms. In this model-dependent method, we expand the power spectra
about the pivot scale to find the series terms as functions of the e-folding
number for some single field models of inflation. Interestingly, for the
viable models studied here, one can sum over all the terms and evaluate the
exact form of the power spectra. This in turn gives more accurate
parametrization of the specific models studied in this work. We finally
compare our results with the recent cosmic microwave background data to find
that our new power spectra are in good agreement with the data.

##  I  Introduction

The recent precision data released by the Planck team favor inflation as a
paradigm for explaining not only the temperature fluctuations in the Cosmic
Microwave Background (CMB) but also large scale structure formation
Ade:2015lrj ; Ade:2015xua ; Ade:2015tva . As it is convenient, the Planck team
considers a power-law form for the spectrum of primordial curvature
perturbations as in (1) below,

$\mathcal{P}_{\mathcal{R}}(k)=\mathcal{P}_{\mathcal{R}}(k_{\ast})\left(\frac{k}{k_{\ast}}\right)^{n_{s}-1}~{},$
(1)

with scalar amplitude, $\mathcal{P}_{\mathcal{R}}(k_{\ast})$, and spectral
index, $n_{s}$, both defined at the pivot scale, $k_{\ast}$. Fitting with the
recent Planck data Ade:2015lrj ; Ade:2015xua  and the BICEP/Keck array
Ade:2015tva ; Array:2015xqh  gives $n_{s}$,
$\mathcal{P}_{\mathcal{R}}(k_{\ast})$, and tensor-to-scalar ratio $r$ at
$k_{\ast}=0.05\textrm{Mpc}^{-1}$ as follows

$\displaystyle
n_{s}=0.9655~{}\pm~{}0.0062~{}~{}~{}(68~{}\%~{}\textrm{CL},\textrm{Planck
TT+lowP}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ln\left(10^{10}\mathcal{P}_{\mathcal{R}}(k_{\ast})\right)=3.089\pm
0.036~{},$ and $\displaystyle
r<0.07\,(95\%\,\textrm{CL},\textrm{BICEP2/Keck})~{}.$ (2)

Considering $n_{s}$ and $r$ as model parameters, the Planck team first
produced the marginalized joint distribution probability regions for these
parameters. They then generated the parametric plot of the $n_{s}$ and $r$
parameters (predicted by different inflationary models) in the $n_{s}-r$ plane
of Planck, whereby they applied the first method to distinguish between the
models of inflation Ade:2015xua . This analysis showed that CMB temperature
fluctuations are consistent with the simplest inflationary models with
plateau-like inflaton potentials with $V_{\phi\phi}<0$ Ade:2015xua . These
models are single fields and predict a nearly Gaussian scale invariant power
spectrum. The Planck team also considered the running of the spectral index,
$d\,n_{s}/d\ln k$, as the next correction to the power law power spectrum and
found that, for $r=0$, the CMB data would favor the power spectrum with a
negligible running Ade:2015xua

$d\,n_{s}/d\ln k=-0.0084\pm 0.0082~{}~{}(68~{}\%~{}\textrm{CL},\textrm{Planck
TT+lowP})~{}.$ (3)

This value just improves the best fit likelihood by
$\Delta\chi^{2}\approx-0.8$ Ade:2015xua . However, considering the running of
running of the spectral index, $d^{2}\,n_{s}/d\ln k^{2}$, the Planck TT+lowP
data gives Ade:2015xua

$\displaystyle n_{s}=0.9569\pm 0.0077~{}~{}(68~{}\%~{}\textrm{CL})~{},$
$\displaystyle d\,n_{s}/d\ln
k=0.011^{+0.014}_{-0.013}~{}~{}(68~{}\%~{}\textrm{CL})~{},$ $\displaystyle
d^{2}\,n_{s}/d\ln
k^{2}=0.029^{+0.015}_{-0.016}~{}~{}(68~{}\%~{}\textrm{CL})~{},$ (4)

which improves the best fit likelihood by $\Delta\chi^{2}\approx-4.8$ at low
multipoles Ade:2015xua .

Another approach explored by the Planck team to disfavor/favor models of
inflation is based on the reconstruction of potential in the neighborhood of
$\phi_{\ast}$, with the value of $\phi$ being determined at the moment when
the mode $k_{\ast}$ exists the horizon for the first time Kinney:2005in ;
Kinney:2006qm ; Peiris:2006ug ; Lesgourgues:2007gp . Using this method, one
can constrain the observable part of the inflaton potential in the interval of
$\phi$ corresponding to the scales of CMB. The observable parameters $n_{s}$,
$d\,n_{s}/d\ln k$, and $r$ are given as a function of the slow-roll parameters
which are calculated at $k_{\ast}$. It will now be possible to include the
slow roll parameters in the parameter estimation process. Using the slow roll
flow equations Liddle:1994dx ; Kinney:2002qn , one can compute their values at
all times or for any values of $k$. The slow roll hierarchy can be truncated
at any given point and constraints can be imposed on the slow roll parameters
while there are no direct constraints on the relevant parameters $n_{s}$, $r$,
and $d\,n_{s}/d\ln k$. However, using the constraint on the slow roll
parameters, one can constrain these parameters to every order in the slow
roll. Moreover, the inflationary potential can be reconstructed to every order
of Taylor series around $\phi_{\ast}$ by assuming a flat prior on the slow
roll parameters Kinney:2005in ; Kinney:2006qm ; Peiris:2006ug ;
Lesgourgues:2007gp .

The third approach proposed in Mortonson:2010er  considers the specific
inflationary models with one or more free parameters. It is assumed that the
reheating phase is also controlled by other free parameters. In this method,
the power spectrum is calculated numerically without assuming slow roll
approximation. This can be performed by adopting numerical codes for an
efficient computation of inflationary perturbations. In order to detail the
end of the inflation, one can consider at least three possibilities according
to different entropy generation mechanisms Planck:2013jfk . This method
significantly reduces the theoretical uncertainty in the parameter space of
inflationary models. However, the comparison between the constraints resulting
from this method and those of the standard analysis with $n_{s}$, $r$, and
$dn_{s}/d\ln k$ parameterizations is not straightforward.

In the present paper, a different approach in utilized to we constrain
inflationary models although it has similarities with the method reported in
Martin:2014vha  which is a model-dependent approach and in which the slow roll
approximation was used to calculate the power spectrum for different
inflationary models while the first few lowest-order terms are preserved. The
slow roll parameters were also calculated in terms of the parameters of each
model and the e-folding number $N$. The authors then compared the inflationary
models with CMB data and scanned the parameter space of each model. This model
dependent approach yielded more accurate parametrization for any model of
inflation. In the present work, we also use the slow roll approximation and
show that at least for some single field models, one can calculate the power
spectrum in terms of $N$ by resumming over all the running terms. The main
advantage of our work is that for these models we can calculate the scalar and
tensor power spectra as a function of $k$ with the e-folding number,
$N(k_{\ast})$, as the free parameter. In fact, we sum over the running terms
to all orders. There are no spectral indices, running, the running of running,
or tensor-to-scalar ratio parameters in the power spectra calculated in our
method. Using the CMB data analysis, one can find the best fit value of
e-folding number, $N(k_{\ast})$, for inflationary models and it will,
therefore, be possible to favor the models of inflation. Additionally, we
study the three types of single field inflation models. After fitting with the
data, the constraint on $N(k_{\ast})$ is determined which will be used to
calculate $n_{s}$ and the parameter $r$. It should noted that $n_{s}$ and $r$
derived in this method are all model-dependent while, in the conventional data
fitting of the Planck team, those parameters characterizing the primordial
power spectra are model-independent. Therefore, the meaning of the parameters
derived by these two methods are certainly different.

The paper is organized as follows: In section II, we review the derivation of
the scalar and tensor power spectra and find a representation for the spectral
indices $n_{s}(k)$ and $n_{t}(k)$ by expanding about the pivot scale
$k_{\ast}$. In section III, we discuss how the e-folding number, $N(k)$, is
generally constrained in the inflationary paradigm. We then turn to the
inflation models with chaotic, hill top, and Starobinsky potentials to find
their power spectra as a function of $N(k_{\ast})$. In section IV, we present
our CMB data analysis and compare the predictions of the new power spectra
with the results obtained from the conventional $\Lambda$CDM model.

##  II The Running of spectral index

We consider a single field inflation with the potential $V(\phi)$. By
convention, the slow roll parameters are defined as follows Lyth:1998xn

$\epsilon_{V}=\frac{M^{2}_{Pl}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\>\>\>\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>\>\>\>\eta_{V}=M^{2}_{Pl}\left(\frac{V^{\prime\prime}}{V}\right)~{},$
(5)

where, the primes denote the derivative with respect to the inflaton field
$\phi$. The slow roll condition is satisfied when $\epsilon_{V}\ll 1$ and
$|\eta_{V}|\ll 1$. The power spectrum of the curvature $\mathcal{R}$ can be
Taylor expanded in terms of $\ln\\!k$ around the pivot scale $k_{\ast}$
Lyth:1998xn

$\ln\mathcal{P}_{\mathcal{R}}(k)=\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})+\frac{d\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})}{d\ln
k}\ln\frac{k}{k_{\ast}}+\frac{1}{2\,!}\frac{d^{2}\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})}{d\ln
k^{2}}\ln^{2}\frac{k}{k_{\ast}}+\frac{1}{3\,!}\frac{d^{3}\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})}{d\ln
k^{3}}\ln^{3}\frac{k}{k_{\ast}}+\cdot\cdot\cdot~{}.$ (6)

Using the second term of this expansion, the scalar spectral index is defined
by $n_{s}(k_{\ast})\equiv 1+d\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})/d\ln k$.
Using the sequence of higher derivative terms, one can define the running and
the running of the running of the spectral index as in (7) and (8),
respectively

$\displaystyle\alpha_{s}(k_{\ast})\equiv\frac{dn_{s}}{d\ln
k}(k_{\ast})=\frac{d^{2}\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})}{d\ln
k^{2}}~{},$ (7) $\displaystyle\beta_{s}(k_{\ast})\equiv\frac{d^{2}n_{s}}{d\ln
k^{2}}(k_{\ast})=\frac{d^{3}\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})}{d\ln
k^{3}}~{},$ (8)

where, the running of running parameter, $\beta_{s}(k_{\ast})$, was first
measured by Planck satellite Planck:2013jfk . Using the same approach, one can
expand the tensor power spectrum $\mathcal{P}_{t}(k)$ about the pivot scale
$k_{\ast}$ as

$\ln\mathcal{P}_{t}(k)=\ln\mathcal{P}_{t}(k_{\ast})+\frac{d\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})}{d\ln
k}\ln\frac{k}{k_{\ast}}+\frac{1}{2\,!}\frac{d^{2}\ln\mathcal{P}_{t}(k_{\ast})}{d\ln
k^{2}}\ln^{2}\frac{k}{k_{\ast}}+\cdot\cdot\cdot~{},$ (9)

where, $\mathcal{P}_{t}(k_{\ast})=r\,\mathcal{P}_{\mathcal{R}}(k_{\ast})$ and
one can thus define the corresponding tensor spectral index
$n_{t}(k_{\ast})\equiv d\ln\mathcal{P}_{t}(k_{\ast})/d\ln k$ and its running
$\alpha_{t}(k_{\ast})\equiv d^{2}\ln\mathcal{P}_{t}(k_{\ast})/d\ln k^{2}$. On
the other hand, the standard calculations in the slow roll regime give the
curvature and tensor power spectra for single field models as follows

$\displaystyle\mathcal{P}_{\mathcal{R}}(k)\simeq\frac{2}{3\pi
M^{6}_{Pl}}\frac{V^{3}}{V^{\prime 2}}~{},$ (10)
$\displaystyle\mathcal{P}_{t}(k)\simeq\frac{16V}{3\pi M^{4}_{Pl}}~{},$ (11)

in which we have considered the power spectra up to the lowest powers of the
slow roll parameters. Accurate computations lead to the power spectra
including the next to the leading order corrections Stewart:1993bc . Using the
following relation

$\frac{d}{d\ln\,k}\simeq-M^{2}_{Pl}\,\frac{V^{\prime}}{V}\,\frac{d}{d\phi}~{},$
(12)

it will be possible to derive the scalar and tensor spectral indices and their
associated running terms in terms of the slow roll parameters up to the first
order in the slow roll parameters. With straightforward calculations, one can
show that

$\displaystyle n_{s}(k_{\ast})\simeq 1-6\epsilon_{V}+2\eta_{V}~{},$ (13)
$\displaystyle\alpha_{s}(k_{\ast})\simeq
16\eta_{V}\epsilon_{V}-24\epsilon_{V}^{2}-2\xi^{2}_{V}~{},$ (14)
$\displaystyle\beta_{s}(k_{\ast})\simeq
192\epsilon^{3}_{V}-192\epsilon_{V}^{2}\eta_{V}+32\epsilon_{V}\eta^{2}_{V}+24\epsilon_{V}\xi^{2}_{V}-2\eta_{V}\xi^{2}_{V}-2\varpi_{V}^{3}~{},$
(15)

where,

$\xi^{2}_{V}=M^{4}_{Pl}\frac{V^{\prime}\,V^{\prime\prime\prime}}{V^{2}}~{},\>\>\>\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>\>\>\>\varpi_{V}^{3}=M^{6}_{Pl}\frac{V^{\prime
2}\,V^{\prime\prime\prime\prime}}{V^{3}}~{},$ (16)

and for the tensor perturbations, we find Gorbunov:2011zzc

$\displaystyle n_{t}(k_{\ast})\simeq-2\epsilon_{V}~{},$ (17)
$\displaystyle\alpha_{t}(k_{\ast})\simeq
4\eta_{V}\epsilon_{V}-8\epsilon_{V}^{2}~{}.$ (18)

Now, the power spectrum of curvature and tensor perturbations can generally be
deduced in the following forms

$\displaystyle\ln\mathcal{P}_{\mathcal{R}}(k)=\ln\mathcal{P}_{\mathcal{R}}(k_{\ast})+(n_{s}(k_{\ast})-1)\ln\frac{k}{k_{\ast}}+\frac{1}{2\,!}\frac{dn_{s}(k_{\ast})}{d\ln
k}\ln^{2}\frac{k}{k_{\ast}}+\frac{1}{3\,!}\frac{d^{2}n_{s}(k_{\ast})}{d\ln
k^{2}}\ln^{3}\frac{k}{k_{\ast}}+\cdot\cdot\cdot~{},$ (19)
$\displaystyle\ln\mathcal{P}_{t}(k)=\ln\mathcal{P}_{t}(k_{\ast})+n_{t}(k_{\ast})\ln\frac{k}{k_{\ast}}+\frac{1}{2\,!}\frac{dn_{t}(k_{\ast})}{d\ln
k}\ln^{2}\frac{k}{k_{\ast}}+\cdot\cdot\cdot~{},$ (20)

where, the expansions (21) and (22) are effectively induced for the scalar and
tensor spectral indices respectively

$n_{s}(k)-1=n_{s}(k_{\ast})-1+\sum_{l=1}^{\infty}\frac{d^{l}n_{s}(k_{\ast})}{d\ln
k^{l}}\frac{\ln^{l}\frac{k}{k_{\ast}}}{(l+1)!}~{},$ (21)

and

$n_{t}(k)=n_{t}(k_{\ast})+\sum_{l=1}^{\infty}\frac{d^{l}n_{t}(k_{\ast})}{d\ln
k^{l}}\frac{\ln^{l}\frac{k}{k_{\ast}}}{(l+1)!}~{}.$ (22)

The parameters $n_{s}(k)$ and $n_{t}(k)$ are given based on the specific model
of inflation. The series can be truncated by putting a cutoff on $l$. As an
example, cutting off the series (21) at $l=2$ is identical to a scalar
spectral index with the running and the running of running corrections. For a
general $\Lambda$CDM model, the parameters such as $n_{s}(k_{\ast})$,
$n_{t}(k_{\ast})$, $\alpha_{s}(k_{\ast})$ and $\beta_{s}(k_{\ast})$ are taken
as free parameters with specific prior assumptions. However, as we will show,
for some specific models of inflation, we can basically calculate these
parameters in terms of the e-folding number $N_{\ast}\equiv N(k_{\ast})$.
Hence, they are not all free parameters of the model. Given that in the
$\Lambda$CDM model a set such as
$(\mathcal{P}_{\mathcal{R}}(k_{\ast}),r,n_{s},\alpha_{s},\beta_{s},\cdot\cdot\cdot)$
may represent the parameter space associated with the primordial
perturbations, the second model-dependent approach gives the free parameters
by the set $(\mathcal{P}_{\mathcal{R}}(k_{\ast}),N,\cdot\cdot\cdot)$ for some
specific models of inflation. Here, the dots denote the basic set of other
post-inflationary cosmological parameters.

##  III  Calculation of the running of spectral index to all orders

As already described in the previous section, for a given inflationary model,
the running of the spectral index can be calculated order by order. By the
term, order by order, we mean that one can calculate the derivatives for each
number of $l$ in the series (21) and (22). Accordingly, by summing over series
terms, one can find the spectral indices with the running corrections to all
orders. In order to find the series terms, we first have to write down the
slow roll parameters and the spectral indices in terms of the e-folding number
$N$. This parameter measures the accelerated expansion of universe during
inflation from a given time $t$ where a mode passes the horizon to time
$t_{e}$ denoting the end of inflation. For a single field model, $N$ is
defined as Lyth:1998xn

$N(\phi)=\int_{t}^{t_{e}}H(t)dt=M^{-2}_{Pl}\int_{\phi_{e}}^{\phi}\frac{V}{V^{\prime}}d\phi~{},$
(23)

where, $\phi_{e}$ is the physical clock that determines the end of inflation.
Here, we consider three different kinds of single field inflationary
potentials belong to the large field, small field, and the Starobinsky-like
models. In the rest of this section, we derive the spectral indices for these
models as a function of $N_{\ast}$. For a given model, we should first
calculate the inflaton field $\phi$ as a function of $N$. It is plausible to
find $\phi(N)$ for a wide range of inflation models Martin:2014vha . The next
step involves the calculation of $n_{s}$, $n_{t}$, and the running terms as a
function of $N$ for each model. One property of the three models studied here
is that one can compute the terms $d^{l}n_{s}(k_{\ast})/d\ln k^{l}$ and
$d^{l}n_{t}(k_{\ast})/d\ln k^{l}$ in (21) and (22) as functions of
$n_{s}(k_{\ast})$, and $n_{t}(k_{\ast})$ respectively. This is the necessary
condition for a given model to be studied by our method. One can factor out
the spectral index terms to sum up the series (21) and (22) in a simple way.

###  III.1  Chaotic potential

To begin with, we consider the chaotic potential $V=g\phi^{n}$ from the large
field model class where $n>0$ and $g$ is the coupling with the dimension
$[g]=(\textrm{mass})^{4-n}$. As has been asserted, comparison with the Planck
data disfavors the chaotic potential with $n=2$ although $n=1$ and $n=2/3$
models have better positions within the $95\%$ CL region Ade:2015xua . For the
epoch $N\sim\mathcal{O}(10)$ before the end of inflation, $\phi\gg\phi_{e}$;
one can, therefore, obtain the following result Peiris:2003ff ;
Gorbunov:2011zzc

$\phi(N)\simeq\sqrt{2nM^{2}_{Pl}N}~{}.$ (24)

On the basis of this approximation and using (5), (10), (11), and (16), we can
calculate the scalar and tensor power spectra as well as the slow roll
parameters as a function $N$

$\displaystyle\mathcal{P}_{\mathcal{R}}(N)\simeq\frac{2g}{3\pi
M^{6}_{Pl}n^{2}}\left(2nM^{2}_{Pl}N\right)^{\frac{n+2}{2}}~{},\>\>\>\>\>\>\>\>\>\mathcal{P}_{t}(N)\simeq\frac{16g}{3\pi
M^{4}_{Pl}}\left(2nM^{2}_{Pl}N\right)^{\frac{n}{2}}~{},$ (25)
$\displaystyle\epsilon_{V}\simeq\frac{n}{4N}~{},\>\>\>\>\>\>\>\>\>\eta_{V}\simeq\frac{n-1}{2N}~{},\>\>\>\>\>\>\>\>\>\xi_{V}^{2}\simeq\frac{(n-1)(n-2)}{4N^{2}}\>\>\>\>\>\textrm{and}\>\>\>\>\>\varpi_{V}^{3}\simeq\frac{(n-1)(n-2)(n-3)}{8N^{3}}~{}.$
(26)

We recall that the scalar and tensor spectral indices are given in terms of
slow roll parameters or equally through the derivative of the power spectrum.
Either way, one can show that

$n_{s}-1\simeq-\frac{n+2}{2N_{\ast}}~{},\>\>\>\>\>\>\>\>\>n_{t}\simeq-\frac{n}{2N_{\ast}}\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>r\simeq\frac{4n}{N_{\ast}}~{},$
(27)

where, the tensor-to-scalar ratio is given by $r=16\epsilon_{V}=-8n_{t}$.
Using the same method, substitution of equation (26) in (14) and (15) gives
$\alpha_{s}$ and $\beta_{s}$ as in the following

$\displaystyle\alpha_{s}\simeq\frac{1!\,(n_{s}-1)}{N_{\ast}}~{},\>\>\>\>\>\>\>\>\>\beta_{s}\simeq\frac{2!\,(n_{s}-1)}{N_{\ast}^{2}}~{}.$
(28)

Using equations (12) and (24) as well as the derivatives of the power spectrum
(7) and (8), we obtain the same results for $\alpha_{s}$ and $\beta_{s}$. We
can continue to derive the higher derivatives of the spectral index using this
approach. After some computations, the following general result is given
Kosowsky:1995aa

$\displaystyle\frac{d^{l}n_{s}}{d\ln
k^{\textit{l}}}\simeq\frac{l\,!\,(n_{s}-1)}{N_{\ast}^{l}}~{}.$ (29)

In a similar manner, for the tensor spectral index, we have

$\frac{d^{l}n_{t}}{d\ln k^{l}}\simeq\frac{l!\,n_{t}}{N_{\ast}^{l}}~{}.$ (30)

Returning to the series (21) and (22), we see that substituting the results
(29) and (30) in the relevant equations and making the summation will yield

$\displaystyle n_{s}(k)-1$ $\displaystyle\simeq$
$\displaystyle(n_{s}(k_{\ast})-1)\sum_{l=0}^{\infty}\frac{x^{l}}{l+1}$ (31)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=\frac{n+2}{2N_{\ast}}\frac{\ln(1-x)}{x}~{},$

and

$\displaystyle n_{t}(k)$ $\displaystyle\simeq$
$\displaystyle\frac{n}{2N_{\ast}}\frac{\ln(1-x)}{x}~{},$ (32)

where, $x\equiv\ln(k/k_{\ast})/N_{\ast}$. The tensor-to-scalar ratio can in
general be determined as a function of $k$; however, we define it at the point
of $k=k_{\ast}$ with no running

$\displaystyle r$ $\displaystyle=$
$\displaystyle-8\,n_{t}(k_{\ast})\simeq\frac{4\,n}{N_{\ast}}~{}.$ (33)

The key ingredient in deriving the parameters $n_{s}(k)$, $n_{t}(k)$, and $r$
is, therefore, the model-dependent parameter $N_{\ast}$ which can practically
be regarded as a free parameter of the model. Compared with the conventional
model-independent approach, there are no spectral indices, running
corrections, or tensor-to-scalar ratio in this case but only the e-folding
number $N_{\ast}$ takes part in the parameter space. This point will be
further investigated by data analysis at the end of this paper.

###  III.2  Starobinsky Model

Another class of inflationary models involves the extension of theories of
gravity up to the $R^{2}$ term known as $R^{2}$ inflation Starobinsky:1980te ;
Mukhanov:1981xt  It has been shown that, in the Jordan frame this model can be
described by the following potential Maeda:1987xf

$V(\phi)=\frac{1}{8\,\alpha}\left(1-e^{-\sqrt{2/3}\,\phi}\right)^{2}~{},$ (34)

where, $\alpha$ is the coupling of $R^{2}$ term. There is an excellent
agreement between this model and the Planck data Ade:2015xua . This is due to
the small value of $r$ predicted by model, which indicates the tiny
gravitational waves. For this potential, one can also show that Lyth:2009zz ;
Hinshaw:2012aka

$\phi(N)\simeq\sqrt{\frac{3}{2}}\ln\frac{4N}{3}~{},\>\>\>\>\>\>\>\>\>\>\>\>\>\>n_{s}-1\simeq-\frac{2}{N_{\ast}}\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>n_{t}\simeq-\frac{3}{2N_{\ast}^{2}}~{}.$
(35)

Employing the method described above yields

$\frac{dn_{s}}{d\ln
k}\simeq-\frac{2}{N_{\ast}^{2}}~{},\>\>\>\>\>\>\>\>\>\frac{d^{2}n_{s}}{d\ln
k^{2}}\simeq-\frac{4}{\,N_{\ast}^{3}}\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>\frac{d^{l}n_{s}}{d\ln
k^{l}}\simeq\frac{l!\,(n_{s}-1)}{\,N_{\ast}^{l}}~{}.$ (36)

Furthermore, we can calculate

$\frac{dn_{t}}{d\ln
k}\simeq-\frac{3}{N_{\ast}^{3}}~{},\>\>\>\>\>\>\>\>\>\frac{d^{2}n_{t}}{d\ln
k^{2}}\simeq-\frac{9}{\,N_{\ast}^{4}}\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>\frac{d^{l}n_{t}}{d\ln
k^{l}}\simeq\frac{(l+1)!\,(n_{t})}{\,N_{\ast}^{l}}~{}.$ (37)

Therefore, using the series (21) and (22), we find

$\displaystyle
n_{s}(k)-1\simeq\frac{2}{N_{\ast}}\frac{\ln(1-x)}{x},\>\>\>\>\>\>\>\>\>\>\>\>\>\>\textrm{
and}\>\>\>\>\>\>\>\>\>\>\>\>\>\>n_{t}(k)\simeq-\frac{3}{2N^{2}_{\ast}}\frac{1}{(1-x)}~{},$
(38)

and the tensor-to-scalar ratio for this case will be $r\approx
12/N^{2}_{\ast}$. Here, the $k$-dependence of the scalar spectral index is the
same as that if the chaotic potential except that new forms are derived for
the $n_{t}(k)$ and $r$. Similar to the previous example, the parameter
$N_{\ast}$ remains as a key element in the parameter space.

###  III.3  Hill top potential

In what follows, an example will be examined from the small field models and
the symmetry breaking hill top inflaton potential will be investigated

$V=\Lambda^{4}\left(1-\frac{\phi^{p}}{\mu^{p}}\right)~{},$ (39)

where, $\Lambda$ is the coupling constant and $\mu$ is the value of inflaton
at the minimum of the potential. We are interested in the potentials with
$p\geq 3$. In this model, the slow roll parameters and scalar spectral tilt
are given in terms of the number of inflationary e-folds as follows
Peiris:2003ff

$\phi(N)\simeq\left(\frac{\mu^{p}}{p(p-2)M^{2}_{Pl}N}\right)^{\frac{1}{p-2}}~{},\>\>\>\>\>\>\>\>\>\epsilon_{V}\simeq\frac{p^{2}}{2}\frac{M^{2}_{Pl}}{\phi^{2}}\left(\frac{\phi}{\mu}\right)^{2p}~{},\>\>\>\>\>\>\>\>\>\eta_{V}\simeq-p(p-1)\frac{M^{2}_{Pl}}{\phi^{2}}\left(\frac{\phi}{\mu}\right)^{p}~{},\>\>\>\>\>\>\>\>\>n_{s}-1\simeq-\frac{p-1}{p-2}\frac{2}{N_{\ast}}~{}.$
(40)

Moreover, one can calculate

$\frac{dn_{s}}{d\ln
k}\simeq-\frac{p-1}{p-2}\frac{2}{N_{\ast}^{2}}~{},\>\>\>\>\>\>\>\>\>\frac{d^{2}n_{s}}{d\ln
k^{2}}\simeq-\frac{p-1}{p-2}\frac{4}{N_{\ast}^{3}}~{},\>\>\>\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>\>\>\>\frac{d^{l}n_{s}}{d\ln
k^{l}}\simeq\frac{l!\,(n_{s}(k_{\ast})-1)}{N_{\ast}^{l}(k_{\ast})}~{}.$ (41)

Therefore, the running of scalar spectral index to all orders is given by

$\displaystyle n_{s}(k)-1$ $\displaystyle\simeq$
$\displaystyle\frac{p-1}{p-2}\frac{2}{N_{\ast}}\frac{\ln(1-x)}{x}~{}.$ (42)

The spectral index of tensor perturbations is given in as the following

$\displaystyle n_{t}$ $\displaystyle\simeq$
$\displaystyle-\frac{p^{2}}{\hat{\mu}^{2}}\left[\frac{\hat{\mu}^{2}}{p(p-2)N_{\ast}}\right]^{\frac{2(p-1)}{p-2}}~{}.$
(43)

where, $\hat{\mu}=\mu/M_{Pl}$. Hence, the running of $n_{t}$ to all orders is
given by

$\displaystyle n_{t}(k)$ $\displaystyle\simeq$ $\displaystyle
n_{t}(k_{\ast})\left[1+\frac{2(p-1)}{2!\,(p-2)}\,x+\frac{2(p-1)(3p-4)}{3!\,(p-2)^{2}}\,x^{2}+\frac{4(p-1)(2p-3)(3p-4)}{4!\,(p-2)^{3}}\,x^{3}\right.$
(44)
$\displaystyle\left.\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>+\frac{4(p-1)(2p-3)(3p-4)(5p-8)}{5!\,(p-2)^{4}}\,x^{4}+\cdot\cdot\cdot\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!=n_{t}(k_{\ast})\frac{p-2}{p\,x}\left[1-(1-x)^{\frac{-p}{p-2}}\right]~{}.$

The tensor to scalar ratio parameter is fixed using the relation
$r=-8n_{t}(k_{\ast})$. Interestingly, the $n_{s}(k)$ parameter predicted by
the hill-top potential is the same as those given by the Starobinsky and
chaotic potentials. However, $n_{t}$ and $r$ in this model have new forms and
contain the dimensionless parameter $\hat{\mu}$ in addition to the $N_{\ast}$.
Therefore, the hill-top model contributes one more free parameter to the
parameter space.

##  IV The CMB data analysis

So far we have computed $n_{s}(k)$, $n_{t}(k)$, and $r$ as a function of
number of e-folding $N_{\ast}$ and the parameters of inflaton potentials.
Despite the possible uncertainties, one can impose upper and lower bounds on
$N_{\ast}$. The main uncertainty is due to the stage of reheating. However,
one can assume an instantaneous reheating to gain a maximum value for
$N_{\ast}$. For a comoving scale $k$ which crosses the Hubble radius during
inflation, the e-folding number $N(k)$ is given by Liddle:2003as ;
Dodelson:2003vq

$N(k)=-\ln\frac{k}{a_{0}H_{0}}+\frac{1}{3(1+w_{\textrm{reh}})}\ln\frac{\rho_{\textrm{reh}}}{\rho_{\textrm{end}}}+\frac{1}{4}\ln\frac{\rho_{\textrm{eq}}}{\rho_{\textrm{reh}}}+\frac{1}{2}\ln\frac{M_{Pl}^{4}}{2\,\rho_{\textrm{eq}}}+\frac{1}{2}\ln
3\pi^{2}r\mathcal{P}_{\mathcal{R}}(k_{\ast})+\ln 219\Omega_{0}h$ (45)

where, $a_{0}H_{0}$ is the present horizon scale, $\rho_{\textrm{end}}$ is the
energy density at the end of inflation, $\rho_{\textrm{reh}}$ is the energy
density of universe at the reheating stage, $H_{\textrm{eq}}$ denotes the
energy scale of equality, and $w_{\textrm{reh}}$ is the effective equation of
state between the energy scales $\rho_{\textrm{end}}$ and
$\rho_{\textrm{reh}}$. For an extreme case where reheating is instantaneous,
$\rho_{\textrm{end}}=\rho_{\textrm{reh}}$, and where there is no significant
reduction in energy density of inflation, so that
$\rho_{\textrm{end}}=3\pi^{2}M_{Pl}^{4}r\mathcal{P}_{\mathcal{R}}(k_{\ast})/2$,
we find a maximum e-folding number as follows

$N(k)=68.3-\ln\frac{k}{a_{0}H_{0}}+\frac{1}{4}\ln
r\mathcal{P}_{\mathcal{R}}(k_{\ast})~{},$ (46)

which gives $N(k_{0})\simeq 63$ for our present Hubble scale
$k_{0}=a_{0}H_{0}$ and with $r=0.2$ and
$\mathcal{P}_{\mathcal{R}}(k_{\ast})=2.2\times 10^{-9}$. Nevertheless, for a
prolonged reheating process, the expected number of reheating will be reduced.
Assuming $0\leq w_{\textrm{reh}}<1/3$ for the reheating phase, we can write

$N(k_{0})=63+\frac{1-3w_{\textrm{reh}}}{12(1+w_{\textrm{reh}})}\ln\frac{\rho_{\textrm{reh}}}{\rho_{\textrm{end}}}~{}.$
(47)

The reheating energy density is defined in terms of reheating temperature as
$\rho_{\textrm{reh}}=\pi^{2}g_{\textrm{reh}}T_{\textrm{reh}}/30$ where
$g_{\textrm{reh}}$, the effective number of relativistic species at the end of
inflation, is usually set to $g_{\textrm{reh}}\simeq 100$. We now rewrite
$N(k_{0})$ in terms of reheating temperature

$N(k_{0})=63+\frac{1-3w_{\textrm{reh}}}{3(1+w_{\textrm{reh}})}\ln\frac{T_{\textrm{reh}}}{10^{16.5}\,\textrm{GeV}}~{},$
(48)

where, for an instantaneous reheating and with $r=0.2$, we obtain the previous
result again. If the reheating process continues to the electroweak era such
that the reheating temperature drops to $T_{\textrm{reh}}\sim
10^{2}\,\textrm{GeV}$, the reduction in the e-folding number will be

$N(k_{0})\simeq 63-\frac{11(1-3w_{\textrm{reh}})}{1+w_{\textrm{reh}}}~{},$
(49)

and for $w_{\textrm{reh}}=0$, we find the minimum value of $N(k_{0})\simeq
52$. In the same way, if reheating lasts to the nucleosynthesis epoch with
$T_{\textrm{reh}}\sim 10^{-3}\,\textrm{GeV}$, we find $N(k_{0})\simeq 48$ by
assuming $w_{\textrm{reh}}=0$. On the other hand, as we can see from (48), the
equation of state $w_{\textrm{reh}}>1/3$ makes the e-folding number larger.
However, this possibility requires somewhat exotic model building. As a
result, for our present Hubble scale, $k_{0}$, the e-folding number is bounded
between $48-63$. For the smaller scales such as the pivot scale
$k_{\ast}=0.05$ which has always been of interest in the literature,
$N_{\ast}$ is confined between $44-59$. This knowledge about the limits on
$N_{\ast}$ is the key ingredient for our next data analysis. In the following,
we compute the Bayesian probability distribution of models in order to obtain
the constraints on the parameters of the models and $N_{\ast}$ by maximizing
the likelihood functions. The bound on $N_{\ast}$ should be consistent with
the reheating constraint.

After computing the $n_{s}(k)$ and $n_{t}(k)$ followed by $r$ for a given
model, it will be possible to parameterize the primordial power spectra of the
curvature and tensor perturbations as in the following

$\mathcal{P}_{\mathcal{R}}(k)=\mathcal{P}_{\mathcal{R}}(k_{\ast})\left(\frac{k}{k_{\ast}}\right)^{n_{s}(k)-1}\>\>\>\>\>\>\>\>\>\>\textrm{and}\>\>\>\>\>\>\>\>\>\>\mathcal{P}_{t}(k)=r\mathcal{P}_{\mathcal{R}}(k_{\ast})\left(\frac{k}{k_{\ast}}\right)^{n_{t}(k)}~{},$
(50)

where, as we already mentioned, the key parameter is the e-folding number
$N_{\ast}$. Therefore, unlike in the conventional method, fitting relation
(50) with the data in this model does not directly constrain the pair
$(n_{s}-r)$. Using the statistical Markov Chain Monte Carlo (MCMC) technique,
it will be possible to evaluate the likelihood function, $\mathcal{L}$, and
estimate the best fit of the cosmological parameters using CMB data. We adopt
the modified version of the CAMB Lewis:1999bs  and CosmoMC Lewis:2002ah  codes
to find the regions of the parameter space with the best fit. The CAMB code
generates the CMB angular power spectra from the primordial curvature and
tensor power spectrum. The modification of CAMB code is implemented in order
to include the power spectra derived throughout this paper. One can then
discriminate between various inflationary models discussed above by analyzing
the data and finding the minimum effective
$\chi^{2}\equiv-2\ln\mathcal{L}_{\textrm{max}}$ for each model. We investigate
the constraints on model parameters, in particular $N_{\ast}$. By fixing
$N_{\ast}$, it will be possible to calculate the spectral index $n_{s}$ or $r$
as a function of $N_{\ast}$. This only meant for comparison with the results
of $\Lambda$CDM model obtained directly from an MCMC analysis where the
$n_{s}-r$ couple is varied in the chain. We perform the numerical analysis for
each inflationary model described above by comparing their predictions with
the Planck TT + lowP and also the BICEP2, the Keck Array, and the Planck (BKP)
data combinations. So far, we have calculated the spectral indexes $n_{s}(k)$
and $n_{t}(k)$ for three different classes of inflationary models. In order to
perform the MCMC fit to the data, we first change the CAMB to include the new
power spectra with $n_{s}(k)$, $n_{t}(k)$ as given in equations (31), (32),
(38), (42), and (44). It should also be noted that for each model, we have to
take into account its relevant $r$ parameter computed above.

The $\Lambda$CDM model is explored by the parameter set
$(\Omega_{b}h^{2},\Omega_{c}h^{2},\tau,100\theta)$ where $\Omega_{b}h^{2}$,
$\Omega_{c}h^{2}$ are the baryon and cold dark matter densities today,
respectively, $\tau$ and $100\theta$ are the optical depth and the angular
size of the sound horizon at the time of last-scattering, respectively. The
primordial perturbations are parameterized by the set
$(\mathcal{P}_{\mathcal{R}}(k_{\ast}),\>N_{\ast})$ in the chaotic and
starobinsky models, and the set
$(\mathcal{P}_{\mathcal{R}}(k_{\ast}),\>N_{\ast},\>\hat{\mu})$ in the hill top
model. In our chains, we consider the flat priors just on the primordial
perturbation parameters. We fix the post-inflationary parameters to the
typical values consistent with the Planck 2015 data, namely
$\Omega_{b}h^{2}=0.022$, $\Omega_{c}h^{2}=0.12$, $\tau=0.1$ and
$100\theta=1.041$ Ade:2015lrj . The primordial Helium fraction is also fixed
to $Y_{P}=0.24$ Ade:2015lrj . We impose a flatness condition $\Omega_{k}=0$,
an adiabatic initial condition for inflation, and a massive neutrino with
$N_{\nu}=3.046$. The MCMC chains are also run with the Gelman and Rubin
criterion $R-1=0.02$ Rubin . We have also fixed the pivot scale to
$k_{\ast}=0.05\>\textrm{Mpc}^{-1}$.

Models |  |  | $-2\ln\,\mathcal{L}_{\textrm{max}}$
---
Planck+lowP BKP
|  | $N(k_{\ast})$
---
Planck+lowP BKP
|  | $n_{s}(k_{\ast})$
---
Planck+lowP BKP
|  | $r(k_{\ast})$
---
Planck+lowP BKP
Chaotic |  |  $n=$2/3
---
$n=$1
$n=$2
| 5635 5657
---
5636 5659
5637 5658
|  |  $32^{+12}_{-8}$ $31^{+15}_{-7}$
---
$32^{+15}_{-8}$ $30^{+16}_{-8}$
$66^{+27}_{-16}$ $34^{+11}_{-10}$
|  |  $0.96\pm 0.01$ $0.96\pm 0.01$
---
$0.95^{+0.02}_{-0.01}$ $0.95^{+0.02}_{-0.01}$
$0.97\pm 0.01$ $0.94^{+0.02}_{-0.02}$
|  |  $<0.1$ $0.1$
---
$<0.1$ $0.1$
$<0.1$ $0.2$
Hilltop |  | p=3
---
p=4
| 5633($\hat{\mu}$=9.5) 5656($\hat{\mu}$=15)
---
5633($\hat{\mu}$=12) 5657($\hat{\mu}$=4.6)
|  |  $115^{+30}_{-26}$ $89^{+42}_{-22}$
---
$85^{+50}_{-19}$ $91^{+45}_{-24}$
|  |  $0.95^{+0.01}_{-0.02}$ $0.96^{+0.01}_{-0.02}$
---
$0.98\pm 0.01$ $0.98\pm 0.01$
|  |  $<0.004$ $<0.001$
---
$<0.01$ $<0.0002$
Starobisky |  | 5633 5657 |  |  $63^{+19}_{-18}$ $58^{+24}_{-14}$ |  |  $0.97\pm 0.01$ $0.97\pm 0.01$ |  |  $<0.003$ $<0.004$
Table 1: Marginalized $1\sigma$ confidence level limits for the parameters of inflationary models in Planck TT + lowP and BKP data set combinations, given with respect to the mean value. Note that the best fit value of $\mathcal{P}_{\mathcal{R}}(k_{\ast})=2.2\times 10^{-9}$ is found for all the models investigated.  Data |  |  | $-2\ln\,\mathcal{L}_{\textrm{max}}$
---
|  | $n_{s}(k_{\ast})$
---
|  | $\alpha_{s}(k_{\ast})$
---
|  | $\beta_{s}(k_{\ast})$
---
|  | $r(k_{\ast})$
---
Planck+lowP |  | 5634 |  | $0.96\pm 0.01$ |  | $\approx 2\times 10^{-2}$ |  | $\approx 3\times 10^{-2}$ |  | $<0.01$
BKP |  | 5657 |  | $0.96\pm 0.01$ |  | $\approx 1\times 10^{-2}$ |  | $\approx 4\times 10^{-2}$ |  | $<0.03$
Table 2: Marginalized $1\sigma$ confidence level limits for the parameters of
the $\Lambda$CDM model in Planck TT + lowP and BKP data set combinations, with
respect to the mean value. For this model, we also find the best fit value of
$\mathcal{P}_{\mathcal{R}}(k_{\ast})=2.2\times 10^{-9}$.

The results of our analysis are summarized in Table I where the $\chi^{2}$ and
constraint on $N_{\ast}$ are shown for different inflationary models. Using
the value of $N_{\ast}$ given for each model, we have computed the $n_{s}$ and
$r$ parameters for comparison with the results of the $\Lambda$CDM model
summarized in Table II. It should be noted that $n_{s}$ and $r$ in Tables I
and II certainly mean different things. In the following, $\Lambda$CDM model
represents a cosmology with the primordial curvature perturbation parameters
$(\mathcal{P}_{\mathcal{R}}(k_{\ast}),r,n_{s}(k_{\ast}),\alpha_{s}(k_{\ast}),\beta_{s}(k_{\ast}))$
and with the post-inflationary parameters as quantified above. In our data
analysis, the best fit value of the amplitude
$\mathcal{P}_{\mathcal{R}}(k_{\ast})$ is found to be
$\mathcal{P}_{\mathcal{R}}(k_{\ast})=2.2\times 10^{-9}$ for all the
inflationary models as well as for the $\Lambda$CDM one. It should be noted
that the results in Table II for $r$ are new in the literature since the
Planck 2015 analysis of $r$ only takes into account, at most, $\alpha_{s}$ but
does not $\beta_{s}$.

The chaotic potential was studied for $n=1,\>2/3,\>2$. The powers $n=1,\>2/3$
are interesting in monodromy inflation models Flauger:2009ab ;
Silverstein:2008sg . The constraint on the number of e-folding $N_{\ast}$ is
obtained using the Planck TT + lowP and the BKP data sets. We find large error
bars on $N_{\ast}$ because we have summed over all the running terms while the
current data cannot constrain the running terms beyond $\beta_{s}$. The best
fit point of $N_{\ast}$ for the $n=2$ model lies outside the expected range of
$44-59$ for the $k_{\ast}$ scale. Moreover, taking into account the goodness
of fit parameter, $\chi^{2}$, the $n=1,\>2/3$ model gives better fits to the
Planck + lowP data. The $n=4$ inflation model is disfavored due to its worse
$\chi^{2}$ value and prediction of large value for $N_{\ast}$; hence is not
reported here. The spectral indices $n_{s}(k_{\ast})$ and $r(k_{\ast})$ can be
calculated using the best fit values of $N_{\ast}$. It should be noted again
that the these parameters are different in nature from those considered by the
$\Lambda$CDM model. We can go one step further and compute the
$\alpha_{s}(k_{\ast})$ and $\beta_{s}(k_{\ast})$ parameters for each model.
Using this procedure, we are now able to compare among these models of
inflation and the best fits of the $\Lambda$CDM model provided in Table II.
The results of the hill top potential have also been reported concisely in
Table I. The parameter space for this model is extended by one more parameter
$\hat{\mu}$, which justified the reduced value of $\chi^{2}$ for this model.
However, the best fit values of $N_{\ast}$ are not within the expected range
$44<N_{\ast}<59$ for either of the $p=3\>\textrm{or}\>4$ models. For the
Starobinsky model, we have a parameter space as large as that of the chaotic
model. However, we find a better fit with the Planck TT + lowP data. This
agreement is due to the the small gravity wave predicted by this model. When
we use the combined BKP data, $\chi^{2}$ becomes comparable with the
corresponding parameter in the other models. It is known that the reheating
temperatures allowed by the Starobinsky model is $T_{\textrm{reh}}\sim 3\times
10^{19}\>\textrm{GeV}$ Gorbunov:2010bn , which gives $N_{\ast}\sim 54.4$.
Considering the $1\>\sigma$ uncertainty, this theoretical prediction is
satisfied by the results reported in Table 1 for the Starobinsky model.

Figure 1: The angular TT power spectrum from the best fit of $\Lambda$CDM
model (solid black curve) in comparison with the Starobinsky potential (dashed
red curve) for the parameter values quoted in Tables I and II given using the
Planck TT + lowP data. The blue diamonds show the Planck 2015 data.

For the $\Lambda$CDM model, we estimate the scalar and tensor power spectra
using equations (19) and (20). With the Planck TT + lowP and BKP data sets,
one can constrain the spectral index $n_{s}(k_{\ast})$, the running of the
spectral index $\alpha_{s}(k_{\ast})$, and the running of running of the
spectral index $\beta_{s}(k_{\ast})$. The results are summarized in Table II.
Clearly, $\chi^{2}$ is better in this case than the corresponding values for
the chaotic potential presented in Table I. This can be explained by the
presence of more free parameters. For instance, in comparison with the chaotic
potential, the $\Lambda$CDM model has 3 additional parameters. Interestingly,
the Starobinsky model shows a better fit with fewer free parameters. Using two
data sets, we have also found negligible values for running,
$\alpha_{s}(k_{\ast})\approx 10^{-2}$, and $\beta_{s}(k_{\ast})\approx
10^{-2}$.

FIG. 1 presents the angular power spectra, $C_{l}$, for the CMB temperature
anisotropy. In this Figure, the solid black curve represents the $C_{l}$ for
the $\Lambda$CDM model with the best fit parameters obtained using the Planck
TT + lowP data. The red dashed curve represents the $C_{l}$ for the
Starobinsky potential. To obtain this curve, we have changed the CAMB to
contain the scalar and tensor power spectrum related to the Starobinsky model
which was calculated in previous section using the best fit value of
$N_{\ast}=63$. We have also plotted the CMB Planck 2015 data for comparison.
As shown in FIG. 1, the suppression of $C_{l}$ predicted by the Starobinsky
model for $l<50$ is the most important factor in reducing the likelihood
compared to the $\Lambda$CDM model.

##  V Conclusion

In inflationary models, it is important to know how the model predictions can
be compared with observational data. A variety of methods have been proposed
in the literature for such comparisons. In this work, we presented a new
method for simple models of inflation. In this method, we first expanded the
logarithm of scalar and tensor power spectra around the pivot scale to found
the series terms for the single field inflation models as a function of
$N_{\ast}$. The expansion terms are denoted by the spectral index, the
running, the running of running, and so on. In order to find the series terms,
we had to solve the equations of motion for the inflaton field, $\phi$, at the
background to find $\phi$ as a function of the e-folding number, $N_{\ast}$,
and to calculate the power spectra as a function of $N_{\ast}$. Using this
approach, we were able to find the series terms in the final step. For the
three types of single field inflation models with simple potentials studied
here, we were able to evaluate the sums and find the spectral indexes and,
thereby, the power spectra as a function of $k$ with one or two free
parameters depending on the model. We compared such new power spectra with the
standard power spectra of the $\Lambda$CDM model consisting of
$n_{s}(k^{\ast})$, $\alpha_{s}(k^{\ast})$, $\beta_{s}(k^{\ast})$, and $r$
variables by using the modified version of CAMB and CosmoMC codes as well as
the recent Planck TT + lowP and BKP data. The goodness of fit of the specific
inflationary models with $N_{\ast}$ as the free parameter was found to be
comparable with that of the $\Lambda$CDM model for Planck TT + lowP and BKP
data sets. In the case of Starobinsky model, the new parametrization yielded a
better fit. Using this analysis, we were also able to constrain the e-folding
number for any inflation model. For an instantaneous reheating stage, one
expects the bound $44<N_{\ast}<59$. We compared the best fit values of
$N_{\ast}$ calculated for different models using this bound. It was found that
the models that predict the worst values of $\chi^{2}$ also predict the
$N_{\ast}$ parameter outside of this bound. We expect that our method of
analyzing the inflationary models can be extended to other models with more
complex potentials.

## Acknowledgment

The author would like to thank H. Firouzjahi, M. S. Movahhed, A. Abolhasani,
S. Baghram, M. H. Namjoo for various comments and useful discussions.

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