# Using relativistic effects in large-scale structure to constrain
astrophysical properties of galaxy populations

Daniel Sobral Blanco  Camille Bonvin  Chris Clarkson  Roy Maartens

###### Abstract

Upcoming large-scale structure surveys will be able to measure new features in
the galaxy two point correlation function. Relativistic effects appear on
large scales as subtle corrections to redshift-space distortions, showing up
as a dipole and octupole when cross-correlating two different tracers of dark
matter. The dipole and octupole are very sensitive to the evolution and
magnification biases of the observed tracers which are hard to model
accurately as they depend upon the derivative of the luminosity function at
the flux limit of the survey. We show that splitting a galaxy population into
bright and faint samples allows us to cross-correlate these and constrain both
the evolution bias and magnification bias of the two samples – using the
relativistic odd multipoles of the correlation function, together with the
even Newtonian multipoles. Although the octupole has much lower signal-to-
noise than the dipole, it significantly improves the constraints by breaking
parameter degeneracies. We illustrate this in the case of a futuristic survey
with the Square Kilometre Array, and demonstrate how splitting the samples in
different ways can help improve constraints. This method is quite general and
can be used on different types of tracers to improve knowledge of their
luminosity functions. Furthermore, the signal-to-noise of the dipole and
octupole peaks on intermediate scales, which means that that they can deliver
a clean measurement of the magnification bias and evolution bias without
contamination from local primordial non-Gaussianities or from systematics on
very large scales.

##  1 Introduction

The coming generation of large-scale structure surveys, like
DESI,111https://www.desi.lbl.gov Euclid,222https://www.euclid-ec.org
LSST333https://rubinobservatory.org and SKA444https://www.skao.int will detect
billions of galaxies over a large part of the history of the Universe. These
observations will provide invaluable information on the properties and
dynamics of galaxies. In particular, they will allow us to determine the
luminosity and redshift distribution of galaxies, which is crucial to
understand how galaxies form and evolve.

Moreover, knowing the luminosity and redshift distribution of galaxies is
necessary to perform some cosmological tests. For example, the large-scale
structure of the Universe can be used to search for primordial non-
Gaussianities in the initial fluctuations generated during inflation [1, 2].
This search is however contaminated by the presence of relativistic effects in
the clustering of galaxies [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16],
whose amplitude directly depends on the astrophysical properties of the
galaxies. A precise knowledge of the luminosity and redshift evolution of
galaxies is therefore essential to model relativistic effects precisely and
subtract the signal from the non-Gaussian one.

A second example is the test of the cosmological principle from the
measurement of the kinematic dipole in the distribution of galaxies. If the
cosmological principle is valid, the kinematic dipole measured from the
temperature of the Cosmic Microwave Background (CMB) and the dipole in the
distribution of astrophysical sources should be in agreement [17]. Current
measurements show tensions between the two dipoles [18]. In this context,
measuring the redshift evolution of the astrophysical dipole could help in
assessing the origin of the tension [19, 20]. Since the amplitude of the
astrophysical dipole is directly governed by the luminosity and redshift
evolution of the sources [21, 22], a precise determination of these quantities
is highly relevant.

The luminosity and redshift distribution of galaxies can be directly measured
by binning the population of galaxies in redshift and luminosity. In this
paper, we show that these measurements can be significantly improved by
measuring, in addition, the two-point cross-correlation function between two
populations of galaxies. More precisely, the luminosity and redshift
distribution of galaxies can be encoded into two parameters: the magnification
bias $s(z)$ which denotes the slope of the cumulative number of sources above
a given luminosity threshold, and the evolution bias $f^{\rm evol}$ which
denotes the slope of the cumulative number of sources with redshift. These two
parameters directly enter in the two-point cross-correlation function, more
precisely in the dipole and octupole of this two-point function. By measuring
these odd multipoles, one can constrain the magnification and evolution
biases, and consequently improve our knowledge on the luminosity and redshift
distribution of galaxies.

The rest of the paper is organized as follow: in Section 2 we introduce the
two-point correlation function of galaxies and we show how the dipole and
octupole depend on the magnification and evolution biases. These quantities
are modeled in Section 3 for a survey like SKA. In Section 4 we discuss the
methodology to constrain the magnification and evolution biases and we present
our results in Section 5.

We work in units in which the speed of light $c=1$. We consider a spatially
flat perturbed Friedmann-Lemaître Universe with metric
$ds^{2}=a^{2}(\eta)[-(1+2\Psi)d\eta^{2}+(1-2\Phi)\delta_{ij}dx^{i}dx^{j}]$,
where $\eta$ denotes conformal time.

##  2 Galaxy clustering observables

Galaxy surveys measure the angular position of galaxies in the sky
$\mathbf{n}$ and their redshift $z$. From this one can construct the galaxy
number count fluctuations

$\Delta(z,\mathbf{n})\equiv\frac{N_{g}(z,\mathbf{n})-\bar{N}_{g}(z)}{\bar{N}_{g}(z)}\,,$
(2.1)

where $N_{g}(z,\mathbf{n})$ denotes the observed number of galaxies at
redshift $z$ and direction $\mathbf{n}$ and $\bar{N}_{g}(z)$ is the mean
number of galaxies at that redshift. At linear order in perturbation theory,
this observable takes the form [23, 24, 25]

$\Delta(z,\mathbf{n})=\Delta^{\mathrm{st}}(z,\mathbf{n})+\Delta^{\rm
lens}(z,\mathbf{n})+\Delta^{\rm rel}(z,\mathbf{n})+\Delta^{\rm
corr}(z,\mathbf{n})\,,$ (2.2)

where

$\displaystyle\Delta^{\mathrm{st}}=$ $\displaystyle
b\,\delta_{m}-\frac{1}{\mathcal{H}}\,\partial_{r}(\mathbf{V}\cdot\mathbf{n})\,,$
(2.3) $\displaystyle\Delta^{\rm lens}=$
$\displaystyle\left(5s-2\right)\int_{0}^{r}dr^{\prime}\frac{r-r^{\prime}}{2rr^{\prime}}\Delta_{\Omega}(\Phi+\Psi)\,,$
(2.4) $\displaystyle\Delta^{\rm rel}=$
$\displaystyle\frac{1}{\mathcal{H}}\partial_{r}\Psi+\frac{1}{\mathcal{H}}\dot{\mathbf{V}}\cdot\mathbf{n}+\Bigg{[}1-\frac{\dot{\mathcal{H}}}{\mathcal{H}^{2}}-\frac{2}{r\mathcal{H}}-5\,s\Bigg{(}1-\frac{1}{r\mathcal{H}}\Bigg{)}+f^{\rm
evol}\Bigg{]}\,\mathbf{V}\cdot\mathbf{n}\,,$ (2.5) $\displaystyle\Delta^{\rm
corr}=$
$\displaystyle\frac{2-5s}{r}\int_{0}^{r}dr^{\prime}(\Phi+\Psi)+\left(3-f^{\rm
evol}\right)\mathcal{H}\nabla^{-2}(\mathbf{\nabla\cdot V})+\Psi+(5s-2)\Phi$
(2.6)
$\displaystyle+\frac{1}{\mathcal{H}}\dot{\Phi}+\left(\frac{\dot{\mathcal{H}}}{\mathcal{H}^{2}}+\frac{2-5s}{r\mathcal{H}}+5s-f^{\rm
evol}\right)\left[\Psi+\int_{0}^{r}dr^{\prime}(\dot{\Phi}+\dot{\Psi})\right]\,.$

Here an overdot denotes derivatives with respect to conformal time,
$\Delta_{\Omega}$ is the angular Laplacian, $r(z)$ is the comoving distance
and $\mathcal{H}(z)=H(z)/(1+z)$ is the conformal Hubble parameter.

The dominant contribution $\Delta^{\mathrm{st}}$ in (2.3) contains two terms:
the first one arises from the galaxy density, and is related to the matter
density $\delta_{m}$ via the linear galaxy bias $b(z)$; the second one
corresponds to the Redshift-Space Distortions (RSD) due to the peculiar
velocity field of galaxies $\mathbf{V}$ [26]. The lensing contribution
$\Delta^{\rm lens}$ in (2.4) is negligible for current spectroscopic surveys,
but it will be relevant for the next generation of surveys at high redshift
[27, 28, 29]. It depends on the time distortion $\Psi$ and spatial distortion
$\Phi$ of the metric. The contribution $\Delta^{\rm rel}$ in (2.5) is
suppressed by one power $\mathcal{H}/k$ compared to the standard contribution.
The first term in $\Delta^{\rm rel}$ is a contribution from gravitational
redshift, which alters the apparent size of the redshift bins due to the time
distortion affecting photons incoming from galaxies located inside potential
wells. The second and third terms are Doppler contributions depending on the
galaxies peculiar velocities $\mathbf{V}$ and its time derivative. Finally,
the last contribution $\Delta^{\rm corr}$ in (2.6) is suppressed by two powers
$\left(\mathcal{H}/k\right)^{2}$ compared to the standard contribution. It
contains Sachs-Wolfe effects, integrated Sachs-Wolfe and time delay.

The contributions $\Delta^{\rm rel}$ and $\Delta^{\rm corr}$ are currently
negligible compared to the standard contributions $\Delta^{\mathrm{st}}$.
However, the coming generation of surveys will observe very large parts of the
sky, allowing us to measure the galaxy power spectrum at very large scales
(i.e. very low $k$). In this regime, $\mathcal{H}/k\sim 1$ and therefore
$\Delta^{\rm rel}$ and $\Delta^{\rm corr}$ become relevant. In particular, one
of the goals of the coming generation of surveys is to measure local
primordial non-Gaussianities by looking at their effect on the galaxy power
spectrum at small $k$, where $\Delta^{\rm rel}$ and $\Delta^{\rm corr}$ are
non-negligible [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. It is
therefore necessary to have a precise modelling of these two contributions,
that act as a contamination to the non-Gaussian signal. From Eqs. (2.5) and
(2.6), we see that these contributions depend on the magnification bias $s(z)$
and the evolution bias $f^{\rm evol}(z)$ of the galaxy population. To properly
model the contaminations to local primordial non-Gaussianities, we need
therefore a precise determination of $s$ and $f^{\rm evol}$. In the following,
we will show that we can use the dipole and octupole, generated by the
$\Delta^{\rm rel}$ contribution at intermediate scales to constrain $s$ and
$f^{\rm evol}$.

The information provided by galaxy surveys is analysed using summary
statistics. In this paper, we focus on the three-dimensional galaxy two-point
correlation function defined as
$\xi\equiv\langle\Delta(\mathbf{n},z)\Delta(\mathbf{n}^{\prime},z^{\prime})\rangle$,
including lensing and all other relativistic effects [30, 31, 32]. This can be
expanded in multipoles of the angle formed by the line of sight and the
relative orientation of the pair [31, 33]

$\xi_{\ell}(z,d)=\frac{2\ell+1}{2}\,\int^{1}_{-1}\,\mathrm{d}\mu\,\xi(z,d,\mu)\,P_{\ell}(\mu)\,,$
(2.7)

where $\mu$ is the cosine of the angle, $d$ is the distance between the two
pixels and $P_{\ell}(\mu)$ are the Legendre polynomials of order $\ell$. We
focus on intermediate scales, i.e. separations $d$ between 20 and 160 Mpc$/h$.
This ensures that linear perturbation theory is valid, and also that local
primordial non-Gaussianities are fully negligible. In this way, we can extract
$s$ and $f^{\rm evol}$ in a robust way. For each of the multipoles of (2.7),
we keep only the contribution which dominates in the range of scales that we
are interested in. These multipoles depend on the population of galaxies
through the galaxy bias $b$ and the magnification and evolution biases, $s$
and $f^{\rm evol}$. Here we consider different populations of galaxies, with
different luminosities, and we account for both auto-correlations and cross-
correlations between different populations. This is essential to measure the
dipole and octupole generated by the $\Delta^{\rm rel}$ contribution, which
vanish in the case of a single population [31, 34, 35, 36, 37, 38, 39, 40].
Moreover, cross-correlating different populations allow us to reduce the error
from cosmic variance, improving the measurement of cosmological parameters
[41].

The standard contributions of Eq. (2.3) generate three even multipoles: a
monopole, a quadrupole and a hexadecapole ($\ell=0,2,4$, respectively). These
have been measured previously for various surveys (see e.g. the eBOSS
collaboration results [42, 43, 44, 45, 46]). Separating the populations into
two provides 7 independent measurable signals, which are given by

$\displaystyle\xi_{\mathrm{LK}}^{(0)}(z,d)$ $\displaystyle=\left[b_{\rm
L}(z)\,b_{\textrm{K}}(z)+\frac{1}{3}\left(b_{\textrm{L}}(z)+b_{\textrm{K}}(z)\right)\,f(z)+\frac{1}{5}\,f^{2}(z)\right]\,\sigma^{2}_{8}(z)\,\mu_{0}(d)\,,$
(2.8) $\displaystyle\xi_{\rm\mathrm{LK}}^{(2)}(z,d)$
$\displaystyle=\left[-\frac{2}{3}\left(b_{\textrm{L}}(z)+b_{\textrm{K}}(z)\right)f(z)-\frac{4}{7}\,f^{2}(z)\right]\,\sigma_{8}^{2}(z)\,\mu_{2}(d)\,,$
(2.9) $\displaystyle\xi^{(4)}(z,d)$
$\displaystyle=\frac{8}{35}\,f^{2}(z)\,\sigma_{8}^{2}(z)\,\mu_{4}(d)\,,$
(2.10)

where $f=\mathrm{d}\ln(\delta_{m})/\mathrm{d}\ln(a)$ is the growth rate and
$\sigma_{8}(z)$ is the amplitude of the (linear) power spectrum on the scale
of $8\;\mathrm{Mpc}/h$. The indices $\textrm{L},\textrm{K}$ label the
population under consideration. In the following we will split the galaxies
according to their luminosity, hence L and K are either Bright (B) or Faint
(F). The functions $\mu_{\ell}(d)$ are defined as

$\mu_{\ell}(d)=\frac{1}{2\pi^{2}}\int\,dk\,k^{2}\;\frac{P_{\delta\delta}(k,z=0)}{\sigma_{8}^{2}(z=0)}\;j_{\ell}(k\,d)\,,\qquad\ell=0,\,2,\,4\,.$
(2.11)

Here $P_{\delta\delta}(k,z=0)$ is the matter power spectrum at $z=0$, which is
computed for any theory of gravity by solving the correspondent second order
evolution equations for the matter density perturbations. In this work, we fix
the cosmology to be $\Lambda$CDM and compute the power spectrum using CAMB
[47, 48]. In addition to their dependence on cosmological parameters, the
multipoles (2.8)- (2.9) depend on the galaxy biases
$b_{\textrm{L},\textrm{K}}(z)$. Those can be modelled for each population of
galaxies, with a number of free parameters that are determined from the data,
as we will describe in detail in Sec. 3.5.

The lensing contribution $\Delta^{\rm lens}$ also contributes to the monopole,
quadrupole and hexadecapole, as shown in [32, 49]. This contribution is
however only relevant at high redshift, above $z=1$ [27, 29]. In the linear
regime, the dipole and octupole are only measurable at redshift below 1 and we
concentrate therefore our analysis on this regime, where we can safely neglect
the lensing contribution. The contributions from the two other terms,
$\Delta^{\rm rel}$ and $\Delta^{\rm corr}$, to the monopole, quadrupole and
hexadecapole are suppressed by at least two powers
$\left(\mathcal{H}/k\right)^{2}$ 555The largest contribution from $\Delta^{\rm
rel}$ is indeed through $\langle\Delta^{\rm rel}\Delta^{\rm rel}\rangle$,
since $\langle\Delta^{\rm st}\Delta^{\rm rel}\rangle$ generates only odd
multipoles, while the largest contribution from $\Delta^{\rm corr}$ is through
$\langle\Delta^{\rm st}\Delta^{\rm corr}\rangle$.. For the range of
separations we are interested in, they are therefore negligible.

The next-to-leading order contribution originates from the cross-correlation
of $\Delta^{\mathrm{st}}$ and $\Delta^{\mathrm{rel}}$, which induces a
breaking of the symmetry in the correlation function, thus generating odd
multipoles: a dipole and an octupole ($\ell=1,\,3$, respectively) [31, 34, 37,
36]. These odd multipoles are nonzero only for the cross-correlation of two
different populations, $\textrm{L}=\textrm{B}$ and $\textrm{K}=\textrm{F}$
[33, 50]. Using the Euler and Poisson equations we obtain

$\displaystyle\xi_{\textrm{B}\textrm{F}}^{(1)}(z,d)$
$\displaystyle=\frac{\mathcal{H}(z)}{\mathcal{H}_{0}}\,\Bigg{[}\frac{3}{5}\left(\beta_{\textrm{B}}(z)-\beta_{\textrm{F}}(z)\right)f^{2}(z)+(b_{\textrm{F}}(z)\,\beta_{\textrm{B}}(z)-b_{\textrm{B}}(z)\,\beta_{\textrm{F}}(z))\,f(z)$
$\displaystyle\qquad\qquad\qquad\qquad+(b_{\textrm{B}}(z)-b_{\textrm{F}}(z))\left(\frac{2}{r\mathcal{H}}+\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}\right)f(z)\Bigg{]}\,\sigma_{8}^{2}(z)\,\nu_{1}(d)\,,$
(2.12) $\displaystyle\xi_{\textrm{B}\textrm{F}}^{(3)}(z,d)$
$\displaystyle=\frac{2}{5}\,\frac{\mathcal{H}}{\mathcal{H}_{0}}\,(\beta_{\textrm{B}}(z)-\beta_{\textrm{F}}(z))\,f^{2}(z)\,\sigma_{8}^{2}(z)\,\nu_{3}(d)\,,$
(2.13)

where we have defined the factor

$\beta_{\textrm{L}}(z)=5\,s_{\textrm{L}}\,\left(\frac{1}{r\mathcal{H}}-1\right)+f^{\rm
evol}_{\textrm{L}}\,,$ (2.14)

containing the magnification and evolution biases. The integrals over the
power spectrum take the form

$\nu_{\ell}(d)=\frac{\mathcal{H}_{0}}{2\pi^{2}}\int\,dk\,k\;\frac{P_{\delta\delta}(k,z=0)}{\sigma_{8}^{2}(z=0)}\;j_{\ell}(k\,d)\,,\qquad\ell=1,\,3\,.$
(2.15)

The odd multipoles are suppressed by one factor $(\mathcal{H}/k)$ with respect
to the standard contribution that generate the even multipoles, as can be seen
comparing Eq. (2.15) with (2.11). The signal-to-noise ratio (SNR) of the odd
multipoles is therefore significantly smaller than that of the even
multipoles. For this reason, the odd multipoles have not been detected yet in
the linear regime [50]. Forecasts for surveys like DESI and the SKA have
however shown that the odd multipoles will be robustly detected in the future,
see e.g. [50, 51, 52, 53, 39, 40]. Importantly, the SNR of the odd multipoles
peaks at intermediate separations, $d\in[20,160]$. It is therefore not
necessary to access very large scales to measure $\Delta^{\rm rel}$. As such,
the dipole and octupole provide a clean way of measuring the magnification
bias and evolution bias from intermediate scales, without contamination from
primodial non-Gaussianities that are relevant only at very large scales.

In the flat-sky approximation, the pure standard terms $\langle\Delta^{\rm
st}\Delta^{\rm st}\rangle$ do not contribute to the odd multipoles. However,
going beyond the flat-sky approximation, one finds wide-angle corrections
arising from RSD that contribute to the dipole and octupole at the same order
as the $\Delta^{\rm rel}$ contribution. These terms need therefore to be
accounted for in the analysis. They can either be included in the modelling,
as done in e.g. [38, 40, 54], or they can be removed from the signal by an
appropriate definition of the estimators [31, 55]. In any case, these terms do
not impact the determination of the magnification bias and evolution bias,
since they do not depend on these quantities. Here we consider the second
approach and remove the wide-angle effects arising in the dipole and octupole.

The cosmological and astrophysical parameters can be constrained using galaxy
clustering information. In particular, measurements of the even multipoles put
constraints on the the galaxy bias $b_{\textrm{L}}$ of each population and on
the growth rate $f$, which is particularly sensitive to modifications of
gravity [56]. The relativistic effects carry additional information through
the gravitational redshift contribution that can be exploited for various
applications, such as measuring the gravitational potential [57], constraining
modified gravity models [58, 59, 60] and testing for additional interactions
in the dark sector [55, 61, 62, 63, 54]. Here we adopt a different
perspective: we assume that the $\Lambda$CDM model is valid, and we use the
even and odd multipoles to determine the cosmological parameters, the galaxy
biases $b_{\textrm{L}}$, the evolution biases $f^{\rm evol}_{\textrm{L}}$ and
the magnification biases $s_{\textrm{L}}$.

##  3 Modeling of the galaxy, magnification and evolution biases

The galaxy, magnification and evolution biases depend on the population of
galaxies as well as on the characteristics of the survey. In this work we
consider the futuristic Square Kilometre Array Phase 2 (SKA2) HI galaxy
survey. We first review the modelling of the number of galaxies and biases
derived in [64] for this survey, and then extend it to the case of two
populations. This modelling is used to determine the fiducial values of the
biases, which control the amplitude of the signal. We then derive fitting
functions for the different biases, and we forecast how well the parameters of
the fitting functions can be determined with a survey like SKA2.

SKA2 is expected to detect around 1 billion HI galaxies over $30,000$ square
degrees in the redshift range $z\in[0.1,\,2.0]$. Following [64] we parametrise
the background (physical) number of sources per redshift $z$ and per solid
angle, detected above flux threshold $\mathcal{F}_{\star}$ as

$\bar{N}_{g}(z,\mathcal{F}_{\star})=10^{c_{1}}\,z^{c_{2}}\,e^{-c_{3}z}\,,$
(3.1)

where the parameters $c_{i}$ depend on $\mathcal{F}_{\star}$ [65, 64]. For
SKA2, the expected flux threshold will depend on redshift, as shown in [64],
Table 2. In order to express $\bar{N}_{g}$ as a function of redshift alone, we
first compute the $c_{i}(\mathcal{F}_{\star})$ using Table 3 of [65].
Secondly, we reconstruct $\mathcal{F}_{\star}(z)$ from Table 2 in [64]. This
allows us to compute $\bar{N}_{g}(z)$ by replacing this
$\mathcal{F}_{\star}(z)$ into the interpolated $c_{i}(\mathcal{F}_{\star})$.
We do this for the redshift interval of $z\in[0.1,\,2]$ using bins of size
$\delta z=0.1$. The resulting number of galaxies as a function of redshift is
plotted in Fig. 1.

Figure 1: Logarithm of the number of galaxies per unit of solid angle and
redshift for different choices of the flux limit. The solid blue line is
obtained by using the $z$-dependent $\mathcal{F}_{*}$ from [64]. The dashed
orange line is obtained by using the fit in [66]. Finally, the dashdot green
line corresponds to the fit from [65] with redshift-independent flux limit
$\mathcal{F}_{*}=5.0\,\mu\mathrm{Jy}$.

###  3.1 The magnification and evolution biases for a single population of
galaxies

The magnification bias $s$ is defined as the logarithmic slope of the
cumulative number of sources above a given luminosity threshold. Since the
luminosity threshold is related to the flux threshold by the luminosity
distance $d_{L}$ (which does not depend on luminosity)

$L_{\star}(z)=4\pi\,\mathcal{F}_{\star}(z)\,d_{L}(z)\,,$ (3.2)

$s$ can equally be expressed as a derivative of
$\bar{N}_{g}(z,\mathcal{F}_{\star})$ with respect to $\mathcal{F}_{\star}$

$\displaystyle
s(z,L_{\star})=-\frac{2}{5}\frac{\partial\log{\bar{N}_{g}(z,L_{\star})}}{\partial\log{L_{\star}}}\Bigg{|}_{z}=-\frac{2}{5}\frac{\partial\log{\bar{N}_{g}(z,\mathcal{F}_{\star})}}{\partial\log{\mathcal{F}_{\star}}}\Bigg{|}_{z}=s(z,\mathcal{F}_{\star})\,.$
(3.3)

The magnification bias is therefore an observable quantity: it can be measured
by binning $\bar{N}_{g}(z,\mathcal{F}_{\star})$ in $\mathcal{F}_{\star}$ and
computing the derivative at the chosen flux threshold. Note that since the
cumulative number of sources can only increase when decreasing
$\mathcal{F}_{\star}$, $s(z,\mathcal{F}_{\star})$ is always positive.

The evolution bias $f^{\rm evol}$ is defined as the partial redshift
derivative of the cumulative comoving number density of galaxies above
$L_{\star}$, $\bar{n}_{g}(z,L_{\star})$, taken at fixed value of the
luminosity threshold

$\displaystyle f^{\rm
evol}(z,L_{\star})=-\frac{\partial\ln\bar{n}_{g}(z,L_{\star})}{\partial\ln(1+z)}\Bigg{|}_{L_{\star}}=-\frac{d\ln\bar{n}_{g}(z,L_{\star}(z))}{d\ln(1+z)}+\frac{\partial\ln\bar{n}_{g}(z,L_{\star})}{\partial\ln
L_{\star}}\,\frac{d\ln L_{\star}(z)}{d\ln(1+z)}\,.$ (3.4)

The comoving number density $\bar{n}_{g}(z,L_{\star})$ is not directly
observable. It can however be related to $\bar{N}_{g}(z,\mathcal{F}_{\star})$
through

$\displaystyle\bar{N}_{g}(z,\mathcal{F}_{\star})=\frac{r^{2}(z)}{H(z)}\,\bar{n}_{g}(z,L_{\star})\,,$
(3.5)

where $H$ is the Hubble parameter (in physical time). Following [64], the
partial derivative with respect to $z$ can be related to a total derivative
with respect to $z$ accounting for the fact that when varying the redshift,
the luminosity threshold also varies. In addition, one needs to take into
account the evolution with redshift of the volume factor, $r^{2}(z)/H(z)$ in
(3.5), as well as the fact that the flux threshold can also vary with
redshift. Putting everything together we obtain

$\displaystyle f^{\rm evol}(z,\mathcal{F}_{\star})=$
$\displaystyle-\frac{d\ln{\bar{N}_{g}(z,\mathcal{F}_{\star})}}{d\ln{(1+z)}}-\frac{d\ln{H(z)}}{d\ln{(1+z)}}+\frac{2(1+z)}{r(z)H(z)}$
(3.6)
$\displaystyle-5\,s(z,\mathcal{F}_{\star})\left[1+\frac{(1+z)}{r(z)H(z)}\right]-\frac{5}{2}\,s(z,\mathcal{F}_{\star})\,\frac{d\ln{\mathcal{F}_{\star}(z)}}{d\ln(1+z)}\,.$

Written in this way, the evolution bias is an observable quantity:
$\bar{N}_{g}$ can indeed be split in bins of redshift and the total derivative
(first term) can be measured. In the following we shall refer to this quantity
as $n^{\rm evol}(z)$

$n^{\rm
evol}(z)\equiv\frac{d\log{\bar{N}_{g}(z,\mathcal{F}_{\star})}}{d\log{(1+z)}}\,.$
(3.7)

The second and third term in (3.6) can either be obtained from measurements of
$H(z)$ and $d_{L}(z)=(1+z)r(z)$ or from measurements of cosmological
parameters in a given model. Finally, the last term in the second line is a
(known) survey-dependent quantity. For surveys with a fixed flux threshold
over the whole redshift range, this last term vanishes.

These results apply for the single-tracer case. Let us now extend them to the
case of multiple populations of galaxies.

###  3.2 Dividing galaxies into bright and faint populations

We now split galaxies into two different populations by introducing an
appropriate flux cut $\mathcal{F}_{c}$. We denote galaxies with fluxes
$\mathcal{F}>\mathcal{F}_{c}$ as _bright_ galaxies, and galaxies with fluxes
$\mathcal{F}_{c}>\mathcal{F}>\mathcal{F}_{\star}$ as _faint_ galaxies

$\displaystyle\bar{N}_{\textrm{B}}(z,\mathcal{F}_{c})$
$\displaystyle\equiv\bar{N}_{g}(z,\mathcal{F}>\mathcal{F}_{c})\,,$ (3.8)
$\displaystyle\bar{N}_{\textrm{F}}(z,\mathcal{F}_{c})$
$\displaystyle\equiv\bar{N}_{g}(z,\mathcal{F}_{c}>\mathcal{F}>\mathcal{F}_{\star})\,.$
(3.9)

We introduce the parameter
$m=\bar{N}_{g}(z,\mathcal{F}_{\star})/N_{\textrm{B}}(z,\mathcal{F}_{c})$, from
which we can obtain the fraction of bright galaxies as $m^{-1}$. For a given
value of $m$ we can find the corresponding $\mathcal{F}_{c}$ in each of the
redshift bin using the fit (3.1) to calculate both
$\bar{N}_{g}(z,\mathcal{F}_{\star})$ and
$\bar{N}_{\textrm{B}}(z,\mathcal{F}_{c})$ and solving numerically the equation

$\log{\bar{N}_{\textrm{B}}(z,\mathcal{F}_{c})}-\log{\bar{N}_{g}(z,\mathcal{F}_{\star})}+\log{m}=0\,.$
(3.10)

This gives $\mathcal{F}_{c}(z)$ as a function of redshift, which is generally
different from $\mathcal{F}_{\star}(z)$. The number of faint galaxies,
$\bar{N}_{\textrm{F}}(z,\mathcal{F}_{c})$, is then simply obtained by
subtracting the number of bright galaxies from the total. Note that the
logarithmic evolution of the galaxies, $n^{\rm evol}$ given in (3.7) is the
same for the total population, and for the bright and faint galaxies, since
$m$ is kept fixed over the redshift bins.

###  3.3 Magnification bias

The dipole (2.12) and octupole (2.13) depend on the magnification bias of the
bright and of the faint populations. These biases multiply the Doppler
contribution. They account for the fact that galaxies that are moving with
respect to the observer do not have the same luminosity distance as galaxies
at the same redshift with no peculiar velocities, as shown in [67]. As a
consequence their flux is magnified or de-magnified by the peculiar velocity
(depending on its direction). This can change the number of bright and faint
galaxies that are observed and consequently impact the correlation function.

For the bright population, galaxies can cross the flux limit
$\mathcal{F}_{c}$, enhancing or decreasing the number of bright galaxies. The
magnification bias of this population is therefore similar to that of the
total population, simply replacing $\mathcal{F}_{\star}$ by $\mathcal{F}_{c}$
in Eq. (3.3)

$s_{\textrm{B}}(z)=s(z,\mathcal{F}_{c})=-\frac{2}{5}\frac{\partial\log{\bar{N}_{g}(z,\mathcal{F}>\mathcal{F}_{c})}}{\partial\log{\mathcal{F}_{c}}}\Bigg{|}_{z}\,.$
(3.11)

The magnification bias for the faint galaxies is however more subtle: the
faint galaxies can indeed either cross the $\mathcal{F}_{c}$ threshold, or the
$\mathcal{F}_{\star}$ threshold. Following [53], we obtain the following
relation for the magnification biases

$\bar{N}_{g}(z,\mathcal{F}_{\star})\,s_{\rm
M}(z)=\bar{N}_{\textrm{B}}(z,\mathcal{F}_{c})\,s_{\textrm{B}}(z)+\bar{N}_{\textrm{F}}(z,\mathcal{F}_{c})\,s_{\textrm{F}}(z)\,,$
(3.12)

where we have defined $s_{\rm M}(z)\equiv s(z,\mathcal{F}_{\star})$ as the
magnification bias of the whole population (bright plus faint). Using that
$\bar{N}_{\textrm{B}}=\bar{N}_{g}/m$, we can rewrite the above equation as

$s_{\textrm{F}}(z)=\frac{m}{m-1}\,s_{\rm
M}(z)-\frac{1}{m-1}\,s_{\textrm{B}}(z)\,.$ (3.13)

Note that since by construction $\bar{N}_{\textrm{B}}<\bar{N}_{g}$ we have
$m>1$.

Figure 2: Predicted values of the magnification biases for the bright
population $s_{\textrm{B}}$, the faint population $s_{\textrm{F}}$, and the
total population $s_{\textrm{M}}$. The left panel shows the magnification
biases for $m=2$ (50% of bright galaxies), and the right panel for $m=10/3$
(30% of bright galaxies).

Contrary to $s_{\rm M}$ and $s_{\textrm{B}}$ that are always positive, since
they represent the slopes of a cumulative number of galaxies, $s_{\textrm{F}}$
can be negative if the second term in Eq. (3.13) dominates over the first one.
This can happen if the slope at the flux cut, $s_{\textrm{B}}$, is steeper
than the slope at the flux threshold of the survey $s_{\rm M}$. In this case,
when the fluxes are magnified by Doppler effects, more galaxies will cross the
flux cut $\mathcal{F}_{c}$ (thus leaving the faint sample to become bright)
than galaxies which will cross the flux threshold of the survey
$\mathcal{F}_{\star}$ to become detectable. In this case, a magnification of
the fluxes leads to decrease in the number of faint galaxies (and vice-versa
for a de-magnification).

From Eq. (3.13) we see that $s_{\textrm{F}}$ can be directly reconstructed
from $s_{\rm M}$ and $s_{\textrm{B}}$. In our forecasts we will therefore
treat $s_{\rm M}$ and $s_{\textrm{B}}$ as free parameters, and compute
$s_{\textrm{F}}$ from these two quantities. In Fig. 2 we show the
magnification bias of the total population, and of the bright and faint
populations, for two different splits: 50% of bright galaxies (left panel) and
30% of bright galaxies (right panel).

###  3.4 Evolution bias

In addition to the dependence on magnification biases, the dipole (2.12) and
octupole (2.13) depend also on the evolution bias of the bright and faint
populations. The evolution bias encodes the fact that the number of galaxies
is evolving with redshift, due to the formation and merging of galaxies. In a
homogeneous Universe, all galaxies observed at fixed redshift $z$ are at the
same distance from the observer and they share therefore the same history.
However, in a inhomogeneous Universe, galaxies that are moving for example
towards the observer, are at a larger distance from the observer than galaxies
at the same redshift with no peculiar velocities. As a consequence, such
moving galaxies live at an earlier time than the ones at rest with the Hubble
flow. If the number of galaxies is evolving with time (for example decreasing
due to mergers) we will see less galaxies at earlier time than closer to us,
which creates fluctuations in the galaxy number density.

In the case of two populations, the evolution bias of each population encodes
how each of the population evolves and it is therefore simply given by [64,
53]

$\displaystyle f^{\rm evol}_{\textrm{B}}(z)$
$\displaystyle=-\frac{\partial\log{\bar{n}_{\textrm{B}}(z,L_{c})}}{\partial\log{(1+z)}}\Bigg{|}_{L_{c}}\,,$
(3.14) $\displaystyle f^{\rm evol}_{\textrm{F}}(z)$
$\displaystyle=-\frac{\partial\log{\bar{n}_{\textrm{F}}(z,L_{\star},L_{c})}}{\partial\log{(1+z)}}\Bigg{|}_{L_{\star},L_{c}}\,.$
(3.15)

Following the same steps as in the case of one population of galaxies, we can
rewrite Eqs. (3.14) and (3.15) in terms of observable quantities

$\displaystyle f^{\rm evol}_{\textrm{B}}(z)=-n^{\rm
evol}(z)-\frac{d\log{H(z)}}{d\log{(1+z)}}+\frac{2(1+z)}{r(z)H(z)}-5\,s_{\textrm{B}}(z)\left[1+\frac{(1+z)}{r(z)H(z)}\right]$
(3.16)
$\displaystyle\qquad\qquad\qquad-\frac{5}{2}s_{\textrm{B}}(z)\,\frac{d\log{\mathcal{F}_{c}}}{d\log{(1+z)}}\,,$
$\displaystyle f^{\rm evol}_{\textrm{F}}(z)=-n^{\rm
evol}(z)-\frac{d\log{H(z)}}{d\log{(1+z)}}+\frac{2(1+z)}{r(z)H(z)}-5\,s_{\textrm{F}}(z)\left[1+\frac{(1+z)}{r(z)H(z)}\right]$
(3.17)
$\displaystyle\qquad\qquad\qquad-\frac{5}{2}\left[\frac{\bar{N}_{g}}{\bar{N}_{\textrm{F}}}\,s_{\rm
M}(z)\frac{d\log{\mathcal{F}_{\star}}}{d\log{(1+z)}}-\frac{\bar{N}_{\textrm{B}}}{\bar{N}_{\textrm{F}}}\,s_{\textrm{B}}(z)\frac{d\log{\mathcal{F}_{c}}}{d\log{(1+z)}}\right]\,,$

where $n^{\rm evol}(z)$ is given by Eq. (3.7). These results are consistent
with Eq. (17) of [53] when both $\mathcal{F}_{\star}$ and $\mathcal{F}_{c}$
are constant. Note that the ratios in the second line of Eq. (3.17) are the
same as those appearing in Eq. (3.13). They are constant by construction and
only depend on the fraction of bright and faint populations, encoded in $m$:

$\frac{\bar{N}_{g}}{\bar{N}_{\textrm{F}}}=\frac{m}{m-1}\,,\qquad\qquad\frac{\bar{N}_{\textrm{B}}}{\bar{N}_{\textrm{F}}}=\frac{1}{m-1}\,.$
(3.18)

Figure 3: Predicted values of the evolution biases for the bright population
$f^{\rm evol}_{\textrm{B}}$, the faint population $f^{\rm evol}_{\textrm{F}}$,
and the total population $f^{\rm evol}$. The left panel shows the evolution
biases for $m=2$ (50% of bright galaxies), and the right panel for $m=10/3$
(30% of bright galaxies).

We have now all the ingredients to compute the evolution bias numerically. As
shown in Sec. 3.2, the first term, $n^{\rm evol}$, in Eqs. (3.16) and (3.17)
is the same for both populations, and equal to that of the total population.
In practice, $n^{\rm evol}$ can be measured directly from the data, by binning
the galaxies in redshift, and computing the total derivative. Here we perform
this derivative numerically (using 5-point stencil) from the number of
galaxies obtained through Eq. (3.1). Similarly, we compute numerically the
redshift derivatives of the flux cuts. The results for the evolution bias of
the whole population, and that of the bright and faint populations are shown
in Fig. 3 for two choices of splitting. We see that the amplitude of the
evolution bias of the faint sample is always larger than that of the bright
sample: faint galaxies therefore evolve faster than bright ones in our
modelling. Note that at low redshift, the evolution biases of the two
populations are quite sensitive to the choice of $m$, providing significantly
different predictions.

From Eqs. (3.11), (3.12), (3.16) and (3.17) we see that the magnification
biases and evolution biases of the bright and faint populations depend only on
three quantities: $s_{\textrm{B}},s_{\textrm{M}}$ and $n^{\rm evol}$.
Interestingly, two of these quantities, $s_{\textrm{M}}$ and $n^{\rm evol}$,
are the same for all splits, since they do not depend on $\mathcal{F}_{c}$. It
is therefore possible to combine different splits (with different values of
$m$) to improve the measurements of $s_{\textrm{M}}$ and $n^{\rm evol}$. These
two quantities fully determine the magnification bias and evolution bias of
the whole population. Hence even though measuring relativistic effects
requires two populations of galaxies, the quantities that can be extracted
from this signal are useful to characterise the whole population. This is
particularly interesting to model the amplitude of the relativistic
contaminations to local primordial non-Gaussianities, as well as the amplitude
of the kinematic dipole, which do not require the use of two populations.

###  3.5 Galaxy bias

Finally, the dipole and octupole depend also on the galaxy biases of the
bright and faint populations. The bias of each population depends directly on
the splitting parameter $m$, i.e. on the fraction of bright and faint
galaxies. Typically, a larger value of $m$ (small fraction of bright galaxies)
implies a larger $\mathcal{F}_{c}$, which leads to a larger bias for the
bright population and therefore a larger bias difference $\Delta b$ between
bright and faint sources. On the contrary a smaller value of $m$ decreases the
bias difference (see e.g. [53] for an example in the case of the survey DESI).

In the following we model the biases of the two populations in terms of the
bias of the total population $b_{\rm T}$, the bias difference $\Delta b$ and
the parameter $m$. The bright and faint biases obey the system of equations

$\displaystyle\frac{\bar{N}_{\textrm{B}}}{\bar{N}_{g}}\,b_{\textrm{B}}+\frac{\bar{N}_{\textrm{F}}}{\bar{N}_{g}}\,b_{\textrm{F}}=b_{\rm
T}\,,$ (3.19) $\displaystyle b_{\textrm{B}}-b_{\textrm{F}}=\Delta b\,.$ (3.20)

Using $\bar{N}_{\textrm{B}}/\bar{N}_{g}=1/m$ and
$\bar{N}_{\textrm{F}}/\bar{N}_{g}=(m-1)/m$ we obtain

$\displaystyle b_{\textrm{B}}=b_{\rm T}+\frac{m-1}{m}\,\Delta b\,,$ (3.21)
$\displaystyle b_{\textrm{F}}=b_{\rm T}-\frac{1}{m}\,\Delta b\,.$ (3.22)

We model the galaxy bias of the total population following [66] as

$b_{\rm T}=b_{1}\,e^{b_{2}\,z}\,,$ (3.23)

where $[b_{1},\,b_{2}]=[0.554,0.783]$. Without precise knowledge of the
luminosity function of the population of galaxies, we do not know the bias
difference $\Delta b$ for a given splitting parameter $m$. In our forecast, we
assume that for $m=2$, $\Delta b=1$, in agreement with the measurements done
in the BOSS survey [50]. We then choose a smaller value of $\Delta b=0.8$ in
the case where we have $70\%$ percent of bright galaxies and a larger value of
$\Delta b=1.2$ when we have $30\%$ of bright galaxies. These choices are
somewhat arbitrary, but they are only necessary because we are doing a
forecast. Once we perform such an analysis on real data, the bright and faint
biases will be considered as free parameters, directly measured from the even
multipoles of the correlation function.

##  4 Methodology

We forecast the capacity of a spectroscopic survey like the SKA2 HI galaxy
survey to measure the magnification and evolution biases using galaxy
clustering information. To this end we first parameterise the different biases
with a set of free parameters, that we then constrain with the dipole and the
octupole.

We use the Fisher formalism and compute the Fisher matrix by performing the
second derivative of the natural logarithm of the likelihood with respect to
the parameters of the model. Assuming Gaussianity and neglecting the
dependence of the covariance on the model parameters, the Fisher matrix can be
expressed as

$\mathrm{F}_{ab}=\sum_{ij}\,\frac{\partial\,\Xi_{i}}{\partial\,\Theta_{a}}\,\left(\mathbb{C}^{-1}\right)_{ij}\,\frac{\partial\,\Xi_{j}}{\partial\,\Theta_{b}}\,,$
(4.1)

where $\vec{\Xi}$ is the data vector, $\mathbb{C}$ is the covariance matrix
and $\Theta$ is the set of parameters. We describe below how we build these
two objects for the specific analysis in this work. Once the Fisher matrix is
computed, we estimate the covariance matrix of the parameters as the inverse
of the Fisher matrix

$\mathrm{C}_{ij}=\left(\mathrm{F}^{-1}\right)_{ij}\,.$ (4.2)

The parameter uncertainties are encoded in the diagonal of $\mathrm{C}$, while
the off-diagonal terms encode the cross-correlations between the parameters.

###  4.1 Survey specifications

Figure 4: Multipoles of the cross-correlation between the bright and faint
populations at redshift $z=0.25$, plotted as a function of separation $d$.

We consider a futuristic survey akin to the HI galaxy survey in Phase 2 of the
Square Kilometer Array, as described in [66, 64]. Such a survey is proposed to
span $30,000$ square degrees ranging from $z=0.1$ to $z=2$, and observing
close to a billion galaxies. In practice, we divide this redshift range into
$19$ bins of size $\Delta z=0.1$ and evaluate all the quantities at the centre
of each $z$-bin. We split the full population of galaxies provided by the
survey into two samples following the prescription described in Sec. 3.2 in
order to have a constant ratio of bright versus faint galaxies in every
$z$-bin. Then, we compute the signal dataset and the covariance matrices at
each $z$-bin in the linear regime, including separations spanning from $d_{\rm
min}=20\,\mathrm{Mpc}/h$ to $d_{\rm max}=160\,\mathrm{Mpc}/h$. We use fiducial
cosmological parameters from Planck [68] (see also Table 6 in Appendix C).

In Fig. 4, as illustration, we show the cross-correlation monopole, dipole,
quadrupole and octupole at $z=0.25$ (second redshift bin) for two different
splits: 50% bright and 50% faint galaxies (blue, solid curves); 30% bright and
70% faint galaxies (red, dashed-dot curves). We see that the monopole and the
quadrupole are significantly larger than the dipole and octupole. The
cumulative SNR over all redshifts and separations for the $50\%-50\%$ split
is: $624$ and $601$ respectively for the bright-faint cross-correlation
monopole and quadrupole; $57$ and $5$ respectively for the dipole and
octupole. Table 4 in Appendix C reports the SNR for all multipoles in the
different splits. Note that the SNR of the dipole and octupole are without
wide-angle effects, since as explained before, one can choose an estimator to
remove them.

###  4.2 Fitting models for galaxy biases, magnification biases and evolution
biases

Our goal is to constrain the various biases from the data. As shown in Sec. 3,
the magnification and evolution biases can be written in terms of three
observable quantities: $s_{\rm M},s_{\textrm{B}}$ and $n^{\rm evol}$, that can
be directly measured from the number of galaxies. Our strategy is therefore
the following: we use our modelling to compute these three quantities. We then
parametrise them with a set of free parameters, with fiducial values given by
our model and with uncertainties expected from future surveys. Typically, we
choose two different values for the uncertainties: 50% uncertainties and 20%
uncertainties. Those are quite large and future surveys may very well provide
tighter constraints, but we prefer to remain conservative here. We then
perform a Fisher forecast to determine the capability of the dipole and the
octupole to improve the precision on these parameters. From this we then
reconstruct $s_{\textrm{B}}(z),s_{\textrm{F}}(z)$, $f^{\rm
evol}_{\textrm{B}}(z)$ and $f^{\rm evol}_{\textrm{F}}(z)$.

####  4.2.1 Parametrisation of $s_{\rm M}(z)$ and $s_{\rm B}(z)$

We parametrise the redshift evolution of $s_{\rm M}(z)$ and
$s_{\textrm{B}}(z)$ with 4 parameters each:

$s_{\rm L}(z)=s_{{\rm L},0}+s_{{\rm L},1}\,z+s_{{\rm L},2}\,\log{z}+s_{{\rm
L},3}\,(\log{z})^{2}\,,$ (4.3)

where ${\rm L}=\textrm{B},\,\textrm{M}$. This expansion recovers well the
redshift evolution plotted in Fig. 2. Thus, we include 8 parameters in the
analysis: $\\{s_{{\rm L},0},\,s_{{\rm L},1},\,s_{{\rm L},2},\,s_{{\rm
L},3}\\}$. The magnification bias of the bright population is sensitive to the
choice of splitting, and we have therefore different fiducial values for the
parameters for different splitting parameter $m$. On the other hand,
$s_{\textrm{M}}$ does not depend on $m$. The fiducial values can be found in
Tables 5 and 6 in Appendix C. As explained above we introduce Gaussian priors
of $50\%$ and $20\%$ around the fiducial values on both set of parameters for
each choice of $m$, to account for our prior knowledge on $s_{\textrm{B}}$ and
$s_{\textrm{M}}$ from direct measurements.

####  4.2.2 Parametrisation of $n^{\mathrm{evol}}(z)$

Since by construction, we split the two populations keeping constant the ratio
of bright and faint galaxies in all redshift bins, the function
$n^{\mathrm{evol}}$ defined in (3.7) is the same for both populations and for
the total population. We parametrise the redshift evolution of this function
with 4 parameters as

$n^{\mathrm{evol}}(z)=n^{\mathrm{evol}}_{0}+n^{\mathrm{evol}}_{1}\,z+n^{\mathrm{evol}}_{2}\,\frac{1}{z}+n^{\mathrm{evol}}_{3}\,e^{-z}\,.$
(4.4)

We have therefore 4 additional free parameters in our analysis
$\\{n^{\mathrm{evol}}_{0},n^{\mathrm{evol}}_{1},n^{\mathrm{evol}}_{2},n^{\mathrm{evol}}_{3}\\}$.
The fiducial values are independent of the value of $m$ and can be found in
Table 6 in Appendix C. Similarly to the magnification bias, we consider
Gaussian priors of $50\%$ and $20\%$ around the fiducial values.

####  4.2.3 Parametrisation of the galaxy biases

We parametrise the evolution of the biases with redshift using (3.21)–(3.23).
For a given splitting we fix the bias difference $\Delta b$ in Eqs. (3.21) and
(3.22). The fiducial values for $b_{1}$ and $b_{2}$ are independent of the
splitting since they encode the evolution of the total population. In our
forecast we vary however these parameters separately when they enter into the
bright or faint biases, since these two quantities can in principle vary
independently. Hence we have 4 additional parameters
$\\{b_{1,\textrm{B}},b_{2,\textrm{B}},b_{1,\textrm{F}},b_{2,\textrm{F}}\\}$,
with fiducial values given in Table 6 in Appendix C. Note that we could have
alternatively used as free parameters $\\{b_{1},b_{2},\Delta b\\}$.

###  4.3 Galaxy samples and data vectors

Since the magnification bias of the total population, $s_{\rm M}$, and the
quantity $n^{\mathrm{evol}}(z)$ are independent of the way galaxies are split,
we explore different cases and compare the constraints. We first perform
Fisher analyses using two populations of galaxies, and then we consider a case
where we have four different populations.

####  4.3.1 Two galaxy populations

We consider three different samples of bright (B) and faint (F) galaxies:

  * •

Analysis with $50\%$ of B and $50\%$ of F galaxies ($50\times 50$): we set
$m=2$ and $\Delta b=1$. This is the standard approach used in previous works,
e.g. Refs. [62, 57, 59].

  * •

Analysis with $30\%$ of B and $70\%$ of F galaxies ($30\times 70$): we set
$m=10/3$ and $\Delta b=1.2$.

  * •

Analysis with $70\%$ of B and $30\%$ of F galaxies ($70\times 30$): we set
$m=10/7$ and $\Delta b=0.8$.

As observables we consider the multipoles of the correlation function
described in Sec. 2. For any value of $m$, we have 7 even multipoles: the
monopoles and quadrupoles of the BB, BF and FF correlations, plus the
hexadecapole of the full population. Additionally, we have 2 odd multipoles
arising from BF cross-correlations, namely a dipole and an octupole. The data
vector at each $z$-bin takes the form

$\vec{\Xi}(z,d)=\left(\xi^{(0)}_{\textrm{B}\textrm{B}},\,\xi^{(0)}_{\textrm{B}\textrm{F}},\,\xi^{(0)}_{\textrm{F}\textrm{F}},\,\xi^{(1)}_{\textrm{B}\textrm{F}},\,\xi^{(2)}_{\textrm{B}\textrm{B}},\,\xi^{(2)}_{\textrm{B}\textrm{F}},\,\xi^{(2)}_{\textrm{F}\textrm{F}},\,\xi^{(3)}_{\textrm{B}\textrm{F}},\,\xi^{(4)}_{\mathrm{T}}\right)\\!\big{|}_{(z,d)}\,.$
(4.5)

We compute the Gaussian covariance matrix of $\vec{\Xi}(z,d)$ including all
the possible cross-correlations between the multipoles, and accounting for
both cosmic variance and shot-noise. We also include the shot-noise
contributions associated to the cross-correlations between the hexadecapole
and the monopoles and quadrupoles, arising from the fact that the B and F
populations overlap with the full population. This contribution is usually
neglected (see e.g. [62, 57, 59]), since cosmic variance dominates the error
budget of the even multipoles for a survey like the one with SKA2. In this
work, we develop a shot-noise model accounting for overlapping populations
(see Appendices A and B) and we find that indeed the resulting contribution is
negligible and has only a small impact on the constraints.

In total we have 21 free parameters: 16 parameters describing the various
biases plus 5 cosmological parameters $h,A_{\rm s},n_{\rm s},\Omega_{\rm
b},\Omega_{\rm m}$, whose fiducial values are also listed in Table 6 in
Appendix C. The derivatives of the multipoles with respect to the cosmological
parameters are computed numerically from CAMB [47, 25] and the FFTLOG
algorithm [69] using the 5-point stencil method.666We used the Python
implementation available in GitHub: 1D FFTlog in Python. The derivatives with
respect to the other parameters can be computed analytically.

####  4.3.2 Four galaxy populations: combining the information of two sample
separations

We explore the possibility of further improving the constraints on the
magnification and evolution biases by performing an additional Fisher analysis
in which we combine two different splits. In this case, we have four
populations of galaxies: two different bright populations (B and b) and two
different faint populations (F and f). These four populations are clearly not
independent, since they are drawn from the same initial population of
galaxies. In the signal, in addition to auto-correlations of each population
with itself, we only consider the cross-correlations of B with F and b with f
(since parts of the galaxies in B are also in b and similarly for the faint).
The data vector is therefore
$\vec{\Xi}_{\mathrm{J}}=\left(\vec{\Xi}_{m},\,\vec{\Xi}_{m^{\prime}}\right)$
with $m=2$ and $m^{\prime}=10/3$.

In the covariance matrix we include all the cross-correlations between the
multipoles of the different splits. In particular, we take into account the
shot-noise contributions due to the fact that we are using the same underlying
population, split in different fractions of B and F galaxies, meaning that
there is certain overlap between the B and F populations of each $m$,
$m^{\prime}$ sub-samples. These contributions are computed in Appendix A. Note
that neglecting the shot-noise contributions between the two different splits
would lead to an overestimation of the parameter constraints.

In this case we have 29 parameters: 5 cosmological parameters, 8 parameters
common to the two different splits (parametrising $n^{\mathrm{evol}}$ and
$s_{\textrm{M}}$) and 16 parameters for
$s_{\textrm{B}},s_{\textrm{b}},b_{\textrm{B}},b_{\textrm{b}},b_{\textrm{F}}$
and $b_{\textrm{f}}$.

##  5 Results

We first consider the cases with two populations of galaxies, and then show
the constraints when combining four populations. We compare the relative
reduction of uncertainties with respect to the priors on the parameters of the
model when including the dipole only versus when adding the octupole, showing
that in this context the octupole contains valuable information and needs to
be included. We also reconstruct the $68\%$ confidence regions for the
magnification and evolution biases as a function of redshift.

###  5.1 Constraints from two galaxy populations

|  $50\times 50$ split
---|---
| $\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$
$s_{\textrm{B},0}$ | 36.81 | 30.02
$s_{\textrm{B},1}$ | 20.01 | 18.47
$s_{\textrm{B},2}$ | 26.36 | 25.03
$s_{\textrm{B},3}$ | 26.18 | 23.81
$s_{\textrm{M},0}$ | 47.70 | 39.29
$s_{\textrm{M},1}$ | 31.80 | 27.83
$s_{\textrm{M},2}$ | 49.26 | 45.07
$s_{\textrm{M},3}$ | 49.99 | 49.94
$n^{\mathrm{evol}}_{0}$ | 44.85 | 41.31
$n^{\mathrm{evol}}_{1}$ | 43.74 | 31.04
$n^{\mathrm{evol}}_{2}$ | 40.39 | 25.55
$n^{\mathrm{evol}}_{3}$ | 33.16 | 30.71

$30\times 70$ split
---
$\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$
41.58 | 32.77
39.44 | 36.00
44.66 | 43.48
24.08 | 21.43
47.97 | 37.71
20.88 | 18.49
49.53 | 46.46
49.98 | 49.96
41.32 | 40.50
37.21 | 28.01
33.77 | 24.09
31.39 | 29.61

$70\times 30$ split
---
$\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$
26.65 | 23.28
17.44 | 16.73
24.12 | 22.87
28.86 | 26.80
46.19 | 37.30
28.56 | 26.66
48.01 | 41.23
49.94 | 49.84
46.17 | 41.65
47.18 | 32.74
32.34 | 26.57
35.96 | 31.62

Table 1: Relative 1$\sigma$ uncertainties for the magnification biases and
number evolution parameters, assuming $50\%$ Gaussian priors. We show the
results for 3 different splits, including only the dipole
($\xi^{(1)}_{\textrm{B}\textrm{F}}$) and including both the dipole and
octupole
($\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$). In all
cases the even multipoles (monopole, quadrupole and hexadecapole) are
included.

Figure 5: Constraints on the redshift evolution of the various functions for
the 50$\times$50 split, starting with a 50% prior on the parameters (grey
region). We show the results when only the dipole is used and when both the
dipole and octupole are included. In all cases the even multipoles (monopole,
quadrupole and hexadecapole) are included. The red functions are those
entering the forecasts, while the blue ones are derived from the red.

In Table 1 we show the relative 1$\sigma$ errors for the 16 parameters
entering the magnification and evolution biases, in the case with 50% priors.
We show the relative errors for the three different splits. We see that the
odd multipoles provide constraints on all parameters, reducing the uncertainty
with respect to the priors. The improvement with respect to the prior differs
significantly depending on the parameters, ranging from a reduction of the
uncertainty of less than a percent (for $s_{\rm M,3}$) to a reduction of a
factor $2-3$ (for $s_{\textrm{B},1},s_{\textrm{B},2}$ and $s_{\textrm{B},3}$
in the $50\times 50$ and $70\times 30$ splits).

We show the relative uncertainty when only the dipole is included, and when
both the dipole and the octupole are used. We see that even though the SNR of
the octupole is 10 times smaller than that of the dipole, adding the octupole
in the analysis has a significant impact on some of the parameters. For
example, adding the octupole improves the constraints on $n_{1}^{\rm evol}$
and $n_{2}^{\rm evol}$ by 20-30% with respect to the constraints from the
dipole only, for all the splits. From Eqs. (2.12) and (2.13), combined with
the expressions for the evolution biases (3.16) and (3.17), we see that the
dipole depends on the magnification bias difference
$s_{\textrm{B}}-s_{\textrm{F}}$, on the combination
$b_{\textrm{F}}s_{\textrm{B}}-b_{\textrm{B}}s_{\textrm{F}}$, as well as on the
product $(b_{\textrm{B}}-b_{\textrm{F}})n^{\rm evol}$. On the contrary, the
octupole depends only on the magnification bias difference
$s_{\textrm{B}}-s_{\textrm{F}}$. As a consequence, there is a strong
degeneracy between $n^{\rm evol}$, $s_{\textrm{B}}$ and $s_{\rm M}$ when only
the dipole is included. Adding the octupole, which does not depend on $n^{\rm
evol}$ partially lifts this degeneracy, improving the constraints on the
parameters. Moreover, the redshift evolution of the dipole and the octupole
differ. The octupole is proportional to $f^{2}(z)(1-1/r(z)\mathcal{H}(z))$,
whereas the dipole contains also other terms that scale with $f(z)$ and with
the biases $b_{\textrm{B}}(z)$ and $b_{\textrm{F}}(z)$. As a consequence,
adding the octupole breaks degeneracies between the different parameters
governing the redshift evolution of $s_{\textrm{B}},s_{\rm M}$ and $n^{\rm
evol}$. Hence even though we find that the diagonal Fisher elements for the
dipole are always significantly larger than for the octupole (by a factor
25-80 depending on the parameters), the octupole significantly helps to
constrain the parameters by breaking degeneracies.

We find similar results when the priors are reduced to 20% (see Table 7 in
Appendix C). In this case the uncertainty on
$s_{\textrm{B},1},s_{\textrm{B},2},s_{\textrm{B},3}$ are reduced by a factor
$\sim 2$, for the $50\times 50$ and $70\times 30$ splits. Generally, we see
that the improvement with respect to the priors are similar for the 3 splits
we have chosen. Some parameters are better constrained in one split than in
another, but there is no systematic trend that would single out one of the
splits.

In Fig. 5, we show the constraints on the redshift evolution of the various
functions for the $50\times 50$ split: the functions entering our forecasts
($n^{\rm evol},s_{\textrm{M}}$ and $s_{\textrm{B}}$ in red), and the derived
ones ($s_{\textrm{F}},f^{\rm evol}_{\textrm{B}}$ and $f^{\rm
evol}_{\textrm{F}}$ in blue). We show the prior in grey (50% in this case), as
well as the uncertainty when only the dipole is included, and when both the
dipole and octupole are included. We see that the improvement on the redshift
evolution is significant for all functions. Adding the octupole helps
constraining all parameters, and the improvement is especially relevant for
$n^{\rm evol}$ and the two evolution biases. Similar results are found using a
20% prior instead of a 50% prior (see Fig. 9 in Appendix C) and for the other
splits (see Fig. 10 in Appendix C).

###  5.2 Constraints from the combined analysis

| $50\times 50$ | Joint | $\Delta_{m=2}(\%)$ | $30\times 70$ | Joint | $\Delta_{m=10/3}(\%)$
---|---|---|---|---|---|---
$s_{\textrm{B},0}$ | 30.02 | 23.72 | 20.96 | 32.77 | 30.89 | 5.74
$s_{\textrm{B},1}$ | 18.47 | 13.35 | 27.73 | 36.00 | 33.36 | 7.33
$s_{\textrm{B},2}$ | 25.03 | 19.53 | 21.97 | 43.48 | 42.91 | 1.33
$s_{\textrm{B},3}$ | 23.81 | 18.75 | 21.22 | 21.43 | 21.18 | 1.19
$s_{\textrm{M},0}$ | 39.29 | 33.61 | 14.45 | 37.71 | 33.61 | 10.87
$s_{\textrm{M},1}$ | 27.83 | 16.42 | 40.99 | 18.49 | 16.42 | 11.20
$s_{\textrm{M},2}$ | 45.07 | 40.30 | 10.58 | 46.46 | 40.30 | 13.25
$s_{\textrm{M},3}$ | 49.94 | 49.82 | 0.26 | 49.96 | 49.82 | 0.28
$n^{\mathrm{evol}}_{0}$ | 41.31 | 39.95 | 3.29 | 40.50 | 39.95 | 1.36
$n^{\mathrm{evol}}_{1}$ | 31.04 | 26.66 | 14.13 | 28.01 | 26.66 | 4.82
$n^{\mathrm{evol}}_{2}$ | 25.55 | 19.57 | 23.40 | 24.09 | 19.57 | 18.76
$n^{\mathrm{evol}}_{3}$ | 30.71 | 27.32 | 11.03 | 29.61 | 27.32 | 7.73
Table 2: Relative 1$\sigma$ uncertainties for the magnification biases and the number evolution, assuming $50\%$ Gaussian priors and including all multipoles. We show the results for a single split and for the joint analysis. The bold numbers are the relative improvement with respect to the results from a single split. Note that since the $s_{\textrm{B},i}$ parameters differ for the two splits, we have two sets of parameters in the joint analysis. Hence we show the constraints for the $50\times 50$ $s_{\textrm{B},i}$ parameters in column 2 and 3, and the constraints for the $30\times 70$ $s_{\textrm{B},i}$ parameters in column 5 and 6. | $50\times 50$ | $30\times 70$ | Joint | $\Delta_{m=2}(\%)$ | $\Delta_{m=10/3}(\%)$
---|---|---|---|---|---
$h$ | 1.1045 | 1.0767 | 1.0536 | 4.61 | 2.15
$\ln{\left(10^{10}\,A_{\rm s}\right)}$ | 0.7080 | 0.6890 | 0.6710 | 5.23 | 2.61
$n_{\rm s}$ | 0.9005 | 0.8715 | 0.8449 | 6.18 | 3.05
$\Omega_{\rm b}$ | 1.1296 | 1.1103 | 1.0928 | 3.26 | 1.58
$\Omega_{\rm m}$ | 0.4963 | 0.4793 | 0.4662 | 6.05 | 2.74
Table 3: Relative 1$\sigma$ uncertainties for the cosmic parameters. We show
the results for two individual splits and for the joint analysis. The bold
numbers show the relative improvement with respect to a single split.

In Table 2 we show the constraints obtained when combining the $50\times 50$
split with the $30\times 70$ split and we compare them with the constraints
from a single split. The improvement is non-negligible for most parameters,
and can even reach $20-40\%$. Hence even if adding a new split adds 4 new
parameters to the Fisher (due to the magnification bias, which differs in
different splits), the common parameters $s_{\textrm{M},i}$ and $n^{\rm
evol}_{i}$ are improved by the additional signals. Interestingly, this
improvement in the common parameters also impacts the individual
$s_{\textrm{B},i}$ that are better measured due to the breaking of
degeneracies brought by the additional split.

In Fig. 6 we show the constraints on the redshift evolution of the functions
entering the forecasts and in Fig. 7 on the derived functions. In general, we
see that the improvement is less significant in the case of the $30\times 70$
split, but they are still non-negligible, especially for the evolution biases.
Note that the wiggles in the redshift evolution of the evolution biases are
due to the reconstruction of these functions from $n^{\rm
evol},s_{\textrm{B}}$ and $s_{\textrm{F}}$.

Finally, in Table 3 we show the constraints on the cosmological parameters. We
find that these parameters are almost purely constrained by the even
multipoles. Adding the odd multipoles only improves the constraints by 0.1% at
most. Note that this is consistent with the results of [10], who showed that
relativistic effects are not useful to constrain cosmological parameters.
Their capacity lies instead in constraining theories beyond $\Lambda$CDM [59,
60, 55, 61, 62, 63, 54], or magnification biases and evolution biases, as we
show here.

We compare the constraints on cosmological parameters using a single split and
using the joint analysis. First we see that the constraints from the $30\times
70$ split are always tighter than those from the $50\times 50$ split. This
suggests that multi-tracer analyses should carefully explore various splits
and determine the optimal one. The $50\times 50$ split is clearly the one
which minimises shot noise in the cross-correlation signals (since one can
form more pairs in this case), but since shot noise will be small in future
surveys due to the very high number density of detected galaxies, this is not
necessarily the optimal choice. The signal may indeed increase in other
choices of splits, improving cosmological constraints. Note that these
conclusions rely on our modelling of the galaxy biases, and may differ for a
different modelling. Finally, we see that the joint analysis provides tighter
constraints than the two single splits. The improvement with respect to the
$30\times 70$ split lies between 1.5% and 3%. This is a somewhat limited
improvement, but it comes at almost no cost. It does not require new data or
different analysis techniques, but simply relies on doubling the size of the
data vector by considering two types of cross-correlation.

Figure 6:  Constraints on the redshift evolution of the functions entering the
forecasts, assuming a 50% prior on the parameters. We show the results for a
single split ($50\times 50$ on the left, $30\times 70$ on the right) and for
the joint analysis. Note that $s_{\textrm{B}}$ is different in the two splits.
In all cases all multipoles are included.

Figure 7: Constraints on the redshift evolution of the derived functions,
assuming a 50% prior on the parameters. We show the results for a single split
($50\times 50$ on the left, $30\times 70$ on the right) and for the joint
analysis. Note that the three functions are different in the two splits. In
all cases all multipoles are included.

##  6 Conclusions

In this work, we showed that exploiting the information carried by the odd
multipoles of the 2-point correlation function can significantly improve our
knowledge of the astrophysical properties of galaxies. We developed a
physically motivated model for the magnification biases and evolution biases
that govern the amplitude of the odd multipoles. These biases are directly
related to the distribution in luminosity and the redshift evolution of
galaxies and can therefore be used to reconstruct the luminosity function of
galaxies. We found that:

  * •

The magnification biases and evolution biases of the various samples can all
be related to three observable quantities: the magnification bias of the whole
(un-split) population $s_{\textrm{M}}$, the magnification bias of the bright
sample $s_{\textrm{B}}$, and the redshift evolution of the number of galaxies
$n^{\mathrm{evol}}$.

  * •

These three quantities cannot be measured from the odd multipoles if we have
no prior knowledge about them, since they are fully degenerate with each
other. However, parametrising their redshift evolution and assuming a prior of
50% on the parameters, the odd multipoles provide constraints that are tighter
than the prior by a factor of up to $2-3$.

  * •

Combining the dipole and the octupole is crucial to obtain tight constraints,
since they do not suffer from the same degeneracies between parameters.

  * •

Splitting the galaxies in different ways provides different constraints, but
there is no significant trend that would single out one split as optimal.

  * •

Combining two different splits further improves the constraints by up to 20%
on the biases and by $\sim 3\%$ on the cosmological parameters $n_{\rm s}$ and
$\Omega_{\rm m}$.

Knowing the astrophysical properties of galaxies is necessary to robustly
constrain local primordial non-Gaussianities from large-scale structure, since
these are contaminated by relativistic effects at very large scales. Since
these effects directly depend on the magnification and evolution biases,
having precise measurements of these quantities is essential. Interestingly,
one can use the odd multipoles to constrain these quantities at intermediate
scales, where local primordial non-Gaussianities are irrelevant, and from this
one can predict the contamination from relativistic effects to the even
multipoles at very large scales, where primordial non-Gaussianities are
measured.

Finally, a good knowledge of the magnification and evolution biases is also
essential to infer the observer velocity from the kinematic dipole, since they
enter as a pre-factor in front of the observer velocity and are therefore
fully degenerate with it. An accurate theoretical model of the kinematic
dipole is critical for robust tests of the Cosmological Principle.

## Acknowledgments

C.B. and D.S.B. acknowledge support from the Swiss National Science
Foundation. C.B. also acknowledges support from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(grant agreement No. 863929; project title “Testing the law of gravity with
novel large-scale structure observables”). R.M. is supported by the South
African Radio Astronomy Observatory and the National Research Foundation
(grant no. 75415).

##  Appendix A Modeling the shot-noise for overlapping populations

In this Appendix, we introduce a model describing the shot-noise contributions
due to the overlap between two populations of galaxies. One usually assumes
that, once the population is split into B and F galaxies, the populations are
independent and therefore there is no shot noise contribution in the cross-
correlation. This is appropriate for a single split, but it is inconsistent
when combining two splits with two different values $m$ and $m^{\prime}$ into
the same analysis pipeline. In this case, there is automatically an overlap
between the populations of the two different splits, as depicted in Fig. 8.
This leads to a non-zero shot noise contribution between the splits.

BFbf Figure 8: Schematic representation of the overlap between two different
splits. In the $m$ split we have more bright galaxies than in the $m^{\prime}$
split, implying that some of the bright galaxies in the former are actually
faint galaxies in the second case.

Let us denote by B and F the bright and faint samples of the first split $m$,
and by b and f those of the second split $m^{\prime}$. From Fig. 8 we see that
some of the B galaxies are the same as some of the f galaxies. Moreover, all b
galaxies are included in the B sample and all F galaxies are included in the f
sample. Mathematically speaking, we can generally define the number density of
each population of the $m^{\prime}$ split as

$\displaystyle\bar{n}_{\mathrm{b}}=\theta_{\mathrm{Bb}}\,\bar{n}_{\mathrm{B}}+\theta_{\mathrm{Fb}}\,\bar{n}_{\mathrm{F}},$
(A.1)
$\displaystyle\bar{n}_{\mathrm{f}}=\theta_{\mathrm{Bf}}\,\bar{n}_{\mathrm{B}}+\theta_{\mathrm{Ff}}\,\bar{n}_{\mathrm{F}},$
(A.2)

where the $\theta$ parameters are numbers between $0$ and $1$ quantifying the
overlap between the populations of each split. We need to find the specific
values for each $m$ and $m^{\prime}$ sub-samples combinations. In particular,
for the case of a joint analysis combining the $50\times 50$ split with the
$30\times 70$ split, we find:

$\theta_{\mathrm{\textrm{B}\textrm{b}}}=\frac{3}{5},\quad\theta_{\mathrm{\textrm{F}\textrm{b}}}=0,\quad\theta_{\mathrm{\textrm{B}\textrm{f}}}=\frac{2}{5},\quad\theta_{\mathrm{\textrm{F}\textrm{f}}}=1.$
(A.3)

We immediately note that there is no overlap between the b and the F
populations.

In general, the fluctuations in the galaxy number count cross-correlations
between the populations of each split have the form

$\langle\Delta_{\textrm{L}}(x_{i})\,\Delta_{\lambda}(x_{j})\rangle=C_{ij}^{\textrm{L}\lambda}+\frac{\theta_{\textrm{L}\lambda}}{\bar{n}_{\lambda}}\,\delta_{ij},$
(A.4)

where $L$, $\lambda$ label the galaxies luminosities in each split and $i$,
$j$ label the pixel sky positions. We have denoted by $\bar{n}_{\lambda}$ the
background number density. The first term represents the cosmic variance
contribution, which is non-zero even if the populations do not overlap, since
they trace the same underlying matter density field. The second term encodes
the shot-noise contribution, which depends directly on the overlapping faction
$\theta_{\mathrm{L}\lambda}$: the less the populations overlap with each
other, the smaller the shot-noise contribution is.

Considering two multipoles of orders $\ell$ and $\ell^{\prime}$, the $m\times
m^{\prime}$ cross-variances will have the form

$\mathrm{COV}\left(\xi^{(\ell)}_{\mathrm{LK}}(z,d),\,\xi^{(\ell^{\prime})}_{\lambda\kappa}(z,d^{\prime})\right)=\mathrm{COV}_{\mathrm{CC}}+\mathrm{COV}_{\mathrm{CP}}+\mathrm{COV}_{\mathrm{P}}\,,$
(A.5)

where L,K $\in\\{\textrm{B},\textrm{F}\\}$ and $\lambda,\kappa\in\\{{\rm
b},{\rm f}\\}$. The last term contains the pure shot-noise contributions. For
the dipole and octupole, the only non-vanishing contribution is when ${\rm
L,K}=\textrm{B},\textrm{F}$ and $\lambda,\kappa={\rm b,f}$ (in all other
cases, the signal itself vanishes) and it reads

$\displaystyle\mathrm{COV}_{\mathrm{P}}\left(\xi^{(\ell)}_{\textrm{B}\textrm{F}}(z,d),\,\xi^{(\ell^{\prime})}_{\textrm{b}\textrm{f}}(z,d^{\prime})\right)=\frac{2\ell+1}{4\pi\,V\bar{N}^{2}_{\rm
tot}l_{\rm
p}}\frac{\delta_{d\,d^{\prime}}\,\delta_{\ell\ell^{\prime}}}{d^{2}}\frac{\theta_{\textrm{B}\textrm{b}}\,\theta_{\textrm{F}\textrm{f}}}{q_{\textrm{b}}\,q_{\textrm{f}}}\,\quad\mbox{for}\quad\ell,\ell^{\prime}=1,3\,.$
(A.6)

For the even multipoles however, there are also other non-vanishing
correlations and one obtains

$\displaystyle\mathrm{COV}_{\mathrm{P}}\left(\xi^{(\ell)}_{\mathrm{LK}}(z,d),\,\xi^{(\ell^{\prime})}_{\lambda\kappa}(z,d^{\prime})\right)=\frac{2\ell+1}{4\pi\,V\bar{N}^{2}_{\rm
tot}l_{\rm
p}}\frac{\delta_{d\,d^{\prime}}\,\delta_{\ell\ell^{\prime}}}{d^{2}}\frac{1}{q_{\lambda}\,q_{\kappa}}\Big{(}\theta_{\textrm{L}\lambda}\,\theta_{\textrm{K}\kappa}+\theta_{\textrm{L}\kappa}\,\theta_{\textrm{K}\lambda}\Big{)}\,$
(A.7)
$\displaystyle\quad\mbox{for}\quad\ell,\ell^{\prime}=0,2,4\quad\mbox{and}\quad{\rm
L,K}\in\\{\textrm{B},\textrm{F}\\},\lambda,\kappa\in\\{{\rm b},{\rm f}\\}\,.$

Here $V$ is the total volume of the survey, $\bar{N}_{\rm tot}$ is the
background total number density of galaxies, $l_{\rm p}$ is the pixel size and
$q_{\lambda}=\bar{N}_{\lambda}/\bar{N}_{g}$ the fraction of galaxies of each
type. In addition, there are also non-zero mixed (CP) contributions:

$\displaystyle\mathrm{COV}_{\mathrm{CP}}(\xi^{(\ell)}_{\mathrm{LK}}(z,d),\,\xi^{(\ell^{\prime})}_{\lambda\kappa}(z,d^{\prime}))=\left(\frac{D_{1}(z)}{D_{1}(0)}\right)^{2}\frac{1}{V\bar{N}_{\rm
tot}}\,(-1)^{\frac{\ell-\ell^{\prime}}{2}}(-1)^{\,\ell+\ell^{\prime}}$
$\displaystyle\quad\times\frac{1}{8}\sum_{a}\Bigg{[}\frac{\theta_{\textrm{K}\kappa}}{q_{\kappa}}\,\alpha_{a}^{\textrm{L}\lambda}+(-1)^{\ell+\ell^{\prime}}\frac{\theta_{\textrm{L}\lambda}}{q_{\lambda}}\,\alpha_{a}^{\textrm{K}\kappa}+(-1)^{\ell^{\prime}}\left(\frac{\theta_{\textrm{K}\lambda}}{q_{\lambda}}\,\alpha_{a}^{\textrm{L}\kappa}+(-1)^{\ell+\ell^{\prime}}\frac{\theta_{\textrm{L}\kappa}}{q_{\kappa}}\,\alpha_{a}^{\textrm{K}\lambda}\right)\Bigg{]}$
$\displaystyle\quad\times\mathcal{G}(\ell,\ell^{\prime},a)\,\mathcal{I}(\ell,\ell^{\prime},d,d^{\prime})\,,$
(A.8)

where $D_{1}(z)$ is the growth function at redshift $z$ and the index $a$ runs
over the values $0$, $2$, $4$. We have defined a set of three functions of the
growth rate and the galaxy biases, denoted by $\alpha_{a}^{\mathrm{LK}}$. They
take the form

$\displaystyle\alpha_{0}^{\mathrm{LK}}=b_{\textrm{L}}\,b_{\textrm{K}}+\frac{f}{3}\,(b_{\textrm{L}}+b_{\textrm{K}})+\frac{f^{2}}{5}\,,$
(A.9)
$\displaystyle\alpha_{2}^{\mathrm{LK}}=\frac{2}{3}\,(b_{\textrm{L}}+b_{\textrm{K}})\,f+\frac{4}{7}\,f^{2}\,,$
(A.10) $\displaystyle\alpha_{4}^{\mathrm{LK}}=\frac{8}{35}\,f^{2}\,.$ (A.11)

In addition, we also have

$\mathcal{G}(\ell,\ell^{\prime},a)=\int^{1}_{-1}d\mu\,P_{\ell}(\mu)\,P_{\ell^{\prime}}(\mu)\,P_{a}(\mu)\,,$
(A.12)

where $\mu=\cos{\beta}$ is the angle formed by the direction of the incoming
photons and the orientation of a pair of galaxies. Finally, we have the
Fourier-Bessel transformations of the power spectrum at $z=0$

$\displaystyle\mathcal{I}(\ell,\ell^{\prime},d,d^{\prime})=\frac{2(2\ell+1)(2\ell^{\prime}+1)}{\pi^{2}}\int\;dk\,k^{2}\;P_{\delta\delta}(k,z=0)\,j_{\ell}(kd)\;j_{\ell^{\prime}}(kd^{\prime})\,.$
(A.13)

Finally, in order to compute the CC contributions, we can use the usual
expressions, replacing the corresponding values for the galaxy biases of each
$m$, $m^{\prime}$ sample [33, 70]. The pure relativistic contributions to the
cosmic variance relevant to the dipole and octupole covariances have been
computed in [53] and are also taken into account in this work.

##  Appendix B Covariance of the total hexadecapole with the monopole and
quadrupole of the bright and faint populations

The hexadecapole of the $2$-point correlation function is independent of the
galaxy bias (see Eq. (2.10)) and it is therefore the same for each of the
populations. As a consequence we only consider the hexadecapole of the whole
population in our analysis. Since the whole population overlaps with the B and
F populations, there is a non-zero shot noise contribution in the covariance
of the hexadecapole with the monopole and quadrupole of the B and F
populations. This can be computed using the formalism described in Appendix B,
where we cross-correlate three populations instead of four (as in Fig. 8). In
this case, we have

$\bar{n}_{\mathrm{T}}=\theta_{\mathrm{TB}}\,\bar{n}_{\mathrm{B}}+\theta_{\mathrm{TF}}\,\bar{n}_{\mathrm{F}},$
(B.1)

where $\theta_{\mathrm{TB}}=\theta_{\mathrm{TF}}=1$. The expressions are
formally the same as Eqs. (A.7)–(A.8). We use $b_{\textrm{B}}$,
$b_{\textrm{F}}$ for the monopoles and quadrupoles ($\ell=0,\,2$) and
$b_{\mathrm{T}}$ for the hexadecapole ($\ell^{\prime}=4$). In this case the P
and CP contributions are only sensitive to the total number density of
galaxies, since the overlap of the B and F populations with the full
population is total.

##  Appendix C Additional figures and tables

In this Appendix, we provide additional figures and tables associated with the
discussion and results developed in this work.

In Table 4 we show the cumulative SNR over all redshifts and separations for
each of the multipoles and three different values of the sample separation
parameter, $m=10/3,\,2.0,\,10/7$.

Tables 5 and 6 contain the fiducial values for all free parameters considered
in the analyses. In Table 5 we show the parameters that differ for each
splits, whereas in Table 6 we show the parameters that do not depend on the
split.

In Table 7 we show the constraints on the various parameters starting with a
$20\%$ prior, for three different splits.

Finally, we include additional plots showing the redshift evolution of the
parameters ($s_{\textrm{B}}$, $s_{\textrm{M}}$, $n^{\mathrm{evol}}$) and the
inferred quantities ($s_{\textrm{F}}$, $f^{\rm evol}_{\textrm{B}}$, $f^{\rm
evol}_{\textrm{F}}$), together with the predicted $68\%$ confidence regions.
In Fig. 9 we show the results for the $50\times 50$ split when assuming a
$20\%$ prior. In Fig. 10 we show the predictions for the $30\times 70$ split
when assuming a $50\%$ prior.

| $\xi^{(0)}_{\textrm{B}\textrm{B}}$ | $\xi^{(0)}_{\textrm{B}\textrm{F}}$ | $\xi^{(0)}_{\textrm{F}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(2)}_{\textrm{B}\textrm{B}}$ | $\xi^{(2)}_{\textrm{B}\textrm{F}}$ | $\xi^{(2)}_{\textrm{F}\textrm{F}}$ | $\xi^{(3)}_{\textrm{B}\textrm{F}}$ | $\xi^{(4)}_{\rm T}$
---|---|---|---|---|---|---|---|---|---
$30\times 70$ | 457.07 | 474.80 | 422.31 | 62.32 | 313.42 | 479.35 | 546.15 | 7.33 |
$50\times 50$ | 521.12 | 426.13 | 301.99 | 55.47 | 391.46 | 522.47 | 489.21 | 4.78 | 74.50
$70\times 30$ | 671.17 | 571.24 | 395.93 | 51.87 | 497.11 | 601.89 | 543.65 | 6.45 |
Table 4: Cumulative SNR over all redshifts and separations for the multipoles considered in the analysis. We show the SNR for three different splits. For the hexadecapole we show only one value corresponding to the whole population. | $30\times 70$ | $50\times 50$ | $70\times 30$
---|---|---|---
$s_{\textrm{B},0}$ | 0.3252 | -0.9011 | -0.9337
$s_{\textrm{B},1}$ | 0.6442 | 1.8230 | 1.8243
$s_{\textrm{B},2}$ | 0.1614 | -1.0188 | -0.9284
$s_{\textrm{B},3}$ | 0.1333 | -0.3034 | -0.2410
Table 5: Fiducial values of the $s_{\textrm{B}}(z)$ fitting parameters. These are generally different for different splits. $s_{\textrm{M},0}$ | -0.1938
---|---
$s_{\textrm{M},1}$ | 1.0747
$s_{\textrm{M},2}$ | -0.0781
$s_{\textrm{M},3}$ | -0.0056
$n^{\mathrm{evol}}_{0}$ | 7.9460
$n^{\mathrm{evol}}_{1}$ | -12.0375
$n^{\mathrm{evol}}_{2}$ | 2.2993
$n^{\mathrm{evol}}_{3}$ | -15.8389
$b_{\textrm{B},1}$ | 0.5540
$b_{\textrm{B},2}$ | 0.7830
$b_{\textrm{F},1}$ | 0.5540
$b_{\textrm{F},2}$ | 0.7830
$h$ | 0.6766
$\ln{(10^{10}A_{\rm s})}$ | 3.0204
$n_{\rm s}$ | 0.9665
$\Omega_{\rm b}$ | 0.0490
$\Omega_{\rm m}$ | 0.3111
Table 6: Fiducial values for the parameters varied in the analysis that are
common to all splits.

|  $50\times 50$ split
---|---
| $\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$
$s_{\textrm{B},0}$ | 15.03 | 13.83
$s_{\textrm{B},1}$ | 10.36 | 9.69
$s_{\textrm{B},2}$ | 10.72 | 10.60
$s_{\textrm{B},3}$ | 11.98 | 11.20
$s_{\textrm{M},0}$ | 19.18 | 17.16
$s_{\textrm{M},1}$ | 14.39 | 13.16
$s_{\textrm{M},2}$ | 19.73 | 18.81
$s_{\textrm{M},3}$ | 19.99 | 19.99
$n^{\mathrm{evol}}_{0}$ | 18.03 | 17.17
$n^{\mathrm{evol}}_{1}$ | 18.42 | 15.00
$n^{\mathrm{evol}}_{2}$ | 16.48 | 14.42
$n^{\mathrm{evol}}_{3}$ | 14.31 | 13.92

$30\times 70$ split
---
$\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$
17.45 | 16.25
17.60 | 16.93
17.93 | 17.73
12.36 | 11.75
19.31 | 17.47
8.86 | 8.37
19.83 | 19.29
19.99 | 19.99
16.85 | 16.65
16.62 | 14.49
14.73 | 13.45
13.44 | 13.29

$70\times 30$ split
---
$\xi^{(1)}_{\textrm{B}\textrm{F}}$ | $\xi^{(1)}_{\textrm{B}\textrm{F}}+\xi^{(3)}_{\textrm{B}\textrm{F}}$
12.50 | 11.71
9.28 | 8.95
10.49 | 10.33
13.11 | 12.60
18.51 | 16.29
14.73 | 13.89
19.28 | 17.76
19.98 | 19.96
18.63 | 17.35
19.36 | 16.14
15.49 | 14.45
15.52 | 14.55

Table 7: Relative 1$\sigma$ uncertainties for the magnification biases and
number evolution parameters, assuming $20\%$ Gaussian priors. We show the
results for 3 different splits, including only the dipole ($\xi_{1}$) and
including both the dipole and octupole ($\xi_{1}+\xi_{3}$). In all cases the
even multipoles (monopole, quadrupole and hexadecapole) are included.

Figure 9: Constraints on the redshift evolution of the various functions for
the 50$\times$50 split, starting with a 20% prior on the parameters (grey
region). We show the results when only the dipole is used and when both the
dipole and octupole are included. In all cases the even multipoles (monopole,
quadrupole and hexadecapole) are included. The red functions are those
entering the forecasts, while the blue ones are derived from the red.

Figure 10: Constraints on the redshift evolution of the various functions for
the 30$\times$70 split, starting with a 50% prior on the parameters (grey
region). We show the results when only the dipole is used and when both the
dipole and octupole are included. In all cases the even multipoles (monopole,
quadrupole and hexadecapole) are included. The red functions are those
entering the forecasts, while the blue ones are derived from the red.

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