# Forecasting the potential of weak lensing magnification to enhance LSST
large-scale structure analyses

Constance Mahony1,2, Maria Cristina Fortuna3, Benjamin Joachimi1, Andreas
Korn1, Henk Hoekstra3, Samuel J. Schmidt4 and the LSST Dark Energy Science
Collaboration
1Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, UK
2Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical
Institute (AIRUB), German Centre for Cosmological Lensing,
44780 Bochum, Germany
3Leiden Observatory, Leiden University, PO Box 9513, Leiden, NL-2300 RA, The
Netherlands
4Department of Physics, University of California, One Shields Ave., Davis, CA
95616, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Recent works have shown that weak lensing magnification must be included in
upcoming large-scale structure analyses, such as for the Vera C. Rubin
Observatory Legacy Survey of Space and Time (LSST), to avoid biasing the
cosmological results. In this work we investigate whether including
magnification has a positive impact on the precision of the cosmological
constraints, as well as being necessary to avoid bias. We forecast this using
an LSST mock catalog and a halo model to calculate the galaxy power spectra.
We find that including magnification has little effect on the precision of the
cosmological parameter constraints for an LSST galaxy clustering analysis,
where the halo model parameters are additionally constrained by the galaxy
luminosity function. In particular, we find that for the LSST gold sample
($i<25.3$) including weak lensing magnification only improves the galaxy
clustering constraint on $\Omega_{\rm{m}}$ by a factor of 1.03, and when using
a very deep LSST mock sample ($i<26.5$) by a factor of 1.3. Since
magnification predominantly contributes to the clustering measurement and
provides similar information to that of cosmic shear, this improvement would
be reduced for a combined galaxy clustering and shear analysis. We also
confirm that not modelling weak lensing magnification will catastrophically
bias the cosmological results from LSST. Magnification must therefore be
included in LSST large-scale structure analyses even though it does not
significantly enhance the precision of the cosmological constraints.

###### keywords:

gravitational lensing: weak – cosmological parameters – large-scale structure
of Universe – methods: analytical – methods: statistical

††pubyear: 2021††pagerange: Forecasting the potential of weak lensing
magnification to enhance LSST large-scale structure analyses–B

##  1 Introduction

As light from distant galaxies travels towards telescopes it is deflected
gravitationally by intervening matter. This means that galaxy images appear
distorted. On average, the distortions to individual galaxy images are very
small, but when combined, they can be used to statistically map the matter
distribution in the universe. This technique is called weak gravitational
lensing.

Weak gravitational lensing distorts both the shape and size of galaxy images.
Statistical measurements of the shape distortions are referred to as cosmic
shear, and statistical measurements of the size distortions are referred to as
magnification. Making a magnification measurement of the matter distribution
in the Universe, which directly uses size information is challenging because
there is a large intrinsic variation in the sizes of galaxies, and it is more
prone to serious systematics (Hoekstra et al., 2017). However, Schmidt et al.
(2011) achieved a simplified magnification measurement using the joint
distribution of galaxy sizes and magnitudes, and there are developing
techniques which anchor the size distribution using the fundamental plane of
galaxies (Huff & Graves, 2013; Freudenburg et al., 2020). Most magnification
analyses therefore focus on making a magnification measurement using galaxy
number density information (Scranton et al., 2005; Myers et al., 2005;
Hildebrandt et al., 2009). In a flux limited survey, distortions to the sizes
of galaxy images affect the observed number density of galaxies for two
reasons:

  1. 1.

Since surface brightness is conserved by lensing if the observed size of a
galaxy is increased, so is its observed flux. This means that galaxies
previously too faint to be observed by a galaxy survey become observable. The
number density of galaxies is increased.

  2. 2.

It is not only the observed size of individual galaxies that is increased by
magnification, but the observed size of the whole patch of sky behind the
lens. This means that the observable separation between galaxies behind the
lens increases and there is a dilution in the number density of galaxies.

These two effects compete and contribute to an overall fluctuation in the
number density of galaxies, as a result of weak lensing magnification (for the
associated equations see section 3.1). Here we are concerned with how
magnification can probe the total matter distribution, but it can also be used
to constrain the mass of galaxy clusters (e.g. Tudorica et al. 2017).

Weak lensing using cosmic shear has been a highly successful technique. In
recent years there have been increasingly precise results using cosmic shear
from galaxy surveys such as the Kilo Degree Survey (KiDS) (van Uitert et al.,
2018; Joudaki et al., 2018; Hildebrandt et al., 2020; Asgari et al., 2021),
the Dark Energy Survey (DES) (Abbott et al., 2018; Troxel et al., 2018; Amon
et al., 2021; Secco et al., 2021) and the Hyper Suprime-Cam Survey (HSC)
(Hikage et al., 2019). Weak lensing magnification has not been included in
standard weak lensing analyses to date. All that has been included is the
sensitivity of results to including a simplified magnification model (Abbott
et al., 2019). The reasoning is that magnification provides similar
information to that of cosmic shear and has a poorer signal-to-noise ratio
(Bartelmann, 2010). However, due to improvements in statistical precision,
recent works have shown that cosmological results from upcoming surveys such
as the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) and
Euclid will be biased if the effects of weak lensing magnification are not
included (Duncan et al., 2014; Cardona et al., 2016; Lorenz et al., 2018;
Thiele et al., 2020).

These works have shown that magnification must be included in future surveys
to avoid bias, but the aim of this work is to determine whether including
magnification as a complementary probe can also improve the final precision of
the LSST weak lensing results. Duncan et al. (2014) and Lorenz et al. (2018)
found no increase in precision from including magnification in a weak lensing
analysis, however LSST is a special case, because it is a very deep ground
based galaxy survey. This means that there will be a lot of very faint, small
and distant galaxies, which will be poorly resolved. It will therefore not be
possible to measure the shape of these galaxies, but it may be possible to
count them for a weak lensing magnification analysis. This means that the
potentially usable sample size for weak lensing magnification is significantly
larger than that for cosmic shear, and as such it is worth investigating
magnification’s potential as a complementary probe in the case of LSST
111Lorenz et al. (2018) also considered LSST specifically, but did not include
systematics or explore departures from the gold sample used for cosmic shear..
Particularly, as Nicola et al. (2020) showed that even with only approximately
100 square degrees, deep samples are already sensitive to magnification.

In summary, we wish to determine the effect of including weak lensing
magnification on the precision of the final constraints from LSST weak
lensing. We determine this using the Fisher matrix formalism introduced in
section 2. We then describe the modelling of the observables (weak lensing
power spectra and the galaxy luminosity function) in sections 3 and 4. We
describe the details of our LSST specific survey modelling in section 5; and
present our results and conclusions in sections 6 and 7. We verify the
stability of our Fisher matrices in appendix B.

##  2 Fisher Analysis

The Fisher Information matrix summarises the expected curvature of the log-
Likelihood function around its maximum,

$F_{ij}=\bigg{\langle}\frac{-\partial^{2}\ln
L}{\partial\theta_{i}\partial\theta_{j}}\bigg{\rangle}\ ,$ (1)

where $L$ is the likelihood and $\theta_{i}$ is a model parameter. If the
likelihood function is sharply peaked for a given parameter, the parameter is
tightly constrained by the data (Dodelson, 2003). The marginal uncertainty on
the model parameter $\theta_{i}$ can be calculated from the Fisher matrix as:

$\Delta\theta_{i}\geq\sqrt{(F^{-1})_{ii}}\ .$ (2)

The greater than or equal relation is in reference to the Cramér-Rao
inequality, which specifies that the Fisher matrix gives the minimum possible
uncertainty on an unbiased model parameter (Tegmark et al., 1997).

The Fisher information matrix can be calculated without data and is therefore
a useful tool for forecasting best case parameter constraints. In the case of
a Gaussian likelihood function and a parameter independent covariance matrix
the Fisher matrix is given by:

$F_{ij}=\sum_{\ell}\frac{\partial C_{\ell}}{\partial\theta_{i}}\
\mathrm{Cov}^{-1}\ \frac{\partial C_{\ell}}{\partial\theta_{j}}\ ,$ (3)

where $C$ is the theory datavector and $\mathrm{Cov}$ is the associated
covariance (Tegmark et al., 1997). In this work we consider two component
Fisher matrices, which we then add together since they concern separate
observables: the Fisher matrix where the theory datavector consists of galaxy
clustering or galaxy clustering and cosmic shear (detailed in section 3), and
the Fisher matrix where the theory datavector consists of the galaxy
luminosity function (detailed in section 4). There may be a small correlation
between the observables due to cosmic variance, but we do not consider this in
this forecast. The associated covariances are detailed in sections 5.4.1 and
5.5 respectively.

##  3 Weak Lensing Observables

The two observable quantities used in this weak lensing analysis are the
shape, often referred to as ellipticity, and the number density of galaxy
images. Since weak lensing is a local effect, the mean ellipticity $\epsilon$
and fluctuation in the number density of galaxies $n$, resulting from weak
lensing, is equal to zero when averaged over large scales in the absence of
systematics. Therefore, the key statistical quantity used in weak lensing
analyses is the two-point correlation function. There are three two-point
correlation functions commonly considered in large-scale structure and weak
lensing analyses: cosmic shear (ellipticity-ellipticity), angular galaxy
clustering (number density-number density) and galaxy-galaxy lensing (number
density-ellipticity). In this work we focus on angular galaxy clustering as an
individual probe (section 6.1 and 6.3), and also consider a combined
clustering and shear analysis, where the analyses occur on separate patches of
sky (section 6.2).

The Fourier space two-point correlation function for angular galaxy clustering
is given by:

$\langle\tilde{n}^{i}(\bm{\ell})\tilde{n}^{j}(\bm{\ell^{\prime}})\rangle=(2\pi)^{2}\delta^{(2)}(\bm{\ell}+\bm{\ell^{\prime}})C_{\mathrm{nn}}^{ij}(\ell)\
,$ (4)

where $\tilde{n}$ is the Fourier transform of the number density contrast,
$\bm{\ell}$ is the angular frequency, $\delta^{(2)}$ is the two-dimensional
Dirac delta function and $C_{\mathrm{nn}}^{ij}$ is the projected number
density power spectrum between redshift bins $i$ and $j$ (Joachimi & Bridle,
2010). It is useful to work in Fourier space because it simplifies linking to
the theory predictions. The galaxy samples used for weak lensing are often
split into redshift bins; a technique called redshift tomography. This binning
enables weak lensing to probe the evolution of the power spectrum with time,
through auto- and cross-correlations between the different redshift bins, and
hence study the expansion of the universe and dark energy.

The Fourier transform of the two-point correlation function for cosmic shear
is given by:

$\langle\tilde{\epsilon}^{i}(\bm{\ell})\tilde{\epsilon}^{j}(\bm{\ell^{\prime}})\rangle=(2\pi)^{2}\delta^{(2)}(\bm{\ell}+\bm{\ell^{\prime}})C_{\mathrm{\epsilon\epsilon}}^{ij}(\ell)\
,$ (5)

where $\tilde{\epsilon}$ is the Fourier transform of the ellipticity and
$C_{\mathrm{\epsilon\epsilon}}^{ij}$ is the projected ellipticity power
spectrum between redshift bins $i$ and $j$ .

###  3.1 2D power spectra

The key quantities in equations 4 and 5, are the two-dimensional (2D) power
spectra $C_{\mathrm{nn}}$ and $C_{\mathrm{\epsilon\epsilon}}$. These are the
observables we model and include in our Fisher matrix theory datavector, see
section 2.

In this work we model the 2D observable power spectra $C_{\mathrm{nn}}$ and
$C_{\mathrm{\epsilon\epsilon}}$ by breaking them down into their constituent
parts. The observed ellipticity of a galaxy comes from a combination of the
intrinsic ellipticity of the galaxy before it is lensed $\epsilon_{I}$ (the
intrinsic alignment, see Joachimi et al. 2015; Troxel & Ishak 2014 for
reviews), the distortion of the shape by weak lensing shear $\gamma_{G}$, and
a random uncorrelated component $\epsilon_{\rm{rnd}}$ which accounts for the
randomness in the intrinsic ellipicity of galaxies,

$\epsilon^{i}(\bm{\theta})=\gamma^{i}_{G}(\bm{\theta})+\epsilon^{i}_{I}(\bm{\theta})+\epsilon^{i}_{\rm{rnd}}(\bm{\theta})\
,$ (6)

where $i$ denotes the redshift bin. The observed number density of galaxies
comes from a combination of the number density fluctuation of galaxies as a
result of galaxy clustering $n_{g}$, the distortion to the number density from
weak lensing magnification $n_{m}$, and a random component $n_{\rm{rnd}}$
which accounts for the shot noise contribution,

$n^{i}(\bm{\theta})=n^{i}_{g}(\bm{\theta})+n^{i}_{m}(\bm{\theta})+n^{i}_{\rm{rnd}}(\bm{\theta})\
.$ (7)

In terms of the Fourier space 2D power spectra the uncorrelated random
components lead to noise power spectra, and separating out the remaining
contributions gives:

$\begin{split}C_{\mathrm{\epsilon\epsilon}}^{ij}(\ell)=C_{\mathrm{GG}}^{ij}(\ell)+C_{\mathrm{IG}}^{ij}(\ell)+C_{\mathrm{IG}}^{ji}(\ell)+C_{\mathrm{II}}^{ij}(\ell)\
,\\\ \
C_{\mathrm{nn}}^{ij}(\ell)=C_{\mathrm{gg}}^{ij}(\ell)+C_{\mathrm{gm}}^{ij}(\ell)+C_{\mathrm{gm}}^{ji}(\ell)+C_{\mathrm{mm}}^{ij}(\ell)\
,\\\ \end{split}$ (8)

where G represents ellipticity from weak lensing shear, I ellipticity from the
intrinsic alignment of galaxies, g number density fluctuations as a results of
intrinsic galaxy clustering and m number density fluctuations as a result of
weak lensing magnification.

We compute all these two-dimensional power spectra $C_{\mathrm{ab}}$ from
their associated three-dimensional power spectra $P_{\mathrm{ab}}$ using the
Limber approximation in Fourier space (Kaiser, 1992):

$\begin{split}&C_{\mathrm{GG}}^{ij}(\ell)=\int_{0}^{\chi_{\rm{hor}}}\mathrm{d}\chi\frac{q^{i}(\chi)q^{j}(\chi)}{f_{K}^{2}(\chi)}P_{\rm{\delta\delta}}\bigg{(}k=\frac{\ell+1/2}{f_{K}(\chi)},\chi\bigg{)}\
,\\\
&C_{\mathrm{IG}}^{ij}(\ell)=\int_{0}^{\chi_{\rm{hor}}}\mathrm{d}\chi\frac{p^{i}(\chi)q^{j}(\chi)}{f_{K}^{2}(\chi)}P_{\rm{I\delta}}\bigg{(}k=\frac{\ell+1/2}{f_{K}(\chi)},\chi\bigg{)}\
,\\\
&C_{\mathrm{II}}^{ij}(\ell)=\int_{0}^{\chi_{\rm{hor}}}\mathrm{d}\chi\frac{p^{i}(\chi)p^{j}(\chi)}{f_{K}^{2}(\chi)}P_{\rm{II}}\bigg{(}k=\frac{\ell+1/2}{f_{K}(\chi)},\chi\bigg{)}\
,\\\
&C_{\mathrm{gg}}^{ij}(\ell)=\int_{0}^{\chi_{\rm{hor}}}\mathrm{d}\chi\frac{p^{i}(\chi)p^{j}(\chi)}{f_{K}^{2}(\chi)}P_{\rm{gg}}\bigg{(}k=\frac{\ell+1/2}{f_{K}(\chi)},\chi\bigg{)}\
,\\\ &C_{\mathrm{gm}}^{ij}(\ell)=2(\alpha^{j}-1)C_{\mathrm{gG}}^{ij}(\ell)\
,\\\
&C_{\mathrm{mm}}^{ij}(\ell)=4(\alpha^{i}-1)(\alpha^{j}-1)C_{\mathrm{GG}}^{ij}(\ell)\
,\\\
&C_{\mathrm{gG}}^{ij}(\ell)=\int_{0}^{\chi_{\rm{hor}}}\mathrm{d}\chi\frac{p^{i}(\chi)q^{j}(\chi)}{f_{K}^{2}(\chi)}P_{\rm{g\delta}}\bigg{(}k=\frac{\ell+1/2}{f_{K}(\chi)},\chi\bigg{)}\
,\\\ \end{split}$ (9)

where $\chi$ is the comoving distance, $f_{K}(\chi)$ is the comoving angular
diameter distance and $p^{i}(\chi)$ is the probability distribution of
galaxies in redshift bin $i$. $q^{i}(\chi)$ is a weight function given by,

$q^{i}(\chi)=\frac{3H_{0}^{2}\Omega_{\rm{m}}}{2c^{2}}\frac{f_{K}(\chi)}{a(\chi)}\int_{\chi}^{\chi_{\rm{hor}}}\mathrm{d}\chi^{\prime}p^{i}(\chi^{\prime})\frac{f_{K}(\chi^{\prime}-\chi)}{f_{K}(\chi^{\prime})}\
,$ (10)

where $H_{0}$ is the Hubble constant, $\Omega_{\mathrm{m}}$ the matter density
parameter, and $a(\chi)$ the scale factor (for further details see Bartelmann
& Schneider 2001). The calculation of the three-dimensional power spectra
$P_{\mathrm{ab}}$ is detailed in the following section.

Equation (9) shows that the 2D power spectra associated with magnification
$C_{\mathrm{gm}}$ and $C_{\mathrm{mm}}$ can be computed from the 2D power
spectra associated with weak lensing shear $C_{\mathrm{gG}}$ and
$C_{\mathrm{GG}}$ using $\alpha^{i}$ the faint end slope of the number counts
in redshift bin $i$. We discuss the galaxy luminosity function in section 4
but detail the relationship between the magnification and shear power spectra
here.

As mentioned previously, weak lensing magnification contributes to
fluctuations in the number density of galaxies $n$. If the number density of
galaxies above the flux limit $f$ is $N_{0}(>f)$, magnification alters the
number density of sources as:

$N(>f)=\frac{1}{\mu}N_{0}(>f/\mu)\ ,$ (11)

where $N(>f)$ is the observed cumulative number density of sources and $\mu$
is the local magnification factor (Bartelmann & Schneider, 2001). If the
cumulative number density of galaxies is assumed to follow a power law
$N_{0}(>f)=kf^{-\alpha}$ near the flux limit of the survey then,

$N(>f)=\frac{1}{\mu}k\bigg{(}\frac{f}{\mu}\bigg{)}^{-\alpha}=N_{0}(>f)\mu^{\alpha-1},$
(12)

where $\alpha$ is equivalent to $\alpha^{i}$ mentioned in the previous
paragraph. This means the fluctuation in the observed number density of
galaxies as a result of magnification $n_{m}$ is given by,

$\begin{split}n_{m}&=\frac{N(>f)-N_{0}(>f)}{N_{0}(>f)}=\mu^{\alpha-1}-1\approx(1+2\kappa)^{\alpha-1}-1\\\
&\approx 2(\alpha-1)\kappa,\end{split}$ (13)

where the weak lensing limit $\mu\approx 1+2\kappa$ has been employed.

###  3.2 3D power spectra

The fundamental ingredient for the construction of all of the 3D power spectra
$P_{\rm{ab}}$ in eq. (9) is the matter power spectrum $P_{\rm{\delta\delta}}$.
It summarises the clustering of matter in the universe and can be derived
numerically using the Boltzmann equations and the primordial power spectrum
predicted by inflation. For the other power spectra, we can only rely on an
effective description, which we detail in this section.

In this work, we compute $P^{\mathrm{lin}}_{\rm{\delta\delta}}$ using the
Boltzmann code CAMB (Lewis et al., 2000; Howlett et al., 2012). To include
non-linear corrections we use HALOFIT (Takahashi et al., 2012). The remaining
power spectra used in this analysis are $P_{\rm{\delta I}}$, $P_{\rm{II}}$,
$P_{\rm{gg}}$ and $P_{\rm{g\delta}}$. $P_{\rm{\delta I}}$ and $P_{\rm{II}}$
are the intrinsic alignment (IA) power spectra, which encode the tendency of
galaxy shapes to point in the direction of a matter overdensity
($P_{\rm{\delta I}}$) or to have an intrinsic coherent alignment with other
galaxy shapes ($P_{\rm{II}}$). $P_{\rm{gg}}$ summarises the clustering of
galaxies, and $P_{\rm{g\delta}}$ the cross-correlations between galaxy
position and gravitational shear. $P_{\rm{g\delta}}$ is linearly related to
the galaxy-magnification power spectrum, which is the quantity of interest in
this work. We employ a halo model formalism to calculate $P_{\rm{gg}}$ and
$P_{\rm{g\delta}}$, while for the IA power spectra we use the empirical Non-
linear Linear Alignment (NLA) model (Hirata & Seljak, 2004; Bridle & King,
2007).

The halo model (e.g. Cooray & Sheth 2002) assumes that dark matter clusters
into dark matter halos and that all dark matter exists within dark matter
halos. We define dark matter haloes as spheres of average density
$\Delta\bar{\rho}_{m}$, with $\Delta=200$ and $\bar{\rho}_{m}$ as the present
day mean matter density of the Universe. Galaxies are then assumed to form
within these dark matter halos, and hence the galaxy distribution traces the
distribution of dark matter. The model relies on two ingredients, the
underlying distribution of dark matter and how galaxies populate dark matter
halos.

The dark matter distribution is summarised by: the halo mass function, which
gives the number density of dark matter halos with mass $M$ at redshift $z$;
the halo bias function, which accounts for dark matter halos being biased
tracers of the underlying dark matter distribution; and the halo density
profile, which summarises how mass is distributed within dark matter halos. In
this work we use the Tinker et al. (2010) functional forms for the halo mass
function and halo bias function, and assume that the density of dark matter
halos follows the Navarro-Frenk-White distribution (Navarro et al., 1996). To
parametrise the concentration-mass relation that enters in the NFW profile, we
follow Duffy et al. (2008). We compute the halo mass function using the
publicly available python package hmf (Murray et al., 2013; Murray et al.,
2021).

We summarise the second ingredient, how galaxies populate dark matter halos,
using the conditional luminosity function (CLF) (Yang et al., 2003; Cacciato
et al., 2013; van den Bosch et al., 2013). The CLF gives the average number of
galaxies with a luminosity $L$ between $L\pm\mathrm{d}L/2$ in a halo of mass
$M$. It is divided into two parts:

$\Phi(L|M)=\Phi_{\rm{c}}(L|M)+\Phi_{\rm{s}}(L|M)\ ,$ (14)

where $\Phi_{\rm{c}}(L|M)$ is the CLF for central galaxies and
$\Phi_{\rm{s}}(L|M)$ is the CLF for satellite galaxies. Central galaxies
reside at the centre of dark matter halos and satellite galaxies orbit around
them. Following the approach detailed in Cacciato et al. (2013) we take the
CLF of central galaxies to be modelled by a lognormal distribution,

$\Phi_{\rm{c}}(L|M)\mathrm{d}L=\frac{\log\mathrm{e}}{\sqrt{2\pi}\sigma_{\rm{c}}}\exp\bigg{[}-\frac{(\log
L-\log L_{\rm{c}})^{2}}{2\sigma_{\rm{c}}^{2}}\bigg{]}\frac{\mathrm{d}L}{L}\ ,$
(15)

where $\sigma_{\rm{c}}$ represents the scatter in the log luminosity of
central galaxies and $L_{\rm{c}}$ is parametrised as:

$L_{\rm{c}}(M)=L_{0}\frac{(M/M_{\rm{1}})^{\gamma_{1}}}{[1+(M/M_{\rm{1}})]^{\gamma_{1}-\gamma_{2}}}\
.$ (16)

$L_{0}=2^{\gamma_{1}-\gamma_{2}}L_{\rm{c}}(M_{1})$ is a normalisation and
$M_{1}$ is a characteristic mass scale. The CLF of satellite galaxies is
modelled by a modified Schechter function,

$\Phi_{\rm{s}}(L|M)\mathrm{d}L=\phi_{\rm{s}}^{*}\bigg{(}\frac{L}{L_{\rm{s}}^{*}}\bigg{)}^{\alpha_{\rm{s}}+1}\exp\bigg{[}-\bigg{(}\frac{L}{L^{*}_{\rm{s}}}\bigg{)}^{2}\bigg{]}\frac{\mathrm{d}L}{L}\
.$ (17)

where $\alpha_{\rm{s}}$ is the faint end slope of the satellite luminosity
function. $\phi_{\rm{s}}^{*}$ is parametrised as:

$\log[\phi_{\rm{s}}^{*}(M)]=b_{0}+b_{1}(\log M_{12})+b_{2}(\log M_{12})^{2},$
(18)

where $M_{12}=M/(10^{12}h^{-1}M_{\odot})$ and $L_{\rm{s}}^{*}$ is parametrised
as:

$L_{\rm{s}}^{*}(M)=0.562L_{\rm{c}}(M)\ .$ (19)

Both of the functional forms in eq. (15) and (17) are derived from the SDSS
galaxy group catalog in Yang et al. (2008). In total we have 9 free parameters
in our CLF model: $\log M_{1}$, $\log L_{0}$, $\gamma_{1}$, $\gamma_{2}$,
$\sigma_{\rm{c}}$, $\alpha_{\rm{s}}$, $b_{0}$, $b_{1}$ and $b_{2}$. We include
all of these parameters in our Fisher matrix.

The Halo Occupation Distribution (HOD) can then be obtained as the integral of
the CLF over the luminosity interval $[L_{1},L_{2}]$:

$\langle
N_{\rm{x}}|M\rangle=\int^{L_{2}}_{L_{1}}\Phi_{\rm{x}}(L|M)\mathrm{d}L\ ,$ (20)

where x can be c, s or g=c+s; $\langle N_{\rm{c}}|M\rangle$ and $\langle
N_{\rm{s}}|M\rangle$ are the average number of central and satellite galaxies
in a halo of mass $M$ within the luminosity interval $[L_{1},L_{2}]$.
Similarly, we can write $\bar{n}_{\rm{g}}$ as the average number density of
galaxies across all halo masses in a given luminosity interval:

$\bar{n}_{\rm{g}}(z)=\int\langle N_{\rm{g}}|M\rangle n(M,z)\mathrm{d}M\ ,$
(21)

where $n(M,z)$ is the halo mass function mentioned above. To keep the notation
compact, we have omitted the redshift dependence of the HOD: it arises as a
consequence of the survey flux-limit: in this case, the luminosity limits
$L_{1}$ and $L_{2}$ in eq. (20) depend on the specific redshift bin under
consideration.

Once we have defined the HOD, we can calculate the 3D power spectra
$P_{\rm{gg}}$ and $P_{\rm{g\delta}}$. First, the power spectra can be split
into contributions from the one-halo (1h) and two-halo (2h) terms. The 1h term
describes the clustering of galaxies on small scales within the same dark
matter halo and the 2h term describes the clustering of galaxies on large
scales between different halos. These contributions can then be split into the
contributions from central c and satellite s galaxies, as with the CLF. This
gives:

$\begin{split}&P_{\rm{gg}}=2P_{\rm{cs}}^{\rm{1h}}+P_{\rm{ss}}^{\rm{1h}}+P_{\rm{cc}}^{\rm{2h}}+2P_{\rm{cs}}^{\rm{2h}}+P_{\rm{ss}}^{\rm{2h}}\
,\\\
&P_{\rm{g\delta}}=P_{\rm{c\delta}}^{\rm{1h}}+P_{\rm{s\delta}}^{\rm{1h}}+P_{\rm{c\delta}}^{\rm{2h}}+P_{\rm{s\delta}}^{\rm{2h}}\
.\end{split}$ (22)

As shown in van den Bosch et al. (2013) these contributions can be calculated
using,

$\begin{split}P_{\rm{xy}}^{\rm{1h}}(k,z)=&\int\mathcal{H}_{\rm{x}}(k,M,z)\mathcal{H}_{\rm{y}}(k,M,z)n(M,z)\mathrm{d}M\
,\\\
P_{\rm{xy}}^{\rm{2h}}(k,z)=&P^{\mathrm{lin}}_{\delta\delta}(k,z)\int\mathrm{d}M_{1}\mathcal{H}_{\rm{x}}(k,M_{1},z)n(M_{1},z)b(M_{1},z)\\\
&\times\int\mathrm{d}M_{2}\mathcal{H}_{\rm{y}}(k,M_{2},z)n(M_{2},z)b(M_{2},z)\
,\end{split}$ (23)

where x and y can be c, s or $\delta$, and $b(M,z)$ is the halo bias. The
function $\mathcal{H}$ encodes the matter or galaxy contribution:

$\begin{split}&\mathcal{H}_{\rm{\delta}}(k,M,z)=\frac{M}{\bar{\rho}_{\rm{m}}}\tilde{u}_{\rm{h}}(k|M,z)\
,\\\ &\mathcal{H}_{\rm{c}}(k,M,z)=\mathcal{H}_{\rm{c}}(M,z)=\frac{\langle
N_{\rm{c}}|M\rangle}{\bar{n}_{\rm{g}}(z)}\ ,\\\
&\mathcal{H}_{\rm{s}}(k,M,z)=\frac{\langle
N_{\rm{s}}|M\rangle}{\bar{n}_{\rm{g}}(z)}\tilde{u}_{\rm{s}}(k|M,z)\
.\end{split}$ (24)

where $\tilde{u}_{\rm{h}}$ is the Fourier transform of the normalised density
distribution of dark matter in a halo of mass $M$ (mentioned above), and
$\tilde{u}_{\rm{s}}$ is the normalised number density distribution of
satellite galaxies in a halo of mass $M$. In his work, we assume satellites to
follow the spatial distribution of the underlying dark matter, i.e.
$\tilde{u}_{\rm{s}}\equiv\tilde{u}_{\rm{h}}$.

To calculate the 3D power spectra $P_{\rm{II}}$ and $P_{\rm{I\delta}}$ we
employ the widely used NLA model. This model links the strength of the tidal
field when a galaxy forms to the intrinsic ellipticity of the galaxy. This
gives,

$\begin{split}&P_{\rm{\delta
I}}(k,z)=-A_{\rm{IA}}C_{1}\rho_{\rm{c}}\frac{\Omega_{\rm{m}}}{D(z)}P_{\rm{\delta\delta}}\
,\\\
&P_{\rm{II}}(k,z)=\bigg{(}A_{\rm{IA}}C_{1}\rho_{\rm{c}}\frac{\Omega_{\rm{m}}}{D(z)}\bigg{)}^{2}P_{\rm{\delta\delta}}\
,\end{split}$ (25)

where $C_{1}$ is a normalisation constant, $\rho_{\rm{c}}$ the critical
density of the Universe today and $D(z)$ the linear growth factor. We set
$C_{1}=5\times 10^{-14}M_{\odot}^{-1}h^{-2}\mathrm{Mpc}^{3}$ based on the IA
amplitude measured at low redshifts using SuperCOSMOS (Brown et al., 2002),
and $A_{\rm{IA}}$ captures the amplitude of the deviation from this reference
case. We take $A_{\rm{IA}}$ as a free parameter in our Fisher matrix. The NLA
model is sufficiently flexible for current studies but can be extended by
including a redshift dependent parameter, or using a halo model formalism to
calculate $P_{\rm{II}}$ and $P_{\rm{I\delta}}$ on small scales. Recently,
Fortuna et al. (2021) explored these options and found that the IA signal in
the one halo regime can be ignored at first order, and that including an extra
redshift dependent parameter is possibly sufficient for LSST. Here we consider
the simplest NLA model, but implementing more complex IA models could be a
future extension of this work.

##  4 Galaxy Luminosity Function

The second part of our Fisher matrix theory datavector, see section 2, is the
galaxy luminosity function. The galaxy luminosity function describes the
distribution of luminosities in a galaxy sample, the number density of
galaxies with a certain luminosity, and is often directly measured from a
galaxy sample. As specified in Cacciato et al. (2013) the galaxy luminosity
function at a given redshift $z$ can be calculated from the CLF detailed in
section 3.2:

$\Phi(L,z)=\int dM\ \Phi(L|M)n(M,z)\ ,$ (26)

where $\Phi(L|M)$ is the CLF and $n(M,z)$ is the halo mass function (see
section 3.2). In this analysis we work with a galaxy sample divided into
redshift bins (labelled $i$ and $j$ previously) so we wish to compute the
galaxy luminosity function for each redshift bin,

$\Phi^{i}(L)=\int dz\ n^{i}(z)\Phi(L,z)\ ,$ (27)

where $\Phi^{i}(L)$ denotes the luminosity function of galaxies in redshift
bin $i$, and $n^{i}(z)$ the normalised redshift distribution in bin $i$. We
include a prediction for the galaxy luminosity function in each redshift bin
in our theory datavector as it helps to constrain the 9 CLF parameters
detailed in section 3.2, and hence is critical for obtaining information from
the small scale clustering. The faint end slope of the number counts is also
required to calculate the magnification 2D power spectra, see eq. (13).

##  5 Survey Modelling

We perform our Fisher forecast using the cosmological parameter estimation
framework CosmoSIS (Zuntz et al., 2015). To calculate the 3D power spectra
detailed in section 3.2 we use our own halo model code, which has been tested
against other halo model codes used in the literature.

In this analysis we define two mock LSST galaxy samples; an ellipticity sample
$\epsilon$-sample and a number density sample n-sample. We use a 440 square
degree mock catalog from the LSST Dark Energy Science Collaboration (DESC)
Data Challenge 2 (DC2) simulations (cosmoDC2 1.1.4; Korytov et al. 2019).
These simulations were designed to enable preliminary LSST DESC analyses, and
the statistical distributions of galaxies have undergone a wide range of
validation tests, for details see Korytov et al. (2019); Kovacs et al. (2021).
The catalog includes photometric redshifts for all galaxies with an i-band
magnitude less than 26.5, up to redshift 3. The photometric redshifts were
calculated using the template fitting code BPZ (Benitez, 2000). The n-sample
is defined as all galaxies in this mock catalog with an i-band magnitude less
than 26.5 and photometric redshift greater than 0.1 and less than 2.0. We set
an upper limit as the photometric redshifts begin to degrade significantly
beyond 1.5, see Fig. 1. The $\epsilon$-sample is defined as a subset of
galaxies in n-sample with $i<25.3$. This corresponds to the LSST gold sample,
which will be used for weak lensing (LSST Science Collaboration, 2009). We do
not apply a separate signal-to-noise cut, but galaxies in the n-sample have a
signal-to-noise ratio $>5$ and galaxies in the $\epsilon$-sample have a
signal-to-noise ratio $>20$.

Figure 1: Photometric redshift point estimate mode against true redshift.
Left-hand panel: number density sample n-sample. Right-hand panel: ellipticity
sample $\epsilon$-sample

###  5.1 Redshift distributions

To compute the 2D power spectra in eq. (9) and the luminosity functions in eq.
(27) we require the redshift distribution of galaxies in each photometric
redshift bin. In this work we split both the galaxy samples, n-sample and
$\epsilon$-sample, into 10 tomographic redshift bins containing equal numbers
of galaxies using their photometric redshifts. Figure 2 shows the resulting
distribution of galaxies with redshift for each tomographic bin, as well as
the tomographic bin boundaries. Figure 2 shows that the photometric redshifts
are close to random for bin 10 of n-sample, so our maximum photometric
redshift cut of 2.0 is well justified.

Figure 2: Number density of galaxies as a function of true redshift for each
photometric bin in the galaxy sample. The dashed lines indicate the
photometric bin boundaries. Left-hand panel: n-sample. Right-hand panel:
$\epsilon$-sample

We compute the number density of galaxies in each tomographic bin to be $12.7\
\mathrm{arcmin}^{-2}$ for n-sample and $4.9\ \mathrm{arcmin}^{-2}$ for
$\epsilon$-sample. However, weak lensing shape measurements typically weight
galaxies by the uncertainty or ability to calibrate the shape measurements,
this would reduce the number density for $\epsilon$-sample, especially at high
redshifts. The LSST science book estimates that the number density of galaxies
in the gold sample will be 55 arcmin-2, with the number density of galaxies
useful for weak lensing approximately 40 arcmin-2 (LSST Science Collaboration,
2009; Chang et al., 2013). This means that our $\epsilon$-sample is slightly
optimistic, with a galaxy number density of 49 arcmin-2.

###  5.2 Faint end number count slopes

The key quantity in determining the amplitude of the fluctuation in the number
density of galaxies as a result of weak lensing magnification is the faint end
slope of the number counts $\alpha$. If $\alpha$ is equal to 1 there is no
overall fluctuation but if $\alpha$ does not equal 1 there is either an
increase or decrease in the number density of galaxies. $\alpha$ can be
represented in terms of magnitudes as,

$\alpha(i_{\rm{mag}})=2.5\frac{\mathrm{d}\log_{10}N(<i_{\rm{mag}})}{\mathrm{d}i_{\rm{mag}}}\
,$ (28)

where $i_{\rm{mag}}$ represents the $i$ band magnitude, and $N(<i_{\rm{mag}})$
the unlensed cumulative number density of galaxies with an $i$ band magnitude
lower (brighter) than $i_{\rm{mag}}$ (e.g. Duncan et al. 2014).

We measure the faint end slopes $\alpha$ from our LSST DC2 mock catalog. We
compute a value $\alpha^{j}$ for each redshift bin $j$, in each mock sample.
To compute $\alpha^{j}$ we vary the $i$ band magnitude in eq. (28) and compute
the cumulative number counts $N(>i_{\rm{mag}})$. We then fit the logarithm of
$N(>i_{\rm{mag}})$ with a straight line, and use the slope to compute
$\alpha^{j}$. Since we are only interested in the slope at the faint end (high
magnitudes) we only fit $\log_{10}N(>i_{\rm{mag}})$ over the last magnitude
before the sample magnitude limit; 25.5-26.5 for n-sample, and 24.3-25.3 for
$\epsilon$-sample. Figure 3 shows that in general this lower fit limit (marked
by a dotted line) captures the value of $\alpha^{j}$ at the faint end of the
sample. Increasing the lower fit limit has little effect on the value of
$\alpha^{j}$ obtained, whereas decreasing the fit limit in general gives a
higher value of $\alpha^{j}$.

Figure 3: The faint end slope of the number counts $\alpha^{i}$ as a function
of the limiting magnitude for each tomographic bin in n-sample (red) and
$\epsilon$-sample (blue). The $\alpha^{i}$ values used in this analysis were
found by fitting the slope of the logarithmic cumulative number counts (see
eq. (28)) between the vertical line and the right hand side of the figure.

Table 1 shows the $\alpha^{j}$ values obtained for each sample and their
associated uncertainties. The uncertainties come from the uncertainty on the
slope coefficient of the least-squares straight line fit detailed above, since
they were found to be much larger than the uncertainties on the values of the
cumulative number counts $N(>i_{\rm{mag}})$ due to the large number of
galaxies in each sample. The uncertainties are very small, and would become
even smaller when using the full 18000 square degree LSST area instead of a
440 square degree mock catalog. We therefore consider the $\alpha^{j}$
parameters as fixed in our forecast, but note that they can be difficult to
measure accurately from real data due to the presence of systematics and
selection effects (see conclusions for further discussion).

We can compare the $\alpha$ values in Table 1 to those found in Duncan et al.
(2014) for the Canada–France–Hawaii Lensing Survey (CFHTLenS). In both cases
$\alpha^{j}$ generally increases with redshift. CFHTLenS reaches an
$\alpha^{j}$ value of approximately 1 at its $i$ band magnitude limit of 24.7,
for its highest redshift bin between 1.02 and 1.3. This roughly corresponds to
$\alpha^{7}$ and $\alpha^{8}$ in $\epsilon$-sample, where the magnitude limit
of 24.7 is included in the $\alpha^{j}$ fit. Table 1 shows that our
$\alpha^{7}$ and $\alpha^{8}$ values for $\epsilon$-sample are consistent with
CFHTLenS.

Table 1: Faint end number count slopes $\alpha^{j}$ for each redshift bin $j$ in n-sample and $\epsilon$-sample, with their associated 1$\sigma$ uncertainties. n-sample |  $\epsilon$-sample
---|---
$\alpha^{1}$ |  $0.445\pm 0.005$ | $\alpha^{1}$ | $0.412\pm 0.005$
$\alpha^{2}$ |  $0.663\pm 0.006$ | $\alpha^{2}$ | $0.624\pm 0.004$
$\alpha^{3}$ |  $0.848\pm 0.006$ | $\alpha^{3}$ | $0.677\pm 0.004$
$\alpha^{4}$ |  $0.781\pm 0.005$ | $\alpha^{4}$ | $0.825\pm 0.006$
$\alpha^{5}$ |  $0.573\pm 0.004$ | $\alpha^{5}$ | $0.97\pm 0.01$
$\alpha^{6}$ |  $0.694\pm 0.006$ | $\alpha^{6}$ | $0.74\pm 0.01$
$\alpha^{7}$ |  $0.74\pm 0.01$ | $\alpha^{7}$ | $0.895\pm 0.006$
$\alpha^{8}$ |  $0.95\pm 0.02$ | $\alpha^{8}$ | $0.99\pm 0.01$
$\alpha^{9}$ |  $1.39\pm 0.01$ | $\alpha^{9}$ | $1.08\pm 0.01$
$\alpha^{10}$ |  $2.24\pm 0.02$ | $\alpha^{10}$ | $1.42\pm 0.01$

###  5.3 Systematics

We include a number of systematics in our analysis using nuisance parameters.
For the fiducial values of these parameters and their associated priors please
see Table 2. To apply a Gaussian prior to a particular parameter in a Fisher
matrix, one simply adds $1/\sigma^{2}_{\rm{prior}}$ to the diagonal element
associated with the parameter (Coe, 2009). In conceptual terms, the priors on
the Fisher matrix parameters can be summarized by a diagonal covariance matrix
with elements $\sigma^{2}_{\rm{prior}}$. This covariance matrix can then be
inverted into a prior Fisher matrix, giving $1/\sigma^{2}_{\rm{prior}}$
diagonal elements, and added to the experimental Fisher matrix.

####  5.3.1 Shear multiplicative bias

Systematic uncertainties in the measuring and averaging of galaxy shapes can
result in a multiplicative scaling of the observed shear. These systematic
effects include: noisy galaxy images, the applicability of the model used to
describe the light profile of galaxies, the details of the galaxy morphology
and selection biases (e.g. Heymans et al. 2006; Mandelbaum et al. 2018; Zuntz
et al. 2018; Kannawadi et al. 2019). We parametrise this multiplicative
scaling using one parameter $m^{i}$ per redshift bin (10 parameters in total),
which scale the cosmic shear and galaxy-galaxy lensing power spectra as:

$\begin{split}&C_{\rm{\epsilon\epsilon}}^{ij}(l)\rightarrow(1+m^{i})(1+m^{j})C_{\rm{\epsilon\epsilon}}^{ij}(l)\
,\\\
&C_{\rm{n\epsilon}}^{ij}(l)\rightarrow(1+m^{j})C_{\rm{n\epsilon}}^{ij}(l)\
.\\\ \end{split}$ (29)

We impose Gaussian priors on these multiplicative parameters, which are guided
by the LSST DESC science requirements (Alonso et al., 2018). These science
requirements forecast the uncertainties LSST will need to achieve in order to
meet their main objectives of significantly improving the constraints on the
dark energy parameters $w_{0}$ and $w_{a}$, compared to previous dark energy
experiments, and obtaining dark energy constraints where the total
calibratable systematic uncertainty is less than the marginalised statistical
uncertainty. For the case of shear multiplicative bias the requirement is that
the ‘systematic uncertainty in the redshift-dependent shear calibration’
should not exceed 0.003 by year 10. We therefore apply a Gaussian prior
centred on zero with a standard deviation of 0.003 to each of our shear
multiplicative bias parameters.

####  5.3.2 Clustering Multiplicative Bias

We parametrise uncertainties in the number count measurement using a similar
approach to that for shear. Systematics which affect the number density of
galaxies include: galactic dust obscuring background galaxies, variable survey
depth impacting the number of sources promoted across the flux limit by
magnification, and stars contaminating the galaxy sample (Hildebrandt, 2015;
Thiele et al., 2020). Usually these effects would be partially absorbed by the
galaxy bias (CLF) parameters, however since we include the galaxy luminosity
function in our analysis the CLF parameters will be tightly constrained. We
therefore felt it was important to include this multiplicative bias
parameterisation for clustering as well as shear.

Analogous to shear multiplicative bias, the observed clustering power spectra
are scaled by a multiplicative factor as,

$\begin{split}&C_{\mathrm{nn}}^{ij}(l)\rightarrow(1+m_{\mathrm{eff}}^{i})(1+m_{\mathrm{eff}}^{j})C_{\mathrm{nn}}^{ij}(l)\
,\\\
&C_{\mathrm{n\epsilon}}^{ij}(l)\rightarrow(1+m_{\mathrm{eff}}^{i})C_{\mathrm{n\epsilon}}^{ij}(l)\
.\end{split}$ (30)

However, since most systematics decrease with signal to noise ratio, we assume
$m^{i}_{\mathrm{eff}}$ has a power law dependence on the signal to noise of
galaxies in redshift bin $i$. This enables us to reduce the number of
clustering multiplicative bias parameters from ten parameters (one
$m^{i}_{\mathrm{eff}}$ per redshift bin) to two parameters $a_{\rm{m}}$ and
$b_{\rm{m}}$. $m^{i}_{\mathrm{eff}}$ is given in terms of $a_{\rm{m}}$ and
$b_{\rm{m}}$ by,

$\begin{split}m^{i}_{\mathrm{eff}}&=m_{\mathrm{step}}-m_{\mathrm{fid}}\\\
&=\frac{1}{N_{i}}\Bigg{[}a_{\rm{m}}\sum_{n=1}^{N_{i}}\Big{(}\frac{S}{N}\Big{)}_{n}^{b_{\rm{m}}}-a_{\mathrm{fid}}\sum_{n=1}^{N_{i}}\Big{(}\frac{S}{N}\Big{)}_{n}^{b_{\mathrm{fid}}}\Bigg{]}\
,\end{split}$ (31)

where $N_{i}$ is the number of galaxies in tomographic bin $i$, the sum is
over the signal-to-noise ratio $S/N$ of all galaxies in tomographic bin $i$,
$a_{\mathrm{fid}}$ is the fiducial value of $a_{\rm{m}}$ and
$b_{\mathrm{fid}}$ is the fiducial value of $b_{\rm{m}}$. We introduce the
$m_{\mathrm{fid}}$ term because if $m_{\mathrm{eff}}=m_{\mathrm{step}}$,
$b_{\rm{m}}$ becomes unconstrained when $a_{\rm{m}}$ is equal to zero, which
breaks the Gaussian Likelihood assumption in the Fisher matrix prediction.

We compute the signal to noise ratio for each galaxy in our samples from the
error on the i band apparent magnitude. Using the signal to noise of every
galaxy in this bias calculation is computationally expensive, since the total
number of galaxies in n-sample and $\epsilon$-sample is of order $10^{7}$ and
$10^{8}$. We therefore use a randomly selected $1\%$ subsample of galaxies in
this calculation. This subsample is representative of the full galaxy sample,
but prevents our bias calculation from being prohibitively slow.

####  5.3.3 Photometric redshift uncertainties

We model uncertainties in the redshift distributions shown in figure 2 by
introducing shift factors $\Delta^{i}$ (Bonnett et al., 2016). $\Delta^{i}$
simply shifts the redshift distribution in bin $i$ so,

$n^{i}(z)\rightarrow n^{i}(z-\Delta^{i}).$ (32)

Since we have two redshift distributions, one for $\epsilon$-sample and one
for n-sample, each divided into 10 bins this results in 20 shift parameters
$\Delta^{i}$. These parameters are likely to be correlated, so we are making a
conservative choice by allowing 20 separate shift parameters, which may
somewhat weaken our final constraints. We impose Gaussian priors on each of
these shift parameters, once again guided by the LSST DESC science
requirements (Alonso et al., 2018). The prior is centred on zero, with a
standard deviation of 0.003 for the n-sample parameters and of 0.001 for the
$\epsilon$-sample parameters.

A future extension of this work could be to include other modes of redshift
uncertainty, such as a change in the width or to the high redshifts tails, as
in Nicola et al. (2020). These may be particularly interesting for
magnification, as they change the level of overlap between different redshift
bins.

###  5.4 Covariances

In this forecast we consider two component Fisher matrices. The Fisher matrix
for the weak lensing observables and the Fisher matrix for the galaxy
luminosity function (see section 2). We therefore require two covariances: the
weak lensing observables covariance and the galaxy luminosity function
covariance.

####  5.4.1 Weak lensing observables covariance

We compute a Gaussian covariance for the observable weak lensing power spectra
($C_{\epsilon\epsilon}$, $C_{\mathrm{nn}}$, $C_{\mathrm{n\epsilon}}$) using
CosmoSIS. The covariance between two power spectra is given by,

$\mathrm{Cov}\big{[}C^{ij}(\ell),\
C^{kl}(\ell^{\prime})\big{]}=\delta_{\ell\ell^{\prime}}\frac{2\pi}{A\ell\Delta\ell}\big{[}\bar{C}^{ik}(\ell)\bar{C}^{jl}(\ell)+\bar{C}^{il}(\ell)\bar{C}^{jk}(\ell)\big{]}\
,$ (33)

where $ijkl$ denote redshift bins, $\delta_{\ell\ell^{\prime}}$ is the
Kronecker delta, $A$ is the survey area and $\Delta\ell$ the size of the
angular frequency $\ell$ bin (Joachimi et al., 2008; Joachimi & Bridle, 2010).
We do not include the non-gaussian contributions to the covariance since their
effect is small, and unlikely to impact our final results (Barreira et al.,
2018). To account for the random terms in equations 6 and 7 we define,

$\bar{C}^{ij}(\ell)=C^{ij}(\ell)+N^{ij}\ ,$ (34)

where $N^{ij}$ is the shot or shape noise contribution. In the case of
$C_{\rm{\epsilon\epsilon}}$,

$N^{ij}=\delta_{ij}\frac{\sigma^{2}_{\epsilon}}{2\bar{n}^{i}}\ ,$ (35)

in the case of $C_{\rm{nn}}$,

$N^{ij}=\delta_{ij}\frac{1}{\bar{n}^{i}}\ ,$ (36)

and in the case of $C_{\rm{n\epsilon}}$, $N^{ij}=0$. Where $\sigma_{\epsilon}$
is the total intrinsic ellipticity dispersion, and $\bar{n}^{i}$ is the
average number density of galaxies in redshift bin $i$ (Bartelmann &
Schneider, 2001). We compute the power spectra covariance for 20 log-spaced
angular frequency $l$ bins from $l_{\rm{min}}=30$, to avoid inaccuracies in
the Limber approximation, to $l_{\rm{max}}=3000$, to avoid the very non-linear
regime.

####  5.4.2 Galaxy luminosity function covariance

We compute the galaxy luminosity function covariance by measuring the galaxy
luminosity functions of our mock LSST galaxy samples and then computing a
bootstrap covariance. Since Fisher forecasts do not require a datavector, only
a covariance, we only use the measured luminosity functions to compute the
covariance and model the galaxy luminosity function in the forecast using the
CLF formalism (see section 4).

To measure the luminosity functions for the n-sample and $\epsilon$-sample we
begin by computing the luminosity of each galaxy from its rest-frame absolute
magnitude in the $i$ band. We then divide our sample into the 10 tomographic
bins described above and scale the luminosity function for each bin $j$ by the
volume of bin $j$, to convert the histogram to a number density. When
calculating the bin volume we assume that the galaxies do not scatter beyond
the tomographic bin boundaries. This is an approximation, which figure 2
shows, is becoming problematic for bin 10.

Ideally, we would use the full range of galaxy luminosities to compute our
bootstrap covariance. However in order to use the low luminosity region we
would need to correct our galaxy samples to be volume complete, for example
through the $1/V_{\mathrm{max}}$ method (Schmidt, 1968; Felten, 1976; Cole,
2011). High luminosity objects can be observed across the full volume of the
survey, but low luminosity objects can only be observed at smaller distances.
This introduces a bias referred to as Malmquist bias, and we therefore only
want to include galaxies that can be observed across the whole volume of the
survey. For the purposes of this work we deemed it sufficient to simply cut
out the low luminosity galaxies to make the sample volume limited, since this
is still a significant step forward compared to previous analyses. For details
of how we determine the volume complete cut see appendix A.

We then compute a bootstrap covariance for our measured galaxy luminosity
functions. First, we sample our dataset with replacement 100 times and compute
the associated datavectors. We then assume that each luminosity bin in each
tomographic bin is independent (each of our datapoints is independent) and
calculate the variance of these 100 samples. This gives us a diagonal
covariance. The variance of the 100 samples is in general small, due to the
very large numbers of galaxies in each sample.

###  5.5 Fiducial values

The Fisher matrix gives the curvature of the log-Likelihood function around
its peak. It does not find the location of the peak, this is defined with a
set of fiducial values (shown in Table 2). The set of parameters required to
calculate the 3D power spectra in section 3.2 are the cosmological parameters
and the CLF parameters. In this work we consider the constraints on a flat
$\Lambda$CDM cosmology, and vary the cosmological parameters;
$\Omega_{\rm{m}}$ the matter density, $h_{0}$ the hubble parameter,
$\Omega_{\rm{b}}$ the baryon density, $n_{\rm{s}}$ the scalar spectral index,
$A_{\rm{s}}/10^{-9}$ the amplitude of primordial fluctuations, and $w$ and
$w_{\rm{a}}$ the dark energy equation of state parameters. We take their
fiducial values from the input values used to generate the simulation for the
LSST DESC mock catalog, or from the values obtained by the Planck satellite
(Aghanim et al., 2020).

Table 2: Fiducial values and priors for the model parameters used to compute the fisher matrices in this work. Flat priors do not contribute to the Fisher matrix so we simply specify flat, and do not include bounds. Parameter | Fiducial Value | Prior
---|---|---
| Survey |
Area | $18000\ \mathrm{deg^{2}}$ | fixed
$\sigma_{e}$ | $0.35$ | fixed
| Cosmology |
$\Omega_{\rm{m}}$ | $0.265$ | flat
$h_{0}$ | $0.71$ | flat
$\Omega_{\rm{b}}$ | $0.0448$ | flat
$n_{\rm{s}}$ | $0.963$ | flat
$A_{\rm{s}}/10^{-9}$ | $2.1$ | flat
$w$ | $-1.0$ | flat
$w_{\rm{a}}$ | $0.0$ | flat
$\Omega_{\rm{k}}$ | $0.0$ | fixed
| CLF |
$\log(M_{1})$ | $11.24$ | flat
$\log(L_{0})$ | $9.95$ | flat
$\gamma_{1}$ | $3.18$ | flat
$\gamma_{2}$ | $0.245$ | flat
$\sigma_{c}$ | $0.157$ | flat
$\alpha_{\rm{s}}$ | $-1.18$ | flat
$b_{0}$ | $-1.17$ | flat
$b_{1}$ | $1.53$ | flat
$b_{2}$ | $-0.217$ | flat
| Intrinsic Alignments |
$A_{\rm{IA}}$ | $1.0$ | flat
| n-sample Photo-z |
$\Delta_{\rm{n}}^{i}$ | $0.0$ | Gauss(0.0, 0.003)
|  $\epsilon$-sample Photo-z |
$\Delta_{\rm{\epsilon}}^{i}$ | $0.0$ | Gauss(0.0, 0.001)
| Shear Bias |
$m^{i}$ | $0.0$ | Gauss(0.0, 0.003)
| Clustering Bias |
$a_{\rm{m}}$ | $0.001$ | flat
$b_{\rm{m}}$ | $0.0$ | flat

We also vary the full set of CLF parameters $\log M_{1}$, $\log L_{0}$,
$\gamma_{1}$, $\gamma_{2}$, $\sigma_{\rm{c}}$, $\alpha_{\rm{s}}$, $b_{0}$,
$b_{1}$ and $b_{2}$, detailed in section 3.2. Here we use the fiducial values
found for SDSS by Cacciato et al. (2013), which have been shown to also be
applicable to higher redshift surveys (Cacciato et al., 2014; van Uitert et
al., 2016).

##  6 Results

###  6.1 Clustering

Figure 4 shows the forecast constraints on the cosmological parameters from
$C_{\mathrm{nn}}$ with and without including magnification terms for n-sample.
In the case of including magnification the observable is
$C_{\mathrm{nn}}=C_{\mathrm{gg}}+C_{\mathrm{gm}}+C_{\mathrm{mm}}$ instead of
$C_{\mathrm{nn}}=C_{\mathrm{gg}}$. Including magnification generally has a
small impact on the cosmological parameter constraints. The greatest change is
the 1$\sigma$ constraint on $\Omega_{m}$, which is improved by a factor of 1.3
from 0.003 to 0.0023.

Figure 4: Constraints on the cosmological parameters used in this analysis
from $C_{\mathrm{nn}}$ and $C_{\mathrm{nn}}$ including magnification terms for
n-sample. Including magnification has only a small impact on the constraints.

The forecast constraints on the cosmological parameters from $C_{\mathrm{nn}}$
and $C_{\mathrm{nn}}$ including magnification terms for $\epsilon$-sample show
that the impact of magnification is reduced compared to the n-sample. The
1$\sigma$ constraint on $\Omega_{m}$ is only improved by a factor of 1.03 from
0.0032 to 0.0031, instead of a factor of 1.3 with the n-sample. This shows
that including magnification has a greater impact for deeper samples.

Figure 5 shows the forecast constraints on the CLF parameters from
$C_{\mathrm{nn}}$ with and without including magnification terms for the
n-sample. Including magnification has little effect on the constraints on the
CLF parameters. This is expected because the CLF constraints are predominantly
determined by the galaxy luminosity function. We focus on the cosmological and
CLF parameters, instead of presenting the full 28 parameter space, for
clarity. The steps taken to ensure the stability of our Fisher matrix are
detailed in appendix B.

Figure 5: Constraints on the Conditional Luminosity Function (CLF) parameters
used in this analysis from $C_{\mathrm{nn}}$ and $C_{\mathrm{nn}}$ including
magnification terms for n-sample. Including magnification has little effect on
the constraints, since they are driven by the galaxy luminosity function not
the weak lensing or clustering observables.

A useful measure of the constraining power of an analysis is the Figure of
Merit (FoM) defined as,

${\rm{FoM}=det([F^{-1}]}_{q})^{\frac{1}{N_{q}}}\ ,$ (37)

where ${\rm{[F^{-1}]}}_{q}$ is the inverse Fisher matrix for the set of
parameters $q$ and $N_{q}$ is the number of parameters $q$ in the set. In this
work we define $q$ as the full set of cosmological parameters, so the FoM
represents the power of the constraints on the cosmological parameters. It is
also common to define a Dark Energy FoM where $q=\\{w,w_{a}\\}$ (Albrecht et
al., 2006).

When magnification is included in the clustering analysis for the n-sample the
FoM is increased by a factor of 1.45. However, when magnification is included
in the clustering analysis for $\epsilon$-sample (the LSST gold sample) the
FoM is increased by a factor of 1.08. This mirrors the conclusions from
looking at the parameter constraints on $\Omega_{m}$ – magnification is more
beneficial for deeper samples with greater numbers of low signal-to-noise
ratio galaxies. Interestingly, there is no increase in the FoM for clustering
without magnification when using the n-sample instead of $\epsilon$-sample.
This implies that it is more beneficial to have a smaller sample of high
signal-to-noise objects than a larger sample including lower signal-to-noise
objects. This is likely due to the additional fainter objects having poorer
photometric redshifts and therefore largely contributing to the tails of the
redshift distribution. Looking back at figure 2 we can see that the redshift
distribution for the $\epsilon$-sample is much cleaner.

###  6.2 Shear calibration

The previous section showed that including weak lensing magnification only has
a small effect on the cosmological parameter constraints from an LSST-like
angular galaxy clustering analysis. In a combined clustering and cosmic shear
analysis the impact of magnification on the cosmological parameter constraints
can only be reduced. This is because magnification predominantly contributes
to the clustering signal and provides very similar information to shear. We
therefore focus on the effect of magnification on the shear multiplicative
bias parameters.

We examine the impact of including magnification on the shear multiplicative
bias parameters for a combined LSST clustering $C_{\mathrm{nn}}$ and shear
$C_{\rm{\epsilon\epsilon}}$ analysis, where the analyses occur on separate
patches of sky so the $C_{\rm{n\epsilon}}$ term is negligible. We are
therefore investigating whether the improved cosmological constraints from
magnification translate into an improved calibration.

Figure 6 shows the forecast constraints on the shear multiplicative bias
parameters from our $C_{\rm{nn}}$ and $C_{\rm{\epsilon\epsilon}}$ analysis,
with and without magnification terms, where $C_{\rm{\epsilon\epsilon}}$ is
calculated for $\epsilon$-sample and $C_{\rm{nn}}$ for the n-sample. Including
magnification only slightly improves the constraints on the shear calibration
parameters, with a greater effect at higher redshift. The 1$\sigma$ constraint
on $m^{1}$ is improved by a factor of 1.06, $m^{6}$ by 1.3 and $m^{10}$ by
1.34 when including magnification. When $C_{\rm{nn}}$ is calculated using the
$\epsilon$-sample the impact is similar, but less pronounced. These results
show that including magnification is not particularly helpful for calibrating
the shear measurement. However, the impact of magnification may be slightly
improved when performing a full ‘3x2pt’ analysis, where the clustering and
shear are measured on the same patch of sky.

Figure 6: Constraints on the shear multiplicative bias parameters from a joint
analysis of $C_{\rm{\epsilon\epsilon}}$ and $C_{\rm{nn}}$ (not including
$C_{\rm{n\epsilon}}$) with and without magnification terms, where we have not
applied the Gaussian prior detailed in Table 2. $C_{\rm{\epsilon\epsilon}}$ is
calculated for $\epsilon$-sample and $C_{\rm{nn}}$ is calculated for n-sample.
Including magnification only slightly improves the constraints, with a greater
impact at higher redshift.

###  6.3 Bias

Recent works have shown that cosmological results from upcoming surveys such
as LSST will be biased if the effects of weak lensing magnification are not
included, due to improvements in statistical precision (Duncan et al., 2014;
Cardona et al., 2016; Lorenz et al., 2018; Thiele et al., 2020). To examine
this for our forecast, figure 7 shows the absolute difference between the
clustering power spectra $C_{\rm{nn}}$ with and without magnification in terms
of the $1\sigma$ uncertainty on the clustering power spectra without
magnification. In this case the clustering power spectra have been calculated
using n-sample. The grey shaded region indicates where $C_{\rm{nn}}$ including
magnification is more than $2\sigma$ away from $C_{\rm{nn}}$ without
magnification. Particularly at high $\ell$ (small scales) $C_{\rm{nn}}$
including magnification significantly diverges from $C_{\rm{nn}}$ without
magnification. It is worth noting that this result is influenced by the very
small uncertainty on the clustering signal for a sample of such great depth.
For a clustering signal with greater uncertainty the difference due to
magnification in terms of the $1\sigma$ uncertainty would be reduced.

For qualitative comparison, we have also shown the impact of changing
$\Omega_{\rm{m}}$ and $A_{\rm{s}}$ by $5\sigma$ in Fig. 7. In all of the
redshift bin combinations shown, the difference from including magnification
is larger than or comparable to the difference from changing $\Omega_{\rm{m}}$
and $A_{\rm{s}}$ by $5\sigma$. This clearly indicates that not including
magnification terms will catastrophically bias cosmological constraints from
LSST. Additionally, the difference from not including magnification seems to
mimic the behaviour of biasing $A_{\rm{s}}$ by $5\sigma$. This implies that
not including magnification could particularly bias the constraints for
$A_{\rm{s}}$, one of the parameters weak lensing is most sensitive to.

Figure 7: Absolute difference between $C_{\rm{nn}}$ with and without
magnification, in terms of the uncertainty on $C_{\rm{nn}}$ without
magnification, for n-sample. The grey shaded region indicates where
$C_{\rm{nn}}$ including magnification is more than $2\sigma$ away from
$C_{\rm{nn}}$ without magnification. The dashed lines show the difference in
$C_{\rm{nn}}$ when $\Omega_{\rm{m}}$ and $A_{\rm{s}}$ are altered by
$5\sigma$.

Figure 8 shows the absolute difference between the clustering power spectra
$C_{\rm{nn}}$ with and without magnification in terms of the $1\sigma$
uncertainty on the clustering power spectra without magnification, where the
clustering power spectra have been calculated using the $\epsilon$-sample. In
this case the difference from including magnification is not as large as for
n-sample, however in most redshift bin combinations is still comparable or
larger than the differences from changing $\Omega_{\rm{m}}$ and $A_{\rm{s}}$
by $5\sigma$.

Figure 8: Absolute difference between $C_{\rm{nn}}$ with and without
magnification, in terms of the uncertainty on $C_{\rm{nn}}$ without
magnification, for $\epsilon$-sample. The grey shaded region indicates where
$C_{\rm{nn}}$ including magnification is more than $2\sigma$ away from
$C_{\rm{nn}}$ without magnification. The dashed lines show the difference in
$C_{\rm{nn}}$ when $\Omega_{\rm{m}}$ and $A_{\rm{s}}$ are altered by
$5\sigma$.

##  7 Conclusions

Previous works have shown that upcoming results from surveys such as LSST and
Euclid will be biased if the effects of weak lensing magnification are not
included (Duncan et al., 2014; Cardona et al., 2016; Lorenz et al., 2018;
Thiele et al., 2020). In this work we forecast whether including weak lensing
magnification as a complementary probe can additionally improve the precision
of the LSST galaxy clustering constraints. We determined this using the Fisher
matrix formalism, where our theory datavector included galaxy clustering and
the galaxy luminosity function. To calculate the galaxy clustering and the
galaxy luminosity function, we employed a halo model, detailed in Cacciato et
al. (2013). We defined two mock LSST galaxy samples from the LSST DC2
simulations (Korytov et al., 2019) for use in our forecast; a sample which
corresponds to the LSST gold sample where the $i$ band magnitude is less than
25.3 (intended to be used for the weak lensing shear measurement), and a very
deep sample where the $i$ band magnitude is less than 26.5.

We found that weak lensing magnification provides little additional
information as a complementary probe for LSST. For a galaxy clustering
analysis using the LSST gold sample we found that including magnification
increased the Figure of Merit (FoM) for the set of cosmological parameters
$\Omega_{\rm{m}}$, $h_{0}$, $\Omega_{\rm{b}}$, $n_{\rm{s}}$,
$A_{\rm{s}}/10^{-9}$, $w$ and $w_{\rm{a}}$ by a factor of 1.08. When using the
deep galaxy sample we found that magnification increased the FoM by a factor
of 1.45. In terms of the precision of the $\Omega_{m}$ constraints, we found
for a galaxy clustering analysis using the LSST gold sample that including
magnification increased the 1$\sigma$ precision by a factor of 1.03, using the
deep sample we found a factor increase of 1.3. These results show that
including magnification is more beneficial for deeper samples.

The effect of including magnification would be smaller in a combined galaxy
clustering and cosmic shear analysis because magnification provides similar
information to that of cosmic shear. However, we investigated the impact of
including magnification on the calibration of the shear measurement. We found
that including magnification only slightly improves the constraints on the
shear calibration parameters.

While this forecast is more realistic than many to date, as it includes LSST
mock catalog data and a flexible galaxy bias model, it still relies on a
number of simplified assumptions about magnification. Firstly, the
magnification modelling assumes that the galaxy sample is purely flux limited.
Often galaxies are also selected based on their signal-to-noise ratio, colours
and morphology which complicates the magnification modelling (Hildebrandt,
2015). Secondly, there are a large number of systematics associated with the
magnification measurement such as dust attenuation, variable survey depth,
star-galaxy separation and the blending of galaxy images (Hildebrandt et al.,
2013; Morrison & Hildebrandt, 2015; Thiele et al., 2020). We included a
multiplicative factor in our modelling of the clustering power spectra in
order to incorporate these effects, but more detailed modelling is likely
required. For example, we could have marginalised over the faint end slopes of
the number counts $\alpha^{i}$, which are required to compute the
magnification power spectra. We chose to fix them, since at least for the gold
sample it should be comparatively easy to explore the luminosity function
beyond the magnitude limit, so measurement errors on $\alpha^{i}$ can be
expected to be very small. This forecast could therefore be considered a best
case scenario for magnification, and even in this scenario we found that
including magnification has little impact. However, we also confirmed that not
including magnification will strongly bias cosmological results from LSST, so
must be modelled.

## Acknowledgements

This paper has undergone internal review in the LSST Dark Energy Science
Collaboration. We would like to thank the internal reviewers David Alonso,
Sukhdeep Singh and Marina Ricci for their insightful comments. We also thank
Hendrik Hildebrandt for comments on the manuscript, and Christopher Duncan and
Harry Johnston for useful discussions. CM was supported by the Spreadbury
Fund, Perren Fund, IMPACT Fund, and by the European Research Council under
grant 770935. CM acknowledges travel support provided by STFC for UK
participation in LSST through grant ST/N002512/1. MCF and HH acknowledge
support from the Netherlands Organisation for Scientific Research (NWO)
through grant 639.043.512. BJ acknowledges support by the UCL Cosmoparticle
Initiative. AK acknowledges support through STFC grant ST/S000666/1. SJS
acknowledges support from DOE grant DESC0009999 and NSF/AURA grant N56981C. We
acknowledge the use of
CosmoSIS222https://bitbucket.org/joezuntz/cosmosis/wiki/Home and
hmf333https://github.com/halomod/hmf, and thank their authors for making these
products public.

The DESC acknowledges ongoing support from the Institut National de Physique
Nucléaire et de Physique des Particules in France; the Science & Technology
Facilities Council in the United Kingdom; and the Department of Energy, the
National Science Foundation, and the LSST Corporation in the United States.
DESC uses resources of the IN2P3 Computing Center (CC-IN2P3–Lyon/Villeurbanne
- France) funded by the Centre National de la Recherche Scientifique; the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded
by UK BIS National E-infrastructure capital grants; and the UK particle
physics grid, supported by the GridPP Collaboration. This work was performed
in part under DOE Contract DE-AC02-76SF00515.

## Data Availability

The cosmoDC2 catalog is publicly available here:
https://portal.nersc.gov/project/lsst/cosmoDC2/_README.html

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##  Appendix A Volume Complete Cut for Galaxy Luminosity Function Covariance

A deeper galaxy sample will be volume complete to lower luminosities, so when
the luminosity function of a shallower sample diverges from the luminosity
function of a deeper sample, we know the shallower sample has ceased to be
volume complete. We can therefore determine the volume complete luminosity cut
for the $\epsilon$-sample by finding where it diverges from the n-sample. Our
divergence condition is

$\frac{|\Phi^{i}_{\epsilon}(L)-\Phi^{i}_{n(\epsilon)}(L)|}{\Phi^{i}_{\epsilon}(L)}>0.2\
,$ (38)

where $\Phi^{i}_{\epsilon}$ is the luminosity function for the
$\epsilon$-sample and $\Phi^{i}_{n(\epsilon)}$ is the luminosity function for
the n-sample, where the n-sample has been binned using the $\epsilon$-sample
tomographic bins. We cut $\Phi^{i}_{\epsilon}$ when there is a difference of
20% from the deeper sample $\Phi^{i}_{n(\epsilon)}$. This value was found to
cut $\Phi^{i}_{\epsilon}$ before it significantly diverged from the deeper
sample whilst allowing for small deviations, see the right panel of Fig. 9.

Figure 9: Measured luminosity functions $\Phi^{i}(L)$ for each photometric
redshift bin in n-sample (left) and $\epsilon$-sample (right), with associated
bootstap errors. The dotted lines show the location of the luminosity cuts to
n-sample (left) and $\epsilon$-sample (right) to make sure they are volume
complete, and do not introduce a bias. $\epsilon$-sample n tomo bins refers to
the $\epsilon$-sample being binned into the n-sample redshift bins, and
n-sample $\epsilon$ tomo bins refers to the n-sample being binned into
$\epsilon$-sample redshift bins. $L\Phi^{i}(L)$ has units of
$h^{3}/\mathrm{Mpc}^{3}$.

Since we did not have a sample deeper than the n-sample available to us, we
made a more stringent volume complete cut on the n-sample luminosity function
based on where the luminosity function of our shallower sample
$\epsilon$-sample diverged. If the shallower sample is volume complete we can
be sure that the deeper sample is also volume complete. In this case our
divergence condition is

$\frac{|\Phi^{i}_{n}(L)-\Phi^{i}_{\epsilon(n)}(L)|}{\Phi^{i}_{n}}>0.2\ ,$ (39)

where $\Phi^{i}_{n}$ is the luminosity function for n-sample and
$\Phi^{i}_{\epsilon(n)}$ is the luminosity function for $\epsilon$-sample,
where $\epsilon$-sample has been binned using the n-sample tomographic bins.
While this luminosity cut enforces that n-sample is volume complete, using a
shallower sample means that the cut is much more conservative than necessary.

##  Appendix B Fisher Matrix Stability

High-dimensional Fisher matrices can be unstable. Here we detail the steps
taken to ensure the stability of our Fisher matrices and hence the robustness
of our results.

The derivatives in eq. (3) are calculated numerically using a method of
numerical differentiation called a 5-pt stencil. This method requires the
pipeline to be evaluated at 4 points around the model parameter’s fiducial
value (5 points including the fiducial value). The separation between these
points is referred to as the step size. If the step size is too large the
Fisher matrix fails to capture the curvature of the likelihood function about
the peak and if it is too small numerical difficulties can arise. Therefore
when using Fisher matrices it is vital to verify whether the step size is
appropriate, otherwise any results are meaningless.

We verify our step sizes in 1 dimension by fixing all but one model parameter.
We then calculate the 1D likelihood using a Fisher matrix with a specified
step size and by sampling the likelihood function directly. If the 1D
likelihoods match we know we are using a reasonable step size when calculating
our Fisher matrix. We sample the likelihood function directly using a
simulated datavector generated at the Fisher matrix fiducial values and a grid
sampler. Grid samplers evaluate the likelihood at a specified set of grid
points. Since we are assuming a Gaussian Likelihood when calculating our
Fisher matrix (eq. (3)) we are only interested in whether the standard
deviation $\sigma$ of the likelihood calculated using the Fisher matrix
matches the $\sigma$ of the likelihood from sampling directly using a grid
sampler.

Figure 10 shows the $\sigma$ of the 1D likelihood calculated using the Fisher
matrix for different choices of step size. These plots show that as the step
size decreases the $\sigma$ of the 1D likelihood reaches a plateau, where the
step size is actually capturing the shape of the likelihood, before becoming
unstable (see subplot for the photometric redshift bias parameter for redshift
bin 10). We therefore select a step size in the range where the $\sigma$ of
the Fisher likelihood is stable. Figure 11 shows the Fisher likelihoods
generated using the selected step sizes overlaid with the likelihood from the
grid sampler to verify that they match. For the case of the magnification bias
parameter $b_{\rm{m}}$ the Fisher and grid likelihoods do not match. This is
because when calculating the Fisher matrix we assume that the likelihood is
Gaussian, and the likelihood of $b_{\rm{m}}$ from direct sampling is clearly
not Gaussian. This is a limitation of the Fisher matrix approach.

We additionally check the Fisher step sizes for the cosmological parameters,
by varying all the cosmological parameters at once and exploring the
multivariate posterior with Markov Chain Monte Carlo (MCMC) sampling444the
MCMC we use is emcee (Foreman-Mackey et al., 2013). Figure 12 shows a
comparison between the constraints obtained from the MCMC and the Fisher
matrix. They match well and show that our Fisher matrix is adequately
capturing the shape of the likelihood.

Figures 10 and 11 show only an example case for the parameters used to
generate the $C_{\mathrm{nn}}$ Fisher matrix for $\epsilon$-sample. However,
the step sizes have been verified using this method for every Fisher matrix
referred to in the results section.

Figure 10: Standard deviation of the 1D Fisher likelihoods against the step
size used to calculate the derivatives in the Fisher matrix. The chosen step
size should be in the plateau region where the Fisher matrix is actually
capturing the shape of the likelihood.

Figure 11: Comparison of the 1D likelihoods from the Fisher matrix calculated
using the selected step size against the 1D likelihoods from sampling the
likelihood directly using a grid sampler.

Figure 12: Comparison of the constraints on the cosmological parameters used
in this analysis when found using and MCMC or a Fisher matrix. All other
parameters have been fixed.