$\Lambda$ polarizing fragmentation function from Belle $e^{+}e^{-}$ data

Umberto D’Alesio1,2$\star$, Francesco Murgia2 and Marco Zaccheddu1,2

1 Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria di
Monserrato, I-09042 Monserrato (CA), Italy

2 INFN, Sezione di Cagliari, Cittadella Universitaria di Monserrato, I-09042
Monserrato (CA), Italy

*<EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

Experimental data from Belle Collaboration for the transverse polarization of
$\Lambda$’s measured in $e^{+}e^{-}$ annihilation processes are used to
extract the polarizing fragmentation function (FF) within a TMD approach. We
consider both associated and inclusive $\Lambda$ production, showing a quite
consistent scenario. Good separation in flavor is obtained, leading to four
independent FFs. Predictions for SIDIS processes at the EIC, crucial for
understanding their universality and evolution properties, are also presented.

##  1 Introduction

Transverse momentum dependent parton distribution and fragmentation functions
(TMDs) play a crucial role in our understanding of the internal structure of
the nucleon as well as the parton fragmentation mechanism into hadrons. In
this contribution we focus on one TMD that, despite its relevance, has been
still poorly explored: the so-called polarizing fragmentation function (pFF),
giving the distribution of a transversely polarized spin-1/2 hadron in the
fragmentation of an unpolarized quark. It is T-odd, but chiral even, allowing
to access it without any unknown, chiral-odd, counterpart.

It was studied phenomenologically long time ago in the context of the puzzling
transverse polarization data for the inclusive production of $\Lambda$’s in
unpolarized hadron-hadron collisions [1], but the lack of additional
experimental information prevented any further theory development. The
situation has significantly changed with the release of new data from the
Belle Collaboration at KEK [2] on transverse $\Lambda/\bar{\Lambda}$ hyperon
polarization in $e^{+}e^{-}$ processes. Indeed, for processes like
$e^{+}e^{-}\to\Lambda\,h+X$, in contrast to the inclusive process
$pp\to\Lambda+X$, a TMD factorization approach has been formally proven [3,
4].

We report here the findings of Ref. [5], showing how Belle data allow for the
first ever extraction of the pFF in a clear way. We will first consider the
associated production ($\Lambda\,h$) data set alone, for which TMD
factorization holds. We then include, within a simplified TMD approach, the
$\Lambda$-jet case, by performing a separate full-data fit. Even if this
configuration presents some caveats both experimentally and theoretically [6,
7], it represents a clear source of information on the explicit $p_{\perp}$
dependence of the TMD pFF.

##  2 Formalism

We consider the processes $e^{+}e^{-}\to h_{1}h_{2}+X$ where $h_{1}$ is a
spin-1/2 hadron and the second (light and unpolarized) hadron, $h_{2}$, is
produced almost back-to-back with respect to $h_{1}$. The complete
calculation, within the helicity formalism, of all accessible spin observables
will be presented elsewhere. Here we discuss in detail the case of the
transverse hyperon polarization, defined generically as

${\cal
P}_{T}=\frac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\uparrow}+d\sigma^{\downarrow}}=\frac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\rm
unp}},$ (1)

where $d\sigma^{\uparrow(\downarrow)}$ is the differential cross section for
the production of a hadron transversely polarized along the up(down) direction
with respect to the production plane and $d\sigma^{\rm unp}$ is the
unpolarized cross section.

As we will show below, this observable is directly sensitive to the polarizing
FF, $\Delta D_{h_{1}^{\uparrow}\\!/q}$, also denoted as $D_{1T}^{\perp q}$
[8]. More precisely, for a hadron with a lightcone momentum fraction $z$,
intrinsic transverse momentum $\bm{p}_{\perp}$ and polarization vector
$\hat{\bm{P}}\equiv\uparrow$, coming from the fragmentation of a quark with
momentum $\bm{p}_{q}$, the pFF is defined as

$\displaystyle\Delta\hat{D}_{h^{\uparrow}\\!/q}(z,\bm{p}_{\perp})\equiv\hat{D}_{h^{\uparrow}\\!/q}(z,\bm{p}_{\perp})-\hat{D}_{h^{\downarrow}\\!/q}(z,\bm{p}_{\perp})=\Delta
D_{h^{\uparrow}\\!/q}(z,p_{\perp})\,\hat{\bm{P}}\cdot(\hat{\bm{p}}_{q}\times\hat{\bm{p}}_{\perp})\,.$
(2)

For associated hadron production the following configuration is adopted: the
produced unpolarized hadron, $h_{2}$, defines the opposite $z$ direction and
the $\widehat{xz}$ plane is determined by the lepton and the $h_{2}$
directions. A second plane is determined by $\hat{\bm{z}}$ and the direction
of the spin-1/2 hadron, $h_{1}$, with three-momentum
$\bm{P}_{h_{1}}=(P_{1T}\cos\phi_{1},P_{1T}\sin\phi_{1},P_{1L})$. This is the
_hadron frame_ configuration and the transverse polarization is measured along
$\hat{\bm{n}}\propto(-\bm{P}_{h_{2}}\times\bm{P}_{h_{1}})$.

The transverse polarization of $h_{1}$, as defined in its helicity frame, has
to be properly projected along $\hat{\bm{n}}$. Moreover, two independent
modulations appear: one in term of the convolution of the pFF with the
unpolarized TMD-FF for $h_{2}$, and another one involving the Collins FF for
$h_{2}$, that, however, vanishes upon integration over $\bm{P}_{1T}$. The
final expression for the polarization along $\hat{\bm{n}}$, so integrated, and
adopting a Gaussian Ansatz for the TMD-FFs reads

$\displaystyle{\cal P}_{n}(z_{1},z_{2})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{e\pi}{2}}\frac{1}{M_{\rm pol}}\frac{\langle
p_{\perp}^{2}\rangle_{\rm pol}^{2}}{\langle p_{\perp
1}^{2}\rangle}\,\frac{z_{2}}{\big{\\{}[z_{1}(1-m_{h_{1}}^{2}/(z_{1}^{2}s))]^{2}\langle
p_{\perp 2}^{2}\rangle+z_{2}^{2}\langle p_{\perp}^{2}\rangle_{\rm
pol}\big{\\}}^{1/2}}$ (3) $\displaystyle\times$
$\displaystyle\frac{\sum_{q}e^{2}_{q}\,\Delta
D_{h_{1}^{\uparrow}/q}(z_{1})D_{h_{2}/\bar{q}}(z_{2})}{\sum_{q}e^{2}_{q}\,D_{h_{1}/q}(z_{1})D_{h_{2}/\bar{q}}(z_{2})}\,,$

where we have used the following parametrization for the TMD-FFs

$\displaystyle D_{h/q}(z,p_{\perp})$ $\displaystyle=$ $\displaystyle
D_{h/q}(z)\,\frac{e^{-p_{\perp}^{2}/\langle p_{\perp}^{2}\rangle}}{\pi\langle
p_{\perp}^{2}\rangle}\,,$ (4) $\displaystyle\Delta
D_{h^{\uparrow}\\!/q}(z,p_{\perp})$ $\displaystyle=$ $\displaystyle\Delta
D_{h^{\uparrow}\\!/q}(z)\frac{\sqrt{2e}\,p_{\perp}}{M_{\rm
pol}}\frac{e^{-p_{\perp}^{2}/\langle p_{\perp}^{2}\rangle_{\rm
pol}}}{\pi\langle p_{\perp}^{2}\rangle},$ (5)

with $\langle p_{\perp}^{2}\rangle_{\rm pol}=\frac{M_{\rm pol}^{2}}{M_{\rm
pol}^{2}+\langle p_{\perp}^{2}\rangle}\,\langle p_{\perp}^{2}\rangle$. By
imposing $|\Delta D(z)|\leq D(z)$ the positivity bound for the pFF, Eq. (2),
is automatically fulfilled.

As mentioned above, we consider also the inclusive $\Lambda$ production case
(within a jet) in a simplified TMD approach. In this case the polarization is
measured orthogonally to the _thrust plane_ , containing the jet and the
spin-1/2 hadron momentum, $\bm{P}_{h_{1}}$. Skipping all details, the
transverse polarization is then given as

$\displaystyle{\cal P}_{T}(z_{1},p_{\perp 1})$ $\displaystyle=$
$\displaystyle\frac{\sum_{q}e^{2}_{q}\,\Delta
D_{h_{1}^{\uparrow}/q}(z_{1},p_{\perp
1})}{\sum_{q}e^{2}_{q}\,D_{h_{1}/q}(z_{1},p_{\perp 1})}\,.$ (6)

For later use we give the first $p_{\perp}$-moment of the pFF:

$\Delta
D_{h^{\uparrow}\\!/q}^{(1)}(z)=\int\\!d^{2}\bm{p}_{\perp}\frac{p_{\perp}}{2zm_{h}}\Delta
D_{h^{\uparrow}\\!/q}(z,p_{\perp})=D_{1T}^{\perp(1)}(z)=\sqrt{\frac{e}{2}}\frac{1}{zm_{h}}\frac{1}{M_{\rm
pol}}\frac{\langle p^{2}_{\perp}\rangle^{2}_{\rm pol}}{\langle
p^{2}_{\perp}\rangle}\Delta D_{h^{\uparrow}\\!/q}(z)\,,$ (7)

where the last expression is obtained by using Eq. (5).

##  3 Results

Two data sets are available: one for the associated production of $\Lambda$
with light hadrons ($\pi$ and $K$), as a function of the energy fractions
$z_{\Lambda}$ and $z_{\pi}(z_{K})$ (128 data points) and one for the inclusive
production as a function of $p_{\perp}$ (the $\Lambda$ transverse momentum
w.r.t. the thrust axis), for different energy fractions, $z_{\Lambda}$ (32
data points).

The $z$-dependent part of the pFF is parameterized as follows

$\Delta
D_{\Lambda^{\uparrow}\\!/q}(z)=N_{q}z^{a_{q}}(1-z)^{b_{q}}\frac{(a_{q}+b_{q})^{(a_{q}+b_{q})}}{a_{q}^{a_{q}}b_{q}^{b_{q}}}D_{\Lambda/q}(z)\,,$
(8)

where $|N_{q}|\leq 1$ and $q=u,d,s,\rm{sea}$. This guarantees that $|\Delta
D(z)|\leq D(z)$.

For the light hadron unpolarized FFs we adopt the DSS07 set [9], and the AKK08
set [10] for $\Lambda$’s. The unpolarized Gaussian widths are taken as
$\langle p^{2}_{\perp}\rangle=0.2$ GeV2 [11], for all hadrons. Evolution
effects do not play any role in this extraction, being performed at a fixed
energy scale. Since all $\Lambda$ FF sets are given for
$\Lambda+\bar{\Lambda}$, including the AKK08 set, we separate the two
contributions assuming

$D_{\bar{\Lambda}/q}=D_{\Lambda/\bar{q}}=(1-z)\,D_{\Lambda/q}\,.$ (9)

Other choices have a very little impact on the fit.

As already stated, we start performing a fit of the associated production data
alone (for a similar analysis see Ref. [12]), including in a second phase also
the inclusive data set. To improve the quality of the fit, we have imposed the
following cuts: $z_{\pi,K}\leq 0.5$ for the associated production and
$z_{\Lambda}\leq 0.5$ for the $\Lambda$-jet data set, leaving us with 96+24 =
120 data points.

Concerning the $z$-dependent part, Eq. (8), we have tried many different
combinations of parameter sets, leading to the following best choice:

$N_{u},\;N_{d},\;N_{s},\;N_{\rm sea},\;a_{s},\;b_{u},\;b_{\rm sea}\,,$ (10)

with all other $a$ and $b$ parameters set to zero. This, together with
$\langle p^{2}_{\perp}\rangle_{\rm pol}$ (Eq. (5)), implies 8 free parameters.

Table 1:  Best values of the 8 free parameters fixing the pFF (Eqs. (5), (8)) for $u,d,s$ and sea quarks, as obtained by fitting the full set of Belle data [2], with their statistical errors (see text).  $N_{u}=0.47^{+0.32}_{-0.20}$ | $N_{d}\;\;=-0.32^{+0.13}_{-0.13}$
---|---
$N_{s}=-0.57^{+0.29}_{-0.43}$ | $N_{\rm sea}=-0.27^{+0.12}_{-0.20}$
$a_{s}\;=2.30^{+1.08}_{-0.91}$ |
$b_{u}\;=3.50^{+2.33}_{-1.82}$ | $b_{\rm sea}\;=2.60^{+2.60}_{-1.74}$
$\langle p^{2}_{\perp}\rangle_{\rm pol}=0.10^{+0.02}_{-0.02}$ GeV2 |

In Table 1 we collect the values of the best-fit parameters for the full-data
analysis. The corresponding estimates, compared against Belle data [2], are
shown in Figs. 1 and 2, respectively for the associated and inclusive
$\Lambda$ production. The fit of the associated production data leads to a
$\chi^{2}_{\rm dof}=1.26$ with $\chi^{2}_{\rm points}=0.8,1.5$ for pion and
kaon data subsets, while the full-data fit gives a $\chi^{2}_{\rm dof}=1.94$,
with $\chi^{2}_{\rm points}=2.75,1.55,1.61$ for jet, pion and kaon data
subsets. The shaded areas, computed following the procedure described in the
Appendix of Ref. [13], correspond to a 2$\sigma$ uncertainty, and result in
the statistical errors quoted in Table 1.

Figure 1: Best-fit estimates, based on the full-data set, of the transverse
polarization for $\Lambda$ and $\bar{\Lambda}$ production in
$e^{+}e^{-}\to\Lambda(\bar{\Lambda})h+X$, vs. $z_{h}$ (of the associated
hadron) for different $z_{\Lambda}$ bins. Data are from Belle [2]. The
statistical uncertainty bands, at 2$\sigma$ level, are also shown. Data for
$z_{\pi,K}>0.5$ are not included in the fit. Figure 2: Best-fit estimates of
the transverse polarization for inclusive $\Lambda$/$\bar{\Lambda}$ production
in $e^{+}e^{-}\to\Lambda(\rm{jet})+X$ vs. $p_{\perp}$ for different
$z_{\Lambda}$ bins, compared against Belle data [2]. The statistical
uncertainty bands, at 2$\sigma$ level, are also shown. Curves for $\Lambda$
and $\bar{\Lambda}$ coincide and data in the rightmost panel are not included
in the fit.

Figure 3: First moments of the pFFs, see Eq. (7), for the up (a), down (b),
strange (c) and sea (d) quarks, as obtained from the full-data fit (red solid
lines) and the $\Lambda$-hadron fit (blue dot-dashed lines). The statistical
uncertainty bands (at 2$\sigma$ level), as well as the positivity bounds
(black dashed lines), are shown.

Even if the best-fit parameters, as obtained in the associated or the full-
data fit, are a bit different, the corresponding first moments, Eq. (7), are
quite stable, as shown in Fig. 3.

Some comments are in order: $i)$ The inclusive production case presents the
largest $\chi^{2}$, on the other hand, while one would expect ${\cal P}_{T}=0$
at $p_{\perp}=0$, as well as ${\cal P}_{T}(\bar{\Lambda})={\cal
P}_{T}(\Lambda)$, the data do not show such features (Fig. 2); $ii)$ For the
associated production data, the charge-conjugation symmetry, ${\cal
P}_{n}(\Lambda h^{+})={\cal P}_{n}(\bar{\Lambda}h^{-})$ is clearly visible
(Fig. 1); iii) At medium $z_{\pi,K}$ values, dominated by the valence
unpolarized FFs, $\Lambda\pi^{-}$, $\Lambda\pi^{+}$ and $\Lambda K^{+}$ data
give direct information on the pFFs respectively for $u$, $d$ and $s$ quarks:
${\cal P}_{n}(\Lambda\pi^{-})>0$ implies a positive pFF of the up quark, while
${\cal P}_{n}(\Lambda\pi^{+})<0$ leads to a negative down quark pFF (see Figs.
1a, 3a,b); $iv)$ For small $z_{\Lambda}$, sea quark FFs start playing some
role, becoming important around $z_{\Lambda}\leq 0.3$; $v)$ The negative
values of ${\cal P}_{n}(\Lambda K^{+}$) at medium $z_{\Lambda}$,is driven by a
sizeable and negative $\Delta D_{\Lambda^{\uparrow}\\!/s}$ (Fig. 3c), coupled
to the leading FF $D_{K^{+}/\\!\bar{s}}$. Similar reasonings apply to the
$\bar{\Lambda}h$ data set.

The so extracted pFFs can be used to give some predictions for the same
observable in SIDIS, in particular for the EIC kinematics. In such a case the
polarization is measured transversely w.r.t. the plane containing the target
and the $\Lambda$ particle. We give here directly the final result as a
function of $x_{\rm B}$ and $z_{h}$ (the standard SIDIS variables) and
adopting a Gaussian parametrization also for the unpolarized TMD parton
distribution:

$P(x_{\rm B},z_{h})=\frac{\sqrt{2e\pi}}{2m_{p}}\frac{\langle
p_{\perp}^{2}\rangle^{2}_{\rm pol}}{\langle
p_{\perp}^{2}\rangle}\frac{1}{\sqrt{\langle p_{\perp}^{2}\rangle_{\rm
pol}+z_{h}^{2}\langle
k_{\perp}^{2}\rangle}}\frac{\sum_{q}e_{q}^{2}f_{q/p}(x_{\rm B})\Delta
D_{\Lambda^{\uparrow}/q}(z_{h})}{\sum_{q}e_{q}^{2}f_{q/p}(x_{\rm
B})D_{\Lambda/q}(z_{h})}\,.$ (11)

The corresponding estimates are shown in Fig. 4, where one can see that the
values are sizeable and could allow for a test of the universality of the pFF
as well as of its flavor separation.

Figure 4: Estimates for the transverse polarization of
$\Lambda$/$\bar{\Lambda}$ in SIDIS for EIC kinematics (see legend). The
statistical uncertainty bands (at 2$\sigma$ level) are shown.

##  4 Conclusions

The recent data from Belle Collaboration for the transverse
$\Lambda/\bar{\Lambda}$ polarization have allowed to extract, for the first
time, the polarizing fragmentation function of $\Lambda$ hyperons. Data favor
a clear separation in flavors, requiring three different valence pFFs, with
their relative sign and size determined quite accurately, as well as a sea-
quark pFF.

### Funding information

This work is supported by the European Union’s Horizon 2020 research and
innovation programme under grant agreement No. 824093 (STRONG2020).

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# Reconstructing k-inflation from $n_{s}(N)$ and reheating constraints

Ramón Herrera<EMAIL_ADDRESS>Michel Housset
<EMAIL_ADDRESS>Constanza Osses<EMAIL_ADDRESS>Nelson Videla<EMAIL_ADDRESS>Instituto de Física, Pontificia
Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla 4059,
Valparaíso, Chile.

###### Abstract

Inspired by the reconstruction scheme of the inflaton field potential
$V(\phi)$ from the attractors $n_{s}(N)$, we investigate the viability of
reconstruct the inflationary potential within the framework of k-inflation for
a non-linear kinetic term $K(X)=k_{n+1}X^{n}$ through three expressions for
the scalar spectral index $n_{s}(N)$, namely: (i) $n_{s}-1=-\frac{2}{N}$, (ii)
$n_{s}-1=-\frac{p}{N}$, and (iii) $n_{s}-1=-\frac{\beta}{N^{q}}$. For each
reconstructed potential, we determine the values of the parameter space which
characterize it by requiring that it must reproduce the observable parameters
from PLANCK 2018 and BICEP/Keck results. Furthermore, we analyze the reheating
era by assuming a constant equation of state, in which we derive the relations
between the reheating duration, the temperature at the end of reheating
together with the reheating epoch, and the number of $e$-folds during
inflation. In this sense, we unify the inflationary observables in order to
narrow the parameter space of each model within the framework of the
reconstruction in k-inflation.

###### pacs:

98.80.Cq

##  I Introduction

Inflation starobinsky1 ; inflation1 ; inflation2 ; inflation3  is the most
successful framework to describe the physics of the very early universe. The
reason for this is twofold. First, the inflationary universe offers a natural
explanation to several long-standing puzzles of the Hot Big-Bang (HBB) model,
such as the horizon, flatness, monopole problems, etc. Besides, since quantum
fluctuations during the inflationary era may give rise to the primordial
density perturbations Starobinsky:1979ty ; R2 ; R202 ; R203 ; R204 ; R205 ,
inflation provides us with a causal interpretation of the origin of the
temperature anisotropies observed on the Cosmic Microwave Background (CMB)
COBE:1992syq ; WMAP:2003elm ; Planck:2013pxb ; WMAP:2012nax ; Planck:2013jfk ;
planck1 ; planck2 ; bicep1 ; bicep2 , while at the same time it comes with a
mechanism to explain the Large-Scale Structure (LSS) of the Universe
Abazajian:2013vfg . The simplest single-field slow-roll inflation model, based
on a canonical kinetic term and a potential $V(\phi)$, predicts a spectrum of
the primordial curvature perturbations which is almost Gaussian and almost
scale-invariant, being confirmed by CMB observations (see Refs.Lidsey:1995np ;
Lyth:1998xn ; Bassett:2005xm ; Baumann:2009ds ; Martin:2013tda ; Renaux-
Petel:2015bja  for reviews). In general, inflationary observables as the
scalar spectral index $n_{s}$ and the tensor-to-scalar ratio $r$ are sensitive
to the shape of the inflaton potential, and they also are strictly constrained
by the Planck data in combination with other cosmological data bicep2 . As far
as tensor-to-scalar ratio is concerned, currently there only exists an upper
bound on it, since the tensor perturbations signature on the CMB as B-mode
polarization has not been measured yet. The recent BICEP/Keck results
BICEP:2021xfz  put a stringent bound on $r$, given by $r_{0.05}<0.036$ at
95$\%$ C.L. These new results have relevant consequences on the development of
inflation models, since e.g., the power-law chaotic $V(\phi)\sim\phi^{p}$ is
completely ruled out for every $p$ in the canonical framework, as well as the
standard version of natural inflation Freese:1990rb ; Adams:1992bn .
Therefore, it would be challenging to build models which predict a tensor-to-
scalar ratio at the current detection limit. For a further discussion
regarding the detection prospects of next generation CMB experiments, see
Ref.Achucarro:2022qrl  (and references therein).

After inflation ends there is a phase of reheating reh1 ; reh2 , where the
universe fills with standard model (SM) particles. These particles interact
and will eventually thermalise to equilibrium at a reheating temperature
$T_{re}$, and then the standard hot big-bang cosmology of radiation dominated
era follows. From the theoretical point of view, the reheating temperature is
assumed to be larger than the temperature of Electroweak (EW) transition
Bassett:2005xm ; reh3 ; reh4 . For some conservative issues, the temperature
for the reheating era must be much larger by several orders of magnitude than
the temperature reached in the big-bang nucleosynthesis (BBN), i.e., above a
MeV bound1 ; bound2 . Historically, reheating was first studied by means
perturbative decays of the inflaton field reh1 ; reh2 . However, the
transition from inflation to the standard hot big-bang cosmology could happen
via very different mechanisms than the perturbative reheating approach. In
particular, parametric resonance effects may be significant under certain
regimes, particularly early in the oscillating regime when the oscillation
amplitude is large Greene:1997fu ; reh4 . As it was shown in Ref.
Podolsky:2005bw , the out-of-equilibrium nonlinear dynamics of fields produces
a sharp variation of the equation-of-state (EoS) parameter during the
reheating phase. Consequently, the physics of reheating is complicated, highly
uncertain, and in addition it cannot be directly probed by observations.
Nevertheless, one may obtain indirect constraints on reheating by assuming for
the fluid a constant equation of state (EoS) $w_{re}$ during reheating. In
this sense, then we find certain relations between the reheating temperature
$T_{re}$ and the duration of reheating $N_{re}$ with $w_{re}$ and inflationary
observables Martin:2010kz ; paper1 ; Martin:2014nya ; paper2 ; Cai:2015soa ;
paper3 ; Rehagen:2015zma ; Ueno:2016dim ; Panotopoulos:2020qzi ;
Mishra:2021wkm ; Osses:2021snt ; Gong:2015qha  On theoretical and
observational grounds, going beyond standard canonical inflation within
General Relativity (GR) has become of a special interest. A more general
scenario is provided by the k-inflation framework, in which a non-linear
function of the kinetic term $K(X)$ is present in the Lagrangian kinflation ;
ps , i.e. $P(\phi,X)=K(X)-V(\phi)$, where
$X=-\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi$ is the canonical
kinetic term. A non-trivial kinetic term can lead a reduced scalar propagation
speed ($c_{s}^{2}<1$). This introduces new features, including a suppression
of the tensor-to-scalar ratio $r$ ps  and at the same time a large amount of
non-Gaussianities (NG) Chen:2006nt ; DeFelice:2011uc ; DeFelice:2011zh . As a
particular case, $K(X)=k_{n+1}X^{n}$ Mukhanov:2005bu ; liddle ;
Unnikrishnan:2012zu , with $n$ taking integer values and $k_{n+1}$ being
constants such that $k_{n+1}X^{n}$ has units of $M_{pl}^{4}$, accounts for a
reduced scalar propagation speed $c_{s}^{2}=\dfrac{1}{2n-1}$ if $n>1$. Thus,
at least in principle, k-inflation becomes phenomenologically distinguishable
from standard inflation, where $c_{s}^{2}=1$. In this regard, in
Refs.Mishra:2022ijb ; Pareek:2021lxz , the authors found that the simplest
power-law potentials, as the chaotic quadratic and quartic ones, become
compatible with current bounds on the tensor-to-scalar ratio in a non-
canonical scenario with $K(X)\propto X^{n}$ if $n>1$. Besides, scalar field
models with a non-trivial kinetic term have been discussed in the context of
k-essence in order to explain the observed speed-up of the universe at present
time Chiba:1999ka ; Armendariz-Picon:2000nqq ; Armendariz-Picon:2000ulo ;
Deffayet:2011gz .

As usual, inflation is studied on the basis of a potential for which, within
the slow-roll approximation, inflationary observables as the spectral index
$n_{s}$ and the tensor-to-scalar ratio $r$ are computed and then compared with
CMB observations through the $n_{s}-r$ plane. However, in Ref.chiba  it was
proposed to reconstruct the potential by assuming a standard canonical
inflaton field from an attractor for the spectral index as a function of the
number of $e$-folds $N$, i.e., $n_{s}(N)$ in the framework of Einstein
gravity. This inverse problem is motivated by the observational data regarding
the spectral index, since the attractor $n_{s}-1=-\,2/N$, which is predicted
from Starobinsky model starobinsky , the $\alpha$-attractor models alpha1 ;
alpha2 ; alpha3 , the quadratic chaotic inflation model inflation3 , and Higgs
inflation with a non-minimal coupling higgs1 ; higgs2  is compatible with
latest data of PLANCK. Similar reconstruction procedures have been also
proposed previously. In Mukhanov:2013tua , the inflaton potential $V(\phi)$ is
reconstructed from a certain function of the slow-roll parameter
$\epsilon(N)$, while in Roest:2013fha  the potential $V(N)$ is obtained when
both slow-roll parameters $\epsilon(N)$ and $\eta(N)$ are given, and then the
tensor-to-scalar ratio is computed. Also, more general expressions
$n_{s}-1=-\,p/N$ and $n_{s}-1=-\,\beta/N^{q}$ have been studied in
Refs.Garcia-Bellido:2014gna ; pattractor  and Huang:2007qz , respectively.
This reconstruction scheme have been also applied in theories beyond the
standard framework, e.g. in k-inflation, when a coupling to the standard
kinetic term $K(\phi)X$ is considered Yi:2021xhw ; Herrera:2020mjh , Randall-
Sundrum II braneworld Herrera:2019xhs ; Bhattacharya:2019ryo , G-inflation
Herrera:2018mvo , and warm inflation Herrera:2018cgi .

The main goal of the present work is to apply the reconstruction scheme from
the scalar spectral index $n_{s}(N)$ in the framework of k-inflation, from a
non-linear kinetic term $K(X)=k_{n+1}X^{n}$, in order to obtain the potential
of the inflaton field $V(\phi)$. In fact, we study how k-inflation modifies
the reconstruction of the inflationary scenario by assuming as attractor the
scalar spectral index $n_{s}(N)$. Furthermore, we discuss the implications for
reheating phase regarding its duration and temperature.

We organize our work as follows: After this introduction, in the next section
we briefly presents the dynamics of inflation and the the reconstruction
scheme from the scalar spectral index $n_{s}(N)$ within the framework of
k-inflation, as well as the basic formulas for determining the duration of
reheating as well as for the reheating temperature after inflation. In
Sections III, IV, and V we analyze the expressions $n_{s}-1=-2/N$,
$n_{s}-1=-\,p/N$, and $n_{s}-1=-\beta/N^{q}$, respectively. For all examples
associated to $n_{s}(N)$, we use the approximation of perfect fluid with a
constant equation of state in order to study reheating after inflation.
Finally, in Section VI we summarize our findings and present the conclusions.
We choose units of $c=\hbar=1$.

##  II k-inflation: Theoretical Framework

###  II.1 Dynamics of a generalized scalar field

Our framework is four-dimensional GR coupled to a single scalar field with a
generalized matter Lagrangian $\mathcal{L}(\phi,X)$, where $\phi$ is the
scalar field and $X=-\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi$ is the
standard kinetic term. is described by the action

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

where $g$ is the determinant of metric tensor $g_{\mu\nu}$, $M_{pl}=2.4\times
10^{18}~{}GeV$ is the reduced Planck mass, and $R$ is the Ricci scalar. For
this matter Lagrangian, the energy-momentum tensor can be recast into the form
of the energy-momentum tensor for a perfect fluid. Accordingly, the pressure
$P$ and energy density $\rho$ are given in terms of $\phi$ and $X$ as follows
ps

$\displaystyle P(\phi,X)$ $\displaystyle=$ $\displaystyle\mathcal{L},$ (2)
$\displaystyle\rho(\phi,X)$ $\displaystyle=$ $\displaystyle
2X\mathcal{L}_{,X}-\mathcal{L},$ (3)

respectively. For a canonical scalar field, we have $P(\phi,X)=X-V(\phi)$,
where $V(\phi)$ is the effective potential associated to the scalar field. In
this work, we restrict ourselves on the case in which the Lagrangian of the
scalar field is of the form

$P(\phi,X)=K(X)-V(\phi),$ (4)

with $K(X)$ being an arbitrary function of the kinetic term. For this
lagrangian, the corresponding energy density is given by

$\rho(\phi,X)=2XK_{,X}-K+V.$ (5)

In the following, the notation $P_{,X}$ corresponds to $\frac{\partial
P}{\partial X}$, $K_{,X}=\frac{\partial K}{\partial X}$,
$V_{,\phi}=\frac{\partial V}{\partial\phi}$,
$P_{,XX}=\frac{\partial^{2}P}{\partial X^{2}}$, etc.

The dynamics of our system is described by the Friedmann equation and the
equation for energy conservation

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\dfrac{\rho}{3M_{pl}^{2}},$ (6) $\displaystyle\dot{\rho}$
$\displaystyle=$ $\displaystyle-3H(\rho+P),$ (7)

where, in order to write Eq.(6) we have assumed a spatially flat Friedmann-
Lemaître-Robertson-Walker (FLRW) metric.

Defining the sound speed $c_{s}$, which describes the properties of the scalar
field in the fluid description, we have

$c_{s}^{2}\equiv\dfrac{P_{,X}}{\rho_{,X}}=\dfrac{P_{,X}}{2XP_{,XX}+P_{,X}}.$
(8)

Also, we can write the energy density conservation as

$\ddot{\phi}+3c_{s}^{2}H\dot{\phi}+\frac{\rho_{,\phi}}{\rho_{,X}}=0,$ (9)

which is a generalized version of the usual Klein-Gordon in a FLRW background.
Here we have considered a homogeneous scalar field i.e.,
$\phi(\vec{r},t)=\phi(t)$.

###  II.2 Inflationary dynamics

In this subsection, we will analyze the slow-roll approximation in order to
describe the inflationary universe.

For k-inflation, the slow-roll conditions read as follows Mukhanov:2005bu

$\displaystyle XK_{,X}\ll V,\,\,K\ll
V,\,\,\big{|}\ddot{\phi}\big{|}\ll\frac{V_{,\phi}}{\rho_{,X}},$ (10)

which ensure an inflationary expansion driven by the potential. Following
Ref.liddle , during the slow-phase, the inflationary dynamics can be
quantified in terms of the three parameters

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle-\dfrac{\dot{H}}{H^{2}},$ (11) $\displaystyle\eta$
$\displaystyle=$ $\displaystyle-\dfrac{\dot{\epsilon}}{H\epsilon},$ (12)
$\displaystyle s$ $\displaystyle=$ $\displaystyle\dfrac{\dot{c_{s}}}{Hc_{s}}.$
(13)

It is assumed that during slow-roll inflation the parameter
$\left(\epsilon,\big{|}\eta\big{|},\big{|}s\big{|}\right)\ll 1$. Therefore, by
considering $P(\phi,X)=K(X)-V(\phi)$, the Friedmann equation (6) and field
equation for the scalar field (7) reduce to

$\displaystyle 3M_{pl}^{2}H^{2}$ $\displaystyle\simeq$ $\displaystyle
V(\phi),$ (14) $\displaystyle 3c_{s}^{2}H\dot{\phi}+\frac{V_{,\phi}}{K_{,X}}$
$\displaystyle\simeq$ $\displaystyle 0,$ (15)

since we neglect the acceleration term $\ddot{\phi}$ in the equation of motion
for the scalar ﬁeld and $V$ is the dominant term in the expression for the
energy density.

For our concrete non-canonical scalar field model without specifying the
potential, in the following we will assume that the Lagrangian density and
energy density are given by

$\displaystyle P(\phi,X)$ $\displaystyle=$ $\displaystyle
k_{n+1}X^{n}-V(\phi),$ (16) $\displaystyle\rho(\phi,X)$ $\displaystyle=$
$\displaystyle(2n-1)k_{n+1}X^{n}+V(\phi).$ (17)

Here, the power $n$ takes integer real values and $k_{n+1}$ is a coupling
constant such that $k_{n+1}X^{n}$ has units of $M_{pl}^{4}$. In particular,
when $n=1$ and $k_{2}=1$, we recover the standard Lagrangian density for a
canonical scalar field. By replacing Eq.(16) into (8), we find that the sound
speed is a constant and given by

$c_{s}^{2}=\dfrac{1}{2n-1}.$ (18)

It should be noted that it can be very small for large $n$, in which $n\gg 1$,
and for the specific case when $n=1$ (canonical field), we have $c_{s}^{2}=1$.

Under the slow-roll approximation, the field equation for the scalar field
$\phi$ from Eq.(15) takes the form

$\dot{\phi}V_{,\phi}\simeq-6nHk_{n+1}X^{n}.$ (19)

From Eqs.(11) and (12), we can introduce the slow-roll parameters
$\epsilon_{V}$ and $\eta_{V}$ in terms of the potential and its derivatives
with respect to the scalar field as liddle

$\displaystyle\epsilon_{V}\simeq-\dfrac{1}{2}\dfrac{V_{,\phi}}{V}\dfrac{\dot{\phi}}{H},$
(20)
$\displaystyle\eta_{V}\simeq-\dfrac{V_{,\phi\phi}}{V_{,\phi}}\dfrac{\dot{\phi}}{H}.$
(21)

Now, by combining $X=\dfrac{\dot{\phi}^{2}}{2}$ and Eqs.(6) and (19), we find
that the ratio $\frac{\dot{\phi}}{H}$ becomes

$\dfrac{\dot{\phi}}{H}=-\alpha(n)\left(\dfrac{V_{,\phi}}{V^{n}}\right)^{\frac{1}{2n-1}},$
(22)

where the quantity $\alpha(n)$ is defined as

$\alpha(n)=\left(\dfrac{6^{n-1}M_{pl}^{2n}}{n\,k_{n+1}}\right)^{\frac{1}{2n-1}}.$
(23)

Then, the slow-roll parameters can be written as

$\displaystyle\epsilon_{V}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha(n)}{2}\left(\dfrac{V_{,\phi}^{2n}}{V^{3n-1}}\right)^{\frac{1}{2n-1}},$
(24) $\displaystyle\eta_{V}$ $\displaystyle=$
$\displaystyle\alpha(n)\left(\dfrac{V_{,\phi\phi}^{2n-1}}{V_{,\phi}^{2n-2}V^{n}}\right)^{\frac{1}{2n-1}}.$
(25)

Notice that for this model the slow-roll parameter $s$ becomes zero, since the
sound speed is a constant. Also, we have that for the case $n=1$ we recover
the usual slow-roll parameters associated to a canonical scalar field.

The amount of inflation is expressed in terms of the number of $e$-folds $N$,
defined by

$N\equiv\ln\left(\dfrac{a(t_{end})}{a(t)}\right)=\int_{t}^{t_{end}}H\,dt=\int_{\phi}^{\phi_{end}}H\,\dfrac{d\phi}{\dot{\phi}}\simeq\dfrac{1}{\alpha(n)}\int_{\phi_{end}}^{\phi}\left(\frac{V^{n}}{V_{,\phi}}\right)^{\frac{1}{2n-1}}\,d\phi,$
(26)

where $t$ and $t_{end}$ correspond to two different values of cosmic time;
$t_{end}$ denotes the end of inflation, and $t$ corresponds to the time when
the cosmological scales cross the Hubble-radius. In order to write $N$ as an
integral over the scalar field, we have used Eq.(22) and we have also assumed
that the number of $e$-folds at the end of inflation is $N_{end}=0$.

###  II.3 Perturbations

In the following we shall brieﬂy review cosmological perturbations in the
model of k-inflation.

Under the slow-roll approximation, the primordial scalar power spectrum was
derived in ps , and it results

$\mathcal{P}_{S}=\dfrac{1}{8\pi^{2}M_{pl}^{2}c_{s}}\dfrac{H^{2}}{\epsilon_{V}},$
(27)

which is evaluated at the time of horizon exit at $c_{s}k=aH$ ($k$ is a
comoving wavenumber). Besides, the scalar spectral index $n_{s}$ is defined as

$n_{s}-1=\left.\frac{d\ln\mathcal{P}_{S}}{d\ln
k}\right|_{c_{s}k=aH}\simeq-2\epsilon-\eta-s.$ (28)

On the other hand, the tensor power spectrum $\mathcal{P}_{T}$ and the
corresponding spectral index $n_{T}$ are given by ps

$\mathcal{P}_{T}=\frac{2H^{2}}{\pi^{2}M_{pl}^{2}},\quad\quad
n_{T}=\left.\frac{d\ln\mathcal{P}_{T}}{d\ln k}\right|_{c_{s}k=aH}.$ (29)

From Eqs.(27) and (29), the tensor-to-scalar ratio is found to be

$r\equiv\frac{\mathcal{P}_{T}}{\mathcal{P}_{S}}=16\epsilon
c_{s}=-8c_{s}n_{T}.$ (30)

As it can be seen from Eq.(30), the consistency relation is modified in
comparison to standard inflation ($c_{s}=1$). Thus, at least in principle,
k-inflation is phenomenologically distinguishable from standard inflation ps .

By replacing Eqs.(14) and (24) in (27), the scalar power spectrum for our
particular Lagrangian density (16) becomes

$\mathcal{P}_{\mathcal{S}}\simeq\dfrac{1}{12\pi^{2}M_{pl}^{4}c_{s}\alpha(n)}\left(\dfrac{V^{5n-2}}{V_{,\phi}^{\,2n}}\right)^{\frac{1}{2n-1}},$
(31)

and its spectral index can be expressed as a function of slow-roll parameters
given by Eqs.(20) and (21) as follows

$n_{s}-1\simeq\dfrac{1}{2n-1}\left[2n\eta_{V}-2(5n-2)\epsilon_{V}\right].$
(32)

For a given potential $V(\phi)$, we can determine the values of the parameters
characterizing the model by requiring that it must reproduce the observable
values of $\mathcal{P}_{S}$, $n_{s}$ and the upper bound on $r$. However, we
are not going to concern ourselves with a particular choice of $V(\phi)$,
since in the present work our main interest is to reconstruct the inflationary
potential from a given $n_{s}(N)$ within the framework of k-inflation.

###  II.4 Reconstructing $V(\phi)$ in k-inflation

Let us to explain how to reconstruct $V(\phi)$ from $n_{s}(N)$ within
k-inflation for a non-linear kinetic term $K(X)=k_{n+1}X^{n}$, following
Refs.liddle  and chiba .

From Eq.(26) an important relation arises between the number of $e$-folds, the
potential and its derivatives with respect to the scalar field, namely

$dN=\dfrac{1}{\alpha(n)}\left(\frac{V^{n}}{V_{,\phi}}\right)^{\frac{1}{2n-1}}\,d\phi.$
(33)

In order to obtain become real quantities, we impose that $V_{,\phi}>0$.
Therefore, we can use Eq.(33) to obtain a relation between the derivative of
the potential with respect to the number of $e$-folds $N$ and the scalar field
such that

$V_{,N}=\alpha(n)\left(\dfrac{V_{,\phi}^{2}}{V}\right)^{\frac{n}{2n-1}}.$ (34)

Note that here we have that $V_{,N}>0$. In order to find the relation between
the scalar field $\phi$ and $N$, we combine Eqs.(33) and (34) such that

$\phi(N)=\alpha(n)^{\frac{2n-1}{2n}}\int\left(\dfrac{V_{,N}}{V^{n}}\right)^{\frac{1}{2n}}\,dN.$
(35)

Now, by replacing Eqs.(33) and (34) in (20) and (21), we rewrite the slow-roll
parameters $\epsilon_{V}$ and $\eta_{V}$ in terms of the derivatives of the
potential with respect to the number of $e$-folds as

$\displaystyle\epsilon_{V}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\dfrac{V_{,N}}{V},$ (36) $\displaystyle\eta_{V}$
$\displaystyle=$
$\displaystyle\left(\dfrac{2n-1}{2n}\right)\dfrac{V_{,NN}}{V_{,N}}+\dfrac{V_{,N}}{2V},$
(37)

respectively. Then, the scalar spectral index (32) is rewritten as

$\displaystyle n_{s}-1$ $\displaystyle=$
$\displaystyle-\dfrac{2V_{,N}}{V}+\dfrac{V_{,NN}}{V_{,N}}=\ln\left(\dfrac{V_{,N}}{V^{2}}\right)_{,N}.$
(38)

We also may write the tensor-to-scalar ratio as

$r=16\epsilon_{V}c_{s}=\frac{8}{\sqrt{2n-1}}\frac{V_{,N}}{V}.$ (39)

In such a way, Eqs.(35) and (38) are the main equations which enables to us to
reconstruct $V(\phi)$ from the attractor $n_{s}(N)$.

###  II.5 Reheating

Here we shall briefly describe how to derive the expressions for the number of
$e$-folds during the reheating epoch $N_{re}$ as well as the reheating
temperature $T_{re}$ for k-inflation considering the reconstruction scheme.
For the derivation of the main formulas, we mainly follow Refs.paper1 ; paper2
; paper3  and also Pareek:2021lxz . First, it is assumed that during reheating
phase the dominant contribution to the energy density of the Universe comes
from a component which has an effective equation-of-state parameter (EoS)
$w_{re}$, and its energy density can be related to the scale factor through
$\rho\propto a^{-3(1+w_{re})}$. Here, we have considered that the EoS
parameter $w_{re}$ is constant. Therefore, we can write down the following
relation

$\frac{\rho_{end}}{\rho_{re}}=\left(\frac{a_{end}}{a_{re}}\right)^{-3(1+w_{re})},$
(40)

where the subscript “$end$ ” denotes the end of inflation and “$re$” the end
of reheating phase. Besides, the number of $e$-folds of reheating $N_{re}$ may
be related to the scale factors at the end of inflation and reheating
according to

$e^{-N_{re}}=\frac{a_{end}}{a_{re}}.$ (41)

Then, by combining Eqs. (40) and (41), we can write the number of $e$-folds
during the reheating scenario as

$N_{re}=\frac{1}{3(1+w_{re})}\ln\left(\frac{\rho_{end}}{\rho_{re}}\right),\,\,\textup{with}\,\,w_{re}\neq-1.$
(42)

On the other hand, by combining the time derivative of Friedmann Eq.(6) and
the conservation equation for energy density (7) along with the slow-roll
parameter $\epsilon$ (11), and then replacing the energy density and pressure
from Eqs. (16) and (17), the slow-roll parameter $\epsilon$ becomes

$\epsilon=\dfrac{3nk_{n+1}X^{n}}{(2n-1)k_{n+1}X^{n}+V}.$ (43)

If we solve the latter equation for $\dfrac{X^{n}}{V}$, we have

$\dfrac{X^{n}}{V}=\dfrac{\epsilon}{k_{n+1}(3n-(2n-1)\epsilon)}.$ (44)

In addition, Eq. (17) can be rewritten as

$\rho=V\left[(2n-1)k_{n+1}\dfrac{X^{n}}{V}+1\right].$ (45)

In such a way, we can express the energy density as a function of the slow-
roll parameter $\epsilon$ and the scalar field potential $V$ by replacing Eq.
(44) into (45)

$\rho=V\left[\dfrac{(2n-1)\epsilon}{3n-(2n-1)\epsilon}+1\right].$ (46)

Accordingly, the relationship between the energy density and the potential at
the end of inflation ($\epsilon(\phi_{end})=1$) is given by

$\rho_{end}=\frac{3n}{n+1}V_{end},$ (47)

where $V(\phi=\phi_{end})=V_{end}$. Note that this result for the energy
density $\rho_{end}$ becomes different from those already used in
Pareek:2021lxz , where $\rho_{end}=\frac{3}{2}V_{end}$, which corresponds to
standard canonical inflation ($n=1$) paper1 ; paper2 ; paper3 .

After replacement of (47) in Eq. (42), we get that $N_{re}$ becomes

$N_{re}=\frac{1}{3(1+w_{re})}\ln\left(\frac{3n}{n+1}\frac{V_{end}}{\rho_{re}}\right).$
(48)

At the end of reheating phase, the energy density of the universe is assumed
to have the form of a relativistic fluid

$\rho_{re}=\frac{\pi^{2}}{30}g_{re}T_{re}^{4},$ (49)

where $g_{re}$ is the number of internal degrees of freedom of relativistic
particles at the end of reheating. Assuming that the degrees of freedom come
from Standard Model (SM) particles, then $g_{re}\sim{\mathcal{O}}(100)$ for a
temperature $T\gtrsim 175$ GeV Husdal:2016haj , while for a Minimal
Supersymmetric Standard Model (MSSM), we have that
$g_{re}\sim{\mathcal{O}}(200)$ Adhikari:2019uaw .

Regarding the entropy, its density is defined as

$s=\frac{2\pi^{2}}{45}gT^{3},$ (50)

where the temperature is inversely proportional to the scale factor for
radiation epoch, i.e., $T\propto a^{-1}$. Then, by replacing the last relation
in Eq.(50), we have that $s\propto a^{-3}$. If we assume the conservation of
entropy, i.e., $gT^{3}a^{3}=const.$, then by applying this conservation
between reheating and the present time, results

$g_{re}T_{re}^{3}a_{re}^{3}=g_{0}T_{0}^{3}a_{0}^{3},$ (51)

where $g_{0}$ denotes the number of internal degrees of freedom of
relativistic particles today, which comes from photons and neutrinos and
$T_{0}=2.725$ K is the temperature of the universe today. Then, Eq.(51)
becomes paper1

$g_{re}T_{re}^{3}=\left(\frac{a_{0}}{a_{re}}\right)^{3}\left[2T_{0}^{3}+\frac{21}{4}T_{\nu
0}^{3}\right].$ (52)

Here, $T_{\nu 0}$ is the neutrino temperature at present time. For the
contribution coming from neutrinos at the right-hand side of (52), we have
used that $T_{\nu 0}=\left(\frac{4}{11}\right)^{1/3}T_{0}$. Besides, the ratio
$\frac{a_{0}}{a_{re}}$ can be rewritten as

$\frac{a_{0}}{a_{re}}=\frac{a_{0}}{a_{eq}}\frac{a_{eq}}{a_{re}},$ (53)

where $\frac{a_{re}}{a_{eq}}$ can be related to the duration in $e$-folds of
the radiation dominated epoch $N_{RD}$ according to
$e^{-N_{RD}}=\frac{a_{re}}{a_{eq}}$. In this way, Eq.(52) is rewritten as

$T_{re}=T_{0}\left(\frac{a_{0}}{a_{eq}}\right)e^{N_{RD}}\left(\frac{43}{11g_{re}}\right).$
(54)

For the ratio $\frac{a_{0}}{a_{eq}}$ we have (see, e.g. Pareek:2021lxz )

$\frac{a_{0}}{a_{eq}}=\frac{a_{0}}{a_{k}}\frac{a_{k}}{a_{end}}\frac{a_{re}}{a_{eq}}\frac{a_{end}}{a_{re}}=\frac{a_{0}H_{k}}{c_{s}k}e^{-N}e^{-N_{re}}e^{-N_{RD}},$
(55)

where we have considered that $\frac{a_{k}}{a_{end}}=e^{-N}$. Also, in order
to obtain the last equation, we have considered as well that the condition for
horizon crossing is defined as $c_{s}\,k=a_{k}H_{k}$. Using this result in
(54) we find

$T_{re}=\left(\frac{43}{11g_{re}}\right)^{1/3}\left(\frac{a_{0}T_{0}}{c_{s}k}\right)H_{k}e^{-N}e^{-N_{re}}.$
(56)

Assuming $g_{re}\sim\mathcal{O}(100)$ and the pivot scale
$\frac{k}{a_{0}}=0.05\textup{ Mpc}^{-1}$ by PLANCK, the number of $e$-folds
during reheating is obtained after replacing of Eq. (56) in Eq. (48), which
results

$N_{re}=\frac{4}{1-3\,w_{re}}\left[61.643-\dfrac{1}{4}\ln\left(\dfrac{3n}{n+1}\right)-\ln\left(\frac{c_{s}\,V_{end}^{\,1/4}}{H_{k}}\right)-N\right],$
(57)

where $H_{k}$ can be written down using the expression for the tensor-to-
scalar ratio $r=P_{T}/P_{S}$. Finally, combining Eqs.(48) and (49), the
reheating temperature is computed as follows

$T_{re}=\left(\frac{30}{g_{re}\,\pi^{2}}\right)^{1/4}\,\left(\frac{3n}{n+1}\,V_{end}\right)^{1/4}\,e^{-\frac{3}{4}(1+w_{re})N_{re}}.$
(58)

The model-dependent part of the main equations for reheating (57) and (58) are
the sound speed $c_{s}$, and therefore the power associated to the kinetic
term $n$. Also, the potential at the end of inflation $V_{end}$, the Hubble
rate when the cosmological scale crosses the Hubble radius $H_{k}$, and the
number of $e$-folds $N_{k}$. Note that both $N_{re}$ and $T_{re}$ implicitly
depend on the observables $\mathcal{P}_{s}$, $r$ and $n_{s}$. It is also
remarkable that the canonical case is recovered when $n=1$.

In the following we will study different relations between the scalar spectral
index $n_{s}$ in terms of the number of $e$-folds $N$ (attractors) in order to
reconstruct the inflationary scenario.

##  III First attractor $n_{s}-1=-\dfrac{2}{N}$

###  III.1 Dynamics of inflation

In order to obtain concrete results, as a first example we consider the
attractor $n_{s}-1=-\dfrac{2}{N}$ chiba . In this case, Eq. (38) becomes

$-\dfrac{2}{N}=\ln\left(\dfrac{V_{,N}}{V^{2}}\right)_{,N},$ (59)

which upon a first integration yields

$\dfrac{A}{N^{2}}=\dfrac{V_{,N}}{V^{2}},$ (60)

where $A$ is an integration constant, which is positive since $V_{,N}>0$. If
we integrate once with respect to the number of $e$-folds, we find the
following expression for the potential $V(N)$

$V(N)=\dfrac{N}{A+BN},$ (61)

with $B$ being a second integration constant, which can be either $B\gtrless
0$ or $B=0$. In order to obtain an expression for the scalar field as a
function of the number of $e$-folds $N$ for any value of the power $n$, we
replace the potential (61) into Eq. (35), yielding

$\displaystyle\phi(N)-C_{1}$ $\displaystyle=$
$\displaystyle\alpha^{\frac{2n-1}{2n}}\int\left(\frac{A}{N^{2}}\right)^{\frac{1}{2n}}\left(\frac{A+BN}{N}\right)^{\frac{n-2}{2n}}dN$
(62) $\displaystyle=$ $\displaystyle\mathcal{F}(N),$ (63)

where $C_{1}$ is a new integration constant and $\mathcal{F}(N)$ is a function
defined as

$\mathcal{F}(N)=\beta(N)\,{}_{2}F_{1}\left[\dfrac{1}{2},1,\dfrac{1}{n},-\dfrac{A}{BN}\right].$
(64)

Here ${}_{2}F_{1}$ corresponds to the hypergeometric function F21  and
$\beta(N)$ is another function given by

$\beta(N)=\dfrac{n\,\alpha^{\footnotesize{\frac{2n-1}{2n}}}}{(n-1)BN}\left(\dfrac{A}{N^{2}}\right)^{\frac{1}{2n}}\left(\dfrac{A}{N}+B\right)^{\frac{3n-2}{2n}}.$
(65)

Accordingly, from Eq.(63), the number of $e$-folds expressed in terms of the
scalar field can be written as

$N=\mathcal{F}^{-1}(\phi-C_{1}),$ (66)

where $\mathcal{F}^{-1}$ represents the inverse function of function (64). In
this way, by replacing Eq.(66) into Eq.(61), the potential being a function of
the scalar field is obtained as follows

$V(\phi)=\dfrac{\mathcal{F}^{-1}(\phi-C_{1})}{A+B\mathcal{F}^{-1}(\phi-
C_{1})}.$ (67)

As a particular case in which $n=1$, i.e., for standard inflation, the
reconstructed potential (67) becomes

$V(\phi)=\dfrac{1}{B}\tanh^{2}\left(\sqrt{\dfrac{B}{\alpha\,A}}\,\dfrac{(\phi-
C_{1})}{2}\right),$ (68)

and this potential corresponds to the T-model inflation studied in tmodel . On
the other hand, for the case $n=2$, the reconstructed potential (67) reduces
to

$V(\phi)=\dfrac{(\phi-C_{1})^{2}}{4A\sqrt{\alpha^{3}A}+B(\phi-C_{1})^{2}}.$
(69)

As asymptotic cases of Eq.(69), first we observe that if
$4A\sqrt{\alpha^{3}A}\gg B(\phi-C_{1})^{2}$, $V(\phi)$ behaves as a quadratic
chaotic potential inflation3

$V(\phi)\simeq\dfrac{(\phi-C_{1})^{2}}{4A\sqrt{\alpha^{3}A}},$ (70)

while if $4A\sqrt{\alpha^{3}A}\ll B(\phi-c)^{2}$, it becomes a constant
potential

$V\simeq\dfrac{1}{B}.$ (71)

For values of the power $n$ such that $n>2$, Eq.(66) cannot be solved
analytically for $N=N(\phi)$, and therefore we are not able to obtain an
analytical expression for $V(\phi)$.

###  III.2 Cosmological perturbations

If we replace Eq.(34) into Eq.(60), and using the expression for the scalar
power spectrum (31), we find that the first integration constant $A$ is
constrained to be

$A=\dfrac{\sqrt{2n-1}N^{2}}{12\pi^{2}M_{pl}^{4}\mathcal{P}_{s}}.$ (72)

On the other hand, upon replacement of Eq.(61) into Eq.(39), and considering
the constraint on $A$ given by Eq.(72), we obtain that the tensor-to-scalar
ratio may be expressed in terms of the number of $e$-folds as follows

$r=\frac{8}{\sqrt{2n-1}N+12B\pi^{2}M_{pl}^{4}\mathcal{P}_{s}}.$ (73)

The allowed values of the integration constants $A$ and $B$ can be found by
using the CMB constraints on the inflationary observables. In particular, we
use the constraint on the amplitude of the scalar spectrum
$\mathcal{P}_{s}\simeq 2.2\times 10^{-9}$ to set $A$, while the allowed values
of $B$ are found from the upper bound on $r$ by the BICEP/Keck data, i.e.,
$r<0.036$ BICEP:2021xfz  for a range $N=50-70$. Therefore, if $r_{0}$ denotes
the current upper limit in the tensor-to-scalar ratio, from Eq.(73) we have
that lower bound on $B$ yields

$B>\frac{8-\sqrt{2n-1}Nr_{0}}{12\pi^{2}M_{pl}^{4}\mathcal{P}_{s}r_{0}}.$ (74)

In Table 1, we summarize the different values associated to the integration
constants $A$ and $B$ for the specific case in which the power $n$ associated
to the kinetic term is $n=2$, from the observational parameters
$\mathcal{P}_{s}$ and $r$. In this sense, we find that the constraints on the
parameters $A$ and $B$ are $A\sim\mathcal{O}\left(10^{10}\right)$ and
$B\gtrsim\mathcal{O}\left(10^{8}\right)$, respectively.

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $1.66\times 10^{10}$ |  $B>5.20\times 10^{8}$
60 | $2.39\times 10^{10}$ |  $B>4.54\times 10^{8}$
70 | $3.26\times 10^{10}$ |  $B>3.88\times 10^{8}$
Table 1: Results for the constraints on the integration constants $A$ and $B$
for the first attractor $n_{s}-1=-\frac{2}{N}$ in k-inflation scenario when
$n=2$. Here we have used $\mathcal{P}_{s}\simeq 2.2\times 10^{-9}$ and
$r_{0}=0.036$

Furthermore, the predictions on the $n_{s}-r$ plane may be generated by
plotting the attractor $n_{s}-1=-\dfrac{2}{N}$ and the tensor-to-scalar ratio
given by Eq. (73), by varying simultaneously the integration constant $B$ in a
wide range and $N$ within the range $N=50-70$. Fig. 1 shows the $n_{s}-r$
plane for $n=2$ considering the two-marginalized joint confidence at 68% and
95% C.L., from the latest BICEP/Keck results.

Figure 1: Plot of the tensor-to-scalar ratio $r$ against the scalar spectral
index $n_{s}$ for the first attractor $n_{s}-1=-\frac{2}{N}$ using $n=2$ along
with the two-dimensional marginalized joint confidence contours for
$(n_{s},r)$ at 68% and 95% C.L. from the latest BICEP/Keck results.

In the next subsection, we will study the reheating constraints by means an
effective EoS parameter $w_{re}$ together with the reconstruction of inflation
(effective potential) in order to see whether the feasible parameter space of
the model can be narrowed or not.

###  III.3 Reheating

Now we are interested in studying the predictions of reheating phase regarding
its duration through the number of $e$-folds $N_{re}$ as well as the
temperature $T_{re}$. In doing so, we plot parametrically Eqs.(57) and (58)
together with the expression for the attractor $n_{s}-1=-\frac{2}{N}$, with
respect to the number of $e$-folds $N$ for several values of the EoS parameter
$w_{re}$ over the range $-\frac{1}{3}\leq w_{re}\leq 1$. In Fig. 2, we show
the plots for the number of $e$-folds of the reheating phase $N_{re}$ (upper
panel) and the reheating temperature $T_{re}$ (lower panel) against the scalar
spectral index $n_{s}$ for the case $n=2$ together with the allowed values of
$A$ and the minimum value of $B$ at $N=50$ (see Table 1). The behaviour of the
reheating predictions become almost the same for the other values of the
constants shown in Table 1. However, we will restrict ourselves to show the
plots that fit better with the current observational data. As a first finding,
we mention that the maximum reheating temperature is given by $T_{re}\simeq
10^{16}$ GeV, which is similar to those found for a canonical scalar field,
where $T_{re}\lesssim 7\times 10^{15}$ GeV paper1 . Secondly, we observe that
when $B$ increases and deviates from its minimum allowed value, the duration
of reheating increases while the reheating temperature decreases from its
maximum value. This behaviour can be inferred from the lower left and right
panels of Fig.2, where the curves shift to the left and the maximum reheating
temperature point decreases, respectively.

Assuming that reheating period can be parametrized by an effective EoS
$P=w_{re}\rho$, it becomes important to distinguish what EoS parameter
$w_{re}$ within the range $-\frac{1}{3}\leq w_{re}\leq 1$ is preferred by
current observational bounds. Consequently, from the reheating temperature
plots we observe the value of $n_{s}$ at which each curve enters the purple
shaded region as well as the value when all curves converge. Thus, we will
obtain the allowed ranges for both the scalar spectral index $n_{s}$ and the
number of $e$-folds $N$ for fixed values of the integration constants $A$ and
$B$. For completeness, the results found from the analysis of upper right and
lower right plots of Fig. 2 are summarized in Table 2. We also mention that
the values we have obtained for the number of $e$-folds during reheating phase
reheating are similar to those found for $A$ and $B$ from Table 1.

Figure 2: Plots of the reheating duration, $N_{re}$, and the reheating temperature $T_{re}$ for the power $n=2$ against the spectral index $n_{s}$ for the attractor $n_{s}-1=\frac{2}{N}$ for different values of the EoS parameter $w_{re}$. The blue, pink, cyan and black curves correspond to the EoS parameter $w_{re}=-1/3,0,2/3,1$, respectively. The gray dotted vertical lines indicates the observational constraints on the spectral index, i.e. $n_{s}=0.965\pm 0.004$. The green dotted horizontal line shows reheating temperatures being below the electroweak scale $T_{re}=10^{2}$ GeV, while the orange dotted line indicates the lower bound on $T_{re}$ from BBN, $T_{re}=10^{-2}$ GeV. For the upper and lower left panels we have used $A=1.66\times 10^{10}\,M_{pl}^{-4}$ and $B=5.20\times 10^{8}\,M_{pl}^{-4}$, while for the upper and lower right panels we have used $A=1.66\times 10^{10}\,M_{pl}^{-4}$ and $B=5.20\times 10^{18}\,M_{pl}^{-4}$.  $w_{re}$ | $N$
---|---
$-1/3$ | 51 - 57
0 | 51 - 57
2/3 | 57 - 64
1 | 57 - 65
Table 2: Summary of the allowed range for the number of $e$-folds for each EoS
parameter $w_{re}$ assuming the power $n$ associated to the kinetic term
$n=2$.

As a consequence, if we restrict the number of $e$-folds $N$ (see Table 2) it
is possible to obtain an allowed range for the reheating temperature for each
parameter of state $w_{re}$: $10^{13}\textup{ GeV}\lesssim T_{re}\lesssim
10^{16}\textup{ GeV}$ for $w_{re}=-1/3$, $10^{7}\textup{ GeV}\lesssim
T_{re}\lesssim 10^{16}\textup{ GeV}$ for $w_{re}=0$, $10^{2}\textup{
GeV}\lesssim T_{re}\lesssim 10^{16}\textup{ GeV}$ for $w_{re}=2/3$, and
$10^{6}\textup{ GeV}\lesssim T_{re}\lesssim 10^{16}\textup{ GeV}$ for
$w_{re}=1$. In order to obtain previous results, we have considered the lower
bound of $B$ already found from the inflationary observables, see Table 1.
Once the number of $e$-folds $N$ becomes narrowed from reheating constraints,
we are able to give the new predictions for this model on the $n_{s}-r$ plane.
These new contour regions are displayed in Fig. 3. The left panel shows the
predictions for both $w_{re}=-1/3$ (blue region) and $w_{re}=1$ (black region)
while the right panel shows the predictions for $w_{re}=0$ (magenta region)
and for $w_{re}=2/3$ (cyan region). It should be noted that for $w_{re}=-1/3$
and $w_{re}=0$ the predictions are the same, whereas for $w_{re}=2/3$ the
region starts at the same point of the black region and it ends almost at the
same point. In such a way, the four EoS parameters considered within the range
$-\frac{1}{3}\leq w_{re}\leq 1$ are preferred by current observational bounds.

Figure 3: Predictions for the tensor-to-scalar ratio $r$ against the scalar
spectral index $n_{s}$ for the first attractor $n_{s}-1=-\frac{2}{N}$ using
$n=2$ after applying reheating constraints. The left panel corresponds to
$w_{re}=-1/3$ and $w_{re}=1$, while the right panel for $w_{re}=0$ and
$w_{re}=2/3$, respectively.

##  IV Second attractor $n_{s}-1=-\dfrac{p}{N}$

###  IV.1 Dynamics of inflation

Now, we consider a little more general attractor $n_{s}-1=-\dfrac{p}{N}$
Garcia-Bellido:2014gna ; pattractor , where $p$ corresponds to a real number.
By replacing in Eq. (38), we have that

$-\dfrac{p}{N}=\ln\left(\dfrac{V_{,N}}{V^{2}}\right)_{,N}.$ (75)

A first integration of this equation with respect to the number of $e$-folds
yields

$\dfrac{A}{N^{p}}=\dfrac{V_{,N}}{V^{2}},$ (76)

where $A$ is a positive integration constant. Integrating again with respect
to $N$ we found that the potential as a function of $N$ can be written as

$V(N)=\dfrac{p-1}{AN^{1-p}+B(p-1)},$ (77)

where $B$ is a new integration constant and $p\neq 1$.

As the case $p=1$ is a singular case, we will analyze it separately. Thus, for
the case $p=1$, and in order to avoid that $V(N)=0$, we must first replace
replace $p=1$ into Eq. (75) and then integrate with respect to the number of
$e$-folds twice to find the potential as a function of the number of $e$-folds
$N$

$V(N)=\dfrac{1}{B-A\,\ln(N)}.$ (78)

We must note that this equation has a pole at $\ln N=\frac{B}{A}$ if $B$ is
positive, so we can only consider the range $\ln\,N<\frac{B}{A}$ for that the
potential to be positive. However, in the following we will assume $\ln
N\ll\frac{B}{A}$ in order to obtain analytical solutions. On the other hand,
analytically invertible expressions for $N(\phi)$ are only found for $n=2$.
Therefore, from Eq.(35) we can integrate to obtain $N(\phi)$ such that

$N(\phi)\simeq\left(\dfrac{3}{4}\right)^{4/3}\dfrac{(\phi-
C_{1})^{4/3}}{(\alpha^{3}A)^{1/3}},$ (79)

where $C_{1}$ is a new integration constant such that $\phi>C_{1}$ in order to
$N$ be real and positive. If we replace the latter equation into (78) and take
the limit $\ln N\ll\frac{B}{A}$, the reconstructed potential $V(\phi)$ becomes

$V(\phi)\simeq\dfrac{1}{B}\left[1+\dfrac{4A}{3B}\ln\left(\frac{3\left(\phi-
C_{1}\right)}{\left(\alpha^{3}A\right)^{1/4}}\right)\right],$ (80)

which is similar in form to the obtained in loop inflation (LI) model, see
Ref.Martin:2013tda .

For the general case in which $p\neq 1$, we have that by replacing Eq.(86) in
(35), the scalar field as a function of the number of $e$-folds becomes

$\displaystyle\phi(N)-C_{1}$ $\displaystyle=$
$\displaystyle\alpha^{\frac{2n-1}{2n}}\int\left[\dfrac{A}{N^{p}}\left(\dfrac{p-1}{AN^{1-p}+B(p-1)}\right)^{2-n}\right]^{\frac{1}{2n}}\,dN$
(81) $\displaystyle=$ $\displaystyle\mathcal{G}(N),$ (82)

where $C_{1}$ is a new integration constant and $\mathcal{G}(N)$ is a function
defined as

$\mathcal{G}(N)=g(N)\,{}_{2}F_{1}\left[1,\dfrac{2-n(5-3p)-p}{2n(p-1)},1+\frac{2n-p}{2n(1-p)},-\dfrac{AN^{1-p}}{B(p-1)}\right].$
(83)

Here ${}_{2}F_{1}$ corresponds to the hypergeometric function F21  and $g(N)$
is another function given by

$g(N)=\frac{\alpha(n)^{\frac{2n-1}{2n}}}{2n-p}2nN\left(AN^{-p}\right)^{\frac{1}{2n}}\left(\frac{p-1}{AN^{1-p}+B(p-1)}\right)^{\frac{2-3n}{2n}}.$
(84)

The number of $e$-folds as a function of the scalar field is obtained from
Eq.(82) as follows

$N=\mathcal{G}^{-1}(\phi-C_{1}),$ (85)

where $\mathcal{G}^{-1}$ represents the inverse function of function (84). In
this way, by replacing Eq.(85) into Eq.(77), the potential being a function of
the scalar field for $p\neq 1$ becomes

$V(\phi)=\dfrac{p-1}{A\left(\mathcal{G}^{-1}(\phi-
C_{1})\right)^{1-p}+B(p-1)}.$ (86)

From Eqs.(85) and (86), we see that $N(\phi)$, and in turns $V(\phi)$, can not
be expressed analytically for values of the power $n>2$ associated to the
kinetic term. So, we will restrict ourselves to the case $p\neq 1$ for $n=2$.

Thus, from Eq.(85) the number of $e$-folds $N(\phi)$ is found to be

$N(\phi)=\gamma(p)\left(\phi-
C_{1}\right)^{\frac{4}{4-p}},\,\,\textup{with}\,p\neq 4,$ (87)

where $\gamma(p)$ is defined as follows

$\gamma(p)=\left[\left(\frac{4-p}{4}\right)^{4}\frac{1}{\alpha^{3}A}\right]^{\frac{1}{4-p}}.$
(88)

Then, by replacing Eq.(87) into (86), it yields the following potential for
any value of the parameter $p$

$V(\phi)=\frac{(p-1)(\phi-
C_{1})^{\frac{4(p-1)}{(4-p)}}}{A\,\gamma(p)^{-(p-1)}+B(p-1)(\phi-
C_{1})^{\frac{4(p-1)}{(4-p)}}}.$ (89)

In order to find concrete expressions for $V(\phi)$, we will consider the
cases $p=3/2$ and $p=3$. At first, for $p=3/2$ Eq.(89) becomes

$V(\phi)=\dfrac{(\phi-
C_{1})^{4/5}}{\left(2^{17}/5^{4}\right)^{1/5}(A^{6}\alpha^{3})^{1/5}+B(\phi-
C_{1})^{4/5}}.$ (90)

As asymptotic limits of Eq.(90), first we observe that if
$\left(2^{17}/5^{4}\right)^{1/5}(A^{6}\alpha^{3})^{1/5}\gg B(\phi-
C_{1})^{4/5}$, $V(\phi)$ behaves as a power law potential

$V(\phi)\simeq\dfrac{(\phi-
C_{1})^{4/5}}{\left(2^{17}/5^{4}\right)^{1/5}(A^{6}\alpha^{3})^{1/5}},$ (91)

while if $\left(2^{17}/5^{4}\right)^{1/5}(A^{6}\alpha^{3})^{1/5}\ll B(\phi-
C_{1})^{4/5}$, it becomes a constant potential

$V\simeq\dfrac{1}{B}.$ (92)

Now, for $p=3$, the potential (89) results

$V(\phi)=\dfrac{(\phi-C_{1})^{8}}{2^{15}(A\alpha^{2})^{3}+B(\phi-C_{1})^{8}}.$
(93)

It can be seen that when $2^{15}(A\alpha^{2})^{3}\gg B(\phi-C_{1})^{8}$, the
potential behaves as

$V(\phi)\simeq\frac{\left(\phi-C_{1}\right)^{8}}{2^{15}(A\alpha^{2})^{3}}.$
(94)

On the contrary, if $2^{15}(A\alpha^{2})^{3}\ll B(\phi-C_{1})^{8}$, the
potential presents a constant behaviour

$V(\phi)\simeq\frac{1}{B}.$ (95)

Thus the effective potential given by Eqs.(94) and (95) correspond to the
limit potential for small and large $\phi$, respectively.

###  IV.2 Cosmological perturbations

Upon replacement of Eq.(76) into the expression for the scalar power spectrum
(31), we find that the constraint on the integration constant $A$ for any
value of $p$ and $n$ is given by

$A=\dfrac{\sqrt{2n-1}N^{p}}{12\pi^{2}M_{pl}^{4}\mathcal{P}_{s}}.$ (96)

However, as we will see, the tensor-to-scalar ratio becomes different and we
will consider the case $p=1$ and $p\neq 1$ separately for the situation in
which $n=2$.

Regarding the case (a) $p=1$ and $n=2$ by using Eqs.(36), the potential (78)
and Eq.(39) along with the constraint on $A$ given by (96), we get that the
tensor-to-scalar ratio becomes

$r=\dfrac{8}{12BM_{pl}^{4}\,\pi^{2}\,\mathcal{P}_{s}-\sqrt{3}\,N\ln N}.$ (97)

On the other hand, for the case (b) $p\neq 1$ and $n=2$, but using the
potential (86) instead, the tensor-to-scalar ratio is expressed as follows

$r=\dfrac{8(p-1)}{12B(p-1)M_{pl}^{4}\,\pi^{2}\,\mathcal{P}_{s}+\sqrt{3}\,N}.$
(98)

The allowed values of the integration constant $A$ are found by evaluating
Eq.(96) by using $\mathcal{P}_{s}\simeq 2.2\times 10^{-9}$ for a range of the
number of $e$-folds in the range $50\leq N\leq 70$, while the upper bound on
the tensor-to-scalar ratio from BICEP2/Keck array (BK14) data sets the lower
bound on $B$. Accordingly, for the case (a) $p=1$ and $n=2$, from Eq.(97) it
is found the following lower limit on $B$

$B>\frac{8-\sqrt{3}r_{0}N\ln N}{12\pi^{2}M_{pl}^{4}\mathcal{P}_{s}r_{0}},$
(99)

where $r_{0}$ denotes the current upper limit on the tensor-to-scalar ratio.
Besides, from Eq.(98), the lower limit of $B$ for the case (b) $p\neq 1$ and
$n=2$ yields

$B>\frac{8(p-1)-\sqrt{3}r_{0}N}{12(p-1)M_{pl}^{4}\,\pi^{2}\,\mathcal{P}_{s}r_{0}}.$
(100)

As before, in order to obtain concretes values, we consider the cases $p=3/2$
and $p=3$.

The resulting allowed values for the integration constant $A$ and the
corresponding lower bounds on $B$ for the cases $p=1$, $p=3/2$ and $p=3$ in
the range $50\leq N\leq 70$ are shown in Table 3.

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $3.32\times 10^{8}$ |  $B>2.15\times 10^{9}$
60 | $3.99\times 10^{8}$ |  $B>2.49\times 10^{9}$
70 | $4.65\times 10^{8}$ |  $B>2.83\times 10^{9}$
(a) $p=1,n=2$

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $2.35\times 10^{9}$ |  $B>1.88\times 10^{8}$
60 | $3.09\times 10^{9}$ |  $B>5.52\times 10^{7}$
70 | $3.89\times 10^{9}$ |  $B>-7.78\times 10^{7}$
(b) $p=3/2,n=2$

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $8.31\times 10^{11}$ |  $B>6.87\times 10^{8}$
60 | $1.44\times 10^{12}$ |  $B>6.53\times 10^{8}$
70 | $2.28\times 10^{12}$ |  $B>6.20\times 10^{8}$
(c) $p=3,n=2$

Table 3: Results for the constraints on the integration constants $A$ and $B$
for the second attractor $n_{s}-1=-\frac{p}{N}$, using $p=1$ (top left panel),
$p=3/2$ (top right panel) and $p=3$ (bottom panel) each for $n=2$ in
k-inflation. Here we have used $\mathcal{P}_{s}\simeq 2.2\times 10^{-9}$ and
$r_{0}=0.036$.

Following the same procedure of Section III, for the case (a) $p=1$ and $n=2$,
the predictions in the $n_{s}-r$ plane may be obtained by plotting the
attractor $n_{s}-1=-\frac{1}{N}$ and Eq.(97), by varying simultaneously the
integration constant $B$ in a wide range and the number of $e$-folds within
the range $N=50-70$. However, we observe that the case $p=1$ yields values for
the scalar spectral index (for $50\leq N\leq 70$) which are greater than its
likelihood value. On the other hand, for the case (b) $p\neq 1$ and $n=2$, we
plot $n_{s}-1=-\frac{p}{N}$ and Eq.(98). In relation to the predictions for
$p\geq 2$, the scalar spectral index now becomes smaller than its preferred
value from current observations. In order to show concrete results, we will
restrict ourselves to the case $p=3/2$. Fig.4 shows the $n_{s}-r$ plots for
$p=3/2$ when $n=2$ considering the two-marginalized joint confidence at 68%
and 95% C.L., from Planck 2018 results. We observe that for the case $p=3/2$,
the purple shaded region lies within the 68% and 95% C.L. from BICEP/Keck for
$50\leq N\leq 60$. These results show that for the power $n=2$ associated to
the kinetic term, the attractor $n_{s}-1=-\frac{p}{N}$ only works for $p=3/2$.

Figure 4: Plots of the tensor-to-scalar ratio $r$ against the scalar spectral
index $n_{s}$ for the second attractor $n_{s}-1=-\frac{p}{N}$ along with the
two-marginalized joint confidence contours for ($n_{s},r$) at 68% and 95% C.L.
from PLANCK 2018. Here we have considered the specific case in which $p=3/2$
for $n=2$.

###  IV.3 Reheating

Since the current CMB constraints only favor the case $p=3/2$, we proceed
further to study the predictions of reheating phase regarding its duration
through the number of $e$-folds $N_{re}$ as well as the temperature $T_{re}$.
In doing so, we follow the same procedure as in Section III. Then, by plotting
parametrically Eqs. (57) and (58) for the attractor $n_{s}-1=-\,\frac{3}{2N}$
for several values of the EoS parameter $w_{re}$ over the range
$-\frac{1}{3}\leq w_{re}\leq 1$, we find the plots of Fig.5. Here, there are
shown the plots for $N_{re}$ (upper plots) and $T_{re}$ (lower plots) against
$n_{s}$. For the upper and lower left plots we have used the allowed value of
$A$, given by $A=2.35\times 10^{9}\,M_{pl}^{-4}$, and the minimum value of $B$
at $N=50$, i.e., $B=1.88\times 10^{8}\,M_{pl}^{-4}$ (see Table 3).

Figure 5: Plots of the reheating duration, $N_{re}$, and the reheating
temperature $T_{re}$ for the power $n=2$ against the spectral index $n_{s}$
for the attractor $n_{s}-1=\frac{p}{N}$ and considering different values of
the EoS parameter $w_{re}$. The blue, pink, cyan and black curves correspond
to the EoS parameter $w_{re}=-1/3,0,2/3,1$, respectively. The gray dotted
vertical lines indicates the observational constraints on the spectral index,
i.e. $n_{s}=0.965\pm 0.004$. The green dotted horizontal line shows reheating
temperatures being below the electroweak scale $T_{re}=10^{2}$ GeV, while the
orange dotted line indicates the lower bound on $T_{re}$ from BBN,
$T_{re}=10^{-2}$ GeV. For the upper and lower left panels we have used
$A=2.35\times 10^{9}\,M_{pl}^{-4}$ and $B=1.88\times 10^{8}\,M_{pl}^{-4}$,
while for the upper and lower right panels we have used $A=2.35\times
10^{9}\,M_{pl}^{-4}$ and $B=1.88\times 10^{18}\,M_{pl}^{-4}$.

On the other side, for the upper and lower right plots we have used the same
value of $A$ and an intermediate value of $B$, $B=1.88\times
10^{18}\,M_{pl}^{-4}$ at $N=50$. Firstly, from the $T_{re}$ against $n_{s}$
plots, it is found that as the value of $B$ increases, the maximum reheating
temperature decreases. Secondly, by performing the same analysis for the EoS
parameter as the previous section, we found that for all the four values of
$w_{re}$ it is required that the number of $e$-folds be $N<50$ in order to
achieve a reheating temperature above the electroweak scale and below the GUT
scale. As a consequence, the scalar spectral index must be lower than $0.967$,
which is not consistent with the predictions of the model on the $n_{s}-r$
plane. Accordingly, in despite the attractor $n_{s}-1=-\,p/N$ becomes
supported by current CMB data only for $p=3/2$, this is not consistent with
reheating constraints.

##  V Third attractor $n_{s}-1=-\dfrac{\beta}{N^{q}}$

###  V.1 Dynamics of inflation

Here, we introduce a more general attractor, namely Huang:2007qz

$n_{s}-1=-\dfrac{\beta}{N^{q}},$ (101)

which is a generalization of the previous expression $n_{s}-1=-\,p/N$. Here,
$\beta$ and $q$ are constant parameters. From Eq.(38), we found

$-\dfrac{\beta}{N^{q}}=\left(\ln\dfrac{V_{,N}}{V^{2}}\right)_{,N}.$ (102)

A first integration of this equation with respect to the number of $e$-folds
$N$ yields

$\dfrac{V_{,N}}{V^{2}}=Ae^{-\beta N^{1-q}/(1-q)},$ (103)

where $A$ is a positive integration constant, since we have assumed that
$V_{,N}>0$. Integrating once again with respect to $N$, we found the following
expression for the potential $V(N)$

$V(N)=\dfrac{1-q}{A\,N\left(\frac{1-q}{\beta\,N^{1-q}}\right)^{\frac{1}{1-q}}\Gamma\left(\frac{1}{1-q},\frac{\beta\,N^{1-q}}{1-q}\right)+B(1-q)},\,\,\textup{with}\,q\neq
1.$ (104)

where $B$ is an arbitrary integration constant and
$\Gamma\left(\frac{1}{1-q},\frac{\beta\,N^{1-q}}{1-q}\right)$ represents the
incomplete gamma function gamma . The expression for the scalar field $\phi$
in terms of the the number of $e$-folds $N$ is obtained by using Eq.(35) and
the potential (104), yielding

$\phi(N)-C_{1}=\int\alpha^{\frac{2n-1}{2n}}\left(Ae^{-\frac{\beta
N^{1-q}}{1-q}}\left(\dfrac{1-q}{A\,N\left(\frac{1-q}{\beta\,N^{1-q}}\right)^{\frac{1}{1-q}}\Gamma\left(\frac{1}{1-q},\frac{\beta\,N^{1-q}}{1-q}\right)+B(1-q)}\right)^{2-n}\right)^{\frac{1}{2n}}\,dN,$
(105)

where $C_{1}$ is a new integration constant. It must be noted that the
integral on the right hand side of Eq.(105) only has analytical solutions for
two cases: (a) $q=0$ and $n=1$; (b) $q=0$ and $n=2$. The case (a) corresponds
to a constant attractor in standard canonical inflation, which has been
studied so far chiba , so we will restrict ourselves to the case (b). For this
latter case, which also corresponds to a constant attractor, Eq.(104) becomes

$V(N)=\dfrac{\beta}{Ae^{-\beta N}+\beta B}.$ (106)

On the other hand, for the case b) i.e., $q=0$ and $n=2$, Eq.(105) can be
inverted in order to find the number $e$-folds as a function of the the scalar
field as follows

$N(\phi)=-\dfrac{4}{\beta}\ln\left(-\dfrac{\beta(\phi-
C_{1})}{4(\alpha^{3}A)^{1/4}}\right).$ (107)

Accordingly, if we replace this expression into Eq.(106), the potential as a
function of the scalar field yields

$V(\phi)=\dfrac{\beta}{\left(-\frac{\beta}{4}\right)^{4}\frac{(\phi-
C_{1})^{4}}{\alpha^{3}}+\beta B}.$ (108)

In particular, if $\left(\frac{-\beta}{4}\right)^{4}\frac{(\phi-
C_{1})^{4}}{\alpha^{3}\beta B}\ll 1$ the potential (108) becomes

$V(\phi)\simeq\dfrac{1}{B}\left[1-\left(-\frac{\beta}{4}\right)^{4}\frac{(\phi-
C_{1})^{4}}{\alpha^{3}\beta B}\right],$ (109)

which has the form of a quartic Hilltop inflationary model Boubekeur:2005zm
as long as $B>0$.

Since we don’t have analytical expressions for the number of $e$-folds as a
function of the scalar field, then the effective potential $V(\phi)$ can’t be
obtained for values $q$ such that $q\neq 0$. However, we can find the
effective potential in terms of $N$. Thus, we can write down the potential as
a function of the number of $e$-folds for $q=2$ and $q=3$ when $n=2$, which
become

$\displaystyle V(N)$ $\displaystyle=$
$\displaystyle\dfrac{1}{A\left(\beta\,\textup{Ei}\left(\frac{\beta}{N}\right)-e^{\frac{\beta}{N}}\right)-B},$
(110) $\displaystyle V(N)$ $\displaystyle=$
$\displaystyle\dfrac{2}{A\left(\sqrt{2\pi\,\beta}\,\textup{erfi}\left(\dfrac{\sqrt{\beta}}{\sqrt{2}\,N}\right)-2e^{\frac{\beta}{2N^{2}}}\,N\right)-2B},$
(111)

respectively. In Eq.(110), $\textup{Ei}(x)$ represents the exponential
integral function defined as Ei

$\textup{Ei}(x)\equiv-\int_{-x}^{\infty}\frac{e^{-y}}{y}dy.$ (112)

Additionally, $\textup{erfi}(x)$ in Eq.(111) corresponds to the imaginary
error function erfi . In this context, we can’t reconstruct the effective
potential as a function of the scalar field. However, for each of these
expressions along with Eq.(106), the power spectrum as well as the tensor-to-
ratio can be computed in order to constraint the parameters by using the CMB
constraints on the inflationary observables.

###  V.2 Cosmological perturbations

By replacing Eq.(103) into the expression for the scalar power spectrum (31)
and solving for the integration constant $A$, it is found that

$A=\dfrac{e^{-\beta N^{1-q}/1-q}}{12\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}}.$
(113)

Furthermore, the several expressions for the tensor-to-scalar ratio for the
potentials $V(N)$, namely (106), (110) and (111), are found by replacing the
former equations and (36) into Eq. (39). Thus, by using the constraint on the
integration constant $A$ given by Eq.(113), it yields

$\displaystyle r$ $\displaystyle=$
$\displaystyle\dfrac{8\,c_{s}\beta}{1+12B\,e^{2N\beta}\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}},$
(114) $\displaystyle r$ $\displaystyle=$ $\displaystyle\dfrac{8\,c_{s}e^{\beta
N}}{12B\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}-e^{\frac{\beta}{N}}N+\beta\textup{Ei}\left(\frac{\beta}{N}\right)},$
(115) $\displaystyle r$ $\displaystyle=$
$\displaystyle\dfrac{16\,c_{s}}{24B\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}-2N+\sqrt{2\pi\beta}e^{-\frac{\beta}{2N^{2}}}\textup{erfi}\left(\frac{\sqrt{\beta}}{\sqrt{2}N}\right)},$
(116)

for the specific cases $q=0$, $q=2$ and $q=3$, respectively. For $q\geq 4$ it
is found that a negative value of $\beta$ is needed for $r$ to be positive,
which implies $n_{s}>1$ and the attractor does not work.

In a similar way as for the attractor (i), the CMB constraints on the
inflationary observables are used to find the allowed values for the parameter
$\beta$ appearing in the expression for this last expression for $n_{s}(N)$,
and also for the integration constants $A$ and $B$. At first, the value of
$\beta$ is constrained so that over the range of the number of $e$-folds
$50\leq N\leq 70$, the scalar spectral index $n_{s}$ becomes consistent with
current observational bounds $n_{s}=0.965\pm 0.004$. In this regard, the
allowed values for $\beta$ are within the ranges $65\leq\beta\leq 190$ and
$3.2\times 10^{2}\leq\beta\leq 9.6\times 10^{2}$ for $q=2$ and $q=3$,
respectively. For the special case $q=0$, i.e. the constant attractor, it is
found that $\beta=0.0351$. Secondly, we use the constraint on the amplitude of
the scalar spectrum $\mathcal{P}_{s}\simeq 2.2\times 10^{-9}$ to set $A$ in
(113), while the allowed values of $B$ are found from the upper bound on $r$
by the BICEP/Keck data, i.e. $0<r<0.036$ BICEP:2021xfz . In doing so, if
$r_{0}$ denotes the upper limit on the tensor-to-scalar ratio, from,
Eqs.(114), (115), and (116), we found the following lower bounds on $B$

$\displaystyle B$ $\displaystyle>$ $\displaystyle\frac{8\,c_{s}\beta-
r_{0}}{12e^{2N\beta}\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}r_{0}},$ (117)
$\displaystyle B$ $\displaystyle>$ $\displaystyle\frac{8\,c_{s}e^{\beta
N}+e^{\frac{\beta}{N}}Nr_{0}-\beta\,r_{0}\textup{Ei}\left(\frac{\beta}{N}\right)}{12\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}r_{0}},$
(118) $\displaystyle B$ $\displaystyle>$
$\displaystyle\frac{16\,c_{s}+2Nr_{0}-\sqrt{2\pi\beta}e^{-\frac{\beta}{2N^{2}}}\textup{erfi}\left(\frac{\sqrt{\beta}}{\sqrt{2}N}\right)}{24\pi^{2}M_{pl}^{4}c_{s}\mathcal{P}_{s}r_{0}},$
(119)

for $q=0$, $q=2$ and $q=3$, respectively. Table 4 summarizes the allowed
values for the integration constants $A$ and $B$ for $n=2$ and several values
of $q$ by considering the CMB constraints on the inflationary observables.

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $3.84\times 10^{7}$ |  $B>6.63\times 10^{8}$
60 | $5.46\times 10^{7}$ |  $B>6.63\times 10^{8}$
70 | $7.76\times 10^{7}$ |  $B>6.63\times 10^{8}$
(a) $q=0,\,n=2,\beta=0.0351$

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $7.37\times 10^{5}$ |  $B>7.21\times 10^{8}$
57 | $9.65\times 10^{5}$ |  $B>7.33\times 10^{8}$
65 | $1.22\times 10^{6}$ |  $B>7.60\times 10^{8}$
(b) $q=2,\,n=2,\beta=110$

$N$ |  $A\,[M_{pl}^{-4}]$ |  $B\,[M_{pl}^{-4}]$
---|---|---
50 | $2.21\times 10^{6}$ |  $B>8.13\times 10^{8}$
55 | $2.68\times 10^{6}$ |  $B>8.42\times 10^{8}$
65 | $3.10\times 10^{6}$ |  $B>8.75\times 10^{8}$
(c) $q=3,\,n=2,\beta=5.5\times 10^{2}$

Table 4: Results for the constraints on the integration constants $A$ and $B$
for the third attractor $n_{s}-1=-\frac{\beta}{N^{q}}$ in k-inflation.

(a) $q=2,\,\beta=110,\,n=2$

(b) $q=3,\,\beta=5500,\,n=2$

Figure 6: Plots of the tensor-to-scalar ratio $r$ against the scalar spectral
index $n_{s}$ for the second attractor $n_{s}-1=-\frac{\beta}{N^{q}}$ along
with the two-marginalized joint confidence contours for ($n_{s},r$) at 68% and
95% C.L. from BICEP3.

In the same way as in Section III, we generate the predictions on the
$n_{s}-r$ plane by plotting the attractor $n_{s}-1=-\frac{\beta}{N^{q}}$ and
the respective equations of $r$ (Eqs. (115) and (116)) by varying
simultaneously the integration constant $B$ in a wide range and $N$ within the
range $N=50-70$. Fig. (6) shows the plots for $q=2$ and $q=3$ both for $n=2$
considering the two-marginalized joint confidence at 68% and 95% C.L., from
BICEP/Keck data. It should be noted that for the constant attractor $q=0$, the
curve on the $n_{s}-r$ plane becomes a vertical line at the central value of
$n_{s}$ (not shown).

Regarding the reheating predictions for this last attractor, the equation for
the slow-roll parameter $\epsilon$ at the end of inflation, i.e.,
$\epsilon(\phi_{end})\equiv 1$ cannot be solved analytically in order to
obtain $\phi_{end}$, and therefore there is no possible to find an analytical
expression for $V_{end}$, which is needed to compute $N_{re}$ and $T_{re}$ as
well. As a consequence, a numerical study of reheating constraints is needed
instead, which is beyond the goals of the present work.

##  VI Conclusions

In this paper we have applied the reconstruction scheme from the attractor
points $n_{s}(N)$ to the framework of k-inflation, in particular for a non-
linear kinetic term $K(X)=k_{n+1}X^{n}$, in order to rebuild the potential
associated to the inflaton field $V(\phi)$. In doing so, we have considered
the following attractors: (i) $n_{s}-1=-\frac{2}{N}$, (ii)
$n_{s}=1-\frac{p}{N}$, and (iii) $n_{s}-1=-\frac{\beta}{N^{q}}$. For simplest
the attractor (i), it was possible to find analytical expressions for the
reconstructed inflaton potential $V(\phi)$ for the non-canonical scalar field
$n=2$. In particular, for $n=1$ we recovered the T-model corresponding to the
standard canonical case. Regarding the case $n=2$, the resulting potential
interpolates between a chaotic quadratic and a constant effective potential,
and the allowed ranges for the parameters characterizing this model were
obtained by comparing its predictions on the $n_{s}-r$ plane with current CMB
constraints on the inflationary observables. As a further analysis, we studied
the reheating constraints regarding the duration of this phase from $e$-folds
$N_{re}$ and the temperature $T_{re}$ in order to narrow the parameter space
of the reconstructed model. In doing so, we considered several values of the
EoS parameter of the fluid into which the inflaton decays over a range
$-\frac{1}{3}\leq w_{re}\leq 1$. In first place, by restricting the number of
$e$-folds $N$, it was possible to find an allowed range for the reheating
temperature $T_{re}$ for each parameter of state $w_{re}$. We also found that
the instantaneous reheating temperature, in which $N_{re}\sim 0$, and it
becomes $T_{re}\simeq 10^{16}$ GeV, which is similar to those found for
standard inflation scenario paper1 . Once the number of $e$-folds was
restricted for each EoS (Table 2), we plot the corresponding predictions in
the $r-n_{s}$ plane, and it was found that all the EoS parameters considered
are preferred by current observational bounds. For the attractor (ii),
analytical solutions for $V(\phi)$ were found for $p=1$, $p=3/2$, and $p=3$
when $n=2$. Specifically, for $p=1$, the inflaton potential obtained is
similar to those from loop inflation (LI) model. On the other hand, for
$p=3/2$ and $p=3$, the resulting potential can be either a power-law or
constant. Regarding its predictions on the $n_{s}-r$ plane, this attractor is
not supported by current observational constraints on the scalar spectral
index $n_{s}$ for $p=1$ and $p=3$, since $n_{s}$ becomes greater and lower
than its likelihood value, respectively. Nevertheless, for the case $p=3/2$,
its predictions are within the 68% and 95% C.L. from BICEP/Keck for $50\leq
N\leq 60$. Although a further analysis regarding the reheating phase is
possible for this attractor, the corresponding constraint on the number of
$e$-folds $N$ becomes small ($N<50$) for any value of the EoS parameter
$w_{re}$, and then this second attractor for $p=3/2$ does not work. For the
attractor (iii), we were able to find analytical solutions for $V(\phi)$ only
for the case $q=0$ and $n=2$, corresponding to a constant attractor. In that
case, the resulting inflaton potential can be approximated to a quartic
hilltop one. For this constant attractor, the parameter $\beta$ can be
restricted in order to reproduce the observational value for the scalar
spectral index. For completeness, we also obtained $V(N)$ analytically for two
additional cases, namely $q=2$ for $n=2$, and $q=3$ for $n=2$ in order to
study their predictions on the $n_{s}-r$ plane. Following the same procedure
for the constant attractor, we also obtained the allowed range for $\beta$ for
the cases $q=2$ and $q=3$. The viable values for the rest of parameters, i.e.,
the integration constants $A$ and $B$ for the three cases, they were found by
using the CMB constraints on the scalar power spectrum and the tensor-to-
scalar ratio, respectively (summarized in Table 4). Concerning the reheating
predictions for this last attractor, we haven’t been able to find analytical
results, then a numerical analysis is needed instead of an analytical one,
which goes beyond the main goals of this work.

As a final remark, we have not addressed the reconstruction scheme from the
tensor-to-scalar ratio $r(N)$ or the slow-roll parameter $\epsilon(N)$, and
neither a more general expression for the Lagrangian density, e.g.
$P(\phi,X)=\sum_{n\geq 0}g_{n}(\phi)X^{n+1}-V(\phi)$ in our analysis. We hope
to be able to address these points in a future work.

## Acknowledgements

C.O. is supported by the Pontificia Universidad Católica de Valparaíso trough
the scholarship “Beca Término de Tesis”.

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