# Observing nulling of primordial correlations via the $21\,\mathrm{cm}$
signal

Shyam Balaji<EMAIL_ADDRESS>Laboratoire de Physique Théorique et
Hautes Energies (LPTHE),
UMR 7589 CNRS & Sorbonne Université, 4 Place Jussieu, F-75252, Paris, France
H. V. Ragavendra<EMAIL_ADDRESS>Department of Physical
Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur, Nadia 741246, India  Shiv K. Sethi<EMAIL_ADDRESS>Raman Research
Institute, C. V. Raman Avenue, Sadashivanagar, Bengaluru 560080, India  Centre
for Strings, Gravitation and Cosmology, Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India  Joseph Silk
<EMAIL_ADDRESS>Institut d’Astrophysique de Paris, UMR 7095 CNRS & Sorbonne
Université, 98 bis boulevard Arago, F-75014 Paris, France  L. Sriramkumar
<EMAIL_ADDRESS>Centre for Strings, Gravitation and Cosmology,
Department of Physics, Indian Institute of Technology Madras, Chennai 600036,
India

###### Abstract

The $21\,\mathrm{cm}$ line emitted by neutral hydrogen (HI) during the Dark
Ages carries imprints of pristine primordial correlations. In models of
inflation driven by a single, canonical scalar field, we show that a phase of
ultra-slow-roll can lead to a null in all the primordial correlations at a
specific wavenumber $k_{\mathrm{dip}}$. We consider scenarios wherein the null
in the correlations occurs over wave numbers $1\lesssim
k_{\mathrm{dip}}\lesssim 10\,\mathrm{Mpc}^{-1}$, and examine the prospects of
detecting such a damping in the HI signal due to the nulls at the level of
power and bi-spectra in future observational missions.

Primordial correlations and $21\,\mathrm{cm}$ observations: Cosmic inflation
remains the most attractive paradigm for the generation of primordial
perturbations. On large scales, e.g. over $10^{-5}\lesssim k\lesssim
1\,\mathrm{Mpc}^{-1}$, the primordial scalar power spectrum as generated in
some of the popular models of slow roll (SR) inflation is remarkably
consistent with the cosmic microwave background (CMB) anisotropies and large-
scale structure (for a comprehensive list of inflationary models consistent
with the Planck data, see Refs. [1, 2]). However, on smaller scales, e.g.
$k\gtrsim 1\,\mathrm{Mpc}^{-1}$, the constraints on the primordial scalar
power spectrum are considerably weaker. Since the discovery of gravitational
waves from merging binary black holes, the weaker constraints over small
scales have been exploited to examine inflationary models which enhance power
on these scales. This leads to significant production of primordial black
holes and generation of secondary gravitational waves of observable strengths
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Often, in single field models of inflation involving the canonical scalar
field, a phase of ultra-slow-roll (USR) is invoked to enhance the scalar power
on small scales. The first SR parameter $\epsilon_{1}$ exponentially decreases
during such a phase, resulting in large values for the second and higher order
SR parameters [17, 18, 4, 5, 6]. Such a departure from SR inflation leads to a
peak in the inflationary scalar power spectrum and, generically, one finds
that the power spectrum rises as $k^{4}$ as it approaches the peak [19, 20,
21]. Interestingly, just before the power spectrum rises towards the peak, a
sharp drop in power occurs [22] and, if the period of USR inflation is
sufficiently long, the scalar power spectrum actually vanishes at a particular
wavenumber, which we denote as $k_{\mathrm{dip}}$ [23]. This occurs because of
the fact that the mode function describing the curvature perturbation
corresponding to the wavenumber $k_{\mathrm{dip}}$ goes to zero at late times
towards the end of inflation. It can immediately be shown that all of the
higher correlations involving the curvature perturbation will also necessarily
vanish at $k_{\mathrm{dip}}$. In cases wherein the duration of USR is not
sufficiently long, although a null does not arise, a sharp dip in the scalar
power spectrum as well as in the higher order correlation functions is still
encountered.

Over the scales $1\lesssim k\lesssim 10\,\mathrm{Mpc}^{-1}$, the
$21\,\mathrm{cm}$ signal of neutral hydrogen (HI) from the Dark Ages carries
the signatures of the primordial spectrum (see, e.g., Refs. [24, 25]). In
contrast to the angular spectra of the CMB, which are a convolution of the
primordial spectra and the transfer function of the photons, the features in
the primordial power and bi-spectra leave direct and distinct imprints in the
HI signal. Therefore, an inflationary feature such as a null or a sharp dip in
the primordial correlations may potentially be observed in HI, if the features
occur over the corresponding scales [26]. In this Letter, we consider specific
scenarios involving a phase of USR inflation and investigate the effects of a
dip on the HI signal at the level of both power and bi-spectra. If a drop in
the scalar power spectrum is to occur over $1\lesssim k\lesssim
10\,\mathrm{Mpc}^{-1}$, we find that the CMB at smaller wavenumbers and the
spectral distortions at higher wavenumbers limit the rise in power on small
scales, and hence the extent of the dip. We calculate the corresponding
observable signatures on the power and bi-spectra of the HI signal and discuss
the prospects of observing them in future missions, such as a lunar array [27,
28, 29]. We also point out challenges that can arise due to Poisson
fluctuations (PF) [30, 31].

Nulls in inflationary correlations: We now demonstrate that nulls in the
inflationary correlations (i.e. in the scalar power spectrum as well as in
higher order correlations) are expected to arise in scenarios involving a
phase of USR inflation. Consider a situation wherein a regime of USR inflation
is sandwiched between two epochs of SR inflation. Let $\eta$ denote the
conformal time coordinate, and let the two transitions between the three
stages occur at the times $\eta_{1}$ and $\eta_{2}$. Also, let the first SR
parameter $\epsilon_{1}$ prior to the first transition be a constant, say
$\epsilon_{1\mathrm{i}}\lesssim 10^{-2}$, while, during the period of USR, it
is given by $\epsilon_{1}=\epsilon_{1\mathrm{i}}\,(\eta/\eta_{1})^{6}$. Since
$\epsilon_{1}\ll 1$ throughout the domains of interest, the Hubble parameter
can be considered to be a constant, say, $H_{{}_{\mathrm{I}}}$, and hence the
scale factor can be assumed to be of the de Sitter form.

Let us focus on the evolution of modes that leave the Hubble radius during the
initial SR regime. In the first domain $\eta<\eta_{1}$, on super-Hubble
scales, the mode function characterizing the curvature perturbation in Fourier
space, say, $f_{k}^{\mathrm{I}}$, can be expressed as

$f_{k}^{\mathrm{I}}(\eta)=C_{k}+\frac{D_{k}}{2}\,\eta^{2}.$ (1)

The constants $C_{k}$ and $D_{k}$ can be determined by matching the super-
Hubble solutions with the complete solution in the SR regime, and they are
found to be
$C_{k}=i\,H_{{}_{\mathrm{I}}}/(\sqrt{4\,k^{3}\,\epsilon_{1\mathrm{i}}}\,M_{{}_{\mathrm{Pl}}})$
and $D_{k}=C_{k}\,k^{2}$. During the USR phase, the modes function, say,
$f_{k}^{\mathrm{II}}$, for modes that are already on super-Hubble scales, can
be expressed as

$f_{k}^{\mathrm{II}}(\eta)=A_{k}+B_{k}\,\left(\frac{1}{\eta^{3}}-\frac{1}{\eta_{1}^{3}}\right).$
(2)

The quantities $A_{k}$ and $B_{k}$ can be determined by matching the mode
functions and their time derivatives at the transition at $\eta_{1}$ to obtain
that $A_{k}=C_{k}\,[1+(k^{2}\,\eta_{1}^{2}/2)]$ and
$B_{k}=-D_{k}\,\eta_{1}^{5}/3$.

When the phase of USR ends, because the wavenumber of interest is on super-
Hubble scales, its amplitude will evidently freeze at its value at the
conformal time $\eta_{2}$. We should clarify that such a behavior can also be
expected if, for $\eta>\eta_{2}$, the parameter $\epsilon_{1}$ begins to grow
leading to the termination of inflation. Hence, the power spectrum is
determined by the value of $f_{k}^{\mathrm{II}}$ at $\eta_{2}$. Upon setting
$f_{k}^{\mathrm{II}}(\eta_{2})$ to be zero, we can immediately determine the
wavenumber $k_{\mathrm{dip}}$ at which the amplitude of the curvature
perturbation vanishes. It is given by

$k_{\mathrm{dip}}\\!=\\!-\frac{1}{\eta_{1}}\left\\{\frac{1}{3}\left[\left(\frac{\eta_{1}}{\eta_{2}}\right)^{3}-1\right]-\frac{1}{2}\right\\}^{-1/2}\\!\\!\\!\simeq\\!\sqrt{3}\,k_{1}\,\mathrm{e}^{-3\,\Delta
N/2},$ (3)

where the final expression has been arrived at by assuming that the epoch of
USR is adequately long so that $\eta_{1}/\eta_{2}\gg 1$, and we have set
$k_{1}=-1/\eta_{1}$ (i.e. the wavenumber that leaves the Hubble radius at the
onset of USR), while $\Delta N$ denotes the duration of USR in e-folds. The
power spectra and all the higher order correlations involve the mode function
$f_{k}$ evaluated towards the end of inflation. Since the mode function
$f_{k}$ corresponding to the wavenumber $k_{\mathrm{dip}}$ vanishes at late
times, any correlation function involving this mode necessarily vanishes as
well. However, if the duration of USR is not long enough (in fact, when
$\mathrm{e}^{\Delta N}\lesssim 5/2$), then the mode function, rather than
vanishing, settles down to a very small value at late times. In such cases, a
sharp dip is produced rather than a null in the correlation functions, and the
relation (3) predicts the location of the dip. Moreover, when the dominant
term in the mode function $f_{k}$ vanishes, the sub-dominant terms can lead to
a small non-zero value at $k_{\mathrm{dip}}$, resulting in a dip as opposed to
a null.

Inflationary models, power and bi-spectra: To illustrate the nulls or dips
that are expected in the correlation functions, we shall consider two models
of inflation driven by a single, canonical scalar field that permit a brief
period of USR. These models should be treated as illustrative examples of
inflationary scenarios generally considered to enhance power on small scales.
We shall also briefly discuss a reconstructed scenario which easily allows us
to achieve the desired background evolution and a power spectrum that is
consistent with the constraints from the CMB on large scales.

The first model we shall consider is a model due to Starobinsky that is
described by a linear potential with a sudden change in its slope [32, 33, 34,
35]. It is one of the simplest models that leads to a regime of USR and a
step-like feature in the scalar power spectrum. The potential describing the
Starobinsky model (ST) is given by [32, 33, 34, 35]

$V(\phi)=\begin{cases}V_{0}+A_{+}\,(\phi-\phi_{0})&\mathrm{for}\leavevmode\nobreak\
\phi>\phi_{0},\\\
V_{0}+A_{-}\,(\phi-\phi_{0})&\mathrm{for}\leavevmode\nobreak\
\phi<\phi_{0},\end{cases}$ (4)

where $V_{0}$ sets the overall energy scale. Evidently, $A_{+}$ and $A_{-}$
determine the slopes of the potential on either side of $\phi_{0}$, and the
slope is discontinuous at this point. Though there are issues in achieving a
natural end to inflation, we consider the model because of its analytical
tractability that helps in illustrating arguments related to features induced
by USR (in this regard, see the supplementary material). In the model, the
epoch of USR occurs when the field crosses $\phi_{0}$ and the duration of this
epoch is determined by the ratio of the slopes of the potential, i.e.
$A_{-}/A_{+}$. It can be shown that, in the model,
$k_{\mathrm{dip}}\simeq\sqrt{3\,(A_{-}/A_{+})}\,k_{0}$, where $k_{0}$ is the
wavenumber that leaves the Hubble radius when the field crosses $\phi_{0}$
[32, 34, 36]. The parameters $V_{0}$ and $A_{+}$ are constrained by COBE
normalization on the CMB scales (for values of the parameters, see the
supplementary material, which includes Refs. [37, 38]). The constraints from
spectral distortions over the wavenumbers $1<k<10^{4}\,\mathrm{Mpc}^{-1}$
limit the extent of enhancement in the power spectrum at small scales [39],
and hence the duration of USR. We choose the parameters $A_{-}$ and $\phi_{0}$
so that the rise in power on smaller scales is consistent with the FIRAS
constraints on $\mu$ distortions [12]. Also, these parameters are chosen such
that the dip in the power spectrum occurs at wavenumbers $k\gtrsim 5\,\rm
Mpc^{-1}$ to evade bounds on the matter power spectrum from the Lyman-$\alpha$
data (see e.g., Refs. [40, 30]). For the parameters we work with, we find that
$k_{\mathrm{dip}}=7.6\,\mathrm{Mpc}^{-1}$. We should clarify that, the
duration of USR in ST is determined by the ratio of $A_{-}/A_{+}$, which, in
turn, determines the height of scalar power at its maximum. Since this
amplitude is constrained by $\mu$ distortion, it imposes a lower bound on this
ratio. Such a bound leads to an inadequate duration of USR, producing a sharp
dip, instead of a null in the power spectrum.

The second model we shall consider is an inflationary scenario driven by the
Higgs field that is coupled non-minimally to gravitation [41, 42, 43, 14]. The
model is known as critical-Higgs inflation (CH), and the effective potential
in this scenario contains a point of inflection, which leads to an epoch of
USR thereby enhancing the scalar power over small scales. The potential
describing the model can be written as

$V(\phi)=V_{0}\,\frac{\left[1+a\,\left(\mathrm{ln}\,z\right)^{2}\right]\,z^{4}}{\left[1+c\,\left(1+b\,\mathrm{ln}\,z\right)\,z^{2}\right]^{2}},$
(5)

where $z=\phi/\phi_{0}$. As in the case of ST, we choose the parameters of the
potential so that the power spectrum is consistent with COBE normalization on
large scales and with the constraints from spectral distortions on smaller
scales. For the values of the parameters we work with (in this regard, see the
supplementary material), we find that a dip in the power spectrum occurs at
around $k_{\mathrm{dip}}=7.6\,\mathrm{Mpc}^{-1}$ and the power reaches its
maximum amplitude at around $5.5\times 10^{9}\,\mathrm{Mpc}^{-1}$. The $\mu$
distortion arising due to this spectrum is found to be about $2.0\times
10^{-5}$, which is within the FIRAS bound [39, 12].

It is known that, in single field models of inflation, if the enhancement in
power is to be achieved over wavenumbers that are close to the CMB scales,
there can arise a tension between the value of the scalar spectral index
$n_{{}_{\mathrm{S}}}$ in the model and the constraint on the parameter from
the CMB data (for recent discussions, see Refs. [44, 45]). In the ST and CH
models, the value of $n_{{}_{\mathrm{S}}}$ at the pivot scale of
$k_{\ast}=0.05\,\mathrm{Mpc}^{-1}$ turns out to be $0.9995$ and $0.78$,
respectively, which are well away from the mean value of $0.96$ from Planck
[46]. One way to circumvent this challenge is to construct inflationary
potentials using the desired behavior of $\epsilon_{1}(N)$ and, interestingly,
it can be shown that these reconstructed potentials too contain a point of
inflection [10]. In other words, using methods of reconstruction, it is
possible to arrive at potentials numerically that are consistent with the CMB
data on large scales and lead to a dip in the power spectrum over $1\lesssim
k\lesssim 10\,\mathrm{Mpc}^{-1}$ (for details, see the supplementary
material).

We find that the features around the dip have the same characteristics in the
reconstructed scenario as in the ST and CH models. Therefore, we shall proceed
by considering these models and examining their imprints on the
$21\,\mathrm{cm}$ signal. We evolve the background and compute the scalar
power and bi-spectra numerically (see the supplementary material for details,
which includes Refs. [47, 48, 49, 50, 51]). In Figs. 1 and 4 (see App. I), we
have presented the inflationary scalar power and bi-spectra, i.e.
$\mathcal{P}_{{}_{\mathrm{S}}}(k)$ and
$\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$, that
arise in the ST and CH models, for the values of the parameters we have worked
with.

Figure 1: The inflationary scalar power spectra arising in ST and CH models
are illustrated for parameters that are consistent with the constraints on
spectral distortions from FIRAS. We have also plotted the nearly scale
invariant spectrum that may be obtained from a typical SR model of inflation.
The inset highlights the dip in the spectra at $k=7.6\,\mathrm{Mpc}^{-1}$, and
we work with parameters such that the dip occurs over wavenumbers where the HI
signal is expected to be most sensitive to the primordial power spectrum.

We note two related points regarding the bispectra. Firstly, in contrast to
the power spectrum which is a positive definite quantity, the bispectra can
cross zero when departures from slow roll arise. Hence, it may vanish at
locations other than $k_{\mathrm{dip}}$. However, these nulls are dependent on
the nature of the integrals involved in the computation of the bispectrum and
may not be observed at the same locations in the higher order correlations.
Therefore, they are not as generic as the dip of interest, which will be
located at $k_{\mathrm{dip}}$ in the higher order correlations as well. Hence,
in models wherein a deviation from slow roll arises due to an epoch of USR,
the bispectra are guaranteed to exhibit a sharp dip at $k_{\mathrm{dip}}$.
Secondly, note that the power and bispectra in the Starobinsky model rise more
sharply than in the Higgs model. This can be attributed to the sharp change in
the slope of the potential in the former model [52, 34]. Moreover, for the
models of interest, the dip in the power spectrum occurs at the linear order
of the perturbations. Though there may arise corrections to the power spectrum
due to higher order correlations, we have checked, for instance, that the
corrections due to the bispectrum are negligible for the parameters
considered.

Imprints on pre-ionization HI signal: We now turn to discuss the imprints of
the inflationary power and bi-spectra with sharp dips that we have obtained on
the $21\,\mathrm{cm}$ signal of HI from the Dark Ages. We briefly outline the
essential points. In the rest frame of a hydrogen atom, the hyperfine
splitting of the ground state causes an energy difference that corresponds to
the wavelength of $\lambda=21.1\,\mathrm{cm}$. The spin temperature
$T_{{}_{\mathrm{S}}}$ of this line is determined by three processes taking
place in the early universe: emission and absorption of CMB photons with a
black body temperature $T_{{}_{\mathrm{CMB}}}$, collisions with atoms, and the
mixing of the two levels caused by Ly-$\alpha$ photons (i.e. the Wouthuysen-
Field effect). The spin temperature $T_{{}_{\mathrm{S}}}$ can be expressed in
terms of $T_{{}_{\mathrm{CMB}}}$, the gas kinetic temperature
$T_{{}_{\mathrm{K}}}$, and the color temperature of the Lyman-$\alpha$ photons
$T_{\alpha}$, as follows [53, 25]:
$T_{{}_{\mathrm{S}}}=(T_{{}_{\mathrm{CMB}}}+y_{\mathrm{c}}\,T_{{}_{\mathrm{K}}}+y_{\alpha}\,T_{\alpha})/(1+y_{\mathrm{c}}+y_{\alpha})$.
In this expression, $y_{\mathrm{c}}$ and $y_{\alpha}$ determine the efficacy
of the collisions between the hydrogen atoms and of the hydrogen atoms with
the Lyman-$\alpha$ photons, respectively. Note that, $y_{\mathrm{c}}\propto
n_{{}_{\mathrm{HI}}}$ and $y_{\alpha}\propto n_{\alpha}$, where
$n_{{}_{\mathrm{HI}}}$ and $n_{\alpha}$ denote the number density of HI and
the Lyman-alpha photons. HI emits or absorbs $21\,\mathrm{cm}$ radiation from
the CMB depending on whether $T_{{}_{\mathrm{S}}}$ is greater than or less
than $T_{{}_{\mathrm{CMB}}}$. This global temperature difference is observable
and can be expressed as [25, 54, 55]

$\Delta T_{b}(z)\simeq
30\,\left(1-\frac{T_{{}_{\mathrm{CMB}}}}{T_{{}_{\mathrm{S}}}}\right)\,\left(\frac{1+z}{10}\right)^{1/2}\,\left(\frac{\Omega_{\mathrm{b}}\,h^{2}}{0.022}\right)\,\mathrm{mK}.$
(6)

The signal is observable at the frequency of $1420\,\mathrm{MHz}/(1+z)$ at a
given redshift $z$.

Before the onset of the era of cosmic dawn, $y_{\alpha}=0$ and the dynamics of
$T_{{}_{\mathrm{S}}}$ is entirely determined by the other two processes. For
$z\gtrsim 50$, collisions dominate and hence $T_{{}_{\mathrm{S}}}\simeq
T_{{}_{\mathrm{K}}}$. As $T_{{}_{\mathrm{K}}}\simeq T_{{}_{\mathrm{CMB}}}$ for
$z\gtrsim 200$, $T_{{}_{\mathrm{S}}}$ relaxes to $T_{{}_{\mathrm{CMB}}}$ in
this redshift range and the observable signal is negligible. At lower
redshifts ($z\lesssim 150$), $T_{{}_{\mathrm{K}}}$ falls adiabatically as
$1/a^{2}$ and, as $T_{{}_{\mathrm{S}}}\simeq T_{{}_{\mathrm{K}}}$, HI is
observable in absorption. At even smaller redshifts, owing to the dilution of
the gas, the collisional coupling becomes progressively weaker and
$T_{{}_{\mathrm{S}}}$ relaxes to $T_{{}_{\mathrm{CMB}}}$, causing the HI
signal to diminish.

We have relegated the details of the computation of the power and bi-spectra
of the HI signal to App. II. In Fig. 2, we present the HI intensity power
spectrum arising in the ST and CH models at the redshifts of $z=27$ and
$z=50$. In the figure, we have also included the results from a typical SR
model, along with the contribution from PF. In Fig. 3, we have illustrated the
HI intensity bispectrum arising in the ST and CH models at two redshifts in
the equilateral, squeezed and flattened limits, along with the contribution
from PF.

Figure 2: The HI intensity power spectra arising from the ST and CH models
have been plotted at the redshifts of $z=27$ and $z=50$. For comparison, we
have also presented the HI intensity power spectra arising in a SR scenario
leading to a nearly scale invariant, power law primordial scalar power
spectrum. We have also included the power spectra due to PF at the
corresponding redshifts.

Our main findings, shown in Figs. 2 and 3, clearly indicate that, in the
presence of an epoch of USR, there arises a significant dip in the HI
intensity power and bi-spectra over the scales $1\lesssim k\lesssim
10\,\mathrm{Mpc}^{-1}$, when compared to a typical SR scenario. The HI signal
arising from the inflationary bispectrum is seen to be smaller than the
contribution from PF by many orders of magnitude. As the Poisson contribution
to the bispectrum depends on the HI power spectrum [cf. Eq. (LABEL:eq:hibpo)],
the detection of this signal would provide further evidence of the presence of
a null or a dip in the inflationary power spectrum.

Figure 3: The HI intensity bispectra arising from the ST and CH models have
been presented in the equilateral, squeezed and the flattened limits with the
same choice of colors as in Fig. 4. We have focused around $k_{\mathrm{dip}}$
and we have set $k_{\mathrm{sq}}=k/100$ to arrive at the behavior in the
squeezed limit. The associated PF have also been indicated.

Sensitivity: We now explore the feasibility of the detection of the HI
intensity power spectrum over scales of interest. As can be seen in Fig. 2,
our main predictions are over the scales $1\lesssim k\lesssim
100\,\mathrm{Mpc}^{-1}$. The signal strength at such scales is of the order of
$10$–$1000\,(\mathrm{mK})^{2}$ in the frequency range $25$–$50\,\mathrm{MHz}$,
for the redshift range $z\simeq 25$–$50$. While the signal at $z\simeq 25$ is
accessible to SKA-Low (see, e.g., Ref. [56]), we expect the signal at $z\simeq
50$ to be more pristine (i.e. less contaminated by astrophysical processes
close to the era of cosmic dawn) and dominant. Such a signal could be explored
by planned lunar missions [27, 28, 29]. Under suitable assumptions (for a
detailed discussion and methodology, see the supplementary material, which
includes Refs. [57, 58, 59]), the brightness temperature sensitivity of
$1$–$10\,(\mathrm{mK})^{2}$ can be achieved for the scales of interest. A
comparison with Fig. 2 immediately suggests that the attainable sensitivity
should allow the detection of the dip due to the epoch of USR in the HI power
spectrum.

Conclusions: In models of inflation driven by a single, canonical scalar
field, an extended phase of USR leads to a null in all the primordial
correlations at a specific wavenumber $k_{\mathrm{dip}}$. We have considered
scenarios in which the null in the primordial correlations occurs over
wavenumbers $1\lesssim k_{\mathrm{dip}}\lesssim 10\,\mathrm{Mpc}^{-1}$. We
show that future experiments should have the sensitivity to detect a damping
of power in the HI signal due to the nulls at the level of the power and bi-
spectra.

Acknowledgements: SB thanks Yi-Peng Wu for helpful discussions. HVR
acknowledges support from the Indian Institute of Science Education and
Research Kolkata through postdoctoral fellowship. LS acknowledges support from
the Science and Engineering Research Board, Department of Science and
Technology, Government of India, through the Core Research Grant
CRG/2018/002200. SB is supported by funding from the European Union’s Horizon
2020 research and innovation programme under grant agreement No. 101002846
(ERC CoG “CosmoChart”) as well as support from the Initiative Physique des
Infinis (IPI), a research training program of the Idex SUPER at Sorbonne
Université.

##  Appendix A Appendix I: Inflationary scalar bispectrum

In this appendix, we have plotted the bispectra in the equilateral (i.e. when
$k_{1}=k_{2}=k_{3}$), squeezed (when ${\bm{k}}_{1}\to 0$ and
${\bm{k}}_{2}=-{\bm{k}}_{3}=\bm{k}$) and flattened (when $k_{1}=k_{2}=k$ and
$k_{3}=2\,k$) limits. In Fig. 4, we have illustrated the dimensionless
quantities such as $k^{6}\,\mathcal{B}_{{}_{\mathrm{S}}}(k)$ in the
equilateral and flattened limits, and
$k_{1}^{3}\,k^{3}\,\mathcal{B}_{{}_{\mathrm{S}}}(k)$ in the squeezed limit. We
find that a dip in the bispectra arises in all the limits at the same location
of the dip (viz. at $k_{\mathrm{dip}}=7.6\,\mathrm{Mpc}^{-1}$) in the power
spectra.

##  Appendix B Appendix II: Computation of power and bi-spectra of $21$ cm
signal

In this appendix, we provide the details of the computation of power and bi-
spectra of the HI intensity signal in terms of the primordial spectra.

At linear order in the perturbations, the density fluctuations in HI follow
the baryonic perturbations. This allows us to express the fluctuating
component of the HI signal in the pre-reionization epoch as $\delta
T({\bm{x}})=\Delta T_{b}(z)\,\delta_{{}_{\mathrm{HI}}}({\bm{x}})$, where
$\delta_{{}_{\mathrm{HI}}}({\bm{x}})$ denotes the inhomogeneities in the
density of the neutral gas. We shall ignore the redshift-space distortions in
our discussion. At small scales, these perturbations are wiped out due to
acoustic damping in the pre-recombination era and are re-generated by dark
matter potential wells in the post-recombination era. In linear theory, the
baryonic perturbations can be expressed in terms of the inflationary scalar
power spectrum as
$\mathcal{P}_{{}_{\mathrm{HI}}}(k,z)=T^{2}(k,z)\,\mathcal{P}_{{}_{\mathrm{S}}}(k)$,
where $T(k,z)$ is the transfer function for baryons defined such that $T(k,z)$
tends to unity for small $k$ (see, for instance, Ref. [60]). This allows us to
write the HI intensity power spectrum at a redshift $z$ in $(\mathrm{mK})^{2}$
as follows:

$\mathcal{P}_{21}(k,z)=\left[\Delta
T_{b}(z)\right]^{2}\,T^{2}(k,z)\,\mathcal{P}_{{}_{\mathrm{S}}}(k).$ (7)

Figure 4: The scalar bispectra arising in the ST and CH models have been
illustrated in the equilateral (eq), squeezed (sq) and flattened (fl) limits.
As highlighted in the insets, the bispectra also exhibit a sharp dip at the
same location (i.e. at $7.6\,\mathrm{Mpc}^{-1}$) as the power spectra in the
previous figure.

At any redshift, a fraction of baryons, say, $f_{c}$, collapse to form halos.
The baryons in these halos remain neutral, since for the parameters of
interest, the masses of collapsed halos are
$\mathcal{O}(10^{6}\,\mathrm{M}_{\odot})$ and the virial temperature of these
halos is less than $1000\,\mathrm{K}$, too small to ionize the gas via
collisional processes (we evade the free-free constraints on the excess matter
power discussed in Ref. [61]; also see Ref. [62]). The HI intensity power
spectrum from PF at a given redshift in $\rm(mK)^{2}$ is given by (in this
context, see, for instance, Ref. [63])

$\mathcal{P}_{21}^{{}^{\mathrm{PF}}}(k,z)=\left[f_{c}\,\Delta
T_{b}(z)\right]^{2}\,\frac{k^{3}}{2\,\pi^{2}}\,\frac{1}{\bar{n}},$ (8)

where $\bar{n}$ is the mean comoving number density of halos. The collapsed
fraction $f_{c}$ and the number density of halos $\bar{n}$ at any redshift can
be computed using the Press-Schecheter formalism. For instance, in the CH
model, at $z=50$, $f_{c}=0.17$ and $\bar{n}=18754\,\mathrm{Mpc}^{-3}$, while
at $z=100$, $f_{c}=7\times 10^{-3}$ and $\bar{n}=1549\,\mathrm{Mpc}^{-3}$. At
both redshifts, the mass function is dominated by halos of mass $M\lesssim
5\times 10^{5}\,\mathrm{M}_{\odot}$. At $z=27$,
$\bar{n}=27638\,\mathrm{Mpc}^{-3}$ and $f_{c}=0.45$, with halos of $M\lesssim
2\times 10^{6}\,\mathrm{M}_{\odot}$ making the most significant contribution
to the mass function. We have plotted the spectra $\mathcal{P}_{21}(k,z)$ and
$\mathcal{P}_{21}^{{}^{\mathrm{PF}}}(k,z)$ for the ST and CH models in Fig. 2.

Given the scalar bispectrum
$\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$
generated during inflation, the HI intensity bispectrum at any redshift can be
expressed in units of $(\mathrm{mK})^{3}$ as

$\displaystyle{\cal B}_{21}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3},z)$
$\displaystyle=$ $\displaystyle\frac{\left[\Delta
T_{b}(z)\right]^{3}}{2\,\pi^{2}}\,T(k_{1},z)\,T(k_{2},z)\,T(k_{3},z)$ (9)
$\displaystyle\times\,\frac{k_{1}^{3}\,k_{2}^{3}\,k_{3}^{3}}{(k_{1}^{3}+k_{2}^{3}+k_{3}^{3})}\,\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3}).\qquad$

Also, the bispectrum from PF of discrete sources is given by (see e.g., Ref.
[63])

$\displaystyle{\cal B}_{21}^{\rm
PF}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3},z)$ $\displaystyle=$
$\displaystyle\frac{f_{c}^{3}\,k_{1}^{3}\,k_{2}^{3}\,k_{3}^{3}\,\Delta
T_{b}(z)}{(k_{1}^{3}+k_{2}^{3}+k_{3}^{3})\,\bar{n}}\,\Biggl{\\{}-\frac{2\,[\Delta
T_{b}(z)]^{2}}{\bar{n}}$
$\displaystyle+\,\frac{\mathcal{P}_{21}(k_{1})}{k_{1}^{3}}+\frac{\mathcal{P}_{21}(k_{2})}{k_{2}^{3}}+\frac{\mathcal{P}_{21}(k_{3})}{k_{3}^{3}}\Biggr{\\}}.$

We have presented the HI intensity bispectrum at redshifts of $z=27$ and $50$
in Fig. 3 along with the corresponding PF computed at $z=50$.

##  Appendix C Supplemental material for
‘Observing nulling of primordial correlations via the $21\,\mathrm{cm}$
signal’

In the following sections, we shall gather material that supplement the
discussions in the manuscript.

###  C.1 Starobinsky functions

The scalar power spectrum in the ST model can be analytically calculated to be
[32, 34, 35]

$\displaystyle\mathcal{P}_{{}_{\mathrm{S}}}(k)$ $\displaystyle=$
$\displaystyle\mathcal{P}_{{}_{\mathrm{S}}}^{0}\,\left(\frac{A_{+}}{A_{-}}\right)^{2}\,\biggl{[}I(k)+I_{c}(k)\,\cos{\left(\frac{2\,k}{k_{0}}\right)}$
(11)
$\displaystyle+\,I_{s}(k)\,\sin{\left(\frac{2\,k}{k_{0}}\right)}\biggr{]},$

where $k_{0}$ is the wavenumber that leaves the Hubble radius when
$\phi=\phi_{0}$, while the quantities $\mathcal{P}_{{}_{\mathrm{S}}}^{0}$ and
$\epsilon_{1_{-}}$ are related to the parameters of the potential by the
relations

$\mathcal{P}_{{}_{\mathrm{S}}}^{0}\simeq\frac{V_{0}}{24\,\pi^{2}\,M_{{}_{\mathrm{Pl}}}^{4}\,\epsilon_{1_{+}}},\quad\epsilon_{1_{+}}\simeq\frac{M_{{}_{\mathrm{Pl}}}^{2}}{2}\left(\frac{A_{+}}{V_{0}}\right)^{2}.$
(12)

The functions $I(k)$, $I_{c}(k)$ and $I_{s}(k)$ that appear in the power
spectrum above are given by

$\displaystyle I(k)$ $\displaystyle=$ $\displaystyle
1+\frac{9}{2}\,\left(\frac{\Delta
A}{A_{+}}\right)^{2}\,\left(\frac{k_{0}}{k}\right)^{2}$ (13a)
$\displaystyle\times\,\left[1+2\,\left(\frac{k_{0}}{k}\right)^{2}+\left(\frac{k_{0}}{k}\right)^{4}\right],$
$\displaystyle I_{c}(k)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\,\frac{\Delta
A}{A_{+}}\,\left(\frac{k_{0}}{k}\right)^{2}\left[3\,\frac{\Delta
A}{A_{+}}-4-3\,\frac{\Delta
A}{A_{+}}\,\left(\frac{k_{0}}{k}\right)^{4}\right],$ $\displaystyle I_{s}(k)$
$\displaystyle=$ $\displaystyle-3\,\frac{\Delta A}{A_{+}}\,\frac{k_{0}}{k}$
$\displaystyle\times\,\biggl{[}1+\left(\frac{3\,\Delta
A}{A_{+}}-1\right)\,\left(\frac{k_{0}}{k}\right)^{2}+3\,\frac{\Delta
A}{A_{+}}\,\left(\frac{k_{0}}{k}\right)^{4}\biggr{]},$

where $\Delta A=A_{-}-A_{+}$. Actually, we need to smoothen the potential to
be able to compute the spectrum numerically (in this context, see Refs. [34,
35, 38]; also see the discussion that follows). We find that the above
analytical form for the power spectrum matches the numerical result very well.

###  C.2 Parameters and background evolution

In the case of the ST model, for numerical analysis, we smoothen the potential
[cf. Eq. (4) of the manuscript] and work with the potential given by [37, 38]

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
V_{0}+\frac{1}{2}\,\left(A_{+}+A_{-}\right)\,\left(\phi-\phi_{0}\right)$ (14)
$\displaystyle+\,\frac{1}{2}\,\left(A_{+}-A_{-}\right)\,\left(\phi-\phi_{0}\right)\,\tanh{\left(\frac{\phi-\phi_{0}}{\Delta\phi}\right)}.\qquad$

We set $V_{0}=2.48\times 10^{-11}\,M_{{}_{\mathrm{Pl}}}^{4}$ and
$A_{+}=V_{0}/(100\,M_{{}_{\mathrm{Pl}}})$ to achieve COBE normalization over
the CMB scales. We choose $A_{-}=V_{0}/(4\times 10^{3}M_{{}_{\mathrm{Pl}}})$
and $\phi_{0}=0.5628\,M_{{}_{\mathrm{Pl}}}$. If we choose the initial value of
the field and the first SR parameter to be
$\phi_{\mathrm{i}}=0.84348\,M_{{}_{\mathrm{Pl}}}$ and
$\epsilon_{1\mathrm{i}}=4.0\times 10^{-5}$, respectively, we find that the
pivot scale leaves the Hubble radius around $6.4$ e-folds before field crosses
$\phi_{0}$. These choices for the parameters leads to a rise in power on
smaller scales that is consistent with the FIRAS constraints on
$\mu$-distortions [12].

In the CH model, we choose the values of the parameters to be
$V_{0}=7.05\times 10^{-8}\,M_{{}_{\mathrm{Pl}}}^{4}$ and
$\phi_{0}=1\,M_{{}_{\mathrm{Pl}}}$. The dimensionless parameters $a$ and $b$
are related to $c$ and the location of the point of inflection, say,
$z_{\mathrm{c}}$. Choosing the parameters $(c,z_{\mathrm{c}})=(2.850,0.820)$
leads to $(a,b)=(1.694,0.601)$. The value of the parameter $b$ is then fine
tuned to achieve the desired duration of the USR epoch so that the rise in
power is consistent with the constraints from FIRAS [12]. For these values of
the model parameters and with the initial value of field
$\phi_{\mathrm{i}}=5.55\,M_{{}_{\mathrm{Pl}}}$ and the first SR parameter
$\epsilon_{1\mathrm{i}}=10^{-3}$, we achieve about $70$ e-folds of inflation.
We set the pivot scale to exit the Hubble radius at $46.5$ e-folds before the
end of inflation. The epoch of USR begins as the field crosses the point
$\phi=0.82\,M_{{}_{\mathrm{Pl}}}$, around $40$ e-folds before the end of
inflation.

###  C.3 Reconstructing a model with a phase of USR

We had pointed out that, at the pivot scale of $0.05\,\mathrm{Mpc}^{-1}$, the
ST and CH models lead to a scalar spectral index that is quite removed from
the mean value of $n_{{}_{\mathrm{S}}}=0.96$ suggested by Planck [46]. We had
also mentioned that, it is challenging to arrive at a model involving a
single, canonical scalar field that leads to a significant enhancement of
power on small scales and is consistent with the constraints from Planck on
the CMB scales [44, 45]. To overcome this issue, one may resort to the
reconstruction of a scenario of inflation using a parametric modeling of the
first SR parameter $\epsilon_{1}$. We find that the following functional form
of $\epsilon_{1}(N)$ leads to the desired behavior of the background and power
spectra (for a detailed discussion, see Ref. [10]):

$\displaystyle\epsilon_{1}(N)$ $\displaystyle=$
$\displaystyle\epsilon_{1a}\,(1+\epsilon_{2a}\,N)\left[1-\tanh\left(\frac{N-N_{1}}{\Delta
N_{1}}\right)\right]+\epsilon_{1b}$ (15)
$\displaystyle+\exp\left(\frac{N-N_{2}}{\Delta N_{2}}\right),$

where the quantities
$(\epsilon_{1a},\epsilon_{2a},\epsilon_{1b},N_{1},N_{2},\Delta N_{1},\Delta
N_{2})$ are parameters which we shall choose suitably. The parameters
$(\epsilon_{1a},\epsilon_{2a})$ help to achieve appropriate values for the
scalar spectral index and the tensor-to-scalar ratio over the CMB scales.
While the parameters $(N_{1},\Delta N_{1},\epsilon_{1a},\epsilon_{1b})$
control the onset and duration of the epoch of USR, the parameters
$(N_{2},\Delta N_{2})$ regulate the behavior of the background after the USR
phase until the termination of inflation. We have chosen the values of these
parameters such that the dip in the power spectrum arises at the desired
location and inflation is terminated after $60$ e-folds. The values of the
parameters we have worked with are as follows: $(\epsilon_{1\rm
a},\epsilon_{2\rm a},N_{1},\Delta N_{1},\epsilon_{1b},N_{2}\Delta
N_{2})=(10^{-4},0.05,16.5,0.3,5\times 10^{-7},60,1)$. In Fig. 5, we have
illustrated the potential arrived at numerically from the above form of the
first SR parameter (i.e. the reconstructed scenario, RS) as well as the
resulting scalar power spectrum.

Figure 5: The scalar power spectrum $\mathcal{P}_{{}_{\mathrm{S}}}(k)$ arising
from the RS involving an epoch of USR has been presented along with the power
spectrum from the ST model (on the left). Notice that both the power spectra
have identical shapes close to $k_{\mathrm{dip}}$ (as also highlighted in the
inset). In the RS, the power spectrum decays at large wavenumbers due to the
termination of inflation. We have also illustrated the form of the potential
$V(\phi)$ describing the RS (on the right). Interestingly, to the extent
permitted by numerical analysis, we find that the reconstructed potential too
contains a point of inflection around $\phi=6.5\,M_{{}_{\mathrm{Pl}}}$
(indicated by the black vertical line). The parameter $V_{0}$ indicated has
been taken to be $3\,H_{\mathrm{i}}^{2}\,M_{{}_{\mathrm{Pl}}}^{2}$, where
$H_{\mathrm{i}}$ is the initial value of the Hubble parameter. We have chosen
$H_{\mathrm{i}}=6.98\times 10^{-6}\,M_{{}_{\mathrm{Pl}}}$ so as to achieve
COBE normalization of the scalar power spectrum over the CMB scales.

Note that the resulting scalar power spectrum from the RS closely resembles
the spectrum in the ST model around $k_{\mathrm{dip}}$.

###  C.4 Computation of scalar power and bi-spectrum

Let ${\mathcal{R}}_{\bm{k}}$ denote the Fourier modes associated with the
curvature perturbation ${\mathcal{R}}$. Recall that the scalar power spectrum
$\mathcal{P}_{{}_{\mathrm{S}}}(k)$ is defined through the relation

$\langle\hat{{\mathcal{R}}}_{\bm{k}}(\eta_{\rm
e})\,\hat{{\mathcal{R}}}_{\bm{k}^{\prime}}(\eta_{\rm
e})\rangle=\frac{(2\,\pi)^{2}}{2\,k^{3}}\,\mathcal{P}_{{}_{\mathrm{S}}}(k)\,\delta^{(3)}(\bm{k}+\bm{k}^{\prime}),$
(16)

where $\eta_{\rm e}$ denotes the conformal time close to the end of inflation
and the expectation value on the left hand side is to be taken in the Bunch-
Davies vacuum. In terms of the positive frequency Fourier modes that describe
the curvature perturbations, say, $f_{k}$, the power spectrum is given by

$\mathcal{P}_{{}_{\mathrm{S}}}(k)=\frac{k^{3}}{2\,\pi^{2}}\,|f_{k}|^{2}.$ (17)

The primordial scalar bispectrum in Fourier space, denoted as
$\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$, is
defined through the relation

$\displaystyle\langle\hat{{\mathcal{R}}}_{{\bm{k}}_{1}}(\eta_{\rm
e})\,\hat{{\mathcal{R}}}_{{\bm{k}}_{2}}(\eta_{\rm
e})\,\hat{{\mathcal{R}}}_{{\bm{k}}_{3}}(\eta_{\rm e})\rangle$ $\displaystyle=$
$\displaystyle(2\,\pi)^{3}\,\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$
(18)
$\displaystyle\times\,\delta^{(3)}({\bm{k}}_{1}+{\bm{k}}_{2}+{\bm{k}}_{3}),\qquad\;\;$

with the expectation value on the left hand side to be evaluated in the
perturbative vacuum [47, 48, 49]. The bispectrum
$\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ can be
calculated using the third order action governing the perturbations and the
standard rules of perturbative quantum field theory [34, 50]. It can be shown
that, in the situations of our interest, the scalar bispectrum can be
expressed as (see, for instance, Refs. [34, 51, 10, 10])

$\displaystyle\mathcal{B}_{{}_{\mathrm{S}}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$
$\displaystyle=$
$\displaystyle\frac{M_{{}_{\mathrm{Pl}}}^{2}}{(2\,\pi)^{9/2}}\,\sum_{C=1}^{6}\biggl{[}f_{k_{1}}(\eta_{\rm
e})\,f_{k_{2}}(\eta_{\rm e})\,f_{k_{3}}(\eta_{\rm e})$ (19)
$\displaystyle\times\,B_{{}_{C}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})+\,\mathrm{complex\leavevmode\nobreak\
conjugate}\biggr{]}$
$\displaystyle+\,\frac{1}{(2\,\pi)^{9/2}}\,\mathcal{B}_{7}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3}),$

where $f_{k}$ are the positive frequency Fourier modes associated with the
curvature perturbation we mentioned above. The quantities
$B_{{}_{C}}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ represent six integrals
that involve the scale factor, the slow roll parameters, the mode functions
$f_{k}$ and their time derivatives $f_{k}^{\prime}$. They correspond to the
six bulk terms appearing in the cubic order action and are described by the
following expressions

$\displaystyle B_{1}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ $\displaystyle=$
$\displaystyle 2\,i\,\int_{\eta_{\rm i}}^{\eta_{\rm
e}}{\mathrm{d}}\eta\;a^{2}\,\epsilon_{1}^{2}\,\biggl{(}f_{k_{1}}^{\ast}\,f_{k_{2}}^{\prime\ast}\,f_{k_{3}}^{\prime\ast}$
(20a) $\displaystyle+\,\mathrm{two\leavevmode\nobreak\
permutations}\biggr{)},$ $\displaystyle
B_{2}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ $\displaystyle=$
$\displaystyle-2\,i\;\left({\bm{k}}_{1}\cdot{\bm{k}}_{2}+\,{\rm
two\leavevmode\nobreak\ permutations}\right)$ (20b)
$\displaystyle\times\,\int_{\eta_{\rm i}}^{\eta_{\rm
e}}{\mathrm{d}}\eta\;a^{2}\,\epsilon_{1}^{2}\,f_{k_{1}}^{\ast}\,f_{k_{2}}^{\ast}\,f_{k_{3}}^{\ast},$
$\displaystyle B_{3}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ $\displaystyle=$
$\displaystyle-2\,i\,\int_{\eta_{\rm i}}^{\eta_{\rm
e}}{\mathrm{d}}\eta\;a^{2}\,\epsilon_{1}^{2}\,\biggl{(}\frac{{\bm{k}}_{1}\cdot{\bm{k}}_{2}}{k_{2}^{2}}\,f_{k_{1}}^{\ast}\,f_{k_{2}}^{\prime\ast}\,f_{k_{3}}^{\prime\ast}$
(20c) $\displaystyle+\,\mathrm{five\leavevmode\nobreak\
permutations}\biggr{)},$ $\displaystyle
B_{4}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ $\displaystyle=$ $\displaystyle
i\,\int_{\eta_{\rm i}}^{\eta_{\rm
e}}{\mathrm{d}}\eta\;a^{2}\,\epsilon_{1}\,\epsilon_{2}^{\prime}\,\biggl{(}f_{k_{1}}^{\ast}\,f_{k_{2}}^{\ast}\,f_{k_{3}}^{\prime\ast}$
(20d) $\displaystyle+\,\mathrm{two\leavevmode\nobreak\
permutations}\biggr{)},$ $\displaystyle
B_{5}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ $\displaystyle=$
$\displaystyle\frac{i}{2}\,\int_{\eta_{\rm i}}^{\eta_{\rm
e}}{\mathrm{d}}\eta\;a^{2}\,\epsilon_{1}^{3}\;\biggl{(}\frac{{\bm{k}}_{1}\cdot{\bm{k}}_{2}}{k_{2}^{2}}\,f_{k_{1}}^{\ast}\,f_{k_{2}}^{\prime\ast}\,f_{k_{3}}^{\prime\ast}$
(20e) $\displaystyle+\,\mathrm{five\leavevmode\nobreak\
permutations}\biggr{)},$ $\displaystyle
B_{6}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ $\displaystyle=$
$\displaystyle\frac{i}{2}\,\int_{\eta_{\rm i}}^{\eta_{\rm
e}}{\mathrm{d}}\eta\,a^{2}\,\epsilon_{1}^{3}\,\biggl{[}\frac{k_{1}^{2}\,\left({\bm{k}}_{2}\cdot{\bm{k}}_{3}\right)}{k_{2}^{2}\,k_{3}^{2}}\,f_{k_{1}}^{\ast}\,f_{k_{2}}^{\prime\ast}\,f_{k_{3}}^{\prime\ast}$
(20f) $\displaystyle+\,\mathrm{two\leavevmode\nobreak\
permutations}\biggr{]}.$

The term $\mathcal{B}_{7}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$ corresponds
to a temporal boundary term in the cubic order action and is given by

$\displaystyle\mathcal{B}_{7}({\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3})$
$\displaystyle=$
$\displaystyle-i\,M_{{}_{\mathrm{Pl}}}^{2}\,\left[f_{k_{1}}(\eta_{\rm
e})\,f_{k_{2}}(\eta_{\rm e})\,f_{k_{3}}(\eta_{\rm e})\right]$ (21)
$\displaystyle\times\,\biggl{[}a^{2}\epsilon_{1}\epsilon_{2}\,f_{k_{1}}^{\ast}(\eta)\,f_{k_{2}}^{\ast}(\eta)\,f_{k_{3}}^{\prime\ast}(\eta)$
$\displaystyle+\,\mathrm{two\leavevmode\nobreak\
permutations}\biggr{]}_{\eta_{i}}^{\eta_{\rm e}}$
$\displaystyle+\,\mathrm{complex\leavevmode\nobreak\ conjugate}.$

In SR scenarios, typically, one evolves the perturbations from well inside the
Hubble radius (say, when $k\simeq 10^{2}\,a\,H$) to adequately late times when
the modes are on super-Hubble scales (say, when $k\simeq 10^{-5}\,a\,H)$, to
arrive at the power spectrum [51]. When there occur departures from SR, in
particular, in the presence of an epoch of USR, it becomes important to evolve
the perturbations until very late times, close to the end of inflation [38,
10]. These arguments broadly apply for the computation of the scalar
bispectrum as well. However, note that, apart from numerically evaluating the
mode functions, to obtain the bispectrum, we also need to carry out integrals
over the mode functions, their time derivatives and the background quantities.
Since the bispectrum, in general, involves three wavenumbers, we need to
evolve the perturbations and carry out the integrals from the time when the
smallest of the three wavenumbers of interest is well inside the Hubble radius
to a time close to the end of inflation (for further details, see Ref. [10]).
It is these procedures that we have adopted to arrive at the power and bi-
spectra we have presented in this work.

###  C.5 Sensitivity of an interferometric array to HI power spectrum

Let us assume that visibilities for a given baseline ${\bm{u}}$,
$V_{\nu}({\bm{u}})$, corresponding to a statistically homogeneous process such
as the HI signal or detector noise, are measured for $n_{\nu}$ discrete
frequency channels. The measured visibilities can be Fourier transformed from
frequency space to the delay space (see e.g. Ref. [24], and references
therein) and correlated to obtain $\langle
V_{\tau}({\bm{u}})\,V_{\tau}({\bm{u}})\rangle$, where $\tau$ is the conjugate
variable of the frequency difference between two maps. In discrete space,
$\tau=n/\Delta\nu$, where $n$ runs between zero and $n_{\nu}/2$, while
$\Delta\nu=\delta\nu\,n_{\nu}$ is the total bandwidth and $\delta\nu$ is the
channel width. The correlation in delay space can be related to the power
spectrum as follows (for a detailed derivation see, for instance, Ref. [57],
and references therein)

$\langle
V_{\tau}({\bm{u}})\,V_{\tau}({\bm{u}})\rangle\simeq\frac{\Delta\nu\,I_{\nu}^{2}\,\Omega_{p}}{r_{\nu}^{2}\,r_{\nu}^{\prime}}\,P(k),$
(22)

where $r_{\nu}$ is the coordinate distance,
$r_{\nu}^{\prime}\equiv{\mathrm{d}}r_{\nu}/{\mathrm{d}}\nu$, $\Omega_{p}$ is
the primary beam, and $I_{\nu}=k\,T_{B}/\lambda_{0}^{2}$. In delay space,
different components of the wave vector
$k=\sqrt{{\bm{k}}_{\perp}^{2}+k_{\parallel}^{2}}$, decomposed as modes on the
plane of the sky ${\bm{k}}_{\perp}$, and along the line of sight
${\bm{k}}_{\parallel}$, are related to the parameters of the radio
interferometer as ${\bm{k}}_{\perp}\simeq 2\,\pi\,{\bf u}_{\nu}/r_{\nu}$ and
$k_{\parallel}=2\,\pi\,\tau/({\mathrm{d}}r_{\nu}/{\mathrm{d}}\nu)$.

Assuming the visibility correlation to be dominated by the detector noise and
a baseline coverage that yields uniform noise 111In an integration time
$t_{\mathrm{int}}$, the total number of baselines is
$N\,(N-1)\,t_{\mathrm{int}}/(2\,t_{\mathrm{acc}})$, where $t_{\mathrm{acc}}$
is the time for one measurement. For high density of baselines, the
instantaneous baseline coverage is complete or it samples each UV pixel for a
range of baselines (generally short baselines, e.g. MWA for baselines shorter
than roughly $200\,\lambda_{0}$ [58]). Therefore, the total number of
independent UV pixels remains $N\,(N-1)/2$ after $t=t_{\mathrm{int}}$. This
yields uniform noise in each pixel in both UV and real space., one can obtain
the error on the brightness temperature fluctuations from the detector noise
(for each primary beam and for non-redundant baselines):

$\delta\left(\frac{T_{B}^{2}\,k^{3}\,P(k)}{2\,\pi^{2}}\right)\simeq\frac{T_{\mathrm{sys}}^{2}\,\Omega_{p}\,r_{\nu}^{\prime}\,r_{\nu}^{2}\,\delta\nu\,k^{3}}{2\,\pi^{2}\,f^{1/2}(k)\,N\,\Delta\nu\,t_{\mathrm{int}}},$
(23)

where $T_{\mathrm{sys}}$ is the system temperature, $N$ is the number of
antennas, and $f(k)$ is the fraction of baseline in the range $k$ and
$(k+{\mathrm{d}}k)$. Current (e.g. LOFAR) and future low-frequency radio
interferometers have the capability to form and image in multi-beams. Multiple
primary beams help to improve both the detector sensitivity and the sample
variance. For $n_{p}$ primary beams, the brightness temperature variance
derived in Eq. (23) decreases by a factor of $n_{p}^{1/2}$.

It is our aim that the interferometer be sensitive to wavenumbers $k\simeq
1$–$10\,\mathrm{Mpc}^{-1}$. We consider two cases: (a) $k_{\perp}\gg
k_{\parallel}$, and (b) $k_{\parallel}\gg k_{\perp}$. When $k_{\perp}\gg
k_{\parallel}$, the baseline is $u\simeq k\,r_{\nu}/(2\,\pi)$. For $z\simeq
50$, this gives a baseline length ($b=u\,\lambda_{0}$) of nearly
$200\,\mathrm{km}$ for $k\simeq 10\,\mathrm{Mpc}^{-1}$. In three dimensions,
$f(k)\propto k^{3}$, which causes the brightness temperature sensitivity to
scale as $k^{3/2}$ (see e.g. Ref. [59]). However, it is not easy to achieve
uniform coverage in this case as the number of telescopes needed to achieve it
scales as $k^{2}$. For instance, if uniform UV coverage could be achieved with
100 telescopes for $k=0.1\,\mathrm{Mpc}^{-1}$ (e.g. MWA configuration, see
Ref. [58]), then nearly $10^{4}$ telescopes would be needed to reach the same
goal for $k=1\,\mathrm{Mpc}^{-1}$. We therefore consider another novel method
which is more suitable for the detection of the HI signal at small scales,
viz. when $k_{\parallel}\gg k_{\perp}$. In this case, $k\simeq k_{\parallel}$
and the baseline length does not play a role in determining the sensitivity of
the array to a given Fourier mode. In this case, $f_{k}=1$ as all baselines
contribute equally to the sensitivity for any scale for which
$k_{\parallel}\gg k_{\perp}$.

We note that better sensitivity might be achieved in case (b) as $f(k)=1$.
However, the sensitivity scales as $k^{3}$ for case (b) whereas it scales as
$k^{3/2}$ for case (a), so case (a) might be the method of choice at large
scales. We choose the following parameters for computing the sensitivity at
$z=50$ for case (b): integration time, $t_{\mathrm{int}}=3\times 10^{7}$;
channel width, $\delta\nu=10\,\mathrm{kHz}$; total bandwidth,
$\Delta\nu=10\,\mathrm{MHz}$; number of antennas, $N=5000$; system
temperature, $T_{\mathrm{sys}}=10^{4}\,\mathrm{MHz}$; primary beam,
$\Omega_{p}=(4/57)^{2}$; number of beams, $n_{p}=100$. This yields a
brightness temperature variance
$\delta(T_{B}^{2}\,k^{3}\,P(k)/(2\,\pi^{2}))\simeq(1\,\mathrm{mK})^{2}$ at
$k=3\,\mathrm{Mpc}^{-1}$. These parameters are within the reach of current
technologies and might be realizable by SKA-II. However, ground-based
telescopes might not work well at $\nu=30\,\mathrm{MHz}$ needed to detect the
signal due to ionospheric effects. Future space-based and lunar radio
interferometers will evade this constraint [27, 28, 29]. We note that the
sensitivity might be improved further with redundant baselines: if each
baseline has redundancy $M$, the total number of independent baselines is
nearly $N^{2}/(2\,M)$. While the non-redundant baselines yield incoherent
addition, the redundant baselines contribute coherently. This results in
sensitivity on the noise power spectrum being better by a factor $M^{-1}$.

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