11institutetext: The Institute of Mathematical Sciences, CIT Campus, Chennai
600113, India. 11email<EMAIL_ADDRESS>22institutetext: Homi Bhabha
National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085,
India. 33institutetext: Department of Physics and Astronomy, University of
Kentucky, KY, USA.33email<EMAIL_ADDRESS>
# Effect of damped oscillations in the inflationary potential

Akhil Antony  1122 Shweta Jain  33

###### Abstract

We investigate the effect of damped oscillations on a nearly flat inflationary
potential and the features they produce in the power-spectrum and bi-spectrum.
We compare the model with the Planck data using Plik unbinned and CamSpec
clean likelihood and we are able to obtain noticeable improvement in fit
compared to the power-law $\Lambda$CDM model. We are able to identify three
plausible candidates each for the two likelihoods used. We find that the best-
fit to Plik and CamSpec likelihoods match closely to each other. The
improvement comes from various possible outliers at the intermediate to small
scales. We also compute the bi-spectrum for the best-fits. At all limits, the
amplitude of bi-spectrum, $f_{NL}\,$is oscillatory in nature and its peak
value is determined by the amplitude and frequency of the oscillations in the
potential, as expected. We find that the bi-spectrum consistency relation
strictly holds at all scales in all the best-fit candidates.

###### Contents

  1. 1 Introduction
  2. 2 Model and data
  3. 3 Results
    1. 3.1 Best-fit Candidates
    2. 3.2 Scalar power-spectrum
    3. 3.3 Scalar bi-spectrum
  4. 4 Conclusion
  5. A Theory of inflation
  6. B Scalar power-spectrum
  7. C Scalar bi-spectrum

##  1 Introduction

Over the last three decades, tremendous advances in precision cosmology have
aided our understanding of the early universe. The Standard Model (SM) has
emerged as the most successful model for describing the evolution of the
universe owing to the support of numerous precise observations. While SM
enjoys widespread acceptance, it does have a few drawbacks, such as horizon
problem, flatness problem etc. Among the numerous candidate theories for the
early universe, inflation Starobinsky:1979ty ; Starobinsky:1980te ; Guth ;
Sato ; Mukhanov ; Linde:1981mu ; Linde  has proven to be the best contender to
account for these problems. Additionally, inflation has been able to account
for the dynamics of primordial fluctuations that seeded the formation of large
scale structures today Kolb:1990vq ; liddle_lyth_2000 . The imprints of these
primordial fluctuations can be best identified in the Cosmic Microwave
Background Radiation (CMB).

From COBE COBE ; COBEnormalization  to the PLANCK 2020 ;
planckcollaboration2020planck ; planckcollaboration2019planck  mission, we
have made significant progress in our understanding of CMB physics. With the
help of CMB data, inflation has emerged as the most promising candidate for
describing the near homogeneous and isotropic nature of the Universe over the
large scale. Numerous inflationary models exist that appear to be consistent
with the CMB observations, in which a near flat potential generates a nearly
scale invariant spectrum of scalar perturbations. While these models appear to
be consistent with the observational data, it has been noted that adding a few
features to these flat power spectra may result in a better fit to the data
Starobinsky:Kink ; Ivanov:1994pa ; Adams:Step0 ; Covi:Step1 ; Allahverdi:PI ;
Ashoorioon:Step2 ; Joy:2007na ; Joy:2008qd ; Jain:PI ; Pahud:Osc2 ;
Hazra:Step3 ; McAllister:Osc0 ; Flauger:Osc1 ; Miranda:Step4 ; Aich:Osc3 ;
Peiris:Osc4 ; Benetti:Step5 ; Meerburg:Osc5 ; Easther:Osc6 ; Bousso:Step8 ;
GallegoCadavid:Step6 ; Chluba:Step7 ; Motohashi:Osc7 ; Miranda:Osc8 . These
additional features are found to fit a few consistent outliers that are being
observed in the data for decades Hannestad_2001 ; Tegmark_2002 ;
Shafieloo_2004 ; Hazra_2013 ; Nicholson_2009 ; Nicholson_2010 ; SL_Bridle ;
Shafieloo_2007 ; Shafieloo_2008 ; Hazra:recon13broad ; Hazra:reconP13 ;
Hunt:2015iua ; Obied:2018qdr ; Braglia:2020fms ; Braglia:2021ckn ;
Hazra:2021eqk ; Braglia:2021rej ; Braglia:2021sun . The persistent existence
of these outliers could be owing to some unknown features of inflationary
dynamics, which could expand our understanding of the early universe. As a
result, it is critical to investigate about these features in the power
spectrum. We provide a single field canonical inflationary potential in this
study that simulates these extra features using damped sinusoidal
oscillations. A noteworthy aspect of this feature is that it could generate
both sharp and resonant featured oscillations, as well as their combinations
Chen:2016zuu . Primordial standard clock Chen:2014cwa  is one model that can
generate a combination of sharp and resonant spectral features, but it is a
two field inflationary model. It is shown in Antony:2022ert  that these
features could account for the additional smoothing in the CMB temperature
spectrum, thus resolving the $A_{lens}$ anomaly. It increases $H_{0}$ while
decreasing $S_{8}$. Additionally, the model fits the One spectrum
Hazra:2022rdl , which resolves many tensions and anomalies in the Planck data.
In this study, we examine alternative best fit candidates for this feature, in
addition to the one mentioned in Antony:2022ert , by analysing the entire
$k$-space. Using Planck CMB data, we identify these features in the primordial
power spectrum using this range of spectra. Later, we will demonstrate that
these features could account for some of the in outliers CMB data that are not
captured by the power-law form of the primordial power-spectrum. We examine a
few possible candidates, sharp featured and resonant featured, that improve
the fit to CMB data noticeably.

This paper is structured as follows. In section 2, we introduce the potential
form and discuss the methodology of work as well as the various datasets used
for it. The results and discussion of various best-fit candidates are provided
in section 3. Finally, we discuss the conclusions and inferences drawn from
the work in section 4. Appendices contain the analytical calculations used in
the work. We used natural units throughout the paper, $\hbar=c=M_{pl}=1$. In
this paper, we use the metric signature (-,+,+,+). For formalising the
equations, we used three sets of coordinates for time, namely cosmic time
$(t)$, conformal time $(\eta)$, and the number of e-folds $({\mathcal{N}})$.
We work in an expanding homogeneous and isotropic universe whose metric is
given by the Friedmann Lemaitre Robertson Walker (FLRW) metric having
perturbations up to linear order. An over-dot and an over-prime denote
differentiation in terms of $t$ and $\eta$, respectively. Differentiation with
respect to ${\mathcal{N}}$ is given by a subscript ${\mathcal{N}}$, i.e.
$f_{{\mathcal{N}}}=\mathrm{d}f/\mathrm{d}{\mathcal{N}}$.

##  2 Model and data

In this paper, we investigate the dynamics of a single canonical scalar field
in an inflationary potential. Our potential is divided into two parts: a
baseline with a slow roll regime where we could begin the inflation, and a
small feature in the form of damped cosine oscillation. The nearly scale
invariant (slightly red titled) power-spectrum is obtained from the baseline
part of the potential. We add features in this baseline potential to capture
extra possible signals in CMB data. The feature we propose could be added to
any baseline potential that could produce a near scale invariant power
spectrum. This was verified using a couple of different base potentials. One
can also directly add features to the power-spectrum and estimate parameters,
but in such cases, it may be difficult to obtain an expression for potential
back from the spectra. Therefore, we work with features in the potential
itself.

We consider the potential to be of the following form,

${V(\phi)=\gamma^{2}\phi\cosh{(\beta_{1}\phi)}+\frac{\alpha\cos{(\omega_{\phi}\phi)}}{[\beta_{2}(\phi-\phi_{0})]^{2}+C}.}$
(1)

Here $\beta_{1}$ and $\beta_{2}$ are fixed and $C$ is an arbitrary constant
added to avoid the divergence at $\phi=\phi_{0}$. It can be any value other
than $0$. The power-spectrum tilt is controlled by $\beta_{1}$. We fix
$\beta_{1}$ by performing a parameter estimation with only baseline potential
parameters, i.e. by varying only two parameters $\gamma$ and $\beta_{1}$. We
allow $\beta_{1}$ to vary between values that result in an approximate tilt of
$0.94-0.98$. This analysis done using Plik bin1. The best-fitting value for
$\beta_{1}$ results in a tilt of $\sim 0.96$. $\beta_{2}$ regulates the amount
of damping applied to the oscillations, ensuring that features remain
localised. We can see from Figure 1 that using the above potential, we can
produce both sharp ($\omega_{\phi}\rightarrow 0$) and resonant features
($\beta_{2}\rightarrow 0$)in the power-spectrum.

Figure 1:  This figure shows two types of features that can be generated by
our model in different limits. Here, the red curve is the sharp featured
oscillation whose peaks are separated linearly in k and purple curve
corresponds to the resonant features whose peaks are separated uniformly in
$\log k$. Note that the x-axis is in $\log$ scale here.

We vary four parameters in the full potential function, one in the baseline
part and three in the feature part. The effect of each parameter on the power-
spectrum could be understood from Figure 2.

Figure 2: Variation of the power spectrum in response to changes in potential
parameters. Here, we varied one parameter by a specific step size while
holding the other three constant. In the upper right corner of each panel, the
varying parameter is specified.

Potential, as described in Equation 1, contains sinusoidal oscillations which
damps as we move both sides from $\phi_{0}$. Assuming a slow roll dynamics
during the initial phase of inflation, given an initial value for the
field($\phi_{i}$), one can obtain other initial conditions required to
calculate the background dynamics. Here we are working with number of e-folds
(${\mathcal{N}}$), defined as $a=a_{i}\mathrm{e}^{\mathcal{N}}$ where $a$ is
the scale factor. Initially, the potential term dominates the kinetic term
therefore one can approximate Hubble parameter as, $3H^{2}\approx V(\phi)$.
Therefore, $(\phi_{\mathcal{N}})_{i}$ for a given $\phi_{i}$ evaluates to be,

$\displaystyle H_{i}=\sqrt{V(\phi_{i})/3},$ (2a)
$\displaystyle{\phi_{\mathcal{N}}}_{i}=-\frac{V_{\phi}(\phi_{i})}{3H_{i}}.$
(2b)

Once we have the initial conditions we solve for the background equations and
get the form of $\phi$, $\phi_{\mathcal{N}}$, and $H$ as a function of
e-folds(${\mathcal{N}}$). We do all these calculations using the FLRW metric.
With aforementioned quantities, we can get the slow roll parameter,
$\epsilon({\mathcal{N}})$, and calculate the end of
inflation($\epsilon({\mathcal{N}}_{end})=1$). To find initial value of scale
parameter, $a_{i}$, we impose that the mode $k=0.05~{}{\rm Mpc^{-1}}$ crosses
the horizon at ${\mathcal{N}}={\mathcal{N}}_{end}-50$ Hamann:2007pa . Once the
background is evaluated completely, we add the perturbations to the fields and
solve for the curvature perturbation as discussed in Appendix A

The field continues to be in the slow roll phase till it feels the effect of
oscillations. Once the effect of oscillations take over, the field accelerates
into an intermediate fast-roll phase which is responsible for the features in
power-spectrum. Inflation continues as the field rolls further down the
potential till $\epsilon$ becomes one. Assuming Bunch-Davies initial condition
Bunch-Davies , one solves for the curvature perturbation (${\mathcal{R}}$),

${\mathcal{R}}_{k}^{\prime\prime}+2\bigg{(}\frac{z^{\prime}}{z}\bigg{)}{\mathcal{R}}_{k}^{\prime}+k^{2}{\mathcal{R}}_{k}=0,$
(3)

and get the power-spectrum from

${\mathcal{P}}_{s}(k)=\frac{k^{3}}{2\pi^{2}}\mathinner{\\!\left\lvert{\mathcal{R}}_{k}\right\rvert}^{2}.$
(4)

One can also use the Mukhanov-Sasaki equation (Equation 13) to get the power-
spectrum.

We calculate the primordial power-spectrum numerically with the help of
publicly available code BINGO Hazra_2013_BINGO . BINGO solves for
${\mathcal{R}}$ for each $k$ to get ${\mathcal{P}}_{s}$ as a function of $k$.
Technically, one needs to integrate curvature perturbation throughout the
inflationary epoch. But it can be safely approximated to a region from
${\mathcal{N}}_{i}$ deep inside Hubble radius($k\gg aH$) and to an
${\mathcal{N}}_{e}$ well outside the Hubble radius($k\ll aH$). This could be
calculated using the following conditions:

$\displaystyle k=$
$\displaystyle{C}_{IC}a({\mathcal{N}}_{i})H({\mathcal{N}}_{i}),$ (5a)
$\displaystyle k=$
$\displaystyle{C}_{SHS}a({\mathcal{N}}_{e})H({\mathcal{N}}_{e}).$ (5b)

For each mode, we calculate the ${\mathcal{N}}_{i}$ and ${\mathcal{N}}_{e}$
using appropriate choice of ${C}_{IC}$ and ${C}_{SHS}$ values Salopek:1988qh .
We fix $C_{SHS}$ but we do vary $C_{IC}$ depending on the oscillations present
at a given scale, i.e. for modes near high frequency oscillations in power-
spectrum, we use a larger value for $C_{IC}$. Typically, it’s value is 200.

We incorporate BINGO into CAMB Lewis_2000 ; CAMB  and calculate the angular
power-spectrum using the Boltzmann equations. We run Markov Chain Monte Carlo
(MCMC) using CosmoMC Lewis_2002 ; COSMOMC  and identify regions in parameter
space which gives an improvement in fit. Starting from the aforementioned
regions, we then run BOBYQA BOBYQA  to obtain the best-fit values for the
parameters.

We performed our analysis using the Planck mission’s most recent CMB
temperature and polarisation anisotropy data. Planck was able to map the CMB
sky over a wide range of multipoles ($\ell=2-2500$) on both small ($\ell\geq
30$) and large ($\ell=2-29)$ scales. We use two sets of likelihood for
high-$\ell$ in our analysis: Plik-bin1-TTTEEE Planck:2019nip ; 2020  and
CamSpec-v12-5-HM-cln-TTTEEE efstathiou2020detailed . We use commander-
dx12-v3-2-29 for low-$\ell$ TT and simall-100x143-offlike5-EE-Aplanck-B for
low-$\ell$ EE. Plik-bin1-TTTEEE represents the completely unbinned TTTEEE
likelihood, and CamSpec represents the newly cleaned CamSpec.
CamSpec-v12-5-HM-cln-TTTEEE employs a sophisticated data analysis pipeline to
generate an improved CamSpec likelihood and also an increased sky fraction for
temperature and polarisation. In our study, we vary the nuisance parameters in
addition to the background parameters, and we include the priors involved, as
indicated in the Planck 2018 and CamSpec likelihood papers 2020 ;
efstathiou2020detailed . The prior used for the potential parameters is given
in Table 1. We perform the analysis across multiple parameter templates and
narrow it down to three best-fit candidates for each likelihood. The following
section will discuss these three candidates in greater detail.

Parameters |  Prior
---|---
$\gamma(\times 10^{4})$ |  0.14 - 0.20
$\alpha(\times 10^{11})$ |  -2.0 - 2.0
$\phi_{0}$ |  12.9 - 13.9
$\omega_{\phi}$ |  0 - 529
Table 1: The table contains the prior range we used for the inflationary
potential parameters CamSpec-v12-5-HM-cln-TTTEEE
Low-T (commander-dx12-v3-2-29)
Low-E (simall-100x143-offlike5-EE-Aplanck-B) |  CamSpec clean(CS)
---|---
Plik-bin1-TTTEEE
Low-T (commander-dx12-v3-2-29)
Low-E (simall-100x143-offlike5-EE-Aplanck-B) |  Plik-bin1(Plik)
Table 2: Short hands used for the combination of Planck likelihoods used for
the analysis. We have used same low $\ell$ (lowT+lowE) likelihoods for both
the sets.

##  3 Results

The following is the nomenclature used to identify the candidates: The
likelihood against which it is tested followed by the candidate number. For
example, Plik-1 specifies that it’s the first candidate analysed against Plik
likelihood.

###  3.1 Best-fit Candidates

We explored the parameter space using MCMC algorithm and were able to identify
multiple regions that could improve the fit to data (Figure 3). We perform the
analysis using Plik-bin1. To identify these regions, we study various points
that lie within $\Delta\chi^{2}=1$ region from the global minima of $\chi^{2}$
value of the MCMC run using BOBYQA. While performing the BOBYQA analysis we
make use of both Plik-bin1 and CamSpec clean likelihoods. Using BOBYQA
analysis, we are able to identify three candidates that gave improvement in
fit for each likelihood. Those are Plik-1, Plik-2, Plik-3 (Plik candidates)
and CS-1, CS-2, CS-3 (CS candidates). We also calculated the bayesian evidence
using MCEvidence2017arXiv170403472H  python package. The evidence only
provided a 0.3-factor weak support for the baseline model. But according to
Jeffrey’s scale, this is an inconclusive evidence.

Figure 3:  The figure contains the triangle plot for the feature model
parameters corresponding to Plik bin1. Here we can see multiple region in
parameter space which could lead improved fit to data using different
combination of these parameters.

The best-fit values for the Plik runs are presented in Table 3. We
investigated three candidates for the Plik-runs based on the $\chi^{2}$
improvement we obtained from the BOBYQA runs. We are comparing it to the
$\chi^{2}$ obtained from the power-law form of the primordial spectrum
(referred to as the power-law model from now on), which has a
$\chi^{2}=24548.5$ value. We can see from Table 3 that we get 10, 8.5, and 6
improvement for the candidates Plik-1, Plik-2, and Plik-3, respectively. The
residual plot from Plik-runs with respect to the power-law model is shown in
Figure 4. The power-law model is represented by the zero line, and the
coloured lines are candidates. We can observe slight power suppression in the
case of Plik-3. Here the improvement comes mainly from the high-$\ell$, 4.3
while the low $\ell$ gives around 1.45 improvement. Plik-3 also gave 2. The TT
residual plot is able to capture the outliers in the range $\ell=1000-1500$,
while the EE and TE residuals could capture comparatively lower multipoles
ranging from 140 to 600. The exact outlying multipole values captured by the
Plik-candidates are given by the Table 5.

Parameters |  Plik-1 |  Plik-2 |  Plik-3
---|---|---|---
$\Omega_{b}h^{2}$ |  0.0223 |  0.0223 |  0.0222
$\Omega_{c}h^{2}$ |  0.1200 |  0.1204 |  0.1207
$100\theta_{MC}$ |  1.0409 |  1.0408 |  1.0409
$\tau$ |  0.0560 |  0.0585 |  0.0510
$\gamma(\times 10^{4})$ |  0.1724 |  0.1729 |  0.1716
$\alpha(\times 10^{11})$ |  -0.1271 |  0.6378 |  -1.223
$\phi_{0}$ |  12.97 |  12.99 |  13.20
$\omega_{\phi}$ |  286.96 |  3.13 |  132.02
$-2\log(\mathcal{L})$ |  24538.47 |  24539.98 |  24542.57
Table 3: Best-fit parameters and likelihood obtained for the candidates of the
Plik-bin1 TTTEEE+lowT+lowE likelihood. First four are the $\Lambda$CDM
background parameters and the next four are the inflationary potential
parameters. Final row gives the $\chi^{2}$ values obtained by the candidates.
Improvement in fit obtained are 10, 8.5, and 6 respectively for the candidates
Plik-1, Plik-2, and Plik-3. Figure 4: Residual plots corresponding best-fit
candidates to Plik-bin1 TTTEEE+lowT+lowE likelihood. Here residual is
calculated from the power-law $\Lambda$CDM model. Zero line corresponds to the
power-law $\Lambda$CDM model and data points are from the 2018 Planck Plik
data residual to the power law best fit.

The best-fit values for the CS runs are given in Table 4. The $\chi^{2}$ value
for power-law primordial power-spectrum is $10211.3$, i.e. the improvement for
candidates CS-1, CS-2, CS-3 are $5.0,3.8$ and $3.7$ respectively. Figure 5
contains the residual plots for CS runs with respective to the power-law
model. Similar to Plik-candidates, residual plot of CS candidates for TT also
captures the outliers in large $\ell$ values and EE captures for $\ell\sim
400$. The complete list $\ell$ values of the outliers captured by the CS
candidates are given in Table 5.

Parameters |  CS-1 |  CS-2 |  CS-3
---|---|---|---
$\Omega_{b}h^{2}$ |  0.0222 |  0.0222 |  0.0221
$\Omega_{c}h^{2}$ |  0.1205 |  0.1200 |  0.1208
$100\theta_{MC}$ |  1.0406 |  1.0410 |  1.0409
$\tau$ |  0.0545 |  0.0602 |  0.0524
$\gamma(\times 10^{4})$ |  0.1723 |  0.1728 |  0.1717
$\alpha(\times 10^{11})$ |  -0.1021 |  0.6394 |  -1.224
$\phi_{0}$ |  12.97 |  12.99 |  13.21
$\omega_{\phi}$ |  294.8 |  4.05 |  132.25
$-2\log(\mathcal{L})$ |  10206.29 |  10207.51 |  10207.55
Table 4: Best-fit parameters and likelihoods obtained for the candidates of the CamSpec clean TTTEEE+lowT+lowE likelihood. First four are the $\Lambda$CDM background parameters and the next four are the inflationary potential parameters. Final row gives the $\chi^{2}$ values obtained by the candidates. Improvement in fit obtained are 5, 3.8, and 3.7 respectively for the candidates CS-1, CS-2 and CS-3. Figure 5: Residual plots corresponding best-fit candidates to CamSpec clean TTTEEE+lowT+lowE likelihood. Here residual is calculated from the power-law $\Lambda$CDM model. Zero line corresponds to the power-law $\Lambda$CDM model and data points are from the 2018 Planck CamSpec data residual to the power law best fit. RUNS | TT | EE | TE
---|---|---|---
Plik-1 | 500, 520 | 60, 420, 440 | 500
Plik-2 | 500,1080,1340,1580 | 60, 140, 440 | 160, 200,240,260,500
Plik-3 | 1080, 1340 | 60 | 240
CS-1 | 520, 960, 1240, 1400 | 420 | 640
CS-2 | 1000, 1140,1420,1520 | 420 | -
CS-3 | - | 340 | -
Table 5: Outliers addressed by the Plik-bin1 (first three) and CamSpec clean
(last three) candidates from the residual plots of TT, EE and TE correlation
in Figure 4 and Figure 5.

###  3.2 Scalar power-spectrum

We present here the local and global best-fits to the data. We saw an
improvement in $\chi^{2}$ at three different points in the parameter space.
One for low frequency, one for intermediate frequency, and one for high
frequency oscillations. CS-1 and Plik-1, which have a high frequency, are the
global best-fit to the data for both CamSpec clean and Plik-bin1 likelihoods.
They are both located in very close proximity in the parameter space. Plik-1,
the best-fitting Plik candidate, has features at a smaller scale, $k\sim
4\times 10^{-2}~{}{\rm Mpc^{-1}}$, whereas Plik-3 has the features at slightly
larger scales that end near $k\sim 10^{-2}~{}{\rm Mpc^{-1}}$. Plik-2 have
features ranging from $k\sim 10^{-3}~{}{\rm Mpc^{-1}}$ to $k\sim 5\times
10^{-2}~{}{\rm Mpc^{-1}}$. We saw a similar pattern with the CamSpec clean
candidates, which are very close to the Plik candidates in the parameter space
and in the same order. The power-spectrum for both sets of candidates has been
plotted in Figure 6 and Figure 7. In Figure 8, a comparison of the global
best-fit for two sets of candidates is given. Both are found in the same
location, and the amplitude and frequency of the oscillations are comparable.

Figure 6: power-spectrum for the best-fit candidates to the Plik-bin1
likelihood Plik-1, Plik-2 and Plik-3. Figure 7: power-spectrum for the best-
fit candidates to the CamSpec clean likelihood CS-1, CS-2 and CS-3.  Figure 8:
power-spectrum for the global best-fit of both CamSpec clean (CS-1) and Plik-
bin1 (Plik-1) likelihood.

###  3.3 Scalar bi-spectrum

Non-Gaussianity in canonical inflationary models that are completely governed
by slow roll dynamics is negligible Maldacena_2003 ; Seery:2005wm . Deviations
from the slow roll nature, on the other hand, produce scale dependent
oscillations in the $f_{NL}(k)$ Chen_2008 ; Chen:Osc00 ; Chen:2010xka ;
Flauger:2010Osc ; Chen:2010Osc ; Martin:2011sn ; Hazra_2013_BINGO ;
Hazra:2012kq ; planck2013_NG ; Adshead:2013zfa ; Achucarro:2013cva ;
planck2015_NG ; Sreenath_2015 ; Martin:2014kja ; Achucarro:2014msa ;
Fergusson:2014hya ; Meerburg:2015owa ; Appleby:2015bpw ; Dias:2016rjq ;
planckcollaboration2019planck . As a result, features in our candidates
produce a significant oscillatory bi-spectrum. In this section, we compute the
$f_{NL}$ for each of the six candidates. We are evaluating bi-spectrum in
three limits: equilateral ($k_{1}=k_{2}=k_{3}$), squeezed ($k_{1}\approx
k_{2}\gg k_{3}$), and scalene (arbitrary triangular configuration). In all
three limits, we use BINGO to evaluate the $f_{NL}$. To calculate the
$f_{NL}$, we use the same method described in subsection 3.2 and substitute in
Appendix C. Figure 9 displays the $f_{NL}$ for all six candidates in the
equilateral limits. One can show that in the squeezed limit $f_{NL}$ reduces
to

$f_{NL}=\frac{5}{12}\left(n_{s}-1\right).$ (6)

This relation is called the consistency condition Maldacena_2003 ; Creminelli
. Figure 10 verifies the consistency relation for all six candidates. The
numerically calculated $f_{NL}$ matches well with the analytical result. In
the scalene limit, we obtain the 2D heat map of
$f_{NL}(k_{1}/k_{3},k_{2}/k_{3})$ by fixing the value of $k_{3}$
Hazra_2013_BINGO . Figure 11 plots the 2D heat map of $f_{NL}$. Top left
corner of 2D map can be identified as the squeezed limit, i.e. $k_{2}=k_{3}\gg
k_{1}$ and the top right corner is the equilateral limit, i.e.
$k_{1}=k_{2}=k_{3}$.

One can locate the maximum non-Gaussianity of three point correlation in
equilateral limits (Figure 9). Here, CS-2 and Plik-2 generates a
$f_{NL}\,$$\sim 6$. This is because of the low frequency oscillations present
in the potential and thereby in the scalar power-spectrum. CS-3 and Plik-3
generates highest $f_{NL}\,$ amongst these candidates which is around $\sim
33$. This is due to the presence high amplitude and relatively larger
frequency of oscillations present in the potential. Even though Plik-1 and
CS-1 have large frequency oscillations, their amplitude is small compared to
other four candidates which results in a $f_{NL}\,$$\sim 19,15$ respectively.

Figure 9: Scalar bi-spectrum evaluated in the equilateral limit for the best-
fit candidates of the both Plik-bin1 (left side) CamSpec clean (right side)
likelihood.  Figure 10: Verifying consistency relation (Equation 6) for the
best-fit candidates to the Plik-bin1 (left side) and CamSpec clean (right
side) likelihood. Figure 11: 2D heat map of $f_{NL}(k_{1}/k_{3},k_{2}/k_{3})$
for the best-fit candidates to the Plik-bin1 (left side) and CamSpec clean
(right side) likelihood. $f_{NL}$ values are given in the color bar. Here
$k_{3}$ is the mode which gave the maximum $f_{NL}$ in the equilateral limit
for the corresponding candidate.

##  4 Conclusion

We have studied the effect of damped oscillations in a nearly flat
inflationary potential and compared the spectra with Planck 2018 data. Our
model is able to identify resonant features and sharp feature separately at
different scales. We are able see that at smaller scales data prefers resonant
features but at the intermediate scales sharp oscillations give better fit. We
have separately used two high $\ell$ likelihoods, namely Plik-bin1 and CamSpec
clean, for the analysis. With the addition of 3 parameters, we are able to get
around $10$ and $5$ improvements for Plik-bin1 and CamSpec clean likelihood
respectively. The Bayesian analysis didn’t give any conclusive evidence for
the model. We have studied three candidates each for both Plik-bin1 and
CamSpec clean likelihoods. They have provided $10,8.5,6$ improvements for the
Plik-bin1 and $5,3.8,3.7$ for the CamSpec clean. This indicates that the
extent of improvement is less in the CamSpec clean compared to the Plik-bin1
for all candidates. While the first candidate has features located at the
smaller scales ($10^{-2}-10^{-1}~{}{\rm Mpc^{-1}}$), the third one has
oscillations at large to intermediate scales ($10^{-3}-10^{-2}~{}{\rm
Mpc^{-1}}$). The second candidate has features at the intermediate to small
scales. Owing to the feature location for different candidates, different
outliers are captured in the $\ell$ space. Scalar bi-spectrum, $f_{NL}\,$, is
evaluated in the squeezed, equilateral limits and also in the scalene
configurations. All candidates provide oscillating $f_{NL}\,$with the maximum
amplitude reaching up to 33. The first and third candidate have produced
relatively higher $f_{NL}\,$amplitude, $\sim 17,33$, while the second one have
generated a maximum $f_{NL}\,$of $\sim 6$ in the equilateral limit. In
squeezed limits, the consistency condition is satisfied at all scales for all
three candidates of both likelihoods. Along with the constraints on
$f_{NL}\,$, we can further narrow down the possible candidates for inflation
in future. The feature candidates have overlap with scales that future Large
Scale Structure (LSS) probes Euclid ; LSST ; Hazra:2012LSS ; Chen:2016LSS ;
Ballardini:2016LSS ; LHuillier:2017lgm ; Beutler:2019LSS ; Ballardini:2019LSS
; Debono20  can explore. Therefore, a joint analysis can help in understanding
the significance of these features better.

## Acknowledgements

We would like to thank Dhiraj Kumar Hazra for the guidance he has given
throughout the whole work. We would also like to thank The Institute of
Mathematical Sciences (IMSc) for providing all resources needed for the
project. All the runs related to this work were done on IMSc’s High
Performance Computing facility Nandadevi (hpc.imsc.res.in).

##  Appendix A Theory of inflation

One could engineer an accelerated expansion of the universe very shortly after
big-bang with the help of a single canonical scalar field called inflaton,
$\phi(t)$, moving in a potential $V(\phi)$. The action governing such a scalar
field is given by

${\mathcal{S}}[\phi]=\int\mathrm{d}^{4}x\sqrt{-g}\big{[}X-V(\phi)\big{]},$ (7)

where $X=-\frac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi$.

Apart from a few simple potential forms, it is not easy to solve the
Einstein’s equations analytically. In such situations, one can resort to
numerical methodologies. For numerical analysis, a better choice of coordinate
will be the number of e-folds denoted by ${\mathcal{N}}$. It tells number of
e-fold times universe expanded in a given cosmic time, i.e.
$a=a_{i}\mathrm{e}^{\mathcal{N}}$. The relation between $t$ and
${\mathcal{N}}$ is given by

$d{\mathcal{N}}=Hdt,$ (8)

where $H$ is the Hubble’s parameter in $t$. Evaluating the Einstein’s equation
corresponding to the FLRW universe (also called Friedmann equations) and the
equation of motion of the field, we get

$\displaystyle H^{2}=$
$\displaystyle\frac{V(\phi)}{3-\phi_{\mathcal{N}}^{2}/2},$ (9a)
$\displaystyle\frac{H_{\mathcal{N}}}{H}=$
$\displaystyle-\frac{\phi_{\mathcal{N}}^{2}}{2},$ (9b)
$\phi_{{\mathcal{NN}}}+\phi_{\mathcal{N}}\Big{(}3+\frac{H_{\mathcal{N}}}{H}\Big{)}+\frac{V_{\phi}}{H^{2}}=0,$
(10)

where subscript $\phi$ denotes the differentiation with respect to $\phi$.

With the help of background equations and linear perturbation theory (Appendix
B), we get the governing equation for the curvature perturbation modes as,

${\mathcal{R}}_{k}^{\prime\prime}+2\bigg{(}\frac{z^{\prime}}{z}\bigg{)}{\mathcal{R}}_{k}^{\prime}+k^{2}{\mathcal{R}}_{k}=0,$
(11)

where $z=a\phi^{\prime}/{\mathcal{H}}$. ${\mathcal{H}}$ is the Hubble
parameter in $\eta$ coordinate. Substituting ${\mathcal{V}}={\mathcal{R}}z$
Sasaki:1986hm ; Mukhanov:1987pv ; MUKHANOV1992203 ; Stewart_1993  one can
obtain the Mukhanov-Sasaki equation.

${\mathcal{V}}_{k}^{\prime\prime}+\bigg{[}k^{2}-\Big{(}\frac{z^{\prime\prime}}{z}\Big{)}\bigg{]}{\mathcal{V}}_{k}=0.$
(12)

Since we will be working in ${\mathcal{N}}$ coordinate, these equations can be
rewritten as,

$\displaystyle{R_{k}}_{{\mathcal{NN}}}+\Big{(}1+\frac{H_{\mathcal{N}}}{H}+2\frac{z_{\mathcal{N}}}{z}\Big{)}{R_{k}}_{\mathcal{N}}$
(13a) $\displaystyle+\frac{k^{2}}{a^{2}H^{2}}R_{k}=0,$
$\displaystyle{{\mathcal{V}}_{k}}_{{\mathcal{NN}}}+\Big{(}1+\frac{H_{\mathcal{N}}}{H}\Big{)}{{\mathcal{V}}_{k}}_{\mathcal{N}}+\bigg{[}\frac{k^{2}}{a^{2}H^{2}}$
(13b)
$\displaystyle-\frac{z_{{\mathcal{NN}}}}{z}-\frac{z_{\mathcal{N}}}{z}\Big{(}1+\frac{H_{\mathcal{N}}}{H}\Big{)}\bigg{]}{\mathcal{V}}_{k}=0.$

Quantizing the above equation, one can calculate the power-spectrum and is
given by the expression,

${\mathcal{P}}_{s}(k)=\frac{k^{3}}{2\pi^{2}}\mathinner{\\!\left\lvert{\mathcal{R}}_{k}\right\rvert}^{2}.$
(14)

Finally one can calculate the angular power-spectrum from the primordial
power-spectrum using the following relation Hu_2004 ,

${\mathcal{C}}_{l}^{XY}=\frac{2}{\pi}\int
k^{2}\mathrm{d}k{\mathcal{P}}_{s}(k){\mathcal{T}}^{Xl}(k){\mathcal{T}}^{Yl}(k),$
(15)

where ${\mathcal{T}}$ is the transfer function which is calculated using the
Boltzmann equations.

The fluctuations in ${\mathcal{R}}$ might not be always Gaussian in nature,
there might be some considerable amount of non-Gaussianities present in it.
With the help of 3-point correlation of curvature perturbation, one can obtain
the measure of non-Gaussianity, $f_{NL}\,$, to be (Appendix C)

$\displaystyle f_{NL}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})=$
$\displaystyle-\frac{10}{3}(2\pi)^{-4}(k_{1}k_{2}k_{3})^{3}G(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle\times[k_{1}^{3}\mathcal{P}_{S}(k_{2})\mathcal{P}_{S}(k_{3})+\mbox{two
permutations}]^{-1}.$ (16)

Using Maldacena formalism Maldacena_2003 , $G(k_{1},k_{2},k_{3})$ can be
expressed as

$\displaystyle G(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=\sum^{6}_{C=1}[\mathcal{R}_{k_{1}}({\mathcal{N}}_{e})\mathcal{R}_{k_{2}}({\mathcal{N}}_{e})\mathcal{R}_{k_{3}}({\mathcal{N}}_{e})]\mathcal{G}_{C}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
(17) $\displaystyle+$
$\displaystyle[\mathcal{R}_{k_{1}}^{*}({\mathcal{N}}_{e})\mathcal{R}_{k_{2}}^{*}({\mathcal{N}}_{e})\mathcal{R}_{k_{3}}^{*}({\mathcal{N}}_{e})]\mathcal{G}_{C}^{*}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})]$
$\displaystyle+$ $\displaystyle
G_{7}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3}).$

##  Appendix B Scalar power-spectrum

Like in any other perturbation theory, we decompose our variable of interest
into background quantity and a variation then equate them separately.
Therefore, the Einstein’s equation at the first order of variation can be
written as

$\delta G_{\mu\nu}=8\pi G\delta T_{\mu\nu}.$ (18)

This is, in fact, a set of linear differential equation that governs the
dynamics of perturbation in metric ($\delta g_{\mu\nu}$). In the FLRW
background, based on the response to a local rotation of spatial coordinates
on constant time hyper-surface, one can decompose the perturbations into
scalar, vector, and tensor transformations. Out of these, only scalar power-
spectrum has re-entered the Hubble horizon during the recombination epoch and
is responsible for the anisotropies present in the universe. Since our goal is
to account for anisotropies in the CMB data, we will be working with scalar
perturbations only. On taking into account the scalar perturbations to
background metric, the FLRW line element can be written as MUKHANOV1992203 ;
Sasaki:1986hm

$\displaystyle{\rm d}s^{2}=-\left(1+2\,A\right)\,{\rm
d}t^{2}+2\,a(t)\,(\partial_{i}{D})\;{\rm d}t\;{\rm d}x^{i}\,$ (19)
$\displaystyle+a^{2}(t)\;\left[(1-2\,B)\;\delta_{ij}+2\,\left(\partial_{i}\,\partial_{j}{E}\right)\right]\,{\rm
d}x^{i}\,{\rm d}x^{j}.$

There are two approaches to study the evolution of perturbations. One is to
construct gauge invariant quantities and study using them while the other is
to choose a specific gauge and work throughout in it. We will be working in a
specific gauge, longitudinal gauge. Because of the coordinate freedom we have,
we set $D=E=0$. Therefore metric in the new gauge can be written as,

${\rm d}s^{2}=-\left(1+2\,A\right)\,{\rm
d}t^{2}+a^{2}(t)\;\left[(1-2\,B)\;\delta_{ij}\right]\,{\rm d}x^{i}\,{\rm
d}x^{j}.$ (20)

Using the above metric, the independent components of the Einstein’s tensor
can be found to be Bassett

$\displaystyle\delta G^{0}_{0}$
$\displaystyle=\;6H(\dot{B}+HA)-\frac{2}{a^{2}}\nabla^{2}B,$ (21a)
$\displaystyle\delta G^{0}_{i}$ $\displaystyle=\;-2\nabla_{i}(\dot{B}+HA),$
(21b) $\displaystyle\delta G^{i}_{j}$
$\displaystyle=\;2\big{[}\ddot{B}+H(3\dot{B}+\dot{A})+(2\dot{H}+3H^{2})A$
$\displaystyle\,\,\;+\frac{1}{a^{2}}\nabla^{2}(A-B)\big{]}\delta^{i}_{j}+\frac{1}{a^{2}}\nabla^{i}\nabla_{j}(B-A).$

If we consider $\delta\phi$ to be the perturbation in scalar field, then the
perturbed energy momentum tensor, up to linear order, can be written as

$\displaystyle\delta T^{0}_{0}=$
$\displaystyle-(\dot{\phi}\dot{\delta\phi}-\dot{\phi}^{2}A+V_{\phi}\delta\phi),$
(22a) $\displaystyle\delta T^{0}_{i}=$
$\displaystyle-\nabla_{i}(\dot{\phi}\delta\phi),$ (22b) $\displaystyle\delta
T^{i}_{j}=$
$\displaystyle(\dot{\phi}\dot{\delta\phi}-\dot{\phi}^{2}A-V_{\phi}\delta\phi)\delta^{i}_{j}.$
(22c)

In the absence of anisotropic stresses $\delta T^{i}_{j}=0$ for $i\neq j$,
i.e. from Equation 18 we get $A=B$. Therefore, we can rewrite the Einstein
equation from Equation 21 and Equation 22 in the conformal coordinates as
follows MUKHANOV1992203 ,

$A^{\prime\prime}+6{\mathcal{H}}A^{\prime}-\nabla^{2}A+(2{\mathcal{H}}^{\prime}+4{\mathcal{H}}^{2})A=0,$
(23)

where ${\mathcal{H}}$ is the Hubble parameter in conformal coordinates. Since
we are analysing a canonical model, speed of perturbation is equal to 1. Also
we assume that perturbations are purely adiabatic in nature Gordon_2000 , i.e.
$\delta T^{0}_{0}=\delta T^{i}_{j}$. Now defining curvature perturbation as
MUKHANOV1992203

${\mathcal{R}}=A+\frac{2\rho}{3{\mathcal{H}}}\frac{A^{\prime}+{\mathcal{H}}A}{\rho+P},$
(24)

and with the help of background equations, Equation 9, we get
${\mathcal{R}}_{k}^{\prime}$ in the Fourier space to be

${\mathcal{R}}_{k}^{\prime}=-\frac{{\mathcal{H}}}{{\mathcal{H}}^{2}-{\mathcal{H}}^{\prime}}k^{2}A_{k}.$
(25)

Differentiating the above equation with respect to the conformal time and with
the help Equation 23 and Equation 24, we get the curvature perturbation
equation given by

${\mathcal{R}}_{k}^{\prime\prime}+2\bigg{(}\frac{z^{\prime}}{z}\bigg{)}{\mathcal{R}}_{k}^{\prime}+k^{2}{\mathcal{R}}_{k}=0,$
(26)

where $z=a\phi^{\prime}/{\mathcal{H}}$. Introducing a new variable
${\mathcal{V}}={\mathcal{R}}z$ Sasaki:1986hm ; Mukhanov:1987pv ;
MUKHANOV1992203 ; Stewart_1993  we have,

${\mathcal{V}}_{k}^{\prime\prime}+\bigg{[}k^{2}-\Big{(}\frac{z^{\prime\prime}}{z}\Big{)}\bigg{]}{\mathcal{V}}_{k}=0.$
(27)

This equation is called the Mukhanov-Sasaki equation. Quantizing the above
equation, one can calculate the power-spectrum and is given by the expression,

${\mathcal{P}}_{s}(k)=\frac{k^{3}}{2\pi^{2}}|{{\mathcal{R}}_{k}}|^{2}.$ (28)

##  Appendix C Scalar bi-spectrum

To obtain the non-Gaussianities present in ${\mathcal{R}}$ one need to
calculate the scalar bi-spectrum,
$\mathcal{B}_{S}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$ Maldacena_2003
, at end of inflation(${\mathcal{N}}_{e}$) which is defined in terms of three
point correlation function of ${\mathcal{R}}$ as Seery_2005 ; Chen_2005 ;
Chen_2008 ; Martin_2012

$\displaystyle\langle\hat{\mathcal{R}}_{\textbf{k}_{1}}\hat{\mathcal{R}}_{\textbf{k}_{2}}\hat{\mathcal{R}}_{\textbf{k}_{3}}\rangle=(2\pi)^{3}\mathcal{B}_{S}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})\delta^{(3)}(\textbf{k}_{1}+\textbf{k}_{2}+\textbf{k}_{3}).$
(29)

One can adopt the following ansatz to study non-Gaussinities Komatsu_2001 ,

$\displaystyle\mathcal{R}({\mathcal{N}},\textbf{k})=\mathcal{R}_{G}({\mathcal{N}},\textbf{k})-\frac{3f_{NL}}{5}\left[\mathcal{R}_{G}^{2}({\mathcal{N}},\textbf{k})-\langle\mathcal{R}_{G}^{2}({\mathcal{N}},\textbf{k})\rangle\right],$
(30)

where $\mathcal{R}_{G}$ is the the Gaussian part and $f_{NL}$ is the non-
Gaussianity parameter. With the help of Wick’s theorem, the 3-point
correlation of the curvature perturbation can be evaluated as

$\displaystyle\langle\hat{\mathcal{R}}_{\textbf{k}_{1}}\hat{\mathcal{R}}_{\textbf{k}_{2}}\hat{\mathcal{R}}_{\textbf{k}_{3}}\rangle$
$\displaystyle=-\frac{3f_{NL}}{10}(2\pi)^{4}(2\pi)^{-3/2}\frac{1}{k_{1}^{3}k_{2}^{3}k_{3}^{3}}$
(31) $\displaystyle\times$
$\displaystyle\delta^{(3)}(\textbf{k}_{1}+\textbf{k}_{2}+\textbf{k}_{3})$
$\displaystyle\times$
$\displaystyle[k_{1}^{3}\mathcal{P}_{S}(k_{2})\mathcal{P}_{S}(k_{3})+\mbox{two
permutations}].$

Using the ansatz
$\mathcal{B}_{S}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})=(2\pi)^{-9/2}G(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$,
Equation 29 and Equation 31, $f_{NL}$ can be determined to be the following

$\displaystyle f_{NL}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=-\frac{10}{3}(2\pi)^{-4}(k_{1}k_{2}k_{3})^{3}G(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle\times[k_{1}^{3}$
$\displaystyle\mathcal{P}_{S}(k_{2})\mathcal{P}_{S}(k_{3})+\mbox{two
permutations}]^{-1}.$ (32)

Using Maldacena formalism Maldacena_2003 , $G(k_{1},k_{2},k_{3})$ can be
expressed as

$\displaystyle G(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=\sum^{6}_{C=1}\Big{[}[\mathcal{R}_{k_{1}}\mathcal{R}_{k_{2}}\mathcal{R}_{k_{3}}]\mathcal{G}_{C}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle+[\mathcal{R}_{k_{1}}^{*}\mathcal{R}_{k_{2}}^{*}\mathcal{R}_{k_{3}}^{*}]\mathcal{G}_{C}^{*}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})\Big{]}$
$\displaystyle+G_{7}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3}),$

where each individual term is given by the following:

$\displaystyle\mathcal{G}_{1}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=2i\int^{{\mathcal{N}}_{e}}_{{\mathcal{N}}_{i}}\mbox{d}{\mathcal{N}}a^{3}\epsilon_{1}^{2}H\mathcal{R}_{k_{1}}^{*}(\mathcal{R}_{k_{2}}^{*})_{\mathcal{N}}(\mathcal{R}_{k_{3}}^{*})_{\mathcal{N}}$
$\displaystyle+\mbox{two permutations}),$
$\displaystyle\mathcal{G}_{2}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=-2i(\textbf{k}_{1}.\textbf{k}_{2}+\mbox{two permutations})$
$\displaystyle\times\int_{{\mathcal{N}}_{i}}^{{\mathcal{N}}_{e}}\mbox{d}{\mathcal{N}}\frac{a\epsilon_{1}^{2}}{H}\mathcal{R}_{k_{1}}^{*}\mathcal{R}_{k_{2}}^{*}\mathcal{R}_{k_{3}}^{*},$
$\displaystyle\mathcal{G}_{3}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=-2i\int_{{\mathcal{N}}_{i}}^{{\mathcal{N}}_{e}}\mbox{d}{\mathcal{N}}a^{3}H\epsilon_{1}^{2}\Bigg{[}\frac{\textbf{k}_{1}.\textbf{k}_{2}}{k_{2}^{2}}\times$
$\displaystyle\mathcal{R}_{k_{1}}^{*}(\mathcal{R}_{k_{2}}^{*})_{\mathcal{N}}(\mathcal{R}_{k_{3}}^{*})_{\mathcal{N}}+\mbox{five
permutations}\Bigg{]},$
$\displaystyle\mathcal{G}_{4}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=i\int_{{\mathcal{N}}_{i}}^{{\mathcal{N}}_{e}}\mbox{d}{\mathcal{N}}a^{3}\epsilon_{1}(\epsilon_{2})_{\mathcal{N}}H\Big{[}\mathcal{R}_{k_{1}}^{*}\mathcal{R}_{k_{2}}^{*}(\mathcal{R}_{k_{3}}^{*})_{\mathcal{N}}$
$\displaystyle+\mbox{two permutations}\Big{]},$
$\displaystyle\mathcal{G}_{5}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=\frac{i}{2}\int_{{\mathcal{N}}_{i}}^{{\mathcal{N}}_{e}}\mbox{d}{\mathcal{N}}a^{3}\epsilon_{1}^{3}H\Bigg{[}\left(\frac{\textbf{k}_{1}.\textbf{k}_{2}}{k_{2}^{2}}\right)\times$
$\displaystyle\mathcal{R}_{k_{1}}^{*}(\mathcal{R}_{k_{2}}^{*})_{\mathcal{N}}(\mathcal{R}_{k_{3}}^{*})_{\mathcal{N}}+\mbox{five
permutations}\Bigg{]}$
$\displaystyle\mathcal{G}_{6}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=\frac{i}{2}\int_{{\mathcal{N}}_{i}}^{{\mathcal{N}}_{e}}\mbox{d}{\mathcal{N}}a^{3}\epsilon_{1}^{3}H\Bigg{[}\left(\frac{k_{1}^{2}(\textbf{k}_{2}.\textbf{k}_{3})}{k_{2}^{2}k_{3}^{2}}\right)\times$
$\displaystyle\mathcal{R}_{k_{1}}^{*}(\mathcal{R}_{k_{2}}^{*})_{\mathcal{N}}(\mathcal{R}_{k_{3}}^{*})_{\mathcal{N}}+\mbox{two
permutations}\Bigg{]},$ $\displaystyle
G_{7}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$
$\displaystyle=\frac{\epsilon_{2}({\mathcal{N}}_{e})}{2}\Big{[}]|\mathcal{R}_{k_{2}}({\mathcal{N}}_{e})|^{2}|\mathcal{R}_{k_{3}}({\mathcal{N}}_{e})|^{2}$
$\displaystyle+\mbox{two permutations}\Big{]}.$

where $\epsilon_{2}$ is the second slow roll parameter that is defined with
respect to the first as follows: $\epsilon_{2}=\mbox{d
ln}\epsilon_{1}/\mbox{d}{\mathcal{N}}$ Liddle_1994 .

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