# Cosmological constraints from the EFT power spectrum and tree-level
bispectrum of 21cm intensity maps

Liantsoa F. Randrianjanahary  Dionysios Karagiannis  Roy Maartens

###### Abstract

We explore the information content of 21cm intensity maps in redshift space
using the 1-loop Effective Field Theory power spectrum model and the
bispectrum at tree level. The 21cm signal contains signatures of dark matter,
dark energy and the growth of large-scale structure in the Universe. These
signatures are typically analyzed via the 2-point correlation function or
power spectrum. However, adding the information from the 3-point correlation
function or bispectrum will be crucial to exploiting next-generation intensity
mapping experiments. The bispectrum could offer a unique opportunity to break
key parameter degeneracies that hinder the measurement of cosmological
parameters and improve on the precision. We use a Fisher forecast analysis to
estimate the constraining power of the HIRAX survey on cosmological
parameters, dark energy and modified gravity.

##  1 Introduction

The effective field theory of large-scale structure (EFTofLSS) is a formalism
developed to accurately and reliably predict the clustering of cosmological
large-scale structure in the mildly nonlinear regime [1, 2, 3, 4, 5]. EFTofLSS
differs significantly from the widely used standard perturbation theory (SPT)
formalism [6]. The fundamental difference arises from the fact that EFTofLSS
considers the impact of nonlinearities on scales that are only mildly
nonlinear by introducing effective stresses into the equations of motion. As a
result, the 1-loop power spectrum is modified by adding ultra-violet
counterterms that account for the influence of short-distance physics on long-
distance scales.

EFTofLSS provides a means to perform analytical computations of LSS
observables with the required precision in the mildly nonlinear regime. [7]
develop efficient implementations of these computations that explore their
dependence on cosmological parameters. [8] investigate the possibility of
constraining parameters of the EFT of inflation with upcoming LSS surveys. [9]
perform the first early dark energy analysis using the full-shape power
spectrum likelihood from the Baryon Oscillation Spectroscopic Survey (BOSS),
based on EFTofLSS. [4] analyses the cosmological parameters in the DR12 BOSS
dataset after validating the approach with various numerical simulations.

EFTofLSS requires several new bias coefficients that contribute to multiple
observables. A set of seven bias parameters adequately characterizes the
manifold statistical properties [10, 11, 12]. EFTofLSS has provided a robust
framework to describe large-scale galaxy distributions without requiring
detailed knowledge of galaxy formation physics. It overcomes the limitations
of the pressureless perfect fluid representation by treating dark matter as an
effective non-ideal fluid. This theory is based on time-sliced perturbation
theory, a non-equilibrium field theory approach that allows consistent
renormalization of cosmological correlation functions and systematic
resummation of large infrared effects. The equations of motion within the
EFTofLSS are derived through a perturbative series expansion of matter fields
in density contrast. The expansion parameter is the ratio of the scale of
interest to the nonlinear scale of matter distribution. EFTofLSS strategically
uses only the relevant degrees of freedom for large-scale dynamics,
effectively parametrizing the impact of unknown short-scale physics. The
framework has undergone extensive validation against data [13, 14],
demonstrating excellent agreement on mildly-nonlinear scales.

The EFTofLSS equations of motion form a set of nonlinear and non-local partial
differential equations, effectively describing the evolution of density and
velocity fields on large scales. This versatile framework facilitates the
prediction of large-scale structure in the universe and enables the study of
various physical processes, such as baryons, neutrinos, and dark energy. The
EFTofLSS framework has become useful for figuring out the mysteries of large-
scale structure in cosmology. It does this by consistently including short-
scale physics, allowing quantifiable comparisons with real-world data or
simulations, and allowing for additional physical processes or modified
gravity models.

In this paper, we apply the EFT model developed by [15] (see also [16]) to
intensity mapping surveys, which measure the emission signal of neutral
hydrogen in galaxies, in order to trace large-scale structure. Then we combine
it with the tree-level bispectrum of 21 intensity mapping, as in [17, 18, 19,
20]. The survey that we consider is similar to that proposed for HIRAX (see
below for details).

The layout of this paper is as follows. Section 2 presents an overview of the
theoretical models used for the summary statistics considered here, i.e. the
power spectrum and the bispectrum. In section 3 we present the methodology
employed for implementing parameter constraints. Subsequently, the results are
presented and elucidated in section 4 and we conclude in section 5.

##  2 HI intensity power spectrum and bispectrum

21cm intensity mapping, also known as neutral hydrogen (HI) intensity mapping
(IM), is emerging as a potentially powerful new way to probe the large-scale
structure of the post-reionization Universe. By measuring the integrated
brightness temperature of the 21cm emission line of HI trapped in galaxies,
this technique evades the need to detect individual galaxies and thus offers a
way to rapidly survey huge volumes of the universe, with exquisite redshift
accuracy. A major cosmological survey is in preparation by the Square
Kilometre Array Observatory (SKAO) mid-frequency dish array [21]. There are
complicated observational systematics, in particular major foreground
contamination, which are being tackled in initial simulations and observations
by precursor experiments (see e.g. [22, 23, 24]).

The SKAO survey will be conducted in single-dish mode, in which the signals
from individual dishes are simply added. An alternative is interferometer
mode, in which the dish signals are correlated, allowing for the measurement
of smaller scales than those accessed by single-dish mode. The HIRAX
experiment is being constructed for cosmological surveys in interferometer
mode (see [25]) and in this paper we will consider a survey that is similar to
that planned for HIRAX. Further details are given in Table 1.

###  2.1 HI IM bias

We follow the bias prescription and models in [17, 18, 19, 20]. The HI IM
clustering bias describes the relation between its distribution and that of
the dark matter. The $i$-th order bias is given by:

$b^{\rm HI}_{i}(z)=\frac{1}{\rho_{\rm HI}(z)}\int_{0}^{\infty}{\rm d}M\,n_{\rm
h}(M,z)b_{i}^{\rm h}(M,z)M_{\rm HI}(M,z),$ (2.1)

where $\rho_{\rm HI}(z)$ is the HI density [26], $n_{\rm h}$ and $b_{i}^{\rm
h}$ are the halo mass function and bias respectively, given by the best-fit
results of [27, 28]. $M_{\rm HI}$ is the average HI mass inside a halo of
total mass $M$ at redshift $z$ and it is given via the halo occupation
distribution (HOD) approach [29] and the model of [30]:

$M_{\rm HI}(M,z)=C(z)(1-Y_{p})\frac{\Omega_{\rm b}}{\Omega_{\rm m}}\,{\rm
e}^{-M_{\rm min}(z)/M}\,M^{q(z)}.$ (2.2)

Details of the HOD parameters are given in [18].

The HI IM overdensity is then given by

$\delta_{\rm HI}(\bm{x},\tau)=b_{1}(\tau)\delta_{\rm
m}(\bm{x},\tau)+\varepsilon(\bm{x},\tau)+\frac{b_{2}(\tau)}{2}\delta_{\rm
m}(\bm{x},\tau)^{2}+\frac{b_{s^{2}}(\tau)}{2}s(\bm{x},\tau)^{2}+\varepsilon_{\delta}(\bm{x},\tau)\,\delta_{\rm
m}(\bm{x},\tau),$ (2.3)

where we dropped label ‘HI’ from $b_{i}$ for brevity. Here $\delta_{\rm m}$ is
the matter overdensity and the $\varepsilon$’s are stochastic terms (see [17]
for details). The first-order and quadratic biases are computed from Equation
2.1. In the redshift range $0.7\lesssim z\lesssim 2.6$, we find the fitting
formulas

$\displaystyle b_{1}(z)$ $\displaystyle=1.333\leavevmode\nobreak\
\,+0.1809\;z+0.05302\;z^{2}-0.0008822\;z^{3}\;,$ (2.4) $\displaystyle
b_{2}(z)$ $\displaystyle=0.1802+0.1711\;z+0.02161\;z^{2}+0.03136\;z^{3}\;.$
(2.5)

The tidal field bias is modelled as $b_{s^{2}}=-4(b_{1}-1)/7$, following [31].

###  2.2 HI IM power spectrum

The HI temperature contrast is $\delta_{\rm HI}={\delta T_{\rm
HI}}/{\bar{T}_{\rm HI}}$, where $T_{\rm HI}$ is the HI brightness temperature
and its background value is modelled as [32]

$\displaystyle\bar{T}_{\rm HI}(z)=188\,h\,\Omega_{\rm
HI}(z)\frac{H_{0}}{H(z)}\leavevmode\nobreak\ \mu{\rm
K}\quad\mbox{with}\quad\Omega_{\rm HI}(z)=4\times 10^{-4}(1+z)^{0.6}\,.$ (2.6)

The redshift space HI IM power spectrum is given by

$P_{\rm HI}(\bm{k},z)=\bar{T}_{\rm HI}(z)^{2}\big{[}P(\bm{k},z)+P_{\rm
SN}(z)\Big{]}+P_{\rm N}(\bm{k},z)\,.$ (2.7)

At linear order $P(\bm{k},z)=P^{11}(\bm{k},z)$, where

$\displaystyle P^{11}(\bm{k},z)=Z_{1}(\bm{k},z)^{2}P_{\rm m}^{\rm
L}(k,z)\quad\mbox{with}$ (2.8) $\displaystyle
Z_{1}(\bm{k},z)=b_{1}(z)+f(z)\mu^{2}\,.$ (2.9)

Here $\mu=\hat{\bm{k}}\cdot\hat{\bm{n}}=k_{\|}/k$ and $\hat{\bm{n}}$ is the
line of sight; $f=-{\rm d}\ln\delta^{\rm L}_{\rm m}/{\rm d}\ln(1+z)$ is the
linear growth rate; $P_{\rm SN}$ is the shot noise term and $P_{\rm N}$ is the
instrumental noise (see below). The linear matter power spectrum $P_{\rm
m}^{\rm L}$ is computed using the numerical Boltzmann code Cosmic Linear
Anisotropy Solving System CLASS [33].

Figure 1: HI IM power spectrum, $P_{\rm HI}/\bar{T}_{\rm HI}^{2}$, for tree-
level and 1-loop EFT models, at $z=0.8$.

In this work we will model the HI IM power spectrum using SPT at next-to-
leading (1-loop) order

$\displaystyle
P^{\text{1-loop}}(\bm{k},z)=P^{11}(\bm{k},z)+P^{22}(\bm{k},z)+P^{13}(\bm{k},z),$
(2.10)

corrected by EFT counterterms [15, 16]:

$\displaystyle P^{\rm EFT}(\bm{k},z)=P^{\text{1-loop}}(\bm{k},z)$
$\displaystyle+\big{(}\alpha_{0}+\alpha_{2}\mu^{2}+\alpha_{4}\mu^{4}\big{)}\Big{(}\frac{k}{k_{*}}\Big{)}^{2}P^{\rm
Zel}_{\rm cb}(\bm{k},z)$ $\displaystyle+N_{0}+N_{2}(\mu k)^{2}+N_{4}(\mu
k)^{4}\,,$ (2.11)

where we substitute $P(\bm{k},z)$ with $P^{\rm EFT}(\bm{k},z)$ in Equation 2.7
and $P_{\rm SN}$ is replaced by the $N_{0}$ term. Moreover, $\alpha_{a}$
represent the counter-terms, while $N_{2}$ and $N_{4}$ encapsulate small-scale
velocities. Following [34], the Zeldovich approximation of the matter power
spectrum to linear order is defined as

$P^{\text{Zel}}_{\text{cb}}(\bm{k},z)=(1+f\mu^{2})^{2}P_{\rm m}^{\rm
L}(k,z)\,.$ (2.12)

The difference between the linear and EFT power spectra is illustrated in
Figure 1. A comprehensive review of the redshift-space counterterms and the
free parameters associated with the EFT 1-loop power spectrum is given in [5].

###  2.3 HI IM bispectrum

For the HI bispectrum, i.e. the Fourier transform of the three-point function,
we use a tree-level model with a phenomenological model of nonlinear RSD,
following e.g. [35, 18]:

$\displaystyle B_{\rm HI}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z)$
$\displaystyle=\bar{T}_{\rm
HI}(z)^{3}\bigg{\\{}D_{\text{fog}}^{B}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z)$
$\displaystyle\times\bigg{[}2Z_{1}(\bm{k}_{1},z)Z_{1}(\bm{k}_{2},z)Z_{2}(\bm{k}_{1},\bm{k}_{2},z)P_{\rm
m}^{\rm L}(k_{1},z)P_{\rm m}^{\rm L}(k_{2},z)+2\leavevmode\nobreak\
\text{perm}\bigg{]}$ $\displaystyle\leavevmode\nobreak\
+2P_{\varepsilon\varepsilon_{\delta}}(z)\Big{[}Z_{1}(\bm{k}_{1},z)P_{\rm
m}^{\rm L}(k_{1},z)+2\leavevmode\nobreak\
\text{perm}\Big{]}+B_{\varepsilon}(z)\bigg{\\}},$ (2.13)

where the second-order redshift space kernels is given by:

$\displaystyle Z_{2}(\bm{k}_{i},\bm{k}_{j})$
$\displaystyle=b_{1}F_{2}(\bm{k}_{i},\bm{k}_{j})+f\mu_{ij}^{2}G_{2}(\bm{k}_{i},\bm{k}_{j})+\frac{b_{2}}{2}+\frac{b_{s^{2}}}{2}S_{2}(\bm{k}_{i},\bm{k}_{j})$
$\displaystyle\leavevmode\nobreak\
+\frac{1}{2}f\mu_{ij}k_{ij}\left[\frac{\mu_{i}}{k_{i}}Z_{1}(\bm{k}_{j})+\frac{\mu_{j}}{k_{j}}Z_{1}(\bm{k}_{i})\right],$
(2.14)

where $\mu_{ij}=(\mu_{i}k_{i}+\mu_{j}k_{j})k_{ij}$,
$k_{ij}^{2}=(\bm{k}_{i}^{2}+\bm{k}_{j}^{2})^{2}$; $F_{2}$ and $G_{2}$ are the
SPT kernels [6]; and the tidal kernel is
$S_{2}(\bm{k}_{1},\bm{k}_{2})=(\hat{\bm{k}}_{1}\cdot\hat{\bm{k}}_{2})^{2}-1/3$
[36, 31]. The RSD ‘fingers of god’ damping factor is [37, 38]

$\displaystyle D_{\text{fog}}^{B}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z)$
$\displaystyle=\exp\Big{[}-\big{(}k_{1}^{2}\mu_{1}^{2}+k_{2}^{2}\mu_{2}^{2}+k_{3}^{2}\mu_{3}^{2}\big{)}\sigma_{B}(z)^{2}\Big{]},$
(2.15)

where $\sigma_{B}$ characterizes the damping scale, with fiducial value given
by the linear velocity dispersion $\sigma_{\upsilon}$. Note that we do not
apply an ‘fog’ damping factor to the power spectrum since nonlinear RSD are
treated by the EFT counter terms. Finally, the fiducial values of the
stochastic terms in Equation 2.13 are [18]:

$P_{\varepsilon\varepsilon_{\delta}}=\frac{b_{1}}{2\overline{n}_{\rm
eff}},\quad B_{\varepsilon}=\frac{1}{\overline{n}_{\rm eff}^{2}},$ (2.16)

where $\overline{n}_{\rm eff}$ is the effective number density and its is
given by the HOD formalism used to derive the HI bias, as in [30].

###  2.4 Alcock-Paczynski effect

The Alcock-Paczynski (AP) effect relates the observed angular and redshift
separations of objects to the actual cosmological distances, providing a means
to investigate the large-scale structure of the universe. The AP effect is
particularly significant in cosmological surveys and clustering analysis, as
it allows us to infer information about the cosmological model and geometry of
the universe. To implement this, we need a rescaling factor in the HI power
spectrum and bispectrum.

The fiducial cosmology wavevector $\bm{k}_{\rm fid}$ is related to the true
wavevector $\bm{k}$ by $\bm{k}_{\rm
fid}=\bm{k}_{\parallel}\alpha_{\parallel}(z)+\bm{k}_{\bot}\alpha_{\bot}(z)$,
where $\alpha_{\parallel}=H_{\rm fid}(z)/H_{\rm true}(z)$ and $\alpha_{\bot}$
as $\alpha_{\bot}=D_{A,\rm true}(z)/D_{A,\rm fid}(z)$. The correlation
function, being a dimensionless quantity, is left unchanged by this rescaling
[15]. Thus $P_{\rm HI}^{\rm obs}(\bm{k}_{\rm fid},z)\,{\rm d}^{3}\bm{k}_{\rm
fid}=P_{\rm HI}^{\rm true}(\bm{k},z)\,{\rm d}^{3}\bm{k}$ leading to:

$\displaystyle P_{\rm HI}^{\rm obs}\left(k,\mu,z\right)$
$\displaystyle=\left(\frac{H_{\rm true}}{H_{\rm
fid}}\right)\left(\frac{D_{A,\rm fid}}{D_{A,\rm true}}\right)^{2}P_{\rm
HI}\left(q,\nu,z\right)\,.$ (2.17)

The fiducial Fourier coordinates $(k,\mu)$ are related to the true values as

$\displaystyle q(k,\mu)$
$\displaystyle=\beta(\mu)\,k\;,\quad\nu(k,\mu)=\frac{\mu}{\beta(\mu)}\,\frac{H_{\rm
true}}{H_{\rm fid}}\quad\mbox{where}$ (2.18) $\displaystyle\beta(\mu)^{2}$
$\displaystyle=\left(\frac{H_{\rm true}}{H_{\rm
fid}}\right)^{2}\mu^{2}+\left(\frac{D_{A,\rm fid}}{D_{A,\rm
true}}\right)^{2}\left(1-\mu^{2}\right)\,.$ (2.19)

Extending Equation 2.18 and Equation 2.19 to 3 dimensions, we find that [39]:

$\displaystyle B^{\rm obs}_{\rm
HI}\left(k_{1},k_{2},k_{3},\mu_{1},\phi,z\right)$
$\displaystyle=\left(\frac{H_{\rm true}}{H_{\rm
fid}}\right)^{2}\left(\frac{D_{A,\rm fid}}{D_{A,\rm true}}\right)^{4}B_{\rm
HI}\left(q_{1},q_{2},q_{3},\nu_{1},\phi,z\right).$ (2.20)

This establishes the relationship between the observed and fiducial bispectra,
accounting for the AP effect by considering the true and fiducial values of
the Hubble parameter $H$ and angular diameter distance $D_{A}$. Such a
formalism is important for making precise measurements and extracting
cosmological information from observations of large-scale structure.

###  2.5 HI IM window

The observational window of HI IM surveys is constrained by strong foreground
contamination. We apply foreground avoidance filters in the
$(k_{\perp},k_{\|})$ plane [17, 15, 20]: (a) a cutoff to excise contaminated
long-wavelength radial modes: $k_{\parallel,\,\min}=0.01\,h/\rm{Mpc}$; and (b)
avoidance of the foreground wedge, where spectrally smooth foregrounds leak
into small-scale transverse modes:

$k_{\parallel}\geq A_{\rm wedge}\,k_{\perp}\quad\text{where}\quad A_{\rm
wedge}=\frac{\chi(z)\,H(z)}{c(1+z)}\sin\big{[}0.61\,N_{\rm w}\,\theta_{\rm
b}(z)\big{]}\,.$ (2.21)

Here $\chi$ is the comoving line-of-sight distance, $N_{\rm w}$ is the number
of primary beams away from the beam centre that contaminate the signal, and
$\theta_{\rm b}(z)=1.2\lambda_{21}(1+z)/D_{\rm dish}$ is the beam, with
$D_{\rm dish}$ the dish diameter. We consider $N_{\rm w}=1$ [20].

Figure 2: The smallest scale (largest $k$) assumed to be accessible as a
function of redshift, using Equation 2.24 and Equation 2.25 for the power
spectrum and Equation 2.22 for the bispectrum.

The minimum wavenumber, which corresponds to the largest scales probed by a
survey of volume $V(z)$ within a redshift bin, is $k_{\rm min}(z)=k_{\rm
f}(z)$, where $k_{\rm f}(z)=2\pi/V(z)^{1/3}$ being the fundamental frequency
of the survey. The maximum wavenumber (corresponding to the smallest scale) is
chosen for the tree-level bispectrum as [17]

$k_{\max}^{B}(z)=0.75\,k_{\rm NL}(z)\,,$ (2.22)

where $k_{\rm NL}$ is the nonlinear scale set by the RMS displacement in the
Zel’dovich approximation:

$k_{\rm NL}(z)^{-2}=\frac{1}{6\pi^{2}}\int_{0}^{\infty}{\rm d}k\,P^{\rm
L}_{\rm m}(k,z).$ (2.23)

The EFT power spectrum probes mildly nonlinear scales with a higher
$k_{\max}$. We assume that the maximum scale evolves as [40, 41]

$k_{\rm max}^{P}(z)=k_{\rm max,0}\,D(z)^{-4/3.3}\,,$ (2.24)

where $D(z)$ is the linear growth factor, normalised to unity at present (i.e.
$D(0)=1$). Following [42, 5, 43], we use

$\displaystyle k_{\rm max,0}=0.25h\leavevmode\nobreak\
\mbox{(pessimistic)},\leavevmode\nobreak\ \leavevmode\nobreak\
\,0.3h\leavevmode\nobreak\ \mbox{(optimistic)}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm Mpc}^{-1}\,.$ (2.25)

The small-scale cuts are shown in Figure 2.

##  3 Fisher forecast formalism

We use the Fisher information matrix [44, 45] to forecast precision on model
parameters attainable by a HIRAX-like survey. The Fisher matrix for the power
spectrum is (e.g. [17, 20])

$F_{\alpha\beta}^{P}(z_{i})=\frac{1}{2}\sum_{k}\displaystyle\int_{-1}^{1}{\rm
d}\mu\,\frac{\partial P_{\rm HI}^{\rm
obs}(\bm{k},_{i})}{\partial\theta_{\alpha}}\frac{\partial P_{\rm HI}^{\rm
obs}(\bm{k},_{i})}{\partial\theta_{\beta}}\frac{1}{\Delta
P^{2}(\bm{k},_{i})}\,$ (3.1)

where the sum is over $k_{\rm min}\leq k\leq k_{\rm max}$. For the bispectrum
it is:

$F_{\alpha\beta}^{B}(z_{i})=\frac{1}{4\pi}\\!\sum_{k_{1},k_{2},k_{3}}\\!\int_{-1}^{1}\\!\\!{\rm
d}\mu_{1}\\!\\!\int_{0}^{2\pi}\\!\\!{\rm d}\phi\frac{\partial B_{\rm HI}^{\rm
obs}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z_{i})}{\partial\theta_{\alpha}}\frac{\partial
B_{\rm HI}^{\rm
obs}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z_{i})}{\partial\theta_{\beta}}\frac{1}{\Delta
B^{2}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z_{i})},$ (3.2)

where the above sum is over all the triangles formed by the wavevectors
$k_{1},k_{2},k_{3}$, while satisfying $k_{\rm min}\leq k_{3}\leq k_{2}\leq
k_{1}\leq k_{\rm max}$. The derivatives are performed over the model
parameters $\theta_{\alpha}$.

We assume that the covariances in the Fisher matrix formalism are Gaussian and
diagonal. Then the variances for the two correlators are [46, 47]

$\displaystyle\Delta P^{2}(\bm{k},z)$
$\displaystyle=\frac{4\pi^{2}}{V(z)\,k^{2}\,\Delta
k(z)}\,P_{\text{HI}}^{\text{obs}}(\bm{k},z)^{2}\,,$ (3.3) $\displaystyle\Delta
B^{2}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},z)$
$\displaystyle=s_{123}\,\pi\,k_{\text{f}}(z)^{3}\,\frac{P_{\text{HI}}^{\text{obs}}(\bm{k}_{1},z)\,P_{\text{HI}}^{\text{obs}}(\bm{k}_{2},z)\,P_{\text{HI}}^{\text{obs}}(\bm{k}_{3},z)}{k_{1}\,k_{2}\,k_{3}\,[\Delta
k(z)]^{3}}\,,$ (3.4)

where the bin size is chosen as $\Delta k=k_{\rm f}$ and $s_{123}=6,2,1$ for
equilateral, isosceles and non-isosceles triangles, respectively. In the case
of the bispectrum, for degenerate configurations, $k_{i}=k_{j}+k_{m}$, we
multiply the variance by a factor of 2 [48, 49]. Note that higher $k^{P}_{\rm
max,0}$ values than those shown in Figure 2 may still give a reasonably
accurate estimate of the power spectrum – but at the cost of increasingly non-
Gaussian contributions to the power spectrum variance [50]. Non-Guassian
contributions to the bispectrum occur more readily than in the power spectrum
case. For this reason, even with our conservative choice of $k^{B}_{\rm
max,0}$, we include higher-order corrections to the variance, as well as
taking account of theoretical errors (see [18] for a discussion).

Property | HIRAX
---|---
redshift $z$ | $0.775-2.55$
$N_{\rm dish}$ | $1024$
$D_{\rm dish}$ [m] | $6$
$D_{\rm max}$ [km] | $0.25$
$S_{\rm area}$ [$\rm{deg}^{2}$] | $15,000$
$t_{\rm survey}$ [hrs] | $17,500$
efficiency $\eta$ | 0.7
Table 1: Specifications for HIRAX interferometer-mode survey (from [25]).

We consider a HI IM survey in interferometer mode, similar to the one proposed
for HIRAX [51, 52, 53, 25] with specifications given in Table 1. The
instrumental noise term that appears in the power spectrum of Equation 2.7, is
assumed to be Gaussian and it is given in the case of an interferometer by
[54, 55, 56]:

$P_{\rm N}(\bm{k}_{\perp},z)=T_{\rm
sys}(z)^{2}\chi(z)^{2}\lambda(z)\frac{(1+z)}{H(z)}\left[\frac{\lambda(z)^{2}}{A_{e}}\right]^{2}\,\frac{S_{\rm
area}}{\theta_{\rm b}(z)^{2}}\,\frac{1}{N_{\rm pol}\,n_{\rm
b}(\bm{k}_{\perp},z)\,t_{\rm survey}}\,,$ (3.5)

where $t_{\rm{survey}}$ is the integration time, $S_{\rm area}$ the sky area,
and $A_{\rm e}$ the effective area, which is determined by the efficiency
$\eta$ via $A_{\rm e}=\eta\pi(D_{\rm dish}/2)^{2}$. For the system temperature
$T_{\rm sys}$, we add the receiver temperature $T_{\rm rx}=50\,{\rm K}$ to the
sky temperature $T_{\rm sky}$, which is obtained from Appendix D of [57]. The
baseline distribution $n_{\rm b}(\bm{k}_{\perp},z)$ presented in Appendix A of
[20] is used here.

For the baseline model, we consider the $\Lambda$CDM cosmological parameters,
the AP parameters, the growth rate, the bias, the amplitudes of the EFT
counter terms and the noise parameters to be free. The counter term and noise
parameters are considered nuisance and are marginalised over for each
correlator. The final parameter vector then is reduced to

$\displaystyle\bm{\theta}(z_{i})=$ $\displaystyle\Big{\\{}\Omega_{\rm
b},\Omega_{\rm c},h,n_{\text{s}},A_{\rm
s};D_{A}(z_{i}),H(z_{i}),f(z_{i}),b_{1}(z_{i}),b_{2}(z_{i}),b_{s^{2}}(z_{i})\Big{\\}}\,,$
(3.6)

We also consider the summed information from the power spectrum and
bispectrum. In this case we sum the two Fisher matrices, without considering
the cross-Fisher between the two correlators, which is assumed to be small,
following [35]:

$\displaystyle
F_{\alpha\beta}^{P+B}(z_{i})=F_{\alpha\beta}^{P}(z_{i})+F_{\alpha\beta}^{B}(z_{i})\,.$
(3.7)

Finally, the total Fisher matrix is a sum over all redshift bins, assuming
that the bins are independent. Note that during the summation we have to be
careful to account for the fact that the redshift-dependent parameters change
– so that they are effectively new parameters for each $z_{i}$. As a
consequence, the dimension of the total Fisher matrix

$\displaystyle F_{\alpha\beta}^{P+B,{\rm
tot}}=\sum_{i}F_{\alpha\beta}^{P+B}(z_{i})\,,$ (3.8)

is much larger than the dimension of a single-bin matrix. This requires care
in the marginalisation. Further details of the marginalisation and projection
in parameter space may be found in [58].

We investigate two further cosmological models that extend $\Lambda$CDM. The
constant cosmological parameters for the 3 models are

$\Lambda$CDM
$\displaystyle:\big{\\{}{\Omega_{\text{b}}},{\Omega_{\text{c}}},h,n_{\text{s}},A_{\rm
s}\big{\\}},$ (3.9) $w_{0}w_{a}{\rm CDM}\,$
$\displaystyle:\big{\\{}{\Omega_{\text{b}}},{\Omega_{\text{c}}},h,n_{\text{s}},A_{\rm
s},w_{0},w_{a}\big{\\}}\quad\mbox{where}\leavevmode\nobreak\
\leavevmode\nobreak\ w_{\rm de}(z)=w_{0}+w_{a}\,\frac{z}{1+z}\,,$ (3.10)
$\gamma{\rm CDM}\,$
$\displaystyle:\big{\\{}{\Omega_{\text{b}}},{\Omega_{\text{c}}},h,n_{\text{s}},A_{\rm
s},\gamma\big{\\}}\quad\mbox{where}\leavevmode\nobreak\ \leavevmode\nobreak\
f(z)=\big{[}\Omega_{\rm m}(z)\big{]}^{\gamma}\,.$ (3.11)

In order to find the final constraints on the above parameters we apply
Jacobian transformations to the Fisher matrix that corresponds to the
parameter space in Equation 3.6, after marginalising over the remaining bias
parameters:

$\widetilde{F}_{AB}=\big{(}J^{\rm T}\,F^{\rm
tot}\,J\big{)}_{AB}=\sum_{\alpha,\beta}J_{A\alpha}\,F_{\alpha\beta}^{\rm
tot}\,J_{\beta B}\qquad\mbox{where}\quad
J_{A\alpha}={\partial\theta_{\alpha}\over\partial\tilde{\theta}_{A}}\,.$
(3.12)

This will provide the final parameter constraints, from each correlator and
their summed signal, in the form
$\sigma(\theta_{A})=\big{[}\big{(}\widetilde{F}^{-1}\big{)}_{AA}\big{]}^{1/2}$
.

The fiducial values used for the cosmological parameters are drawn from Planck
cosmic microwave background (CMB) measurements [59]:
$\Omega_{\text{b}}h^{2}=0.02237$, $\Omega_{\text{c}}h^{2}=0.12$, $h=0.6736$,
$n_{\text{s}}=0.9649$, and $10^{9}A_{\rm s}=2.1$. Fiducial values for the
extended model parameters are $\gamma=0.55$, $w_{0}=-1$, and $w_{a}=0$.

The Fisher matrix results are combined with the data on cosmological
parameters obtained from the Planck satellite [59]. For this, we use the
Markov chain that samples the posterior distribution, obtained from the Planck
webpage111http://pla.esac.esa.int/pla/#cosmology. The covariance matrix is
computed from the chains, specifically for the subset of parameters being
analyzed. This matrix is then inverted to obtain the Planck 2018 Fisher
matrix. The Fisher matrices of the power spectrum and bispectrum and their
combination are then summed with the latter. The Planck likelihood is treated
as a multivariate Gaussian, appropriate for the free cosmological parameters
being studied.

The cosmological parameter results from Planck data include various
cosmological models and provide results from MCMC exploration chains, best
fits, and tables. Specifically, we use the $\Lambda$CDM chains with baseline
likelihoods based on plikHM_TTTEEE_lowl_lowE, which we sample using
Getdist222https://getdist.readthedocs.io/en/latest/.

##  4 Results

Here we present and analyse the results derived from the Fisher matrices for
the three cosmological models.

Figure 3: Constraints on $\Lambda$CDM cosmological parameters from the
combination of the EFT power spectrum and tree-level bispectrum of a HIRAX-
like survey for $k_{\max,0}=0.25h/$Mpc in Equation 2.24. All three cases
presented here consider Planck priors (see section 3) and utilise the signal
from the whole redshift range.

###  4.1 $\Lambda$CDM

We expect the bispectrum to provide complementary information to the power
spectrum. For $\Lambda$CDM, Figure 3 shows that bispectrum constraints on
cosmological parameters are weaker than from the EFT power spectrum.
Nevertheless there are advantages from combining the 1-loop power spectrum and
tree-level bispectrum as shown in Table 2. This combination enhances parameter
constraints for the cosmological parameters $\Omega_{b}$, $h$ and $n_{s}$. The
constraints achieved with the combined analysis are notably more stringent
than those derived from Planck 2018 CMB data.

We also find that the 1-loop power spectrum exhibits different correlations
between parameters than the bispectrum. This indicates the possibility to
break degeneracies and tightly constrain these parameters. The significant
advantages in combining the two correlators is not evident here, due to the
relatively small values considered for $k_{\max}^{B}(z)$ in the tree-level
bispectrum analysis, compared to those used for the power spectrum EFT.

Figure 4: As in Figure 3, for the $\gamma{\rm CDM}\,$model.

###  4.2 Modified gravity

The $\gamma{\rm CDM}\,$model provides a simple test of gravity: the fiducial
value $\gamma=0.55$ corresponds to the $\Lambda$CDM and is a good
approximation for simple non-clustering models of dark energy in general
relativity. This means that a value of $\gamma$ that deviates significantly
from 0.55 indicates either a breakdown in general relativity on large scales,
or a more complicated dark energy within the Friedmann-Lemaître-Robertson-
Walker model (see e.g. [60]).

The 1-loop EFT power spectrum gives reasonable constraints for all considered
cosmological parameters, while the bispectrum provides notably less stringent
constraints for cosmological parameters and $\gamma$.

Constraints on the cosmological parameters in the $\gamma{\rm CDM}\,$model,
are naturally weakened, compared to $\Lambda$CDM, by the presence of a new
free parameter which affects both the power spectrum and bispectrum. The EFT
power spectrum provides stronger constraints on the growth rate
$f(z)=\Omega_{\rm m}(z)^{\gamma}$ compared to the bispectrum, as we show below
in Figure 6. However the $f(z)$ constraints do not simply translate to
$\gamma$ constraints, since $\gamma$ arises in different ways in the two
correlators, and since $\Omega_{\rm m}(z)$ involves the cosmological
parameters. Figure 4 shows that in fact, the EFT power spectrum gives better
results than the tree-level bispectrum in constraining $\gamma$, while their
combination delivers a notable improvement as shown in Table 2.

Figure 4 reveals the possibility to break parameter degeneracies, displayed as
different orientation directions in the elliptical contours. Pushing the
bispectrum analysis into smaller scales would have improved constraints, once
the EFT power spectrum and tree-level bispectrum are combined.

Figure 5: As in Figure 3, for the $w_{0}w_{a}{\rm CDM}\,$model.

###  4.3 Dynamical dark energy

The $w_{0}w_{a}{\rm CDM}\,$model is a simple model often used to test for
deviations from a cosmological constant that are due to dynamical (or
evolving) dark energy (see e.g. [59]). Instead of choosing specific dynamical
dark energy models, a two-parameter dark energy equation of state $w(z)$ is
used, providing some level of model independence.

With two additional parameters $w_{0},w_{a}$ relative to $\Lambda$CDM, we
expect the constraints on cosmological parameters to worsen. This is confirmed
by Figure 5. It is apparent that bispectrum constraints degrade even more than
those from the power spectrum, where the latter provides more robust
constraints to the dark energy parameters than the first. Despite the weakness
of the bispectrum constraints, the combination with the power spectrum breaks
degeneracies and leads to a noteworthy improvement in constraining
$w_{0},w_{a}$, as shown in Table 2.

###  4.4 Summary of results on $\Lambda$CDM, $\gamma$CDM and $w_{0}w_{a}$CDM

HIRAX $[\%]$ P B P+B $\Lambda$CDM $\sigma(\Omega_{\rm b})/\Omega_{\rm b}$ 0.29
(0.27) 0.52 0.27 (0.25) $\sigma(\Omega_{\rm c})/\Omega_{\rm c}$ 0.12 (0.11)
0.62 0.12 (0.11) $\sigma(h)/h$ 0.06 (0.05) 0.19 0.05 (0.05) $\sigma(n_{\rm
s})/n_{\rm s}$ 0.16 (0.15) 0.24 0.15 (0.13) $\sigma(A_{\rm s})/A_{\rm s}$ 0.25
(0.24) 0.96 0.25 (0.24) $\gamma{\rm CDM}\,$ $\sigma(\Omega_{\rm
b})/\Omega_{\rm b}$ 0.30 (0.28) 0.53 0.27 (0.26) $\sigma(\Omega_{\rm
c})/\Omega_{\rm c}$ 0.12 (0.11) 0.65 0.12 (0.11) $\sigma(h)/h$ 0.06 (0.05)
0.19 0.05 (0.05) $\sigma(n_{\rm s})/n_{\rm s}$ 0.17 (0.16) 0.25 0.15 (0.15)
$\sigma(A_{\rm s})/A_{\rm s}$ 0.26 (0.25) 1.15 0.25 (0.24)
$\sigma(\gamma)/\gamma$ 1.38 (1.30) 8.68 1.27 (1.22) $w_{0}w_{a}{\rm CDM}\,$
$\sigma(\Omega_{\rm b})/\Omega_{\rm b}$ 0.32 (0.29) 1.79 0.27 (0.26)
$\sigma(\Omega_{\rm c})/\Omega_{\rm c}$ 0.31 (0.27) 2.85 0.20 (0.18)
$\sigma(h)/h$ 0.11 (0.10) 0.91 0.09 (0.08) $\sigma(n_{\rm s})/n_{\rm s}$ 0.17
(0.15) 0.80 0.15 (0.14) $\sigma(A_{\rm s})/A_{\rm s}$ 0.26 (0.25) 3.36 0.25
(0.24) $\sigma(w_{0})/|w_{0}|$ 1.39 (1.22) 7.53 1.02 (0.93) $\sigma(w_{a})$
4.45 (3.91) 35.69 3.80 (3.41)

Table 2: For $\Lambda$CDM, $\gamma{\rm CDM}\,$and $w_{0}w_{a}{\rm CDM}\,$:
1$\sigma$ constraints on cosmological parameters and model parameters, from
the whole redshift range, using the 1-loop EFT power spectrum (P) and tree-
level bispectrum (B) and their combination (P+B). We use $k_{\max}^{B}$ as in
Equation 2.22 and $k_{\max}^{P}$ as in Equation 2.24, with $k_{\max,0}=0.25h$
(pessimistic case). Constraints in brackets are those from the opitimistic
case, where $k_{\max,0}=0.3h$ is used in Equation 2.24.

Table 2 summarises the 1$\sigma$ constraints on the 3 models, $\Lambda$CDM,
$\gamma{\rm CDM}\,$, and $w_{0}w_{a}{\rm CDM}\,$. It demonstrates that
incorporating the bispectrum enhances the constraints on some parameters
compared to using the power spectrum alone. This improvement is mainly evident
for the $\gamma{\rm CDM}\,$and $w_{0}w_{a}{\rm CDM}\,$model parameters (i.e.
$\gamma$, $w_{0}$ and $w_{\alpha}$), while for cosmology the gain is minimal.
To improve on the latter one would require to extract the bispectrum
information from smaller scales, comparable to those used for the power
spectrum analysis, by utilising the EFT framework.

Table 2 shows that there are only small changes in the power spectrum
constraints on the cosmological parameters $\Omega_{\rm b}$, $\Omega_{\rm c}$,
$h$, $n_{\rm s}$ and $A_{s}$ when moving from $\Lambda$CDM to $\gamma{\rm
CDM}\,$and $w_{0}w_{a}{\rm CDM}\,$. By contrast, there is a significant
increase in error on $A_{s}$ (by a factor 1.73) for the bispectrum constraints
in $\gamma{\rm CDM}\,$. In the $w_{0}w_{a}{\rm CDM}\,$model, the increase in
bispectrum errors is even greater, and affects all 5 parameters. The increase
in parameter uncertainties when transitioning from $\Lambda$CDM to $\gamma{\rm
CDM}\,$and $w_{0}w_{a}{\rm CDM}\,$arises due to the additional correlations
introduced among variables as we increase the number of parameters in the
model.

Table 2 shows the improvement in the power spectrum constraint when using the
optimistic case $k_{\max,0}=0.3h/$Mpc compared to the pessimistic one (i.e.
$k_{\max,0}=0.25/h$Mpc). This is due to the information that resides within
the additional small scale modes. However, this improvement is marginal in the
case of $\Lambda$CDM and $\gamma{\rm CDM}\,$, unlike the dark energy model
$w_{0}w_{a}{\rm CDM}\,$, which shows a more notable enhancement in the
constraints. This is an indication that the information from the power
spectrum saturates, as we venture to smaller scales, due to the thermal noise
in the a HIRAX-like instrument.

###  4.5 Distance and growth rate errors

In this section, we study the constraints on the angular diameter distance
$D_{A}(z)$ and the Hubble rate $H(z)$ via the AP effect in the standard
$\Lambda$CDM model. We also compute errors on the growth rate $f(z)=-{\rm
d}\ln D(z)/{\rm d}\ln(1+z)$.

The constraints are derived after marginalising over the nuisance parameters
for both correlators, while the remaining model parameters are considered free
(Equation 3.6). Figure 6 (upper panels) presents the results for the
marginalized relative errors as a function of redshift.

The power spectrum delivers remarkably precise measurements, with an
uncertainty of less than 1% across all redshift bins for the angular diameter
distance $D_{A}$ and Hubble parameter $H$, and within 1-2% for the growth rate
$f$. The bispectrum alone shows a lower precision, yielding relative errors
within 10-20% for $D_{A}$ and $H$, while for the growth rate the precision is
$\sim$5%, across the redshift range. Consequently, incorporating the
bispectrum into the analysis contributes a small improvement in precision on
the BAO distance scale parameters and the growth rate.

###  4.6 Bias parameters

Clustering biases affect both the power spectrum and the bispectrum. Here we
provide the constraints on biases as a function of redshift before
marginalising them to get the cosmology constraints of the previous sections.
The results are shown in Figure 6 (lower panels).

Figure 6: Upper panels: Forecasts of marginalized relative errors on $D_{A}$,
$H$, and $f$ from the power spectrum (red), bispectrum (blue), and their
combination (orange), in the pessimistic case (Equation 2.25). Lower panels:
Similarly, for the bias parameters in Equation 2.3.

The power spectrum provides a precision of $0.5-1\%$ in most of the redshift
range for the HI IM linear bias $b_{1}$. The joint constraint from the two
correlators is at the same level of precision as the power spectrum,
indicating the poor performance of the bispectrum. The latter provides a
$\sim$5% precision on $b_{1}$ constraints, which saturates beyond $z\sim 1.5$.
For the quadratic bias $b_{2}$ and the tidal bias $b_{s^{2}}$, the bispectrum
provides a precision of $\sim$20% and $\sim$50% respectively, where the
information saturates beyond $z\sim 2$. The power spectrum, on the other hand,
exhibits an improvement in the constraints, reaching a precision smaller than
$\sim$10% for $b_{2}$, at high redshift bins, while for $b_{s^{2}}$ the
relative errors saturate at $\sim$15% for $z>1.5$. Combining the two
correlators offers a significant improvement for these higher order bias
parameters, reaching $\sim$4% and $\sim$5% at high redshifts for $b_{2}$ and
$b_{s^{2}}$ respectively. This indicates the great potential offered by a
joint EFT power spectrum and bispectrum analysis.

##  5 Conclusions

This work presents an analysis to estimate the cosmological precision possible
with a HIRAX-like neutral hydrogen intensity mapping survey, using the 1-loop
effective field theory power spectrum and the tree-level bispectrum. In
addition to constraints on standard cosmological parameters, we estimate the
uncertainties on modified gravity ($\gamma$) and dynamical dark energy
($w_{0},w_{a}$) parameters. Furthermore, we find the errors on the angular
diameter distance $D_{A}(z)$, the Hubble rate $H(z)$ and on the growth rate
$f(z)$. Finally, we show the constraints on the linear and higher order
clustering biases.

In order to remain within the domain of validity of the tree-level bispectrum
and 1-loop EFT power spectrum, we have imposed appropriate scale cuts, as
illustrated in Equation 2.22–Equation 2.25. For the EFT power spectrum, our
baseline is a ‘pessimistic’ $k^{P}_{\rm max}$ cut, with errors also computed
for a more optimistic choice.

The 1$\sigma$ marginalised errors of our Fisher forecasts, from the whole
redshift range, are summarised in Table 2. For $\Lambda$CDM, the precision
from P+B is sub-percent on the cosmological parameters $\Omega_{\rm b}$,
$\Omega_{\rm c}$, $h$, $n_{\rm s}$, and $A_{\rm s}$. For the extended models,
$\gamma{\rm CDM}\,$and $w_{0}w_{a}{\rm CDM}\,$, these constraints naturally
degrade, but the precision on the model parameters is high: 1.27% on the
growth index $\gamma$ and (1%, 3.8%) on ($w_{0}$, $w_{a}$).

Compared to the tree-level power spectrum from our previous work [20], we find
that EFT model improves precision on $\Omega_{\rm b}$, $\Omega_{\rm c}$, and
$h$, due to the additional information from small scales, as probed by the
1-loop power spectrum. However, this is not the case for $A_{s}$ and $n_{s}$,
which are less sensitive to this regime. The comparison with [20] also
demonstrates the potential of the EFT power spectrum in breaking parameter
degeneracies. This is clearly seen in the contour plots of Figure 3–Figure 5.

Finally, for the redshift-dependent parameters, we find a sub-percent
precision for most redshifts on $D_{A},H$, and within 1-2% for most redshifts
on $f$ (Figure 6, upper panels). The precision on the linear clustering bias
$b_{1}$ is sub-percent, for most redshifts, while for the quadratic biases it
reaches $\sim$5% from the joint analysis at high redshifts (Figure 6, lower
panels).

The results presented in this work have relied on certain simplifications
that, in a subsequent work, could be relaxed for a more comprehensive
analysis. Introducing the 1-loop power spectrum covariance would make the
constraints more realistic for future surveys, while extending the reach of
the bispectrum, by using the 1-loop Effective Field Theory model (see e.g.
[61]), would improve significantly its role in this joint analysis and
subsequently the cosmological constraints.

## Acknowledgements

The authors are supported by the South African Radio Astronomy Observatory and
the National Research Foundation (Grant No. 75415).

## References

  * [1] L. Senatore and M. Zaldarriaga, _Redshift Space Distortions in the Effective Field Theory of Large Scale Structures_ , 1409.1225.
  * [2] R. E. Angulo, S. Foreman, M. Schmittfull and L. Senatore, _The one-loop matter bispectrum in the effective field theory of large scale structures_ , 1406.4143.
  * [3] S. Foreman and L. Senatore, _The EFT of large scale structures at all redshifts: analytical predictions for lensing_ , _Journal of Cosmology and Astroparticle Physics_ 2016 (2016) 033 [1503.01775].
  * [4] G. d'Amico, J. Gleyzes, N. Kokron, K. Markovic, L. Senatore, P. Zhang et al., _The cosmological analysis of the SDSS/BOSS data from the effective field theory of large-scale structure_ , _Journal of Cosmology and Astroparticle Physics_ 2020 (2020) 005 [1909.05271].
  * [5] M. M. Ivanov, _Effective Field Theory for Large-Scale Structure_ , 2212.08488.
  * [6] F. Bernardeau, S. Colombi, E. Gaztañaga and R. Scoccimarro, _Large-scale structure of the Universe and cosmological perturbation theory_ , _Phys. Rep._ 367 (2002) 1 [astro-ph/0112551].
  * [7] M. Cataneo, S. Foreman and L. Senatore, _Efficient exploration of cosmology dependence in the EFT of LSS_ , _JCAP_ 04 (2017) 026 [1606.03633].
  * [8] A. Naskar, S. Choudhury, A. Banerjee and S. Pal, _EFT of Inflation: Reflections on CMB and Forecasts on LSS Surveys_ , 1706.08051.
  * [9] M. M. Ivanov, E. McDonough, J. C. Hill, M. Simonović, M. W. Toomey, S. Alexander et al., _Constraining Early Dark Energy with Large-Scale Structure_ , _Phys. Rev. D_ 102 (2020) 103502 [2006.11235].
  * [10] R. Angulo, M. Fasiello, L. Senatore and Z. Vlah, _On the statistics of biased tracers in the effective field theory of large scale structures_ , _Journal of Cosmology and Astroparticle Physics_ 2015 (2015) 029 [1503.08826].
  * [11] A. Perko, L. Senatore, E. Jennings and R. H. Wechsler, _Biased tracers in redshift space in the eft of large-scale structure_ , 1610.09321.
  * [12] T. Fujita, V. Mauerhofer, L. Senatore, Z. Vlah and R. Angulo, _Very massive tracers and higher derivative biases_ , _Journal of Cosmology and Astroparticle Physics_ 2020 (2020) 009 [1609.00717].
  * [13] P. Zhang and Y. Cai, _BOSS full-shape analysis from the EFTofLSS with exact time dependence_ , _Journal of Cosmology and Astroparticle Physics_ 2022 (2022) 031 [2111.05739].
  * [14] P. Carrilho, C. Moretti and A. Pourtsidou, _Cosmology with the EFTofLSS and BOSS: dark energy constraints and a note on priors_ , _Journal of Cosmology and Astroparticle Physics_ 2023 (2023) 028 [2207.14784].
  * [15] N. Sailer, E. Castorina, S. Ferraro and M. White, _Cosmology at high redshift — a probe of fundamental physics_ , _Journal of Cosmology and Astroparticle Physics_ 2021 (2021) 049 [2106.09713].
  * [16] A. Pourtsidou, _Interferometric H i intensity mapping: perturbation theory predictions and foreground removal effects_ , _Mon. Not. Roy. Astron. Soc._ 519 (2023) 6246 [2206.14727].
  * [17] D. Karagiannis, A. Lazanu, M. Liguori, A. Raccanelli, N. Bartolo and L. Verde, _Constraining primordial non-Gaussianity with bispectrum and power spectrum from upcoming optical and radio surveys_ , _MNRAS_ 478 (2018) 1341 [1801.09280].
  * [18] D. Karagiannis, A. Slosar and M. Liguori, _Forecasts on Primordial non-Gaussianity from 21 cm Intensity Mapping experiments_ , _JCAP_ 11 (2020) 052 [1911.03964].
  * [19] D. Karagiannis, J. Fonseca, R. Maartens and S. Camera, _Probing primordial non-Gaussianity with the power spectrum and bispectrum of future 21 cm intensity maps_ , _Phys. Dark Univ._ 32 (2021) 100821 [2010.07034].
  * [20] D. Karagiannis, R. Maartens and L. F. Randrianjanahary, _Cosmological constraints from the power spectrum and bispectrum of 21cm intensity maps_ , _Journal of Cosmology and Astroparticle Physics_ 2022 (2022) 003 [2206.07747].
  * [21] SKA collaboration, D. J. Bacon et al., _Cosmology with Phase 1 of the Square Kilometre Array: Red Book 2018: Technical specifications and performance forecasts_ , _Publ. Astron. Soc. Austral._ 37 (2020) e007 [1811.02743].
  * [22] J. Wang et al., _H i intensity mapping with MeerKAT: calibration pipeline for multidish autocorrelation observations_ , _Mon. Not. Roy. Astron. Soc._ 505 (2021) 3698 [2011.13789].
  * [23] S. Cunnington et al., _H i intensity mapping with MeerKAT: power spectrum detection in cross-correlation with WiggleZ galaxies_ , _Mon. Not. Roy. Astron. Soc._ 518 (2022) 6262 [2206.01579].
  * [24] S. Cunnington et al., _The foreground transfer function for H i intensity mapping signal reconstruction: MeerKLASS and precision cosmology applications_ , _Mon. Not. Roy. Astron. Soc._ 523 (2023) 2453 [2302.07034].
  * [25] D. Crichton et al., _Hydrogen Intensity and Real-Time Analysis Experiment: 256-element array status and overview_ , _J. Astron. Telesc. Instrum. Syst._ 8 (2022) 011019 [2109.13755].
  * [26] F. Villaescusa-Navarro, M. Viel, K. K. Datta and T. R. Choudhury, _Modeling the neutral hydrogen distribution in the post-reionization Universe: intensity mapping_ , _JCAP_ 09 (2014) 050 [1405.6713].
  * [27] J. Tinker, A. V. Kravtsov, A. Klypin, K. Abazajian, M. Warren, G. Yepes et al., _Toward a Halo Mass Function for Precision Cosmology: The Limits of Universality_ , _ApJ_ 688 (2008) 709 [0803.2706].
  * [28] T. Lazeyras, C. Wagner, T. Baldauf and F. Schmidt, _Precision measurement of the local bias of dark matter halos_ , _J. Cosmology Astropart. Phys_ 2016 (2016) 018 [1511.01096].
  * [29] A. Cooray and R. Sheth, _Halo models of large scale structure_ , _Phys. Rep._ 372 (2002) 1 [astro-ph/0206508].
  * [30] E. Castorina and F. Villaescusa-Navarro, _On the spatial distribution of neutral hydrogen in the Universe: bias and shot-noise of the HI power spectrum_ , _MNRAS_ 471 (2017) 1788 [1609.05157].
  * [31] T. Baldauf, U. Seljak, V. Desjacques and P. McDonald, _Evidence for quadratic tidal tensor bias from the halo bispectrum_ , _Phys. Rev. D_ 86 (2012) 083540 [1201.4827].
  * [32] R. A. Battye, I. W. A. Browne, C. Dickinson, G. Heron, B. Maffei and A. Pourtsidou, _H I intensity mapping: a single dish approach_ , _MNRAS_ 434 (2013) 1239 [1209.0343].
  * [33] D. Blas, J. Lesgourgues and T. Tram, _The cosmic linear anisotropy solving system (CLASS). part II: Approximation schemes_ , _Journal of Cosmology and Astroparticle Physics_ 2011 (2011) 034 [1104.2933].
  * [34] S.-F. Chen, Z. Vlah and M. White, _The reconstructed power spectrum in the Zeldovich approximation_ , _JCAP_ 09 (2019) 017 [1907.00043].
  * [35] V. Yankelevich and C. Porciani, _Cosmological information in the redshift-space bispectrum_ , _Mon. Not. Roy. Astron. Soc._ 483 (2019) 2078 [1807.07076].
  * [36] P. McDonald and A. Roy, _Clustering of dark matter tracers: generalizing bias for the coming era of precision LSS_ , _J. Cosmology Astropart. Phys_ 8 (2009) 020 [0902.0991].
  * [37] J. A. Peacock and S. J. Dodds, _Reconstructing the linear power spectrum of cosmological mass fluctuations_ , _Mon. Not. Roy. Astron. Soc._ 267 (1994) 1020 [astro-ph/9311057].
  * [38] W. E. Ballinger, J. A. Peacock and A. F. Heavens, _Measuring the cosmological constant with redshift surveys_ , _Mon. Not. Roy. Astron. Soc._ 282 (1996) 877 [astro-ph/9605017].
  * [39] Y.-S. Song, A. Taruya and A. Oka, _Cosmology with anisotropic galaxy clustering from the combination of power spectrum and bispectrum_ , _J. Cosmology Astropart. Phys_ 8 (2015) 007 [1502.03099].
  * [40] T. Baldauf, M. Mirbabayi, M. Simonović and M. Zaldarriaga, _LSS constraints with controlled theoretical uncertainties_ , 1602.00674.
  * [41] A. Chudaykin, M. M. Ivanov, O. H. E. Philcox and M. Simonović, _Nonlinear perturbation theory extension of the Boltzmann code CLASS_ , _Phys. Rev. D_ 102 (2020) 063533 [2004.10607].
  * [42] T. Steele and T. Baldauf, _Precise Calibration of the One-Loop Trispectrum in the Effective Field Theory of Large Scale Structure_ , _Phys. Rev. D_ 103 (2021) 103518 [2101.10289].
  * [43] D. J. H. Chung, M. Münchmeyer and S. C. Tadepalli, _Search for Isocurvature with Large-scale Structure: A Forecast for Euclid and MegaMapper using EFTofLSS_ , 2306.09456.
  * [44] M. Tegmark, _Measuring cosmological parameters with galaxy surveys_ , _Physical Review Letters_ 79 (1997) 3806 [astro-ph/9706198].
  * [45] M. Tegmark, A. J. S. Hamilton, M. A. Strauss, M. S. Vogeley and A. S. Szalay, _Measuring the galaxy power spectrum with future redshift surveys_ , _The Astrophysical Journal_ 499 (1998) 555 [astro-ph/9708020].
  * [46] E. Sefusatti, M. Crocce, S. Pueblas and R. Scoccimarro, _Cosmology and the bispectrum_ , _Phys. Rev. D_ 74 (2006) 023522 [astro-ph/0604505].
  * [47] E. Sefusatti and E. Komatsu, _Bispectrum of galaxies from high-redshift galaxy surveys: Primordial non-Gaussianity and nonlinear galaxy bias_ , _Phys. Rev. D_ 76 (2007) 083004 [0705.0343].
  * [48] K. C. Chan and L. Blot, _Assessment of the information content of the power spectrum and bispectrum_ , _Phys. Rev. D_ 96 (2017) 023528 [1610.06585].
  * [49] V. Desjacques, D. Jeong and F. Schmidt, _Large-scale galaxy bias_ , _Phys. Rep._ 733 (2018) 1 [1611.09787].
  * [50] K. C. Chan and L. Blot, _Assessment of the Information Content of the Power Spectrum and Bispectrum_ , _Phys. Rev. D_ 96 (2017) 023528 [1610.06585].
  * [51] L. B. Newburgh et al., _HIRAX: A Probe of Dark Energy and Radio Transients_ , _Proc. SPIE Int. Soc. Opt. Eng._ 9906 (2016) 99065X [1607.02059].
  * [52] B. R. B. Saliwanchik, A. Ewall-Wice, D. Crichton, E. R. Kuhn, D. Ölçek, K. Bandura et al., _Mechanical and optical design of the HIRAX radio telescope_ , in _Ground-based and Airborne Telescopes VIII_ , vol. 11445 of _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_ , p. 114455O, Jan., 2021, 2101.06338.
  * [53] E. R. Kuhn, B. R. B. Saliwanchik, M. Harris, M. Aich, K. Bandura, T.-C. Chang et al., _Design and implementation of a noise temperature measurement system for the Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX)_ , 2101.06337.
  * [54] M. Zaldarriaga, S. R. Furlanetto and L. Hernquist, _21 Centimeter Fluctuations from Cosmic Gas at High Redshifts_ , _ApJ_ 608 (2004) 622 [astro-ph/0311514].
  * [55] M. Tegmark and M. Zaldarriaga, _Fast Fourier transform telescope_ , _Phys. Rev. D_ 79 (2009) 083530 [0805.4414].
  * [56] P. Bull, P. G. Ferreira, P. Patel and M. G. Santos, _Late-time cosmology with 21cm intensity mapping experiments_ , _Astrophys. J._ 803 (2015) 21 [1405.1452].
  * [57] Cosmic Visions 21 cm collaboration, R. Ansari et al., _Inflation and Early Dark Energy with a Stage II Hydrogen Intensity Mapping experiment_ , 1810.09572.
  * [58] Euclid collaboration, A. Blanchard et al., _Euclid preparation: VII. Forecast validation for Euclid cosmological probes_ , _Astron. Astrophys._ 642 (2020) A191 [1910.09273].
  * [59] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., _Planck 2018 results. VI. Cosmological parameters_ , _A &A_ 641 (2020) A6 [1807.06209].
  * [60] M. Ishak, _Testing General Relativity in Cosmology_ , _Living Rev. Rel._ 22 (2019) 1 [1806.10122].
  * [61] G. D’Amico, Y. Donath, M. Lewandowski, L. Senatore and P. Zhang, _The one-loop bispectrum of galaxies in redshift space from the Effective Field Theory of Large-Scale Structure_ , 2211.17130.