# Enhancing inflationary model predictions via refined slow-roll dynamics

Debottam Nandi, Simran Yadav, and Manjeet Kaur

###### Abstract

The inflationary paradigm not only addresses early Universe puzzles but also
aligns well with the observational constraints, with slow-roll inflationary
models fitting best. Evaluating these model predictions requires considering
both slow-roll inflationary dynamics and the subsequent reheating epoch. This
involves the quantitative analysis that takes into account the effective
equation of state (EoS) and duration of reheating, connecting these with the
perturbations generated deep during the inflationary era. Given the
complexities involved, many approximations are often used for simplification.
However, as future observations are expected to improve the accuracy of these
observables significantly, this work takes a different approach. Instead of
relying on approximations, we consider the near-accurate solutions of
inflationary models and compare our results with the pre-existing ones. It
mainly incorporates two improvements: the first is the accurate dynamics of
the slow-roll evolution, and, thus, the end of inflation; and the second is
the higher-order slow-roll corrections to the perturbed observables. Our
findings indicate that, by implementing these corrections, the theoretical
predictions improve significantly. It also indicates that seemingly minor
corrections can have significant effects on the perturbed observables and
these refined predictions can be compared with future observations to
potentially rule out models and help resolve the degeneracy problem of the
inflationary paradigm.

##  1 Introduction

The inflationary paradigm [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], which
describes a brief period of accelerated expansion, is the most successful
framework for explaining the dynamics of the early Universe. Within this
paradigm, slow-roll inflation involves field(s) slowly rolling down the
potential, resulting in near-exponential expansion and providing an
explanation for the observational constraints [32, 33, 34, 35]. As the
field(s) approaches the bottom of the potential, inflation ends, and shortly
afterward, the field(s) begins to oscillate around this minimum. This phase,
known as the reheating epoch, involves the inflaton field(s) coupling with and
decaying into other standard particles, leading to the onset of the radiation-
dominated era [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58].

Theoretical constraints derived from an inflationary model, however, not only
depend on the dynamics of inflation itself but also depend on the reheating
era, and therefore, studying reheating dynamics is crucial for accurately
predicting the early Universe behavior. There are two main approaches to this:
the first is the qualitative analysis [42, 36, 37, 38, 39, 41, 40, 43, 46]
which examines the exact decay process of the inflaton field through the
parametric resonance, leading to the understanding of the microphysics
involved during the reheating epoch. The second approach, which is the main
focus of this manuscript, is the quantitative analysis [59, 54, 60, 44, 61,
45, 47, 49, 52, 62, 63]. This approach looks at the macrophysics of the
reheating epoch by constraining the duration ($N_{\rm re}$) and the equation
of state (EoS) parameter ($w_{\rm re}$) during the reheating era using
perturbations generated deep in the early (inflation) Universe.

In this analysis, the Hubble horizon and the associated length scales are used
to constrain the duration when the pivot scale exits the Hubble horizon from
the end of inflation, i.e., $N_{k}$, and the reheating parameters ($N_{\rm
re},\ w_{\rm re}$). Given any model, using $N_{k}$, one can determine both the
scalar and tensor perturbations associated with the pivot scale and compare
this with the observations. Thus, the theoretically obtained perturbed
observables, namely the scalar-spectral index $n_{s}$ and the tensor-to-scalar
ratio $r$ are heavily dependent on the reheating parameters $N_{\rm re}$ and
$w_{\rm re}.$ Since the perturbations ($n_{s}$ and $r$) are highly sensitive
to $N_{k}$, and consequently, to ($N_{\rm re},\ w_{\rm re}),$ even small
corrections or deviations in $N_{k}$ or reheating parameters can significantly
improve $n_{s}.$ Ongoing experiments [32, 33, 34, 35] already measure $n_{s}$
with an accuracy of $\Delta n_{s}\sim 10^{-2}$ and upcoming experiments like
PRISM [64], EUCLID [65], cosmic 21-cm surveys [66], and CORE experiments [67]
are projected to improve this accuracy to $\Delta n_{s}\sim 10^{-3}$.
Therefore, taking advantage of this, in this work, we focus on refining the
analysis itself to obtain tighter constraints on the inflationary paradigm, as
the analysis relies on numerous approximations. This refined approach, in
turn, can help resolve the degeneracy problem of the inflationary paradigm
[68, 25, 69, 70] by eliminating more inflation models.

Improving the analysis can again be performed in two main ways. The first one
involves a detailed and careful consideration of the reheating analysis that
can accurately depict the reheating parameters ($N_{\rm re},w_{\rm re}$).
However, since it entails dealing with various distinct microphysical
processes, i.e., the qualitative analysis, we have opted not to pursue this
approach as it is beyond the scope of this work. Instead, we focus on the
second, more straightforward improvement: the slow-roll methodology and
approximations. Recent findings in Refs. [71, 72] indicate that a near-
accurate analytical solution can significantly enhance background predictions
near the end of inflation, thereby improving $N_{k}$ [71]. For example, in the
case of Starobinsky inflation, the improvement $\Delta N_{k}\sim 1.5$ suggests
that

$n_{s}\sim 1-\frac{2}{N_{k}},\quad\Delta n_{s}\sim\frac{\Delta
N_{k}}{N_{k}^{2}}\sim 10^{-3}.$

In addition to leading-order slow-roll approximations, higher-order slow-roll
corrections [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87] can
be considered to further improve the evaluation of perturbed observables such
as $n_{s}$ and $r$. Therefore, the objective of this work is to enhance the
model predictions by implementing both the improved near-accurate dynamics and
higher-order slow-roll corrections and comparing our results with the existing
literature. As an example, we consider the most popular inflationary model:
the Starobinsky model of inflation [1, 2, 8, 88, 89]. Using the conventional
leading order analytical approximation, the constraint on $n_{s}$ is:
$0.9613\leq n_{s}\leq 0.9641,$ which leads to further constrain $N_{\rm
re}\leq 16$ (equivalently, the energy at the end of reheating, i.e., $T_{\rm
re}\geq 10^{10}$ GeV, assuming $w_{\rm re}=0$). However, after implementing
the accurate dynamics as well as the higher-order slow-roll corrections, the
bounds on these parameters change significantly, and the new constraint on
$n_{s}$ for numerical solution becomes: $0.9613\leq n_{s}\leq 0.9649,$ which
leads to further constrain $N_{\rm re}\leq 21$ (and subsequently, $T_{\rm
re}\geq 10^{8}$ GeV), while for higher-order corrections these become:
$0.9613\leq n_{s}\leq 0.9653$, constraints on reheating parameters are,
$N_{\rm re}\leq 24$ and $T_{\rm re}\geq 10^{7}$, which has the potential to
rule out the model from future observations. The result is unexpected, as one
might anticipate the corrections to be negligible. Thus, instead of relying on
the conventional approximation method, our results demonstrate the critical
need for corrections due to the accurate dynamics in each model of inflation,
which may lead to a significant change in theoretical predictions and
potentially rule out models from observations. This implies that even
seemingly minor improvements to the background dynamics can have a substantial
impact on the theoretically obtained perturbed observables, which must not be
ruled out without verification.

The article is structured as follows: the following section 2 presents the
model and the subsequent dynamics for the early Universe dynamics. In this
section, we also demonstrate the slow-roll conditions as well as the slow-roll
approximations. In the subsequent section 3, we study the quantitative
analysis of reheating linking the inflationary perturbed variables to the
observable constraints. This section, as an example, also examines the popular
Starobinsky model of inflation, and with the help of the reheating analysis
along with leading-order slow-roll approximations, we carefully obtain the
observational consequences of the model. In Sec. 4, we highlight the dire need
for the corrections to the dynamics, subsequently implement these changes, and
obtain the improved constraint of the model, which turns out to be
significant. Finally, we conclude our work in Sec. 5.

A brief explanation about some terms and symbols we’ll be using. In our work,
we use natural units where $\hbar=c=k_{B}=1$. We also define the reduced
Planck mass as $M_{\rm pl}\equiv(8\pi G)^{-1/2}=1$. The metric signature we
use is $(-,+,+,+)$. Greek indices are contracted with the metric tensor
$g_{\mu\nu}$, while Latin indices will be contracted with the Kronecker delta
$\delta_{ij}$. We use symbols $\partial$ and $\nabla$ for partial and
covariant derivatives, respectively. Additionally, the dots and primes above
symbols represent derivatives for cosmic and conformal time, respectively,
related to the Friedmann-Lemaître-Robertson-Walker (FLRW) line-element.

##  2 General equations and slow-roll conditions

In this section, we discuss the general equations and solutions for a slow-
roll inflationary Universe. For that, let us consider the simplest action with
a single canonical scalar field $\phi$ minimally coupled to gravity with a
potential $V(\phi)$, which can be written as

$\displaystyle S=\frac{1}{2}\int{\rm
d}^{4}x\sqrt{-g}(R-g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-2V(\phi)).$
(2.1)

Here, $R$ is the Ricci scalar. The corresponding equations of motion, i.e.,
Einstein’s equations and the equation of the scalar field, can be written as

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle T_{\mu\nu(\phi)},$ (2.2)
$\displaystyle\nabla_{\mu}T^{\mu\nu}_{(\phi)}$ $\displaystyle=$ $\displaystyle
0,$ (2.3)

where $T^{\mu}_{\,\nu{(\phi)}}$ is the stress-energy tensor corresponding to
the $\phi$ field and can be written as

$T_{\mu\nu{(\phi)}}=\partial_{\mu}\phi\ \partial_{\nu}\phi-
g_{\mu\nu}\left(\frac{1}{2}\partial_{\lambda}\phi\
\partial^{\lambda}\phi+V(\phi)\right).$ (2.4)

Using the flat FLRW line element describing the homogeneous and isotropic
Universe at large scales, i.e.,

$\displaystyle{\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t){\rm d}\mathbf{x}^{2},$ (2.5)

where, $a(t)$ is the scalar factor, Eqs. (2.2) and (2.3) can be reduced to

$\displaystyle 3H^{2}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (2.6)
$\displaystyle\dot{H}=-\frac{1}{2}\dot{\phi}^{2},$ (2.7)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0.$ (2.8)

where, $H\equiv\dot{a}/a$ is the Hubble parameter, and $A_{x}\equiv\partial
A/\partial{x}$. These equations can be re-defined in terms of the e-fold
variable $N\equiv\int H{\rm d}t=\ln\left(a/a_{0}\right)$ as

$\displaystyle
H=\sqrt{\frac{V}{3-\frac{1}{2}\phi_{N}^{2}}},\qquad\frac{H_{N}}{H}=-\frac{1}{2}\phi_{N}^{2},$
(2.9)
$\displaystyle\phi_{NN}+\left(3-\frac{1}{2}\phi^{2}_{N}\right)\left(\phi_{N}+\frac{V_{\phi}}{V}\right)=0,$
(2.10)

where, $A_{xx}=\partial^{2}A/\partial{x}^{2}.$ Given a potential $V(\phi)$,
one can solve the above equations to correctly obtain the dynamics of the
early Universe. Once the background equations are solved, one can in
principle, solve the scalar $(\mathcal{R}_{k})$ and tensor $(h_{k})$
perturbations and evaluate the power spectra:

$\displaystyle\mathcal{P}_{\mathcal{R}}=\frac{k^{3}}{2\pi^{2}}|\mathcal{R}_{k}|^{2}\equiv
A_{\mathcal{R}}\left(\frac{k}{k_{*}}\right)^{n_{s}-1},\quad\mathcal{P}_{T}=2.\frac{k^{3}}{2\pi^{2}}|h_{k}|^{2}\equiv
A_{T}\left(\frac{k}{k_{*}}\right)^{n_{T}}$ (2.11)

where, $k_{*}$ is the pivot scale, $A_{\mathcal{R}},~{}A_{T}$ are the scalar
and tensor spectral amplitude,

$n_{s}\equiv\frac{d\,{\rm ln}\mathcal{P}_{\mathcal{R}}}{d\,{\rm
ln}k}\bigg{|}_{k=k_{*}},~{}~{}~{}~{}~{}~{}~{}~{}~{}n_{T}\equiv\frac{d\,{\rm
ln}\mathcal{P}_{T}}{d\,{\rm ln}k}\bigg{|}_{k=k_{*}}$

are the scalar and tensor spectral index and and

$r\equiv\frac{A_{T}}{A_{\mathcal{R}}}$

is defined as the tensor-to-scalar ratio, which we compare with the
observations. The current observations (PLANCK [32, 33, 35], BICEP/Keck [34] )
suggest that, at the pivot scale ($k_{*}=0.05\,\text{Mpc}{}^{-1}$), the
amplitude of the scalar power spectrum is $A_{\mathcal{R}}\simeq
2.101^{+0.031}_{-0.034}\times 10^{-9}(68\%\,\,\text{CL})$ with the scalar
spectral index being $n_{{}_{\rm s}}=0.9672\pm 0.0059\,(68\%\,\,\text{CL})$,
while the ratio $(r)$ of the tensor-to-scalar perturbation amplitudes is
bounded by $r<0.028\,(95\%\,\,\text{CL})$. These observations agree remarkably
well with the predictions of a near-scale-invariant spectrum of density
perturbations in the inflationary paradigm. In the next section, we will
discuss in detail, how, without explicitly evaluating the perturbations, one
can obtain these observables using the background solutions in the case of
slow-roll inflation.

### Slow-roll inflation

Let us first define the two slow-roll parameters $\epsilon_{1}$ and
$\epsilon_{2}$ as

$\displaystyle\epsilon_{\rm{1}}\equiv-\frac{\dot{H}}{H^{2}}=\frac{1}{2}\phi_{N}^{2},\qquad\epsilon_{\rm{2}}\equiv\frac{\dot{\epsilon_{\rm{1}}}}{H\epsilon_{\rm{1}}}=2\frac{\phi_{NN}}{\phi_{N}}.$
(2.12)

The first variable $\epsilon_{1}<1$ ensures that the Universe is accelerating.
On the other hand, the second variable $\epsilon_{2}\ll 1$ ensures the
sufficient duration of the inflationary phase. In the case of slow-roll, both
these parameters are extremely small, i.e.,

$\displaystyle\epsilon_{\rm{1}}\ll 1,\quad\epsilon_{\rm{2}}\ll 1.$ (2.13)

These two condition are also referred to as slow-roll conditions. Under this
conditions, Eqs. (2.9) and (2.10) become

$\displaystyle
H^{2}\simeq\frac{1}{3}V,\quad\phi_{N}\simeq-\frac{V_{\phi}}{V},$ (2.14)

and the slow-roll parameters can be re-expressed in terms of shape of
potential as

$\displaystyle\epsilon_{1}\simeq\frac{1}{2}\left(\frac{V_{\phi}}{V}\right)^{2},\quad\epsilon_{2}\simeq
2\left(\frac{V^{2}_{\phi}}{V^{2}}-\frac{V_{\phi\phi}}{V}\right).$ (2.15)

Eqs. (2.14) are referred to as the slow-roll equations. By solving these
simplified equations, one can obtain the approximate analytical solution of
the Hubble parameter as well as the slow-roll parameters, and thus the
dynamics of the Universe in slow-roll regime.

In order to check for consistency with the observation, one can, in principle,
obtain the observables corresponding to the perturbations associated with the
pivot scale $k$ in terms of slow-roll parameters as

$\displaystyle
A_{\mathcal{R}}\simeq\frac{H^{2}}{8\pi^{2}\epsilon_{1}},~{}~{}~{}~{}\quad
n_{s}\simeq 1-2\epsilon_{1}-\epsilon_{2},~{}~{}~{}~{}\quad r\simeq
16\epsilon_{1}.$ (2.16)

The above expressions are obtained using leading-order slow-roll
approximation, assuming the higher-order contributions are heavily suppressed.
Please note that these observables depend on the time variable $N_{k}$,
representing the duration between the end of inflation and the epoch when the
pivot scale $k$ leaves the Hubble horizon. Typically, this duration is
considered to be $N_{k}\sim 50-60$. However, it is a model-dependent quantity
and in order to obtain the exact duration $N_{k}$, one must consider the post-
inflationary dynamics, i.e., the reheating epoch. In the very next section, we
will focus on this, which is also known as the quantitative analysis of
reheating.

##  3 Quantitative analysis of reheating

As mentioned in the previous section, in the slow-roll regime, the slow-roll
parameters are very small, i.e., $\epsilon_{1},\ \epsilon_{2}\ll 1$. However,
as the field rolls down the potential, these slow-roll parameters increase,
and the inflation ends as soon as this condition is violated, i.e.,
$\epsilon_{1}=1$. Briefly, after the end of inflation, the field reaches the
bottom and begins to oscillate around it and couples with the other (standard)
particles. As a consequence, at this stage, the time average of kinetic energy
is the same as the average of the potential energy, and the field decays. This
epoch is known as the reheating phase.

To quantify the effect of reheating, let us first simplify it by parametrizing
the reheating epoch using an effective EoS parameter $w_{\rm re}$, such that
the energy density during reheating effectively behaves as

$\displaystyle\rho\propto a^{-3(1+w_{\rm re})}.$ (3.1)

We also define the duration of reheating $N_{\rm re}$ as

$\displaystyle N_{\rm re}\equiv{\rm ln}\left(\frac{a_{\rm re}}{a_{\rm
end}}\right)$ (3.2)

where $a_{\rm end},\ a_{\rm re}$ are the scale factor solutions at the end of
the inflation and reheating epoch, respectively. In order to calculate the
duration $N_{\rm re}$, let us compare the mode of interest with the present
scale as

$\displaystyle\frac{k}{a_{0}H_{0}}=\frac{a_{k}H_{k}}{a_{0}H_{0}}=\frac{a_{k}}{a_{\rm
end}}\frac{a_{\rm
end}}{a_{\rm{re}}}\frac{a_{\rm{re}}}{a_{\rm{eq}}}\frac{a_{\rm{eq}}}{a_{0}}\frac{H_{k}}{H_{\rm{eq}}}\frac{H_{\rm{eq}}}{H_{0}},$
(3.3)

where $k$ is the comoving scale, and (${\rm eq}$) and ($0$) denote the
quantities at the matter-radiation equality and the present epoch,
respectively. Then the above equation leads to:

$\displaystyle{\rm ln}\left(\frac{k}{a_{0}H_{0}}\right)=-N_{k}-N_{\rm
re}-N_{\rm{RD}}+{\rm
ln}\left(\frac{a_{\rm{eq}}H_{\rm{eq}}}{a_{0}H_{0}}\right)+{\rm
ln}\left(\frac{H_{k}}{H_{\rm{eq}}}\right),$ (3.4)

where $N_{k}\equiv{\rm ln}\left(\frac{a_{k}}{a_{\rm end}}\right),\,N_{\rm
RD}\equiv{\rm ln}\left(\frac{a_{\rm re}}{a_{\rm eq}}\right).$ At the end of
reheating, the energy density is

$\displaystyle\rho_{\rm{re}}=\frac{\pi^{2}}{30}g_{\rm{re}}\,T_{\rm{re}}^{4}$
(3.5)

where, $g_{\rm{re}}$ is the effective number of relativistic species upon
thermalization. Radiation drives the ensuing expansion, with non-relativistic
matter and dark energy making recent additions. Assuming a negligible entropy
shift following $T_{\rm{re}}$, the current CMB and neutrino background both
retain the reheating entropy, which leads to the relation,

$\displaystyle
g_{\rm{s,re}}\,T_{\rm{re}}^{3}=\left(\frac{a_{0}}{a_{\rm{re}}}\right)^{3}\left(2T_{0}^{3}+6\times\frac{7}{8}T_{\nu
0}^{3}\right),$ (3.6)

where the present (CMB) temperature, $T_{0}\simeq 2.73$ K, the neutrino
temperature, $T_{\nu 0}=(4/11)^{1/3}\,T_{0}$, and $g_{\rm s,re}$ is the
effective number of species for entropy at reheating. Therefore, $T_{\rm{re}}$
can be written as the present temperature $T_{0}$ as

$\displaystyle\frac{T_{\rm
re}}{T_{0}}=\frac{a_{0}}{a_{\rm{eq}}}\frac{a_{\rm{eq}}}{a_{\rm{re}}}\left(\frac{43}{11g_{\rm{s,re}}}\right)^{1/3},$
(3.7)

and the Eq. (3.4) can be re-written as

$\displaystyle N_{\rm re}=\frac{4}{1-3w_{\rm
re}}\left(-N_{k}-\frac{1}{4}\ln\left(\frac{30}{\pi^{2}g_{\rm{re}}}\right)-\frac{1}{3}\ln\left(\frac{11g_{\rm{s,re}}}{43}\right)-\ln\left(\frac{k}{a_{0}T_{0}}\right)-\frac{1}{4}\ln\rho_{\rm{end}}+\ln
H_{k}\right).$

and $T_{\rm re}$ can be reduced to the following form:

$\displaystyle
T_{\rm{re}}=\frac{a_{0}T_{0}}{k}\left(\frac{43}{11g_{\rm{s,re}}}\right)^{1/3}H_{k}e^{-N_{\rm
k}-N_{\rm{re}}}.$ (3.9)

where $H_{k}$ can be written as

$\displaystyle H_{k}=\frac{\pi\sqrt{rA_{\mathcal{R}}}}{\sqrt{2}}.$ (3.10)

Please note that the Hubble parameter $H_{k}$ is a function of duration
$N_{k},$ and therefore, $N_{\rm re}$ is eventually a function of $N_{k}$. As
spectral index $n_{s}$ (or any time-dependent parameters like the field $\phi$
or tensor-to-scalar ratio $r$) depends on $N_{k}$, considering $\\{g_{\rm
re},g_{\rm s,re}\\}\sim 100$, one can plot $n_{s}$ vs. $N_{\rm re}$ (or
$T_{\rm re}$ from the above expression) and obtain strict constraint on
$N_{\rm re}$ [59, 25, 47, 53, 90, 55, 91] as $n_{s}$ is strictly bounded. This
is referred to as the quantitative analysis of reheating. In the next section,
we will elaborately discuss this analysis by implementing it on the
Starobinsky model of inflation.

###  3.1 Starobinsky inflation

One of the most successful models of slow-roll inflation is the Starobinsky
model [1, 2, 8, 88, 89] where the potential in the Einstein frame is given by:

$\displaystyle
V(\phi)=\frac{3}{4}M^{2}\left(1-e^{-\sqrt{2/3}\phi}\right)^{2},$ (3.11)

where $M$ is the mass of the scalar field $\phi$.

Figure 1: The Starobinsky potential given in Eq. (3.11).

Assuming the slow-roll conditions are satisfied, we can immediately obtain the
two slow-roll equations (i.e., Eqs. (2.14)) as

$\displaystyle
H\simeq\frac{M}{2}\left(1-e^{-\sqrt{2/3}\phi}\right),\quad\phi_{N}\simeq{-2}\sqrt{\frac{2}{3}}\frac{e^{-\sqrt{2/3}\phi}}{\left(1-e^{-\sqrt{2/3}\phi}\right)},$
(3.12)

and the slow-roll parameters can be expressed in terms of the field $\phi$ as
(i.e., Eqs. (2.15)):

$\displaystyle\epsilon_{1}\simeq\frac{4e^{-2\sqrt{2/3}\phi}}{3\left(1-e^{-\sqrt{2/3}\phi}\right)^{2}},\qquad\quad\epsilon_{2}\simeq\frac{8e^{-\sqrt{2/3}\phi}}{3\left(1-e^{-\sqrt{2/3}\phi}\right)^{2}}.$
(3.13)

With the help of Eq. (3.12), one can easily solve the scalar field solution in
terms of the e-folding number $N_{k},$ and the solutions of the slow-roll
parameters at the leading order can be obtained as:

$\displaystyle\epsilon_{1}\simeq\frac{3}{4N_{k}^{2}},\qquad\quad\epsilon_{2}\simeq\frac{2}{N_{k}},$
(3.14)

and, using the leading order slow-roll approximation, i.e., Eqs. (2.16), we
can quickly evaluate the scalar spectral index and the tensor-to-scalar ratio
in terms of the e-folding number as:

$\displaystyle n_{s}\simeq 1-\frac{2}{N_{k}},\qquad\quad
r\simeq\frac{12}{N_{k}^{2}}.$ (3.15)

Figure 2:  We plot the duration of reheating $N_{\rm re}$ and reheating
temperature $T_{\rm re}$ given in Eqs. (3) and (3.9) as function of the scalar
spectral index $n_{s}$ given by Eq. (3.15) parametrically for the case of
Starobinsky inflation. Please note that different colors represent dynamics
corresponding to different effective EoS parameter $w_{\rm re}$ as indicated
in the figure. The blue-shaded region represents the $1\sigma$ constraint on
the value of $n_{s}$ using ongoing observations [32, 33, 34, 35] with
$n_{s}=0.9672\pm 0.0059$. The dark blue region shows the future projected
bound on $n_{s}$ with a sensitivity of $10^{-3}$, assuming its central value
remains unchanged. The temperature below the lighter red region is excluded
due to the constraint from the electroweak scale, which is taken to be $100$
GeV. Figure 3:  We plot the inflationary observables, tensor-to-scalar ratio
($r$) as a function of scalar spectral index ($n_{s}$) for the Starobinsky
model using slow-roll approximations given by Eq. (3.15). This evolution is
presented considering the bound on duration $N_{k}$ using the reheating
regime. We consider different EoS parameters during reheating $w_{\rm re}$ and
observe the constraint on the duration $N_{\rm k}$. Correspondingly, in the
figure, blue line corresponds to the bound for $w_{\rm re}<1/3$, the gray line
for $w_{\rm re}>1/3$ and the purple line for the future observational bound of
$n_{\rm s}$. The blue-shaded region represents the $1\sigma$ constraint on the
value of $n_{\rm s}$ using ongoing observations [32, 33, 34, 35] with $n_{\rm
s}=0.9672\pm 0.0059$. The dark blue region shows the future projected bound on
$n_{\rm s}$ with a sensitivity of $10^{-3}$, assuming its central value
remains unchanged.

Using the above equation along with Eqs. (3) and (3.9), one can obtain
parametric dependence of $N_{\rm re}$ and $T_{\rm re}$ on $n_{s}$. In the case
of the Starobinsky model, this is shown in Fig. 2. Keep in mind that, $N_{\rm
re}\geq 0$, with $N_{\rm re}\ll 1$ is referred to as instantaneous reheating
[92, 91]. It immediately leads to the constraint on the value of $n_{s}.$ For
example, considering that during reheating, the evolution of the Universe
effectively behaves as matter-expansion, i.e., $w_{\rm re}=0$. Thus, in the
$1\sigma$ domain, we get the bound on $N_{k}$ as

$\displaystyle 52\leq N_{k}\leq 56$ (3.16)

and correspondingly, the bound on $n_{s}$ becomes:

$\displaystyle 0.961\leq n_{s}\leq 0.964$ (3.17)

In fact, as can be seen, for $w_{\rm re}<1/3,$ it is obvious that there is an
upper bound on $N_{k}$, which implies a bound on $n_{s}$ as well, i.e.,
$N_{k}\leq 56$ and $n_{s}<0.964.$ It also immediately puts a tight constraint
on $N_{\rm re}$ and $T_{\rm re}$ (for $w_{\rm re}=0$) as

$\displaystyle N_{\rm re}\leq 16,~{}~{}~{}\quad T_{\rm re}\geq
10^{10}\,\text{GeV}.$ (3.18)

However, during reheating, if we assume $w_{\rm re}>1/3,$ these constraints
flips and in the $1\sigma$ domain, there arises a new bound with a lower value
of $N_{k}$ as well as in $n_{s}$ as

$\displaystyle 56\leq N_{k}\leq 74,~{}~{}~{}~{}\quad 0.964\leq n_{s}\leq
0.973.$ (3.19)

Consequently, the bound on reheating parameters becomes (for $w_{\rm re}=1$)

$\displaystyle N_{\rm re}\leq 37,~{}~{}~{}~{}\quad T_{\rm re}\geq
10^{-9}\,\text{GeV}.$ (3.20)

Considering these bounds into account, we plot $r$ as a function of $n_{s}$ in
Fig. 3. Further, instead of looking in $1\sigma$ limit, if we consider the
observational constraints proposed by the forthcoming experiments such as
PRISM [93] and EUCLID [94], cosmic 21-cm surveys [66], and CORE experiments
[67], which offer a precision enhancement of $10^{-3}$ in $n_{s}$, the
constraints on $N_{k}$, $N_{\rm re}$ and $T_{\rm re}$ can be substantially
improved111The central value of $n_{s}$ is assumed to be $0.9672$ [35]. Then
one can immediately refer from the Fig. 2 and 3 that for $w_{\rm re}<1/3,$ the
scalar spectral index does not satisfy the constraints in $1\sigma$ level
posed by future observations. Therefore, under these conditions, this model
can be ruled out. The model can be survived if one assumes $w_{\rm re}>1/3,$
and the bound on $N_{\rm re}$ and $T_{\rm re}$ for the constraints posed by
future observations becomes

$\displaystyle 7\leq N_{\rm re}\leq 14,~{}~{}~{}~{}~{}\quad
10^{9}\,\text{GeV}\geq T_{\rm re}\geq 10^{6}\,\text{GeV}.$ (3.21)

Even in that case, if the future observations detect instantaneous reheating
[92, 91], i.e., $N_{\rm re}\ll 1,$ $n_{s}$ appears to be outside of the
contour from the future observations. Therefore, unless $w>1/3$ with the
prolonged reheating era, the analytical approximation may lead to ruling out
the Starobinsky inflation model. However, keeping aside the future
observations, in the next section, we will see a significant improvement in
these constraints by implementing meaningful corrections to the dynamics
rather than the analytical approximations.

##  4 Numerical improvements

In the previous section, we discussed the CMB constraints on various
inflationary models, which depend on the effective EoS parameter $w_{\rm re}$
as well as the duration of the reheating epoch $N_{\rm re}$. There are two
issues associated with this analysis. First, around and at the end of
inflation, the slow-roll approximation breaks down. Yet, in evaluating
duration $N_{k}$ as well as energy density at end of inflation $\rho_{\rm
end}$ in Eq. (3), we rely upon slow-roll approximations. Second, in evaluating
perturbed observables in Eqs. (2.16), we consider only the leading-order
contributions and ignore higher-order contributions, assuming their impact to
be negligible. However, as anticipated in. Refs. [71, 72], considering
accurate dynamics rather than the slow-roll approximations may lead to notable
enhancements in these predictions. Additionally, one can consider the onset of
reheating not at the end of inflation but at the bottom of the potential, as
will be discussed in the subsequent section. The implementation and,
therefore, the resulting corrections due to these considerations can be
categorized into three parts:

  1. 1.

Implementation of accurate inflationary dynamics through numerical methods
instead of slow-roll approximations.

  2. 2.

Implementation of higher-order slow-roll approximations instead of leading-
order slow-roll approximations.

  3. 3.

Implementation of onset of reheating as the bottom of the potential instead of
the end of inflation.

In the following sections, we will conduct a detailed study of the impact of
each of these corrections on the observational constraints. We will also
compare the cumulative effects of these corrections with the existing
predictions presented in the previous section.

###  4.1 Implementation of accurate inflationary dynamics through numerical
methods

As mentioned earlier, this epoch of the slow-roll regime lasts as long as both
of the slow-roll parameters are small, i.e., ($\epsilon_{1}\ll
1,~{}\epsilon_{2}\ll 1$). On the other hand, $\epsilon_{1}=1$ signals the end
of inflation. It turns out that even before the end of inflation, these slow
approximations break down. Despite this, we still rely on (leading-order)
slow-roll approximations even at the end of inflation to determine the
analytical solution of the background variables, assuming the contribution due
to accurate dynamics in evaluating perturbed observables is negligible.
However, it is crucial, as in this section, we shall learn the significance of
obtaining the accurate values of not only the background parameters but also
the end of inflation. To illustrate this, let’s now discuss this in detail
with an example: the Starobinsky inflation.

Figure 4:  We plot the duration of reheating $N_{\rm re}$ and reheating
temperature $T_{\rm re}$ given by Eqs. (3) and (3.9) as function of the scalar
spectral index $n_{s}$ parametrically for both analytical and numerical
solution, given by Eqs. (3.15), (2.16) respectively. The solid lines are for
the numerical solution, and the dashed lines are for the analytically
approximated solution. Please note that different colors represent dynamics
corresponding to different effective equations of state parameter $w_{\rm re}$
as indicated in the figure. The blue-shaded region represents the $1\sigma$
constraint on the value of $n_{s}$ using ongoing observations [32, 33, 34, 35]
with $n_{s}=0.9672\pm 0.0059$. The dark blue region shows the future projected
bound on $n_{s}$ with a sensitivity of $10^{-3}$, assuming its central value
remains unchanged. The temperature below the lighter red region is excluded
due to the constraint from the electroweak scale, which is taken to be $100$
GeV.

Using leading-order slow-roll approximations and solving the slow-roll
equations (viz. Eqs. (2.14)):

$H^{2}\simeq\frac{V}{3},\quad\phi_{N}\simeq-\frac{V_{\phi}}{V},$

with the initial condition $\phi_{i}=5.8,$ we obtain the following results:

$N_{\rm end}\simeq 80.86,\quad\phi_{\rm end}\simeq 0.94,\quad H_{\rm
end}\simeq 0.267\,M,\quad\rho_{\rm end}\simeq 0.215\,M^{2},$

along with the slow-roll solutions given in Eqs. (3.14). Here, $N_{\rm end}$
is the duration of the field rolling from the initial value $\phi_{i}$ to
$\phi_{\rm end},$ i.e., the duration of inflation. These values are essential
for the analysis in Eqs. (3) and (3.9) which has been shown in the previous
section with Fig. 2.

To improve the analysis, in this section, we now consider, rather than the
slow-roll, the full inflationary equations (viz. Eqs. (2.9) and (2.10)):

$H=\sqrt{\frac{V}{3-\frac{1}{2}\phi_{N}^{2}}},\quad\phi_{NN}+\left(3-\frac{1}{2}\phi_{N}^{2}\right)\left(\phi_{N}+\frac{V_{\phi}}{\phi}\right)=0.$

Solving the highly non-linear differential equation with identical initial
condition $\phi_{i}=5.8$, we now obtain the numerical solutions of the
background variables such as the Hubble parameter and the slow-roll parameters
along with:

$N_{\rm end}\simeq 82.48,\quad\phi_{\rm end}\simeq 0.61,\quad H_{\rm
end}=0.241\,M,\quad\rho_{\rm end}\simeq 0.175\,M^{2}.$

Thus, there is a significant improvement not only in the numerical solution of
the background variables but also in $H_{\rm end},$ $\rho_{\rm end},$ and
$N_{\rm end}.$ These improvements directly impact the reheating analysis in
two ways. The first is the direct influence of $\rho_{\rm end}$ in Eq. (3).
The second, and more crucial, impact is the significant improvement in $\Delta
N_{\rm end}=\Delta N_{k}\sim 1.5,$ which in turn enhances the accuracy of the
background variables. For example, in the case of leading order
approximations, $\\{\epsilon_{1},\epsilon_{2}\\}\simeq\\{0.0003,0.040\\}$ for
$N_{k}=50,$ whereas, numerically we obtain these values as
$\\{\epsilon_{1},\epsilon_{2}\\}\simeq\\{0.00027,0.038\\},$ and thus $\Delta
n_{s}\simeq 2\times 10^{-3},$ as also discussed in Ref. [71] 222Please note
that, for a multi-parameter model, numerical evaluation for the entire range
of the parameters can be challenging. In that case, one can rely upon a semi-
analytical solution presented in Ref. [71], where analytically obtained
solutions are nearly comparable to the numerical solutions..

Figure 5:  We plot the inflationary observables, tensor-to-scalar ratio ($r$)
as a function of scalar spectral index ($n_{s}$) for the Starobinsky model
using numerical solution given by Eq. (2.16). This evolution is presented
considering the bound on duration $N_{k}$ using the reheating regime. We
consider different EoS parameters during reheating $w_{\rm re}$ and observe
the constraint on the duration $N_{\rm k}$. Correspondingly, in the figure,
blue line corresponds to the bound for $w_{\rm re}<1/3$, the gray line for
$w_{\rm re}>1/3$ and the purple line for the future observational bound of
$n_{\rm s}$. The blue-shaded region represents the $1\sigma$ constraint on the
value of $n_{\rm s}$ using ongoing observations [32, 33, 34, 35] with $n_{\rm
s}=0.9672\pm 0.0059$. The dark blue region shows the future projected bound on
$n_{\rm s}$ with a sensitivity of $10^{-3}$, assuming its central value
remains unchanged.

We now examine the impact of this on the reheating analysis and compare the
results with the results obtained using slow-roll approximations. This is
illustrated in Fig. 4. As anticipated, the improvement is significant. In the
figure, one can see that numerical predictions show a significant shift to the
right in the estimates of $N_{\rm re}$ and $T_{\rm re}.$ Due to this shift,
the bound on $N_{k}$, and subsequently $n_{s}$ also changes, with an
improvement being $\Delta n_{s}\sim 10^{-3}$. For example, in the case of
leading-order slow-roll approximation with $w_{\rm re}<1/3,$ the upper bound
on $n_{s}$ is: $n_{s}\leq 0.964,$ whereas, in this improved case, the bound
becomes $n_{s}\leq 0.965.$ The improvements are explicitly shown in the Table
1. This confirms our anticipation that the accurate dynamics help improve the
theoretical predictions, which is one of the main results of this work.

|  Analytical approximations |  Numerical solution
---|---|---
$w_{\rm re}<1/3$ |  $52\leq N_{k}\leq 56$, |  $50\leq N_{k}\leq 56$,
| $0.9613\leq n_{s}\leq 0.9641$ |  $0.9613\leq n_{s}\leq 0.9649$,
|  $N_{\rm re}\leq 16$ ($w_{\rm{re}}=0$), |  $N_{\rm re}\leq 21$ ($w_{\rm{re}}=0$),
|  $T_{\rm re}\geq 10^{10}\ \rm{GeV}$ ($w_{\rm{re}}=0$) |  $T_{\rm re}\geq 10^{8}\ \rm{GeV}$ ($w_{\rm{re}}=0$)
$w_{\rm re}>1/3$ |  $56\leq N_{k}\leq 74$, |  $56\leq N_{k}\leq 73$,
|  $0.9641\leq n_{s}\leq 0.9731$, |  $0.9649\leq n_{s}\leq 0.9731$,
|  $N_{\rm re}\leq 37$ ($w_{\rm{re}}=1$), |  $N_{\rm re}\leq 34$ ($w_{\rm{re}}=1$),
|  $T_{\rm re}\geq 10^{-9}$ GeV ($w_{\rm{re}}=1$) |  $T_{\rm re}\geq 10^{-8}$ GeV ($w_{\rm{re}}=1$)
future observations |  $59\leq N_{k}\leq 63$, |  $58\leq N_{k}\leq 62$,
($w_{\rm re}>1/3$: allowed) |  $0.9662\leq n_{s}\leq 0.9682$, |  $0.9662\leq n_{s}\leq 0.9682$,
($w_{\rm re}<1/3$: not |  $7\leq N_{\rm re}\leq 14$ ($w_{\rm{re}}=1$), |  $4\leq N_{\rm re}\leq 12$ ($w_{\rm{re}}=1$),
allowed) |  $10^{9}\ \rm{GeV}\geq T_{\rm re}\geq 10^{6}\ \rm{GeV}$ ($w_{\rm{re}}=1$) |  $10^{12}\ \rm{GeV}\geq T_{\rm re}\geq 10^{7}\ \rm{GeV}$ ($w_{\rm{re}}=1$)
Table 1: Starobinsky Inflation: The bounds on variables,
$N_{k},~{}n_{s},~{}N_{\rm re},~{}\text{and}~{}T_{\rm re}$ for different values
of $w_{\rm re}$ corresponding to analytically (slow-roll) approximated and the
numerical solutions.

###  4.2 Implementation of higher-order slow-roll approximations

In the previous section, we estimated the reheating parameters and e-folding
number using the numerical background solution with the leading order
inflationary observables, i.e., Eqs. (2.16). However, as mentioned earlier,
these expressions are obtained using leading-order slow-roll, and the full
solutions of the amplitude of scalar power spectrum, scalar spectral index as
well as the tensor-to-scalar ratio in terms of the slow-roll parameter are
given as [73, 25]:

$\displaystyle A_{s}$ $\displaystyle=$ $\displaystyle
1-2(C+1)\epsilon_{1}-C\epsilon_{2}+\left(2C^{2}+2C+\frac{\pi^{2}}{2}-f\right)\epsilon^{2}_{1}+\left(C^{2}-C+\frac{7\pi^{2}}{12}-g\right)\epsilon_{1}\epsilon_{2}$
(4.1)
$\displaystyle+\left(\frac{1}{2}C^{2}+\frac{\pi^{2}}{8}-1\right)\epsilon^{2}_{2}+\left(-\frac{1}{2}C^{2}+\frac{\pi^{2}}{24}\right)\epsilon_{2}\epsilon_{3},$
$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle
1-2\epsilon_{1}-\epsilon_{2}-2\epsilon^{2}_{1}-(3+2C)\epsilon_{1}\epsilon_{2}-C\epsilon_{2}\epsilon_{3},$
(4.2) $\displaystyle r$ $\displaystyle=$ $\displaystyle
16\epsilon_{1}(1+2C\frac{\epsilon_{2}}{2}),$ (4.3)

where $C=-2+\ln{2}+\gamma$, where $\gamma$ is the Euler constant, $f=5$ and
$g=7$. In this section, instead of using Eqs. (2.16), along with the numerical
solution, we consider the above expressions for the observable, repeat the
reheating analysis, and compare our results with the numerical results
obtained in the previous section, which has been illustrated in Figs. 6 and 7
as well as in Table 2.

Figure 6: We plot the duration of reheating $N_{\rm re}$ and reheating
temperature $T_{\rm re}$ given by Eqs. (3) and (3.9) as function of the scalar
spectral index $n_{s}$ parametrically comparing numerical solution with
leading-order approximations and numerical solution with higher-order
approximations given by Eq. (2.16), (4.2) respectively. The solid lines are
for the higher-order approximations and the dashed lines are for the leading-
order approximations. Please note that different colors represent dynamics
corresponding to different effective equations of state parameter $w_{\rm re}$
as indicated in the figure. The blue-shaded region represents the $1\sigma$
constraint on the value of $n_{s}$ using ongoing observations[32, 33, 34, 35]
with $n_{s}=0.9672\pm 0.0059$. The dark blue region shows the future projected
bound on $n_{s}$ with a sensitivity of $10^{-3}$, assuming its central value
remains unchanged. The temperature below the lighter red region is excluded
due to the constraint from the electroweak scale, which is taken to be $100$
GeV.

As again can be seen in Fig. 6, there is a further shift in reheating
parameters, $N_{\rm re}$ and $T_{\rm re}$ compared to the improvements
mentioned in the previous section. We obtain an additional improvement of
$\Delta n_{s}\sim 4\times 10^{-4},$ resulting a new bound on $n_{s}$ becomes
$n_{s}\leq 0.9653$ for $w_{\rm re}<1/3.$ Thus, considering higher-order slow-
roll approximations instead of the leading-order can significantly improve the
estimation of the observables by an order of $\sim 10^{-3}$. The detailed
bounds on parameters are shown in the Table 2.

Figure 7:  We plot the inflationary observables, tensor-to-scalar ratio ($r$) as a function of scalar spectral index ($n_{s}$) for the Starobinsky model using higher-order approximations given by Eq. (4.2), (4.3). This evolution is presented considering the bound on duration $N_{k}$ using the reheating regime. We consider different EoS parameters during reheating $w_{\rm re}$ and observe the constraint on the duration $N_{\rm k}$. Correspondingly, in the figure, blue line corresponds to the bound for $w_{\rm re}<1/3$, the gray line for $w_{\rm re}>1/3$ and the purple line for the future observational bound of $n_{\rm s}$. The blue-shaded region represents the $1\sigma$ constraint on the value of $n_{\rm s}$ using ongoing observations [32, 33, 34, 35] with $n_{\rm s}=0.9672\pm 0.0059$. The dark blue region shows the future projected bound on $n_{\rm s}$ with a sensitivity of $10^{-3}$, assuming its central value remains unchanged. |  Numerical solution with |  Numerical solution with
---|---|---
| leading-order $n_{s}$, $r$ | higher-order $n_{s}$, $r$
| expressions | expressions
$w_{\rm re}<1/3$ |  $50\leq N_{k}\leq 56$, |  $50\leq N_{k}\leq 56$,
| $0.9613\leq n_{s}\leq 0.9649$ |  $0.9613\leq n_{s}\leq 0.9653$,
|  $N_{\rm re}\leq 21$$(w_{\rm{re}}=0)$, |  $N_{\rm re}\leq 24$$(w{{re}}=0)$,
|  $T_{\rm re}\geq 10^{8}\ \rm{GeV}$$(w_{\rm re}=0)$ |  $T_{\rm re}\geq 10^{7}\ \rm{GeV}$ $(w_{\rm re}=0)$
$w_{\rm re}>1/3$ |  $56\leq N_{k}\leq 73$, |  $56\leq N_{k}\leq 72$,
|  $0.9649\leq n_{s}\leq 0.9731$, |  $0.9653\leq n_{s}\leq 0.9731$,
|  $N_{\rm re}\leq 34$$(w_{\rm re}=1)$, |  $N_{\rm re}\leq 33$$(w_{\rm re}=1)$,
|  $T_{\rm re}\geq 10^{-8}$ GeV $(w_{\rm re}=1)$ |  $T_{\rm re}\geq 10^{-7}\ \rm{GeV}$ $(w_{\rm re}=1)$
future observations |  $58\leq N_{k}\leq 62$, |  $57\leq N_{k}\leq 61$,
($w_{\rm re}>1/3$: allowed) |  $0.9662\leq n_{s}\leq 0.9682$, |  $0.9662\leq n_{s}\leq 0.9682$,
($w_{\rm re}<1/3$: not |  $4\leq N_{\rm re}\leq 12$ $(w_{\rm re}=1)$, |  $3\leq N_{\rm re}\leq 10$ $(w_{\rm re}=1)$,
allowed) |  $10^{12}\ \rm{GeV}\geq T_{\rm re}\geq 10^{7}\ \rm{GeV}$ $(w_{\rm re}=1)$ |  $10^{13}\ \rm{GeV}\geq T_{\rm re}\geq 10^{8}\ \rm{GeV}$ $(w_{\rm re}=1)$
Table 2: Starobinsky Inflation: The bounds on variables,
$N_{k},~{}n_{s},~{}N_{\rm re},~{}\text{and}\ T_{\rm re}$ for different values
of $w_{\rm re}$ corresponding to leading-order approximations and the higher-
order approximations. Note that, in both cases, the background solutions are
numerically obtained.

###  4.3 Implementation of the onset of reheating as the bottom of the
potential

Till now, we were estimating $N_{\rm re}$ and $T_{\rm re}$, considering the
onset of the reheating era after the end of inflation. However, in Ref. [62],
it has been shown that, instead of considering minimal gravity theory, if one
considers modified gravity theories, the onset of reheating as the end of
inflation can a make a significant change in the analysis. This is because,
$\epsilon_{1}$ is not a invariant under conformal transformation. Therefore,
in a meaningful manner, one can choose the onset of reheating as the bottom of
the potential, as this point is conformally invariant, and can be used even
for general modified gravity theories. Therefore, in this section, we
introduce corrections to the reheating analysis, and thus the observables, by
considering that the reheating era commences after the epoch at which the
potential touches its minima (bottom of the potential). In that case, Eq. (3)
can be re-written as

$\displaystyle N_{\text{r}e}$ $\displaystyle=$
$\displaystyle\frac{4}{1-3w_{\text{re}}}\left(-N_{k}-N_{\text{eb}}-\frac{1}{4}\log\left(\frac{30}{\pi^{2}g_{\text{reh}}}\right)-\frac{1}{3}\log\left(\frac{11g_{\text{s,re}}}{43}\right)-\log\left(\frac{k}{a_{0}T_{0}}\right)\right.$
(4.4) $\displaystyle\left.-\frac{1}{4}\log\rho_{\text{b}}+\log H_{k}\right),$

and the reheating temperature, i.e., Eq. (3.9) can also be evaluated as:

$\displaystyle
T_{\text{re}}=\frac{a_{0}T_{0}}{k}\left(\frac{43}{11g_{\text{s,re}}}\right)e^{-N_{\text{k}}-N_{\text{eb}}-N_{\text{re}}}.$
(4.5)

Here, $N_{\rm eb}\equiv a_{\rm b}/a_{\rm end}$ and $\rho_{\rm b}\equiv 3H_{\rm
b}^{2},$ where $(b)$ denotes the bottom of the potential. Again, we perform
our analysis considering the numerically obtained solution with higher-order
slow-roll corrections and the onset of reheating as the bottom of the
potential. However, to our surprise, we find that this implementation does not
improve the accuracy to a significant amount, as $\Delta n_{s}\sim 10^{-5}.$

Figure 8: We plot the duration of reheating $N_{\rm re}$ and reheating
temperature $T_{\rm re}$ given by Eqs. (3) and (3.9) as function of the scalar
spectral index $n_{s}$ parametrically comparing analytically approximated
solution and numerical solution with higher-order approximations, given by
Eqs. (3.15), (4.2) respectively. The solid lines are for the onset of
reheating as the bottom of the potential and the dashed lines are for the
analytically approximated solution. Please note that different colors
represent dynamics corresponding to different effective equations of state
parameter $w_{\rm re}$ as indicated in the figure. The blue-shaded region
represents the $1\sigma$ constraint on the value of $n_{s}$ using ongoing
observations [32, 33, 34, 35] with $n_{s}=0.9672\pm 0.0059$. The dark blue
region shows the future projected bound on $n_{s}$ with a sensitivity of
$10^{-3}$, assuming its central value remains unchanged. The temperature below
the lighter red region is excluded due to the constraint from the electroweak
scale, which is taken to be $100$ GeV.

To summarize our cumulative result, in Fig. 8, we plot the results obtained
through the analytical approximations mentioned in the first section and the
improvement obtained through the implementation of three corrections to the
dynamics, and Table 3 summarizes all the results. Compared to the conventional
analytical case, our corrections improve the accuracy in $n_{s}$ to $\Delta
n_{s}\sim 1.2\times 10^{-3},$ which is significant. Note that, if the central
value of $n_{s}$ is assumed to be $\sim 0.9672$ even in this case, then even
with the improvements, the Starobinsky model can indeed be ruled out at
$1\sigma$ level from the expected future observation. These are the main
results of this work.

|  Analytical |  Higher-order
---|---|---
| approximations | approximations
$w_{\rm re}<1/3$ |  $52\leq N_{k}\leq 56$, |  $50\leq N_{k}\leq 56$,
|  $0.9613\leq n_{s}\leq 0.9641$, |  $0.9613\leq n_{s}\leq 0.9653$,
|  $N_{\rm re}\leq 16$ $(w_{\rm re}=0)$, |  $N_{\rm re}\leq 24$ $(w_{\rm re}=0)$,
|  $T_{\rm re}\geq 10^{10}\ \rm{GeV}$ $(w_{\rm re}=0)$ |  $T_{\rm re}\geq 10^{7}\ \rm{GeV}$ $(w_{\rm re}=0)$
$w_{\rm re}>1/3$ |  $56\leq N_{k}\leq 74$, |  $56\leq N_{k}\leq 72$,
|  $0.9641\leq n_{s}\leq 0.9731$, |  $0.9653\leq n_{s}\leq 0.9731$,
|  $N_{\rm re}\leq 37$ $(w_{\rm re}=1)$, |  $N_{\rm re}\leq 33$ $(w_{\rm re}=1)$,
|  $T_{\rm re}\geq 10^{-9}\ \rm{GeV}$ $(w_{\rm re}=1)$ |  $T_{\rm re}\geq 10^{-7}\ \rm{GeV}$ $(w_{\rm re}=1)$
future observations |  $59\leq N_{k}\leq 63$, |  $57\leq N_{k}\leq 61$,
($w_{\rm re}>1/3$: allowed) |  $0.9662\leq n_{s}\leq 0.9682$, |  $0.9662\leq n_{s}\leq 0.9682$,
($w_{\rm re}<1/3$: not |  $7\leq N_{\rm re}\leq 14$ $(w_{\rm re}=1)$, |  $3\leq N_{\rm re}\leq 10$ $(w_{\rm re}=1)$,
allowed) |  $10^{9}\ \rm{GeV}\geq T_{\rm re}\geq 10^{6}\ \rm{GeV}$ $(w_{\rm re}=1)$ |  $10^{13}\ \rm{GeV}\geq T_{\rm re}\geq 10^{8}\ \rm{GeV}$ $(w_{\rm re}=1)$
Table 3: Starobinsky Inflation: The bounds on variables,
$N_{k},~{}n_{s},~{}N_{\rm re},~{}\text{and}~{}T_{\rm re}$ for different values
of $w_{\rm re}$ corresponding to analytical approximated parameters, and
numerical solution with higher-order approximations.

##  5 Summary and conclusions

In this article, we consider a single canonical scalar field model minimally
coupled to the gravity with a potential $V(\phi)$ that drives the evolution of
the early Universe, including both slow-roll inflation and oscillatory
solution around the minima of the potential, also known as the reheating
epoch. In obtaining the solution, we often use different sets of
approximations in both regimes. In the slow-roll inflationary regime, where
$\epsilon_{1},\epsilon_{2}\ll 1,$ we rely upon the slow-roll approximations.
In contrast, the reheating epoch is, for simplicity, quantitatively
characterized by the effective EoS parameter $w_{\rm re}$ and the duration of
this epoch $N_{\rm re}$ (or the reheating temperature $T_{\rm re}$). However,
these approximations may lead to a significant discrepancy in the theoretical
estimation of the observables ($n_{s}$ and $r$). While accurately defining the
reheating epoch (viz. the qualitative analysis) is challenging, this work
focuses on examining the impact of a more accurate dynamics, rather than the
slow-roll approximations, on the perturbed observables.

In order to do this, we specifically considered the following improvements in
the dynamics:

  1. 1.

Numerical Solution: In this method, we solve the complete background Eq.
(2.10) using the numerical method, which provides us with an accurate solution
for the dynamics of the Universe in its early phase. Using this solution, we
estimate the observables and compare the results with the solutions obtained
using slow roll approximation.

  2. 2.

Higher-order approximations: This method includes higher-order slow-roll
corrections to the inflationary observables, Eqs. (4.1), (4.2) and (4.3),
rather than Eqs. (2.16). Here also, we used complete numerical solution and
estimated the observables using higher order slow-roll corrections and
estimated the discrepancy in the observables with respect to the analytical
solution obtained using slow-roll approximation.

  3. 3.

Onset of the reheating as the bottom of the potential: Typically, the end of
inflation is considered the onset of the reheating epoch. However, in this
method, we show that if reheating begins at the bottom of the potential, the
reheating parameters are modified, as given by Eq. (4.4) and Eq. (4.5). This
modification can provide a new bound on the duration $N_{k}$, leading to a
discrepancy in the scalar spectral index.

After implementing these changes, we found an improvement in the theoretical
bounds on the observables. As seen in Fig. 8, there is an improvement of
$\Delta n_{s}\sim 1.2\times 10^{-3}.$ These numerical predictions indicate
that for $w_{\rm re}<1/3$, the Starobinsky model has an upper bound as
$n_{s}\leq 0.9653.$ This can be ruled out from the future observational
constraint if the central value is $n_{s}\simeq 0.9672$ [35]. However, for
$w_{\rm re}>1/3$, the reheating parameters fall within the future
observational bounds, though, at the cost of sufficient reheating e-folding
number ($\sim\mathcal{O}(1-2)$). This scenario can again be ruled out if
future observations detect instantaneous reheating [92, 91].

The result is compelling, as now it has been established that corrections due
to accurate dynamics or higher-order slow-roll approximations can not be
ignored. This also implies that even seemingly minor corrections must be
considered and implemented to study their implications on perturbed
observables. Therefore, when obtaining observational bounds on any
inflationary models, one must consider all these corrections, which have been
reserved for future endeavors.

In addition to these methods, other improvements can also be made to further
enhance the accuracy. For example, in this work, we only considered the slow-
roll dynamics, but one could also improve the reheating analysis by
incorporating qualitative analysis of the reheating and implementing the
changes into the quantitative analysis. Second, higher-order correlation such
as non-Gaussianity (see, for instance, Refs. [95, 96, 97, 98, 99]) can play a
pivotal role in the observables. Third, the implications of primordial black
holes and primordial gravitational waves may also play a crucial role in
inflationary and post-inflationary dynamics. Fourth, exploring the non-minimal
sector instead of the minimal theory of gravity is another avenue. In fact, in
our work, while the corrections due to the onset of reheating as the bottom of
the potential did not play a significant role, they could be significant in
the non-minimal sector [100, 62]. Finally, extending the analysis to non-
inflationary dynamics, such as viable classical bounces, is also possible.
Recent studies [101, 102, 99, 103, 104, 105, 106, 107] show that these
alternatives can also fit observations well. Therefore, one must consider all
alternatives to slow-roll inflation and carefully study the analysis with all
effects, even those that seem minor, to determine the observable constraints
of the model.

## References

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# A novel deep learning-based approach for sleep apnea detection using single-
lead ECG signals

Anh-Tu Nguyen13, Thao Nguyen13, Huy-Khiem Le1, Huy-Hieu Pham12, and Cuong Do12
1 College of Engineering & Computer Science, VinUniversity, Hanoi, Vietnam
2 VinUni-Illinois Smart Health Center, VinUniversity, Hanoi, Vietnam
3 These authors contributed equally to this work.

###### Abstract

Sleep apnea (SA) is a type of sleep disorder characterized by snoring and
chronic sleeplessness, which can lead to serious conditions such as high blood
pressure, heart failure, and cardiomyopathy (enlargement of the muscle tissue
of the heart). The electrocardiogram (ECG) plays a critical role in
identifying SA since it might reveal abnormal cardiac activity. Recent
research on ECG-based SA detection has focused on feature engineering
techniques that extract specific characteristics from multiple-lead ECG
signals and use them as classification model inputs. In this study, a novel
method of feature extraction which based on the detection of S peaks is
proposed to enhance the detection of adjacent SA segments using a single-lead
ECG. In particular, ECG features collected from a single lead (V2) are used to
identify SA episodes. On the extracted features, a CNN model is trained to
detect SA. Experimental results demonstrate that the proposed method detects
SA from single-lead ECG data is more accurate than existing state-of-the-art
methods, with 91.13% classification accuracy, 92.58% sensitivity, and 88.75%
specificity. Moreover, the further usage of features associated with the S
peaks enhances the classification accuracy by 0.85%. Our findings indicate
that the proposed machine learning system has the potential to be an effective
method for detecting SA episodes.

##  1 Introduction

Sleep apnea (SA) is the most prevalent breathing problem associated with sleep
[1]. It causes people to repeatedly stop and start breathing during sleep.
There are various types of sleep apnea, but obstructive sleep apnea is the
most prominent, which occurs when upper airway muscles relax during sleep and
obstruct the airway, preventing adequate airflow [1]. Patients’ breathing may
stop for 10 seconds or longer before the reflexes kick in and then resume
breathing. Sleep apnea affects approximately 3% of normal-weight adults, but
over 20% of obese individuals, and in general, men are more susceptible to
sleep apnea than women [2].

Sleep apnea is frequently associated with heart disease and metabolic
disorders such as diabetes [3]. Several studies have demonstrated a link
between sleep apnea and health issues such as type 2 diabetes, strokes, heart
attacks, and even a shorter lifespan [3, 4, 5, 6]. It is essential to diagnose
and detect sleep apnea in order to avoid long-term health repercussions. The
questionnaire, which includes the STOP-Bang Questionnaire [7, 8] and Berline
Questionnaire [9], is one method for screening patients at risk for SA
syndrome. The gold standard for diagnosing sleep-related breathing disorders,
however, is polysomnography — a method for collecting physiologic parameters
during sleep [10]. A polysomnogram (PSG) is a diagnostic test that uses
electroencephalogram (EEG), electrooculogram (EOG), electromyogram (EMG),
electrocardiogram (ECG), and pulse oximetry, in addition to airflow and
respiratory effort, to identify the underlying reasons of sleep abnormalities
[10]. However, this diagnostic procedure is time-consuming, expensive, and
inconvenient. Patients must be connected to at least 22 electrodes for several
nights in order to measure 11 channels of sleep signals in the laboratory
using specialized equipment [11]. This is a barrier for patients to
independently install and use the device at home. Moreover, to diagnose a
patient, physicians must spend considerable time monitoring and interpreting
that data. Consequently, the PSG-based technique is costly, complicated, and
cumbersome. That trigger a need for a simple, affordable, and user-friendly
alternative.

ECG is a viable tool for diagnosing and detecting sleep apnea, which has
lately generated considerable interest. While some study groups have
demonstrated that a patient’s cardiovascular activity changes during sleep
apnea, resulting in typical ECG readings [12, 13], other studies have
demonstrated a link between patients with SA and other cardiovascular
disorders [14]. Thus, the Apnea-ECG database [15] was developed by PhysioNet
to spot abnormalities in patients’ ECG signals when they have SA. Various
research use ECG data to identify SA episodes, some of which employ
traditional machine learning techniques based on feature engineering. [16, 17,
18, 19, 20, 21, 22] and others applying deep learning techniques [23, 17] for
their classification tasks. In [24], Li et al. suggested a technique for
detecting SA using ECG data, which is based on a deep neural network and a
Hidden Markov model (HMM). The approach used a sparse auto-encoder to learn
features, which were then fed into two types of classifiers (SVM and ANN). In
the per-segment SA detection, they achieve 84.5% classification accuracy. In
[22], a Sgolay filter was applied to extract the Heart Rate Variability (HRV)
and the ECG-Derived Respiration (EDR), then they were used for the training
procedure to achieve 82.2% accuracy. Shen et al. introduced a method that
based on a weighted-loss time-dependent (WLTD) classification model and a
multiscale dilation attention 1-D convolutional neural network (MSDA-1DCNN)
[18]. That study obtained 89.4% accuracy, 89.8% sensitivity and 89.1%
specificity. A novel method for diagnosing SA using a pre-trained AlexNet
model is reported in [21], in which per-minute segments of a single-lead ECG
recording are decomposed using continuous wavelet transform (CWT) and
subsequently 2D scalogram images are created. Following that, a CNN based on a
deep learning algorithm is used to improve classification performance.

While recent studies have sought to utilize ECG signals as direct input to
deep learning models, research using handcrafted features offers the potential
for development because of its transparency and interpretability. However,
recent research targeted at identifying SA by feature engineering only
consider the position of the R peak in the ECG, neglecting the remaining four
peaks (P, Q, S, and T). In this work, we not only identify the R peak, but
also determine the position of the S peak and demonstrate that using the S
peak enhances the model’s performance. To reduce noise and signal artifacts,
we first applied a Finite Impulse Response (FIR) band-pass filter to the
signal. Afterward, features are extracted based on the detection of the R and
S peaks. We employ the feature extraction approach given by Wang et al. [25],
in which an ECG record is divided into 5 minute-long segments, then the R peak
is detected. In this work, the detection of S peak is added, followed by cubic
interpolation to generate 900 values for each feature. The extracted features
were used as input to the SE-ResNext 50 model to classify ECG signals with and
without SA. Our model achieved 91.13% classification accuracy, 92.58%
sensitivity, and 88.75% specificity. It has been demonstrated that using
additional features extracted from S peaks improves accuracy by 0.85% compared
to using only R-peak-related features, which may indicate anomalies in the S
peak morphology during SA.

The rest of this paper is structured as follows. Methods for ECG signal
preprocessing, feature extraction, and classification are introduced in
section 2. Section 3 describes the dataset and experimental setups. In Section
3 and 5 we report experimental results and summarize the key findings of this
work.

##  2 Methods

Methods for preprocessing the ECG signal, extracting hand-crafted features,
and developing a classifier for ECG with and without SA are described in this
section. The proposed system’s schematic is depicted in Fig. 1.

Figure 1: An overview of the proposed approach for detecting sleep apnea.

###  2.1 Pre-processing data

ECG signal is frequently contaminated by a variety of noise sources, such as
50/60 Hz interference from power lines, EMG signal from muscles, motion
artefacts, and variations in electrode-skin contact. Therefore, a band-pass
filter with a frequency range of 8 to 12 Hz was applied to remove noise and
artifacts while maintaining the ECG signal’s QRS complex properties [26]. The
signals before and after passing the band-pass filter are shown in Fig. 2a and
Fig. 2b, respectively.

(a)

(b)

Figure 2: (a) The original signal of V2 lead ECG and (b) The component falls
within the 8 to 20 Hz frequency range of the ECG signal.

###  2.2 Feature extraction

In order to detect ECG segments in patients with SA, abnormal characteristics
of the ECG signal are extracted and fed into a machine learning classifier. As
low QRS voltage and a leftward shift of the electrical axis may be associated
with certain ECG abnormalities of SA patients [27], features associated to the
QRS complex of the ECG signal can be utilized to detect SA episodes
efficiently. Previous studies have classified ECG signals with and without SA
based on characteristics associated to R peaks, which has limitations as other
peaks are neglected (P, Q, S, T). To the best of our knowledge, this study is
the first to recover features linked to S peaks and then then feed them into a
CNN model in order to distinguish between SA and non-SA ECG. We use the
following features for the classification purpose:

  * •

Amplitude of R peaks

  * •

Amplitude of S peaks

  * •

RR interval (duration between two consecutive R peaks)

  * •

SS interval (duration between consecutive S peaks)

The distributions of R-peak-related features and S-peak-related features are
depicted in Fig. 3 and Fig. 4, respectively. It seems that the RR and SS
intervals of SA ECG are the same as those of Non-SA ECG. It is noteworthy that
R peaks in SA ECG have significantly lower amplitude than those in Non-SA ECG.
In contrast, S peaks in SA ECG have a higher amplitude than those in non-SA.
These analyses suggest that these features can be used to distinguish between
SA and Non-SA.

(a)

(b)

Figure 3: (a) The histogram of R peaks and (b) The histogram of RR intervals.

(a)

(b)

Figure 4: (a) The histogram of S peaks and (b) The histogram of SS intervals.

To determine the positions of the R and S peaks, the following steps are
taken:

  * •

_R peaks detection:_ In order to determine the positions of the R peaks and
calculate the RR intervals, we first used the Hamilton algorithm [28] to
locate the R peaks. The amplitude of the R peaks are then extracted, and their
positions are utilized to estimate the RR intervals. To remove redundant R
peaks due to false detection, a local median filter proposed in [29] is
applied. As the suspected irregular RR intervals can be caused by either false
R peaks or missed R peaks, a lower bound and an upper bound are defined based
on the physiological range of RR intervals in order to distinguish between
these two types of uninterpretable data points. For abnormal RR intervals
caused by false R peaks detection, the RR intervals in a sliding window are
compared to the lower bound and rectified using either averaging or merging
procedures. For irregular RR intervals caused by a missed R wave detection,
the RR intervals in the current window are divided into several equal values
or averaged with the neighboring window based on the specified criteria.

  * •

_S peaks detection:_ A method for detecting S peaks based on the position of R
peaks is proposed, in which, the resulting S peak is the first negative peak
(in the case of a positive R peak). S peak detection algorithm is specified in
algorithm 1.

function find_S_peaks(data$\\_$ecg, R$\\_$peaks)

num$\\_$peaks $\leftarrow$ R$\\_$peaks.shape[0]

S$\\_$peaks $\leftarrow$ list()

$N$ $\leftarrow$length(num$\\_$peaks)

for $index\leftarrow 0$ to $N$ do

$i$ $\leftarrow$ R$\\_$peaks[index]

cnt $\leftarrow$ $i$

if cnt+1 $>=$ data$\\_$ecg.shape[0] then

break

end if

while data$\\_$ecg[cnt] $>$ data$\\_$ecg[cnt+1] do

cnt $\leftarrow$ cnt+1

if cnt $>=$ data$\\_$ecg.shape[0] then

break

end if

end while

S$\\_$peaks.append(cnt)

end for

return S$\\_$peaks

end function

Algorithm 1 Algorithm for determining the ECG S peaks

Fig. 5a and Fig. 5b display the location of the R and S peaks found by the
algorithm described above on the filtered signal and the corresponding raw
signal, respectively.

(a)

(b)

Figure 5: Positions R and S peaks are detected on the original signal (a) and
on the filtered signal (b).

The effectiveness of the method for detecting R and S peaks is evaluated by
manually labeling the positions of R and S peaks, followed by a comparison
with the algorithm’s return values. We randomly labeled 200 R peaks and 200 S
peaks in both classes (with and without SA); the accuracy and F1-score of R
and S peaks detection algorithms are reported in Table 1. All the statistics
are performed on the PhysioNet Apnea–ECG dataset.

Table 1: R and S peaks detection results for filtered signal. |  | #beats
---
TP | TN | FP | FN | Acc | F1-score
R peaks |  | 200
---
196 | 0 | 0 | 4 | 98% | 98.99%
S peaks |  | 200
---
196 | 0 | 0 | 4 | 98% | 98.99%

Due to the disparity in units (amplitude in millivolts and time interval in
seconds), feature values are normalized to the normal distribution before
feeding them into the neural network. Based on the findings on study [25],
cubic interpolation yielding 900 points of each feature every 5-minute segment
is utilized as an effective data augmentation technique.

###  2.3 Performance metrics

In this study, to evaluate the performance of the machine learning classifier,
we adopt the accuracy, sensitivity, specificity and F1-score as evaluation
metrics.

  * •

Sensitivity: Sensitivity of a class is defined as the ratio of correctly
classified samples to total number of samples actually belonging to that
class.

$\text{\emph{Sensitivity}}=\frac{TP}{TP+FN}$ (1)

  * •

Specificity: Specificity is used to measure the proportion of negatives that
are correctly identified. It is defined as the ratio of true negatives
predicted to total number of samples which belong to negative class.

$\text{\emph{Specificity} }=\frac{TN}{TN+FP}$ (2)

  * •

Accuracy: It is defined as the ratio of number of correctly classified samples
to that of total samples.

$\text{\emph{Accuracy}}=\frac{TP+TN}{TP+FN+TN+FP}$ (3)

  * •

F1-score: It is the harmonic mean of the precision and recall.

$\text{\emph{F1-score}}=\frac{2TP}{2TP+FP+FN}$ (4)

where _TP_ : True Positive, _TN_ : True Negative, _FP_ : False Positive, _FN_
: False Negative.

##  3 Experiments

###  3.1 Dataset

The PhysioNet Apnea–ECG dataset used to study SA was made available by
Philipps University [15]. The dataset consists of 70 single-lead ECG records
(35 recordings from the public set and 35 from the withheld set). Each
recording ranges in length from just under 7 hours to over 10 hours and
consists of a digitized ECG signal and a set of apnea annotations, which is
derived by human experts on the basis of simultaneously recorded respiration
and related signals. The ECG signal is separated into 1-minute segments and
labeled; a signal segment is identified as having SA if an apnea event occurs
during that minute. Table 2 provides details on the number of 1-minute ECG
signal segments identified with and without SA in the training and test sets.

Table 2: Number of 1-minute ECG signal segments labeled with and without SA on training and test sets. | Training set | Test set
---|---|---
SA | 6,473 (38.74%) | 6,490 (38.30%)
Non SA | 10,236 (61.26%) | 10,455 (61.79%)
Total | 16,709 | 16,945

Previous research has demonstrated that adjacent segments provide valuable
information for SA detection [30, 25]. We adopt the sampling method in [25],
in which, each 1-minute signal segment and the surrounding 2-minute signal are
used to form a 5-minute signal segment, which is then used for preprocessing
and classification.

###  3.2 Implementation details and training methodology

After being preprocessed, the extracted features will be utilized to train the
1D CNN model. The NVIDIA GeForce RTX 3080 Ti GPU, 31 GB of RAM, and Intel Core
i9-10900X processor operating at 3.70 GHz are used in all experiments. With a
mini-batch size of 256, we train a specific model for 100 epochs, evaluating
each model after every epoch. The optimal model for each training procedure
will be the checkpoint with the highest F1-score. To assess the impact of
employing features associated with S peaks, we set up two experiments as
follows:

  1. 1.

Only R peak amplitude and RR intervals, which are features related to R peaks,
are used.

  2. 2.

Use a combination of features related to R and S peaks, including amplitudes
and intervals of R and S peaks.

The ECG signal classifier is constructed using the SE-ResNext 50 [31] model as
its backbone. The SE-ResNext 50 [32] model is a variant of ResNet 50 with the
replacement of the identity connection with a Squeeze-and-Excitation block. It
enables feature recalibration, allowing the network to learn how to use global
information to selectively emphasize informative characteristics and suppress
less helpful ones. This model has demonstrated its effectiveness with 1D data
and ECG signals in particular [33].

##  4 Results and Discussion

###  4.1 Model performance

As previously noted, in order to test the efficacy of using S peak-related
features, we conducted experiments with and without these features. Table 3
shows the classification results of the model employing solely R-related
features versus the model employing all types of features. Our approach (RR
intervals + R Amplitude) reports an accuracy of 90.28%, a specificity of
90.44%, and a sensitivity of 90.00%. In terms of F1-score, we report an
F1-score of 86.85 (for SA class) and 92.30 for non SA class. According to the
results, using features associated to the S peak enhances the F1-score by up
to 1.5% in the SA class and 0.54% in the Non-SA class. That finding indicates
that the characteristics associated with the S peak may represent an
alteration in the ECG signal of individuals with SA and may contribute to the
capacity to classify SA and Non-SA.

###  4.2 Comparison to state-of-the-art

Table 4 show the comparison between the proposed method and current state-of-
the-art approaches to SA detection. We show that our approach surpassed almost
all of the competitors, with the largest disparity in accuracy is up to 9.1%.
These results indicate the robustness of the propose method.

Table 3: Experimental results using different features. Results
---
| Feature
---
combination
Classifier | Acc (%) | Spe (%) | Sen (%) | F1-score (%)
|  |  |  |  | SA |  | Non
---
SA
| RR intervals
---
R Amplitude
| SE
---
ResNext
50
90.28 | 90.44 | 90.00 | 86.85 | 92.30
| RR intervals
---
R Amplitude
SS intervals
S Amplitude
| SE
---
ResNext
50
91.13 | 92.58 | 88.75 | 88.35 | 92.84

Table 4: Comparison with state-of-the-art approaches.

Comparison
---
Study | Classifer | Accuracy | Sensitivity | Specificity
| Li
---
et al.[24]
| Decision
---
fusion
83.80% | 88.90% | 88.40%
| André Pinho
---
et. al.[22]
ANN | 82.12% | 88.41% | 89.10%
| Mahsa Bahrami
---
et. al.[25]
LeNet-5 | 87.6% | 83.10% | 90.30%
| Shen
---
et. al. [18]
| 1DCNN
---
WLTD
89.40% | 89.80% | 89.10%
| Mahsa Bahrami
---
et. al. [17]
| ZFNet
---
GRU
88.13% | 84.26% | 92.27%
| Singh
---
et. al. [21]
Scalogram | 86.22% | 90.00% | 83.82%
| Kaicheng Feng
---
et. al. [16]
FSSAE | 85.10% | 86.20% | 84.40%
Ours [a] |  | SE-ResNext
---
50
90.28% | 90.44% | 90.00%
Ours [b] |  | SE-ResNext
---
50
91.13% | 92.58% | 88.75%

a: The model solely uses features relating to R peaks (RR intervals and R
amplitude).
b: The model uses additional S-peak-related features (RR intervals, R
amplitude, SS intervals and S amplitude).

##  5 Conclusions

In this study, we introduce a novel method for detecting sleep apenea based on
SS intervals and S amplitudes. Our experimental results demonstrate that the
proposed technique is effective for SA detection, and its performance
outperforms state-of-the-art works. Although showing a high-level of
performance, our approach has several disadvantages. For example, the lack of
access to SA patients’ ECG data and the large volume of data are the primary
limitations of this study. Although proposed method has been tested with a
number of experimental setups, it should be evaluated on a larger dataset.

In the future, we expect to integrate a SA detection module into a mobile
application, which will alert patients suffering from SA to wake them up.
Additionally, we also consider muti-model learning approaches [34], e.g.
integrating demographic data and clinical data [35] to boost model
performance. An example can be the image and encoded text (in the form of a
graph neural network or a knowledge graph) fusion for better representation
learning [36, 37, 38, 39]. Not only that, we will use the explainable method
to indicate which segments are at risk of disease and improve the accuracy of
the model [40]. Specifically, we will highlight which segments of the ECG
signal are suspect in the ECG signal obtained using an eXplainable. As a
result, this application may save the patient’s life by restoring normal
breathing and consciousness without complex setups.

##  6 Acknowledgement

The authors would like to thank the VinUni-Illinois Smart Health Center and
the source of funding : VinUni Seed Grant 2020. For more information, please
visit the website at https://vinuni.edu.vn/vinuni-research/seed-funding-
program/. Beside, this work was also funded by Vingroup Joint Stock Company
(Vingroup JSC), Vingroup, and supported by Vingroup Innovation Foundation
(VINIF) under project code VINIF.2021.DA00128.

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