# Inflation Story: slow-roll and beyond

Dhiraj Kumar Hazra  Daniela Paoletti  Ivan Debono  Arman Shafieloo  George F.
Smoot  Alexei A. Starobinsky

###### Abstract

We present constraints on inflationary dynamics and features in the primordial
power spectrum of scalar perturbations using the Cosmic Microwave Background
temperature, polarization data from Planck 2018 data release and updated
likelihoods. We constrain the slow-roll dynamics using Hilltop Quartic
Potential and Starobinsky $R+R^{2}$ model in the Einstein frame using the
Planck 2018 binned Plik likelihood. Using the Hilltop as base potential, we
construct Whipped Inflation potential to introduce suppression in the scalar
power spectrum at large angular scales. We notice marginal (68% C.L.)
preference of suppression from the large scale temperature angular power
spectrum. However, large-scale E-mode likelihood based on high frequency
instrument cross spectrum, does not support this suppression and in the
combined data the preference towards the suppression becomes negligible. Based
on the Hilltop and Starobinsky model, we construct the Wiggly Whipped
Inflation potentials to introduce oscillatory features along with the
suppression. We use unbinned data from the recently released CamSpec v12.5
likelihood which updates Planck 2018 results. We compare the Bayesian
evidences of the feature models with their baseline slow-roll potentials. We
find that the complete slow-roll baseline potential is moderately preferred
against potentials which generate features. Compared to Planck 2015 PlikHM
bin1 likelihood, we find that the significance of sharp features has decreased
owing to the updates in the data analysis pipeline. We also compute the
bispectra for the best fit candidates obtained from our analysis.

##  1 Introduction

Predictions of different models of inflation for primordial spectra of scalar
and tensor perturbations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] are most
effectively tested by Cosmic Microwave Background (CMB) surveys. Importantly,
a space-based CMB survey can probe the signature of inflationary dynamics
thanks to the full sky coverage that provides access to the largest angular
scale information. Using CMB observations we can constrain the power spectra
of the scalar and the tensor perturbations generated during inflation. If
inflation is driven by a scalar field following strict slow roll dynamics (in
a nearly flat potential), then both the scalar and tensor power spectra can be
locally approximated by a power law form parametrized by the respective
amplitudes ($A_{S}(k),A_{T}(k)$) and tilts ($n_{S}(k),n_{T}(k)$). The ratio
$A_{T}/A_{S}$ relates to the scale of the inflaton potential while the other
parameters refer to the overall shape of the potential. First measurements of
the scalar amplitude $A_{S}$ was reported with COBE [13, 14] and the spectral
tilt $n_{S}$ with WMAP [15]. The current state of the art results are provided
by the Planck 2018 data [16]. Planck has performed a cosmic variance limited
measurement of the temperature anisotropies in the region where CMB is
dominant. It has covered from the largest scales up to $\ell\sim 2500$,
corresponding to a 5′ resolution. Observation of temperature and polarization
anisotropies in the CMB sky enabled Planck to constrain the cosmological
parameters such as baryon and cold dark matter densities, the angular diameter
distance to the last scattering surface and the parameters for the spectrum of
primordial fluctuations to the sub-percent level, with only the optical depth
remaining above this uncertainty. Planck data has rejected the hypothesis of
complete scale-invariant spectra $n_{S}=1$ with $8\sigma$ significance [17].
This discovery confirms the overall shape of the inflationary potential.
However, since the primordial B-modes are not yet detected, it can only put an
upper bound on $A_{T}/A_{S}$ [18] and therefore on the scale of inflation.

The Planck 2018 release brought two major improvements concerning our
analysis. Firstly we refer to the improvement in the E-mode polarization on
large angular scales due to the adoption of the cross power spectrum of
$100\times 143$ GHz channels from the High Frequency Instrument (HFI) [19].
The use of the cross spectrum has lowered the noise to the point of halving
the uncertainty on the integrated optical depth together with moving it
towards lower central values providing $\tau=0.054\pm 0.007$ [17] at 68% C.L.
compared to $\tau=0.066\pm 0.016$ [20] in Planck 2015. The CMB angular power
spectra are proportional to $A_{s}e^{-2\tau}$ causing a full degeneracy among
the scalar amplitude and the optical depth. The improvement in the E-mode
polarization allows us to measure the optical depth from the reionization
bump, reducing by consequence the error induced on the amplitude of scalar
fluctuations. In second instance the improvement in the characterization of
the temperature to polarization leakage and polar efficiency has allowed a
cosmology-grade high-$\ell$ polarization which reduces the uncertainty on the
scalar spectral index from $0.006$ in Planck 2015 (where only temperature
could be used) to $0.004$ in Planck 2018 release. Therefore simply from the
perspective of slow roll inflation, Planck 2018 brings new bounds on the
potential.

Together with this overall shape of the potential, the Planck data, with
accurate measurements of temperature and polarization anisotropies also hint
at possible detailed structure of the inflationary potential. A closer look at
the data subtracted from the best fit model with power law perturbation
spectrum reveals localized outliers [21] at different scales with some having
up to 2-3$\sigma$ significance. If the outliers do not represent statistical
fluctuations, then a simple solution can be addressed from a non-trivial
inflationary dynamics. The characteristics of the outliers, and equivalently
the reconstructions [22, 23, 24, 25] point towards two main types of features,
namely large scale suppression and bursts of oscillations at certain scales
***for earlier works on reconstruction with the pre-Planck data, see [26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Oscillating features are also
present in the polarization data [21]. Therefore it is important to explore
whether these oscillations have a single primordial origin.

In single field models of inflation, the presence of a (smoothed)
step/discontinuity, kink or an inflection point [40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54] in a nearly flat potential, or its sinusoidal
modification [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66], can lead to
departures from strict slow roll dynamics that results in a burst of
oscillations or persistent oscillations in the primordial power spectra (PPS).
Two field models of inflation can also generate sharp feature signals [67, 68,
69, 70] or resonant signals [71, 72, 73, 74, 75, 76, 77, 78] in the PPS that
help fitting the data better than standard slow roll models. Change in the
speed of sound during inflation also can source these types of features in the
PPS [79]. Although none of these models is favoured with respect to the
standard model by Bayesian indicators, the analyses of these features remain a
crucial point in current and future data. In fact, some of the large and
intermediate scale features have been consistently present in the data since
WMAP [80] with ever increasing precision. In some models with features, the
fitting of these outliers in both temperature and polarization is robust
against the increase of data points [81, 76].

We compare the updated Planck likelihood testing it against slow roll
inflation and a set of models that allow variation in the form of change in
slope (low-$\ell$ power suppression in temperature anisotropy) and oscillatory
behaviour to fit to observed excursions of the CMB anisotropy power spectrum.
A framework that allows these kinds of behaviour is Wiggly Whipped Inflation
(WWI) (see, [82, 83, 81, 84, 85, 86]). Scalar primordial spectra generated
from WWI contain two characteristic features: at large scales scalar power
spectra is suppressed, at the intermediate and small scales the power spectra
exhibits oscillations. They include different features relevant to the CMB
data and its baseline configuration allows us to compare with the strict slow
roll scenarios. Therefore in this single framework we can address constraints
on inflation with standard slow-roll and with initial or intermediate fast-
roll phases. The baseline slow-roll part of the WWI framework is constrained
with the Planck power spectrum data on temperature, E-mode polarization,
lensing and B-mode data from BICEP-KECK-Planck joint analysis [18].

Since WWI framework generates a suppression of the scalar spectrum at the
large scales, it is supported by the CMB temperature data [81] for fitting the
well known lack of power observed in the low multipole region since WMAP. The
new increased accuracy of E-mode polarization data from HFI opens the
possibility to investigate the possible presence of a suppression at large
scales also in this channel (although likely more difficult due to the
reionization bump). After the Planck official release in 2018 (followed by the
likelihood release in 2019), an updated CamSpec 12.5HMcln likelihood for
temperature and polarization data was publicly released where the authors have
used improved data analysis pipeline to produce an upgraded CamSpec
likelihood. Since WWI framework had already been tested against Plik bin1
(unbinned) likelihood (from Planck 2015) [81], in this paper we have used the
new CamSpec likelihood for the intermediate and small scale features. We also
compare the samples with better likelihoods in different temperature and
polarization combinations to examine their overlap. Our analysis is more
conservative than Planck [21] feature analysis where the nuisance parameters
were fixed to the Planck baseline best fit values. Apart from estimating
inflationary parameters, we carry out a Bayesian model comparison between the
models within the WWI framework. We search for the best fits and identify a
few representative spectra that provide improvement in fit to the data
compared to standard slow roll models.

Three point correlation functions of the primordial perturbations are known as
the bispectra and are commonly quantified with a parameter $f_{\rm NL}$ which
is a suitable dimensionless ratio of the bispectra and the product of the
power spectra. Therefore $f_{\rm NL}$ is one of the quantifier of the
primordial non-Gaussianity in a model. Using temperature and polarization data
Planck reports constraints [87] on $f_{\rm NL}$ as, $f_{\rm NL}^{\rm
local}=-0.9\pm 5.1$, $f_{\rm NL}^{\rm eq}=-26\pm 47$, $f_{\rm NL}^{\rm
orth}=-38\pm 24$ at local, equilateral and orthogonal limits respectively. The
68% uncertainties indicate that the results are consistent with Gaussian
nature of primordial fluctuations. Note that these constraints are obtained
assuming a scale independent bispectrum amplitude. Slow roll models of
inflation in a canonical Lagrangian generate negligible non-Gaussianity [88,
89]. However departure from slow rolls generate scale dependent oscillations
in the $f_{\rm NL}(k)$ [90, 55, 91, 59, 60, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103]. Therefore apart from the power spectrum, features
considered in our analysis generate oscillatory and non-negligible bispectra.
In the context of WWI models, the bispectra were discussed in [83, 81]. With
the updated constraints on the potential parameters with the final Planck
data, in this paper we also compute the $f_{\rm NL}$ for the best fit
candidates.

The paper is organized as follows: in section 2 we discuss the slow-roll and
beyond slow-roll models we plan to compare against the data. In section 3 we
provide a detailed discussion of the data and the likelihoods used in the
analyses. We also discuss the choice of priors and the Bayesian model
comparison methodology. Following that, in section 4 we compare the models
with the marginal likelihoods, present the posterior distribution of the
parameters, discuss the best fit primordial power spectra and compute the
bispectrum amplitude. Finally, we summarise our conclusions in section 5. For
readers who are familiar with modelling and Planck data analysis, we would
like to suggest that they can skip section 3 and move to section 4 directly
after section 2.

##  2 Framework

A power law primordial spectrum is largely consistent with Planck data at all
scales [104, 23, 21]. The stringent constraint on the spectral tilt translates
to a strong constraint on the slope of the inflationary potential [21].
Moreover, recent results from BICEP-KECK-Planck (BK15) [18] joint analysis
have bounded the tensor-to-scalar ratio ($r$) to $r<0.056$ at 95% C.L. [21].
Therefore, within these data combinations, the slow roll inflationary models
that we will consider as baseline models must have sufficient freedom to allow
variations of the spectral tilt and produce a tensor-to-scalar ratio
compatible with the current tight constraints †††BK18 data was released after
the release of the paper. Although it provides stronger constraints with
$r<0.036$ at 95% C.L. we do not re-analyse our models with this data as the
tensor-to-scalar ratio remains well within the BK18 bound..

For feature models we use the Wiggly Whipped Inflation framework which we
introduced in [82, 83]. This was also analyzed in [81, 84] for the constraints
using Planck combined data from temperature and polarization and the forecasts
for future CMB space-based observations. We note that since certain WWI best
fit power spectra to the Planck 2015 data contain non-local oscillations
extending to small scales, future large scale structure (LSS) surveys such as
Euclid [105], LSST [106] are also expected to play an important role in
constraining such models. We examined these constraints for WWI with Euclid in
[86]. In this work, since our goal is to use the latest available Planck
likelihood, we make some adjustments in the analysis of the models while
keeping the main framework unchanged.

We categorize the potential into 2 different classes of potentials wherein
with different limits on the parameters we obtain different features in the
primordial power spectrum.

###  2.1 WWI-potential:

The base slow roll potential used here is the Hilltop potential [107]. The
WWI-potential is expressed as,

$V({\phi})=V_{i}\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)+\Theta(\phi_{\rm
T}-\phi)V_{i}\left(\gamma(\phi_{\rm T}-\phi)^{q}+\phi_{0}^{q}\right),~{}$
(2.1)

In the limit of $\gamma=\phi_{0}=0$ the potential reduces to,

$V({\phi})=V_{i}\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right),~{}$ (2.2)

as the base slow roll potential for the WWI model [83]. We denote it as WWI-
base. We fix $p=4$ for our analyses with which this model generates different
spectral tilt and $r$ as discussed in [108]. We therefore allow $\mu$ to vary
along with $V_{i}$. Since this potential allows only strict slow roll, the
primordial power spectrum is featureless.

In the limit of $\phi_{0}=0$, the potential reduces to,

$V({\phi})=V_{i}\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)+\Theta(\phi_{\rm
T}-\phi)V_{i}\left(\gamma(\phi_{\rm T}-\phi)^{q}\right),~{}$ (2.3)

that allows the scalar field to start rolling down a steeper slope potential
ending up on the WWI-base potential with a continuous transition. We will
refer this potential as the Whipped Inflation potential [82].

Note that $\phi_{\rm T}$ is the value of the field where the transition occurs
and $\gamma$ determines the departure from slow roll. We use $q=2$. However,
in such cases, the slow roll parameter $\epsilon_{\rm
H}=-\frac{\dot{H}}{H^{2}}$ remains much less than 1 and does not refer to the
completely kinetic-dominated regime ‡‡‡For inflation with initial kinetic
domination see [109, 110, 111]. Such dynamics suppress the power at large and
intermediate scales without affecting the smaller-scale part of the spectrum.
This potential is useful to determine the significance of the power
suppression of the CMB T-mode angular power spectrum at lower multipoles. We
refer to this potential as WI-potential.

The general potential in Equation 2.1 allows the field to transit to the slow
roll part through a discontinuity in the potential. Apart from the large scale
power suppression, this potential also creates bursts of oscillations either
localized to a particular cosmological scale or persistent at all scales
depending on the sharpness of the transition $\Delta$ when modelled with a
step function. $\phi_{0}^{q}$ determines the extent of the discontinuity. This
general potential will be referred as the WWI-potential.

###  2.2 WWIP-potential:

The base slow roll potential used here is the $\alpha$-attractor [112, 113,
114] model. WWIP-potential (Wiggly Whipped Inflation Prime potential that
contains discontinuities in the derivatives of the potential) is expressed as:

$V({\phi})=\Theta(\phi_{\rm
T}-\phi)V_{i}\left(1-\exp\left[-\alpha\kappa\phi\right]\right)+\Theta(\phi-\phi_{\rm
T})V_{ii}\left(1-\exp\left[-\alpha\kappa(\phi-\phi_{0})\right]\right),~{}$
(2.4)

which reproduces Starobinsky’s $R+R^{2}$ model [2] of inflation for
$\alpha=\sqrt{2/3}$, $\phi_{0}\to\ln 2/(\alpha\kappa)$, $V_{i}=V_{ii}$ and
$\kappa\phi\gg 1$. For $V_{i}=V_{ii}$,

$V({\phi})=V_{i}\left(1-\exp\left[-\alpha\kappa\phi\right]\right).~{}$ (2.5)

We use it as a base potential for WWIP and we denote it as WWIP-base. In this
potential the spectral tilt and the tensor-to-scalar ratio are fixed.
Therefore in this potential we allow the $N_{*}$ of the pivot mode to vary. In
our analysis we fix the initial scale factor by demanding the pivot mode
$0.05~{}{\rm Mpc}^{-1}$ leaves the Hubble radius $N_{*}$ e-folds before the
end-of-inflation.

When $\phi_{0}\neq 0$, note that we have two different slopes at two different
field regions separated by $\phi_{\rm T}$. $V_{i}$ and $V_{ii}$ are related by
demanding the potential to be continuous at the transition. While the
potential is continuous, its derivatives are discontinuous. This potential
also produces suppression of scalar power at the large and intermediate scales
and wide oscillations at the intermediate to small scales.

This framework, constructed by these potentials allows us to explore the
constraints on slow roll inflation and different types of departure from slow-
roll. We use the power spectrum calculator from the BI-spectra and Non-
Gaussianity Operator (BINGO) [93, 97] to solve the inflationary dynamics.
First we solve the Klein-Gordon equation assuming initial slow roll with
$3H\dot{\phi}=-V_{\phi}$. The initial value of the scalar field is solved
using a shooting algorithm. We assume the inflation ends at 70 e-folds as it
is necessary to have at least 70 e-folds to ensure that the largest scale
participating in the convolution integral of power spectrum and transfer
function stays well inside the Hubble radius at the onset of inflation. As
mentioned before, we normalize the initial scale factor ($a(N)=a_{i}$ at
$N=0$) by fixing the time when pivot mode $0.05~{}\mathrm{Mpc}^{-1}$ departs
Hubble radius at $N=20$. Note that this is exactly 50 e-folds before the end
of inflation as we have fixed $N_{\mathrm{end}}$. We integrate the curvature
perturbation equations (and also the tensor perturbation equations) for each
mode-$k$ from sub-Hubble $k=100aH$ using Bunch-Davies initial condition to
super-Hubble $k=10^{-5}aH$ scales. For sharp change in the potential due to
the presence of discontinuity, sub-Hubble evolution becomes important and
therefore $k=100aH$ is not sufficient choice for the convergence of the
integral. We therefore dynamically change this condition at smaller scales in
such a way that the solutions to the differential equations converge. We use
CAMB [115] to compute the angular power spectrum. For WWI and WWIP models we
compute the angular power spectra for temperature and polarization for all
multipoles avoiding interpolation.

##  3 Data and analysis

In this section we will discuss the properties of the data we are going to use
in our analysis. We also discuss the prior ranges on the parameters, the
methodology of model comparison and the method to search the best fit
primordial power spectra.

###  3.1 Planck data: likelihoods we use

The Planck 2018 data release delivered the temperature and polarization
likelihoods on both large and small angular scales. Large angular scales are
provided by the lowT likelihood in temperature based on the commander
component-separated temperature map and the Simall E-mode polarization
likelihood based on the $100\times 143\,\mathrm{GHz}$ cross-spectrum. As for
the small angular scales, two different likelihoods have been delivered, based
on different algorithms and data treatments, both based on Half-Mission data
auto and cross power spectra for 100, 143 and 217 GHz [116]. For some cases we
also add to the analysis the lensing likelihood which is derived from the
four-point correlation function on Planck CMB maps [117]. In Table 1 we list
all the likelihoods we use in our analysis. Note that depending on the nature
of feature models we use different likelihood combinations. For all the
models, we use lowT likelihood for TT and Simall likelihood for EE at
low-$\ell$. The both cover multipoles $2-29$. At higher multipoles the ranges
depend on the channels considered (either TT or EE). These are provided in the
table. Using the keys mentioned in the table we will denote the combinations
of Planck temperature and polarization likelihoods used in our analysis. For
the base models we only use binned likelihoods from Plik, since base models do
not have features. For base models we add also the B-mode measurements from
the BICEP-Keck 2015 data [18] at 95-150 and 220 GHz. The likelihood covers the
intermediate multipoles between 50 and 300 and it includes also some of the
WMAP and Planck channels and a mitigation of the foreground contamination. For
the base model constraints we use the Plik-TTTEEE+lowT+lowE+Bk15+lensing
likelihood jointly. We set $\mu$ in WI and WWI and $N_{*}$ in WWIP case to
their best-fit values obtained in base model analyses. Since WI-model produces
only suppression at large scales, we use three combinations: Plik-TT+lowT,
Plik-EE+lowE and Plik-TTTEEE+lowT+lowE. We keep one temperature-only
likelihood separate where we do not use any low-$\ell$ polarization
information to understand the extent of support towards low-multipole power
suppression from large-scale polarization data. Since WWI and WWIP models have
sharp oscillatory features, for these cases we use the unbinned likelihood.
However, instead of the unbinned bin1 Plik likelihood used in the Planck
inflationary feature analysis, we use the recently released Camspec 12.5HMcl
likelihood [118] obtained from a reanalysis of Planck data. This likelihood is
based on an alternative data pipeline that excludes the 100 GHz temperature
autospectrum and cleans the 143 and 217 GHz channels using the 545 GHz one. It
applies a different mask, and the version we use, which applies the most
aggressive masks, enlarges the sky fraction up to 80% in temperature and
polarization. We denote this likelihood as Clean CamSpec. In [76, 77] (where
one of the author of this paper has co-authored) this likelihood is termed as
EG20. For the WWI and WWIP feature models, we use this likelihood for
high-$\ell$. In both these models we use TT+lowT+lowE, EE+lowE, TE+lowE and
TTTEEE+lowT+lowE combinations for constraints. In the complete
TTTEEE+lowT+lowE likelihood we have 9 nuisance parameters, where 6 are
foreground and 3 are calibration parameters.

Likelihood type | Key |  | Multipole
---
range ($\ell$)
| Likelihood name
---
in official release
Commander | lowT | 2-29 | $\tt{commander\\_dx12\\_v3\\_2\\_29}$
Simall | lowE | 2-29 | $\tt{simall\\_100x143\\_offlike5\\_}$
|  |  | $\tt{EE\\_Aplanck\\_B}$
| PlikHM
---
binned
| Plik-TT
---
Plik-EE
Plik-TTTEEE
| TT: 30-2508
---
EE: 30-1996
TE: 30-1996
| $\tt{plik\\_rd12\\_HM\\_v22\\_TT}$
---
$\tt{plik\\_rd12\\_HM\\_v22\\_EE}$
$\tt{plik\\_rd12\\_HM\\_v22b\\_TTTEEE}$
Clean CamSpec (EG20) |  | TT
---
EE
TTTEEE
| TT: 30-2500
---
EE: 30-2000
TE: 30-2000
| $\tt{v12.5HMclnTT}$
---
$\tt{v12.5HMclnEE}$
$\tt{v12.5HMclnTE}$
$\tt{v12.5HMclnTTTEEE}$
BICEP-KECK 2015 | BK15 | 50-300 | $\tt{BK15\\_bandpowers\\_20180920}$
Table 1:  Likelihood codes based on Planck data that we use in this work. As
explained in the text, we use different combinations of likelihoods for our
analysis. Note that Clean CamSpec here is the same as EG20 as termed in [76,
77].

###  3.2 Priors on parameters

In Table 2 we tabulate the priors used for background and inflationary
parameters. Since we use nested sampling, we use only relevant parameter
ranges as uniform priors. For the parameters common to all the models, listed
in the first row of the table, the priors ensure that the posterior
distributions for the least constraining data (EE+lowE in most cases) are not
prior-dominated in the baseline case. The round brackets show the priors
adopted where lowE is not used. Since the reionization optical depth and
therefore the amplitude of primordial spectrum is not well constrained without
lowE, we use sufficiently larger priors to obtain converged two-tailed
distributions. Inflationary parameters are divided into two classes: standard
parameters that control the amplitude and the tilt and feature parameters that
define the characteristics of the features. In the Hilltop and $R+R^{2}$
models, $V_{0}$ represents the amplitude of the primordial scalar power
spectrum. In all feature models, the parameter $\phi_{T}$ is allowed to vary
in the range that generates the features within the cosmological scales probed
by Planck. In WI model the parameter $\gamma$ determines the extent of
suppression, and a negative value implies amplification of scalar power at
large scales. In this table, we provide the likelihood or likelihood
combinations used in constraining each model in this paper.

Model | Parameter | Prior range | Likelihood
---|---|---|---
Common parameters | $\Omega_{\mathrm{b}}h^{2}$ | $[0.02,0.0265]$ |
| $\Omega_{\mathrm{c}}h^{2}$ | $[0.1,0.135]$ |
| $100\theta$ | $[1.03,1.05]$ |
| $\tau$ | $[0.03,0.08](0.03,0.18)$ |
Hilltop | $\ln(10^{10}V_{i})$ | $[0.8,3]$ | Plik-TTTEEE + lowT
| $\mu$ | [10,30] | +lowE + BK15
Starobinsky-$R+R^{2}$ | $\ln(10^{10}V_{i})$ | $[-2,2]$ | Plik-TTTEEE + lowT
$\alpha$-attractor | $N_{*}$ | $[50,60]$ | +lowE + BK15
| $\alpha$ | $\sqrt{2/3}$ |
WI | $\ln(10^{10}V_{i})$ | $[2.2,2.35](2.2,2.5)$ | Plik-TT+lowT
| $\gamma$ | $[-0.005,0.2]$ | Plik-EE+lowE
| $\phi_{T}$ | $[13.05,13.3]$ | Plik-TTTEEE+lowT+lowE
| $\mu$ | 21 |
WWI | $\ln(10^{10}V_{i})$ | $[2.2,2.35]$ | [Clean CamSpec for high-$\ell$]
| $\gamma$ | $[0,0.07]$ | TT+lowT+lowE
| $\phi_{0}$ | $[0,0.05]$ | TE+lowE
| $\phi_{T}$ | $[13.05,13.3]$ | EE+lowE
| $\ln\Delta$ | [-12,-3] | TTTEEE+lowT+lowE
| $\mu$ | 21 |
WWIP | $\ln(10^{10}V_{i})$ | $[0,0.3]$ | [Clean CamSpec for high-$\ell$]
| $\phi_{0}$ | $[0,1]$ | TT+lowT+lowE
| $\alpha$ | $\sqrt{2/3}$ | TE+lowE
| $\phi_{T}$ | $[4.45,4.57]$ | EE+lowE
| $N_{*}$ | 51 | TTTEEE+lowT+lowE
Table 2:  Prior ranges for the cosmological parameters. The numbers in the
round parentheses denote the ranges used when lowE is not included in the
analysis. We use uniform priors over the ranges mentioned in the table.
Foreground and calibration nuisance parameters have the same ranges and priors
as were used in the Planck 2018 and Clean CamSpec publications. The fourth
column indicates the likelihood or combination of likelihoods used in
constraining the model.

The range of $\gamma$ is chosen to ensure two outcomes: first, that the
potential remains positive, which may not be true below the chosen threshold,
with a lower limit of $\gamma\sim-0.005$; second, that the upper limit implies
a power suppression on large scales already ruled out by data to consider all
possible scenarios allowed. The parameter $\phi_{0}$ is related to the
amplitude of wiggles. In WWI and WWIP we keep the lower limit as 0 to include
the no-wiggle scenario. Note that the upper limit of the $\phi_{0}$ for WWI
should be selected in relation to $\Delta$ as they are correlated. Exploring
this degeneracy prior to the data comparison, we select the upper limits that
are expected to be ruled out for all values of $\Delta$. We use a logarithmic
prior on the sharpness ($\Delta$) parameter of the step and allow $\ln\Delta$
to vary between -12 and -3. Large $\ln\Delta$ corresponds to wider features
and is expected to be allowed only at large scales. Very small $\ln\Delta$
represents sharp features with higher frequencies. With small amplitudes,
sharp features can persist over a large range of scales, but CMB data is not
able to distinguish between sharp features beyond Planck’s resolution. The
wiggles in WWIP model are only parametrized by the location and amplitude.
Similar to WWI model, the upper limit of amplitude ensures that the posterior
probability becomes negligible well within the limit. We vary all the nuisance
parameters required by the different likelihoods together with the
cosmological and inflationary parameters. For calibration parameters and
amplitude of galactic dust parameters we use Gaussian priors as discussed in
[116, 118].

We would like to outline the logic of using different likelihoods at
high-$\ell$s for different potential. For potentials with featureless
primordial spectrum as described by Equation 2.2 and Equation 2.5 and for
potentials with featureless spectrum at intermediate and small scales
described by Equation 2.3 we use Plik binned likelihoods for high-$\ell$s. In
these cases, the use of unbinned likelihoods do not provide additional
information but require too much resources. For potentials with features at
intermediate and small scales (e.g. Equation 2.1 and Equation 2.4) we use
Clean CamSpec likelihood that is unbinned. Of course for marginal likelihoods
we compare the feature models with their baseline for the same dataset
combination.

###  3.3 Model selection

We investigate whether the updated CamSpec 12.5HMcln likelihood and the PlikHM
likelihood based on Planck 2018 data support the variants of Wiggly Whipped
Inflation, and we quantify the degree of support. This is a model selection or
model comparison problem within the framework of Bayesian inference.

Parameter estimation methods involve the calculation of the maximum
likelihood. This is the average of the likelihood over the prior distribution.
The maximum likelihood is therefore larger for a model if more of its
parameter space is likely, and smaller for a model with a large parameter
space in which the likelihood is low (known as ‘wasted parameter volume’).
This holds even if the likelihood function has very high peaks, because these
peaks must compensate for volumes in parameter space where the likelihood is
extremely low. A simpler theory with a more compact parameter space will have
a larger maximum likelihood than a more complicated model, unless the latter
is significantly better at explaining the data. A simple comparison of
$\chi^{2}$ values obtained from parameter estimation techniques (such as MCMC)
are not sufficient, since they do not take into account the prior volume of
the models being compared. We must therefore calculate the model evidence for
each model, and compare the model evidence using the Bayes factor.

Bayesian inference is based on the logic of probability theory. The product
rule gives us Bayes’ theorem:

$p(\boldsymbol{\theta}|d,M)=\frac{p(d|\boldsymbol{\theta},M)p(\boldsymbol{\theta}|M)}{p(d|M)}.$
(3.1)

The left-hand side is the posterior probability for the vector of unknown
model parameters $\boldsymbol{\theta}$ of length $n$ given the data $d$ under
model $M$. The prior probability distribution function
$p(\boldsymbol{\theta}|M)$ is an expression of our state of knowledge before
observing the data. This defines the prior available parameter space under the
model $M$. The denominator $p(d|M)$ is the Bayesian evidence or model
likelihood. This is the probability of observing the data $d$ given that the
model $M$ is correct.

Model selection usually involves the calculation of the evidence or
marginalised likelihood. This may be expressed as the multidimensional
integral of the likelihood over the prior, or the parameter space under $M$:

$Z\equiv p(d|M)=\int
p(d|\boldsymbol{\theta},M)p(\boldsymbol{\theta}|M)\mathrm{d}\boldsymbol{\theta},$
(3.2)

where $\boldsymbol{\theta}$ is in general multidimensional, and $d$ is a
collection of measurements (current or future).

The Bayes factor is then the ratio of the evidence for two competing models
$M_{0}$ and $M_{1}$, or the ratio of posterior odds. If the competing models
have equal prior probability, then:

$B_{01}\equiv\frac{p(d|M_{0})}{p(d|M_{1})}=\frac{p(M_{0}|d)}{p(M_{1}|d)}$
(3.3)

We use PolyChord[119, 120] to sample the parameter space and calculate the
marginalized likelihoods for our models §§§In particular, we use the PolyChord
integration of CosmoMC [121] via CosmoChord. Getdist [122] is used for post-
processing.. PolyChord is a nested sampling algorithm, optimized for high-
dimensional spaces. As such it is designed to calculate the marginalised
likelihoods rather than the posterior, although posterior samples may still be
obtained by using the dead and live points.

We compare the evidence for each model with the evidence for the respective
featureless base models. In the base models, we fix the feature parameters to
zero, and we keep the priors on all the other parameters unchanged. The
comparison is made against the same dataset combination, as indicated earlier.

##  4 Results

###  4.1 Model comparison

The results of our Bayesian model comparison are shown in Table 3. The
logarithm of the Bayes factor $\ln B$ is the difference of $\ln Z$ for the
feature model and the featureless model.

The interpretation of the value of $\ln B$ depends to a large extent on the
experiment being carried out. We use the Jeffreys scale to interpret the
values in terms of the strength of evidence for the competing model (see [123]
and [124]). An experiment for which $|\ln B|<1$ is usually deemed
inconclusive, that is, the alternative model cannot be distinguished from the
null hypothesis.

We note that $\ln B$ for all models falls between $0.12$ and $-2.8$. The only
positive value is obtained for WI against the featureless model using TT+lowT.
However, this is still within the inconclusive range. The TT+lowT+lowE,
EE+lowE, and TTTEEE+lowT+lowE data sets give positive evidence against all
three feature models. With TE+lowE, the evidence against WWIP is even
stronger.

| Model |  |
---|---|---|---
Data set | WI | WWI | WWIP
TT+lowT | 0.12 | NA | NA
TT+lowT+lowE | NA | -2.3 | -2.5
TE+lowE | NA | -2.3 | -2.8
EE+lowE | -1.8 | -2.4 | -2.1
TTTEEE+lowT+lowE | -1.4 | -2.5 | -1.61
Table 3:  Logarithm of the Bayes factor for the feature models compared to
their respective slow-roll baseline model for the same dataset combinations.
The data sets are as follows: for WI, high-$\ell$ TT, TE, EE and TTTEEE refer
to PlikHM binned data, while for WWI and WWIP they refer to Clean CamSpec
data. NA for a model indicates that model has not been compared with the
corresponding dataset.

###  4.2 Parameter estimation

From the samples gathered using the nested sampling through PolyChord we
explore the posterior distributions of the parameters in all the models within
the WWI family and for all dataset combinations considered in this analysis.

In Figure 1 we present the constraints on the background and inflationary
potential parameters for Hilltop inflation discussed in Equation 2.2. Here we
use the complete Planck temperature, polarization and lensing datasets and
BK15 likelihood. Compared to the Planck baseline results we do not notice
changes in the background parameter constraints, as expected. The potential
$V_{i}$ is strongly constrained by the overall normalization. However,
compared to the constraints on $A_{s}$ in a power law primordial spectrum, the
posterior distribution is found to be strongly skewed for $V_{i}$. This is an
artefact of the degeneracy with $\mu$. In Table 4 the $1\sigma$ constraints
and unidirectional 95% limits (for distributions without two tails) are
mentioned. Our analysis constrains
$\mu>13.4~{}\mathrm{M}_{{}_{\mathrm{Pl}}}{}$ at 95% C.L. We use the best fit
from this analysis and fix $\mu=21\mathrm{M}_{{}_{\mathrm{Pl}}}{}$ for the WI
and WWI case, where the Hilltop quartic represents the potential in the slow-
roll phase. Constraints on $R+R^{2}$ are plotted in Figure 2 and tabulated in
Table 4. The data are not able to constrain the pivot scale $N_{*}$ beyond
$1\sigma$ upper bound. This is expected as the change in the constraint in the
spectral tilt is not substantial, and therefore the posterior remains prior-
dominated. We find the best-fit value of $N_{*}\sim 51$, which we use for the
runs with features. Here too we find degeneracies between the potential
$V_{i}$ and the pivot scale that substantially relax the constraint on
$V_{i}$.

|  PlikHM binned TTTEEE | \+ lowT + lowE + lensing+BK15
---|---|---
Param | Hilltop | Starobinsky $R+R^{2}$
$\Omega_{b}h^{2}$ | $0.0224\pm 0.0001$ | $0.0224\pm 0.0001$
$\Omega_{c}h^{2}$ | $0.120\pm 0.001$ | $0.120\pm 0.001$
$100\theta_{MC}$ | $1.0409\pm 0.0003$ | $1.0409\pm 0.0003$
$\tau$ | $0.0546_{-0.0070}^{+0.0064}$ | $0.0559_{-0.0068}^{+0.0069}$
$V_{i}$ | $2.21_{-0.21}^{+0.53}$ | $0.0671_{-0.0802}^{+0.1297}$
$\mu$ | $>13.4$ | NA
$N_{*}$ | NA | -
Table 4:  Constraints on Hilltop quartic potential and Starobinsky $R+R^{2}$ model obtained using PlikHM binned TTTEEE + lowT + lowE, Planck lensing and BK15 likelihood. We compare both scalar and tensor power spectrum in these two models against the data. Note that in the Hilltop model, $\mu$ allows the variation of spectral tilt and the tensor-to-scalar ratios, while in the Starobinsky model $N_{*}$ determines the spectral tilt. In the runs for beyond slow-roll we use the best fit values of $\mu$ and $N_{*}$ from these analyses. Figure 1: Constraints in Hilltop slow roll inflation using Planck 18 TTTEEE (PlikHM binned) + lowT + Simall + lensing + BK15 likelihood. We compare both primordial scalar and tensor power spectra with the data. Figure 2: Triangle plot for Starobinsky ($R+R^{2}$) slow roll potential in Einstein frame. Here too, we compare both primordial scalar and tensor power spectra with the data. |  |  PlikHM binned |
---|---|---|---
Param | TTTEEE | TT | EE
| lowT+lowE | +lowT | +lowE
$\Omega_{b}h^{2}$ | $0.0224\pm 0.0001$ | $0.0223\pm 0.0002$ | $0.0232\pm 0.0009$
$\Omega_{c}h^{2}$ | $0.120\pm 0.0010$ | $0.120\pm 0.001$ | $0.119\pm 0.003$
$100\theta_{MC}$ | $1.0409\pm 0.0003$ | $1.0409\pm 0.0004$ | $1.0397\pm 0.0008$
$\tau$ | $0.0590_{-0.0095}^{+0.0080}$ | $0.1138_{-0.0261}^{+0.0259}$ | $0.0571_{-0.0120}^{+0.0092}$
$\ln[10^{10}V_{i}]$ | $2.271_{-0.019}^{+0.017}$ | $2.379\pm 0.052$ | $2.278_{-0.0265}^{+0.0232}$
$\gamma$ | $<0.068$ | $<0.069$ | $<0.166$
$\phi_{T}$ | $<13.19$ | $<13.22$ | $<13.23$
Table 5:  Results for the WI potential. We report the $1\sigma$ uncertainties
($2\sigma$ upper bounds). Here we consider PlikHM binned likelihood for
TT+lowT, EE+lowE and TTTEEE+lowT+lowE

Constraints on Whipped Inflation potential are shown in Figure 3. Here we plot
constraints from different datasets. Since we have fixed $\mu$ to the best-fit
value in the base slow roll potential, and the best fit is obtained using
B-mode observation data from BK15, we do not evaluate and compare the B-mode
anisotropy spectrum. Since the present upper bound on the tensor power
spectrum indicates no noticeable effect in the large-scale temperature
anisotropy spectrum, we do not compute the tensor spectra either. The
constraints plotted here are from PlikHM binned TT+lowT, EE+lowE and
TTTEEE+lowT+lowE. Therefore, for the WI model, all the likelihoods that we
discuss will belong to PlikHM. Here we do not consider TE data separately,
since the suppression in the scalar power spectrum is constrained at large
scales. In the combination of temperature data (TT+lowT) we do not include
lowE to differentiate the support for the suppression from different available
data at the largest scales. In this model $\gamma,\phi_{T}$ represent the
extent and the location of the suppression of power. From the marginalized
posterior distributions, a support between 1 and $2\sigma$ for the suppression
can be noticed in TT+lowT. Since the low multipole temperature power spectrum
data is consistently lower than the baseline model prediction, we find that a
suppression provides improvement in fit compared to the baseline model, making
it marginally favoured. As discussed before, we find positive but inconclusive
Bayesian evidence for this model only in TT+lowT case. When we consider
EE+lowE datasets, we do not find any support for large-scale features, which
suggests that lowE data does not allow any broad features in the power
spectrum. Due to the lack of support, additional prior volume from
${\gamma,\phi_{T}}$ is penalized in the marginal likelihood and we find
inconclusive preference for baseline model over the WI model. When we combine
all datasets, we find the preference for the WI model reduced due to the
addition of lowE. As expected, the value of $\ln B$ is found to be between the
value obtained with TT+lowT and EE+lowE, which means that the evidence for the
baseline over WI is slightly weakened with the combined datasets. It is also
important to note the role of the lowE likelihood in constraining the scale of
the potential $V_{i}$. Since in the constraints from temperature likelihood
the amplitude and the optical depth are highly degenerate, the constraints on
the potential and therefore the optical depth are weak. The lowE data breaks
the degeneracy by putting stringent bounds on the optical depth, which in turn
constrains $V_{i}$ significantly better than TT+lowT. The uncertainties on the
parameters are tabulated in Table 5, where the drastic improvement in the
bounds on the potential can be noticed. A comparison of $\tau$-constraints
between Table 5 and Table 4 or as obtained in Planck analysis ¶¶¶see Planck
Params table: PLA: Planck parameters table ($\tau=0.0544^{+0.0070}_{-0.0081}$)
indicates a 10% increase in uncertainties in both direction around an
increased mean optical depth. A broad suppression in the power spectrum at
large scales is marginally degenerate with the optical depth. This degeneracy
increases the uncertainties and a preference of suppression in turn prefers an
increase in the optical depth in the lowE data. Note that to compensate for
the suppression in the lowE data, $\tau$ must increase to match the
reionization bump in EE.

In Figure 4 we plot the improvement in $\chi^{2}$ w.r.t. the baseline best fit
from the PolyChord samples for different data combinations. The samples are
plotted as a functions of the transition field value ($\phi_{T}$) which
represents the location of the feature, and they are colored according to the
value of $\gamma$ representing the extent of suppression. Here the higher
$\phi_{T}$ represents a power spectrum with the suppression extending to
smaller scales. Top right, left plots and the plot at the bottom correspond to
TT+lowT, EE+lowE and TTTEEE+lowT+lowE respectively. For TT+lowT we find
several samples providing better fit to the data compared to the baseline. A
clear correlation is visible here. With higher $\phi_{T}$ Planck data only
allows smaller $\gamma$ in order to provide the improved fit. For EE+lowE we
do not obtain samples that improve the fit to the data. A few samples are
located with negligible improvement ($\Delta\chi^{2}<0.5$). For the complete
datasets we find marginally lower improvement compared to the temperature only
data. The samples with higher $\gamma$ relocate to larger scales (higher
$\phi_{T}$) as well.

Figure 3: Constraints on Whipped Inflation from different data combinations
PlikHM+ binned TT + lowT, PlikHM binned EE + LowE and PlikHM binned TTTEEE +
lowT + lowE. In temperature-only constraints we note marginal support for the
large scale suppression. However since polarization-only likelihood
(essentially LowE) does not support large-scale scalar suppression, in the
complete likelihood we find decreased significance for the suppression. Large
uncertainties in the determination of the amplitude and the reionization
optical depth in the TT+lowT case occur owing to the absence of large-scale
polarization data.

Figure 4:  The samples with improvement in $\chi^{2}$ in the Whipped Inflation
compared to its featureless baseline model. Note that we have compared the
$\chi^{2}$ from the PolyChord analysis and therefore the maximum value does
not necessarily mean global best-fit sample. We have plotted the samples
against $\phi_{T}$ and the $\gamma$ parameter representing the location and
magnitude of the large scale suppression in the scalar power spectrum. While
TT+lowT data supports such transition with certain improvement in the
$\chi^{2}$ compared to the baseline model we notice no such improvement in the
EE+lowE data. Finally in the combined datasets we find less support towards
large scale suppression compared to temperature only data.

In our analysis, wiggles in the primordial spectrum are compared against the
Clean CamSpec likelihood. Therefore in the context of WWI and WWIP potentials,
mention of high-$\ell$ dataset combinations refer to Clean CamSpec 12.5
likelihood.

Figure 5 plots the constraints on the WWI model. Given the increased
degeneracy with 4 extra parameters compared to its slow roll base Hilltop
quartic potential, we do not get significant determination of any feature
parameters. Table 3 containing evidence also indicates preference towards slow
roll baseline model compared to the WWI model. Marginalized posteriors on the
suppression parameter $\gamma$ and the wiggle amplitude $\phi_{0}$ peak at
zero values. We get a secondary plateau for $\phi_{0}$ parameter around
$0.01\mathrm{M}_{{}_{\mathrm{Pl}}}{}$ for TT+lowT+lowE and TTTEEE+lowT+lowE.
Also for these two datasets, we find marginal preference on the sharpness of
the transition parameter $\ln\Delta\sim-5$. The EE+lowE and TE+lowE datasets
however do not show any preference on feature parameters in the marginalized
posteriors. Constraints on the parameter values are listed in Table 6. While
for TE+lowE and EE+lowE we are unable to provide any significant bounds on
$\gamma$ and $\ln\Delta$, we obtain upper bounds in these parameters when
adding the low-$\ell$ temperature likelihood for the TT+lowT+lowE and
TTTEEE+lowT+lowE combinations. We find the mild preference in the $\ln\Delta$
is not statistically significant at all in any combination of data as we are
not able to constrain the parameter at 95% in either direction around the
mean. Compared to WI potential, if we compare Table 5 and Table 6 in WWI model
the bounds on $\gamma$ and $\phi_{T}$ are tighter. Note that here we can only
compare the EE+lowE and TTTEEE+lowT+lowE case as in WI model we have not
considered TT+lowT+lowE. The tighter constraints even with extra degrees of
freedom are due to the occurrence of wiggles along with the suppression in
this model, i.e. a non-zero $\gamma$ and $\phi_{0}$ imply both suppression and
wiggles which are not allowed to extend below certain cosmological scales
unless the features are sharp and sufficiently small in amplitude (as can be
noticed in the correlation between $\gamma$, $\phi_{0}$ and $\ln\Delta$).
Comparison of the $\Omega_{b}h^{2}$ posteriors in Figure 3 and Figure 5 and
the mean values in Table 5 and Table 6 reveals that for WWI, the
$\Omega_{b}h^{2}$ posterior from EE+lowE shifts towards TT+lowT+lowE. This
does not occur for WI. This is expected because the wiggles in the WWI are
marginally degenerate with the baryon acoustic oscillations [81] causing a
shift in WWI of the polarization-only results towards temperature data,
increasing the agreement between the two datasets.

Figure 5: Constraints on WWI potential for four different combination of datasets, namely Clean CamSpec TT+lowT+lowE, EE+lowE, TE+lowE and TTTEEE+lowT+lowE. |  | Clean CamSpec |  |
---|---|---|---|---
Param | TTTEEE | TT | TE | EE
| lowT+lowE | lowT+lowE | lowE | lowE
$\Omega_{b}h^{2}$ | $0.0222\pm 0.0001$ | $0.0221\pm 0.0002$ | $0.0223\pm 0.0001$ | $0.0226\pm 0.0010$
$\Omega_{c}h^{2}$ | $0.120\pm 0.001$ | $0.119\pm 0.001$ | $0.118\pm 0.002$ | $0.120_{-0.003}^{+0.004}$
$100\theta_{MC}$ | $1.0410\pm 0.0003$ | $1.0407\pm 0.0004$ | $1.0415\pm 0.0004$ | $1.0408\pm 0.0008$
$\tau$ | $0.0564_{-0.0077}^{+0.0079}$ | $0.0550_{-0.0078}^{+0.0079}$ | $0.0537_{-0.0088}^{+0.0083}$ | $0.0530\pm 0.0091$
$\ln[10^{10}V_{i}]$ | $2.264\pm 0.016$ | $2.260\pm 0.017$ | $2.255\pm 0.022$ | $2.264\pm 0.025$
$\phi_{0}$ | $<0.021$ | $<0.019$ | $<0.029$ | $<0.028$
$\gamma$ | $<0.059$ | $<0.056$ | $-$ | $-$
$\phi_{T}$ | $<13.15$ | $<13.15$ | $<13.19$ | $<13.21$
$\ln\Delta$ | $-$ | $-$ | $-$ | $-$
Table 6: Results for WWI potential. Here for high-$\ell$ likelihood we have
used the Clean CamSpec unbinned angular power spectrum data. We note that no
feature parameter is detected and we only obtain upper bounds in these
parameters.

While the marginal likelihood prefers the slow roll model over WWI model, we
find several local best fit primordial features that fit the data better the
baseline slow roll model. These features are important for discussion as
future cosmic variance-limited polarization data or joint analysis with LSS
can potentially detect one of these features with high significance if that
represents the true model of the Universe. In order to identify the patches in
the parameter space with better likelihood, we plot the samples in Figure 6
where the improvement in $\chi^{2}$ from PolyChord is plotted as a function of
$\phi_{T}$. The samples are colored according to the wiggle amplitude
$\phi_{0}$. In samples from TT+lowT+lowE we notice warmer colored samples
(higher amplitude wiggles) are located at large scales (low $\phi_{T}$). For
TE+lowE and EE+lowE, the better likelihood samples are distributed nearly at
all scales (as we have discussed in the marginalized posteriors before).
TTTEEE+lowT+lowE has samples with better likelihood that are dominated by
TT+lowT+lowE and TE+lowE. Importantly the combined data analysis has samples
located sharply around $\phi_{T}=13.15\mathrm{M}_{{}_{\mathrm{Pl}}}{}$, which
is also visible in the samples from temperature and in the cross-correlation
data. We can investigate the possibility of overlap of favored parameters
spaces between different datasets in Figure 7. In this plot we show all
samples that provide features with improvement to the data compared to strict
slow roll model. The samples indicate the location and amplitude of the
features that are colored by the sharpness/frequency of the features. The
extent of improvement in this plot cannot be visualized here but can be
compared with Figure 6. We note that the red samples near
$\phi_{T}=13.075\mathrm{M}_{{}_{\mathrm{Pl}}},\phi_{0}=[0.01-0.02]\mathrm{M}_{{}_{\mathrm{Pl}}}$
in temperature panel (top left) also appear in the combined data panel (bottom
right). We notice samples overlap in the top left and top right panel around
$\phi_{T}=13.1\mathrm{M}_{{}_{\mathrm{Pl}}}$ and as expected they appear in
the combined dataset too. While there are several parameter spaces favored by
different datasets, substantial overlap between parameter spaces in different
panels can not be noticed. Evidently a feature supported by multiple datasets
has the highest chance of representing the true model of the inflationary
dynamics.

Figure 6:  The improvements in $\chi^{2}$ (from PolyChord) when WWI potential
is used compared to its featureless baseline model are plotted against the
position ($\phi_{T}$) and extent ($\phi_{0}$) of the discontinuity in the
potential. Different locations and colors in the plot denotes different types
of features from the WWI potential.

Figure 7: Same samples are plotted as in Figure 6. Here all points represent samples that fit the data better than the baseline slow-roll potential and therefore they provide the map of the favored feature parameters. The plots for different datasets do not reveal any wide overlap within the parameter space. While in Figure 6 for TT+lowT+lowE and TE+lowE show we find samples at $\phi_{T}\sim 13.1\mathrm{M}_{{}_{\mathrm{Pl}}}$ the same samples in this map reveal that these features do have different frequencies determined by $\Delta\phi$. |  | Clean CamSpec |  |
---|---|---|---|---
Param | TTTEEE | TT | TE | EE
| lowT+lowE | lowT+lowE | +lowE | +lowE
$\Omega_{b}h^{2}$ | $0.0222\pm 0.0001$ | $0.0221\pm 0.0002$ | $0.0223\pm 0.0002$ | $0.0224_{-0.0010}^{+0.0008}$
$\Omega_{c}h^{2}$ | $0.120\pm 0.001$ | $0.120\pm 0.001$ | $0.119\pm 0.002$ | $0.121\pm 0.003$
$100\theta_{MC}$ | $1.0410\pm 0.0003$ | $1.0407\pm 0.0004$ | $1.0415\pm 0.0004$ | $1.0407\pm 0.0008$
$\tau$ | $0.0556_{-0.0085}^{+0.0071}$ | $0.0537_{-0.0080}^{+0.0079}$ | $0.0533_{-0.0086}^{+0.0088}$ | $0.0548_{-0.0104}^{+0.0085}$
$\ln[10^{10}V_{i}]$ | $0.190_{-0.017}^{+0.015}$ | $0.185_{-0.017}^{+0.018}$ | $0.179_{-0.022}^{+0.020}$ | $0.199_{-0.028}^{+0.023}$
$\phi_{0}$ | $<0.144$ | $<0.136$ | $<0.270$ | $<0.415$
$\phi_{T}$ | $-$ | $-$ | $-$ | $-$
Table 7:  Results for WWIP potential. Here for high-$\ell$ likelihood we have
used the Clean CamSpec unbinned angular power spectrum data. Figure 8:
Constraints on WWIP potential for four different combination of datasets,
namely Clean CamSpec TT+lowT+lowE, EE+lowE, TE+lowE and TTTEEE+lowT+lowE.

Figure 9: Same as in Figure 6 but for WWIP potential. Here the colorplots explore the complete parameter space of WWIP potential with improvements in fit as the potential has two extra parameters w.r.t. the baseline model. We notice samples near $\phi_{T}=4.53\mathrm{M}_{{}_{\mathrm{Pl}}}$ that are preferred by both TT+lowT+lowE and TE+lowE data. We find similar samples in the EE+lowE data but with higher $\phi_{0}$ values. Finally TTTEEE+lowT+lowE selects $\phi_{T}=4.53\mathrm{M}_{{}_{\mathrm{Pl}}}$ samples as the region of maximum likelihood. Figure 10:  Primordial power spectra that are local best fit to different combinations of datasets in WWI potential. We have plotted spectra that are visually different, though they provide similar improvement in fit to the data. We plot two such local best fits to TT+lowT+lowE (top panel), TE+lowE (middle panel) and EE+lowE (bottom panel). Note that though there are other local best fits to the data as shown in Figure 7 here we show only two for each of the combinations. Figure 11:  Primordial power spectra that are local best fit to the complete Clean CamSpec TTTEEE+lowT+lowE dataset in WWI potential. We have plotted four spectra that are visually different. Parameters | WWI-TTTEEE-A | WWI-TTTEEE-B | WWI-TTTEEE-C | WWI-TTTEEE-D
---|---|---|---|---
$\ln[10^{10}V_{i}]$ | 2.272 | 2.265 | 2.263 | 2.262
$\phi_{0}$ | 0.0033 | 0.0045 | 0.0030 | 0.0021
$\gamma$ | 0.0210 | 0.0077 | 0.0129 | 0.0012
$\phi_{T}$ | 13.10 | 13.07 | 13.15 | 13.22
$\ln\Delta$ | -11.09 | -7.98 | -8.25 | -7.31
$\Delta\chi^{2}$ | 7.7 | 6.9 | 9.25 | 6.3
Table 8:  Parameter values for the four different bestfits from WWI model. The
final row represents the improvement in $\chi^{2}$ compared to the baseline
model. Figure 12:  Best fit angular power spectra residual to the power law
baseline best fit in WWI potential. These spectra correspond to the PPS
plotted in Figure 10. Figure 13:  Best fit angular power spectra residual to
the power law baseline best fit. Here we plot the residuals for two local best
fits to TTTEEE + lowT+lowE data, namely WWI-TTTEEE-A and WWI-TTTEEE-B (PPS
plotted in Figure 11). Figure 14:  Angular power spectra residual to the power
law baseline best fit. Here we plot the residuals for two local best fits to
TTTEEE + lowT+lowE data, namely WWI-TTTEEE-C and WWI-TTTEEE-D (PPS plotted in
Figure 11).. Figure 15:  Primordial power spectra that are best fits to
different dataset combinations in WWIP potential. Note that here we plot only
one representative spectrum from the pool of best fits to each datasets since
in Figure 9 we notice that most of the improvement is coming from large
$\phi_{T}$ region. Figure 16:  Best fit angular power spectra residual to the
power law baseline best fit. Here we plot the residuals for the best fits
obtained from WWIP potential (PPS plotted in Figure 15).

The results for WWIP model are plotted in Figure 8 and the parameter mean
values with uncertainties are provided in Table 7. Here we have two parameters
that define the features, namely $\phi_{0},\phi_{T}$. The marginalized
posterior of $\phi_{T}$ peaks at large scales (higher $\phi_{T}$). EE+lowE and
TE+lowE data show a peak around $\phi_{0}=0\mathrm{M}_{{}_{\mathrm{Pl}}}$ and
a plateau around $\phi_{0}=0.25\mathrm{M}_{{}_{\mathrm{Pl}}}$. Combined
dataset does not show the plateau as it is not supported by the temperature
auto-correlation data. However, unlike TT+lowT+lowE where $\phi_{0}$ posterior
sharply drops away from 0, the distribution obtained from the combined data is
somewhat flat near the peak. However, we do not find any statistically
significant support for the features. The evidence in Table 3 also shows weak
preference for the baseline slow roll model. $\ln B$ in WWIP is comparatively
higher w.r.t. WWI in TTTEEE+lowT+lowE. Due to two parameters less in the WWIP
model, the marginal likelihood is less penalized compared to the WWI model in
the combined dataset. The mean values on baryon density from temperature and
polarization here too are in better agreement compared to slow roll potential
as noticed in WWI potential.

The samples with improvement in fit are plotted in Figure 9. The improvements
in $\chi^{2}$ are plotted against the position and amplitude parameter for the
samples with features. As indicated by the marginalized posteriors, here we
can notice most of the favored samples are located at large scales
($\phi_{T}\sim 4.53\mathrm{M}_{{}_{\mathrm{Pl}}}$). However compared to
temperature data, the polarization data analysis prefers samples with
sufficiently larger amplitude (warm colored samples). Larger uncertainties in
the EE data allow large-amplitude wiggles in the primordial spectrum that are
ruled out by temperature data.

###  4.3 Best fit power spectra

Apart from parameter estimation and comparing marginal likelihoods, we also
search for the best fits to each model and explore the possibility of the best
fits to address temperature and polarization data individually and in
combination. We use BOBYQA [125] method for the minimization. The optimization
is not very efficient in the large prior volume consisting of background,
inflationary and nuisance parameters. Given the starting point of the best-fit
search, the method usually converges to a local minima. To locate the local
best fits we start the search for best fit from the parameter chains that
represent the peaks in the $\Delta\chi^{2}$ plots presented in Figure 6 and
Figure 9. Note that in similar way we had obtained four local minima to the
Planck 2015 data using WWI in [81]. In [76] three methods of best fit search
have been presented. However given the smaller number of nuisance parameters
in CamSpec 12.5HMcln likelihood compared to PlikHM likelihood we use the most
conservative method where we vary the nuisance parameters including the
$\chi^{2}$ from priors in the total $\chi^{2}$ (method III in [76]).

Since the WI model just provides suppression at large scales, we do not
discuss the best fits here. In Figure 10 we plot the local best fits to the
individual datasets in WWI model. In the top, middle and bottom panels
primordial power spectra that are local best fits are plotted from
TT+lowT+lowE, TE+lowE and EE+lowE respectively. Corresponding to these best
fit primordial spectra, the angular power spectra residual to TTTEEE+lowT+lowE
featureless best fit angular power spectra are plotted in Figure 12. In each
case we plot two power spectra that can be distinguished in terms of
frequency, amplitude or position of oscillations in the primordial spectra.
WWI-TT-A has a suppression at large scales followed by oscillations extending
to the small scales (till 0.1 $\mathrm{Mpc^{-1}}$). WWI-TT-B on the other hand
has a pronounced dip around $k=0.002~{}\mathrm{Mpc^{-1}}$ that improved the
fit around $\ell=22-30$. Given sharper transition in WWI-TT-A compared to WWI-
TT-B we notice the former primordial spectrum having oscillations with higher
frequency than the latter. This allows the spectrum to provide better fit to
the data around $\ell=200-300$ and smaller scales. WWI-TE-A and WWI-TE-B both
have lower amplitude oscillations in the primordial power spectra. In WWI-
TE-A, the features start appearing at small scales. WWI-TE-B on the other hand
has a suppression at large scales and persistent oscillations continuing to
smallest scales probed by Planck. WWI-EE-A an WWI-EE-B represent two best fits
to the EE data. The former contains primordial features located between
$k=0.001-0.08~{}\mathrm{Mpc}^{-1}$ and the latter exhibits non-local
oscillations from $k=0.001~{}\mathrm{Mpc}^{-1}$ that extend to the smallest
scales in the Planck window. The middle and bottom panels of Figure 12 which
plot the residual TE and EE angular power spectrum from TE and EE best fits
also show similar localized and non-local oscillations in the residuals. While
individual fits to the temperature and polarization data are not significant
compared to the combined datasets, it is important to note that the best fits
to individual datasets do not share similar features, since the amplitude,
frequency and the scale of the oscillations are different. The WWI best fits
to the combined data are plotted in Figure 11. We present 4 best fits (WWI-
TTTEEE-[A-D]) that are different in characteristics of the features. The
inflationary parameters for these four best fits are tabulated in Table 8. The
residual angular power spectra are plotted in Figure 13 and in Figure 14. WWI-
TTTEEE-A contains large scale suppression followed by oscillations extending
to the smallest Planck-CMB scales. WWI-TTTEEE-B represents an envelope of
oscillations that decays and merges to power law spectrum at
$k=0.1~{}\mathrm{Mpc}^{-1}$ similar to WWI-TT-A. These two best fits to the
full data provide nearly 8 and 7 improvements in fit compared to the baseline.
The high-frequency oscillations at intermediate to small scales are reflected
in the residual plots where we notice envelope of oscillations at different
scales. For the best fit WWI-TTTEEE-A we notice oscillations persist till the
smallest scales probed by Planck while for WWI-TTTEEE-B we find the
oscillations decay, as expected from the PPS. The data residual to the
baseline model are plotted in fine bins that seem to be addressed by the
features around $\ell\sim 100-400$ in TT and $\ell\sim 100-500$ in TE. WWI-
TTTEEE-C and WWI-TTTEEE-D best-fit spectra have oscillations with lower
amplitude and frequency. WWI-TTTEEE-C represents the global best fit (9.25
improvement in fit compared to power law) from the model. Data in the residual
plot are binned with larger bin widths. Residuals show that the global best-
fit addresses the outliers in TT and TE datasets. The oscillations in the WWI-
TTTEEE-D best fit is even wider and can also be noticed in the residual plots.
Angular spectra to this particular PPS seem to address TE and EE outliers at
multipoles less than 1000. However, due to larger errors, the improvement in
fit is only 6.3. Overall the improvement in fit drops 3 in $\chi^{2}$ compared
to Planck 2015 data (Plik likelihood) [81].

WWIP best-fit PPS for different datasets are plotted in Figure 15. Since this
model generates one type of feature, we get single local best-fit (that is
also the global best-fit) to each data. Note that the extent of suppression at
the largest scales that fits the low-multipole anomalies determines the
amplitude of the oscillation feature at intermediate and small scales. The
residuals are plotted in Figure 16. In the top panel we plot TT residuals for
WWIP-TT and WWIP-TTTEEE best fits. In the middle panel we plot TE residuals
for WWIP-TE and WWIP-TTTEEE and the panel at the bottom contains EE residuals
for WWIP-EE and WWIP-TTTEEE best fits. We keep the same color convention used
in Figure 15. As expected, best fit to individual data fits the outliers of
the corresponding data. For example WWIP-TT fits the oscillatory residuals
around $\ell\sim 100-400$ and the dip around $\ell\sim 750$ and $1500$. WWIP-
TE allows significantly larger amplitude oscillations in the PPS compared to
WWIP-TT (with similar peak positions) and the residual plots captures its
attempts to fit oscillatory TT residuals within $\ell=50-700$. Oscillations in
WWIP-EE best fit PPS are phase shifted (the first peak being shifted to
smaller scales) compared to WWIP-TE with the oscillations. Since they are
phase-shifted, the same power spectrum can not address outliers in the TE and
EE spectra. WWIP-EE best fit PPS addresses the residual data around $\ell\sim
400$ and between $600-800$. WWIP-TTTEEE best fit PPS is very similar to WWIP-
TT with certain shift in the overall amplitude. Unlike WWIP-TT, here the
residual plot does not seem to address the TT residual data around $\ell\sim
750$ and $1500$. It fits the TT oscillatory residuals around $\ell\sim
100-400$ and TE residuals with lower amplitude around $\ell=50-700$. WWIP-
TTTEEE provides an improvement in fit by 6.5 in $\chi^{2}$ to the full data
compared to featureless model. The best fit parameters are
$\left[\ln(10^{10}V_{i},),\phi_{0},\phi_{\rm
T}\right]=\left[0.1973,0.091,4.528\right]$. Compared to [81] we notice an
overall 6 decrease in the $\Delta\chi^{2}$.

There are two reasons for this decrease in both WWI and WWIP models. Firstly,
with the WI potential we find that while the large scale scalar suppression is
supported by the TT data, EE data from HFI prefer no suppression. Therefore
the extent of fit to the data decreases. Secondly, we are using the Clean
CamSpec likelihood instead of Plik. It has been shown in [76] that the
improvement in fit to the data is less in Clean CamSpec compared to Plik2018.
Overall, this decrease definitely points to the need to explore features that
are in agreement with both temperature and polarization data.

###  4.4 Non-Gaussianity: bispectrum

We compute the bispectra for the best fit potential parameters. We use BINGO
[93, 97] to numerically evaluate the $f_{\rm NL}$ parameter as functions of
wave-numbers ($k$). Note that the bispectrum amplitude $f_{\rm
NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ is defined as,

$f_{\rm
NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})=-\frac{10}{3}\frac{\sqrt{2\pi}k_{1}^{3}k_{2}^{3}k_{3}^{3}{\cal
B}_{\zeta}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})}{k_{1}^{3}{\cal
P}_{\zeta}(k_{2}){\cal P}_{\zeta}(k_{3})+k_{2}^{3}{\cal P}_{\zeta}(k_{3}){\cal
P}_{\zeta}(k_{1})+k_{3}^{3}{\cal P}_{\zeta}(k_{1}){\cal P}_{\zeta}(k_{2})},$
(4.1)

with ${\cal
B}_{\zeta}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ denoting
the bispectra and ${\cal P}_{\zeta}(k)$ denoting the power spectrum.

We compute all the terms contributing to the bispectra and combine them to
calculate $f_{\rm NL}$ as discussed in [92, 93, 97]. In Figure 17 we plot the
$f_{\rm NL}$ for the four WWI best fit potentials to the Planck data plotted
in Figure 11. We obtain oscillatory bispectra for all the best fits. Sharpness
of the transition determines frequency of the oscillations in the power
spectrum and that is reflected too in the bispectra. As we have discussed
earlier, the discontinuity in the potential and its derivatives are modelled
with a step function and its derivatives.

Figure 17:  [Left] $f_{\rm NL}(k)$ computed for four best fits (PPS plotted in
Figure 11) to the Planck data in WWI potential in the equilateral limit.
[Right] $f_{\rm NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$
plotted for the same best fits as on the left, but for arbitrary triangular
configuration as a function of $k_{2}/k_{1}$ and $k_{2}/k_{1}$ where $k_{1}$
is chosen to be a particular small scale mode.

When the step is sharp enough, we notice divergent $f_{\rm NL}$ in the
equilateral limit. Therefore while the power spectrum displays oscillations of
equal amplitude, the $f_{\rm NL}$ follows a power law divergence. In WWI-
TTTEEE-B, the oscillations in the PPS decay at small scales as the step is not
steep. The equilateral $f_{\rm NL}$ here grows till $0.08\rm M_{\rm
pc}{}^{-1}$ and then decays. Note that WWI-TTTEEE-D shows a growing $f_{\rm
NL}$ till the smallest observed cosmological scales by Planck. It decays at
smaller scales. Here we notice the divergence pattern at scales smaller
$0.08\rm M_{\rm pc}{}^{-1}$ (corresponding to WWI-TTTEEE-B). Although the
sharpness and magnitude of the transition in potential for the best fit WWI-
TTTEEE-B are more than WWI-TTTEEE-D, since the transition occurs at smaller
scales in the latter we do not notice the decay of $f_{\rm NL}$ there. In the
panel at the right we plot the density plots of $f_{\rm
NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ (represented by
the colorbar). Here a small scale mode is fixed as $k_{1}$ and the $f_{\rm
NL}$ is plotted as a function of $k_{2}/k_{1}$ and $k_{3}/k_{1}$.

In Figure 18 for the best fit WWIP-TTTEEE we plot the $f_{\rm NL}(k)$ in the
equilateral limit (left panel) and $f_{\rm
NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ with fixed
$k_{1}$ (right panel). As has been discussed [126, 98] the $f_{\rm NL}(k)$ is
divergent. Similar to WWI best fits here we also notice similar $f_{\rm
NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ in the plot at
the right panel. Note that the Dirac delta function in the second derivative
of the potential (the potential is continuous with discontinuous first
derivative) is contributing to the divergent $f_{\rm NL}$. Unlike the WWI
model, here the same power spectrum can provide different bispectra depending
on the sharpness of the transition in the derivative of the potential. While
for the power spectrum we do not get divergences from these discontinuities,
divergences in the bispectrum will be constrained well from the bispectrum
data that will add further constraints on transition width and possibly can
rule our some of the best fit candidates.

Figure 18:  [Left] $f_{\rm NL}(k)$ computed for WWIP-TTTEEE best fit (PPS
plotted in Figure 15) to the Planck data in the equilateral limit. [Right]
$f_{\rm NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ plotted
for arbitrary triangular configuration as a function of $k_{2}/k_{1}$ and
$k_{2}/k_{1}$ where $k_{1}$ is chosen to be a particular small scale mode.

In general the BINGO output for $f_{\rm
NL}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3})$ with
$\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3}$ satisfying a
triangle should be compared with the bispectrum from the CMB observation along
with the power spectrum. Since such a joint likelihood for a generic power
spectrum and bispectra is not available this estimation is beyond the scope of
this paper.

##  5 Discussions

Using the baseline and feature variants of Wiggly Whipped Inflation we compare
inflationary slow-roll dynamics and its departure with the data from Planck
2018 release. Planck binned power spectrum data for temperature, polarization
anisotropy, lensing are used jointly with BICEP-Keck-Planck B-mode likelihood
to constrain the baseline strict slow roll inflationary model. Once the
baseline model is constrained with the full Planck+BK15 baseline, we explore
the possible features of our models, appearing at different characteristics
scales, and we compare the results for temperature and polarization (taken
separately or in combination). Our main results are listed below:

  * •

Using the full Planck2018+BK15 baseline combination (binned Plik TTTEEE + lowT
+ lowE + lensing) we find, for Hilltop quartic inflation, the lower limit (at
95%) on the field value at the end of inflation ($\mu$) is
$13.4\mathrm{M}_{{}_{\mathrm{Pl}}}$.

  * •

In Whipped Inflation, when TT+lowT data combination is used, a marginal (68%)
significance towards suppression is found with $\Delta\chi^{2}\sim 3$ compared
to the baseline. This significance decreases when lowE data is used in the
complete data combination. A separate analysis with EE+lowE suggests no
support for the suppression. For TT+lowT data we find positive but
inconclusive preference for the WI model over the baseline with $\ln B=0.12$.
TTTEEE+lowT+lowE data analysis does not show such preference as we find the
marginal likelihood to be $\ln B=-1.8$. Though we considered some specific
sub-class of possible potentials with features in our paper, our results
reflect the following general tendency: taking CMB polarization data into
account diminishes statistical importance of anomalies seen in the CMB
temperature anisotropy for $\ell<30$.

  * •

When Clean CamSpec data is used to constrain the wiggles in the WWI
(discontinuous potential) model, we find localized peaks in the posterior
distribution of feature position. These peaks do not have overlaps between
posteriors from different datasets. We find $\Delta\chi^{2}=9.25$ w.r.t.
baseline for CamSpec TTTEEE+lowT+lowE datasets. Here extra parameters are
penalized more and we find $\ln B=-2.5$ w.r.t. the baseline.

  * •

WWIP model (with the discontinuity in the derivatives of a continuous
potential) best fits for TT+lowT+lowE and EE+lowE show a mismatch in phases of
the oscillations. Here for CamSpec TTTEEE+lowT+lowE analysis we find $\ln
B=-1.6$. Best fit spectrum provides nearly 6.5 improvement in fit compared to
the baseline. Although WWIP shows lower improvement in fit compared to the
WWI, the Bayesian evidence here is better because lesser parameters for
features decrease the prior volume.

  * •

Overall we have noticed, compared to Planck 2015 data, a decrease in the
improvement in fit to the data when HFI large scale polarization data is used
and when Clean CamSpec with higher sky fraction and updated cleaning
strategies are implemented.

  * •

$f_{\rm NL}$ computed for the best fit candidates in both WWI and WWIP
potentials show oscillatory bispectra. Provided a joint power spectrum and
bispectra likelihood, in future joint estimation for feature models is
possible.

To summarize, comparing the Bayesian evidences while we find a marginal favor
of large scale suppression by the temperature anisotropy power spectrum,
large-scale E-model polarization likelihood based on high frequency instrument
cross spectrum does not support this suppression and in the combined data, the
preference towards the suppression becomes negligible. Oscillatory features
that require more parameters are not favored compared to the baseline model
although they provide upto 10 improvement in fit to the combined data. This is
in agreement with other findings [21, 76, 79]. However, given the low signal-
to-noise ratios in TE and EE datasets, we expect that a conclusive evidence
for or against the features such as suppression and wiggles can be provided
using other data from observations such as cosmic variance-limited
polarization [127, 128, 129] and large-scale-structure [130, 105, 106, 131,
132, 133, 85, 134, 135, 86].

## Acknowledgements

The authors acknowledge the use of computational resources at the Institute of
Mathematical Science’s High Performance Computing facility (hpc.imsc.res.in)
[Nandadevi] and at the Istituto Nazionale di Fisica Nucleare’s CNAF cluster.
The authors would like to thank Matteo Braglia, Xingang Chen and Fabio Finelli
for their comments on the manuscript. DP acknowledges financial support by ASI
Grant 2016-24-H.0 and the agreement n. 2020-9-HH.0 ASI-UniRM2 “Partecipazione
italiana alla fase A della missione LiteBIRD”. DP acknowledges the computing
centre of Cineca and INAF, under the coordination of the “Accordo Quadro MoU
per lo svolgimento di attivitá congiunta di ricerca Nuove frontiere in
Astrofisica: HPC e Data Exploration di nuova generazione”, for the
availability of computing resources and support with the project INA17-C5A42.
AAS has been partially supported by the project number 0033-2019-0005 of the
Russian Ministry of Science and Higher Education and by the Kazan Federal
University Strategic Academic Leadership Program.

## References

  * [1] A. A. Starobinsky, _Spectrum of relict gravitational radiation and the early state of the universe_ , _JETP Lett._ 30 (1979) 682.
  * [2] A. A. Starobinsky, _A New Type of Isotropic Cosmological Models Without Singularity_ , _Phys. Lett. B_ 91 (1980) 99.
  * [3] A. H. Guth, _The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems_ , _Phys. Rev. D_ 23 (1981) 347.
  * [4] K. Sato, _First Order Phase Transition of a Vacuum and Expansion of the Universe_ , _Mon. Not. Roy. Astron. Soc._ 195 (1981) 467.
  * [5] V. F. Mukhanov and G. V. Chibisov, _Quantum Fluctuations and a Nonsingular Universe_ , _JETP Lett._ 33 (1981) 532.
  * [6] A. D. Linde, _A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems_ , _Phys. Lett. B_ 108 (1982) 389.
  * [7] A. Albrecht and P. J. Steinhardt, _Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking_ , _Phys. Rev. Lett._ 48 (1982) 1220.
  * [8] S. W. Hawking, _The Development of Irregularities in a Single Bubble Inflationary Universe_ , _Phys. Lett. B_ 115 (1982) 295.
  * [9] A. A. Starobinsky, _Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations_ , _Phys. Lett. B_ 117 (1982) 175.
  * [10] A. H. Guth and S. Y. Pi, _Fluctuations in the New Inflationary Universe_ , _Phys. Rev. Lett._ 49 (1982) 1110.
  * [11] A. A. Starobinsky, _The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy_ , _Sov. Astron. Lett._ 9 (1983) 302.
  * [12] A. D. Linde, _Chaotic Inflation_ , _Phys. Lett. B_ 129 (1983) 177.
  * [13] COBE collaboration, _Structure in the COBE differential microwave radiometer first year maps_ , _Astrophys. J. Lett._ 396 (1992) L1.
  * [14] M. J. White and E. F. Bunn, _The COBE normalization of CMB anisotropies_ , _Astrophys. J._ 450 (1995) 477 [astro-ph/9503054].
  * [15] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley et al., _Nine-year wilkinson microwave anisotropy probe ( wmap ) observations: Cosmological parameter results_ , _The Astrophysical Journal Supplement Series_ 208 (2013) 19.
  * [16] Planck collaboration, _Planck 2018 results. I. Overview and the cosmological legacy of Planck_ , _Astron. Astrophys._ 641 (2020) A1 [1807.06205].
  * [17] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [18] BICEP2, Keck Array collaboration, _BICEP2 / Keck Array x: Constraints on Primordial Gravitational Waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season_ , _Phys. Rev. Lett._ 121 (2018) 221301 [1810.05216].
  * [19] Planck collaboration, _Planck 2018 results. III. High Frequency Instrument data processing and frequency maps_ , _Astron. Astrophys._ 641 (2020) A3 [1807.06207].
  * [20] Planck collaboration, _Planck 2015 results. XIII. Cosmological parameters_ , _Astron. Astrophys._ 594 (2016) A13 [1502.01589].
  * [21] Planck collaboration, _Planck 2018 results. X. Constraints on inflation_ , _Astron. Astrophys._ 641 (2020) A10 [1807.06211].
  * [22] D. K. Hazra, A. Shafieloo and G. F. Smoot, _Reconstruction of broad features in the primordial spectrum and inflaton potential from Planck_ , _JCAP_ 12 (2013) 035 [1310.3038].
  * [23] D. K. Hazra, A. Shafieloo and T. Souradeep, _Primordial power spectrum from Planck_ , _JCAP_ 11 (2014) 011 [1406.4827].
  * [24] P. Hunt and S. Sarkar, _Search for features in the spectrum of primordial perturbations using Planck and other datasets_ , _JCAP_ 12 (2015) 052 [1510.03338].
  * [25] G. Obied, C. Dvorkin, C. Heinrich, W. Hu and V. Miranda, _Inflationary versus reionization features from $Planck$ 2015 data_, _Phys. Rev. D_ 98 (2018) 043518 [1803.01858].
  * [26] S. Hannestad, _Reconstructing the inflationary power spectrum from CMBR data_ , _Phys. Rev. D_ 63 (2001) 043009 [astro-ph/0009296].
  * [27] A. Shafieloo and T. Souradeep, _Primordial power spectrum from WMAP_ , _Phys. Rev. D_ 70 (2004) 043523 [astro-ph/0312174].
  * [28] P. Mukherjee and Y. Wang, _Model-independent reconstruction of the primordial power spectrum from WMAP data_ , _Astrophys. J._ 599 (2003) 1 [astro-ph/0303211].
  * [29] S. L. Bridle, A. M. Lewis, J. Weller and G. Efstathiou, _Reconstructing the primordial power spectrum_ , _Mon. Not. Roy. Astron. Soc._ 342 (2003) L72 [astro-ph/0302306].
  * [30] N. Kogo, M. Sasaki and J. Yokoyama, _Constraining cosmological parameters by the cosmic inversion method_ , _Prog. Theor. Phys._ 114 (2005) 555 [astro-ph/0504471].
  * [31] S. M. Leach, _Measuring the primordial power spectrum: Principal component analysis of the cosmic microwave background_ , _Mon. Not. Roy. Astron. Soc._ 372 (2006) 646 [astro-ph/0506390].
  * [32] D. Tocchini-Valentini, Y. Hoffman and J. Silk, _Non-parametric reconstruction of the primordial power spectrum at horizon scales from wmap data_ , _Mon. Not. Roy. Astron. Soc._ 367 (2006) 1095 [astro-ph/0509478].
  * [33] A. Shafieloo and T. Souradeep, _Estimation of Primordial Spectrum with post-WMAP 3 year data_ , _Phys. Rev. D_ 78 (2008) 023511 [0709.1944].
  * [34] P. Paykari and A. H. Jaffe, _Optimal Binning of the Primordial Power Spectrum_ , _Astrophys. J._ 711 (2010) 1 [0902.4399].
  * [35] G. Nicholson and C. R. Contaldi, _Reconstruction of the Primordial Power Spectrum using Temperature and Polarisation Data from Multiple Experiments_ , _JCAP_ 07 (2009) 011 [0903.1106].
  * [36] C. Gauthier and M. Bucher, _Reconstructing the primordial power spectrum from the CMB_ , _JCAP_ 10 (2012) 050 [1209.2147].
  * [37] J. A. Vazquez, M. Bridges, M. P. Hobson and A. N. Lasenby, _Model selection applied to reconstruction of the Primordial Power Spectrum_ , _JCAP_ 06 (2012) 006 [1203.1252].
  * [38] D. K. Hazra, A. Shafieloo and T. Souradeep, _Primordial power spectrum: a complete analysis with the WMAP nine-year data_ , _JCAP_ 07 (2013) 031 [1303.4143].
  * [39] P. Hunt and S. Sarkar, _Reconstruction of the primordial power spectrum of curvature perturbations using multiple data sets_ , _JCAP_ 01 (2014) 025 [1308.2317].
  * [40] A. A. Starobinsky, _Spectrum of adiabatic perturbations in the universe when there are singularities in the inflation potential_ , _JETP Lett._ 55 (1992) 489.
  * [41] P. Ivanov, P. Naselsky and I. Novikov, _Inflation and primordial black holes as dark matter_ , _Phys. Rev. D_ 50 (1994) 7173.
  * [42] J. A. Adams, B. Cresswell and R. Easther, _Inflationary perturbations from a potential with a step_ , _Phys. Rev. D_ 64 (2001) 123514 [astro-ph/0102236].
  * [43] L. Covi, J. Hamann, A. Melchiorri, A. Slosar and I. Sorbera, _Inflation and WMAP three year data: Features have a Future!_ , _Phys. Rev. D_ 74 (2006) 083509 [astro-ph/0606452].
  * [44] A. Ashoorioon and A. Krause, _Power Spectrum and Signatures for Cascade Inflation_ , hep-th/0607001.
  * [45] M. Joy, V. Sahni and A. A. Starobinsky, _A New Universal Local Feature in the Inflationary Perturbation Spectrum_ , _Phys. Rev. D_ 77 (2008) 023514 [0711.1585].
  * [46] M. Joy, A. Shafieloo, V. Sahni and A. A. Starobinsky, _Is a step in the primordial spectral index favored by CMB data ?_ , _JCAP_ 06 (2009) 028 [0807.3334].
  * [47] D. K. Hazra, M. Aich, R. K. Jain, L. Sriramkumar and T. Souradeep, _Primordial features due to a step in the inflaton potential_ , _JCAP_ 10 (2010) 008 [1005.2175].
  * [48] V. Miranda, W. Hu and P. Adshead, _Warp Features in DBI Inflation_ , _Phys. Rev. D_ 86 (2012) 063529 [1207.2186].
  * [49] M. Benetti, _Updating constraints on inflationary features in the primordial power spectrum with the Planck data_ , _Phys. Rev. D_ 88 (2013) 087302 [1308.6406].
  * [50] A. Gallego Cadavid and A. E. Romano, _Effects of discontinuities of the derivatives of the inflaton potential_ , _Eur. Phys. J. C_ 75 (2015) 589 [1404.2985].
  * [51] J. Chluba, J. Hamann and S. P. Patil, _Features and New Physical Scales in Primordial Observables: Theory and Observation_ , _Int. J. Mod. Phys. D_ 24 (2015) 1530023 [1505.01834].
  * [52] R. Bousso, D. Harlow and L. Senatore, _Inflation After False Vacuum Decay: New Evidence from BICEP2_ , _JCAP_ 12 (2014) 019 [1404.2278].
  * [53] R. Allahverdi, K. Enqvist, J. Garcia-Bellido and A. Mazumdar, _Gauge invariant MSSM inflaton_ , _Phys. Rev. Lett._ 97 (2006) 191304 [hep-ph/0605035].
  * [54] R. K. Jain, P. Chingangbam, J.-O. Gong, L. Sriramkumar and T. Souradeep, _Punctuated inflation and the low CMB multipoles_ , _JCAP_ 01 (2009) 009 [0809.3915].
  * [55] X. Chen, R. Easther and E. A. Lim, _Generation and Characterization of Large Non-Gaussianities in Single Field Inflation_ , _JCAP_ 04 (2008) 010 [0801.3295].
  * [56] L. McAllister, E. Silverstein and A. Westphal, _Gravity Waves and Linear Inflation from Axion Monodromy_ , _Phys. Rev. D_ 82 (2010) 046003 [0808.0706].
  * [57] R. Flauger, L. McAllister, E. Pajer, A. Westphal and G. Xu, _Oscillations in the CMB from Axion Monodromy Inflation_ , _JCAP_ 06 (2010) 009 [0907.2916].
  * [58] C. Pahud, M. Kamionkowski and A. R. Liddle, _Oscillations in the inflaton potential?_ , _Phys. Rev. D_ 79 (2009) 083503 [0807.0322].
  * [59] R. Flauger and E. Pajer, _Resonant Non-Gaussianity_ , _JCAP_ 01 (2011) 017 [1002.0833].
  * [60] X. Chen, _Folded Resonant Non-Gaussianity in General Single Field Inflation_ , _JCAP_ 12 (2010) 003 [1008.2485].
  * [61] M. Aich, D. K. Hazra, L. Sriramkumar and T. Souradeep, _Oscillations in the inflaton potential: Complete numerical treatment and comparison with the recent and forthcoming CMB datasets_ , _Phys. Rev. D_ 87 (2013) 083526 [1106.2798].
  * [62] H. Peiris, R. Easther and R. Flauger, _Constraining Monodromy Inflation_ , _JCAP_ 09 (2013) 018 [1303.2616].
  * [63] P. D. Meerburg and D. N. Spergel, _Searching for oscillations in the primordial power spectrum. II. Constraints from Planck data_ , _Phys. Rev. D_ 89 (2014) 063537 [1308.3705].
  * [64] R. Easther and R. Flauger, _Planck Constraints on Monodromy Inflation_ , _JCAP_ 02 (2014) 037 [1308.3736].
  * [65] H. Motohashi and W. Hu, _Running from Features: Optimized Evaluation of Inflationary Power Spectra_ , _Phys. Rev. D_ 92 (2015) 043501 [1503.04810].
  * [66] V. Miranda, W. Hu, C. He and H. Motohashi, _Nonlinear Excitations in Inflationary Power Spectra_ , _Phys. Rev. D_ 93 (2016) 023504 [1510.07580].
  * [67] S. Cremonini, Z. Lalak and K. Turzynski, _Strongly Coupled Perturbations in Two-Field Inflationary Models_ , _JCAP_ 1103 (2011) 016 [1010.3021].
  * [68] A. Achucarro, J.-O. Gong, S. Hardeman, G. A. Palma and S. P. Patil, _Features of heavy physics in the CMB power spectrum_ , _JCAP_ 01 (2011) 030 [1010.3693].
  * [69] M. Braglia, D. K. Hazra, L. Sriramkumar and F. Finelli, _Generating primordial features at large scales in two field models of inflation_ , _JCAP_ 08 (2020) 025 [2004.00672].
  * [70] M. Braglia, D. K. Hazra, F. Finelli, G. F. Smoot, L. Sriramkumar and A. A. Starobinsky, _Generating PBHs and small-scale GWs in two-field models of inflation_ , _JCAP_ 08 (2020) 001 [2005.02895].
  * [71] X. Chen, _Primordial features as evidence for inflation_ , _JCAP_ 01 (2012) 038 [1104.1323].
  * [72] X. Chen and C. Ringeval, _Searching for Standard Clocks in the Primordial Universe_ , _JCAP_ 08 (2012) 014 [1205.6085].
  * [73] X. Chen, M. H. Namjoo and Y. Wang, _Models of the Primordial Standard Clock_ , _JCAP_ 1502 (2015) 027 [1411.2349].
  * [74] X. Chen and M. H. Namjoo, _Standard Clock in Primordial Density Perturbations and Cosmic Microwave Background_ , _Phys. Lett. B_ 739 (2014) 285 [1404.1536].
  * [75] M. Braglia, X. Chen and D. K. Hazra, _Probing Primordial Features with the Stochastic Gravitational Wave Background_ , _JCAP_ 03 (2021) 005 [2012.05821].
  * [76] M. Braglia, X. Chen and D. K. Hazra, _Comparing multi-field primordial feature models with the Planck data_ , _JCAP_ 06 (2021) 005 [2103.03025].
  * [77] M. Braglia, X. Chen and D. K. Hazra, _Uncovering the History of Cosmic Inflation from Anomalies in Cosmic Microwave Background Spectra_ , 2106.07546.
  * [78] M. Braglia, X. Chen and D. K. Hazra, _Primordial Standard Clock Models and CMB Residual Anomalies_ , 2108.10110.
  * [79] G. Cañas Herrera, J. Torrado and A. Achúcarro, _Bayesian reconstruction of the inflaton’s speed of sound using CMB data_ , _Phys. Rev. D_ 103 (2021) 12 [2012.04640].
  * [80] WMAP collaboration, _First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for inflation_ , _Astrophys. J. Suppl._ 148 (2003) 213 [astro-ph/0302225].
  * [81] D. K. Hazra, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, _Primordial features and Planck polarization_ , _JCAP_ 1609 (2016) 009 [1605.02106].
  * [82] D. K. Hazra, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, _Inflation with Whip-Shaped Suppressed Scalar Power Spectra_ , _Phys. Rev. Lett._ 113 (2014) 071301 [1404.0360].
  * [83] D. K. Hazra, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, _Wiggly Whipped Inflation_ , _JCAP_ 1408 (2014) 048 [1405.2012].
  * [84] D. K. Hazra, D. Paoletti, M. Ballardini, F. Finelli, A. Shafieloo, G. F. Smoot et al., _Probing features in inflaton potential and reionization history with future CMB space observations_ , _JCAP_ 1802 (2018) 017 [1710.01205].
  * [85] B. L’Huillier, A. Shafieloo, D. K. Hazra, G. F. Smoot and A. A. Starobinsky, _Probing features in the primordial perturbation spectrum with large-scale structure data_ , _Mon. Not. Roy. Astron. Soc._ 477 (2018) 2503 [1710.10987].
  * [86] I. Debono, D. K. Hazra, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, _Constraints on features in the inflationary potential from future Euclid data_ , _Mon. Not. Roy. Astron. Soc._ 496 (2020) 3448 [2003.05262].
  * [87] Planck collaboration, _Planck 2018 results. IX. Constraints on primordial non-Gaussianity_ , _Astron. Astrophys._ 641 (2020) A9 [1905.05697].
  * [88] J. M. Maldacena, _Non-Gaussian features of primordial fluctuations in single field inflationary models_ , _JHEP_ 05 (2003) 013 [astro-ph/0210603].
  * [89] D. Seery and J. E. Lidsey, _Primordial non-Gaussianities in single field inflation_ , _JCAP_ 06 (2005) 003 [astro-ph/0503692].
  * [90] X. Chen, R. Easther and E. A. Lim, _Large Non-Gaussianities in Single Field Inflation_ , _JCAP_ 06 (2007) 023 [astro-ph/0611645].
  * [91] X. Chen, _Primordial Non-Gaussianities from Inflation Models_ , _Adv. Astron._ 2010 (2010) 638979 [1002.1416].
  * [92] J. Martin and L. Sriramkumar, _The scalar bi-spectrum in the Starobinsky model: The equilateral case_ , _JCAP_ 01 (2012) 008 [1109.5838].
  * [93] D. K. Hazra, L. Sriramkumar and J. Martin, _BINGO: A code for the efficient computation of the scalar bi-spectrum_ , _JCAP_ 05 (2013) 026 [1201.0926].
  * [94] D. K. Hazra, J. Martin and L. Sriramkumar, _The scalar bi-spectrum during preheating in single field inflationary models_ , _Phys. Rev. D_ 86 (2012) 063523 [1206.0442].
  * [95] P. Adshead, W. Hu and V. Miranda, _Bispectrum in Single-Field Inflation Beyond Slow-Roll_ , _Phys. Rev. D_ 88 (2013) 023507 [1303.7004].
  * [96] A. Achúcarro, V. Atal, P. Ortiz and J. Torrado, _Localized correlated features in the CMB power spectrum and primordial bispectrum from a transient reduction in the speed of sound_ , _Phys. Rev. D_ 89 (2014) 103006 [1311.2552].
  * [97] V. Sreenath, D. K. Hazra and L. Sriramkumar, _On the scalar consistency relation away from slow roll_ , _JCAP_ 02 (2015) 029 [1410.0252].
  * [98] J. Martin, L. Sriramkumar and D. K. Hazra, _Sharp inflaton potentials and bi-spectra: Effects of smoothening the discontinuity_ , _JCAP_ 09 (2014) 039 [1404.6093].
  * [99] A. Achucarro, V. Atal, B. Hu, P. Ortiz and J. Torrado, _Inflation with moderately sharp features in the speed of sound: Generalized slow roll and in-in formalism for power spectrum and bispectrum_ , _Phys. Rev. D_ 90 (2014) 023511 [1404.7522].
  * [100] J. R. Fergusson, H. F. Gruetjen, E. P. S. Shellard and M. Liguori, _Combining power spectrum and bispectrum measurements to detect oscillatory features_ , _Phys. Rev. D_ 91 (2015) 023502 [1410.5114].
  * [101] P. D. Meerburg, M. Münchmeyer and B. Wandelt, _Joint resonant CMB power spectrum and bispectrum estimation_ , _Phys. Rev. D_ 93 (2016) 043536 [1510.01756].
  * [102] S. Appleby, J.-O. Gong, D. K. Hazra, A. Shafieloo and S. Sypsas, _Direct search for features in the primordial bispectrum_ , _Phys. Lett. B_ 760 (2016) 297 [1512.08977].
  * [103] M. Dias, J. Frazer, D. J. Mulryne and D. Seery, _Numerical evaluation of the bispectrum in multiple field inflation—the transport approach with code_ , _JCAP_ 12 (2016) 033 [1609.00379].
  * [104] Planck collaboration, _Planck 2013 results. XXII. Constraints on inflation_ , _Astron. Astrophys._ 571 (2014) A22 [1303.5082].
  * [105] G. D. Racca, R. Laureijs, L. Stagnaro, J.-C. Salvignol, J. Lorenzo Alvarez, G. Saavedra Criado et al., _The Euclid mission design_ , in _Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave_ , H. A. MacEwen, G. G. Fazio, M. Lystrup, N. Batalha, N. Siegler and E. C. Tong, eds., vol. 9904 of _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_ , p. 99040O, July, 2016, DOI [1610.05508].
  * [106] LSST collaboration, _LSST: from Science Drivers to Reference Design and Anticipated Data Products_ , _Astrophys. J._ 873 (2019) 111 [0805.2366].
  * [107] L. Boubekeur and D. H. Lyth, _Hilltop inflation_ , _JCAP_ 07 (2005) 010 [hep-ph/0502047].
  * [108] G. Efstathiou and S. Chongchitnan, _The search for primordial tensor modes_ , _Prog. Theor. Phys. Suppl._ 163 (2006) 204 [astro-ph/0603118].
  * [109] C. R. Contaldi, M. Peloso, L. Kofman and A. D. Linde, _Suppressing the lower multipoles in the CMB anisotropies_ , _JCAP_ 07 (2003) 002 [astro-ph/0303636].
  * [110] L. T. Hergt, W. J. Handley, M. P. Hobson and A. N. Lasenby, _Constraining the kinetically dominated Universe_ , _Phys. Rev. D_ 100 (2019) 023501 [1809.07737].
  * [111] H. V. Ragavendra, D. Chowdhury and L. Sriramkumar, _Suppression of scalar power on large scales and associated bispectra_ , 2003.01099.
  * [112] R. Kallosh and A. Linde, _Universality Class in Conformal Inflation_ , _JCAP_ 07 (2013) 002 [1306.5220].
  * [113] R. Kallosh and A. Linde, _Multi-field Conformal Cosmological Attractors_ , _JCAP_ 12 (2013) 006 [1309.2015].
  * [114] R. Kallosh, A. Linde and D. Roest, _Superconformal Inflationary $\alpha$-Attractors_, _JHEP_ 11 (2013) 198 [1311.0472].
  * [115] A. Lewis, A. Challinor and A. Lasenby, _Efficient computation of CMB anisotropies in closed FRW models_ , _Astrophys. J._ 538 (2000) 473 [astro-ph/9911177].
  * [116] Planck collaboration, _Planck 2018 results. V. CMB power spectra and likelihoods_ , _Astron. Astrophys._ 641 (2020) A5 [1907.12875].
  * [117] Planck collaboration, _Planck 2018 results. VIII. Gravitational lensing_ , _Astron. Astrophys._ 641 (2020) A8 [1807.06210].
  * [118] G. Efstathiou and S. Gratton, _A Detailed Description of the CamSpec Likelihood Pipeline and a Reanalysis of the Planck High Frequency Maps_ , 1910.00483.
  * [119] W. J. Handley, M. P. Hobson and A. N. Lasenby, _POLYCHORD: next-generation nested sampling_ , _Mon. Not. Roy. Astron. Soc._ 453 (2015) 4384 [1506.00171].
  * [120] W. J. Handley, M. P. Hobson and A. N. Lasenby, _PolyChord: nested sampling for cosmology_ , _Mon. Not. Roy. Astron. Soc._ 450 (2015) L61 [1502.01856].
  * [121] A. Lewis and S. Bridle, _Cosmological parameters from CMB and other data: A Monte Carlo approach_ , _Phys. Rev. D_ 66 (2002) 103511 [astro-ph/0205436].
  * [122] A. Lewis, _GetDist: a Python package for analysing Monte Carlo samples_ , 1910.13970.
  * [123] H. Jeffreys, _Theory of Probability_. Oxford University Press: Oxford, 1961\.
  * [124] R. Trotta, _Applications of Bayesian model selection to cosmological parameters_ , _”Mon. Not. Roy. Astron. Soc.”_ 378 (2007) 72 [arXiv:astro-ph/0504022].
  * [125] M. J. Powell, _The bobyqa algorithm for bound constrained optimization without derivatives_ , .
  * [126] F. Arroja, A. E. Romano and M. Sasaki, _Large and strong scale dependent bispectrum in single field inflation from a sharp feature in the mass_ , _Phys. Rev. D_ 84 (2011) 123503 [1106.5384].
  * [127] M. Hazumi, P. A. Ade, A. Adler, E. Allys, K. Arnold, D. Auguste et al., _Litebird satellite: Jaxa’s new strategic l-class mission for all-sky surveys of cosmic microwave background polarization_ , _Space Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter Wave_ (2020) .
  * [128] “CMB Bharat – Assessing the prospects for frontier CMB space experiments from India.”
  * [129] K. Young, M. Alvarez, N. Battaglia, J. Bock, J. Borrill, D. Chuss et al., _Optical design of pico, a concept for a space mission to probe inflation and cosmic origins_ , 2018.
  * [130] DESI collaboration, _The DESI Experiment Part I: Science,Targeting, and Survey Design_ , 1611.00036.
  * [131] D. K. Hazra, _Changes in the halo formation rates due to features in the primordial spectrum_ , _JCAP_ 03 (2013) 003 [1210.7170].
  * [132] X. Chen, C. Dvorkin, Z. Huang, M. H. Namjoo and L. Verde, _The Future of Primordial Features with Large-Scale Structure Surveys_ , _JCAP_ 11 (2016) 014 [1605.09365].
  * [133] M. Ballardini, F. Finelli, C. Fedeli and L. Moscardini, _Probing primordial features with future galaxy surveys_ , _JCAP_ 10 (2016) 041 [1606.03747].
  * [134] F. Beutler, M. Biagetti, D. Green, A. Slosar and B. Wallisch, _Primordial Features from Linear to Nonlinear Scales_ , _Phys. Rev. Res._ 1 (2019) 033209 [1906.08758].
  * [135] M. Ballardini, R. Murgia, M. Baldi, F. Finelli and M. Viel, _Non-linear damping of superimposed primordial oscillations on the matter power spectrum in galaxy surveys_ , _JCAP_ 04 (2020) 030 [1912.12499].