# Speed of sound and scalar spectral index: Reconstructing inflation and
reheating in a non-canonical theory

Ramón Herrera<EMAIL_ADDRESS>Instituto de Física, Pontificia
Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla 4059,
Valparaíso, Chile.  Carlos Ríos<EMAIL_ADDRESS>Instituto de Física,
Pontificia Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla
4059, Valparaíso, Chile.  Departamento de Enseñanza de las Ciencias Básicas,
Universidad Católica del Norte, Larrondo 1281, Coquimbo, Chile.

###### Abstract

In this article we analyze the reconstruction of inflation in the framework of
a non-canonical theory. In this sense, we study the viability of
reconstructing the background variables assuming a non-lineal kinetic term
given by $K(X,\phi)=X+g(\phi)X^{2}$, with $X$ the standard kinetic term
associated to the scalar field $\phi$ and $g(\phi)$ an arbitrary coupling
function. In order to achieve this reconstruction in the context of inflation,
we assume the slow-roll approximation together with the parametrization of the
scalar spectral index $n_{s}$ and the speed of sound $c_{s}$ as a function of
the number of $e-$folds $N$. By assuming the simplest parametrizations for
$n_{s}-1=-2/N$ and $c_{s}\propto N^{-\beta}$ with $\beta$ a constant, we find
the reconstruction of the effective potential $V(\phi)$ and the coupling
function $g(\phi)$ in terms of the scalar field. Besides, we study the
reheating epoch by considering a constant equation of state parameter, where
we determine the temperature and number of $e-$folds during the reheating
epoch in terms of the reconstructed variables and the observational
parameters. In this way, the parameter-space related to the reconstructed
inflationary model are constrained during the epochs of inflation and
reheating by assuming the current astronomical data from Planck and BICEP/Keck
results.

##  I Introduction

It is well known that the dynamics and evolution of the early epoch of the
universe can be specified by the hot big bang model, nevertheless, this model
posses different cosmological problems (flatness, horizon, monopole, etc.)
that the inflationary scenario resolves assuming an accelerated expansion
before to the radiation era 1A ; 2a ; 2b ; 3A . Here, the great importance of
introducing the inflationary epoch is that this stage gives an account of the
large-scale structure, as well it delivers a causal description of the
anisotropies observed in the cosmic microwave background (CMB) radiation 5 ; 6
; 7 .

In order to describe the inflationary epoch, we have different models that
give an account of an accelerated expansion during the early universe. In this
sense, we can mention those inflationary models in which the inflationary
phase is driven for a canonical or also a non-canonical scalar field 2a ; 8 ;
9 . In this respect, we can distinguish the k-essence inflation, in which the
action (or Lagrangian density) associated to k-essence introduces a
nonstandard higher order kinetic term or non-canonical term related with the
scalar field or inflaton field, see Refs.10 ; 11 . In this respect, a non-
lineal kinetic term characterized for an arbitrary function $K(X,\phi)$ which
is present in the Lagrangian density $\mathcal{L}(X,\phi)=K(X,\phi)-V(\phi)$,
where $X$ corresponds to the canonical kinetic term associated to scalar field
$\phi$ and $V(\phi)$ is the effective potential. In particular the simplest
non-lineal kinetic term is given by $K(X)=k_{n+1}X^{n}$ Mukhanov:2005bu ;
Li:2012vta , where $n$ corresponds to an integer number and the parameter
$k_{n+1}$ is a constant such that the quantity $k_{n+1}X^{n}$ has units of
$M_{p}^{4}$, with $M_{p}$ the Planck mass. The situation in which the non-
lineal kinetic term is given by $K(X,\phi)=K(\phi)X+X^{2}$, with $K(\phi)$ an
arbitrary function of the inflaton field was studied in the framework of warm
inflation in Ref.Peng:2016yvb . Besides, in order to study the k-inflation
different effective potentials related with the scalar field have been
analyzed in the literature in the framework of the slow-roll dynamics SL1 . It
is well known that in the framework of a non-canonical theory a non lineal
kinetic term introduces a reduced speed of sound $c_{s}$ smaller than the
speed of light ($c_{s}<1$). This reduction on $c_{s}$ generates a suppression
of the tensor to scalar ratio 11  as well as a large amount of non-
Gaussianities during inflation, see Refs. Chen:2006nt ; DeFelice:2011uc .
Thus, any non-canonical theory has an evolving speed of sound, being the
simplest the case in which this speed is a constant, see e.g.,
Refs.Sergijenko:2014pwa ; Kunz:2015oqa . In this sense, the sound speed should
always be subluminal and also a real quantity, since we do not expect that
whichever physical information to propagate faster than the speed of light in
a vacuum, with which $0<c_{s}^{2}\leq 1$. In particular, for the simplest non-
lineal kinetic term $K(X)=k_{n+1}X^{n}$, the speed of sound is given by
$c_{s}^{2}=\frac{1}{2n-1}=$ constant and it is reduced if the power $n$
becomes $n>1$, see e.g.,Pareek:2021lxz ; Herrera:2023mas . In relation to the
speed of sound, we can find in the literature some expressions, in order to
describe the early universe. In particular in Ref.Shi:2021tmn  the authors
utilize the ansatz for the speed of sound given by $c_{s}\propto(-\tau)^{s}$,
in which $\tau$ corresponds to the conformal time and the power $s$ a
constant, to study the duality of cosmological models during inflation. The
situation in which the speed of sound becomes $c_{s}^{2}\propto\rho^{1-2n}$,
with $n$ a constant associated to the non lineal kinetic term into the
Lagrangian was developed in Magueijo:2008pm  and the specific case in which
$n=-1/2$ the speed $c_{s}$ is proportional to the energy density $\rho$. In
Ref.Cai:2009hw  the authors analized different sound speeds $c_{s_{I}}\propto
H/t$ in a model of multiple Dirac-Born-Infeld (DBI) type actions with multiple
scalar fields. For another interesting variations of the speed of sound
associated to the early universe, see e.g., Tolley:2009fg ; Canas-
Herrera:2020mme ; Achucarro:2010da . In this form, the k-essence models allow
that the speed of sound associated to a non lineal kinetic term is smaller
than speed of light or equal to speed of light, see e.g.10 ; Deffayet:2011gz ;
Babichev:2007dw . In this context, in the literature these models become
attractive to describe the inflationary epoch as well the present era
characterized by the dark energy Chimento:2003ta ; Melchiorri:2002ux .

Additionally, in relation to the k-models an important consideration to take
into account is related with the speed gravitational waves and it is equal to
the speed of light. Thus, the speed gravitational waves in the k-models
coincides with the speed found from the detection made by GW170817 and the
$\gamma$-ray burstGW .

In relation to the early universe and in particular to the inflationary epoch,
the reconstruction of the background quantities, such as the effective
potential and the coupling functions using the observational variables, such
as the scalar spectrum, the scalar spectral index, and the tensor to scalar
ratio, have been studied by different authors in the modern cosmologyAu1 ; Au2
; Chiba .

In this respect, we can mention that the procedure of reconstruction of the
inflation, through the parametrization of the observational variables as a
function of the number of $e-$folds $N$ is developed under the slow-roll
approximation. Under this formalism, using the parametrization of the
observational parameters, we can say that the scalar spectral index $n_{s}$ in
terms of the number of $e-$ folds $N$ i.e., $n_{s}(N)$ is the most utilized.
Thus, the simplest parametrization for the spectral index $n_{s}(N)$
corresponds to $n_{s}(N)=1-2/N$ for large-$N$ and this index is well confirmed
by Planck results, in the case in which the number $N\simeq 55-70$. Here large
$N$ corresponds to values of the number
$N\sim\mathcal{O}(10)\sim\mathcal{O}(10^{2})$ generated during the slow roll
stage Chiba . In this respect, we assume that the number of $e-$folds $N\simeq
55-70$ denotes to the comoving scale that crossed the Hubble radius during the
inflationary era. In the frame of the general relativity (GR), the
parametrization of the scalar spectral index $n_{s}(N)=1-2/N$ gives rise to
the reconstruction of the different effective potentials associated to the
scalar field. In this sense, in the frame of GR the parametrization or
attractor characterized by the spectral index $n_{s}(N)=1-2/N$ gives rise to
the reconstruction of the effective potential as a function of the scalar
field such as; T-model, the E-model, $R^{2}-$model, the chaotic model, etc.
see Refs.Chiba ; M1 ; M2 . For the reconstruction of two variables associated
to the background inflation as in the case of the reconstruction of warm
inflation, it was required to utilize two parameters; the scalar spectral
index $n_{s}(N)$ and the tensor to scalar ratio $r(N)$ Herrera:2018cgi , see
also the reconstruction (two background variables) of $G-$inflation in
Ref.Herrera:2018mvo .

However, there are other parametrizations in terms of the number of $e-$folds
$N$ used in the literature, in order to rebuild the background variables
associated to the inflationary epoch in the context of slow-roll
approximation. For example, we mention the parametrization utilized on the
slow roll parameter $\epsilon(N)$, as a function of the number of $e-$ folds
$N$ Au2 , see also Es2 ; Es2a ; Es3 . Besides, in Ref.Roe  was studied the
reconstruction of the effective potential and the scalar spectral index
considering the parametrization the two slow-roll parameters; $\epsilon(N)$
and the another parameter $\eta(N)$. Similarly, the reconstruction of the
effective potential $V(\phi)$ (with $\phi$ the scalar field), utilizing the
speed of the scalar field in terms of $N$ as ansatz was analyzed in Ref.Seba .
For a review of another reconstructions using different quantities, see
Refs.Es4 ; Es8 ; Es9 ; Gonzalez-Espinoza:2021qnv ; Herrera:2022kes .

On the other hand, at the end of the inflationary era we have a phase of
reheating of the universe, in order to connect with the standard big-bang
model, see St1 . In this scenario of reheating of the universe, the matter and
the radiation are produced via the decay of the inflaton field or other
fields, whereas the temperature of the universe during this era increases in
magnitude and then the universe enters to the radiation dominated era. In
order to describe the reheating of the universe, there are different
mechanisms or reheating models to grow the temperature during the early
universe. In this sense, we have the mechanism related to the perturbative
decay of the scalar field from an oscillating process at the end of inflation
in which particles are producedR1 . Also, we have a mechanism associated to a
non-perturbative analysis as the model of instant reheatingR4  or as
parametric resonance decay of the inflaton or another fieldR2 . Besides, we
have some models of inflation in which the inflaton field does not oscillate
around of the minimum of the potential and these models are called non-
oscillating models (NO models) and the process of reheating takes place from
the decay of another field called curvaton R5 ; yo2 , see also Ref.R3  for the
reheating from tachyonic instability.

In relation to the reheating epoch, we have some important parameters that
characterize this scenario such as; the reheating temperature
$T_{\text{reh}}$, an equation of state (EoS) denoted as
$w_{\text{reh}}=p/\rho$ related to the matter content during the reheating era
and the duration of this stage which is distinguished by the number of
$e-$folds occurred during this scenario $N_{\text{reh}}$. Here, the quantities
$p$ and $\rho$ denote the pressure and energy density of the fluid associate
to the matter.

In the case of the reheating temperature, we have a lower bound imposed by
primordial nucleosynthesis (BBN), where the temperature in this period
$T_{\text{BBN}}\sim$ 10 MeV, see Ref.MeV . In relation to the EoS parameter
$w_{\text{reh}}$, several numerical analyzes were studied for the matter
content during the reheating epoch, in which different interactions are
realized between the scalar field and other fields during the reheating. In
general, we have that this EoS parameter is a function of the time for the
different process in the reheating era. Thus, in particular for the situation
in which we have a power law potential, the EoS parameter at the end of the
inflationary stage becomes $w_{\text{reh}}=0$ and then increase to positive
values where the parameter achieves the value $w_{\text{reh}}\sim 0.3$, see
Ref.Felder:2000hq . In the situation in which we have a massive field, the
authors in Ref.Felder:2000hq  found that the EoS parameter takes a negative
valor at the end of inflationary stage $w_{\text{reh}}=-1/3$ and then increase
to the value $w_{\text{reh}}=0$ for a later time, see also Dodelson:2003vq ;
Munoz:2014eqa . As a first approximation and following Ref.Munoz:2014eqa , we
will assume that the EoS parameter can be assumed independent of time i.e., a
constant parameter during this period, in order to find analytical expressions
for the temperature and $e-$folds during the reheating.

The main goal of the present research is to apply the reconstruction
methodology under the slow-roll approximation in the early universe,
considering as attractor the parametrization of the the scalar spectral index
$n_{s}(N)$ together with the speed of sound $c_{s}(N)$, in terms of the number
of $e-$folds $N$. In this context, from the framework of a non-canonical
theory, we will study a non-linear kinetic term defined as
$K(X,\phi)=X+g(\phi)X^{2}$, to rebuild the potential associated to the
inflaton field $V(\phi)$ and the coupling function $g(\phi)$. In this way, in
the context of this theory, we will study how the reconstruction of the
background variables are modified assuming the parametrizations on the
spectral index $n_{s}(N)$ and the speed $c_{s}(N)$, respectively. In this
sense, considering a general procedure, we will rebuild the effective
potential $V(\phi)$ and the coupling function $g(\phi)$ giving $n_{s}(N)$ and
the speed of sound $c_{s}(N)$.

Besides, we will analyze the reheating epoch and how the parameters during
this stage are modified from the reconstruction (background variables) found
during the slow-roll inflation. In relation to these reheating parameters, we
will determine the reheating temperature and the duration of the reheating
characterized through of the number of $e-$folds during the reheating and how
considering the observational parameters from Planck measurements these
reheating quantities are constrained.

We organize our article as follows: In Section II we give a brief description
of the inflationary phase in a non-canonical theory. Here we show the
background equations in the framework of the slow-roll approximation together
with the cosmological perturbations in this theory. In Section III we find
under a general formalism, expressions for the effective potential and the
coupling function associated to the non-lineal kinetic term as a function of
the number of $e-$folds $N$ to obtain the reconstruction from any
parametrization related to the scalar spectral index $n_{s}(N)$ as well as the
speed of sound $c_{s}(N)$. In Section IV, we consider a specific example for
both the scalar spectral index $n_{s}=n_{s}(N)$ given by $n_{s}=1-2/N$ and the
speed of sound $c_{s}(N)\propto N^{-\beta}$ with $\beta$ a constant parameter,
in order to reconstruct both the effective potential $V(\phi)$ as well as the
coupling function associated to the non-lineal kinetic term $g(\phi)$.
Besides, we obtain the constrains on the space-parameter for the different
quantities or parameters associated to our model considering the present
astronomical data from Planck and BICEP/Keck results. In Section V, we analyze
the reheating stage considering the background variables found in the previous
section. In this section, we find the reheating temperature together with the
number of $e-$folds during the reheating epoch. Finally, in Section VI we give
our conclusions and remarks. We chose units in which $c=\hbar=1$.

##  II The inflationary phase in a non-canonical theory

In this section we will give a brief description of theory of gravitation
described by a non-canonical kinetic term in the framework of the GR. In this
context, we start with the four dimension action $S$ given by K1 ; K2

$S=\int\left[\frac{R}{2\kappa}+\mathcal{L}(X,\phi)\right]\sqrt{-g}\,d^{4}\,x,$
(1)

in which $\kappa=8\pi G=M_{p}^{-2}$, where recalled that $M_{p}$ is the Planck
mass, $R$ denotes the scalar Ricci and $g$ is the determinant of the metric
$g_{\mu\nu}$. Besides, the quantity $\mathcal{L}(X,\phi)$ corresponds to the
Lagrangian density related with the scalar field $\phi$ and the kinetic energy
of the scalar field $X$ defined as
$X=(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi)/2$.

From the effective action given by Eq.(1), we can identify that the energy
density $\rho$ and the pressure $p$ as a function of the scalar field $\phi$
and $X$ assuming a perfect fluid for the matter becomeK1 ; K2

$\rho=2X\frac{\partial\mathcal{L}(X,\phi)}{\partial
X}-\mathcal{L}(X,\phi),\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,p=\mathcal{L}(X,\phi),$
(2)

respectively. Note that for the specific case in which
$\mathcal{L}(X,\phi)=X-V(\phi)$, with $V(\phi)$ the effective potential, these
quantities are reduced to the standard expressions for the energy density
$\rho$ and the pressure $p$ associated to a scalar field in the framework of a
canonical theory.

In relation to the Lagrangian density associated to the scalar field this
Lagrangian can be written as an expansion of the form K1

$\mathcal{L}(X,\phi)=\sum_{n\geq 0}g_{n}(\phi)X^{n+1}-V(\phi),$ (3)

where the quantities $g_{n}(\phi)$ correspond to arbitrary functions of the
scalar field $\phi$. As before, for the case in which we take $n=0$ and
$g_{0}(\phi)=1$, it is reduced to a canonical theory.

In the following we will consider the specific values of $n=0$ and $n=1$,
where the functions $g_{n}(\phi)$ are defined as $g_{0}(\phi)=1$ and
$g_{1}(\phi)=g(\phi)$, respectively. Here, just for simplicity and in order to
obtain analytical expressions for the reconstruction of the background
variables, we choose these two terms in the expansion given by Eq.(3). In this
way, the Lagrangian density associated to the inflaton field takes the
structure analyzed in Ref.K1  (see also Refs.KK1 ; KK2 )

$\mathcal{L}(X,\phi)=K(X,\phi)-V(\phi)=X+g(\phi)X^{2}-V(\phi),$ (4)

where $K(X,\phi)=X+g(\phi)X^{2}$ corresponds to an arbitrary function
associated to a non-lineal kinetic term and the function $g(\phi)$ has units
of $\kappa^{2}=M_{p}^{-4}$. In this form, from Eq.(4), we have that the energy
density and pressure are defined as

$\rho=X+3g(\phi)X^{2}+V(\phi),\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,p=X+g(\phi)X^{2}-V(\phi),$
(5)

respectively.

In order to study the early universe and its dynamics, we consider a spatially
flat Friedmann Robertson Walker metric, along with a homogeneous scalar field
i.e., $\phi=\phi(t)$. Thus, using the Friedmann equation given by
$H^{2}=(\kappa/3)\rho$, where $H=\dot{a}/a$ represents the Hubble parameter
and $a=a(t)$ denotes the scale factor, we have

$H^{2}=\frac{\kappa}{3}\left[\frac{1}{2}\dot{\phi}^{2}+\frac{3}{4}g\dot{\phi}^{4}+V\right].$
(6)

In the following, the dots represent differentiation with respect to the time
$t$ and $g(\phi)=g$.

Additionally, from the continuity equation defined as
$\dot{\rho}+3H(\rho+p)=0$ and considering Eqs.(2) and (4), this continuity
equation can be rewritten as

$\dot{X}=\frac{\sqrt{2X}c_{s}^{2}}{p_{X}}\left(2Xp_{X\phi}-p_{\phi}-3H\sqrt{2X}p_{X}\right),$
(7)

or equivalently

$\ddot{\phi}+\frac{3H\dot{\phi}(1+g\dot{\phi}^{2})}{1+3g\dot{\phi}^{2}}-\left[\frac{3/4g_{\phi}\dot{\phi}^{4}-V_{\phi}}{1+3g\dot{\phi}^{2}}\right]=0.$
(8)

Here the quantity $c_{s}^{2}$ is defined as

$c_{s}^{2}\equiv\frac{p_{X}}{\rho_{X}}=\left(1+\frac{2Xp_{XX}}{p_{X}}\right)=\frac{1+g\dot{\phi}^{2}}{1+3g\dot{\phi}^{2}},$
(9)

and it corresponds to the adiabatic speed of sound squared and this speed
depends excursively of the Lagrangian density associated to scalar field. In
our model, we note that the speed of sound depends on the coupling function
$g$ associated to the non lineal kinetic term and the speed of scalar field
$\dot{\phi}$. Also, note that in the limit in which $g\rightarrow 0$, this
speed is reduced to the standard canonical field theory in which
$c_{s}\rightarrow 1$ (speed of light). In the following, we will utilize that
the notation $p_{X}=\partial p/\partial X$, $p_{\phi}$ denotes
$p_{\phi}=\partial p/\partial\phi$, $p_{XX}$ to $p_{XX}=\partial^{2}p/\partial
X^{2}$, etc.

From Eq.(9), we find that the coupling function $g$ can be expressed in term
of the speed of sound and the speed of the scalar field as

$g=\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\frac{1}{\dot{\phi}^{2}},\,\,\,\,\,\mbox{with}\,\,\,\,\,c_{s}^{2}\neq\frac{1}{3}.$
(10)

Here we note that from the specific Lagrangian density defined by Eq.(4), the
above equation shows that the speed of sound presents an upper and a lower
limit given by $1\geqslant c_{s}^{2}>1/3$, in order to have a continuous
coupling function $g(\phi)$ for all values of the field $\phi$. As before, we
note that in the specific value in which $c_{s}\rightarrow 1$, it reduces to
the standard canonical field theory in which the function $g\rightarrow 0$.

In relation to the expansion of the universe during the inflationary era, we
can introduce the number of $e-$folds defined as

$\Delta
N=N-N_{f}=\ln[a(t_{f})/a(t)]=\int^{t_{f}}_{t}H\,dt=\kappa\int^{\phi_{f}}_{\phi}(H/\dot{\phi})d\phi,$
(11)

where $N$ is the value of the number of $e$-folds at the cosmological time “t”
and $N_{f}$ denotes the number of $e$-folds at the end of inflation. In
addition, in order to analyze the inflationary epoch, it is also useful to
define the following slow-roll parameters K1 ; K2

$\epsilon=-\frac{\dot{H}}{H^{2}},\,\,\,\,\,\,\eta=\frac{\dot{\epsilon}}{H\epsilon},\,\,\,\,\,\,\,\,\,s=\frac{\dot{c}_{s}}{Hc_{s}},$
(12)

in which the parameters $\epsilon$, $\eta$ and $s$ are much less the unity.
Thus, the inflationary scenario occurs when the $\epsilon$ parameter is less
than unity, which is equivalent to the fact that the $\ddot{a}>0$ (accelerated
phase). Besides, the inflationary epoch ends when the parameter $\epsilon=1$
or equivalently $\ddot{a}=0$.

In this context, the equations of motion assuming the set of slow-roll
conditions are reduce to K1 ; K2

$H^{2}\simeq\frac{\kappa}{3}\,V,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,3H\dot{\phi}\,(g\dot{\phi}^{2}+1)+V_{\phi}\simeq
0,$ (13)

respectively. Here the equation for the scalar field in the slow roll
approximation differs from the canonical case due to the existence of the term
$3Hg\dot{\phi}^{3}$. In this way, we can observe two limits; in the case in
which $1\gg g\dot{\phi}^{2}$ and it corresponds to the standard canonical
theory and the situation in which $g\dot{\phi}^{2}\gg 1$ predominates and then
the dynamics of the canonical theory is modified.

On the other hand, for a non canonical theory the power spectrum of the
curvature perturbations $A_{s}$, together with the spectral index $n_{s}$ as a
function of the slow roll parameters are given by K1 ; K2

$A_{s}=\frac{\kappa}{8\pi^{2}}\frac{H^{2}}{c_{s}\epsilon},$ (14)

and

$n_{s}-1=\frac{dA_{s}}{d\ln k}\simeq-(2\epsilon+\eta+s),$ (15)

respectively.

Besides, the generation of tensor perturbation $A_{t}$ during the inflationary
period is not modified in the framework of non canonical theory and then its
expression coincides with the canonical theory, with which the tensor
perturbation $A_{t}=(2\kappa H^{2}/\pi^{2})$. However, the quantity associated
to the tensor to scalar ratio $r$ under the slow roll approximation is
modified by the factor $c_{s}$ and it becomesK1 ; K2

$r=\frac{A_{t}}{A_{s}}=16c_{s}\epsilon.$ (16)

Thus, in the frame of a canonical theory in which the speed of sound
$c_{s}=1$, the tensor to scalar ratio is reduced to the standard expression
$r=16\epsilon$.

In the following we will study the reconstruction of inflation in the
framework of the a non canonical theory. In this sense, for the reconstruction
the background variables we will determine the effective potential $V$
together with the coupling function $g$ in terms of the scalar field $\phi$
i.e., $V(\phi)$ and $g(\phi)$, respectively.

##  III General reconstruction

In this section, we will realize the procedure to the reconstruction of the
background variables, such as; the effective potential and the coupling
function $g$ as a function of the scalar field, in the context of a non
canonical theory, considering the scalar spectral index $n_{s}$ and the speed
of sound $c_{s}$ in terms of the number of $e-$ folds $N$. In this context, we
will rewrite the spectral index $n_{s}$ and the coupling function $g$ as a
function of the potential and of the speed of sound in terms of the number of
$e-$ folds $N$ and its derivatives i.e., $V_{N}=\partial V/\partial N$,
$V_{NN}=\partial^{2}V/\partial N^{2}$, ${c_{s}}_{N}=\partial c_{s}/\partial
N$, etc. In this way, from these quantities and giving the spectral index
$n_{s}=n_{s}(N)$ and the speed of sound $c_{s}=c_{s}(N)$, we should find the
effective potential $V=V(N)$ together with the coupling function $g=g(N)$ as a
function of the number of $e-$ folds $N$. Subsequently, from the relation
between the number $N$ and the scalar field $\phi$ given by the differential
equation (11), we should obtain the number $N$ in terms of $\phi$, i.e., the
relation $N=N(\phi)$ and then we should rebuild the effective potential
$V(\phi)$ and the function $g(\phi)$ associated to the non-lineal kinetic term
$gX^{2}$ of the Lagrangian density given by Eq.(4).

In this sense, we begin by rewriting the function coupling and the scalar
spectral index given by Eqs. (10) and (15) in terms of the number $N$. In this
form, the slow roll parameters $\epsilon$ and $\eta$ can be rewritten as a
function of the number of $e-$ folds $N$ using the fact that

$V_{\phi}=\frac{\partial V}{\partial\phi}=\frac{\partial V}{\partial
N}\,\frac{\partial
N}{\partial\phi}=V_{N}\,N_{\phi}=-V_{N}\left(\frac{H}{\dot{\phi}}\right),\,\,\,\,\,\mbox{where}\,\,\,\,\,N_{\phi}=-\frac{H}{\dot{\phi}}.$
(17)

Here we have considered that the quantity $dN=-Hdt=-Hd\phi/\dot{\phi}$ Chiba ;
Herrera:2018cgi .

Thus, we find that the first slow roll parameter $\epsilon$ can be written as

$\epsilon=-\frac{1}{2}\left(\frac{V_{\phi}}{V}\right)\left(\frac{\dot{\phi}}{H}\right)=\frac{1}{2}\left(\frac{V_{N}}{V}\right).$
(18)

Similarly, the slow roll parameter $\eta$ defined as
$\eta=\frac{\dot{\epsilon}}{H\epsilon}=-\frac{d\ln\epsilon}{dN}$ becomes

$\eta=-\frac{V_{NN}}{V_{N}}+\frac{V_{N}}{V},$ (19)

and the slow roll parameter associated to speed of sound $s$ results

$s=\frac{\dot{c}_{s}}{Hc_{s}}=-\frac{d\ln c_{s}}{dN}.$ (20)

In this form, using the above equations, we can rewrite the scalar spectral
index given by Eq.(15) in terms of the effective potential and the speed of
sound together with its derivatives with respect to $N$ as

$n_{s}-1\simeq-2\frac{V_{N}}{V}+\frac{V_{NN}}{V_{N}}+\frac{d\ln c_{s}}{dN}.$
(21)

Thus, we obtain that the effective potential in terms of the number of $e-$
folds can be written as

$V(N)=V=\left[-\int\left(\exp\int\left[(n_{s}-1)-\frac{d\ln
c_{s}}{dN}\right]dN\right)dN\right]^{-1}.$ (22)

In the same form, for the coupling function $g$ we have

$g(N)=g=\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\frac{1}{\dot{\phi}^{2}}=3\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\left[1+\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\right]\,\frac{1}{V_{N}},$
(23)

or equivalently

$g(N)=3\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\left[1+\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\right]\,\frac{\exp\left(\int\left[(1-n_{s})+\frac{d\ln
c_{s}}{dN}\right]dN\right)}{V^{2}}\,\,=$
$3\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\left[1+\left(\frac{1-c_{s}^{2}}{3c_{s}^{2}-1}\right)\right]\,\exp\left(\int\left[(1-n_{s})+\frac{d\ln
c_{s}}{dN}\right]dN\right)\left[\int\left(\exp\int\left[(n_{s}-1)-\frac{d\ln
c_{s}}{dN}\right]dN\right)dN\right]^{2},$ (24)

where we have used Eqs.(13) and (17), respectively. Here we note that in the
special case in which the speed of sound is a constant equal to unity,
Eqs.(22) and (24) reduce to the standard reconstruction in the framework of a
canonical theory, i.e., $V(N)$ corresponds to Ref.(Chiba ) and $g=0$.

In this way, we mention that the Eqs.(22) and (24) are the essential equations
to obtain the effective potential and the coupling function in terms of the
number of $e-$ folds $N$. In this sense, in order to obtain these background
variables as a function of $N$, we need to give the parametrization of the
scalar spectrum index $n_{s}=n_{s}(N)$ together with the parametrization of
the speed of sound $c_{s}=c_{s}(N)$.

For the power spectrum of the curvature perturbations from Eq.(14) we have

$A_{s}\simeq\frac{\kappa^{2}}{12\pi^{2}}\,\left[\frac{V^{2}}{c_{s}\,V_{N}}\right],$
(25)

and for the tensor to scalar ratio in terms of the potential and speed of
sound results

$r=8c_{s}\,\left(\frac{V}{V_{N}}\right).$ (26)

Additionally, in order to find the relation between the number of $e-$ folds
$N$ and the scalar field $\phi$ we get

$\,\left[\frac{V_{N}\,(3c_{s}^{2}-1)}{V\,c_{s}^{2}}\right]^{1/2}dN=\sqrt{2\kappa}\,d\phi,$
(27)

where we have used Eqs.(10) and (11), respectively.

Here, we mention that the system of equations given by Eqs.(22), (24) and (27)
are the basic equations to rebuild the effective potential $V$ and the
coupling function $g$, in terms of the scalar field $\phi$, i.e., $V=V(\phi)$
and $g=g(\phi)$, respectively.

In the following, we will assume a specific example for the scalar spectral
index as well as for the speed of sound in terms of the number of $e-$ folds
$N$, to reconstruct the effective potential $V(\phi)$ and the coupling
function $g(\phi)$, respectively.

##  IV Reconstruction from $n_{s}=n_{s}(N)$ and $c_{s}=c_{s}(N)$

In this section we will apply the methodology previously described, assuming a
parametrization for both the scalar spectral index and the sound of speed in
terms of the number of $e-$ folds $N$, in order to rebuild the background
variables; the effective potential $V$ and the coupling function $g$ as a
function of the scalar field $\phi$.

In this sense, we can assume the simplest parametrization or attractor for the
scalar spectral index $n_{s}$ as a function of $N$ given by Chiba:2015zpa

$n_{s}(N)=1-\frac{2}{N},\,\,\,\,\,\,\mbox{in which }\,\,\,\,\,\,\,\,N\neq 0,$
(28)

and for the speed of sound propagation of the scalar perturbations we consider
the parametrization

$c_{s}(N)=c_{s_{f}}\left(\frac{N_{f}}{N}\right)^{\beta}=\tilde{c}_{s_{f}}N^{-\beta},$
(29)

where the quantity $c_{s_{f}}$ is a constant and it corresponds to the value
of the speed of sound propagation at the end of inflation, the quantity
$\tilde{c}_{s_{f}}$ is defined as $\tilde{c}_{s_{f}}=c_{s_{f}}N_{f}^{\beta}$
and the power $\beta$ is a constant. In relation to the parametrization of the
speed of sound given by Eq.(29) in terms of the number of $e-$folds $N$, we
mention that a study associated to the reconstruction of an inflationary stage
using this parametrization does not exist in the literature. In this form, the
motivation to consider the simplest parametrization of the speed of sound
$c_{s}=c_{s}(N)$ described by Eq.(29) comes from the speed of sound analyzed
in Ref.Shi:2021tmn  in which a power law type dependence on time
$c_{s}\propto(-\tau)^{s}$ is considered or with the energy density
$c_{s}^{2}\propto\rho^{1-2n}$ as in Magueijo:2008pm .

In this context, the present work is the first step towards that direction
using this physical parameter and this dependence with the number of $e-$folds
during inflation, in order to rebuild the background variables during the
early universe.

By assuming Eq.(29) and as the speed propagation satisfies the relation
$1\geqslant c_{s}^{2}>1/3$ (see Eq.(10)), then we find that the upper and
lower bounds for $\beta$ becomes

$\frac{\ln(c_{s_{f}})}{\ln(N/N_{f})}\leq\beta<\frac{\ln(3^{1/2}\,c_{s_{f}})}{\ln(N/N_{f})}.$
(30)

However, we observe that as the number of $e-$folds
$N\sim\mathcal{O}(10^{2})\gg N_{f}\sim\mathcal{O}(1)$ and the speed of sound
$c_{s_{f}}\sim\mathcal{O}(1)$ (since $1\geqslant c_{s_{f}}^{2}>1/3$), then
from Eq.(30) we have that the parameter $\beta$ will always be less than unity
i.e., $\beta<1$.

On the other hand, in order to find the potential in terms of the number of
$e-$ folds $N$, we replace Eqs.(28) and (29) into Eq.(22) resulting

$V(N)=\left(C+\frac{\alpha}{1-\beta}N^{\beta-1}\right)^{-1},$ (31)

where $\alpha$ and $C$ are integration constants and these constants have
units of $M_{p}^{-4}$. Similarly, from Eq.(24) we find that the coupling
parameter $g$ as a function of the number of $e$-folds becomes

$g(N)=\frac{3}{\alpha}\left(\frac{1-\tilde{c}_{s_{f}}^{\,2}N^{-2\beta}}{1-3\,\tilde{c}_{s_{f}}^{\,2}N^{-2\beta}}\right)^{2}\left(C+\frac{\alpha}{1-\beta}N^{\beta-1}\right)^{2}N^{2-\beta}.$
(32)

Therefore, to obtain the reconstruction of the effective potential $V(\phi)$
and the coupling function $g(\phi)$, we need to find the relation between the
number of $e-$ folds $N$ and the scalar field $\phi$ given by Eq.(27).

In order to find the relation between the number of $e-$ folds $N$ and the
scalar field $\phi$ i.e., $N=N(\phi)$ from Eq.(27), we need to replace the
potential $V(N)$ and $c_{s}(N)$ given by Eqs.(29) and (31) into (27). However,
we cannot analytically solve this equation. Thus, to solve the Eq.(27), we can
assume that during the inflationary epoch the integration constant $C$ is a
positive quantity and $C\gg\alpha N^{\beta-1}/[(1-\beta)]$. Under this
approximation, we find that the effective potential as a function of the
number $N$ given by Eq.(31) can be approximate to

$V(N)\simeq C^{-1}\left(1-\frac{\alpha}{(1-\beta)C}N^{\beta-1}+\dotsb\right).$
(33)

In this form, the differential equation associated to the number of $e-$folds
and the scalar field given by Eq.(27) is reduced to

$\left(\frac{2\kappa
C}{\alpha}\right)^{1/2}\simeq\left(\frac{3\tilde{c}_{s_{f}}^{\,2}N^{-2\beta}-1}{\tilde{c}_{s_{f}}^{\,2}N^{-2\beta}}\right)^{1/2}\left(1+\frac{\alpha}{2(1-\beta)C}N^{\beta-1}\right)N^{\beta/2-1}N_{\phi}.$
(34)

Solving this differential equation we obtain that the number of $e$-folds as a
function of the field $\phi$ can be written as

$N(\phi)=N=\mathcal{F}^{-1}(\phi),$ (35)

where the function $\mathcal{F}^{-1}(\phi)$ represents the inverse function of

${\mathcal{F}}(\phi)=2\sqrt{3}\left(\sqrt{\frac{8\kappa
C^{3}}{\alpha}}(1-\beta)\phi+\gamma\right)^{\beta/2}\left[\mathcal{F}_{1}(\phi)+\mathcal{F}_{2}(\phi)\right],$
(36)

where the functions $\mathcal{F}_{1}(\phi)$ and $\mathcal{F}_{2}(\phi)$ are
defined as

$\displaystyle\mathcal{F}_{1}(\phi)$ $\displaystyle=$
$\displaystyle\frac{2(1-\beta)C}{\beta}\,\,_{2}F_{1}\left[-\frac{1}{2},\frac{1}{4},\frac{5}{4},\frac{\left(\sqrt{\frac{8\kappa
C^{3}}{\alpha}}(1-\beta)\phi+\gamma\right)^{2\beta}}{3\tilde{c}_{s_{f}}^{\,2}}\right],\,\,\,\,\,\,\,\,\,\,\,\mbox{and}$
(37) $\displaystyle\mathcal{F}_{2}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\alpha\left(\sqrt{\frac{8\kappa
C^{3}}{\alpha}}(1-\beta)\phi+\gamma\right)^{\beta-1}}{3\beta-2}\,\,_{2}F_{1}\left[-\frac{1}{2},\frac{1}{4}\left(3-\frac{2}{\beta}\right),\frac{1}{4}\left(7-\frac{2}{\beta}\right),\frac{\left(\sqrt{\frac{8\kappa
C^{3}}{\alpha}}(1-\beta)\phi+\gamma\right)^{2\beta}}{3\tilde{c}_{s_{f}}^{\,2}}\right],$
(38)

respectively. Besides, the parameter $\gamma$ is an integration constant and
the quantity $\,{}_{2}F_{1}$ corresponds to the hypergeometric functionLibro .

In this form, replacing the number of $e-$ folds in terms of the scalar field
given by Eq.(35) into the potential (33), we find that the reconstruction of
the effective potential as a function of the scalar field becomes

$V(\phi)\simeq
C^{-1}\left(1-\frac{\alpha}{(1-\beta)C}\left[\mathcal{F}^{-1}(\phi)\right]^{\beta-1}\right).$
(39)

Analogously, for the coupling function $g(\phi)$, we obtain that the
reconstruction for this background variable results

$g(\phi)\simeq\frac{3C^{2}}{\alpha}\left(\frac{1-\tilde{c}_{s_{f}}^{\,2}\left[\mathcal{F}^{-1}(\phi)\right]^{-2\beta}}{3\,\tilde{c}_{s_{f}}^{\,2}\left[\mathcal{F}^{-1}(\phi)\right]^{-2\beta}-1}\right)\left[1+\left(\frac{1-\tilde{c}_{s_{f}}^{\,2}\left[\mathcal{F}^{-1}(\phi)\right]^{-2\beta}}{3\,\tilde{c}_{s_{f}}^{\,2}\left[\mathcal{F}^{-1}(\phi)\right]^{-2\beta}-1}\right)\right]\left[\mathcal{F}^{-1}(\phi)\right]^{2-\beta}.$
(40)

Additionally, we have that the speed of sound as a function of the scalar
field can be written as

$c_{s}(\phi)\simeq\tilde{c}_{s_{f}}\left[\mathcal{F}^{-1}(\phi)\right]^{-\beta},$
(41)

here in the reconstruction of $c_{s}(\phi)$, we have considered Eq.(29).

Figure 1:  The behavior of the number of $e-$ folds $N$ on the scalar field
$\phi$ (upper panel) and the reconstructed effective potential dimensionless
$\tilde{V}=C\,V$ versus the scalar field $\phi$ (lower panel), for different
values of the speed of sound $c_{s_{f}}$ at the end of inflationary epoch. In
these plots we have used the values $C\simeq 3.04\times 10^{10}\kappa^{2}$,
$\alpha\simeq 1.80\times 10^{10}\kappa^{2}$ and $N_{f}=0.8$.

Figure 2:  The upper panel shows the reconstruction of the coupling function
$g$ as a function of the scalar field and the lower panel shows the speed of
sound in terms of the scalar field i.e., $c_{s}(\phi)$. As before we have
considered different values of the speed of sound at the end of inflation
$c_{s_{f}}$. Besides, we have used the values $C\simeq 3.04\times
10^{10}\kappa^{2}$, $\alpha\simeq 1.80\times 10^{10}\kappa^{2}$ and
$N_{f}=0.8$, respectively.

In Fig.1, the upper panel shows the evolution of the number of $e-$ folds $N$
versus the scalar field $\phi$, while the lower panel shows the reconstructed
effective potential $\tilde{V}$ in terms of the scalar field. In both panels,
we consider the situation in which the speed sound at the end of inflation
$c_{s_{f}}$ has three different values. Besides, in these plots we have
assumed the values of $C\simeq 3.04\times 10^{10}\kappa^{2}$, $\alpha\simeq
1.80\times 10^{10}\kappa^{2}$ and $N_{f}=0.8$, respectively. In order to write
down values of the number of $e-$ folds $N(\phi)$ and the effective potential
$\tilde{V}(\phi)$, we have used Eqs.(35) and (39), respectively. From the
upper panel we observe that the end of the inflationary epoch in which
$N_{f}\sim\mathcal{O}(1)$ the scalar field takes the values
$\phi_{f}\sim\mathcal{O}(0)$. Also, we observe that the number of $e-$folds
$N$ takes the values $N\sim 60$ when the scalar field is approximately
$\phi\sim\mathcal{O}(M_{p})$. From the lower panel we note that for values of
$\phi>3M_{p}$, the reconstructed potential becomes constant equal to $V\sim C$
and independent of the value of the sound seed at the end of inflation
$c_{s_{f}}$.

In Fig.2, the upper panel shows the reconstruction of the coupling function
versus the scalar field $\phi$, while the lower panel shows the reconstructed
speed of sound in terms of the scalar field. As before, in both panels we
consider the case in which the speed sound at the end of inflation $c_{s_{f}}$
has three different values. Also, in these plots we have assumed the values of
$C\simeq 3.04\times 10^{10}\kappa^{2}$, $\alpha\simeq 1.80\times
10^{10}\kappa^{2}$ and $N_{f}=0.8$, respectively. From Fig.2 we note that the
coupling function increases very fast for large $N$ (see relation between
$\phi$ and $N$ of Fig.1). Besides, from the lower panel we observe that the
speed of sound presents a small slope in relation to the scalar field. Here we
note that during the inflationary scenario the cases in which $c_{s_{f}}=0.99$
(green line) or $c_{s_{f}}=0.6$ (blue line) the evolution of the speed
$c_{s}(\phi)$ increases up to the specified value of $c_{s_{f}}$. However, for
the situation in which $c_{s_{f}}=0.99$ (red line) this dependence with the
scalar field decreases when the field moves from $N\sim\mathcal{O}(10^{2})$ to
the end of inflation where $N_{f}\sim\mathcal{O}(1)$.

On the other hand, to constrain the integration constant $\alpha$, we can
consider the expression of the scalar power spectrum given by Eq.(14). Thus,
using the Eqs.(14) and (29) we obtain that the power spectrum $A_{s}$ is given
by

$A_{s}\simeq\frac{\kappa^{2}}{12\alpha\pi^{2}\tilde{c}_{s_{f}}}N^{2}\left[1-\frac{2\alpha
N^{\beta-1}}{(1-\beta)C}\right],$ (42)

and then the integration constant $\alpha$ becomes

$\alpha=\frac{\kappa^{2}\,N^{2}}{12\pi^{2}\tilde{c}_{s_{f}}\,A_{s}}\left[1+\frac{\kappa^{2}N^{2}}{6\pi^{2}A_{s}C\,c_{s_{f}}N_{f}\,x}\right]^{-1},\,\,\,\,\,\,\mbox{where}\,\,\,\,\,\,\,\,x=(1-\beta)\left(\frac{N}{N_{f}}\right)^{(1-\beta)}.$
(43)

Besides, from the effective potential given by Eq.(33) together with Eq.(26),
we obtain that the tensor to scalar ratio as a function of the number of
$e$-folds $N$ yields

$r\simeq\frac{8\alpha\,\tilde{c}_{s_{f}}}{C\,N^{2}}\left[1-\frac{\alpha}{(1-\beta)C}N^{\beta-1}\right]^{-1}.$
(44)

In this way, from the tensor to scalar ratio (44) and considering Eq.(43), we
find an equation for the quantity $x$ (defined in Eq.(43)) given by

$x=(1-\beta)\left(\frac{N}{N_{f}}\right)^{(1-\beta)}=\frac{\alpha_{0}\alpha_{1}}{(1-\alpha_{1}C)}=\alpha_{2}\,\,\,\,\,\mbox{with}\,\,\,\,\,\alpha_{1}C\neq
1.$ (45)

Here, the quantities $\alpha_{0}$ and $\alpha_{1}$ become

$\alpha_{0}=\frac{\tilde{\alpha}_{0}}{N_{f}}=\frac{\kappa^{2}\,N^{2}}{12\pi^{2}\,A_{s}\,c_{s_{f}}N_{f}},\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\alpha_{1}=\frac{3\pi^{2}\,A_{s}\,r}{2\kappa^{2}}\,,$
(46)

respectively. In this form, the solution of Eq.(45) for the parameter $\beta$
can be written in terms of the observational parameters $A_{s}$, $r$ and the
number of $e-$ folds $N$ as

$\beta=1-\frac{\mbox{ProductLog}\left[\alpha_{2}\,\ln(N/N_{f})\,\right]}{\ln(N/N_{f})}.$
(47)

Here the ProductLog function corresponds to a product logarithm, also called
the Omega function or Lambert W function, see e.g., Refs.87 ; 88 .

Now inserting Eq.(47) into Eq.(30), we find that the second integration
constant $C$ is constrained by an upper and lower bounds from the
observational parameters $A_{s}$, $r$ and the number of $e-$ folds $N$
together with the parameters at the end of inflation $c_{s_{f}}$ and $N_{f}$
as

$\frac{F_{1}-1}{\alpha_{1}\,F_{1}}<C\leq\frac{F_{2}-1}{\alpha_{1}\,F_{2}},$
(48)

where the quantities $F_{1}$ and $F_{2}$ are defined as

$F_{1}=\frac{(N/N_{f})}{\sqrt{3}\alpha_{0}\alpha_{1}c_{s_{f}}}\left[1-\frac{\ln(\sqrt{3}c_{s_{f}})}{\ln(N/N_{f})}\right]\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,F_{2}=\frac{(N/N_{f})}{\alpha_{0}\alpha_{1}c_{s_{f}}}\left[1-\frac{\ln(c_{s_{f}})}{\ln(N/N_{f})}\right].$
(49)

On the other hand, in order to obtain the number of $e-$folds at the end of
inflation $N_{f}$, we can consider the slow roll parameter $\epsilon$ defined
by Eq.(18) together with the reconstructed potential (33). In this way,
assuming that inflation ends when $\epsilon(N=N_{f})=1$ (or equivalently
$\ddot{a}=0$), we have that

$N_{f}^{3}-p\,N_{f}-\frac{2p^{2}}{(1-\beta)}=0,\,\,\,\,\,\,\mbox{where}\,\,\,\,\,\,\,\,\,p=\frac{\tilde{\alpha}_{0}}{2(2/\alpha_{1}-C)}.$
(50)

Here the parameter $p$ is a function of the $N_{f}$, and then the Eq.(50) does
not have an analytical solution. However, we can solve the Eq.(50)
numerically. Thus, in particular considering the values of $N=60$,
$A_{s}=2.2\times 10^{-9}$, $r=0.04$, $c_{s_{f}}=1$ and assuming the upper
bound of Eq.(48) for $C$ (that depends of $N_{f}$), then numerically we find
that the number of $e-$folds at the end of inflation from Eq.(50) becomes
$N_{f}\simeq 5.08$ and for the case in which $c_{s_{f}}=0.6$ we have
$N_{f}\simeq 6.96$. Now for the case in which we consider the tensor to scalar
ratio $r=0.001$ together with the lower and upper bounds of speed of sound at
the end of inflation $1/\sqrt{3}<c_{s_{f}}\leq 1$, we find that the number of
$e-$folds at the end of inflationary epoch $N_{f}$ is in the range $0.62\leq
N_{f}<0.84$. In this sense, we find that the number of $e-$folds at the end of
inflation $N_{f}$ becomes $N_{f}\sim\mathcal{O}(1)$.

In relation to the consistency relation i.e., $r=r(n_{s})$, combining Eqs.(28)
and (44), we find that this relation can be written as

$r(n_{s})\simeq\frac{2\alpha\,\tilde{c}_{s_{f}}(1-n_{s})^{2}}{C}\left[1-\frac{2^{\beta-1}\,\alpha}{(1-\beta)C}(1-n_{s})^{1-\beta}\right]^{-1}.$
(51)

In figure 3 we show the evolution of the tensor to scalar ratio $r$ on the
scalar spectral index $n_{s}$ (consistency relation) for different values of
the number of $e-$folds at the end of inflation $N_{f}$, from the latest
BICEP/Kerck data BICEP:2021xfz . For this plot we have fixed the value of the
speed of sound at the end of the inflationary phase $c_{s_{f}}=0.6$. In this
figure, the solid, dotted and dashed lines correspond to the pairs ($C\simeq
6.01\times 10^{8}\kappa^{2}$,$\alpha\simeq 2.22\times 10^{10}\kappa^{2}$),
($C\simeq 2.87\times 10^{9}\kappa^{2}$,$\alpha\simeq 2.48\times
10^{10}\kappa^{2}$), and ($C\simeq 5.94\times 10^{9}\kappa^{2}$,$\alpha\simeq
2.42\times 10^{10}\kappa^{2}$), respectively. Besides, considering
Ref.BICEP:2021xfz , two-dimensional marginalized constraints on the tensor to
scalar ratio and the scalar spectral index fixed at $k_{0}=0.002$Mpc-1. The
latest BICEP/Keck results places stronger limits on the ratio $r$ shown in
blue (at 68% blue region and 95% light blue region levels of confidence) and
the green color corresponds to the two-dimensional marginalized constraints
found in Ref.Planck:2018vyg .

We noted from Fig.(3) that the reconstruction of our model is well supported
by the latest BICEP/Keck data when the speed of sound at the end of inflation
$c_{s_{f}}=0.6$ and the number of $e-$folds at the end of inflation
$N_{f}\sim\mathcal{O}(1)$. Similarly, the model is well supported by the
Planck data (plane $r=r(n_{s})$) when the speed of sound at the end of
inflation $c_{s_{f}}$ lies in the range $1\sqrt{3}<c_{s_{f}}\leq 1$ and the
number of $e-$folds at the end of inflation $N_{f}\sim\mathcal{O}(1)$ (figure
not shown).

Figure 3:  The figure shows the consistency relation $r=r(n_{s})$ for different values of the number of $e-$folds at the end of the inflationary phase $N_{f}$, for the case in which we have fixed the speed sound at the end of inflation $c_{s_{f}}=0.6$. In this panel, the solid, dotted and dashed lines correspond to the pairs ($C\simeq 6.01\times 10^{8}\kappa^{2}$, $\alpha\simeq 2.22\times 10^{10}\kappa^{2}$) for the case in which $N_{f}=6.74$, ($C\simeq 2.87\times 10^{9}\kappa^{2}$, $\alpha\simeq 2.48\times 10^{10}\kappa^{2}$) for $N_{f}=3.19$ and ($C\simeq 5.94\times 10^{9}\kappa^{2}$,$\,\alpha\simeq 2.42\times 10^{10}\kappa^{2}$) when $n_{f}=2.12$. Here we have considered the two-dimensional marginalized joint confidence contours for the plane $(n_{s},r)$ at 68% and 95% levels of confidence from BICEP/Keck results BICEP:2021xfz .  $c_{s_{f}}$ |  $C\,[\kappa^{2}]$ |  $\alpha\,[\kappa^{2}]$ |  $N_{f}$ |  $\beta$
---|---|---|---|---
$0.6$ |  $3.030\times 10^{10}<C\leq$3.050$\times 10^{10}$ |  $2.22\times 10^{10}\leq\alpha<$2.25$\times 10^{10}$ |  $0.82\leq N_{f}<$0.84 | $-0.12\leq\beta<0.0090$
0.8 |  $3.027\times 10^{10}<C\leq$3.048$\times 10^{10}$ | 1.67$\times 10^{10}$ $\leq\alpha<$ 1.72$\times 10^{10}$ | $0.70\leq N_{f}<0.71$ | $-0.050\leq\beta<0.074$
0.99 |  $3.025\times 10^{10}<C\leq$ 3.047$\times 10^{10}$ |  $1.36\times 10^{10}\leq\alpha<$ 1.42$\times 10^{10}$ | $0.62\leq N_{f}<0.63$ | $-0.0022\leq\beta<0.12$
Table 1: Summary of the allowed range for the integration constants $C$ and
$\alpha$ associated to the reconstructed model together with the range for the
parameters $N_{f}$ and $\beta$, when we fix different values of the speed of
sound at the end of inflation $c_{s_{f}}$.

The resulting allowed values for the integration constants $C$, $\alpha$,
$N_{f}$ and $\beta$ for the different values of the speed of sound at the end
of the inflationary epoch are shown in Table 1. Here we have considered the
values $N_{k}=60$, $A_{s_{k}}=2.2\times 10^{-9}$ and $r_{k}=0.001$. The Table
I shows that the integration constants $C$ and $\alpha$ become similar and the
order of $\mathcal{O}(10^{10})\kappa^{2}$. Also, we observer from Table I that
the number of $e-$folds at the end of the inflationary epoch
$N_{f}\sim\mathcal{O}(1)$ and the parameter $\beta$ associated to the power of
the speed of sound is close to zero. Besides from the Table I, we note that
the constraints on the parameters $C$, $\alpha$ and $N_{f}$ do not depend
strongly on the value of the speed of sound at the end of inflationary epoch,
since the values of these parameters are similar for all the values of
$c_{s_{f}}$. Additionally, as we note from this table, the range associated to
the parameters; $C$, $\alpha$ and $N_{f}$ is very narrow. However, as we
notice the range for the parameter $\beta$ is a little larger, when we
consider different values of the parameters $C$, $\alpha$, $N_{f}$ and
$c_{s_{f}}$. Also, we note that the parameter $\beta$ can take negative values
(lower bound) depending on the values of the speed of sound at the end of
inflationary epoch.

##  V Reheating

In this section we will study the reheating epoch considering the background
reconstructed variables found in the previous section. In this context, we
will utilize the potential $V$ and the coupling function $g$ as a function of
the scalar field given by Eqs.(39) and (40), respectively.

In the framework of a non canonical theory, we can assume that the ratio
between the comoving Hubble scale crosses the horizon during the inflationary
epoch $c_{s}k=a_{k}H_{k}$ and the actual wave number $k_{0}$ related with the
actual Hubble scale $H_{0}$ through $c_{s_{0}}k_{0}=k_{0}=a_{0}H_{0}$, becomes

$\frac{c_{s}k}{k_{0}}=\frac{a_{k}H_{k}}{a_{0}H_{0}}=\left(\frac{a_{k}}{a_{f}}\right)\left(\frac{a_{f}}{a_{\text{reh}}}\right)\left(\frac{a_{\text{reh}}}{a_{\text{eq}}}\right)\left(\frac{a_{\text{eq}}H_{\text{eq}}}{a_{0}H_{0}}\right)\left(\frac{H_{k}}{H_{\text{eq}}}\right),$
(52)

where we have assumed that at the present the speed of sound
$c_{s_{0}}=1$Liddle:2003as . Also, the notation “$f$” corresponds to the end
of inflation, “reh” denotes to the reheating and “eq” is the radiation-matter
equality. Besides, we recalled that the quantities with the notation $k$ are
evaluated at horizon exit during the inflationary epoch, e.g.,
$N\,|_{c_{s}k=a_{k}H_{k}}=N_{k}$.

Additionally, we can express the number of $e$-folds $N$ in each epoch, in
terms of the scale factor $a$, as follows
$N_{k}=\ln\left[\frac{a_{f}}{a_{k}}\right]$,
$N_{\text{reh}}=\ln\left[\frac{a_{\text{reh}}}{a_{f}}\right],$ and
$N_{RD}=\ln\left[\frac{a_{\text{eq}}}{a_{\text{reh}}}\right]$, in which
$N_{\text{reh}}$ is the number of $e$-folds during reheating era and
$N_{\text{RD}}$ corresponds to the number of $e$-folds in the radiation
dominance.

Thus, considering the different $e-$folds and Eq.(52) we get

$\text{ln}\left(\frac{c_{s}k}{a_{0}H_{0}}\right)=-N_{k}-N_{\text{reh}}-N_{\text{RD}}+\text{ln}\left(\frac{a_{\text{eq}}H_{\text{eq}}}{a_{0}H_{0}}\right)+\text{ln}\left(\frac{H_{k}}{H_{\text{eq}}}\right).$
(53)

Furthermore, we can write that the ratio between the energy density at the end
of inflationary epoch $\rho_{f}$ and the energy density at the end of the
reheating regime $\rho_{\text{reh}}$ can be associated from an Equation of
State (EoS) parameter $\omega_{\text{reh}}$ related to the reheating regime.
In this form, the ratio $\rho_{\text{reh}}/\rho_{f}$ yields

$\frac{\rho_{\text{reh}}}{\rho_{f}}=e^{-3N_{\text{reh}}(1+\omega_{\text{reh}})},$
(54)

in which we have assumed that during the reheating scenario the energy density
decays as a function of the scale factor as $\rho\propto
a^{-3(1+\omega_{\text{reh}})}$, with $\omega_{\text{reh}}=$ constant.

In relation to the energy density at the end of inflationary epoch $\rho_{f}$
from Eq.(5), we have that

$\rho_{f}=X_{f}+3g_{f}X_{f}^{2}+V_{f}=\left(1+\lambda\right)V_{f},$ (55)

where the parameter $\lambda=(X_{f}+3g_{f}X_{f}^{2})/V_{f}$. Besides,
considering the definition of the slow-roll parameter given by Eq.(12), we
find that the parameter $\epsilon$ at the end of the inflationary stage
becomes

$\epsilon_{f}=3\frac{X_{f}+2g_{f}X_{f}^{2}}{X_{f}+3g_{f}X_{f}^{2}+V_{f}}.$
(56)

As before considering that the end of inflation takes place when
$\epsilon_{f}=1$ (or equivalently $\ddot{a}=0$), and then replacing the value
of $X_{f}$ into the parameter $\lambda$ we get

$\lambda=1+\frac{1}{3g_{f}V_{f}}\left(1\pm\sqrt{1+3g_{f}V_{f}}\right).$ (57)

Thus, from Eq.(55) we find that the energy density at the end of inflation can
be written as

$\rho_{f}=\left[\frac{3}{2}+\frac{1}{3g_{f}V_{f}}\left(1+\frac{3g_{f}V_{f}}{2}\pm\sqrt{1+3g_{f}V_{f}}\right)\right]V_{f}.$
(58)

In the following we will consider the negative signs of Eqs.(57) and (58), in
order to recover the expressions of $\lambda$ and $\rho_{f}$ in the limit of a
canonical theory in which the quantity $g_{f}\rightarrow 0$. Thus, in this
limit the Eqs.(57) and (58) with the negative signs are reduced to
$\lambda=1/2$ and $\rho_{f}=3/2V_{f}$, respectively Munoz:2014eqa ;
Dai:2014jja ; Cook:2015vqa .

In order to determine the reheating temperature $T_{\text{reh}}$, we can
assume the entropy conservation, where the reheating entropy is preserved in
the CMB together with the neutrino background at the present Dai:2014jja . In
this sense, from the entropy conservation we can write Dai:2014jja

$g_{\text{s,reh}}a^{3}_{\text{reh}}T^{3}_{\text{reh}}=a_{0}^{3}\left(2T_{0}^{3}+\frac{21}{4}T_{\nu
0}^{3}\right),$ (59)

in which the parameter $g_{\text{s,reh}}$ denotes the effective number of
relativistic degrees of freedom for entropy at reheating, $T_{0}\simeq 2.7$K
is the current CMB temperature and the temperature $T_{\nu 0}$ corresponds to
the present neutrino temperature. However, we know that the relation between
the neutrino temperature $T_{\nu 0}$ and $T_{0}$ becomes $T_{\nu
0}=\left(\frac{4}{11}\right)^{1/3}T_{0}$ Dai:2014jja , and then we can
associate the scale factors during the reheating era and at the present epoch
through
$a_{\text{reh}}/a_{0}=\left(\frac{43}{11g_{\text{s,reh}}}\right)^{1/3}T_{0}/T_{\text{reh}}.$

Additionally, we can assume that the energy density at the end of reheating
$\rho_{\text{reh}}$ is equal to the hot radiation and then this density yields

$\rho_{\text{reh}}=\frac{\pi^{2}}{30}g_{\star\text{,reh}}T^{4}_{\text{reh}}\,.$
(60)

Here the quantity $g_{\star\text{,reh}}$ corresponds to the effective number
of relativistic degrees of freedom at the end of reheating epoch.

Thus, considering the above relations, we obtain that the reheating
temperature $T_{\text{reh}}$ in terms of the parameters $V_{f}$, $g_{f}$,
$w_{\text{reh}}$ and the number $N_{\text{reh}}$ results

$T_{\text{reh}}=\left(\frac{3}{10\pi^{2}}\right)^{1/4}\left[\frac{3}{2}V_{f}+\frac{1}{3g_{f}}\left(1+\frac{3g_{f}V_{f}}{2}-\sqrt{1+3g_{f}V_{f}}\right)\right]^{1/4}\text{exp}\left[-\frac{3}{4}(1+\omega_{\text{reh}})N_{\text{reh}}\right].$
(61)

Here the number of $e$-folds during the reheating scenario is given by

$N_{\text{reh}}=\frac{4}{1-3\omega_{\text{reh}}}\left[-N_{k}-\text{ln}\left(\frac{c_{s}k}{a_{0}T_{0}}\right)-\frac{1}{3}\text{ln}\left(\frac{11g_{\text{s,reh}}}{43}\right)-\frac{1}{4}\text{ln}\left(\frac{30\kappa^{2}\rho_{f}}{g_{\text{reh}}\pi^{2}}\right)+\frac{1}{2}\text{ln}\left(\frac{\pi^{2}rA_{s}}{2}\right)\right].$
(62)

In order to find the number $N_{\text{reh}}$ given by Eq.(62), we have used
that the Hubble parameter $H_{k}$ satisfies the relation between the tensor to
scalar ratio $r$ and the scalar power spectrum $A_{s}$ through
$H_{k}=\sqrt{\frac{\pi^{2}}{2\kappa}rA_{s}}$. Here we mention that this
relation for the Hubble parameter $H_{k}$ is still valid in the framework of a
non canonical theory.

Additionally, we can rewrite Eq. (61) and (62) in function of spectral index
$n_{s}$. Thus, we find that the reheating temperature and the number of
$e-$folds during the reheating period as a function of the scalar spectral
index $n_{s}$ become

$T_{\text{reh}}=\left[\frac{3\rho_{f}(n_{s})}{10\pi^{2}}\right]^{1/4}\text{exp}\left[-\frac{3}{4}(1+\omega_{\text{reh}})N_{\text{reh}}(n_{s})\right],$
(63)

and

$N_{\text{reh}}=\frac{4}{1-3\omega_{\text{reh}}}\left[\frac{2}{(n_{s}-1)}-\text{ln}\left(\frac{c_{s}k}{a_{0}T_{0}}\right)-\frac{1}{3}\text{ln}\left(\frac{11g_{\text{s,reh}}}{43}\right)-\frac{1}{4}\text{ln}\left(\frac{30\kappa^{2}\rho_{f}(n_{s})}{g_{\text{reh}}\pi^{2}}\right)+\frac{1}{2}\text{ln}\left(\frac{\pi^{2}r(n_{s})A_{s}(n_{s})}{2}\right)\right],$
(64)

respectively. Here the quantity $\rho_{f}(n_{s})$ is defined as

$\rho_{f}(n_{s})=\left[2+z(n_{s})^{-1}\left(1-\sqrt{1+z(n_{s})}\,\right)\right]\left[\frac{1}{C}-\frac{\alpha
N_{f}^{-y(n_{s})}}{y(n_{s})C^{2}}\right],$

where the function $z(n_{s})$ and $y(n_{s})$ are given by

$\displaystyle z(n_{s})$ $\displaystyle=$
$\displaystyle\frac{18C}{\alpha}\frac{c_{s_{f}}^{2}\left(1-c_{s_{f}}^{2}\right)}{(3c_{s_{f}}^{2}-1)^{2}}\left[N_{f}^{1+y(n_{s})}-\frac{\alpha
N_{f}}{y(n_{s})C}\right],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{and}$ (65)
$\displaystyle y(n_{s})$ $\displaystyle=$
$\displaystyle\frac{\text{ProductLog}[\alpha_{2}\ln(2(1-n_{s})^{-1}N_{f}^{-1})]}{\ln[2(1-n_{s})^{-1}N_{f}^{-1}]}.$
(66)

Besides, the tensor to scalar ratio in function of the scalar spectral index
(consistency relation) $r=r(n_{s})$ is given by Eq.(51) and the power spectrum
$A_{s}(n_{s})$ from Eq.(42) becomes

$A_{s}(n_{s})=\frac{\kappa^{2}}{3\alpha\pi^{2}\tilde{c}_{s_{f}}(1-n_{s})^{2}}\left[1-\frac{2^{\beta}\alpha(1-n_{s})^{1-\beta}}{(1-\beta)C}\right].$
(67)

Let us now consider the fact that the energy density of reheating has to be
less than the energy density at the end of the inflationary epoch. In this
way, we can write the following equation Chiba:2015zpa

$\frac{\pi^{2}}{30}g_{\star\text{,reh}}T^{4}_{\text{reh}}<\rho_{f}=\frac{\rho_{f}}{V}\frac{3\pi^{2}rA_{s}}{2\kappa^{2}},$
(68)

where we have considered that the effective potential
$V=3\pi^{2}rA_{s}/(2\kappa^{2})$.

Thus, if we consider the upper limit of this inequality in which
$(\pi^{2}/30)g_{\star\text{,reh}}T_{\text{reh}}^{\text{c}^{4}}=\rho_{f}$, we
can obtain the so-called critical temperature $T_{\text{reh}}^{\text{c}}$ of
the model and this temperature becomes

$T_{\text{reh}}^{\text{\,c}}\simeq\left[\frac{3}{2}+\frac{1}{3g_{f}V_{f}}\left(1+\frac{3g_{f}V_{f}}{2}-\sqrt{1+3g_{f}V_{f}}\right)\right]^{1/4}\,\left[1-\frac{(1-n_{s})^{2y(n_{s})}}{2^{2y(n_{s})}y(n_{s})^{2}}\left(\frac{\alpha}{C}\right)^{2}\right]^{1/4}\,\left[\frac{30\,V_{f}}{g_{\star\text{,reh}}\pi^{2}}\right]^{1/4},$
(69)

where we have used Eqs.(33), (51) and (67), respectively. In particular
considering the special case in which $c_{s_{f}}=0.8$ together with the values
of the parameters of the Table I, we get

$T_{\text{reh}}^{\text{\,c}}\simeq 1.94\times
10^{15}\text{GeV}\left[1-\frac{(1-n_{s})^{2y(n_{s})}}{2^{2y(n_{s})}y(n_{s})^{2}}\left(\frac{\alpha}{C}\right)^{2}\right]^{1/4},$
(70)

where we have assumed that $g_{\star\text{,reh}}=106.75$.

In Fig.4 we show the evolution the number of $e-$folds during reheating
$N_{\text{reh}}$ (upper panel) and the reheating temperature $T_{\text{reh}}$
(lower panel) in terms of the scalar spectral index $n_{s}$. Here different
values have been used for the EoS parameter during the reheating period;
$\omega_{\text{reh}}=\\{-1/3,0,2/3,1\\}$ together with some values of the
speed of sound at the end of the inflationary era;
$c_{s_{f}}=\\{0.6,0.8,0.99\\}$. In this context, for each value of the EoS
parameter $\omega_{\text{reh}}$, we have graphed three propagation speed of
sound at the end of inflation $c_{s_{f}}$.

The different regions of the Fig.4 correspond to; the light blue shaded region
denotes the maximum and minimum limits of the scalar spectral index
$n_{s}=0.9649\pm 0.0042$ from data Planck at 1$\sigma$ bounds. Besides, the
blue shaded corresponds to a projected sensitivity from the central value of
the scalar spectral index at $n_{s}=0.9649$ of $\pm 10^{-3}$ in the test
studied in Ref. -3 . Further, the pink shaded region denotes the electroweak
scale in which the temperature is approximately $T_{\text{EW}}\sim
100\text{GeV}$ and the purple shaded region with temperatures below 10 MeV,
and this region is rejected by primordial nucleosynthesis. From Fig.4, the
point in which all the lines converge corresponds to the instantaneous
reheating where the number of $e-$folds $N_{\text{reh}}\rightarrow 0$. Here we
note that the EoS parameter $\omega_{\text{reh}}$ does not play an important
role, since all lines related to the EoS parameter $\omega_{\text{reh}}$
converge to the same point. Additionally, we note from Fig.4 that the speed of
sound at the end of the inflationary epoch $c_{s_{f}}$ does not play a
significant role in the different parameters related to the reheating era,
since all lines associated to a certain value of EoS parameter
$\omega_{\text{reh}}$ are similar.

From Fig.4, we note that the reconstructed model agrees with Planck’s
1$\sigma$ bounds on the spectral index $n_{s}$ for the distinct values of the
EoS parameters $\omega_{\text{reh}}$, when we consider high reheating
temperatures $T_{\text{reh}}>10^{13}$ GeV. In particular for the case in which
the EoS parameter $\omega_{\text{reh}}>0$, we observe that this compatibility
with the Planck-data occurs even at low reheating temperature. Besides, we
note that any reheating temperature between the primordial nucleosynthesis
bound and the instantaneous reheating value is permitted inside the Planck’s
1$\sigma$ bound regardless of the values of EoS parameter excepting the
specific value $\omega_{\text{reh}}=-1/3$, since this values of
$\omega_{\text{reh}}$ presents high temperature. Also, we note that the
reconstructed model predicts a small number of $e-$ folds during the reheating
epoch ($N_{\text{reh}}<40$) in the range of Planck-data on the scalar spectral
index.

Figure 4:  The figure shows the number of $e-$ folds during the reheating
scenario $N_{\text{reh}}$ (upper panel) and the reheating temperature
$T_{\text{reh}}$ (lower panel) versus the scalar spectral index $n_{s}$ for
different speed of sound at the end of inflationary epoch $c_{s_{f}}$. For the
value $c_{s_{f}}=0.6$ corresponds to the red line, $c_{s_{f}}=0.8$ to blue
line and $c_{s_{f}}=0.99$ to green line, respectively. Besides, in both panels
we have assumed different values of the EoS parameter $w_{\text{reh}}$;
$w_{\text{reh}}=-1/3,0,2/3$ and $w_{\text{reh}}=1$, respectively. Also we have
used the parameters shown in Table I (upper bounds).

##  VI Conclusions and Remarks

In this article we have first studied the reconstruction of the background
variables in the framework of a non-canonical theory, assuming the
parametrization on the cosmological parameters which are; the scalar spectral
index $n_{s}=n_{s}(N)$ and the speed of sound $c_{s}=c_{s}(N)$, in terms of
the number of $e$-folds $N$. In the context of the slow roll approximation, we
have obtained a general formalism of reconstruction for the background
variables; the effective potential and coupling function as a function of $N$.
In this general analysis, we have obtained from the parametrization of the
scalar spectral index $n_{s}(N)$ and the speed of sound $c_{s}(N)$ different
integral relationships for the effective potential and the coupling function
as a function of the number of $e$-folds $N$.

As a specific example to obtain the reconstructions of the scalar potential
and the coupling function in terms of the scalar field, we have considered the
simplest parametrizations for both the scalar spectral index given by
$n_{s}=1-2/N$ and the speed of sound $c_{s}\propto N^{-\beta}$. In fact, we
have used our general expressions given by Eqs.(22), (23) and (27), in order
to find $V(\phi)$ and $g(\phi)$. Thus, from the relation between the scalar
field and the number of $e$-folds $N$, we have obtained the reconstruction of
the effective potential and the coupling function in terms of the scalar
field, i.e., $V(\phi)$ and $g(\phi)$. In this respect, Figs.(1) and (2) show
the reconstruction of the background variables in terms of the scalar field
from the simplest parametrizations $n_{s}(N)$ and $c_{s}(N)$, as well the
relation between the number of $e-$folds $N$ and the scalar field $\phi$,
i.e., $N(\phi)$ (see upper panel of Fig.1). Additionally, we have found the
consistency relation $r=r(n_{s})$ given by Eq.(51), in which we have
considered different number of $e-$folds at the end of the inflationary
scenario. Here we have obtained that the reconstruction of our model is well
supported by Planck data when the number of $e-$ folds at the end of inflation
$N_{f}\sim\mathcal{O}(1)$, see Fig(3). Besides, we have found the parameter-
space for the different integration constants obtained from the reconstruction
mechanism used, as well the constraints on the number of $e-$folds $N_{f}$ and
the power $\beta$ assuming a specific value of the speed of sound at the end
of the inflation. These results are shown in Table I. From this table, we have
noted that the integration constants $C$ and $\alpha$ are similar and the
constraints on these constants do not depend strongly on the value of sound
speed $c_{s_{f}}$. In the same form, we have found that the number of $e-$
folds at the end of inflation $N_{f}\sim\mathcal{O}(1)$ and it does not depend
strongly on the value of sound speed. However, we have found that the power
$\beta$ related to the ansatz of the speed of sound presents a small variation
between $-0.1<\beta<0.1$ for the different values of sound speed $c_{s_{f}}$
studied.

Regarding the analysis during the reheating era in the framework of a non-
canonical theory, we have obtained that it is feasible to describe this epoch
in terms of the different reheating parameters as; the reheating temperature
$T_{\text{reh}}$, number of $e-$folds associated to the reheating
$N_{\text{reh}}$ and the EoS parameter $w_{\text{reh}}$. Thus, we have found
the fundamental reheating parameters given by Eqs.(63) and (64), which are
temperature and the number of $e-$folds during the reheating epoch in terms of
parameters associated to the model and these are; the sound of speed, the EoS
parameter $w_{\text{reh}}$ together with the observational parameters such as;
$A_{s}$, $n_{s}$, and $r$. In relation to this point, from Fig(4) we have
found that the number of $e-$folds together with the temperature during the
reheating epoch do not depend strongly of the value of the speed of sound at
the end of inflationary scenario. Here the red, blue and green lines that
indicate the different values of the speed of sound $c_{s_{f}}$ are similar
for a given value of the EoS parameter $w_{\text{reh}}$.

In relation to the Planck data at 1$\sigma$ bound on the scalar spectral index
$n_{s}$, we have obtained that for the situations in which the EoS parameter
$w_{\text{reh}}\leq 0$, the reconstructed model shows higher values associated
to the reheating temperatures, in particular for the case in which the EoS
parameter $w_{\text{reh}}=-1/3$, we have found that $T_{\text{reh}}\gtrsim
10^{12}$GeV. Additionally, we have found that the duration of the reheating
epoch distinguished by the number of $e$-folds $N_{\text{reh}}$ turns out to
be small that $N_{\text{reh}}\lesssim 35$ for the different values of the EoS
parameter $w_{\text{reh}}$ from the constraints given by Planck data at
1$\sigma$ bound on $n_{s}$, see Fig.(4).

Finally in this work, we have not addressed the reheating study in numerical
form assuming an EoS parameter $w_{\text{reh}}$ that depends on time, to
analyze the reheating parameters; $T_{\text{reh}}$ and $N_{\text{reh}}$. In
this point, we hope to return to this study in the near future.

##  VII acknowledgments

C. R. thanks to the Vicerrectoría de Investigación Creación e Innovación,
Pontificia Universidad Católica de Valparaíso, Postgraduate Scholarship PUCV
2024.

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