capbtabboxtable[][]

CPPC-2022-09

# Constraining primordial tensor features with the anisotropies of the Cosmic
Microwave Background

Jan Hamann and Ameek Malhotra

###### Abstract

It is commonly assumed that the stochastic background of gravitational waves
on cosmological scales follows an almost scale-independent power spectrum, as
generically predicted by the inflationary paradigm. However, it is not
inconceivable that the spectrum could have strongly scale-dependent features,
generated, e.g., via transient dynamics of spectator axion-gauge fields during
inflation. Using the temperature and polarisation maps from the Planck and
BICEP/Keck datasets, we search for such features, taking the example of a log-
normal bump in the primordial tensor spectrum at CMB scales. We do not find
any evidence for the existence of bump-like tensor features at present, but
demonstrate that future CMB experiments such as LiteBIRD and CMB-S4 will
greatly improve our prospects of determining the amplitude, location and width
of such a bump. We also highlight the role of delensing in constraining these
features at angular scales $\ell\gtrsim 100$.

##  1 Introduction

The inflationary paradigm [1, 2, 3, 4, 5] is not only a compelling solution to
the horizon and flatness problems of hot big bang cosmology, but also provides
a means to naturally generate the seeds of the observed Cosmic Microwave
Background (CMB) anisotropies and the Large Scale Structure (LSS) [6, 7, 8, 9,
10] via quantum fluctuations. Cosmological observations are consistent with a
power spectrum of scalar (density) fluctuations that is adiabatic, Gaussian
and nearly (but not exactly) scale-invariant, in excellent agreement with the
predictions of the simplest single field slow roll (SFSR) models [11]. Another
universal prediction of inflation is the existence of a stochastic
gravitational wave background produced during this epoch [12, 13, 14, 15].

Although such primordial gravitational waves (PGW) have not yet been detected,
their effects could show up in a wide variety of cosmological observables.
Most notably, PGW contribute to both the temperature and polarisation
anisotropies of the CMB [15, 16, 17, 18, 19, 20, 21, 22], and this fact can be
used to constrain their amplitude [23, 24, 25, 26]. Indeed, the most stringent
bounds on the amplitude of these PGW come from the CMB which constrains the
tensor-scalar ratio to $r<0.032$ [24] at $95\%$ CL for a nearly scale-
invariant power spectrum of tensor fluctuations, as expected in SFSR models.

Interferometric detection with next generation detectors like ET [27] and LISA
[28] may also be a possibility, but only for inflationary models departing
strongly from SFSR dynamics on direct detection scales [29, 30]. Additionally,
PGW could also be detected via Pulsar Timing arrays, through their imprints on
large scale structure (‘tensor fossils’), spectral distortions of the CMB, as
well as through the gravitational lensing effects of PGW, see [31] for an
overview.

The $B$-mode polarisation of the CMB remains the most promising avenue to
detect these primordial tensor perturbations, keeping in the mind the
projected sensitivities as well the foreground/noise sources affecting the
various probes mentioned above. A detection of these primordial tensor
perturbations would be extremely significant since their amplitude in SFSR
models is directly related to the energy scale of inflation
$V_{\mathrm{inf}}\simeq 3\pi^{2}A_{\mathrm{s}}M_{\mathrm{pl}}^{4}\,r/2$ [31]
and would allow a precise reconstruction of the inflaton potential in the
observable window [32, 33, 34]. Interestingly, certain well-motivated single
field inflationary models are currently in excellent agreement with the CMB
data [11], e.g. the Starobinsky model [35, 36, 37], and predict values of $r$
within the reach of next generation of CMB experiments. For these reasons, it
is easy to understand why the search for PGWs is an important science goal for
future probes like the BICEP array [38], Simons Observatory [39], CMB-S4 [40]
and LiteBIRD [41].

While a detection of PGW in itself would be extremely valuable, additional
information on the production mechanism could be gleaned from measuring the
shape of the GW spectrum. For PGW generated from vacuum fluctuations, the
shape of the spectrum is a power law with spectral index related to the
tensor-scalar ratio as $n_{\mathrm{t}}\simeq-r/8$. This ‘tensor consistency
relation’ is valid for PGW arising from SFSR dynamics [42] and in the event of
a $B$-mode detection, provides a way to confirm whether the observed PGW are
arising from vacuum fluctuations or not. Unfortunately however, testing this
relation appears out of reach with CMB data alone, even with the sensitivity
of CMB-S4 [40] or LiteBIRD [43, 26]. On the other hand, large deviations from
the consistency relation could still be observed with these experiments. Such
deviations would signal a departure from SFSR dynamics at CMB scales which is
possible in inflationary models with additional fields capable of sourcing
PGW, e.g. [44, 45, 46, 47, 48, 49, 50, 51, 43, 52, 53].

In this paper we study one such example of a deviation from a power-law
spectrum of tensor perturbations, namely a bump-like PGW feature at CMB
scales. Such a feature is typical of GW sourced from spectator axion-gauge
fields during inflation, leading to a strong scale-dependence. Parameterising
the spectrum of these sourced PGW as a log-normal function, we present
constraints on the amplitude, width and location of the peak of the log-normal
using temperature and polarisation data from the Planck and BICEP/Keck
datasets and forecast the discovery potential of CMB-S4 and LiteBIRD for this
scenario.

The paper is organised as follows: in Section 2 we describe the tensor power
spectrum parameterisation and discuss possible inflationary models where such
a spectrum may arise. We also describe the effects of such a tensor power
spectrum shape on the CMB temperature and polarisation anisotropies. In
Section 3 we present constraints on the model parameters from current data and
forecast sensitivities of future CMB experiments. Finally, we conclude in
Section 4.

##  2 Tensor modes from Inflation

In this section we first present the tensor power spectrum parameterisation
used to describe the bump-like feature. We then discuss the effect of such a
tensor spectrum shape on the CMB temperature and polarisation anisotropies and
conclude this section with a discussion of inflationary models where one can
expect such power spectrum shapes.

###  2.1 Log-normal spectrum

We parameterise the tensor power spectrum on CMB scales as a log-normal with

$\displaystyle\mathcal{P}_{\mathrm{t}}(k)=r_{\mathrm{pk}}A_{\mathrm{s}}\exp\left[-\frac{\left(\ln\left(k/k_{\rm
pk}\right)\right)^{2}}{2\sigma^{2}}\right].$ (2.1)

Here $\sigma$ denotes the width of the log-normal, $k_{\rm pk}$ the location
of the peak and $r_{\mathrm{pk}}$ the rescaled amplitude of the tensor power
spectrum at the peak scale in terms of the scalar amplitude
$A_{\mathrm{s}}=2.09\times 10^{-9}$ at the pivot scale
$k_{\mathrm{p}}=0.05\,\text{Mpc}^{-1}$. When $\sigma\gg 1$, the power spectrum
becomes degenerate with a flat spectrum on scales

$\displaystyle-\sigma\lesssim\log k/k_{\rm pk}\lesssim\sigma.$ (2.2)

For comparison, the power spectrum of tensor perturbations from vacuum
fluctuations during inflation has the following form,

$\displaystyle\mathcal{P}_{\mathrm{t}}^{\rm
vac}(k)=rA_{\mathrm{s}}\left(\frac{k}{k_{\mathrm{p}}}\right)^{n_{\mathrm{t}}}$
(2.3)

where $r$ denotes the vacuum tensor to scalar ratio and $n_{\mathrm{t}}$ the
tensor spectral tilt. The two power spectra are plotted in the left panel of
Figure 1 taking $r=0.03$ and $n_{\mathrm{t}}$ given by the consistency
relation for the SFSR spectrum and $r_{\mathrm{pk}}=0.1,\;\sigma=1$ and
$k_{\rm pk}=10^{-3}$ for the log-normal.

Figure 1: Left : Power spectra with tensor spectrum parameters
$r_{\mathrm{pk}}=0.1,~{}\sigma=1$ and $k_{\rm pk}=10^{-3}$. Right: The
temperature and $B$-mode anisotropies for the same value of the tensor power
spectrum parameters.

####  2.1.1 CMB anisotropies

Much like the scalar perturbations, tensors also contribute to the CMB
anisotropies and this contribution can be expressed as

$\displaystyle C_{\ell,\rm t}^{XY}=4\pi\int d\ln
k\,\mathcal{P}_{\mathrm{t}}(k)\,\Delta_{\ell,\rm t}^{X}(k)\Delta_{\ell,\rm
t}^{Y}(k).$ (2.4)

Here $X,Y=T,E,B$ and $\Delta_{\ell,\rm t}^{X}$ denotes the corresponding
transfer function for the tensor source. The analytical expressions for these
transfer functions can be found in [19, 20, 21, 22], and they can be
numerically computed using Boltzmann codes such as CLASS [54] or CAMB [55].

Physically, tensor modes generate a quadrupolar anisotropy in the radiation
density field at last scattering and reionisation, and thus tensor
perturbations lead to anisotropies in both temperature and polarisation of the
CMB. Note that the correlations $\langle TB\rangle$ and $\langle EB\rangle$
vanish due to the fact that $B$-modes are parity-odd whereas $T$ and $E$ are
not [56].

In general, the tensor contribution is relevant mainly on the largest angular
scales since tensor modes decay rapidly on sub-horizon scales ($\ell\gtrsim
100$ at recombination). For the temperature anisotropies, the tensor
contribution is smaller than the scalar one by a factor of the tensor-scalar
ratio $r$ on these large scales. As for the $B$-mode polarisation, for
$r\gtrsim 0.001$ this is dominated by the primordial tensor perturbations on
these scales since scalars cannot source $B$-modes at linear order.

However, scalars do generate $B$-mode polarisation at second order via the
lensing of the $E$-modes [57] and these lensing-induced $B$-modes dominate the
primordial $B$-mode spectrum at smaller scales ($\ell\gtrsim 100$). In the
$B$-mode polarisation angular power spectrum one can also see the reionization
bump ($\ell\leq 10$) [21] and the recombination bump ($\ell\sim 80$) [58],
corresponding to scales that re-enter the horizon at those times.

We plot the tensor contribution to the temperature and $B$-mode angular power
spectra in the right panel of Figure 1. The difference between the SFSR
prediction and the log-normal spectrum can also be understood from the same
figure. The additional power on scales close to the peak scale $k=10^{-3}$
leads to an enhancement of the temperature as well as the polarisation
anisotropies on large angular scales. Away from the peak scales, the anistropy
power spectrum in the log-normal case falls off rapidly relative to the SFSR
one.

The effect of varying the log-normal parameters on the $B$-mode spectrum is
illustrated in Figure 2. The location of the peak scale $k_{\rm pk}$ can be
directly related to the angular scale at which the anisotropies are enhanced,
whereas the peak width $\sigma$ controls the range of scales around the peak
where this happens. Roughly speaking, peaks at $k_{\rm
pk}=10^{-4},10^{-3},10^{-2}$ can be mapped to an enhancement of power centered
around angular scales of $\ell\sim 2,\,10\;\text{and}\;100$ respectively.

Figure 2: Effect of varying $\sigma$ and $k_{\rm pk}$ on $C_{\ell}^{BB}$
taking $r_{\mathrm{pk}}=0.1$. The black dotted lines show the $B$-modes from
lensing.

####  2.1.2 Inflationary Models

A peaked primordial tensor power spectrum is characteristic of transient
phenomena occurring during inflation that source gravitational waves on scales
of order the horizon size at that time. The prototypical examples of this type
are inflationary models with a spectator sector involving an axion coupled to
a gauge field [47, 48, 49, 50, 51, 43, 52, 53]. The spectator sector
Lagrangian in this case can be written as

$\displaystyle\mathcal{L}_{\mathrm{spec}}=-\frac{1}{2}(\partial\chi)^{2}-U(\chi)-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu,\nu}+\frac{\lambda\,\chi}{f}F_{\mu\nu}\tilde{F}^{\mu,\nu}$
(2.5)

where $\chi$ represents the axion-like field, $U(\chi)$ is the axion
potential, $f$ the decay constant, $F_{\mu\nu}$ the field strength tensor of
the gauge field and $\lambda$ a coupling constant. The specific shape of the
GW spectrum in such models arises from the fact that one of the gauge field
helicities experiences a transient instability at horizon crossing and gets
enhanced relative to the other. Thus, in general the GW spectrum sourced in
these models is chiral, i.e. $\mathcal{P}_{R}\neq\mathcal{P}_{L}$. Although
one can test for chirality through the observation of non-zero $\langle
TE\rangle$ or $\langle EB\rangle$ cross-spectra [56, 59, 50], in our analysis
we only concern ourselves with the overall shape and amplitude of the tensor
power spectrum and neglect these parity violating correlations in obtaining
the constraints in Section 3. In general the signal is much weaker in the
cross-spectra than in the corresponding $\langle BB\rangle$ spectrum, making
them harder to detect [50].

##  3 Constraints on model parameters

Constraints on the axion gauge-field model parameters of equation (2.5),
specifically upper bounds on the effective coupling between the axion and the
gauge field were also obtained in [53] for two different models of the axion
potential $U(\chi)$. The analysis of [53] was carried out for fixed values of
the peak width and location using the profile likelihood approach. However,
our analysis cannot be directly compared with theirs since varying the
effective coupling also affects the sourced scalar spectra whereas here we
work at the level of the log-normal parameters which affects only the
primordial tensor power spectrum through equation (2.1).

###  3.1 Constraints from Planck \+ BK18

For the analysis of our model comprising the base $\Lambda$CDM parameters plus
our additional tensor power spectrum parameters $\\{r_{\mathrm{pk}},k_{\rm
pk},\sigma\\}$, we use the following data: Firstly, the Planck low-$\ell$
temperature+polarisation and high-$\ell$ $TTTEEE$ likelihood [60] and the
Planck lensing likelihood [61]. We shall refer to this combination as Planck
hereafter. Secondly, we also use the most recent BICEP/Keck data release (BK18
hereafter) which contains polarisation data in the multipole range
$20<\ell<330$ obtained from the BICEP2, Keck Array and BICEP3 experiments
[23].

We impose flat priors on the base $\Lambda$CDM parameters as well as flat
priors on $r_{\mathrm{pk}},\sigma$ and $\log_{10}k_{\rm pk}$ in the ranges
shown in Table 1. The CMB power spectra are computed with CAMB [55] and the
parameter space is explored using a Markov Chain Monte Carlo sampler [62, 63],
through its interface with Cobaya [64]. The resulting chains are analysed with
GetDist [65].

Note that we assume in our analysis that the scale-independent vacuum
contribution to the tensor power spectrum has a much lower amplitude than the
log-normal sourced one, so we do not include the variation of $r$ and instead
set it to zero. Current data do not indicate the presence of such vacuum
tensor perturbations and for values of $r$ much smaller than $r_{\mathrm{pk}}$
the constraints on $r_{\mathrm{pk}}$ will not be significantly affected. We
will revisit this assumption when doing the Fisher forecasts in Section 3.2.

Parameter | range
---|---
$r_{\mathrm{pk}}$ | $[0,1/5]$
$\sigma$ | $[1/10,10]$
$\log_{10}k_{\rm pk}$ | $[-5,-1]$
Table 1: Prior ranges for the log-normal tensor power spectrum parameters.

The joint posterior distributions for the scalar and tensor power spectrum
parameters are shown in Figure 3 (for posterior contours of all 9 parameters
we refer the reader to Figure 7 in the Appendix). The tensor spectrum
parameters are found to be mostly uncorrelated with the scalar spectrum ones.
We also see that the strongest constraints on $r$ come from the region
$10^{-3}<k_{\rm pk}<10^{-2}$ which is not surprising since features at these
scales mainly affect anisotropies in the multipole range
$20\lesssim\ell\lesssim 300$, i.e., exactly the range covered by the BK18
$B$-mode data.

For large $\sigma$ the constraints on $r$ reduce to the flat/power-law
constraints, as expected. Figure 4 also shows that large values of $r\gtrsim
0.1$ are only allowed in the region $k_{\rm pk}\lesssim 10^{-4}$ or $k_{\rm
pk}\gtrsim 10^{-2}$ and for $\sigma\lesssim 2$. The parameter limits are
presented in Table 2. The best fit point is found to be
$\\{r_{\mathrm{pk}}=0.04,\,\sigma=0.43,\,k_{\rm pk}=2\times 10^{-2}\\}$ with a
$\Delta\chi^{2}$ relative to base $\Lambda{\text{CDM}}$ of
$\Delta\chi^{2}=-0.22$. This result is fully compatible with the absence of
tensor modes in the Planck+BK18 data.

Figure 3: Marginalised $68\%$ and $95\%$ contours for the scalar and tensor power spectrum parameters. For comparison we also show the same contours obtained using only the Planck data, this mainly relaxes the constraints on $r_{\mathrm{pk}}$. Figure 4: Distribution for the tensor log-normal parameters using Planck+BK18 data. Parameter | 68% limit
---|---
$\sigma$ | $<4.83$
$r_{\mathrm{pk}}$ | $<0.0460$
Table 2: Parameter limits for the tensor log-normal parameters from
Planck+BK18 data.

###  3.2 Forecasts with LiteBIRD + CMB-S4

In this section we present forecasts of the ability of next generation CMB
experiments to constrain the log-normal tensor parameters. We first take the
example of LiteBIRD which is a proposed space-based CMB experiment led by JAXA
with the goal of mapping the temperature and polarisation anisotropies of the
CMB in the multipole range $2\leq\ell\leq 200$ [41]. Thus, the reionization
bump ($\ell\lesssim 10$) as well as the recombination bump ($\ell\sim 80$)
will both be accessible to LiteBIRD. Note that a similar analysis of the
detectability of such signals with LiteBIRD was also performed in Ref. [50],
but only for values of $\sigma$ in the range $2\leq\sigma\leq 10$ and taking
$k_{\rm pk}=7\times 10^{-5},\,5\times 10^{-3}\text{Mpc}^{-1}$.

To estimate the constraining power of LiteBIRD, we perform a simple Fisher
matrix forecast111The results presented in this paper are based on only
varying the tensor parameters in the Fisher forecast. We checked that
including the $\Lambda{\text{CDM}}$ parameters does not significantly affect
the estimates for the tensor parameters’ expected uncertainties. to evaluate
the detection prospects of the log-normal tensor spectrum. The Fisher matrix
for the parameters $\vec{\theta}=(r_{\mathrm{peak}},\sigma,k_{\rm pk})$ can be
written as [66],

$\displaystyle F_{ij}=f_{\rm sky}\sum_{\ell=2}^{\ell_{\rm
max}}\frac{2\ell+1}{2}\Tr\left[\mathbf{C}_{\ell}^{-1}\frac{\partial\mathbf{C}_{\ell}}{\partial\theta_{i}}\mathbf{C}_{\ell}^{-1}\frac{\partial\mathbf{C}_{\ell}}{\partial\theta_{j}}\right]$
(3.1)

where the matrix $\mathbf{C}_{\ell}$ is,

$\displaystyle\begin{pmatrix}C_{\ell}^{TT}+N_{\ell}^{TT}&C_{\ell}^{TE}&0\\\
C_{\ell}^{TE}&C_{\ell}^{EE}+N_{\ell}^{EE}&0\\\
0&0&C_{\ell}^{BB}+N_{\ell}^{BB}\end{pmatrix}$ (3.2)

We also have

$\displaystyle\frac{1}{N_{\ell}^{X,\mathrm{inst}}}=\sum_{\nu_{i}}\frac{1}{N_{\ell,\nu_{i}}^{X}}$
(3.3)

where the instrument noise spectra at an observing frequency $\nu$ can be
written as,

$\displaystyle
N_{\ell,\nu}^{X}=\Delta^{X}_{\nu}\exp\left[\ell(\ell+1)\frac{\theta^{2}_{\rm
FWHM}}{8\log 2}\right]$ (3.4)

with $X=TT,EE,BB$. We take $f_{\rm sky}=0.65,\,\ell_{\rm max}=200$ and adopt
the LiteBIRD instrument noise specifications given in Eq. (3.1) of Ref. [67]
for temperature. For the $B$-mode polarisation we also include foregrounds in
addition to the instrument noise and then use the residual foregrounds plus
post component separation noise as described in [43].222The corresponding
noise data were made available by the authors of Ref. [43] at
https://github.com/pcampeti/SGWBProbe. The lensing $B$-modes also act as a
noise component in the search for the primordial signal and are included in
the calculation of the Fisher matrices.

The marginalised $1\sigma$ error on the parameter $\theta_{i}$ is then given
by

$\displaystyle\sigma_{i}=\sqrt{\left(F^{-1}\right)_{ii}}.$ (3.5)

In the left panel of Figure 5 we present the forecast for the parameter set
$\\{r_{\mathrm{pk}}=0.04,\,\sigma=2,\,k_{\rm pk}=10^{-3}\\}$. Although this
parameter set is quite close to the upper limits from the Planck+BK18 data, it
is useful to understand the constraining ability of LiteBIRD.

One can see that the tightest constraints can be achieved for
$r_{\mathrm{pk}}$ whereas precise measurements of $\sigma$ require a
significant amount of delensing. This should be clear from the fact that to
measure the width of the primordial spectrum accurately, we need to detect the
primordial $B$-mode signal over a larger range of scales. However, significant
delensing using CMB data is not possible with LiteBIRD alone since it requires
information at small scales which are inaccessible to the experiment. This
issue could however be overcome by using presently existing or future external
datasets such as those from Planck or CMB-S4.

Figure 5: Fisher forecast for the tensor parameters assuming fiducial models
with tensor features that have a chance of being detected by LiteBIRD and
CMB-S4, taking $r_{\mathrm{pk}}=0.05,\,\sigma=2,\,k_{\rm pk}=10^{-3}$ (left)
and $r_{\mathrm{pk}}=0.04,\,\sigma=0.5,\,k_{\rm pk}=2\times 10^{-2}$ (right).
The blue ellipses show the estimates in the case where the CMB-S4 data are
delensed with $A_{\mathrm{delens}}=0.8$.

In the right panel of Figure 5, we also present forecasts for the case of
$\\{r_{\mathrm{pk}}=0.04,\,\sigma=0.5,\,k_{\rm pk}=2\times 10^{-2}\\}$,
corresponding to a parameter set which produces features on scales $\ell\sim
100$. In this case, using LiteBIRD data alone, the constraints on
$r_{\mathrm{pk}}$ are weaker since this corresponds to scales where the
$B$-mode signal is lensing dominated. Significant delensing needs to be
achieved to precisely constrain this scenario. For this purpose, we consider
the possibility of delensing using the high-resolution capabilities of CMB-S4
[40, 68].

CMB-S4 forecasts for $r$ typically assume $f_{\mathrm{sky}}=0.03$ and we take
the same value here for the Fisher forecasts of the lognormal parameters. This
specific choice of $f_{\mathrm{sky}}$ arises from an optimisation procedure
which takes into account delensing requirements, foreground mitigation and
reproducibility of results across the sky [68]. We also take a delensing
fraction $A_{\mathrm{delens}}=0.8$ which represents an $80\%$ reduction in the
lensing $B$-mode power spectra. The analysis of [68] suggests that achieving
this value is not entirely unrealistic. The CMB-S4 multipole range for the
delensed $B$-mode data is taken to be $30<\ell<300$ and the instrument noise
specifications are those corresponding to the ‘SAT’ configuration and are
available at this
link333https://cmb-s4.uchicago.edu/wiki/index.php/Delensing_sensitivity_-
_updated_sensitivities,_beams,_TT_noise. The resulting estimates for this
setup are shown in the same figure. We can see that a significant increase in
constraining power is obtained with the addition of CMB-S4 and a delensing
level $A_{\mathrm{delens}}=0.8$.

We conclude this section with two more examples.

Figure 6: Left: Fisher forecast for the tensor parameters taking
$r_{\mathrm{pk}}=0.04,\,\sigma=2,\,k_{\rm pk}=10^{-3}$ for three different
values of the vacuum tensor scalar ratio $r$. No delensing assumed for this
case. Right: $r_{\mathrm{pk}}=0.02,\,\sigma=0.5,\,k_{\rm pk}=2\times 10^{-2}$
assuming $A_{\mathrm{delens}}=0.8$. $r=0$ for this case.

The left panel of Figure 6 presents forecasts assuming the presence of the
vacuum contribution to the tensor power spectrum with $r=0.03,\,0.01$ which
effectively acts as a noise for the log-normal tensor spectrum detection. For
the $r=0.01$ scenario, the forecasts are not very different from the case of
$r=0$ which is expected since in this case the sourced tensor spectrum is much
larger than the vacuum contribution on the relevant scales. In the right panel
of the same figure, we present a scenario with a smaller amplitude of the
sourced tensors $r_{\mathrm{pk}}=0.02$, $\sigma=0.5$ and $k_{\rm pk}=2\times
10^{-2}$. In this case the signal is detectable only if a significant amount
of delensing can be achieved. For this value of $r_{\mathrm{pk}}$, the
$B$-mode signal is entirely lensing dominated on angular scales $\ell>100$.

##  4 Conclusions

In the event of a future $B$-mode detection associated to primordial tensor
perturbations, the natural next step would be to understand the production
mechanism behind these perturbations. This information is contained in the
amplitude, shape, chirality and non-Gaussianity of the primordial tensor
spectrum.

Our focus here has been on the shape of the tensor spectrum, taking the
example of a bump-like feature typical of GW sourced by axion-gauge fields
during inflation. This shape deviates sharply from the SFSR prediction which
is bound by the tensor consistency relation to be a power law with a slightly
red tilted spectrum.

Naturally, a detection of such a feature in the primordial tensor spectrum
would hint to inflationary dynamics richer than those of SFSR and could be
used to constrain the parameter space of such axion-gauge field models. On the
other hand, even the non-detection of such a spectrum would be quite
informative since that would strengthen our confidence in the SFSR model,
especially given the difficulty in directly verifying the SFSR tensor
consistency relation with future CMB probes.

In this paper, we first searched for the presence of such features using the
temperature and polarisation anisotropy data from the Planck \+ BICEP/Keck
experiments. While the current data do not provide any evidence for the
presence of such features, they do place constraints on the tensor power
spectrum parameters. In particular, strong constraints on the peak amplitude
are obtained in the region $10^{-3}\lesssim k\lesssim 10^{-2}$ which is main
sensitivity range of the BICEP/Keck $B$-mode data.

We also presented forecasts of the ability of two future CMB experiments,
namely LiteBIRD and CMB-S4, to detect such features. LiteBIRD’s unprecedented
accuracy in measuring polarisation at large angular scales gives it an
excellent sensitivity to the amplitude of such features and will improve upon
current sensitivity by at least a factor of two. However, accurate
measurements of the width will require a significant amount of delensing,
which could be provided by high-resolution CMB experiments such as CMB-S4.

If primordial tensor power spectrum features were realised in Nature, their
detection would represent a fascinating window into the earliest moments of
the Universe, and help us get closer to a more complete understanding of the
physics of inflation. With the CMB experiments coming online in the next
decade, we might just be able to take a peek.

## Acknowledgements

This research includes computations using the computational cluster Katana
supported by Research Technology Services at UNSW Sydney [69]. AM would also
like to thank Giovanni Pierobon for useful discussions regarding Katana.

## References

  * [1] A. H. Guth, _The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems_ , _Phys. Rev._ D23 (1981) 347.
  * [2] K. Sato, _First Order Phase Transition of a Vacuum and Expansion of the Universe_ , _Mon. Not. Roy. Astron. Soc._ 195 (1981) 467.
  * [3] A. D. Linde, _A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems_ , _Phys. Lett._ 108B (1982) 389.
  * [4] A. Albrecht and P. J. Steinhardt, _Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking_ , _Phys. Rev. Lett._ 48 (1982) 1220.
  * [5] A. D. Linde, _Chaotic Inflation_ , _Phys. Lett. B_ 129 (1983) 177.
  * [6] V. F. Mukhanov and G. V. Chibisov, _Quantum Fluctuations and a Nonsingular Universe_ , _JETP Lett._ 33 (1981) 532.
  * [7] S. Hawking, _The development of irregularities in a single bubble inflationary universe_ , _Physics Letters B_ 115 (1982) 295.
  * [8] A. H. Guth and S. Y. Pi, _Fluctuations in the New Inflationary Universe_ , _Phys. Rev. Lett._ 49 (1982) 1110.
  * [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] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, _Spontaneous Creation of Almost Scale - Free Density Perturbations in an Inflationary Universe_ , _Phys. Rev. D_ 28 (1983) 679.
  * [11] Planck collaboration, _Planck 2018 results. X. Constraints on inflation_ , [1807.06211].
  * [12] L. P. Grishchuk, _Amplification of gravitational waves in an istropic universe_ , _Zh. Eksp. Teor. Fiz._ 67 (1974) 825.
  * [13] A. A. Starobinskiǐ, _Spectrum of relict gravitational radiation and the early state of the universe_ , _Soviet Journal of Experimental and Theoretical Physics Letters_ 30 (1979) 682.
  * [14] V. A. Rubakov, M. V. Sazhin and A. V. Veryaskin, _Graviton Creation in the Inflationary Universe and the Grand Unification Scale_ , _Phys. Lett. B_ 115 (1982) 189.
  * [15] R. Fabbri and M. d. Pollock, _The Effect of Primordially Produced Gravitons upon the Anisotropy of the Cosmological Microwave Background Radiation_ , _Phys. Lett. B_ 125 (1983) 445.
  * [16] R. K. Sachs and A. M. Wolfe, _Perturbations of a Cosmological Model and Angular Variations of the Microwave Background_ , _ApJ_ 147 (1967) 73.
  * [17] L. Abbott and M. B. Wise, _Constraints on generalized inflationary cosmologies_ , _Nuclear Physics B_ 244 (1984) 541.
  * [18] A. A. Starobinskii, _Cosmic Background Anisotropy Induced by Isotropic Flat-Spectrum Gravitational-Wave Perturbations_ , _Soviet Astronomy Letters_ 11 (1985) 133.
  * [19] M. Kamionkowski, A. Kosowsky and A. Stebbins, _A Probe of primordial gravity waves and vorticity_ , _Phys. Rev. Lett._ 78 (1997) 2058 [astro-ph/9609132].
  * [20] U. Seljak and M. Zaldarriaga, _Signature of gravity waves in polarization of the microwave background_ , _Phys. Rev. Lett._ 78 (1997) 2054 [astro-ph/9609169].
  * [21] M. Zaldarriaga and U. Seljak, _An all sky analysis of polarization in the microwave background_ , _Phys. Rev. D_ 55 (1997) 1830 [astro-ph/9609170].
  * [22] M. Kamionkowski, A. Kosowsky and A. Stebbins, _Statistics of cosmic microwave background polarization_ , _Phys. Rev. D_ 55 (1997) 7368 [astro-ph/9611125].
  * [23] BICEP/Keck Collaboration collaboration, _Improved constraints on primordial gravitational waves using planck, wmap, and bicep/keck observations through the 2018 observing season_ , _Phys. Rev. Lett._ 127 (2021) 151301 [2110.00483].
  * [24] M. Tristram et al., _Improved limits on the tensor-to-scalar ratio using BICEP and Planck data_ , _Phys. Rev. D_ 105 (2022) 083524 [2112.07961].
  * [25] G. Galloni, N. Bartolo, S. Matarrese, M. Migliaccio, A. Ricciardone and N. Vittorio, _Updated constraints on amplitude and tilt of the tensor primordial spectrum_ , [2208.00188].
  * [26] D. Paoletti, F. Finelli, J. Valiviita and M. Hazumi, _Planck and BICEP/Keck Array 2018 constraints on primordial gravitational waves and perspectives for future B-mode polarization measurements_ , [2208.10482].
  * [27] M. Punturo et al., _The Einstein Telescope: A third-generation gravitational wave observatory_ , _Class. Quant. Grav._ 27 (2010) 194002.
  * [28] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender et al., _Laser interferometer space antenna_ , [1702.00786].
  * [29] M. Maggiore et al., _Science Case for the Einstein Telescope_ , _JCAP_ 03 (2020) 050 [1912.02622].
  * [30] LISA Cosmology Working Group collaboration, _Cosmology with the Laser Interferometer Space Antenna_ , [2204.05434].
  * [31] M. Guzzetti, N. Bartolo, M. Liguori and S. Matarrese, _Gravitational waves from inflation_ , _Riv. Nuovo Cim._ 39 (2016) 399 [1605.01615].
  * [32] E. J. Copeland, E. W. Kolb, A. R. Liddle and J. E. Lidsey, _Reconstructing the inflaton potential: Perturbative reconstruction to second order_ , _Phys. Rev. D_ 49 (1994) 1840 [astro-ph/9308044].
  * [33] H. Peiris and R. Easther, _Recovering the Inflationary Potential and Primordial Power Spectrum With a Slow Roll Prior: Methodology and Application to WMAP 3 Year Data_ , _JCAP_ 07 (2006) 002 [astro-ph/0603587].
  * [34] J. Hamann, J. Lesgourgues and W. Valkenburg, _How to constrain inflationary parameter space with minimal priors_ , _JCAP_ 04 (2008) 016 [0802.0505].
  * [35] A. A. Starobinsky, _A New Type of Isotropic Cosmological Models Without Singularity_ , _Phys. Lett._ B91 (1980) 99.
  * [36] 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.
  * [37] A. Kehagias, A. Moradinezhad Dizgah and A. Riotto, _Remarks on the Starobinsky model of inflation and its descendants_ , _Phys. Rev. D_ 89 (2014) 043527 [1312.1155].
  * [38] L. Moncelsi et al., _Receiver development for BICEP Array, a next-generation CMB polarimeter at the South Pole_ , _Proc. SPIE Int. Soc. Opt. Eng._ 11453 (2020) 1145314 [2012.04047].
  * [39] Simons Observatory collaboration, _The Simons Observatory: Science goals and forecasts_ , _JCAP_ 02 (2019) 056 [1808.07445].
  * [40] CMB-S4 collaboration, _CMB-S4 Science Book, First Edition_ , [1610.02743].
  * [41] LiteBIRD collaboration, _Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey_ , [2202.02773].
  * [42] E. J. Copeland, E. W. Kolb, A. R. Liddle and J. E. Lidsey, _Observing the inflaton potential_ , _Phys. Rev. Lett._ 71 (1993) 219 [hep-ph/9304228].
  * [43] P. Campeti, E. Komatsu, D. Poletti and C. Baccigalupi, _Measuring the spectrum of primordial gravitational waves with CMB, PTA and Laser Interferometers_ , _JCAP_ 01 (2021) 012 [2007.04241].
  * [44] J. L. Cook and L. Sorbo, _Particle production during inflation and gravitational waves detectable by ground-based interferometers_ , _Phys. Rev. D_ 85 (2012) 023534 [1109.0022].
  * [45] N. Barnaby, E. Pajer and M. Peloso, _Gauge Field Production in Axion Inflation: Consequences for Monodromy, non-Gaussianity in the CMB, and Gravitational Waves at Interferometers_ , _Phys. Rev. D_ 85 (2012) 023525 [1110.3327].
  * [46] N. Barnaby, J. Moxon, R. Namba, M. Peloso, G. Shiu and P. Zhou, _Gravity waves and non-Gaussian features from particle production in a sector gravitationally coupled to the inflaton_ , _Phys. Rev. D_ 86 (2012) 103508 [1206.6117].
  * [47] A. Maleknejad, _Axion Inflation with an SU(2) Gauge Field: Detectable Chiral Gravity Waves_ , _JHEP_ 07 (2016) 104 [1604.03327].
  * [48] E. Dimastrogiovanni, M. Fasiello and T. Fujita, _Primordial Gravitational Waves from Axion-Gauge Fields Dynamics_ , _JCAP_ 01 (2017) 019 [1608.04216].
  * [49] J. Garcia-Bellido, M. Peloso and C. Unal, _Gravitational waves at interferometer scales and primordial black holes in axion inflation_ , _JCAP_ 12 (2016) 031 [1610.03763].
  * [50] B. Thorne, T. Fujita, M. Hazumi, N. Katayama, E. Komatsu and M. Shiraishi, _Finding the chiral gravitational wave background of an axion-SU(2) inflationary model using CMB observations and laser interferometers_ , _Phys. Rev. D_ 97 (2018) 043506 [1707.03240].
  * [51] V. Domcke and K. Mukaida, _Gauge Field and Fermion Production during Axion Inflation_ , _JCAP_ 11 (2018) 020 [1806.08769].
  * [52] T. Fujita, K. Imagawa and K. Murai, _Gravitational waves detectable in laser interferometers from axion-SU(2) inflation_ , _JCAP_ 07 (2022) 046 [2203.15273].
  * [53] P. Campeti, O. Özsoy, I. Obata and M. Shiraishi, _New constraints on axion-gauge field dynamics during inflation from Planck and BICEP/Keck data sets_ , _JCAP_ 07 (2022) 039 [2203.03401].
  * [54] D. Blas, J. Lesgourgues and T. Tram, _The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes_ , _JCAP_ 1107 (2011) 034 [1104.2933].
  * [55] A. Lewis, A. Challinor and A. Lasenby, _Efficient computation of CMB anisotropies in closed FRW models_ , _Astrophys. J._ 538 (2000) 473 [astro-ph/9911177].
  * [56] A. Lue, L.-M. Wang and M. Kamionkowski, _Cosmological signature of new parity violating interactions_ , _Phys. Rev. Lett._ 83 (1999) 1506 [astro-ph/9812088].
  * [57] M. Zaldarriaga and U. Seljak, _Gravitational lensing effect on cosmic microwave background polarization_ , _Phys. Rev. D_ 58 (1998) 023003 [astro-ph/9803150].
  * [58] M. Zaldarriaga, _Polarization of the microwave background in reionized models_ , _Phys. Rev. D_ 55 (1997) 1822 [astro-ph/9608050].
  * [59] L. Sorbo, _Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton_ , _JCAP_ 06 (2011) 003 [1101.1525].
  * [60] Planck collaboration, _Planck 2018 results. V. CMB power spectra and likelihoods_ , _Astron. Astrophys._ 641 (2020) A5 [1907.12875].
  * [61] Planck collaboration, _Planck 2018 results. VIII. Gravitational lensing_ , _Astron. Astrophys._ 641 (2020) A8 [1807.06210].
  * [62] A. Lewis and S. Bridle, _Cosmological parameters from CMB and other data: A Monte Carlo approach_ , _Phys. Rev._ D66 (2002) 103511 [astro-ph/0205436].
  * [63] A. Lewis, _Efficient sampling of fast and slow cosmological parameters_ , _Phys. Rev._ D87 (2013) 103529 [1304.4473].
  * [64] J. Torrado and A. Lewis, _Cobaya: Code for Bayesian Analysis of hierarchical physical models_ , _JCAP_ 05 (2021) 057 [2005.05290].
  * [65] A. Lewis, _GetDist: a Python package for analysing Monte Carlo samples_ , [1910.13970].
  * [66] W. L. K. Wu, J. Errard, C. Dvorkin, C. L. Kuo, A. T. Lee, P. McDonald et al., _A Guide to Designing Future Ground-based Cosmic Microwave Background Experiments_ , _Astrophys. J._ 788 (2014) 138 [1402.4108].
  * [67] D. K. Hazra, D. Paoletti, F. Finelli and G. F. Smoot, _Reionization in the dark and the light from Cosmic Microwave Background_ , _JCAP_ 09 (2018) 016 [1807.05435].
  * [68] CMB-S4 collaboration, _CMB-S4: Forecasting Constraints on Primordial Gravitational Waves_ , _Astrophys. J._ 926 (2022) 54 [2008.12619].
  * [69] D. Smith and L. Betbeder-Matibet, _Katana_ , 2010.  doi:10.26190/669X-A286.

##  Appendix A $\Lambda$CDM parameter estimates

Figure 7: Marginalised $68\%$ and $95\%$ contours for $\Lambda$CDM and tensor spectrum parameters. Parameter | 68% limits
---|---
$\log(10^{10}A_{\mathrm{s}})$ | $3.045\pm 0.014$
$n_{\mathrm{s}}$ | $0.9649\pm 0.0040$
$H_{0}$ | $67.3\pm 0.53$
$\Omega_{\mathrm{b}}h^{2}$ | $0.02237\pm 0.00014$
$\Omega_{\mathrm{c}}h^{2}$ | $0.1201\pm 0.0012$
$\tau_{\mathrm{reio}}$ | $0.054\pm 0.0073$
Table 3: $\Lambda{\text{CDM}}$ parameter means and $68\%$ limits for the log-
normal model using Planck+BK18 data.