# Quantifying the redshift space distortion of the bispectrum III : Detection
prospects of the multipole moments

Arindam Mazumdar1, Debanjan Sarkar2, Somnath Bharadwaj1,3
1Centre for Theoretical Studies, Indian Institute of Technology Kharagpur,
Kharagpur - 721302, India
2Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva -
84105, Israel
3Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur -
721302, India
arindam.mazumdar@iitkgp.ac.indebanjan<EMAIL_ADDRESS>
###### Abstract

The redshift space anisotropy of the bispectrum is generally quantified using
multipole moments. The possibility of measuring these multipoles in any survey
depends on the level of statistical fluctuations. We present a formalism to
compute the statistical fluctuations in the measurement of bispectrum
multipoles for galaxy surveys. We consider specifications of a Euclid like
galaxy survey and present two quantities: the signal-to-noise ratio (SNR)
which quantifies the detectability of a multipole, and the rank correlation
which quantifies the correlation in measurement errors between any two
multipoles. Based on SNR values, we find that Euclid can potentially measure
the bispectrum multipoles up to $\ell=4$ across various triangle shapes,
formed by the three k vectors in Fourier space. In general, SNR is maximum for
the linear triangles. SNR values also depend on the scales and redshifts of
observation. While, $\ell\leq 2$ multipoles can be measured with ${\rm SNR}>5$
even at linear/quasi-linear ($k\lesssim 0.1\,{\rm Mpc}^{-1}$) scales, for
$\ell>2$ multipoles, we require to go to small scales or need to increase bin
sizes. For most multipole pairs, the errors are only weakly correlated across
much of the triangle shapes barring a few in the vicinity of squeezed and
stretched triangles. This makes it possible to combine the measurements of
different multipoles to increase the effective SNR.

###### keywords:

methods: statistical – cosmology: theory – large-scale structures of Universe.

††pubyear: 2022††pagerange: Quantifying the redshift space distortion of the
bispectrum III : Detection prospects of the multipole moments–LABEL:lastpage

##  1 Introduction

Observations of the Cosmic Microwave Background (CMB) (Fergusson et al., 2012;
Oppizzi et al., 2018; Akrami et al., 2020; Shiraishi, 2019) and galaxy
clustering (Feldman et al., 2001; Scoccimarro et al., 2004; Liguori et al.,
2010; Ballardini et al., 2019) indicate that the primordial density
fluctuations are consistent with the the simplest models of inflation which
predict these to be a Gaussian random field (Baumann, 2009). The power
spectrum is sufficient to quantify the statistics of a Gaussian random field
for which all the higher order statistics are predicted to be zero. However,
several inflationary scenarios also predict the primordial fluctuations to be
non-Gaussian (primordial non-Gaussianity; Bartolo et al. 2004). Further, the
non-linear evolution of initially Gaussian density fluctuations and also the
non-linear biasing of the tracer fields (e.g. galaxies) both introduce
‘induced non-Gaussianity’ (Fry, 1984; Bernardeau et al., 2002). The
bispectrum, which is the Fourier transform of the three-point correlation
function, is the lowest order statistics which is sensitive to the non-
Gaussianity. Second order perturbation theory predicts that measurements of
the bispectrum in the weakly non-linear regime can be used to constrain the
bias parameters (Matarrese et al., 1997), and this framework has been applied
in several galaxy surveys to extract the galaxy bias parameters (Feldman et
al., 2001; Scoccimarro et al., 2001; Verde et al., 2002; Nishimichi et al.,
2007; Gil-Marín et al., 2015). Further, measurements of the bispectrum allow
us to break the degeneracy between the matter density parameter $\Omega_{m}$
and the linear bias parameter $b_{1}$, something which is not possible by
using the power spectrum alone (Scoccimarro et al., 1999). A precise
measurement of the Baryon Acoustic Oscillations (BAO) in the bispectrum allows
us to constrain the expansion rate of the universe Pearson & Samushia (2018).
Upcoming galaxy surveys like Euclid (Blanchard et al., 2020), promises to
measure the bispectrum with high precision, and thereby constrain the above
parameters with unprecedented accuracy.

Redshift space distortion (RSD) is an important effect in galaxy redshift
surveys (Kaiser, 1987; Jackson, 1972; Hamilton, 1998). The line-of-sight (LoS)
anisotrpy in the redshift space power spectrum contains a wealth of
cosmological information. For example, this can been used to: $(i)$ measure
the growth of structures $f$ on linear scales (Loveday et al., 1996; Peacock
et al., 2001; Hawkins et al., 2003; Guzzo et al., 2008), $(ii)$ constrain the
total density from massive neutrinos (Hu et al., 1998; Upadhye, 2019), and
$(iii)$ test dark energy and modified gravity theories (Linder, 2008; Song &
Percival, 2009; de la Torre et al., 2017; Johnson et al., 2016; Mueller et
al., 2018).

The bispectrum also is affected by RSD, and the redshift space bispectrum also
contains a wealth of cosmological information (Scoccimarro et al., 1999;
Yankelevich & Porciani, 2019; Hahn & Villaescusa-Navarro, 2021), and it is
important to accurately model and quantify this. Hivon et al. (1995) and Verde
et al. (1998) have formulated the initial theoretical framework for
calculating the bispectrum in redshift space. However, they mainly focused on
measuring the large scale bias parameter and the cosmological parameters, and
they have not quantified the RSD anisotropy in general. Later, Scoccimarro et
al. (1999) have quantified the anisotropy of the redshift space bispectrum
utilizing spherical harmonics. Their work, however, was restricted only to the
monopole and one quadrupole component $(\ell=2,m=0)$. The work of Hashimoto et
al. (2017) also was limited to a single quadrupole component of the redshift
space bispectrum. Nan et al. (2018) have calculated approximate analytical
expressions for the higher angular multipole moments up to $\ell=4$ based on
the halo model, and their analysis is only limited to a few triangle
configurations. Yankelevich & Porciani (2019) and Gualdi & Verde (2020) have
analysed the combined ability of the redshift space power spectrum and
bispectrum to constrain the cosmological parameters. Desjacques et al. (2018)
have utilised effective field theory to model the redshift space bispectrum on
quasi non-linear scales. The theoretical analysis of Clarkson et al. 2019 and
de Weerd et al. (2020) suggests that relativistic effects will introduce a
dipole anisotropy in the redshift space bispectrum on very large length-
scales.

There has been some work towards developing fast estimators which quantify the
anaistropy of the redshift space bispectrum. Slepian & Eisenstein (2017, 2018)
have introduced a technique to expand the redshift space three-point
correlation function in terms of the products of two spherical harmonics. On
the other hand, Sugiyama et al. (2019) have proposed a tri-polar spherical
harmonic decomposition to quantify the anisotropy of the redshift space
bispectrum, and as a demonstration they applied this to the Baryon Oscillation
Spectroscopic Survey (BOSS) Data Release 12.

Our recent work (Bharadwaj et al. 2020, hereafter Paper I) presents a
formalism to quantify the anisotropy of the redshift space bispectrum
$B^{s}(\mathbf{k_{a}},\mathbf{k_{b}},\mathbf{k_{c}})$ by decomposing it into
multipole moments $\bar{B}^{m}_{\ell}(k_{1},\mu,t)$. Here $k_{1}$, the length
of the largest side, and $(\mu,t)$ respectively quantify the size and shape of
the triangle $(\mathbf{k_{a}},\mathbf{k_{b}},\mathbf{k_{c}})$. We have
illustrated this formalism by quantifying the anisotrpoy due to linear RSD of
the bispectrum arising from primordial non-Gaussianity. We have found that
only the first four even $\ell$ multipoles are non-zero, for which we have
presented explicit analytical expressions. These results are expected to be
important to constrain $f_{\text{NL}}$ using the bispectrum measured from
future redshift surveys. In a subsequent work Mazumdar et al. (2020)
(hereafter Paper II), the same formalism was used to quantify the anisotropy
of the induced redshift space bispectrum arising from Gaussin initial
conditions. We present analytical expressions for all the multipole moments
which are predicted to be non-zero ($\ell\leq 8,m\leq 6$ ) at second order
perturbation theory. Considering triangles of all possible shapes, we have
analysed the shape dependence of all the multipoles holding $k_{1}=0.2\,{\rm
Mpc}^{-1},\beta_{1}=1,b_{1}=1$ and $\gamma_{2}=0$ fixed. Here
$\beta_{1},b_{1}$ and $\gamma_{2},$ quantify the linear redshift distortion
parameter, linear bias and quadratic bias respectively. For most multipoles we
find that the maxima or minima, or both, occur very close to the squeezed
limit. Further, the absolute values $\mid\bar{B}^{m}_{\ell}\mid$ are found to
decrease rapidly if either $\ell$ or $m$ are increased. We have also provided
rough estimates for measuring the various multipoles using upcoming galaxy
redshift surveys. The present paper presents a detailed analysis for the
prospects of measuring the various multipole moments using upcoming galaxy
redshift surveys.

The ability to measure any particular bispectrum multipole
$\bar{B}^{m}_{\ell}$ primarily depends on its amplitude, the galaxy number
density and the extent of the survey volume. Until recently, measurements of
the bispectrum were limited by the shot noise from the limited galaxy number
density and the cosmic variance due to the limited volume of the galaxy
redshift surveys (Scoccimarro et al., 2001; Verde et al., 2002; Croton et al.,
2004; Jing & Boerner, 2004; Kulkarni et al., 2007; Gaztanaga et al., 2009;
Marin, 2011). However, Gil-Marín et al. (2015, 2017); Philcox & Ivanov (2022)
have recently measured the isotropic component (monopole) of the galaxy
bispectrum using the Baryon Oscillation Spectroscopic Survey at a relatively
high level of precision. They showed that it is possible to improve the
constraints on the cosmological parameters by including the bispectrum along
with the power spectrum. Upcoming galaxy surveys will cover unprecedented
large volumes with high galaxy number density, and it is anticipated that this
will enable us to precisely measure the higher multipoles of the bispectrum.
For example, the Euclid space telescope (Blanchard et al., 2020) is expected
to complete a wide survey that will measure $\sim\\!10^{8}$ galaxy redshifts
over $15,000$ square degrees on the sky in the range $z\lesssim 2.5$. Precise
measurements of the galaxy bispectrum using Euclid are expected to place tight
constraints on: $(i)$ primordial non-Gaussianity (Fedeli et al., 2011; Dai &
Xia, 2020), $(ii)$ neutrino masses (Chudaykin & Ivanov, 2019; Hahn &
Villaescusa-Navarro, 2021), $(iii)$ modified gravity models (Bose et al.,
2020), and $(iv)$ all the other currently constrained standard cosmological
parameters (Agarwal et al., 2021). The above estimates mostly rely on the
real-space bispectrum or its redshift space monopole. It is important and
interesting to also consider the higher multipoles of the bispectrum in order
to utilize the full reach of this mission (Gualdi & Verde, 2020).

In this paper we developed a formalism to calculate the statistical
fluctuations expected in the bispectrum multipoles measured from any given
galaxy redshift survey. We quantify these statistical fluctuations through the
error covariance of the different multipoles. Here we have considered the
specifications of the Euclid galaxy redshift survey for which we present the
signal-to-noise ratio (SNR) with which the different multipoles are expected
to be measured. We also present results for the correlations expected between
the different multipole moments. The paper is organised as follows. In Section
2, we present the formalism for calculating the error covariance between
different bispectrum multipoles. In Section 3 we present our main results,
which include the dependence of bispectrum multipoles on triangle shapes,
sizes and redshifts, possibility of measuring the different multipoles, and
error covariance between the pair of multipoles in terms of rank correlation.
Finally in Section 4, we summarize our findings and outline some future
directions. In this paper, we use the Boltzmann code CLASS (Lesgourgues 2011;
Blas et al. 2011) to calculate the input matter power spectrum for our
analysis and assumed cosmological parameters from Planck 2015 (Planck
Collaboration et al., 2016) throughout the work.

##  2 Formalism

We first consider $\delta^{s}(\mathbf{x})$ which represents the density
contrast of a smooth cosmological field in redshift space. This can be
decomposed into Fourier modes using

$\delta^{s}(\mathbf{x})={1\over
V}\sum_{k}e^{i\mathbf{k}.\mathbf{x}}\Delta^{s}(\mathbf{k})\,,$ (1)

for which, in the absence of Poisson noise, the power spectrum is defined as

$\langle\Delta^{s}(\mathbf{k_{1}})\Delta^{s}(\mathbf{k_{2}})\rangle=V\delta_{\mathbf{k_{1}}+\mathbf{k_{2}},0}P^{s}(k_{1},\mu_{1})\,,$
(2)

where $\mu_{1}$ is the cosine of the angle between the Fourier mode
$\mathbf{k_{1}}$ and the line of sight (LoS) direction ${\bf\hat{n}}$. We have
considered ${\bf\hat{n}}=\hat{z}$ here. The bispectrum is similarly defined as

$\langle\Delta^{s}(\mathbf{k_{1}})\Delta^{s}(\mathbf{k_{2}})\Delta^{s}(\mathbf{k_{3}})\rangle=V\delta_{\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}},0}B^{s}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\,.$
(3)

Throughout this paper we assume that the three vectors
$(\mathbf{k_{a}},\mathbf{k_{b}},\mathbf{k_{c}})$, which form a closed
triangle, are ordered such that $k_{1}\geq k_{2}\geq k_{3}$. In the absence of
redshift space distortion the bispectrum depends only on the shape and size of
the triangle formed by the three vectors
$(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$. Here, following Paper I, we
parametrize this using $(k_{1},\mu,t)$ where $k_{1}$ the length of the largest
edge quantifies the size of the triangle while $\mu=\cos\theta$ (fig. 1) and
$t=k_{2}/k_{1}$ together quantity the shape of the triangle. Note that $\mu$
and $t$ are both bounded within the range $0.5\leq\mu,t\leq 1$ and $2\mu t\geq
1$. In the presence of redshift space distortion the bispectrum also depends
on $\mu_{a}=\mathbf{k_{a}}\cdot{\bf\hat{n}}/k_{a}$ where $a=1,2,3$. Therefore,
the bispectrum now depends on how the triangle is oriented with respect to
$\hat{n}$.

It is necessary to consider triangles of all possible orientations in order to
quantify the anisotropy of the redshift space bispectrum. Here we start from a
reference triangle in the $x-z$ plane (fig. 1) for which we have

$\displaystyle\mathbf{k_{1}}$ $\displaystyle=$ $\displaystyle k_{1}\hat{z}\,,$
$\displaystyle\mathbf{k_{2}}$ $\displaystyle=$ $\displaystyle
k_{1}t[-\mu\hat{z}+\sqrt{1-\mu^{2}}\hat{x}]\,,$ (4)

and $\mathbf{k_{3}}=-\mathbf{k_{1}}-\mathbf{k_{2}}$. It is possible to obtain
all possible orientations of the triangle by applying different rotations
$\hat{\cal{R}}$ to the reference triangle. We parameterize these rotations
using $(\alpha,\beta,\gamma)$ the Euler angles which refer to successive
rotations along the $z$, $y$ and $z$ axes respectively (Sakurai, 1994). We
then have

$\displaystyle\mu_{1}$ $\displaystyle=$ $\displaystyle\ p_{z}\,$
$\displaystyle\mu_{2}$ $\displaystyle=$
$\displaystyle-\mu\,p_{z}+\sqrt{1-\mu^{2}}\,p_{x}$ $\displaystyle\mu_{3}$
$\displaystyle=$ $\displaystyle\frac{-[(1-\mu
t)p_{z}+t\sqrt{1-\mu^{2}}p_{x}]}{\sqrt{1-2\mu t+t^{2}}}$ (5)

where we have introduced an unit vector
$\mathbf{\hat{p}}=\hat{\cal{R}}^{-1}{\bf\hat{n}}$ which has components
$p_{z}=\cos(\beta)$ and $p_{x}=-\sin(\beta)\,\cos(\gamma)$ respectively. Note
that these expressions are independent of $\alpha$ i.e. the axis of the first
rotation ($z$) coincides with ${\bf\hat{n}}$, and it does not affect the
redshift space distortion. In summary, the redshift space bispectrum
$B^{s}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$ can be completely
parameterized using $B^{s}(k_{1},\mu,t,\mathbf{\hat{p}})$ where
$(k_{1},\mu,t)$ quantify the size and shape dependence of the triangle while
$\mathbf{\hat{p}}$ quantifies its orientation with respect to ${\bf\hat{n}}$.

The redshift space anisotropy of the bispectrum is quantified using multipole
moments defined as

$\bar{B}^{m}_{\ell}(k_{1},\mu,t)=\sqrt{\frac{(2\ell+1)}{4\pi}}\int[Y^{m}_{\ell}(\mathbf{\hat{p}})]^{*}B^{s}(k_{1},\mu,t,\mathbf{\hat{p}})\,d\Omega_{\mathbf{\hat{p}}}$
(6)

where the integral over the solid angle $d\Omega_{\mathbf{\hat{p}}}$ accounts
for all possible orientations of the triangle. As discussed in Paper I, only
the even $\ell$ multipoles of the bispectrum are non-zero and these are all
real valued.

In reality, we encounter a finite number of Fourier modes distributed on a
regular grid. Here we define the bispectrum multipole estimator as a discrete
sum over triangle configurations in Fourier space,

$\hat{B}_{\ell}^{m}(k_{1},\mu,t)=\sum_{n}\frac{w_{l}^{m}(\mathbf{\hat{p}}_{n})}{2V}[\Delta^{s}({\mathbf{k}}_{n})\Delta^{s}(\bar{\mathbf{k}}_{n})\Delta^{s}(\tilde{\mathbf{k}}_{n})+c.c.]$
(7)

where $\mathbf{k}_{n}+\bar{\mathbf{k}}_{n}+\tilde{\mathbf{k}}_{n}=0$ form a
closed triangle, each value of $n$ represents a different closed triangle,
$c.c.\equiv$ complex conjugate, and

$w_{\ell}^{m}(n)={\rm Re}\Big{[}\sqrt{2\ell+1\over
4\pi}{Y_{\ell}^{m}(\hat{p}_{n})\over\sum_{n_{1}}|Y_{\ell}^{m}(\mathbf{\hat{p}}_{n_{1}})|^{2}}\Big{]}\,.$
(8)

The sum over $n$ covers triangles of all possible orientations, with shape and
size in a bin of extent $(\Delta k_{1},\Delta\mu,\Delta t)$ around
$(k_{1},\mu,t)$. The exact binning scheme used for the present work is
discussed later in this paper.

###  2.1 The Error Covariance

Figure 1: This shows a reference triangle
($\mathbf{k_{1}}\mathbf{k_{2}},\mathbf{k_{3}}$) in $x-z$ plane which is used
to define various parameters used here. $\bf dk_{2}$, $d\theta$ represents the
binning of the bi-spectrum triangle discussed in Section 2.3.

We assume that the grid spacing in $\mathbf{k}$ is adequately fine so that
various triangle orientations are adequately sampled which implies that
$\langle\hat{B}_{\ell}^{m}(k_{1},\mu,t)\rangle=\bar{B}^{m}_{\ell}(k_{1},\mu,t)$
where $\langle...\rangle$ denotes an ensemble average of different
realisations of the random density field $\Delta(\mathbf{k})$. In the present
work we are interested in the statistical fluctuations (error)

$\Delta\hat{B}_{\ell}^{m}=\hat{B}_{\ell}^{m}(k_{1},\mu,t)-\bar{B}^{m}_{\ell}(k_{1},\mu,t)$
(9)

which we quantify through the covariance

$C_{\ell\ell^{\prime}}^{mm^{\prime}}=\langle\Delta\hat{B}_{\ell}^{m}\,\Delta\hat{B}_{\ell^{\prime}}^{m^{\prime}}\rangle\,.$
(10)

We calculate this using

$\displaystyle
C_{\ell\ell^{\prime}}^{mm^{\prime}}+\bar{B}^{m}_{\ell}\bar{B}^{m^{\prime}}_{\ell^{\prime}}=(2V)^{-2}\sum_{n_{1},n_{2}}w_{\ell}^{m}(\mathbf{\hat{p}}_{n_{1}})w_{\ell^{\prime}}^{m^{\prime}}(\mathbf{\hat{p}}_{n_{2}})\times$
$\displaystyle[\langle\Delta^{s}({\mathbf{k}}_{n_{1}})\Delta^{s}(\bar{\mathbf{k}}_{n_{1}})\Delta^{s}(\tilde{\mathbf{k}}_{n_{1}})\Delta^{s}({\mathbf{k}}_{n_{2}})\Delta^{s}(\bar{\mathbf{k}}_{n_{2}})\Delta^{s}(\tilde{\mathbf{k}}_{n_{2}})\rangle+$
$\displaystyle c.c.+$
$\displaystyle\langle\Delta^{s}({\mathbf{k}}_{n_{1}})\Delta^{s}(\bar{\mathbf{k}}_{n_{1}})\Delta^{s}(\tilde{\mathbf{k}}_{n_{1}})\Delta^{s*}({\mathbf{k}}_{n_{2}})\Delta^{s*}(\bar{\mathbf{k}}_{n_{2}})\Delta^{s*}(\tilde{\mathbf{k}}_{n_{2}})\rangle$
$\displaystyle+c.c.]\,.$ (11)

We have evaluated this using

$\displaystyle\langle\Delta^{s}({\mathbf{k}}_{n_{1}})\Delta^{s}(\bar{\mathbf{k}}_{n_{1}})\Delta^{s}(\tilde{\mathbf{k}}_{n_{1}})\Delta^{s}({\mathbf{k}}_{n_{2}})\Delta^{s}(\bar{\mathbf{k}}_{n_{2}})\Delta^{s}(\tilde{\mathbf{k}}_{n_{2}})\rangle=$
$\displaystyle[1+3\delta_{n_{1},n_{2}}]V^{2}B^{s}(\mathbf{k}_{n_{1}},\bar{\mathbf{k}}_{n_{1}},\tilde{\mathbf{k}}_{n_{1}})B^{s}(\mathbf{k}_{n_{2}},\bar{\mathbf{k}}_{n_{2}},\tilde{\mathbf{k}}_{n_{2}})$
$\displaystyle+\delta_{n_{1},-n_{2}}V^{3}P^{s}(\mathbf{k}_{n_{1}})P^{s}(\bar{\mathbf{k}}_{n_{1}})P^{s}(\tilde{\mathbf{k}}_{n_{1}})$
(12)

which assumes that the signal is weakly non-Gaussian whereby contributiosn
from $T^{s}$ the trispectrum and $P^{s}_{6}$ the sixth order polyspectrum can
be neglected. Here $\delta_{n_{1},n_{2}}$ is a Kronecker delta which has value
one only when $n_{1}$ and $n_{2}$ refer to the same triangle and is zero
otherwise. Further, $-n_{2}$ refers to the same triangle as $n_{2}$, but the
two are exactly oppositely oriented ie.
$\mathbf{k}_{-n_{1}}=-\mathbf{k}_{n_{1}}$, etc. Also note that
$\Delta^{s}({\mathbf{k}}_{-n_{2}})=\Delta^{s*}({\mathbf{k}}_{n_{2}})$. Using
these in eq. (10) we obtain

$\displaystyle
C_{\ell\ell^{\prime}}^{mm^{\prime}}=\sum_{n}w_{\ell}^{m}(\mathbf{\hat{p}}_{n})w_{\ell^{\prime}}^{m^{\prime}}(\mathbf{\hat{p}}_{n})\times$
$\displaystyle\\{3[B^{s}(\mathbf{k}_{n},\bar{\mathbf{k}}_{n},\tilde{\mathbf{k}}_{n})]^{2}+VP^{s}(\mathbf{k}_{n})P^{s}(\bar{\mathbf{k}}_{n})P^{s}(\tilde{\mathbf{k}}_{n})\\}$
(13)

which provides an estimate of the error covariance in the absence of Poisson
noise.

###  2.2 Poisson noise

We now consider a discrete tracer (e.g. galaxies) whose density represents a
Poisson sampling of the smooth filed which we have been discussing till now.
We use $\Delta_{g}(\mathbf{k})$ to represent the corresponding density
contrast in Fourier space. The discrete sampling of the smooth field gives
rise to a shot noise (or Poisson noise) whose contribution to the power
spectrum and the bi-spectrum are well studied (Peebles, 1980; Scoccimarro et
al., 2001; Smith, 2009). Using $\Delta_{g}(\mathbf{k})$ instead of
$\Delta(\mathbf{k})$ in eq. (2) to estimate the power spectrum, we have

$V^{-1}\langle\Delta_{g}^{s}(\mathbf{k_{1}})\Delta_{g}^{s*}(\mathbf{k_{1}})\rangle=P^{s}(k_{1},\mu_{1})+n_{g}^{-1}\,,$
(14)

where, $P^{s}(k_{a},\mu_{a})$ is the power spectrum of the smooth field
$\Delta(\mathbf{k})$ and $n_{g}$ is the number density of galaxies. Similarly
considering the bispectrum we have

$\displaystyle
V^{-1}\langle\Delta_{g}^{s}(\mathbf{k_{1}})\Delta_{g}^{s}(\mathbf{k_{1}})\Delta_{g}^{s}(\mathbf{k_{3}})\rangle=B^{s}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+$
$\displaystyle
n_{g}^{-1}[P^{s}(k_{1},\mu_{1})+P^{s}(k_{2},\mu_{2})+P^{s}(k_{3},\mu_{3})]+n_{g}^{-2}$
(15)

where it has been assumed that
$\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}}=0$. It is now necessary to
consider how eq. (13) for the bispectrum error covariance is modified if we
account for the discrete sampling. We first consider the ensemble average of
the product of six $\Delta_{g}(\mathbf{k})$s which is required to calculate
the covariance (eq. 10).Here we use the shortened notation
$\Delta_{ag}\equiv\Delta_{g}(\mathbf{k_{a}})$,
$\Delta_{a}\equiv\Delta(\mathbf{k_{a}})$,
$\Delta_{ab}\equiv\Delta(\mathbf{k_{a}}+\mathbf{k_{b}})$,
$\Delta_{abc}\equiv\Delta(\mathbf{k_{a}}+\mathbf{k_{b}}+\mathbf{k_{c}})$, etc.
We then have

$\displaystyle\langle[\Delta_{ag}\Delta_{bg}\Delta_{cg}\Delta_{dg}\Delta_{eg}\Delta_{fg}]-[\Delta_{a}\Delta_{b}\Delta_{c}\Delta_{d}\Delta_{e}\Delta_{f}]\rangle=$
$\displaystyle
Vn_{g}^{-5}+n_{g}^{-4}\langle[\Delta_{a}\Delta_{bcdef}+\Delta_{ab}\Delta_{cdef}+\Delta_{abc}\Delta_{cdef}]\rangle+$
$\displaystyle
n_{g}^{-3}\langle[\Delta_{a}\Delta_{b}\Delta_{cdef}+\Delta_{a}\Delta_{bc}\Delta_{def}+\Delta_{ab}\Delta_{cd}\Delta_{ef}]\rangle+$
$\displaystyle
n_{g}^{-2}\langle[\Delta_{a}\Delta_{b}\Delta_{c}\Delta_{def}+\Delta_{a}\Delta_{b}\Delta_{cd}\Delta_{ef}]\rangle]$
$\displaystyle
n_{g}^{-1}\langle[\Delta_{a}\Delta_{b}\Delta_{c}\Delta_{d}\Delta_{ef}]\rangle+{\rm
combinations}$ (16)

where all the terms in the r.h.s. arise due to the discrete sampling, and for
each term it is necessary to also consider all the distinct combinations of
the indices $a,b,c,...$. Instead of embarking on a detailed calculation of the
r.h.s., we present it in a schematic form which can be used for an order of
magnitude estimate. Using $[C_{\ell\ell^{\prime}}^{mm^{\prime}}]_{g}$ and
$C_{\ell\ell^{\prime}}^{mm^{\prime}}$ to represent the error covariance
respectively in the presence and absence of shot noise, we have

$\displaystyle[C_{\ell\ell^{\prime}}^{mm^{\prime}}]_{g}-C_{\ell\ell^{\prime}}^{mm^{\prime}}\sim
V^{-1}[n_{g}^{-5}+n_{g}^{-4}P^{s}+n_{g}^{-3}B^{s}]$
$\displaystyle+n_{g}^{-2}[P^{s}]^{2}+n_{g}^{-1}P^{s}B^{s}$ (17)

where $C_{\ell\ell^{\prime}}^{mm^{\prime}}\sim V[P^{s}]^{3}+3[B^{S}]^{2}$ (eq.
13).Here we consider estimates for the upcoming Euclid survey (Laureijs et
al., 2011) which is expected to cover $V\sim 10^{9}\,{\rm Mpc}^{3}$. Following
Yankelevich & Porciani (2019) we adopt $n_{g}\sim 10^{-3}{\rm Mpc}^{-3}$ and a
linear bias parameter $b\sim 1$ at $z=0.7$. We have used Blas et al. (2011) to
estimate $P^{s}\sim 10^{4}\,{\rm Mpc}^{3}$ and $B^{s}\sim[P^{s}]^{2}$ at
$k\sim$ $0.2{\rm Mpc}^{-1}$ where we expect second order perturbation to be
reasonably valid. Using these, we first consider eq. (14) where we have
$P^{s}\sim 10^{4}{\rm Mpc}^{3}$ and $n_{g}^{-1}\sim 10^{3}{\rm Mpc}^{3}$.
Similarly in eq. (15) we have $B^{s}\sim 10^{8}{\rm Mpc}^{6}$, whereas
$n_{g}^{-1}P^{s}\sim 10^{7}{\rm Mpc}^{6}$ and $n_{g}^{-2}\sim 10^{6}{\rm
Mpc}^{6}$. These estimates show that it is necessary to account for the shot
noise in order to correctly estimate the power spectrum and the bispectrum. We
now consider the error covariance for which we have $V[P^{s}]^{3}\sim
10^{21}{\rm Mpc}^{12}$ and $[B^{s}]^{2}\sim 10^{16}{\rm Mpc}^{12}$ whereby
$C_{\ell\ell^{\prime}}^{mm^{\prime}}\sim\sim 10^{21}{\rm Mpc}^{12}$ i.e. the
error in the estimated bispectrum is dominated by the terms $\sim
V[P^{s}]^{3}$. Considering the shot noise, we see that we have the largest
contribution from $n_{g}^{-1}P^{s}B^{s}\sim 10^{15}{\rm Mpc}^{12}$ and the
magnitude decreases with increasing power of $n_{g}^{-1}$ with the smallest
contribution coming from $V^{-1}n_{g}^{-5}\sim 10^{6}{\rm Mpc}^{12}$. Even if
we consider a lower galaxy number density $n_{g}\sim 10^{-4}{\rm Mpc}^{-3}$,
we find that the terms $V^{-1}[...]$ are all $\sim 10^{11}{\rm Mpc}^{12}$ and
the two remaining terms are $\sim 10^{16}{\rm Mpc}^{12}$, all of which are
several orders of magnitude smaller than the predicted value of
$C_{\ell\ell^{\prime}}^{mm^{\prime}}$. We conclude that it is quite reasonable
to ignore the shot noise contribution, and we use
$C_{\ell\ell^{\prime}}^{mm^{\prime}}$ for the error estimates presented in the
subsequent analysis.

###  2.3 Binning

We finally discuss the binning scheme which we have considered here. For the
purpose of analytic predictions, it is convenient to assume that $V$ is very
large whereby $dN_{k}$ the number of $\mathbf{k}$ modes in the interval
$d^{3}k$ is given by $dN_{k}=(2\pi)^{-3}V\,d^{3}k$. The number of triangles
$dN_{tr}$ is then given by

$dN_{tr}=(2\pi)^{-6}V^{2}\,d^{3}k_{1}^{3}\,d^{3}k_{2}^{3}\,,$ (18)

and we replace the sum in eq. (13) using $\sum_{n}\rightarrow\int dN_{tr}$.
Starting from $\mathbf{k_{1}}=k_{1}\hat{x}$ (fig. 1) we obtain all other
possible vectors $\mathbf{k_{1}}$ by changing the length $k_{1}$ or rotations
through the Euler angles $\alpha$ and $\beta$, and we have

$d^{3}k_{1}=k_{1}^{2}\,dk_{1}\,d\beta\,\sin\beta\,d\alpha\,.$ (19)

Considering $\mathbf{k_{2}}$, a change in length could occur through either a
change in $k_{1}$ or in $t$, whereas a change in orientation is associated
with either a change in $\mu=\cos\theta$ or $\gamma$ the third Euler angle. We
then have

$d^{3}k_{2}=k_{1}^{2}t^{2}\,(t\,dk_{1}+k_{1}\,dt)\,d\mu\,d\gamma\,.$ (20)

Here we have considered bins of extent $(\Delta k_{1},\Delta\mu,\Delta t)$
centered around $(k_{1},\mu,t)$. Using eq. (19) and eq. (20), we have

$dN_{tr}=({8\pi^{2}})^{-1}N_{tr}\,d\alpha\,\sin\beta\,d\beta\,d\gamma$ (21)

where $N_{tr}$ the number of triangles in any particular bin is given by

$N_{tr}=(8\pi^{4})^{-1}\,(Vk_{1}^{3})^{2}\,t^{2}[\Delta\ln
k_{1}\,(t\,\Delta\ln k_{1}+\Delta t)\,\Delta\mu]$ (22)

Using these we have

$\sum_{n_{1}}\mid
Y^{m}_{\ell}(\mathbf{\hat{p}}_{n_{1}})\mid^{2}=(4\pi)^{-1}N_{tr}$ (23)

which we use in the expression for $w_{\ell}^{m}$ (eq. 8) to write eq. (13) as

$\displaystyle
C_{\ell\ell^{\prime}}^{mm^{\prime}}(k_{1},\mu,t)=\frac{\sqrt{(2\ell+1)(2\ell^{^{\prime}}+1)}}{N_{tr}}\int
d\Omega_{\mathbf{\hat{p}}}\times$ $\displaystyle{\rm
Re}[Y_{\ell}^{m}(\mathbf{\hat{p}})]Re[Y_{\ell^{^{\prime}}}^{m^{^{\prime}}}(\mathbf{\hat{p}})]\Big{\\{}3[B^{s}(k_{1},\mu,t,\mathbf{\hat{p}})]^{2}$
$\displaystyle+VP^{s}(k_{1},\mu_{1})P^{s}(k_{2},\mu_{2})P^{s}(k_{3},\mu_{3})\,.\Big{\\}}$
(24)

Note that $k_{2},k_{3},\mu_{1},\mu_{2},\mu_{3}$ can all be calculated from
$(k_{1},\mu,t,\mathbf{\hat{p}})$ using eq. (2) and eq. (5). We have used eq.
(24) to calculate the error estimates presented subsequently in this paper
considering bins of extent $(\Delta\ln k_{1}=0.1,\Delta\mu=0.05,\Delta
t=0.05)$.

###  2.4 Second Order Induced Bispectrum

The induced bispectrum for any tracer in redshift space from second-order
perturbation theory (2LPT) can be written as

$B_{\rm
2PT}^{s}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=2b_{1}^{-1}(1+\beta_{1}\mu_{1}^{2})(1+\beta_{1}\mu_{2}^{2})\Big{\\{}F_{2}(\mathbf{k_{1}},\mathbf{k_{2}})+{\gamma_{2}\over
2}+\mu_{3}^{2}\beta_{1}G_{2}(\mathbf{k_{1}},\mathbf{k_{2}})-b_{1}\beta_{1}\mu_{3}k_{3}\Big{[}{\mu_{1}\over
k_{1}}(1+\beta_{1}\mu_{2}^{2})+{\mu_{2}\over
k_{2}}(1+\beta_{1}\mu_{1}^{2})\Big{]}\Big{\\}}P(k_{1})P(k_{2})+{\rm
cyc...}\,,$ (25)

where $\beta_{1}$ is the linear redshift distortion parameter,
$\gamma_{2}=b_{2}/b_{1}$ where $b_{1}$ is the linear bias and $b_{2}$ is the
quadratic bias, and

$G_{2}(k_{1},k_{2})={3\over 7}+{\mathbf{k_{1}}\cdot\mathbf{k_{2}}\over
2}\left({1\over k_{1}^{2}}+{1\over k_{2}^{2}}\right)+{4\over
7}{(\mathbf{k_{1}}\cdot\mathbf{k_{2}})^{2}\over k_{1}^{2}k_{2}^{2}},$ (26)

refers to the second order kernel for the divergence of the peculiar velocity.
Here we find it convenient to use the following notation

$F_{2}(\mathbf{k_{1}},\mathbf{k_{2}})=G_{2}(\mathbf{k_{1}},\mathbf{k_{2}})+\Delta
G(\mathbf{k_{1}},\mathbf{k_{2}})$ (27)

where

$\Delta G(\mathbf{k_{1}},\mathbf{k_{2}})={2\over
7}\left[1-{(\mathbf{k_{1}}\cdot\mathbf{k_{2}})^{2}\over
k_{1}^{2}k_{2}^{2}}\right]\,.$ (28)

Note that, the notations used here are taken explicitly from Paper I and Paper
II.

In Paper II, we have quantified the anisotropy of the induced redshift-space
2LPT bispectrum (eq. 25) in terms of the multipole moments
$\bar{B}^{m}_{\ell}(k_{1},\mu,t)$ defined in eq. (6). There we have also
presented the formulas needed to calculate all the multipole moments that are
predicted to be non-zero at second-order perturbation theory. Further, we have
analysed the $\mu-t$ dependence of $\bar{B}^{m}_{\ell}(k_{1},\mu,t)$ at fixed
$k_{1}=0.2\,\,{\rm Mpc}^{-1}$ and for the parameter values $\beta_{1}=1$,
$b_{1}=1$, $\gamma_{2}=0$. We find that the even multipole moments
$\bar{B}^{0}_{0}$, $\bar{B}^{0}_{2}$ and $\bar{B}^{0}_{4}$ show very similar
behaviour where their values are positive in the whole $\mu-t$ plane, the
smallest values occur near the equilateral triangles and the largest values
are found for the linear triangles. Two other even multipole moments
$\bar{B}^{2}_{2}$ and $\bar{B}^{4}_{4}$ show positive values in the whole
$\mu-t$ plane, however, their shape dependence is different from the
multipoles discussed above. For the rest of the multipole moments, we find
that they exhibit negative values at some parts of or through the entire
$\mu-t$ plane. Their contour patterns are also different from each other. We
broadly see that, although the higher multipoles ($\ell>2$) show rich variety
of shape dependence, their amplitude $\mid\bar{B}^{m}_{\ell}\mid$ fall off
sharply as $\ell$ and $m$ are increased. Note that, the analysis in Paper II
ignores the Finger-of-God (FoG) effect. At large scales linear perturbation
theory and its second order extension are expected to provide a reasonably
accurate description of the clustering of matter in real space. This however
does not hold in redshift space where it is found that large peculiar
velocities arising from highly non-linear small scale structures cause the
large scale clustering pattern to appear elongated along the LoS. This is
known as the Finger of God (FoG) effect, and it is important to include this
in any realistic analysis of the redshift space power spectrum and bispectrum.

###  2.5 Finger of God Effect

In case of the power spectrum, the FoG effect is generally incorporated as an
ad-hoc damping profile which multiplies the linear redshift space power
spectrum. A number of profiles, namely Lorentzian, Gaussian, Lorentzian-
squared etc., have been considered in the literature. Although the Lorentzian
profile works well for the simulated data (Davis & Peebles, 1982; Hamilton,
1997; Hatton & Cole, 1999; Seljak, 2001; White, 2001; Sarkar & Bharadwaj,
2018, 2019), the Gaussian profile is expected to occur naturally (Bharadwaj,
2001; Scoccimarro, 2004; Hikage & Yamamoto, 2013; Hikage et al., 2013; Okumura
et al., 2015; Hikage & Yamamoto, 2016). In this work, we consider a Gaussian
profile for the FoG damping and model the FoG redshift-space power spectrum as
(Peacock, 1992)

$P^{s}_{\rm
FoG}(k_{1},\mu_{1})=\exp{\left[(-k_{1}^{2}\mu_{1}^{2}){\sigma_{p}^{2}\over
2}\right]}\times P^{s}_{\rm 1PT}(k_{1},\mu_{1})$ (29)

where $P^{s}_{\rm 1PT}(k_{1},\mu_{1})=(1+\beta_{1}\mu_{1}^{2})^{2}\,P(k_{1})$
is the linear redshift space power spectrum, and $\sigma_{p}$ parametrizes the
pairwise velocity dispersion is in units of comoving ${\rm Mpc}$. We can
equivalently use $[\sigma_{p}\,a\,H(a)]$ in units of ${\rm km\,s^{-1}}$. Note
that on very large scales $k_{1}\sigma_{P}\ll 1$, the damping factor
$\exp{\left[-k_{1}^{2}\mu_{1}^{2}{\sigma_{p}^{2}/2}\right]}\approx 1$, and the
Kaiser effect $(1+\beta_{1}\mu_{1}^{2})^{2}$ suffices to describe the RSD
effect.

We similarly incorporate the FoG effect for the bispectrum as

$\displaystyle B^{s}_{\rm FoG}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$
$\displaystyle=$
$\displaystyle\exp\left[-(k_{1}^{2}\mu_{1}^{2}+k_{2}^{2}\mu_{2}^{2}+k_{3}^{2}\mu_{3}^{2}){\sigma_{p}^{2}\over
2}\right]$ (30) $\displaystyle\times B_{\rm
2PT}^{s}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$

where we use eq. (25) to calculate $B_{\rm 2PT}$. Due to the FoG damping term,
we do not have closed form analytic expressions for the various multipole
moments of $B^{s}_{\rm FoG}$. Here we have numerically integrated eq. (6) to
compute the various multipole moments for which the results are presented in
the following section.

##  3 Results

In this section we present the shape and size dependence of the redshift space
bispectrum, and also the corresponding statistical errors for an Euclid like
survey. As discussed in Section 2, we have used the largest side $k_{1}$ to
quantify the size of the triangle, and we have used $\mu=\cos\theta$ and
$t=k_{2}/k_{1}$ (fig. 1) to quantify the shape. Following Fry (1984), we have
defined the dimensionless bispectrum multipoles ())

$Q_{\ell}^{m}(k_{1},\mu,t)=b_{1}{\bar{B}^{m}_{\ell}(k_{1},\mu,t)\over
3P(k_{1})^{2}}\,,$ (31)

which we have used to exhibit the results presented here.

Galaxy surveys like Euclid expect to cover a broad redshifts range from
$0.001$ to $2.5$ (Ilić et al., 2021; Pozzetti et al., 2016), which can be
divided into various redshift bins for the subsequent analysis. For the
present work, we use the $z$ bins as used in Yankelevich & Porciani (2019) and
we also adopt their predicted values for the bias parameters $b_{1}$ and
$\gamma_{2}$, the survey volume $V$ and the pairwise velocity dispersion
$\sigma_{p}$. We have used the Boltzmann code CLASS (Lesgourgues, 2011; Blas
et al., 2011) to compute the real space matter power spectrum and the growth
rate $f$ for each $z$ bin.

We have considered the $z=0.7$ redshift bin as the fiducial value for which
most of our results are shown. Following Yankelevich & Porciani (2019), we
have adopted the values $b_{1}=1.18$, $\gamma_{2}=-0.9$, $V=8.97\,{\rm
Gpc}^{3}$ and $\sigma_{p}=7.09$Mpc for this particular $z$ bin. Figure 2 shows
$Q_{\ell}^{m}$ as functions of $\mu$ and $t$ at $k_{1}=0.2{\rm Mpc^{-1}}$ for
the fiducial redshift $z=0.7$. Here $k_{1}=0.2{\rm Mpc^{-1}}$ is a
sufficiently large length-scale where we may expect a combination of 2PT and
FoG to provide a reasonably good description of the redshift space bispectrum.
The cosmic variance is expected to increase if we consider small values of
$k_{1}$, whereas non-linear effects increase if we consider larger $k_{1}$.
Guided by this, we have mainly shown the results for $k_{1}=0.2{\rm
Mpc^{-1}}$. We now briefly discuss the shapes of the triangle corresponding to
different values of $\mu$ and $t$, the reader is referred to Paper I for
further details. Considering any panel of fig. 2, the right boundary $\mu=1$
corresponds to linear triangles where the three sides are co-linear. Here, the
bottom right corner $(\mu,t)=(1,0.5)$ corresponds to stretched triangles where
$k_{2}=k_{3}=k_{1}/2$, and the top right corner $(1,1)$ corresponds to
squeezed triangles where $k_{1}=k_{2},k_{3}\rightarrow 0$. The top boundary
$t=1$ corresponds to L-isosceles triangles where the two larger sides have
equal length $(k_{1}=k_{2})$, and the bottom boundary $2\mu t=1$ corresponds
to S-isosceles triangles where the two smaller sides have equal length
$(k_{2}=k_{3})$. The top left corner $(0.5,1)$ corresponds to equilateral
triangles. The diagonal $\mu=t$ corresponds to right-angled triangles, and the
regions $t>\mu$ and $t<\mu$ correspond to acute and obtuse triangles
respectively.

###  3.1 Predicted Multipole Moments

Figure 2: This shows $Q_{\ell}^{m}$ as functions of $\mu$ and $t$ at
$k_{1}=0.2{\rm Mpc^{-1}}$ for the fiducial redshift $z=0.7$.
$Q_{\ell}^{m}$[FoG] means that the effect of FoG has been considered in
bispectrum according to eq. (30). In other plots, where FoG is not mentioned
in the parenthesis, the bispectrum is considered only up to the 2PT level (eq.
25). From the plots it is evident that FoG suppress the bispectrum
significantly at $k_{1}=0.2{\rm Mpc}^{-1}$.

In fig. 2, we have shown $Q_{\ell}^{m}$ for $\ell=0,2$ and $4$. As mentioned
earlier, only the even $\ell$ multipoles are expected to be non-zero. Further,
in 2PT we expect non-zero multipoles up to $\ell=8$ (Paper II), however the
value of $Q_{\ell}^{m}$ falls with increasing $\ell$ and $m$, and we have
shown the main results only up to $\ell=4,m=3$. As shown later in this
section, we do not expect to have a statistically significant measurement of
the higher multipoles for the survey parameters considered here. Note that
$Q_{\ell}^{m}$ and $Q_{\ell}^{m}$[FoG] respectively denote the results without
and with the FoG. In all cases we find that the magnitude of
$Q_{\ell}^{m}$[FoG]s is highly suppressed compared to $Q_{\ell}^{m}$ due to
the FoG effect. Considering $Q_{0}^{0}$, we find that the the minima $(0.33)$
occurs for the equilateral triangles and the the maxima (52.83) occurs close
to the squeezed limit ($\mu\rightarrow 1,t=0.865$). The minimum and maximum
values fall to $0.09$ and $23.50$ for $Q_{0}^{0}$[FoG]. We notice that the
drop in value due to the FoG is more pronounced for equilateral triangles
where the minima occurs. However, the overall patterns are visually very
similar for $Q_{\ell}^{m}$ and $Q_{\ell}^{m}$[FoG].

The contour pattern of $Q_{2}^{0}$ shows features similar to those of
$Q_{0}^{0}$. Its peak value ($86.83$) is larger than the peak value of
$Q_{0}^{0}$ and it occurs at $\mu=0.995,t=0.795$. We observe that the FoG
effect causes a larger suppression for the quadrupole than in monopole, and
the maximum value of $Q_{2}^{0}$[FoG] drops to 11.95 which occurs at a
slightly different location (at $\mu=0.995,t=0.735$). It is interesting to
note that in the absence of the FoG damping $Q_{2}^{0}$ is larger than than
all the other multipoles including the monopole. However as soon as we apply
the FoG effect, the peak value of $Q_{2}^{0}$[FoG] drops and and
$Q_{0}^{0}$[FoG] overtakes it to become the multipole with the largest value.
The entire contour pattern of $Q_{2}^{0}$[FoG] is quite different from that of
$Q_{2}^{0}$. Unlike $Q_{2}^{0}$ which is positive everywhere, we see that
$Q_{2}^{0}$[FoG] is negative over a region near the squeezed limit. The
minimum values of $Q_{2}^{0}$ (0.23) and $Q_{2}^{0}$[FoG] (-3.88) respectively
occurs at the equilateral and squeezed limits, however the magnitude of
$Q_{2}^{0}$[FoG] is still minimum for equilateral triangles.

Considering $Q_{2}^{1}$, we find that this has negative values for most of the
acute triangles and positive values for obtuse triangles, with a zero crossing
somewhat above the line $\mu=t$ which corresponds to right-angled triangles.
The minima (-0.43) and maxima (19.24) respectively occur at
($\mu=0.965,t\rightarrow 1$) and ($\mu\rightarrow 1,t=0.915$) which are both
very close to the squeezed limit. The FoG effect substantially changes the
contour pattern, and we see that $Q_{2}^{1}$[FoG] is positive valued
throughout. The maxima (9.61) now occurs near the squeezed limit while the
minima ($\approx 0$) occurs for equilateral triangles.

In case of $Q_{2}^{2}$, we find that the minimum value (0.08) is near the
stretched limit and the maximum value (6.45) occurs close to the squeezed
limit ($\mu\rightarrow 1,t=0.975$). The maximum (4.84) and minimum (0.04)
values in case of $Q_{2}^{2}$[FoG] do not change position, only the amplitudes
drop. Overall, $Q_{2}^{2}$ and $Q_{2}^{2}$[FoG] show similar contour patterns.

Considering $Q_{4}^{0}$, we see that it shows similar shape dependence as
$Q_{0}^{0}$ and $Q_{2}^{0}$, only its maximum (21.5) and minimum (0.02) values
are smaller. Comparing $Q_{4}^{0}$[FoG] with $Q_{4}^{0}$, we see that the
patterns are visually similar, however the values of $Q_{4}^{0}$[FoG] are all
negative and they are roughly an order of magnitude smaller. Considering
$Q_{4}^{1}$, we see that its minima (0.01) occurs for equilateral triangles
and the maxima (9.23) is at ($\mu\rightarrow 1,t=0.905$) which is somewhat
below the squeezed limit. Comparing $Q_{4}^{1}$[FoG] with $Q_{4}^{1}$, we see
that the patterns are quite different, and the values of $Q_{4}^{1}$[FoG] are
roughly an order of magnitude smaller. We see that $Q_{4}^{1}$[FoG] is
negative near the squeezed limit where $\mid Q_{4}^{1}\mid$[FoG] has a maxima
(1.33), whereas the minima (0.30) occurs for equilateral triangles. In summary
we note that the FoG effect has a significant impact on the redshift space
bispectrum at the length-scales and redshifts of our interest, and we have
included this in our subsequent analysis.

It is interesting to compare the $Q_{\ell}^{m}$ values (without FoG) shown in
fig. 2 with those shown in Paper II. We note that the bias parameters and
redshift $z=0.7$ used here are different from those used in Paper II. We find
that the contour patterns of $Q_{0}^{0},Q_{2}^{0}$ and $Q_{4}^{0}$ look very
similar in both fig. 2 and Paper II, however their values are $\geqslant 5$
times larger in Paper II which considers $z=0$. On the other hand, the odd
multipoles are found to be very different. For example, in Paper II the minima
and maxima of $Q_{2}^{1}$ lie along the $\mu=1$ and $t=1$ lines respectively
whereas it is exactly the opposite in fig. 2. This is also true for
$Q_{4}^{1}$ where the contour patterns show completely opposite trends in
Paper II and fig. 2. These differences are mainly due to the choice of the
non-linear bias parameter $\gamma_{2}$ which is set to zero in Paper II
whereas we have used $\gamma_{2}=-0.9$ for fig. 2.

The entire discussion has been restricted to $k_{1}=0.2{\rm Mpc}^{-1}$ till
now. It is worth noting that the impact of the FoG effect falls exponentially
as we move to lower $k$ (large scales), and it causes $\lesssim 5\%$change in
the bispectrum at $k_{1}=0.05{\rm Mpc^{-1}}$ for $z=0.7$. Second order
perturbation theory (eq. 25) alone is adequate to model the redshift space
bispctrum at $k_{1}<0.05{\rm Mpc^{-1}}$, however we do not consider these
small $k_{1}$ values in our work as most of the bispectrum multipoles become
undetectable due to cosmic variance (as shown later).

###  3.2 SNR predictions

Figure 3: This shows the signal to noise ${[\rm SNR]}_{\ell}^{m}$ (eq. 32) for
different multipoles with $k_{1}=0.2{\rm Mpc}^{-1}$ and $z=0.7$ fixed. The
regions where ${[\rm SNR]}_{\ell}^{m}<1$ have been masked out from the plots.
We do not show the multipoles for which ${[\rm SNR]}_{\ell}^{m}$ never exceeds
unity. The vertical grey lines in the color bars mark the values ${[\rm
SNR]}_{\ell}^{m}=5$.

We now quantify the prospects of detecting the various bispectrum multipole
moments $B_{\ell}^{m}$ using the signal-to-noise ratio (SNR) which is defined
as

$\displaystyle{[\rm
SNR]}_{\ell}^{m}={|B_{\ell}^{m}|\over\sqrt{C_{\ell\ell}^{mm}}}\,.$ (32)

A value ${[\rm SNR]}_{\ell}^{m}\geq 5$ (or possibly ${[\rm
SNR]}_{\ell}^{m}\geq 3$) would be considered as a statistically significant
detection of the particular multipole moment. As mentioned earlier, we have
considered bins of width $(\Delta\ln k_{1}=0.1,\Delta\mu=0.05,\Delta t=0.05)$.
We note that the signal to noise ratio will increase if we consider bins of
larger widths. Considering a situation where our analysis predicts ${[\rm
SNR]}_{\ell}^{m}\approx 1$, it may still be possible to achieve a
statistically significant detection by widening the bin width using
$(\Delta\ln k_{1}=0.2,\Delta\mu=0.1,\Delta t=0.1)$ which still retains a
significant amount of the information regarding the length-scale and shape
dependence of the bispectrum. Further the values of the various parameters
used in our predictions are rather uncertain, and it is possible that the
actual parameter values could result in a larger ${[\rm SNR]}_{\ell}^{m}$.
Motivated by these factors, we have shown the results for ${[\rm
SNR]}_{\ell}^{m}\geq 1$ where the signal and noise have equal amplitude, and
we have also included this in the discussion.

Figure 3 shows ${[\rm SNR]}_{\ell}^{m}$ for different multipoles with
$k_{1}=0.2{\rm Mpc}^{-1}$ and $z=0.7$ fixed. The regions where ${[\rm
SNR]}_{\ell}^{m}<1$ have been masked out from the plots, and the multipoles
where ${[\rm SNR]}_{\ell}^{m}$ never exceeds unity have not been shown.
Broadly, the ${[\rm SNR]}_{\ell}^{m}$ values decrease with increasing $\ell$,
and for a fixed $\ell$ they decrease with increasing $m$. However, we find an
exception that the maximum value of ${[\rm SNR]}_{2}^{1}$ exceeds that of
${[\rm SNR]}_{2}^{0}$. For all ${[\rm SNR]}_{\ell}^{m}$ we find that the
maximum value occurs for linear triangles $(\mu=1)$, however the value of $t$
corresponding to the maxima varies depending on $\ell$ and $m$. Further, in
most cases ${[\rm SNR]}_{\ell}^{m}$ decreases as the shape of the triangle is
changed from linear $(\mu=1)$ to other obtuse triangles $(\mu>t)$ and then
acute triangles $(\mu<t)$ and finally the equilateral triangle
$(\mu,t)=(0.5,1)$. Considering $B_{0}^{0}$, we see that this can be detected
at a high level of precision $({[\rm SNR]}_{0}^{0}>7)$ for all triangle
configurations, and ${[\rm SNR]}_{0}^{0}$ is maximum $(\approx 140)$ around
$(1,0.75)$ . Considering $B_{2}^{0}$, we see that the SNR exceeds unity for
most triangle configurations, however ${[\rm SNR]}_{2}^{0}<5$ over a
considerable region where $t>\mu$. We also find a small region with ${[\rm
SNR]}_{2}^{0}<1$ near the squeezed limit where the values of $B_{2}^{0}$ have
a zero crossing (fig. 2). The maximum value ${[\rm SNR]}_{2}^{0}=28.6$ occurs
around $(1,0.6)$ which is near the stretched limit. Considering ${[\rm
SNR]}_{2}^{1}$, we see that the maximum value $(\approx 33.4)$ occurs around
$(1,0.85)$ and we have ${[\rm SNR]}_{2}^{1}>5$ in a region surrounding this.
As mentioned earlier, the maximum value of ${[\rm SNR]}_{2}^{1}$ exceeds that
of all the other multipole moments barring ${[\rm SNR]}_{0}^{0}$, however the
condition ${[\rm SNR]}_{2}^{1}>1$ is satisfied for a relatively small range of
shapes compared to ${[\rm SNR]}_{2}^{0}$ and ${[\rm SNR]}_{2}^{2}$. The
maximum value of ${[\rm SNR]}_{2}^{2}$ ($\approx 22.7$) occurs for squeezed
triangles, and we have ${[\rm SNR]}_{2}^{2}>5$ in a region around this. The
condition ${[\rm SNR]}_{2}^{2}>1$ is satisfied over most of the $(\mu,t)$
space except for a small region near the stretched triangles. ${[\rm
SNR]}_{4}^{0}$ is very similar to ${[\rm SNR]}_{2}^{1}$, except that the
values are smaller and the maximum value now is $\approx 7.1$. The higher
multipoles with $\ell=4$ and $m=1,2$ and $3$ do not have ${[\rm
SNR]}_{\ell}^{m}>5$ anywhere, however there is a rather large region where
${[\rm SNR]}_{4}^{2}>1$. The maxima of ${[\rm SNR]}_{4}^{2}$ $(\approx 3.6)$
occurs at the squeezed limit, whereas for $(m=1,3)$ the maxima $(2.9,1.8)$
occur very close to the squeezed limit. The two latter multipoles satisfy
${[\rm SNR]}_{\ell}^{m}>1$ in only a small region around the maxima.

Figure 4: This shows the variation of ${[\rm SNR]}_{\ell}^{m}$ (eq. 32) with
$k_{1}$, the largest wavevector (see fig. 1), at fixed $z=0.7$. We show
results for three specific triangle shapes: squeezed (left-hand panel),
stretched (central panel) and equilateral (right-hand panel). The horizontal
dotted-black lines show the ${[\rm SNR]}_{\ell}^{m}=1$ and $5$ levels
respectively. Note that we show ${[\rm SNR]}_{\ell}^{m}$ for the same
multipoles as in fig. 3.

We now consider the $k_{1}$ dependence of ${[\rm SNR]}_{\ell}^{m}$ with
$z=0.7$ fixed, as shown in fig. 4. The smallest value of $k_{1}$ roughly
corresponds to the linear extent of the survey. The results are shown for only
three different triangle shapes namely squeezed, stretched and equilateral.
However, we can combine these with fig. 3 to qualitatively infer the $k_{1}$
dependence expected for other triangle configurations. The number of triangles
in any bin scales as $N_{tr}\propto k_{1}^{3}$ (eq. 22), and we expect the
cosmic variance to scale as $k_{1}^{-3}$ (eq. 24) as $k_{1}$ is increased.
Based on this we may expect ${[\rm SNR]}_{\ell}^{m}$ to increase monotonically
with increasing $k_{1}$. This is broadly true for all the triangles in fig. 4,
except for the sharp dips which are seen to occur for a few of the multipoles
at certain values of $k_{1}$. As seen in fig. 2, some of the multipoles have
both positive and negative values, and the dips in fig. 4 correspond to the
zero crossings which are also reflected in ${[\rm SNR]}_{\ell}^{m}$. We now
discuss the prospects of detecting the various multipoles at different values
of $k_{1}$. Here we use $k_{1}[5]$ to denote the smallest value of $k_{1}$
where the condition ${[\rm SNR]}_{\ell}^{m}>5$ is satisfied, and generally we
expect the $B_{\ell}^{m}$ to be detectable for all $k_{1}\geq k_{1}[5]$.
Considering $B_{0}^{0}$ we see that $k_{1}[5]=0.05{\rm Mpc}^{-1}$ for the
squeezed and stretched triangles, and $k_{1}[5]=0.1{\rm Mpc}^{-1}$ for
equilateral triangles. Overall, we expect to detect $B_{0}^{0}$ for all linear
triangles and some obtuse triangles at $k_{1}\geq 0.05{\rm Mpc}^{-1}$, and for
all triangle at $k_{1}\geq 0.1{\rm Mpc}^{-1}$. Note that $B^{0}_{0}$ is
predicted to be negative at large $k_{1}$ for equilateral triangles. We find a
small $k_{1}$ range around $k_{1}\approx 0.4{\rm Mpc}^{-1}$ where we do not
expect to detect $B_{0}^{0}$ for equilateral and possibly some of the acute
triangles. Considering $B_{2}^{0}$, we have $k_{1}[5]=0.05{\rm Mpc}^{-1}$ and
$0.1{\rm Mpc}^{-1}$ for stretched and squeezed triangles respectively. The SNR
falls off towards equilateral triangles where we have ${\rm SNR}\geq 1$ for
$k_{1}\geq 0.1{\rm Mpc}^{-1}$ and $k_{1}[5]=0.6{\rm Mpc}^{-1}$. Overall we
expect $B_{2}^{0}$ to be detectable in the vicinity of the stretched limit for
$k_{1}\geq 0.05{\rm Mpc}^{-1}$, and for several other linear and obtuse
triangles for $k_{1}\geq 0.1{\rm Mpc}^{-1}$. Considering acute and equilateral
triangles, it will be possible to detect $B_{2}^{0}$ at $k_{1}\geq 0.1{\rm
Mpc}^{-1}$ if we increase the bin widths. For all shapes, $B_{2}^{0}$ is
predicted to be negative at large $k_{1}$. The transition from positive to
negative values occurs at $k_{1}\approx 0.1{\rm Mpc}^{-1}$ for squeezed
triangles, and then spreads to other shapes (stretched and then equilateral)
as $k_{1}$ is increased. As seen in fig. 3, it will not be possible to detect
$B_{2}^{0}$ in a few $\mu-t$ bins where the zero crossing occurs. The exact
$\mu-t$ locations of these bins will shift with $k_{1}$. Considering
$B_{2}^{1}$ and $B_{2}^{2}$ together, we see that $k_{1}[5]\approx 0.1{\rm
Mpc}^{-1}$ for squeezed triangles, and we expect to detect these multipoles
for several linear and obtuse triangles for $k_{1}\geq 0.1{\rm Mpc}^{-1}$ .
Further, in all three cases (fig. 4) $B^{2}_{2}$ satisfies ${\rm SNR}\geq 1$
for $k_{1}>0.1{\rm Mpc}^{-1}$. In this $k_{1}$ range we expect to detect
$B^{2}_{2}$ across the entire $\mu-t$ plane if we consider larger bin widths.
We find that the rest of $B_{\ell}^{m}$ shown here ($\ell=4,m=0,1,2,3$) all
have $k_{1}[5]$ in the range $0.2-0.3{\rm Mpc}^{-1}$ for squeezed triangles,
and we expect to detect these multipoles for several linear and obtuse
triangles near the squeezed limit. We note that $B_{4}^{0}$ also has
$k_{1}[5]$ in the range $0.2-0.3{\rm Mpc}^{-1}$ for stretched triangles, and
the region of $\mu-t$ space where this will be detected is relative large
compared to the other $\ell=4$ multipoles. We also note that $B^{0}_{4}$ and
$B_{4}^{2}$ both satisfy ${\rm SNR}\geq 1$ for the three triangle shapes shown
in fig. 4 for $k_{1}\geq 0.3{\rm Mpc}^{-1}$, and we expect to detect these
over the entire $\mu-t$ space if we increase the bin widths.

Figure 5: This shows the variation of ${[\rm SNR]}_{\ell}^{m}$ (eq. 32) with
redshift at fixed $k_{1}=0.2\,{\rm Mpc}^{-1}$. We show results for three
specific triangle shapes: squeezed (left-hand panel), stretched (central
panel) and equilateral (right-hand panel). The horizontal dotted-black lines
show the ${[\rm SNR]}_{\ell}^{m}=1$ and $5$ levels respectively. Note that we
show ${[\rm SNR]}_{\ell}^{m}$ for the same multipoles as in fig. 3. Filled
circles represent the $z$ bins adopted from Yankelevich & Porciani (2019) to
show our results.

Figure 5 shows how ${[\rm SNR]}_{\ell}^{m}$ varies with redshift with
$k_{1}=0.2\,{\rm Mpc}^{-1}$ fixed. Note that the bias parameters
$b_{1},\gamma_{2}$, as well as the survey volume $V$ change with changing
redshift, and we have used the redshift dependent values from Table 1 of
(Yankelevich & Porciani, 2019). We see that the survey volume $V$ increases
with $z$, and we expect the cosmic variance at a fixed $k_{1}$ to decrease
with increasing $z$. As a result of this, ${[\rm SNR]}_{\ell}^{m}$ is expected
to increase monotonically with redshift. However, there are two more effects
that determine ${[\rm SNR]}_{\ell}^{m}$ values at each $z$: $(i)$ the
amplitude of the bispectrum decreases with increasing $z$, and $(ii)$ the FoG
suppression also decreases with increasing $z$. The bias parameters also
affect the $z$ dependence of ${[\rm SNR]}_{\ell}^{m}$. We see that for many of
the multipoles ${[\rm SNR]}_{\ell}^{m}$ increase slightly with increasing $z$
whereas there are some where the opposite occurs, and for a few cases we also
have dips in ${[\rm SNR]}_{\ell}^{m}$ at some values of $z$. As discussed
earlier, these dips correspond to the zero crossings of $B_{\ell}^{m}$. We now
discuss the prospects of detecting the different multipoles at the various
redshifts shown here. Considering $B_{0}^{0}$, we see that this is detectable
with ${[\rm SNR]}_{0}^{0}>5$ at all the redshifts for all the three triangles
considered. $B_{2}^{0}$ is detectable with ${[\rm SNR]}_{2}^{0}>5$ at all
redshifts for stretched triangles. For squeezed triangles, the above is true
for $z<1.5$, whereas for equilateral triangles $1<{[\rm SNR]}_{2}^{0}<5$ at
all redshifts. Considering $B_{2}^{1}$ and $B_{2}^{2}$, we see that they will
be detected with ${[\rm SNR]}_{\ell}^{m}>5$ at all the redshifts only for
squeezed triangles. For the stretched and equilateral triangles, ${[\rm
SNR]}_{2}^{1}$ and ${[\rm SNR]}_{2}^{2}$ values are below $5$ at all
redshifts. All the $\ell>2$ multipoles cannot be observed with ${[\rm
SNR]}_{\ell}^{m}>5$ at any redshift and for the triangle shapes considered.
Note that, the above discussion is true only for for three specific triangle
shapes at $k_{1}=0.2\,{\rm Mpc}^{-1}$. However, as we have discussed earlier,
it is possible to detect the higher order multipoles with ${[\rm
SNR]}_{\ell}^{m}>5$ by considering triangles of other shapes or somewhat
larger $k_{1}$ values or larger bin widths.

###  3.3 Correlation between different multipoles

Figure 6: This shows rank correlations ${\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}$ (eq. 33) between different multipoles at
fixed $k_{1}=0.2{\rm Mpc}^{-1}$ and $z=0.7$. Note that, we do not show rank
correlations for which $\mid{\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}\mid<0.1$ throughout the $\mu-t$ plane.

It is important to note that the estimated values of the different multipole
moments are expected to be correlated. We have quantified the correlations in
the measurement errors between any two multipoles using the rank correlation

$\displaystyle{\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}={C_{\ell\ell^{\prime}}^{mm^{\prime}}\over\sqrt{C_{\ell\ell}^{mm}}\sqrt{C_{\ell^{\prime}\ell^{\prime}}^{m^{\prime}m^{\prime}}}}\,.$
(33)

with $-1\leq{\cal R}_{\ell\ell^{\prime}}^{mm^{\prime}}\leq 1$. Here positive
and negative values indicate that the errors are correlated and anti-
correlated respectively. We show the rank correlations between different
multipoles in fig. 6. We find that for most multipole pairs, the errors are
only weakly correlated ($\mid{\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}\mid<0.1$ ) across much of the $\mu-t$
plane. However, there are a few exceptions which we highlight below. We first
consider ${\cal R}_{02}^{00}$ which refers to $B_{0}^{0}$ and $B_{2}^{0}$
which have the highest ${\rm SNR}$. We see that the measurement errors in
these two multipoles are correlated for obtuse triangles and anti-correlated
for acute triangles. ${\cal R}_{02}^{00}$ shows the highest correlation
$(0.42)$ in the stretched limit, and the highest anti-correlation $(-0.1)$ in
the squeezed limit. Considering ${\cal R}_{02}^{02}$, we see that the
measurement errors of the pair ($B_{0}^{0}$,$B_{2}^{2}$) are correlated with
values $\sim 0.25$ in the vicinity of the squeezed limit. Considering ${\cal
R}_{22}^{02}$, we see that this show maximum anti-correlation $(-0.18)$ in the
squeezed limit. ${\cal R}_{24}^{00}$ shows the highest correlation $(0.33)$
and anti-correlation $(-0.1)$ in the stretched and squeezed limits
respectively. We have $\mid{\cal R}_{\ell\ell^{\prime}}^{mm^{\prime}}\mid<0.1$
across the entire $\mu-t$ plane for all the other multipoles not shown here.
Overall, the errors in the various multipoles are weakly correlated for most
triangle shapes barring a few in the vicinity of squeezed and stretched
triangles.

Figure 7: This shows the redshift dependence of the rank correlations ${\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}$ (eq. 33) between different multipoles
for a fixed $k_{1}=0.2\,{\rm Mpc}^{-1}$. We show results for three specific
triangle shapes: squeezed (left-hand panel), stretched (central panel) and
equilateral (right-hand panel). Note that, we have considered the same rank
correlations as shown in fig. 6. Filled circles represent the $z$ bins adopted
from Yankelevich & Porciani (2019) to show our results.

Figure 7 shows the redshift dependence (from $z=0.7$ to $2$) of the four rank
correlations shown in fig. 6 for three specific triangle shapes with
$k_{1}=0.2\,{\rm Mpc}^{-1}$ fixed. Considering ${\cal R}_{02}^{00}$ (${\cal
R}_{24}^{00}$) we see that the peak correlation, which occurs for stretched
triangles, increases from $0.42$ ($0.32$) to $\sim 0.5$ ($0.4$) whereas the
value changes from $\sim-0.1$ to $\sim 0.1$ for squeezed triangles.
Considering ${\cal R}_{02}^{02}$, the peak correlation, which occurs for
squeezed triangles, drops from $\sim 0.25$ to $0.2$ across $z$. Considering
${\cal R}_{22}^{02}$, for squeezed triangles its value changes from $-0.18$ to
$-0.11$ across $z$. The other cases shown in the figure all have $\mid{\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}\mid<0.1$, with the exception of ${\cal
R}_{02}^{02}$ which has values in the range $0.1$ to $0.2$ at $z\geq 1$ for
equilateral triangles.

Figure 8: This shows the $k_{1}$ dependence of the rank correlations ${\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}$ (eq. 33) between different multipoles
for a fixed $z=0.7$. We show results for three specific triangle shapes:
squeezed (left-hand panel), stretched (central panel) and equilateral (right-
hand panel). Note that, we have considered the same rank correlations as shown
in fig. 6.

Figure 8 shows the $k_{1}$ dependence of the four rank correlations shown in
fig. 6 for for three specific triangle shapes with $z=0.7$ fixed. Considering
all the panels, we see that the rank correlations are practically independent
of $k_{1}$ for $k_{1}\lesssim 0.05\,{\rm Mpc}^{-1}$. We have already mentioned
that the FoG effect can be neglected at $k_{1}\lesssim 0.05\,{\rm Mpc}^{-1}$.
The numerator and denominator of eq. (33) are both dominated by the power
spectrum contribution, and the $k_{1}$ dependent factors cancel out to make
${\cal R}_{ll^{\prime}}^{mm^{\prime}}$ independent of $k_{1}$ in the absence
of the FoG effect. However, we see a strong $k_{1}$ dependence for
$k_{1}>0.1\,{\rm Mpc}^{-1}$ where the FoG is important. We see that ${\cal
R}_{02}^{00}$ and ${\cal R}_{24}^{00}$ exhibit very similar $k_{1}$ dependence
in all the three panels where we have positive correlations at
$k_{1}<0.05\,{\rm Mpc}^{-1}$, the correlations decline with increasing $k_{1}$
and cross zero around $k_{1}\sim 0.2$ and we have nearly complete anti-
correlation $(\sim-1)$ for $k_{1}>0.5\,{\rm Mpc}^{-1}$. Considering
$k_{1}<0.05\,{\rm Mpc}^{-1}$ we have ${\cal R}_{02}^{00}>{\cal R}_{24}^{00}$,
with the maximum correlation $(0.5-0.75)$ occurring for stretched triangles,
followed by values in the range $0.3-0.5$ for squeezed triangles and $0.2-0.3$
for equilateral triangles. The $k_{1}$ dependence of ${\cal R}_{02}^{02}$ is
very similar to that of ${\cal R}_{02}^{00}$ and ${\cal R}_{24}^{00}$ for
equilateral triangles. For squeezed (stretched) triangles ${\cal
R}_{02}^{02}\sim 0.25\,(0.0)$ for $k_{1}\leq 0.4\,{\rm Mpc}^{-1}$ beyond which
both sharply declines to $\sim-0.25$ at $k_{1}=1\,{\rm Mpc}^{-1}$. In
contrast, ${\cal R}_{22}^{02}$ shows exactly the opposite $k_{1}$ dependence
in that we have anti-correlations for small $k_{1}$ and correlations at large
$k_{1}$, with a zero crossing whose $k_{1}$ value depends on the shape of the
triangle. Considering squeezed, stretched and equilateral triangles we
respectively have ${\cal R}_{22}^{02}\sim-0.1,0.0$ and $-0.2$ at small $k_{1}$
$(\leq 0.1\,{\rm Mpc}^{-1})$ , whereas these values are $\sim 0.25,0.25$ and
$1.0$ for large $k_{1}$ $(\leq 0.1\,{\rm Mpc}^{-1})$.

##  4 Discussion and Conclusion

The redshift space anisotropy of the bispectrum is generally quantified using
multipole moments as defined in eq. (6). The possibility of measuring these
multipoles in any survey depends on the level of statistical fluctuations. In
this paper, we developed a formalism to compute the statistical fluctuations
in the measurement of bispectrum multipoles for galaxy surveys. We quantify
the fluctuations through the covariance as defined in eq. (24) which assumes
the flat sky approximation. We consider the specifications of a Euclid like
galaxy survey to present our results. We mainly consider two quantities: $(i)$
the signal-to-noise ratio(SNR) ${[\rm SNR]}_{\ell}^{m}$ which quantifies the
detectability of a multipole $B_{\ell}^{m}$ (eq. 32), and $(ii)$ the rank
correlation which quantifies the correlation in measurement errors between any
two multipoles (eq. 33). We show, how these quantities depend on the triangle
configurations in ${\bf k}$ space, as well as their evolution in redshift. We
also show how our results change as we introduce the FoG effect.

We find that the FoG effect plays a crucial role at length scales
$k>0.05\,{\rm Mpc}^{-1}$ for $z=0.7$. This suppresses the values of bispectrum
multipoles and the suppression is in general stronger for the higher order
multipoles. The amplitude of the monopole $B_{0}^{0}$ predicted by 2PT is
reduced by half when FoG effect is introduced. For $B_{4}^{m}$, the amplitude
is suppressed by almost an order of magnitude in presence of FoG effect. This
FoG suppression, however, is not very important at small $k$ or at high $z$.

We see that in general ${[\rm SNR]}_{\ell}^{m}$ values decrease with
increasing $\ell$, and for a fixed $\ell$ they decrease with increasing $m$.
Considering all ${[\rm SNR]}_{\ell}^{m}$, we find that the maximum value
occurs for linear triangles $(\mu=1)$, however the value of $t$ corresponding
to the maxima varies depending on $\ell$ and $m$. Also, in most of the cases
${[\rm SNR]}_{\ell}^{m}$ decreases as the shape of the triangle is changed
from linear $(\mu=1)$ to other obtuse triangles $(\mu>t)$ and then acute
triangles $(\mu<t)$ and finally the equilateral triangle $(\mu,t)=(0.5,1)$. We
note that, at large, ${[\rm SNR]}_{\ell}^{m}$ increases with $k_{1}$ and $z$.

Considering individual multipoles, we expect to detect $B_{0}^{0}$ for all the
triangles at $k_{1}>0.1\,{\rm Mpc}^{-1}$ across the redshifts considered here.
On the other hand, we expect $B_{2}^{0}$ to be detectable in the vicinity of
the stretched limit for $k_{1}>0.05\,{\rm Mpc}^{-1}$ at all the redshifts, and
for several other linear and obtuse triangles for $k_{1}\geq 0.1$ up to $z\sim
1.5$. Detection of $B_{2}^{0}$ for acute and equilateral triangles, as well as
for obtuse triangles at $z>1.5$, is possible if we increase the bin size.
Detection of $B_{2}^{1}$ and $B_{2}^{2}$ at all redshifts is possible only for
squeezed triangles at $k_{1}\geq 0.1\,{\rm Mpc}^{-1}$. For other triangle
shapes, detection is possible at higher $k_{1}$. Considering $B_{4}^{m}$
multipoles, we see that the detection is possible across the redshift range
only for a few linear and obtuse triangles near the squeezed limit at
$k_{1}>0.2\,{\rm Mpc}^{-1}$. Note that, for all the multipoles, the
possibility of detection, or the ${[\rm SNR]}_{\ell}^{m}$ values, can be
increased by increasing the bin size in $k_{1},\mu$ or $t$.

Considering ${\cal R}_{\ell\ell^{\prime}}^{mm^{\prime}}$, we find that for
most multipole pairs the errors are only weakly correlated (with $\mid{\cal
R}_{\ell\ell^{\prime}}^{mm^{\prime}}\mid<0.1$ ) across much of the $\mu-t$
plane barring a few in the vicinity of squeezed and stretched triangles. For a
fixed triangle shape, the ${\cal R}_{\ell\ell^{\prime}}^{mm^{\prime}}$ values
are practically independent of $k_{1}$ for $k_{1}\lesssim 0.05\,{\rm
Mpc}^{-1}$ and show a strong $k_{1}$ dependence for $k_{1}>0.1\,{\rm
Mpc}^{-1}$ where FoG effect is really important. For a fixed triangle shape
and $k_{1}$, ${\cal R}_{\ell\ell^{\prime}}^{mm^{\prime}}$ evolve moderately
with redshift.

We, therefore, conclude that the future surveys like Euclid can potentially
measure the higher order redshift space bispectrum multipoles (up to
$\ell=4$), beyond the isotropic component (monopole), across various triangle
shapes and sizes. The signal-to-noise or significance of these measurements,
however, depend on the scales and redshifts of observation. The signal-to-
noise also determines the information content of the individual multipoles.
Significant measurements of $\ell\leq 2$ multipoles are possible even at
$k_{1}\lesssim 0.1\,{\rm Mpc}^{-1}$ across the $z$ range $0.7-2$. These scales
are particularly important as the FoG suppression is minimum here. Here, we
expect to measure the $\ell\leq 2$ multipoles with highest signal-to-noise for
linear and obtuse triangles. For $\ell>2$ multipoles, we require to go to
large $k_{1}$ for significant detection or we need large bin size to increase
the signal-to-noise. Due to the weak correlation (at $k_{1}<0.2\,{\rm
Mpc}^{-1}$) between the errors of multipole pairs for most of the triangle
shapes, it is possible to combine different multipoles to increase the
information content. This becomes particularly important when we try to
extract cosmological parameters from bispectrum measurements. Following our
analysis, we expect to reduce the errors on the cosmological parameters when
we combine the higher multipoles along with the monopole results (Gil-Marín et
al., 2015, 2017; Philcox & Ivanov, 2022). This we plan to study in future.
Finally, we reiterate that in Euclid like surveys, we expect to measure
bispectrum multipoles up to $\ell=4$ by suitably choosing the scale and
redshifts of observation.

## DATA AVAILABILITY

The data generated during this work will be made available upon reasonable
request to the authors.

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