# Propagating spatially-varying multiplicative shear bias to cosmological
parameter estimation for stage-IV weak-lensing surveys

Casey Cragg,1 Christopher A. J. Duncan,1 Lance Miller1 and David Alonso1
1Department of Physics, University of Oxford, Denys Wilkinson Building, Keble
Road, Oxford OX1 3RH, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We consider the bias introduced by a spatially-varying multiplicative shear
bias ($m$-bias) on tomographic cosmic shear angular power spectra. To compute
the bias in the power spectra, we estimate the mode-coupling matrix associated
with an $m$-bias map using a computationally-efficient pseudo-$C_{\ell}$
method. This allows us to consider the effect of the $m$-bias to high $\ell$.
We then conduct a Fisher matrix analysis to forecast resulting biases in
cosmological parameters. For a Euclid-like survey with a spatially-varying
$m$-bias, with zero mean and rms of 0.01, we find that parameter biases reach
a maximum of $\sim 10\%$ of the expected statistical error, if multipoles up
to $\ell_{\mathrm{max}}=5000$ are included. We conclude that the effect of the
spatially-varying $m$-bias may be a sub-dominant but potentially non-
negligible contribution to the error budget in forthcoming weak lensing
surveys. We also investigate the dependence of parameter biases on the
amplitude and angular scale of spatial variations of the $m$-bias field, and
conclude that requirements should be placed on the rms of spatial variations
of the $m$-bias, in addition to any requirement on the mean value. We find
that, for a Euclid-like survey, biases generally exceed $\sim 30\%$ of the
statistical error for $m$-bias rms $\sim 0.02-0.03$ and can exceed the
statistical error for rms $\sim 0.04-0.05$. This allows requirements to be set
on the permissible amplitude of spatial variations of the $m$-bias that will
arise due to systematics in forthcoming weak lensing measurements.

###### keywords:

gravitational lensing: weak – cosmology: cosmological parameters, large-scale
structure of the Universe – methods: numerical, statistical

††pagerange: Propagating spatially-varying multiplicative shear bias to
cosmological parameter estimation for stage-IV weak-lensing surveys–References

##  1 Introduction

A key area of study in cosmology is that of the statistics of the large-scale
matter distribution, which can provide constraints on the evolution of cosmic
structures, the expansion history of the Universe, the nature and behaviour of
dark matter and dark energy, the conditions in the very early Universe, and
alternatives to the $\Lambda$CDM cosmological model. One promising probe of
the large-scale structure is the phenomenon of weak gravitational lensing, in
which the images of distant galaxies are distorted due to the gravitational
perturbation of the paths of light rays by matter between the source and
observer (Jain & Seljak, 1997; Kilbinger, 2015). Current and future weak
lensing surveys aim to measure the spin-2 cosmic shear $\bm{\gamma}$,
estimated by the ellipticities of source galaxy images, into which a
spatially-coherent signal is introduced due to weak lensing. By studying the
cosmic shear we can make inferences about the large-scale structure.

Cosmic shear measurements have already been made by previous and ongoing
cosmological surveys, including CFHTLenS (Heymans et al., 2012), KiDS (Giblin
et al., 2021) and DES (DES Collaboration, 2021), and shear measurement is one
of the main science goals of the forthcoming stage IV surveys, including the
Rubin Observatory’s Legacy Survey of Space and Time (LSST) (LSST Science
Collaboration, 2009), Euclid (Laureijs et al., 2011) and the Roman telescope
(Green et al., 2012). Cosmic shear is a very subtle effect, with distortions
introduced of the order of a per cent of the intrinsic galaxy ellipticities
(Kilbinger, 2015). As additional data is contributed by each successive
survey, the resulting reduction in statistical uncertainty means that
systematic errors need to be more tightly controlled. While previous weak
lensing surveys have accounted for systematic effects, forthcoming surveys
will require understanding and mitigation of systematics to an unprecedented
level of precision, including effects which have not previously been
considered. One significant effect of astrophysical and instrumental
systematics is the introduction of biases into the measurement of the cosmic
shear inferred from galaxy shapes. This can be expressed with a linear bias
model:

$\widetilde{\bm{\gamma}}=\bm{\gamma}(1+m)+c\ ,$ (1)

with $\widetilde{\bm{\gamma}}$ the biased shear field, $m$ the multiplicative
“$m$-bias” and $c$ the additive “$c$-bias”. These biases can arise from a
number of systematic effects, including galaxy shape measurement errors,
selection effects, as well as point spread function (PSF) estimation errors
and other instrumental effects (Huterer et al., 2006; Kilbinger, 2015;
Mandelbaum, 2018; Taylor & Kitching, 2018; Pujol et al., 2020). Different
systematic effects can separately introduce additive and multiplicative
biases; for example, a $c$-bias can arise from anisotropy in the modelled PSF,
while an $m$-bias can result from PSF size errors (Huterer et al., 2006).
While it is desirable to estimate and correct for these biases theoretically
(eg. from PSF modelling), it may also be possible to (at least partially)
detect the $c$-bias empirically, eg. using null tests (van Uitert & Schneider,
2016). However, the latter is not possible for the $m$-bias, which must be
predicted, eg. from image simulations (Kannawadi et al., 2019; Pujol et al.,
2019), or inferred from self-calibration during the measurement process (Huff
& Mandelbaum, 2017; Sheldon & Huff, 2017; Sheldon et al., 2020). For
forthcoming stage-IV surveys, it is expected that the shape biases will need
to be constrained to the level of $c\lesssim 10^{-4}$, $m\lesssim 2\times
10^{-3}$ in order to obtain the required sensitivity (Massey et al., 2013).

In this paper, we investigate the effect of a spatially-varying multiplicative
bias in the shear measurement on cosmological parameters estimated from the
cosmic shear angular power spectrum, $C_{\ell}^{\gamma\gamma}$. In forthcoming
surveys we will be concerned with residual shear biases, i.e. biases arising
due to imperfect calibration of systematics as a result of uncertainties in
systematics models. Such a bias could arise e.g. when there are position-
dependent errors in the PSF model. In particular, multiplicative bias is
expected to occur due to errors in the assumed size of the PSF (Paulin-
Henriksson et al., 2008). In optical weak-lensing surveys the typical imaging
field size is $\lesssim 1^{\circ}$, corresponding to multipole $\ell\gtrsim
200$, which provides a natural characteristic scale on which the $m$-bias may
vary (Euclid Collaboration, 2020a). Additionally, there are expected to be PSF
variations, and hence the possibility of PSF modelling errors, down to
arcminute scales ($\ell\sim 10^{3}$ $-$ $10^{4}$), motivating us to study the
effect of a spatially-varying $m$-bias on a wide range of scales. There may
also be spatially-varying $m$-bias arising from other effects in the shear
measurement, such as may arise from variations in galaxy density or
morphology.

Previous work has been conducted in this area, in particular by Kitching et
al. (2019), Kitching et al. (2020), in which it was found that a spatially-
varying $m$-bias has a negligible effect on the shear power spectrum compared
with the mean of the $m$-bias field, and should be negligible also when the
$m$-bias field averages to zero across the sky. However, Kitching et al.
(2019) did not extend the fully non-linear analytic calculation beyond a
maximum multipole of $\ell_{\mathrm{max}}=$ 32 (64 in Kitching et al. (2020)).
In Kitching et al. (2020), a linear approximation of the bias in
$C_{\ell}^{\gamma\gamma}$ was also calculated up to $\ell_{\mathrm{max}}=2048$
and compared with a forward model, with differences found that were $\sim 4$
orders of magnitude smaller than the cosmic variance. Kitching et al. (2020)
also calculated the bias in the estimated amplitude of the shear power
spectrum resulting from marginalisation over a prior on the mean $m$, and
placed requirements on the bias and variance of that distribution with respect
to the true mean $m$. That work did not consider the parameter bias arising
due to the convolutive effect on the shear $C_{\ell}$ arising due to the
spatial variations of the $m$-bias field.

Kitching et al. (2019), Kitching et al. (2020) propagated shear biases into
biases in the $C_{\ell}^{\gamma\gamma}$ using a fully general calculation
which allowed for an $m$-bias field which is a complex-valued spin field (i.e.
the m-bias field was allowed to have a rotation), resulting in a rapid scaling
of the computational time required as $\ell_{\mathrm{max}}^{5}$. We show that
the resulting stringent upper limit on the range of multipoles that can be
included in the analysis is insufficient, because a spatially-varying $m$-bias
with some characteristic scale (which may be large or small) will bias the
$\smash{C_{\ell}^{\gamma\gamma}}$ down to small scales due to mode-mode
coupling. In this work, we restrict our analysis to consider a real-valued
spin-0 $m$-bias, as is commonly assumed in shear analyses and is expected to
result from realistic shear systematics; the real part was predicted to be
dominant compared to the imaginary part of the $m$-bias field by Kitching et
al. (2019). We apply a computationally efficient pseudo-$C_{\ell}$ formalism
taking advantage of the resulting symmetries to determine the effect on the
shear power spectrum of a spatially-varying $m$-bias. Exploiting the
assumption of a spin-0 $m$-bias field, this method scales as
$\ell_{\mathrm{max}}^{3}$ (Alonso et al., 2019). This allows us to compute the
bias in the $C_{\ell}^{\gamma\gamma}$ down to scales corresponding to
$\ell_{\mathrm{max}}\sim 10^{3}-10^{4}$. The calculation to high $\ell$ is
tractable because, in the case of a spin-0 $m$-bias, orthogonality of the
Wigner 3-j symbols involved in the calculation of the mode-coupling matrix
reduces the dimensionality of the calculation. We consider the effect of a
random $m$-bias map generated according to an angular power spectrum with some
characteristic scale defined by a peak multipole $\ell_{\mathrm{peak}}$ and
width $\sigma_{m}$. In our analysis we consider the bias in the
$\smash{C_{\ell}^{\gamma\gamma}}$ arising due to a range of values for
$\ell_{\mathrm{peak}}$ and $\sigma_{m}$ and subsequently utilise a Fisher-
matrix analysis to predict the sensitivity of cosmological parameters to the
spatially-varying $m$-bias for a typical next-generation survey. While
previous works have propagated shear biases into biases in cosmological
parameters using a Fisher matrix analysis, these have either ignored the
spatial variations of the $m$-bias (eg. Amara & Réfrégier (2008)), relied on
assumptions about the $\ell$-coupling of the $C_{\ell}$s (eg. Massey et al.
(2013); Kitching et al. (2016)), or did not carry out the full propagation of
the shear bias from angular space (eg. Taylor & Kitching (2018), which
propagated the shear bias power spectra into dark energy figures of merit);
this work is the first to calculate the full mode-coupling matrix
corresponding to $m$-bias maps and propagate into biases in cosmological
parameters. We also compute the $BB$ power spectra which result from the mode-
mixing effect arising due to the spatially-varying $m$-bias and compare it to
the bias in the $EE$ power spectra. Note that we do not consider the additive
shear bias in the present work, as its form is expected to be strongly
dependent on the particular systematics from which it results, whereas here we
consider a generic form for the $m$-bias which may be broadly relevant to
future surveys. We are primarily interested in the effect of the $m$-bias at
high multipoles in the $C_{\ell}^{\gamma\gamma}$, and defer investigation of
the $c$-bias to future work.

This paper is organised as follows. In section 2 we describe our methods,
including the generation of our fiducial $C_{\ell}^{\gamma\gamma}$s, $m$-bias
maps, the pseudo-$C_{\ell}$ formalism used to calculate the biased
$C_{\ell}^{\gamma\gamma}$s, and the Fisher-matrix analysis used to forecast
resulting biases in cosmological parameters. In section 3 we show our results,
including the residual $C_{\ell}^{\gamma\gamma}$s, and biases and the ratios
of biases to the respective $1\sigma$ uncertainty in cosmological parameter
estimates. In section 4 we discuss the results of our analysis, and in section
5 we summarise the findings of our work.

##  2 Methods

###  2.1 Pseudo-$C_{\ell}$ formalism

Consider a spin-$s_{a}$ vector field $\bm{a}(\bm{\Omega})$, where
$\bm{\Omega}=(\theta,\phi)$ is the angular unit vector on the sphere, which is
observed with a mask (or set of weights) $W_{a}(\bm{\Omega})$, such that the
observed field
$\widetilde{\bm{a}}(\bm{\Omega})=\bm{a}(\bm{\Omega})W_{a}(\bm{\Omega})$. We
may calculate the spherical harmonic expansion coefficients of the masked
field as follows (Hivon et al., 2002; Kogut et al., 2003):

$\displaystyle\widetilde{\bm{a}}_{\ell m}$
$\displaystyle=\int_{\bm{\Omega}}d\bm{\Omega}\mathsf{Y}_{\ell
m}^{s_{a}{\dagger}}(\bm{\Omega})\bm{a}(\bm{\Omega})W_{a}(\bm{\Omega})$ (2)
$\displaystyle=\sum_{\ell\textquoteright
m\textquoteright}\int_{\bm{\Omega}}d\bm{\Omega}W_{a}(\bm{\Omega})\left(\mathsf{Y}_{\ell
m}^{s_{a}{\dagger}}(\bm{\Omega})\cdotp\mathsf{Y}_{\ell\textquoteright
m\textquoteright}^{s_{a}}(\bm{\Omega})\right)\cdotp\bm{a}_{\ell\textquoteright
m\textquoteright}$ (3) $\displaystyle\equiv\sum_{\ell\textquoteright
m\textquoteright}\mathsf{K}_{\ell m\ell\textquoteright
m\textquoteright}^{s_{a}}[W_{a}]\cdotp\bm{a}_{\ell\textquoteright
m\textquoteright}\ ,$ (4)

where $\mathsf{Y}_{\ell m}^{s_{a}}$ is a $2\times 2$ matrix defined in terms
of the spin-$s_{a}$-weighted spherical harmonic of degree $\ell$ and order $m$
(Zaldarriaga & Seljak, 1997; Alonso et al., 2019), and superscript ${\dagger}$
here denotes conjugate transposition; and $\mathsf{K}_{\ell
m\ell\textquoteright m\textquoteright}^{s_{a}}[W]$ is the spin-$s_{a}$ mode-
mixing kernel describing the mode-mode coupling of a spin-$s_{a}$ field due to
$W_{a}$, and is also a $2\times 2$ matrix. If we also consider a spin-$s_{b}$
vector field $\bm{b}(\bm{\Omega})$ observed with mask $W_{b}(\bm{\Omega})$,
then the $2\times 2$ matrix of angular power spectra of vector fields
$\widetilde{\bm{a}}(\bm{\Omega})$ and $\widetilde{\bm{b}}(\bm{\Omega})$ is
estimated according to the following pseudo-$C_{\ell}$ formalism (Hivon et
al., 2002; Kogut et al., 2003; Alonso et al., 2019):

$\displaystyle\mathrm{vec}$
$\displaystyle\left[\langle\widetilde{\mathsf{C}}_{\ell}^{\widetilde{a}\widetilde{b}}\rangle\right]=\frac{1}{2\ell+1}$
$\displaystyle\times\sum_{\ell^{\prime}}\sum_{mm^{\prime}}\left(\mathrm{vec}\left[\mathsf{K}_{\ell
m\ell^{\prime}m^{\prime}}^{s_{a}}[W_{a}]\right]\cdotp\mathrm{vec}\left[\mathsf{K}_{\ell
m\ell^{\prime}m^{\prime}}^{s_{b}}[W_{b}]\right]^{{\dagger}}\right)$
$\displaystyle\;\cdotp\mathrm{vec}\left[\langle\mathsf{C}_{\ell^{\prime}}^{ab}\rangle\right]$
(5)
$\displaystyle\equiv\sum_{\ell^{\prime}}\mathsf{M}_{\ell\ell^{\prime}}^{s_{a}s_{b}}\mathrm{vec}\left[\langle\mathsf{C}_{\ell^{\prime}}^{ab}\rangle\right]\
,$ (6)

where the vectorisation operation $\mathrm{vec}[\mathsf{x}]$ creates an
$ab\times 1$ vector from an $a\times b$ matrix $\mathsf{x}$ by transposing
rows and concatenating them into a single column vector (Hamimeche & Lewis,
2008). Note that the angle brackets denote an ensemble average; we will
henceforth omit these for brevity.
$\mathsf{M}_{\ell\ell\textquoteright}^{s_{a}s_{b}}$ is the mode-coupling
matrix which convolves the true power spectra
$\mathsf{C}_{\ell\textquoteright}^{ab}$ with the cross power spectrum of the
masks, and is calculated from the spin-weighted spherical harmonic
decompositions of $W_{a}(\bm{\Omega})$ and $W_{b}(\bm{\Omega})$. Considering
equation $(1)$, we can consider $1+m(\bm{\Omega})$ as a non-binary mask that
is applied to the true shear field $\bm{\gamma}(\bm{\Omega})$, and can
therefore calculate the pseudo-$C_{\ell}$ of the biased shear field using
equation $(6)$. Note that for a spin-$2$ field such as the cosmic shear, the
mode-coupling matrix also mixes $E$\- and $B$-modes, generating $BB$-power
which is otherwise expected to be subdominant in the intrinsic shear field.

As equation (5) includes a sum over three indices of a product that is
integrated over the whole sphere, calculation of the mode-coupling matrix is
computationally expensive in the general case. However, it is possible to
express the mode-mixing kernels in terms of Wigner 3-$j$ symbols (Edmonds,
1957; Hivon et al., 2002; Kogut et al., 2003). In the case that the $m$-bias
is isotropic such that it affects both components of the shear equally, as we
consider here and is likely to be the case in real surveys, the Wigner 3-$j$
symbols obey an orthogonality relation that allows us to collapse the
summation in indices $m_{1},m_{2}$, greatly simplifying the calculation to a
sum over the angular cross-power spectrum of the masks weighted by 3-$j$
symbols (Kogut et al., 2003). NaMaster employs this method (and additionally
replaces in the calculation the $C_{\ell}$ of the masks with their
pseudo-$C_{\ell}$, which is more computationally efficient to calculate)
(Alonso et al., 2019). This allows for a significant reduction in the
computational cost of the calculation of the angular power spectra of masked
fields, reducing the scaling of the calculation by a factor of
$\ell_{\mathrm{max}}^{2}$, thereby allowing calculation beyond the maximum
multipoles that would be of interest for stage IV weak-lensing experiments, an
improvement (subject to our assumption of spin-0 $m$) on the scaling found in
the general calculation by Kitching et al. (2019).

The mode-coupling matrix $\mathsf{M}_{\ell\ell\textquoteright}^{s_{a}s_{b}}$
depends only on the power spectrum of the mask (Alonso et al., 2019). As the
clustering of the $m$-bias map considered here is by construction fully
described by its angular power spectrum, it is not necessary to consider
multiple realisations of the $m$-bias map for each of our values of
$\ell_{\mathrm{peak}}$, $\sigma_{m}$ and $m$-bias rms. It should be noted
however that NaMaster takes as its input the mask and computes an estimate of
the mask $C_{\ell}$, which is used to compute the mode-coupling matrix. In our
case, this mask is a noisy realisation of the $m$-bias map, which is a random
field. This results in a noisy estimate of the $m$-bias map $C_{\ell}$ which
is used to calculate the $\mathsf{M}_{\ell\ell\textquoteright}^{s_{a}s_{b}}$.
While this results in a loss of optimality of the
$\mathsf{M}_{\ell\ell\textquoteright}^{s_{a}s_{b}}$ estimate, this is not
expected to be a significant effect when the smallest scales $\gtrsim 10^{4}$
are neglected, and so we did not modify NaMaster to take the $m$-bias map
$C_{\ell}$ as its input directly in order to avoid this effect. This
modification would be trivial in a practical application. Additionally, the
$m$-bias $C_{\ell}$ will not be known in a real survey, and in a realistic
case propagation from angular space (rather than $\ell$-space) will be
necessary; this work is also intended to serve as a proof of concept that the
full propagation from angular space to $m$-bias maps to cosmological parameter
biases is tractable.

In the case of a spatially varying multiplicative bias as considered here, the
biased shear field is:

$\widetilde{\bm{\gamma}}(\bm{\Omega})=\bm{\gamma}(\bm{\Omega})(1+m(\bm{\Omega}))\
.$ (7)

We will henceforth refer to $1+m(\bm{\Omega})$ as the $m$-bias map. We can
therefore apply a pseudo-$C_{\ell}$ approach to describe the effect of a
spatially-varying $m$-bias on the shear $C_{\ell}$s. This will allow us to
compute the bias in the cosmic shear angular power spectra due to
$m({\bf{\Omega}})$ with a weaker scaling with $\ell_{\mathrm{max}}$ than was
obtained in previous work in this area (eg. Kitching et al. (2019, 2020)),
enabling us to probe the impact of the spatially-varying $m$-bias out to much
higher multipoles.

Note that here we have neglected both the $c$-bias and any separate spatially
constant $m$-bias. While $m(\bm{\Omega})$ could in general contain a spatially
constant term, we are seeking specifically to investigate the impact of the
spatial variations which have previously been claimed to have negligible
effect. Moreover, since the shear is a spin-2 vector field, in general it can
be decomposed into $E$\- and $B$-modes, and there are therefore in principle
four power spectra, $C_{\ell}^{EE}$, $C_{\ell}^{BB}$, $C_{\ell}^{BE}$ and
$C_{\ell}^{EB}$; however, in the absence of certain systematic effects
(including intrinsic alignments, selection effects and some PSF correction
effects), the $B$-mode field is expected to be highly sub-dominant (Kilbinger,
2015) (though not vanishing, as source clustering is expected to introduce
some $B$-modes (Schneider et al., 2002)). We will therefore not consider any
intrinsic $B$-mode field. In our analysis, we calculate the biased $EE$ power
spectrum as well as the $BB$ power spectrum generated by the mode-coupling
arising due to $m(\bm{\Omega})$. No $E$-$B$ cross-correlation is expected to
be generated in the presence of a spin-0 $m$-bias because the parity-odd $EB$,
$BE$ power spectra are not mixed with the parity even $EE$, $BB$ power spectra
(Alonso et al., 2019). As the shear field is expected to be approximately
$B$-mode-free, any observed $B$-mode power may be used to detect the presence
of and potentially calibrate residual systematic effects (Asgari et al.,
2021). In particular, the $BB$ power spectra contain information about the
$m$-bias field which may in principle be used for self-calibration (Kitching
et al., 2020). However, we do not consider this in our analysis due to
uncertainty in the viability of self-calibration of the $m$-bias using the
$BB$ power, arising due to potential degeneracies in the $B$-modes generated
by different systematics, according to our systematics-agnostic approach.

We do not consider any redshift-dependence of the $m$-bias, and apply the same
$m$-bias to all of the tomographic shear power spectra we consider. In a real
survey there will be a redshift-dependence of the $m$-bias, eg. due to the
decrease in observed galaxy size with redshift; as we here consider a generic
form for the $m$-bias, without reference to a specific bias model, we defer
consideration of this effect to future work. We also conduct a full-sky
analysis and so do not consider the mode-coupling effect due to any survey
mask boundary.

We consider a randomly-drawn $m$-bias map which is distributed according to an
angular power spectrum which has a Gaussian profile as a function of $\ell$.
We choose a certain peak multipole for the $m$-bias $C_{\ell}$,
$\ell_{\mathrm{peak}}$, and a standard deviation, $\sigma_{m}$. The $m$-bias
$C_{\ell}$ is normalised such that the resulting $m$-bias map has r.m.s. value
equal to 0.01, a nominal fiducial value which is a reasonable expectation for
a forthcoming stage-IV survey such as Euclid. We also investigate how our
results vary with this value in order to allow requirements to be set on the
amplitude of spatial variations of the $m$-bias. We consider $m$-bias
$C_{\ell}$s with $\ell_{\mathrm{peak}}$ in the range $50-5000$ and
$\sigma_{m}$ in the range $64-2048$.

###  2.2 Survey model

For the fiducial shear power spectra, we assume a redshift distribution of
galaxies $n(z)$ of the following form (Smail et al., 1994):

$n(z)\propto\bigg{(}\frac{z}{z_{0}}\bigg{)}^{2}\exp\Bigg{[}-\bigg{(}\frac{z}{z_{0}}\bigg{)}^{\frac{3}{2}}\Bigg{]}\
,$ (8)

which is normalised to unity, and where $z_{0}=z_{m}/\sqrt{2}$ with $z_{m}$
the median redshift of the distribution (Euclid Collaboration, 2020b). We
divide this distribution into 10 equipopulated redshift bins up to a maximum
redshift of 3, and convolve the distribution in each bin $n_{i}(z)$ with a
probability density function with two slightly offset Gaussian terms, given by
Euclid Collaboration (2020b), to account for the effect of photometric
redshift errors with catastrophic outliers. The resulting $n_{i}(z)$ are shown
in fig. 1.

Figure 1: Gaussian-convolved $n(z)$ for the 10 tomographic redshift bins.

We calculate fiducial shear auto- and cross- ($EE$) power spectra for each
pair of bins using CCL (Chisari et al., 2019), denoted
$\smash{C_{\ell}^{\gamma_{i}\gamma_{j}}}$ for bin pair $(i,j)$. We assume
spatially-flat $\Lambda$CDM with the Planck 2018 best-fit cosmological
parameter values as our fiducial values for cosmological parameters (see table
1, Planck Collaboration (2020)). We do not include massive neutrinos in our
fiducial cosmological model. In our Fisher-matrix analysis, we consider dark
energy with a dynamical equation of state in which the parameters $w_{0}$ and
$w_{a}$ are allowed to vary, with fiducial values of $-1$ and $0$
respectively.

Table 1: Fiducical $\Lambda$CDM cosmological parameter values - cold dark matter density parameter $\Omega_{c}$; baryonic matter density parameter $\Omega_{b}$; reduced Hubble constant $h$; amplitude of the matter power spectrum $\sigma_{8}$; scalar spectra index $n_{s}$ (values from Planck Collaboration (2020), table 2, final column); dark energy equation-of-state parameters $w_{0}$ and $w_{a}$. Cosmological parameter | Fiducial value
---|---
$\Omega_{c}$ | $0.2607$
$\Omega_{b}$ | $0.0490$
$h$ | $0.6766$
$\sigma_{8}$ | $0.810$
$n_{s}$ | $0.9665$
$w_{0}$ | $-1$
$w_{a}$ | $0$

To each of the auto-power spectra $C_{\ell}^{\gamma_{i}\gamma_{i}}$ we add a
constant shape-noise term $n$, given by (Euclid Collaboration, 2020b) for the
case of equipopulated redshift bins:

$n=\frac{\sigma_{e}^{2}N_{z}}{n_{\mathrm{gal}}}\ ,$ (9)

where $\sigma_{e}^{2}$ is the intrinsic ellipticity variance of source
galaxies, $N_{z}$ is the number of tomographic redshift bins and
$n_{\mathrm{gal}}$ is the sky number density of source galaxies; following
Euclid Collaboration (2020b), we take these to be
$\sigma_{e}^{2}=0.3^{2}=0.09$, $N_{z}=10$ and $n_{\mathrm{gal}}=30\
\mathrm{arcmin}^{-2}$. For each $m$-bias map under study, we use NaMaster to
compute the mode-coupling matrix $\mathsf{M}_{\ell\ell\textquoteright}^{22}$
and couple this with each $\smash{C_{\ell}^{\gamma_{i}\gamma_{j}}}$ to obtain
biased shear power spectra
$\widetilde{C}_{\ell}^{\widetilde{\gamma}_{i}\widetilde{\gamma}_{j}}$. We then
calculate residual $C_{\ell}$s:

$\Delta
C_{\ell}^{\gamma_{i}\gamma_{j}}=C_{\ell}^{\gamma_{i}\gamma_{j}}-\widetilde{C}_{\ell}^{\widetilde{\gamma}_{i}\widetilde{\gamma}_{j}}\
.$ (10)

Residual shear $C_{\ell}$s are computed using NaMaster up to
$\ell_{\mathrm{cut}}=12288$ (this is set by the resolution of the input
Healpix map, where we choose $N_{\mathrm{side}}=\ell_{\mathrm{cut}}/3=4096$).
We calculate the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$ up to higher
multipoles than we consider in our subsequent Fisher matrix analysis in order
to obtain more accurate results for the mode-mixing at the scales of interest.

###  2.3 Fisher matrix analysis

We conduct a Fisher matrix analysis to forecast biases in cosmological
parameters propagated from the bias in the shear $C_{\ell}$s due to the
spatially-varying $m$-bias. Given a data vector $\bm{D}_{\ell}$ with non-zero
mean consisting of angular power spectra at a particular $\ell$ and parameter-
independent data covariance matrix $\mathsf{Cov}(\ell)$, the Fisher matrix can
be approximated as (Tegmark et al., 1997; Duncan et al., 2014):

$F_{\eta\tau}=\sum_{\ell}(\partial_{\eta}\bm{D}_{\ell})(\mathsf{Cov}^{-1})(\ell)\partial_{\tau}\bm{D}_{\ell}\
,$ (11)

where $F_{\eta\tau}$ is the Fisher matrix and $\partial_{\tau}$ denotes
partial differentiation with respect to parameter $\theta_{\tau}$. We take for
our data vector at each $\ell$ the vector composed of our fiducial
$\smash{C_{\ell}^{\gamma_{i}\gamma_{j}}}$s, which we denote as $\bm{C}_{\ell}$
for brevity; i.e.
$\bm{C}_{\ell}=\\{C_{\ell}^{\gamma_{0}\gamma_{0}},C_{\ell}^{\gamma_{0}\gamma_{1}},\dotsb,C_{\ell}^{\gamma_{(N_{z}-1)}\gamma_{(N_{z}-1)}}\\}$,
where the redshift bin indices run from 0 to 9, and $\mathsf{Cov}(\ell)$ is
the $55\times 55$ covariance matrix of these power spectra at multipole
$\ell$, i.e. a submatrix of the $55N_{\ell}\times 55N_{\ell}$ full block-
diagonal covariance matrix, with $N_{\ell}$ the number of $\ell$-modes
considered.

The derivatives of the data vector $\partial_{\tau}\bm{C}_{\ell}$ are
approximated with a simple two-step derivative:

$\partial_{\tau}\bm{C}_{\ell}\approx\frac{\bm{C}_{\ell}(\theta_{\tau,\mathrm{fid}}+\delta\theta_{\tau})-\bm{C}_{\ell}(\theta_{\tau,\mathrm{fid}}-\delta\theta_{\tau})}{2\delta\theta_{\tau}}\
,$ (12)

where $\theta_{\tau,\mathrm{fid}}$ is the fiducial value of parameter
$\theta_{\tau}$ and $\delta\theta_{\tau}$ is a small derivative step in
$\theta_{\tau}$. We conduct our analysis with two sets of cosmological
parameters for which cosmic shear is expected to be sensitive. In our base
parameter set, we consider the cold-dark-matter density parameter,
$\Omega_{c}$, that is the ratio of the density of cold dark matter to that of
the critical density at the present epoch; $\sigma_{8}$, the amplitude of the
linear matter power spectrum on scales of 8$h^{-1}\mathrm{Mpc}$; and the
constant dark energy equation-of-state parameter, $w_{0}$. In our extended
parameter set, we consider the base parameter set in addition to a parameter
capturing the redshift-variation of the dark energy equation-of-state,
$w_{a}$, where the evolution of the equation of state parameter is
parametrised by $w(z)=w_{0}+w_{a}z/(1+z)$. We take for our derivative steps in
these parameters $\\{\delta\Omega_{c}=0.01,\delta\sigma_{8}=0.005,\delta
w_{0}=0.055,\delta w_{a}=0.058\\}$. Numerical stability of the derivatives was
ensured by selecting values of the derivative steps such that all the Fisher
matrix elements were stable with respect to perturbations in the region of the
step choice. We consider these parameters to be "free" in our analysis,
considering the impact on their inference from the
$\smash{C_{\ell}^{\gamma\gamma}}$ due to bias in the shear power spectra,
whereas the other parameters listed in table $(1)$ ($h$, $\Omega_{b}$,
$n_{s}$) are assumed to be well-known and fixed to their fiducial values. In
our results, we present the bias in $\Omega_{m}$, the total matter density
parameter, which in $\Lambda$CDM is the sum of $\Omega_{c}$ and $\Omega_{b}$
(the density parameter of baryonic matter); since we only vary $\Omega_{c}$
and fix the value of $\Omega_{b}$ to the Planck Collaboration (2020) value,
the Fisher matrix elements and the resulting biases that are obtained by
varying $\Omega_{c}$ are also the correct results for $\Omega_{m}$ .

The covariance between a pair of shear power spectra
$(C_{\ell}^{\gamma_{i}\gamma_{j}},C_{\ell\textquoteright}^{\gamma_{p}\gamma_{q}})$
in the Gaussian approximation is given by (Joachimi & Bridle, 2010):

$\mathrm{Cov}[C_{\ell}^{\gamma_{i}\gamma_{j}},C_{\ell\textquoteright}^{\gamma_{p}\gamma_{q}}]\\\
=\delta_{K}^{\ell\ell\textquoteright}\frac{2\pi}{(2\ell+1)f_{sky}\Delta\ell}\times\\{C_{\ell}^{\gamma_{i}\gamma_{p}}C_{\ell}^{\gamma_{j}\gamma_{q}}+C_{\ell}^{\gamma_{i}\gamma_{q}}C_{\ell}^{\gamma_{j}\gamma_{p}}\\}\\\
\equiv\mathrm{Cov}^{ijpq}(\ell)\ ,$ (13)

where $\delta_{K}^{\ell\ell\textquoteright}$ is the Kronecker delta, $f_{sky}$
is the fraction of the sky covered by the survey, and $\Delta\ell$ is the
width of multipole bandpowers. We take $f_{sky}$ to be 0.36, corresponding to
a 15,000 $\mathrm{deg}^{2}$ survey such as Euclid (Euclid Collaboration,
2020b). In a real survey analysis, the sky mask will introduce off-diagonal
covariance on scales corresponding to the mask window function. In this case,
it is necessary to bin the power spectra and covariance matrix into bandpowers
in order for the diagonal covariance assumption to hold. While we do not apply
our analysis to the cut sky, we incorporate its effect at the Fisher level by
binning our data vector and covariance into bandpowers with a nominal value of
$\Delta\ell=10$, and scaling the covariance by $f_{sky}$.

Given our Fisher matrix, fiducial angular power spectra and bias in the power
spectrum $\Delta C_{\ell}^{I}$ corresponding to a tomographic bin pair
$I\equiv(\gamma_{i},\gamma_{j})$, the resulting contribution to the bias in
some cosmological parameter $\theta_{\eta}$ is given by the following (Amara &
Réfrégier, 2008; Thiele et al., 2020):

$\displaystyle\Delta\theta_{\eta}^{I}$
$\displaystyle=\sum_{\tau}(\mathsf{F}^{-1})_{\eta\tau}\sum_{J}\sum_{\ell}(\partial_{\tau}C_{\ell}^{J})(\mathsf{Cov}^{-1})^{J,I}(\ell)\Delta
C_{\ell}^{I}\ ,$ (14)

where the covariance matrix $\mathrm{Cov}^{J,I}(\ell)$ is given by (13). The
bias on parameter $\theta_{\eta}$ is then obtained by summing the
contributions from each bin:

$\Delta\theta_{\eta}=\sum_{I}\Delta\theta_{\eta}^{I}\ .$ (15)

The marginalised $1\sigma$ uncertainty of parameter $\theta_{\alpha}$ is given
by:

$\sigma_{\alpha}=\sqrt{\left(\mathsf{F}^{-1}\right)_{\alpha\alpha}}\ ,$ (16)

and we calculate the ratio of our calculated bias in parameter
$\theta_{\alpha}$ to the corresponding 1-dimensional $1\sigma$ uncertainty as:

$\mathrm{Sig}[\Delta\theta_{\alpha}]=\frac{\Delta\theta_{\alpha}}{\sigma_{\alpha}}\
.$ (17)

Henceforth, we refer to the multipole at which we truncate the calculation of
the mode-coupling matrix as $\ell_{\mathrm{cut}}$ and the maximum multipole
considered in our Fisher analysis as $\ell_{\mathrm{max}}$. In our Fisher
analysis, we consider a low $\ell$-cut of $\ell_{\mathrm{min}}=10$. We apply a
range of high $\ell$-cuts in the range $\ell_{\mathrm{max}}=1500-5000$ at the
Fisher level, in order to see the effect of including or excluding different
scales on the Fisher matrix and parameter biases. Such a cut will be necessary
in any real survey, since at high multipoles non-linear modes of the power
spectrum contaminate the $C_{\ell}$s, biasing them with respect to the linear
theory prediction and degrading the resulting cosmological parameter
constraints. According to Euclid Collaboration (2020b),
$\ell_{\mathrm{max}}=1500$ corresponds to a pessimistic $\ell$-cut, whereas
$\ell_{\mathrm{max}}=5000$ is expected to be optimistic. These cuts are
applied both in the computation of the Fisher matrix (11), and of the
parameter biases, (14). We apply our analysis to $\smash{\Delta
C_{\ell}^{\gamma\gamma}}$s computed with a range of values for the the peak
multipole of the $m$-bias $C_{\ell}$, in the range
$\ell_{\mathrm{peak}}=50-5000$, and fix $\sigma_{m}$, the width of the
$m$-bias $C_{\ell}$, to $\sigma_{m}=64$. Additionally, we conduct our analysis
keeping $\ell_{\mathrm{max}}$ fixed, and vary $\sigma_{m}$ in the range
$64-2048$ and $\ell_{\mathrm{peak}}$ in the range $100-1100$, in order to test
the validity of our choice of $\sigma_{m}$. In our Fisher matrix analysis we
consider first the parameter set $\\{\Omega_{m},\sigma_{8},w_{0}\\}$ (varying
$\Omega_{c}$ with $\Omega_{b}$ fixed at the Planck Collaboration (2020) value)
in addition to the extended parameter set
$\\{\Omega_{m},\sigma_{8},w_{0},w_{a}\\}$ and compare the results.

##  3 Results

###  3.1 Residual power spectra

Residual power spectra $\smash{\Delta C_{\ell}^{\gamma_{i}\gamma_{j}}}$
arising due to the spatially-varying $m$-bias are computed by applying the
pseudo-$C_{\ell}$ formalism described in section $(2.1)$. Examples of the
fractional residual power spectra $\Delta
C_{\ell}^{\gamma_{i}\gamma_{j}}/C_{\ell}^{\gamma_{i}\gamma_{j}}$ are shown in
fig. 2, with $\ell_{\mathrm{peak}}$ in the range $50-5000$. As illustrated in
fig. 2, the residual power spectra are generally at least $\sim 4$ orders of
magnitude below the fiducial $C_{\ell}$s. As can be seen, the structure of the
residual $C_{\ell}$s is highly dependent on the scale of the $m$-bias
$C_{\ell}$, $\ell_{\mathrm{peak}}$. There is typically a minimum in the
$\Delta C_{\ell}^{\gamma\gamma}/C_{\ell}^{\gamma\gamma}$ at $\ell\lesssim 10$
with an enhancement at higher $\ell$. In the auto-spectra, this enhancement
has a gentle gradient and plateau with increasing $\ell$, whereas for the
cross-spectra with high $\ell_{\mathrm{peak}}$ the enhancement is more sharply
peaked. There is also a smaller drop at still higher $\ell$, with this
structure shifting to higher $\ell$ and increasing in strength with increasing
$\ell_{\mathrm{peak}}$. The high-$\ell$ smaller minimum of the fractional
residual $C_{\ell}$s is typically located at $\ell$ greater than
$\ell_{\mathrm{peak}}$ by approximately 500. For lower values of
$\ell_{\mathrm{peak}}$ the residual $C_{\ell}$s are enhanced across all
$\ell$, and they also vary more smoothly than for higher
$\ell_{\mathrm{peak}}$.

Figure 2: Ratio of residual to fiducial angular power spectra as a function of
peak multipole of $m$-bias $C_{\ell}$, with $\sigma_{m}=64$ and $m$-bias rms
$=0.01$. Black dashed line: Maximum value of $\ell_{\mathrm{max}}$ considered
in the Fisher analysis. Top panel: autocorrelation of first tomographic
redshift bin $\Delta
C_{\ell}^{\gamma_{0}\gamma_{0}}/C_{\ell}^{\gamma_{0}\gamma_{0}}$. Bottom
panel: cross-correlation of first and fifth tomographic redshift bins $\Delta
C_{\ell}^{\gamma_{0}\gamma_{4}}/C_{\ell}^{\gamma_{0}\gamma_{4}}$.

####  3.1.1 $BB$ power arising due to mode mixing

We also calculate the $BB$ power spectra arising due to the mode-mixing effect
of the spatially-varying $m$-bias, and examples of the $BB$ power spectra are
shown in fig. 3, for $\ell_{\mathrm{max}}=5000$ and $\ell_{\mathrm{peak}}=500$
and $5000$, alongside the corresponding $\Delta C_{\ell}^{EE}$, divided by the
fiducial $C_{\ell}^{EE}$. For high $\ell_{\mathrm{peak}}$, the $BB$ power
spectra are of the same order of magnitude and follow closely the $\Delta
C_{\ell}^{EE}$; for lower $\ell_{\mathrm{peak}}$ the generated $BB$ power
spectra follow the $\Delta C_{\ell}^{EE}$ less closely at high and low $\ell$.
The observed $BB$ power spectra may therefore be a useful diagnostic of a
residual spatially-varying $m$-bias, especially with a small characteristic
scale. While this will be complicated by the fact that there are many
systematics which may produce measurable shear $B$-modes, such that it will be
difficult to isolate the signal arising due to any given systematic in the
measured $B$-mode signal, it may be a useful diagnostic if different
systematics result in different signature effects in the resulting $BB$ power
spectra. As we consider a generic form for the $m$-bias map, we cannot make
further conclusions about the viability of the measured $B$-mode signal as a
systematics diagnostic in this work.

Figure 3: Comparison of $BB$ power spectra generated by the mode coupling
arising due to the spatially-varying $m$-bias $\widetilde{C}_{\ell}^{BB}$ with
the residual $EE$ power spectra $\Delta C_{\ell}^{EE}$, up to
$\ell_{\mathrm{max}}=5000$. Top panel: autocorrelation of first tomographic
redshift bin $\Delta
C_{\ell}^{\gamma_{0}\gamma_{0}}/C_{\ell}^{\gamma_{0}\gamma_{0}}$. Bottom
panel: cross-correlation of first and fifth tomographic redshift bins $\Delta
C_{\ell}^{\gamma_{0}\gamma_{4}}/C_{\ell}^{\gamma_{0}\gamma_{4}}$.

####  3.1.2 Numerical effects in the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$

As a test of our methods, we vary the multipole at which the calculation of
the mode-coupling matrix is truncated, $\ell_{\mathrm{cut}}$. In the top panel
of fig. 2, at high $\ell$ and for high $\ell_{\mathrm{peak}}$, the fractional
auto-$\Delta C_{\ell}$ curves can be seen to have sharp reductions at
$\ell\sim 8\times 10^{3}-10^{4}$. In order to determine whether this is a
numerical effect arising due to the truncation of the calculation of the mode-
coupling matrix at $\ell_{\mathrm{cut}}=12288$, the calculation of the $\Delta
C_{\ell}$s was also carried out with $\ell_{\mathrm{cut}}=6144$ (corresponding
to $N_{\mathrm{side}}=\ell_{\mathrm{cut}}/3=2048$). We show in fig. 4 the
effect of this on examples of the auto- and cross-power spectra for a range of
values of $\ell_{\mathrm{peak}}$. As can be seen in the top panel, for the
auto-$\Delta C_{\ell}$s, the gentle minimum at intermediate $\ell$ followed by
a rise and plateau at higher $\ell$ is robust against changes to
$\ell_{\mathrm{cut}}$ and is therefore likely to be a real feature. The same
is also true of the sharp peak in the cross-$\Delta C_{\ell}$s. However, for
$\ell_{\mathrm{peak}},\ell\gtrsim 1.25N_{\mathrm{side}}$, the auto-$\Delta
C_{\ell}$s contain a spurious sharp minimum due to the truncation of the
calculation of the mode-coupling matrix. Indeed, when $\ell_{\mathrm{peak}}$
exceeds $\sim 1.25N_{\mathrm{side}}$ by more than a factor of $\sim 2$, the
calculated $\Delta C_{\ell}$s are biased down to smaller $\ell$ (as can be
seen in the highest $\ell_{\mathrm{peak}}$ curves in the disparity between the
two $\ell_{\mathrm{cut}}$ cases). For $\ell_{\mathrm{cut}}=12288$,
$1.25N_{\mathrm{side}}$ corresponds to $\ell\sim 5000$ (indicated with a
vertical red dashed line in each panel). This justifies the maximum multipole
$\ell_{\mathrm{max}}$ at which we truncate our residual power spectra in our
Fisher-matrix analysis; in addition to corresponding to the optimistic Euclid
setting (Euclid Collaboration, 2020b), this also prevents contamination of our
analysis by edge effects. Due to this effect, we also neglect
$\ell_{\mathrm{peak}}>5000$ in our Fisher-matrix analysis; this corresponds to
scales of $\lesssim 2$ arcminutes, so that we still probe scales on which
spatially-varying $m$-bias might be expected to arise due to PSF errors. This
effect was also tested for the other tomographic power spectra with similar
results.

Figure 4: Ratio of residual to fiducial angular power spectra as a function of
peak multipole of $m$-bias $C_{\ell}$, with $\sigma_{m}=64$ and $m$-bias rms
$=0.01$, showing the effect of the truncation of the calculation of the mode-
coupling matrix at a particular multipole $\ell_{\mathrm{cut}}$. Bold lines:
$\ell_{\mathrm{cut}}=3N_{\mathrm{side}}=12288$. Thin lines:
$\ell_{\mathrm{cut}}=3N_{\mathrm{side}}=6144$. Vertical dashed lines
correspond to the maximum threshhold values of $\ell_{\mathrm{max}}$ that
could be considered in a subsequent Fisher analysis, equal to
$1.25N_{\mathrm{side}}$; blue for the $\ell_{\mathrm{cut}}=6144$ case and red
for the $\ell_{\mathrm{cut}}=12288$ case which we consider. Top panel:
autocorrelation of first tomographic redshift bin $\Delta
C_{\ell}^{\gamma_{0}\gamma_{0}}/C_{\ell}^{\gamma_{0}\gamma_{0}}$. Bottom
panel: cross-correlation of first and fifth tomographic redshift bins $\Delta
C_{\ell}^{\gamma_{0}\gamma_{4}}/C_{\ell}^{\gamma_{0}\gamma_{4}}$.

###  3.2 Bias in cosmological parameters

Using the Fisher matrix analysis described in section $(2.3)$, biases due to
shifts in the $\smash{C_{\ell}^{\gamma_{i}\gamma_{j}}}$ are forecast for the
base cosmological parameter set $\\{\Omega_{m},\sigma_{8},w_{0}\\}$ (varying
$\Omega_{c}$ in our Fisher-matrix analysis, with $\Omega_{b}$ fixed at the
Planck Collaboration (2020) value). We apply a high-$\ell$ cut
$\ell_{\mathrm{max}}$ in both the computation of the Fisher matrix (11) and of
the parameter biases (14), and repeat the analysis for a range of values of
$\ell_{\mathrm{max}}$. In fig. 5 we show the ratio of our calculated parameter
biases to the $1\sigma$ uncertainty as a function of $\ell_{\mathrm{peak}}$
and $\ell_{\mathrm{max}}$ with $\sigma_{m}$ fixed at 64. We next consider the
extended parameter set $\\{\Omega_{m},\sigma_{8},w_{0},w_{a}\\}$, and in fig.
6 show the ratio of our parameter biases to the $1\sigma$ uncertainty as a
function of $\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$ with
$\sigma_{m}$, the width of the $m$-bias $C_{\ell}$, fixed at 64. We also
conducted our analysis keeping $\ell_{\mathrm{max}}$ fixed, and varying
$\sigma_{m}$ in the range $64-2048$, and $\ell_{\mathrm{peak}}$ in the range
$100-1100$. We found that for all parameters considered, the biases and their
ratios to the $1\sigma$ uncertainty were insensitive to the value of
$\sigma_{m}$ for $\sigma_{m}\lesssim 500$, justifying our choice of
$\sigma_{m}=64$ for our main analysis. Larger values of $\sigma_{m}$ for a
given value of $m$-bias rms resulted in reduced cosmological parameter bias.

In Table 2, we report the maximum absolute values of the bias-uncertainty
ratios obtained in parameter biases, for the base parameter set of
$\\{\Omega_{m},\sigma_{8},w_{0}\\}$. We report bias-uncertainty ratios for
three values of $\ell_{\mathrm{max}}$, corresponding to pessimistic and
optimistic settings for a Euclid-style survey (Euclid Collaboration, 2020b)
(note that for the 3-parameter case, the highest ratio of bias to 1$\sigma$
uncertainty in $\sigma_{8}$ of 0.032 is achieved for
$\ell_{\mathrm{max}}=3000$). In table 3, we report the same, for the extended
parameter set of $\\{\Omega_{m},\sigma_{8},w_{0},w_{a}\\}$.

For both parameter sets, there are two clear trends; firstly, that the
parameter bias-uncertainty ratio is greater when higher multipoles $\ell$ are
included in the Fisher analysis; and secondly, that the $m$-bias generally has
a greater influence when it is peaked at smaller $\ell$. Furthermore, the
inclusion of the parameter $w_{a}$ in the Fisher matrix has resulted in more
significant biases for all parameters, due to the additional parameter
degeneracy in the $C_{\ell}$s arising when more parameters are included in the
Fisher analysis. There is additionally a trend of subsidiary peaks in the
bias-uncertainty ratios at higher $\ell_{\mathrm{peak}}$ for all parameters,
which is typically strongest where $\ell_{\mathrm{peak}}$ is less than
$\ell_{\mathrm{max}}$ by $\sim$ 500, though these peaks become suppressed for
the highest values of $\ell_{\mathrm{max}}$. These occur because a given value
of $\ell_{\mathrm{max}}$ will roughly coincide with the high-$\ell$ minimum of
the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$ for which
$\ell_{\mathrm{peak}}\sim\ell_{\mathrm{max}}-500$. Therefore, for smaller
values of $\ell_{\mathrm{peak}}$ than this, for which the structure in the
$\smash{\Delta C_{\ell}^{\gamma\gamma}}$ is shifted to lower $\ell$, less of
the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$ that is included in the bias
calculation will consist of the low-$\ell$ plateau, and more will be in the
higher-$\ell$ trough; whereas for larger values of $\ell_{\mathrm{peak}}$, the
$\smash{\Delta C_{\ell}^{\gamma\gamma}}$ is generally suppressed. In both
cases, the resulting impact in the inferred cosmology is smaller, so we get a
peak in parameter bias at this value of $\ell_{\mathrm{peak}}$. At the lowest
values of $\ell_{\mathrm{peak}}$, the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$
is generally enhanced, resulting in the low-$\ell_{\mathrm{peak}}$ rise in
parameter bias and bias-uncertainty ratio.

Table 2: Maximum absolute bias$/1\sigma$ for pessimistic (1500) and optimistic (5000) values of $\ell_{\mathrm{max}}$, base parameter set (2 s.f.). | Bias$/1\sigma$
---|---
Parameter | $\ell_{\mathrm{max}}=1500$ | $\ell_{\mathrm{max}}=5000$
$\Omega_{m}$ | $0.0059$ | $0.052$
$\sigma_{8}$ | $0.020$ | $0.017$
$w_{0}$ | $0.013$ | $0.068$
Table 3: Maximum absolute bias$/1\sigma$ for pessimistic (1500) and optimistic (5000) values of $\ell_{\mathrm{max}}$, extended parameter set (2 s.f.). | Bias$/1\sigma$
---|---
Parameter | $\ell_{\mathrm{max}}=1500$ | $\ell_{\mathrm{max}}=5000$
$\Omega_{m}$ | $0.014$ | $0.10$
$\sigma_{8}$ | $0.0057$ | $0.063$
$w_{0}$ | $0.025$ | $0.13$
$w_{a}$ | $0.021$ | $0.11$

Figure 5: Parameter bias-uncertainty ratio in the case that
$\\{\Omega_{m},\sigma_{8},w_{0}\\}$ are included in the Fisher matrix. Upper
panel: Bias-uncertainty ratio for $\Omega_{m}$ as a function of
$\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$. Middle panel: Bias-
uncertainty ratio for $\sigma_{8}$ as a function of $\ell_{\mathrm{peak}}$ and
$\ell_{\mathrm{max}}$. Lower panel: Bias-uncertainty ratio for $w_{0}$ as a
function of $\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$.

|
---|---
|

Figure 6: Parameter bias-uncertainty ratio in the case that
$\\{\Omega_{m},\sigma_{8},w_{0},w_{a}\\}$ are included in the Fisher matrix.
Upper-left panel: Bias-uncertainty ratio for $\Omega_{m}$ as a function of
$\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$. Upper-right panel: Bias-
uncertainty ratio for $\sigma_{8}$ as a function of $\ell_{\mathrm{peak}}$ and
$\ell_{\mathrm{max}}$. Lower-left panel: Bias-uncertainty ratio for $w_{0}$ as
a function of $\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$. Lower-right
panel: Bias-uncertainty ratio for $w_{a}$ as a function of
$\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$.

###  3.3 Dependence of parameter bias on $m$-bias rms

We also investigate how our forecast parameter biases vary with the amplitude
of the variations of the $m$-bias map. In fig. 7 we show the $m$-bias rms
which results in parameter bias-uncertainty ratios of
$\Delta\theta/\sigma=0.1$, $0.3$ and $1$ as a function of
$\ell_{\mathrm{peak}}$, for the base parameter set of
$\\{\Omega_{m},\sigma_{8},w_{0}\\}$, with $\ell_{\mathrm{max}}=5000$. In fig.
8 we show this for the extended parameter set of
$\\{\Omega_{m},\sigma_{8},w_{0},w_{a}\\}$. As can be seen, a given bias-
uncertainty ratio results from a smaller rms for smaller values of
$\ell_{\mathrm{peak}}$, and has a weakly increasing dependence on the number
of parameters included in the Fisher matrix. For both parameter sets, the
bias-uncertainty ratio is $\sim 0.1$ when the rms$=0.01$, $\sim 0.3$ when the
rms$=0.02-0.03$, and the parameter biases reach the statistical errors when
the rms$=0.03-0.06$, depending on $\ell_{\mathrm{peak}}$.

Figure 7: $m$-bias rms which results in selected values of parameter bias-
uncertainty ratio for the base parameter set
$\\{\Omega_{c},\sigma_{8},w_{0}\\}$, as a function of $\ell_{\mathrm{peak}}$.
Solid line: $m$-bias rms which results in bias-uncertainty ratio of $0.1$.
Dashed line: $m$-bias rms which results in bias-uncertainty ratio of $0.3$.
Dotted line: $m$-bias rms which results in bias-uncertainty ratio of $1$.
Figure 8: $m$-bias rms which results in selected values of parameter bias-
uncertainty ratio for the extended parameter set
$\\{\Omega_{c},\sigma_{8},w_{0},w_{a}\\}$, as a function of
$\ell_{\mathrm{peak}}$. Solid line: $m$-bias rms which results in bias-
uncertainty ratio of $0.1$. Dashed line: $m$-bias rms which results in bias-
uncertainty ratio of $0.3$. Dotted line: $m$-bias rms which results in bias-
uncertainty ratio of $1$.

##  4 Discussion

###  4.1 Residual power spectra

As can be seen in the fractional $\smash{\Delta C_{\ell}^{\gamma\gamma}}$
shown in fig. 2, an $m$-bias with some particular $\ell_{\mathrm{peak}}$ can
introduce a bias in the $C_{\ell}$s with generally consistent magnitude on a
range of scales, due to mode-mode coupling; therefore, any analysis of the
effect of an $m$-bias on the $C_{\ell}$s must not be limited only to the
scales on which the $m$-bias varies. Note that as the effect of the spatially-
varying $m$-bias on the power spectrum is more complicated than a constant
scaling, the impact on the cosmological parameters due to marginalisation over
a constant bias parameter will not be an effective means of mitigation. In a
real survey, the $C_{\ell}$ of the $m$-bias map will not be not known a
priori, and will probably not be easily describable with a small number of
parameters as in the case studied here. As a result, it will be difficult to
marginalise over the effect of a spatially-varying $m$-bias in cosmological
parameter estimation, because the number of required nuisance parameters will
be too large. Due to this we have not considered the effect of $m$-bias
marginalisation on cosmological parameter biases or figures of merit in the
present work.

The fractional residual shear angular power spectra we find reach a maximum of
$\sim 10^{-4}$. Kitching et al. (2019, 2020) also calculated residuals between
the full analytic shear $C_{\ell}$s, which included spatially-varying shear
bias, and an analytic linear approximation, which only included mean and
spatially-constant terms, up to a maximum multipole of 64. Kitching et al.
(2020) also calculated residuals between forward-modelled shear $C_{\ell}$s,
which included spatially-varying shear bias, and the linear approximation, up
to a maximum multipole of 2048, and found differences which were also $\sim$4
orders of magnitude smaller than the fiducial power spectra. We therefore find
similar results at the order-of-magnitude level in the regions of $\ell$-space
in which our results overlap.

###  4.2 Bias in cosmological parameters

As shown in figs. 5 and 6, the parameter bias-uncertainty ratio generally
increases with decreasing $\ell_{\mathrm{peak}}$ and increasing
$\ell_{\mathrm{max}}$. The increase in bias-uncertainty ratio (and indeed in
bias) with $\ell_{\mathrm{max}}$ occurs because the $C_{\ell}$s are biased on
all scales and so including contributions from additional scales will increase
the size of the parameter bias. Indeed, while including higher multipoles in
the analysis results in a smaller parameter uncertainty, this effect is small
compared with the increase in parameter bias with $\ell_{\mathrm{max}}$. The
greater influence of an $m$-bias with small $\ell_{\mathrm{peak}}$ arises due
to the generally larger $\smash{\Delta C_{\ell}^{\gamma\gamma}}$ for smaller
$\ell_{\mathrm{peak}}$. $m$-bias maps with different $\ell_{\mathrm{peak}}$
also affect each parameter differently; each parameter exhibits different
variation in the parameter bias with $\ell_{\mathrm{peak}}$ and
$\ell_{\mathrm{max}}$, due to sensitivity of different parts of the shear
power spectra to each parameter. The scale-dependence of the parameter biases
which result from the spatial variations of the $m$-bias field, which come in
through the $\ell_{\mathrm{peak}}$-dependence, demonstrates that the bias in
shear power spectra and in cosmological parameters varies with the shape of
the power spectrum of $m$-bias field. This shows that the impact of spatial
variations of the $m$-bias on cosmological parameter estimation can not be
fully mitigated by marginalisation, unless a parametric form of power spectrum
of the $m$-bias field is known, which will be unlikely in practice.

Maximum values of parameter bias-uncertainty ratio obtained for the
pessimistic and optimistic settings for $\ell_{\mathrm{max}}$ (Euclid
Collaboration, 2020b) are shown in tables $2$ and $3$, for the base and
extended parameter sets respectively. Larger parameter bias-uncertainty ratios
are obtained when more free parameters are included in the Fisher analysis. In
the case of the optimistic cutoff multipole for a Euclid-style survey,
corresponding to $\ell_{\mathrm{max}}=5000$, the maximum bias-uncertainty
ratios are at about $0.05$ and $0.1$ for the base and extended parameters
sets, respectively. In the case of $w_{0}$, for which the maximum bias-
uncertainty ratios for $\ell_{\mathrm{max}}=5000$ is $0.12$, this may still
represent a non-negligible contribution to the total systematic error budget;
this is potentially relevant in light of the fact that one of the main science
goals of Euclid and other stage-IV surveys is the investigation of dark energy
and the accelerated expansion (Laureijs et al., 2011). In the pessimistic
setting of $\ell_{\mathrm{max}}=1500$, bias-uncertainty ratios do not exceed
$0.025$ in any case. In general, the systematic bias introduced by the
spatially-varying $m$-bias on cosmological parameter estimates is expected to
be sub-dominant, but not wholly negligible in certain cases. This is a
refinement of previous work in this area, for example Kitching et al. (2019,
2020) which found that the impact on the shear power spectra due to the
spatially-varying $m$-bias should be small compared with the impact of the
mean term; while we assume an $m$-bias with zero spatial mean and as such do
not make comparative conclusions, we have quantified explicitly that the
cosmological parameter bias induced by the spatially-varying $m$ may be a non-
negligible contribution to the error budget. While the mean $m$ term may
present the dominant contribution to the error budget from the $m$-bias, if
this is well-constrained then the spatial variations may still result in a
non-negligible bias in inferred cosmological parameter values.

###  4.3 Dependence of parameter bias on $m$-bias rms

The amplitude of variations of the $m$-bias map which gives rise to parameter
bias-uncertainty ratios of $0.1,0.3$ and $1$ is shown in figs. 7 and 8 for the
3-parameter and 4-parameter cases respectively, as a function of
$\ell_{\mathrm{peak}}$, and for the optimistic high $\ell$-cut of
$\ell_{\mathrm{max}}=5000$. We also find that the dependence of bias-
uncertainty ratio on $m$-bias rms is close to quadratic, with the exponent
departing from 2 (negatively, by a few per cent) for
$\ell_{\mathrm{peak}}\lesssim 500$. The quadratic relationship is is expected
as the mode-coupling matrix is proportional to the $m$-bias angular power
spectrum which scales as the variance of the $m$-bias map. The parameter bias-
uncertainty ratio obtained from an $m$-bias map with a given rms depends on
the scale of $m$-bias variations, and for an $m$-bias with a large
characteristic scale the parameter bias can exceed the statistical uncertainty
$\sigma$ for rms $\sim 0.04-0.05$; biases exceed $30\%$ of the statistical
uncertainty for rms $\sim 0.02-0.03$ for $m$-biases peaked across the range of
scales examined. This shows that the importance of the spatially-varying
$m$-bias is sensitive to the rms of the $m$-bias map, and hence that the
potential impact of the spatially-varying $m$-bias on cosmological parameter
constraints from cosmic shear cannot be neglected unless the expected
amplitude of variations of the $m$-bias map is known from predicted
constraints on the shear systematics which produce the spatially-varying
$m$-bias. This allows requirements to be set on the allowed amplitude of
variations of the spatially-varying $m$-bias, and hence on constraints of
weak-lensing systematics from which the spatially-varying $m$-bias originates.
In the case that the requirement on the amplitude of spatial variations of the
$m$-bias field is satisfied, such that the resulting bias on inferred
cosmological parameters is deemed to be negligible, then it will be possible
to neglect the convolutive effect of the spatial variations of the $m$-bias on
the shear power spectra, and consider the effect of the $m$-bias on the shear
$C_{\ell}$s as a constant scaling arising due to the mean $m$. In this case it
will be acceptable to mitigate the effect of the $m$-bias on cosmological
parameter estimation by marginalising over a prior on the mean $m$, and the
dominant effect of the $m$-bias on parameter inference will be the
marginalisation effect considered by Kitching et al. (2020).

It should also be noted that while requirements exist on the value of the mean
$m$-bias (the requirement from Massey et al. (2013) is that $m\lesssim 2\times
10^{-3}$) no such requirement has been placed on the amplitude of spatial
variations of the $m$-bias (the rms of the $m$-bias field). In forthcoming
surveys such as Euclid, we will be interested in residual shear biases, i.e.
biases which result from uncertainties in our knowledge of the observational
systematics that give rise to shear biases. Therefore, even if the mean
$m$-bias is well calibrated subject to requirements, this does not by itself
mean that the spatial variations of the residual $m$-bias must necessarily
also be so; a key conclusion of this work is that it is separately necessary
to place requirements on the amplitude of the spatial variations of the
residual $m$-bias. This will allow requirements to be placed on models of the
observational systematics that produce shear biases (eg. requirements on the
precision of PSF models) and on priors of these systematics.

##  5 Summary

We have considered the effect of a spatially-varying multiplicative shear bias
on the estimation of cosmological parameters using the cosmic shear angular
power spectrum. We have applied a computationally efficient pseudo-$C_{\ell}$
formalism to determine the bias in the cosmic shear power spectra
$\smash{C_{\ell}^{\gamma\gamma}}$ arising due to an $m$-bias field obeying a
$C_{\ell}$ with a Gaussian profile, by considering the $m$-bias map as acting
as a mask that introduces mode mixing into the shear $C_{\ell}$s, and
computing the mode-mixing matrix which convolves the
$\smash{C_{\ell}^{\gamma\gamma}}$ with the power spectrum of the $m$-bias map.
This has allowed us to consider the effect of the $m$-bias at high $\ell$,
down to arcminute-scales corresponding to the expected field-of-view PSF
variations in a Euclid-like photometric survey. We considered the settings for
a stage-IV style lensing survey such as Euclid (Euclid Collaboration, 2020b)
and computed fiducial tomographic shear $C_{\ell}$s using CCL (Chisari et al.,
2019). Note that while we consider Euclid as an example, our indicative
results are likely to be informative for other stage-IV weak-lensing surveys.
We repeat this for a number of different $m$-bias maps characterised by
different values for the mean and width of the $m$-bias map,
$\ell_{\mathrm{peak}}$ and $\sigma_{m}$ respectively. We also calculate the
$BB$ shear power spectra generated by the mode-coupling effect of the
spatially-varying $m$-bias and finds that it follows the residual $EE$ shear
power spectra closely, especially for a spatially-varying $m$-bias with a
characteristic scale of arcminutes. We conclude that it may in principle be
possible to use the induced $BB$ power to self-calibrate the spatially-varying
$m$-bias, in agreement with previous work in this area (eg. Kitching et al.
(2020)), though this will be complicated by the fact that the $BB$ power
induced by different systematics may be degenerate, and so we do not consider
this in our analysis.

We find fractional residual $C_{\ell}$s which reach a maximum of $\sim
10^{-4}$. This is similar to the magnitude of fractional residuals found by
Kitching et al. (2019, 2020) between analytic calculations involving the
spatially-varying $m$-bias and only the constant and mean terms up to a
maximum multipole of 64, and between a numerical forward model including the
spatially-varying $m$-bias and an analytic calculation involving only the
constant and mean terms up to a maximum multipole of 2048, though we have
extended the analytic calculation involving the spatially-varying $m$-bias to
high $\ell$.

We then employed a Fisher matrix analysis to forecast biases on cosmological
parameters inferred from the $\smash{C_{\ell}^{\gamma\gamma}}$ due to the bias
in the power spectra $\smash{\Delta C_{\ell}^{\gamma\gamma}}$ resulting from
the spatially-varying $m$-bias, with no spatially constant term, i.e. assuming
perfect calibration of the shear field for the spatially invariant $m$-bias.
We find that the ratio of the parameter biases to the $1\sigma$ uncertainty
depends strongly on $\ell_{\mathrm{max}}$ (a cut-off scale applied at the
Fisher level) and $\ell_{\mathrm{peak}}$, with the parameter biases typically
peaking both at high and low values of $\ell_{\mathrm{peak}}$, due to the
dynamic nature of the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$ as a function
of $\ell_{\mathrm{peak}}$ and $\ell_{\mathrm{max}}$. We find that the
parameter bias-uncertainty ratios are higher when more free parameters are
included in the Fisher analysis. For realistic values of $\ell_{\mathrm{max}}$
for a Euclid-style survey, and for $m$-bias map rms $=0.01$, the bias in
cosmological parameters resulting from a spatially-varying multiplicative
shear bias reach a maximum of $\gtrsim 10\%$ of the forecast statistical
error. Whereas Kitching et al. (2019, 2020) concluded that the effect of a
spatially-varying m-bias should be small compared with that of the mean
$m$-bias, and negligible in the case of an $m$-bias field with zero mean, we
find that the effect of the spatially-varying m-bias can be neglected only
subject to requirements on the properties of the $m$-bias field, in particular
the rms and characteristic scale of spatial variations. This caveat arises
because we consider the impact of the $\smash{\Delta C_{\ell}^{\gamma\gamma}}$
up to significantly higher multipoles, which allows us to explicitly calculate
the effect on the inference of cosmological parameters. This requires that the
spatially-varying $m$-bias be a spin-0 field, though this is known to be a
realistic assumption (for example, Kitching et al. (2019) found that the
effect on the shear power spectra due to the imaginary part of the $m$-bias
should be very small compared with the real part). This simplification allows
us to make use of an orthogonality relation in the computation of the $m$-bias
mode-coupling matrix which significantly reduces the scaling of the
calculation with the maximum multipole considered compared with the formalism
of Kitching et al. (2019).

We have also investigated the variation of the parameter biases with the rms
of the $m$-bias map, and find that the relationship is close to quadratic. We
find that biases exceed $30\%$ of the statistical error for rms $\sim
0.02-0.03$ across the range of $\ell_{\mathrm{peak}}$ examined, and exceed the
statistical error for rms $\sim 0.04-0.05$ for small $\ell_{\mathrm{peak}}$.
This allows requirements to be set on the permissible amplitude of variations
of the $m$-bias, and hence on models of the relevant shear systematic effects
that will be used for systematics control in forthcoming surveys. If the
spatial variations of the $m$-bias field satisfy such a requirement, then it
may be possible to mitigate the effect of the $m$-bias on parameter inference
by marginalisation, as considered by Kitching et al. (2020).

In this work, we have considered a simple model for spatially-varying
$m$-bias, in which the $m$-bias field exhibits power on a specific scale. This
allows us to consider the sensitivity of inferred cosmological parameters to
generic $m$-bias, arising due to systematics with power on a specific physical
scale. However, a more realistic model may consider the potential redshift
dependence of the $m$-bias, or a more realistic prescription for the
statistics of the $m$-bias field, including the $m$-bias $C_{\ell}$. In
particular, it will be of interest to consider the effect of specific
systematics which may contribute to the shear bias, including PSF and other
instrumental effects, as well as effects relating to target selection, galaxy
shape measurement, etc. In considering such effects it may also be of interest
to consider the effect of the additive shear bias in a related analysis, which
may potentially be considered in the pseudo-$C_{\ell}$ framework.

## Acknowledgements

We thank T. D. Kitching for useful discussions. CC is supported by an STFC
doctoral studentship. CAJD acknowledges support from the Beecroft trust. DA
acknowledges support from the Beecroft trust and from STFC through an Ernest
Rutherford Fellowship, grant reference ST/P004474/1. We are thankful to the
authors of NaMaster, CCL and CLASS for developing and making their codes
publicly available.

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