# The Impact of Quadratic Biases on Cosmic Shear

T. D. Kitching1,† A. C. Deshpande1<EMAIL_ADDRESS>© 2021. All rights
reserved. 1Mullard Space Science Laboratory, University College London,
Holmbury St Mary, Dorking, Surrey RH5 6NT, UK

###### Abstract

In this paper we revisit potential biases in cosmic shear power spectra caused
by bias terms that multiply up to quadratic powers of the shear. Expanding the
multiplicative bias field as a series of independent spin-$s$ fields we find
terms $m_{s}$ that multiply integer and half-integer powers of the shear. We
propagate these biases into shape measurement statistics and the cosmic shear
power spectrum. We find that such biases can be measured by performing
regression on calibration data. We find that for integer powers of shear the
impact of quadratic order terms on the power spectrum is an additional
bispectrum dependency; ignoring quadratic terms can lead to biases in
cosmological parameters of up to $2(m_{2}+m_{-2}-m_{6})0.4\sigma$ for Stage-IV
dark energy experiments, but that the susceptibility to them can be decreased
by using methods to remove small-scale sensitivity. We also find, for half-
integer powers of the shear that, for a Stage-IV experiment, biases are
required to be known to better than approximately
$\sigma[m_{0}+15(m_{1}+m_{3})+0.1(m_{-1}+m_{5})]\leq 0.01$. In future, Stage-
IV dark energy experiments should seek to measure and minimise such all such
bias terms.

##  1 Introduction

The weak lensing effect on galaxies images is to induce a change in the third
eccentricity or third flattening, known as shear. But the inference of shear
from data can be biased with respect to the true shear by many effects, for
example inaccuracies in the algorithms used to measure galaxy shapes (Heymans
et al., 2006; Massey et al., 2007; Bridle et al., 2010; Kitching et al., 2012;
Mandelbaum et al., 2015), the size of the point spread function (PSF)
(Hoekstra et al., 2017; Kannawadi et al., 2019; Gatti et al., 2021), detector
effects (Antilogus et al., 2014), or detection effects (Hoekstra et al., 2015;
Hoekstra, 2021). To linear order the two biases that relate the
inferred/measured shear to the true shear are a multiplicative bias (that
multiplies the true shear by a value) and an additive bias (that adds a value
to the true shear). The propagation of such biases into cosmic shear power
spectra is shown in Kitching et al. (2019, 2020, 2021), that include
multiplicative and additive biases and the impact of masked data sets.

In the majority of weak lensing studies to date (see e.g. Bridle et al., 2010;
Mandelbaum et al., 2015; Hildebrandt et al., 2020; DES Collaboration et al.,
2021) a linear multiplicative bias has been assumed (Guzik & Bernstein, 2005;
Huterer et al., 2006). This may appear to be a reasonable approximation,
because the amplitude of shear is typically small and so quadratic terms may
be even smaller. However, going beyond this assumption Heymans et al. (2006)
and Kitching et al. (2012) both fitted a quadratic term to the available
methods at the time, and found values for a coefficient describing this effect
of between $0.01-1$, depending on the method in question. In this paper we
revisit the impact of such quadratic terms.

As discussed in Kitching et al. (2021) the observed shear field, including
biases, noise, can be written to linear order, including only integer powers
of shear, in multiplicative bias terms like

$\displaystyle\widetilde{\gamma}(\mathbf{\Omega})$ $\displaystyle=$
$\displaystyle[1+m_{0}(\mathbf{\Omega})]\gamma(\mathbf{\Omega})+m_{4}(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})+c(\mathbf{\Omega}),$
(1)

where each of the quantities are dependent on angular coordinates
$\mathbf{\Omega}=(\theta,\phi)$, where $\theta$ and $\phi$ are latitude and
longitude (or R.A. and dec). The true spin-2 shear (in this nomenclature,
spin-$s$ means spin positive $s$) is $\gamma(\mathbf{\Omega})$, and the
measured spin-2 shear is $\widetilde{\gamma}(\mathbf{\Omega})$.
$m_{0}(\mathbf{\Omega})$ and $m_{4}(\mathbf{\Omega})$ are spin-0 and spin-4
position-dependent multiplicative bias terms respectively. We note here that
each term needs to be spin-2, shear is spin-2 so at linear order only a spin-0
multiplicative bias or a spin-4 multiplied by a spin-(-2) field can contribute
to linear order. $c(\mathbf{\Omega})$ is a spin-2 position-dependent additive
bias. Here we do not include the un-lensed uncorrelated galaxy ellipticity or
a survey mask but refer to Kitching et al. (2021) for these generalisations;
also any intrinsic alignment terms are captured in the
$\gamma(\mathbf{\Omega})$ term. ∗ is a complex conjugate. In general we use
$\widetilde{x}$ to mean an observed quantity.

This can be generalised beyond linear order, and beyond integer powers of
shear, including new multiplicative bias terms. To do this we create a
geometric series of powers of shear that result in integer-spin fields;
because shear is spin-2 this means taking either integer or half-integer
powers111Taking any other power would result in fields that were not symmetric
under rotations of $2\pi$.. This results in

$\displaystyle\widetilde{\gamma}(\mathbf{\Omega})$ $\displaystyle=$
$\displaystyle\gamma(\mathbf{\Omega})+\left[\sum_{i=0}^{M}\sum_{j=0}^{N}m^{i,j}_{2(1-i/2+j/2)}(\mathbf{\Omega})\gamma^{i/2}(\mathbf{\Omega})\gamma^{*,{j/2}}(\mathbf{\Omega})\right],$
(2)

where $i$ and $j$ are integers, and the multiplicative bias sum runs up to
order $N$ and $M$ respectively. We do not consider $i<0$ or $j<0$ because
$\gamma_{i}\in[-1,1]$ hence a negative power would lead to divergence as
$\gamma_{i}\rightarrow 0$. The subscript in $m^{i,j}_{s}(\mathbf{\Omega})$
refers to a series of multiplicative bias terms with spin-$2(1-i/2+j/2)$,
where $s=2(1-i/2+j/2)\in\mathbb{Z}$ (i.e. integer); the superscript is a label
for the term corresponding to a multiplication of
$\gamma^{i/2}(\mathbf{\Omega})\gamma^{*,{j/2}}(\mathbf{\Omega})$, not a power.
We note that $c(\mathbf{\Omega})$ corresponds to
$m^{0,0}_{2}(\mathbf{\Omega})$. In creating this geometric expansion terms
with power $i/2$ should not be equated with the $(i-2)$ edth derivative of the
shear field (see Castro et al., 2005), although both do have the same spin-
weight. We assume that non-integer powers relate to the positive root of the
complex value in question i.e. we assume no additional phases of $\pi(ik)$
where $k\in\mathbb{Z}$ (see Markushevich & Silverman, 2005), and assume any
sign information of each term in the series is captured by the biases
$m^{i,j}_{s}$. In addition to a geometric expansion of shear with spin-
weighted bias coefficients (as in equation 2), one may also want to expand
each bias as a function of shear under the assumption that the biases are
functions of the shear field $m^{i,j}_{s}(\gamma,\gamma^{*})$ (e.g. as a
Taylor expansion), but we leave this for a future analysis.

If we expand equation (2) to second order in shear we find that

$\displaystyle\widetilde{\gamma}(\mathbf{\Omega})$ $\displaystyle=$
$\displaystyle\gamma(\mathbf{\Omega})$ (3) $\displaystyle+$ $\displaystyle
m^{4,0}_{-2}(\mathbf{\Omega})\gamma^{2}(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle m^{3,0}_{-1}(\mathbf{\Omega})\gamma^{3/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{3,1}_{0}(\mathbf{\Omega})\gamma^{3/2}(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{2,2}_{0}(\mathbf{\Omega})\gamma(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle m^{1,0}_{1}(\mathbf{\Omega})\gamma^{1/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{2,1}_{1}(\mathbf{\Omega})\gamma(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{2,2}_{2}(\mathbf{\Omega})\gamma(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{1,1}_{2}(\mathbf{\Omega})\gamma^{1/2}(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{0,1}_{3}(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle
m^{1,2}_{3}(\mathbf{\Omega})\gamma^{1/2}(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{0,2}_{4}(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle
m^{1,3}_{4}(\mathbf{\Omega})\gamma^{1/2}(\mathbf{\Omega})\gamma^{*,3/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m^{0,3}_{5}(\mathbf{\Omega})\gamma^{*,3/2}(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle m^{0,4}_{6}(\mathbf{\Omega})\gamma^{*,2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle c(\mathbf{\Omega}),$

where $m^{i,j}_{2}(\mathbf{\Omega})$ are spin-$2$ multiplicative bias fields
and similarly for the other terms. We define quadratic/second order as
$\gamma^{i}\gamma^{*,j}$ where $i+j\leq 2$. For the remainder of this paper we
do not include the $\mathbf{\Omega}$ in each equation for clarity, however an
angular dependence of each quantity should be assumed unless otherwise stated.
Terms above second order $\mathcal{O}(\gamma^{3})$ should be small (we note
the irony of saying this in a paper that is revisiting
$\mathcal{O}(\gamma^{2})$ that have been thought to have been unimportant),
and we leave investigation of the impact of these (if any exist) to future
work. To simplify the analysis for the remainder of this paper we make the
assumption that $m^{i,j}_{s}=m^{k,l}_{s}$ for any $i,j,k,l$ i.e. that for each
spin-weight there is a single multiplicative bias field; this assumption could
be dropped in future. In this case we have

$\displaystyle\widetilde{\gamma}(\mathbf{\Omega})$ $\displaystyle=$
$\displaystyle\gamma(\mathbf{\Omega})$ (4) $\displaystyle+$ $\displaystyle
m_{-2}(\mathbf{\Omega})\gamma^{2}(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle m_{-1}(\mathbf{\Omega})\gamma^{3/2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle
m_{0}(\mathbf{\Omega})[\gamma(\mathbf{\Omega})+\gamma^{3/2}(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})]$
$\displaystyle+$ $\displaystyle
m_{1}(\mathbf{\Omega})[\gamma^{1/2}(\mathbf{\Omega})+\gamma(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})]$
$\displaystyle+$ $\displaystyle
m_{2}(\mathbf{\Omega})[\gamma(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})+\gamma^{1/2}(\mathbf{\Omega})\gamma^{*,1/2}(\mathbf{\Omega})]$
$\displaystyle+$ $\displaystyle
m_{3}(\mathbf{\Omega})[\gamma^{*,1/2}(\mathbf{\Omega})+\gamma^{1/2}(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})]$
$\displaystyle+$ $\displaystyle
m_{4}(\mathbf{\Omega})[\gamma^{*}(\mathbf{\Omega})+\gamma^{1/2}(\mathbf{\Omega})\gamma^{*,3/2}(\mathbf{\Omega})]$
$\displaystyle+$ $\displaystyle
m_{5}(\mathbf{\Omega})\gamma^{*,3/2}(\mathbf{\Omega})$ $\displaystyle+$
$\displaystyle m_{6}(\mathbf{\Omega})\gamma^{*,2}(\mathbf{\Omega})$
$\displaystyle+$ $\displaystyle c(\mathbf{\Omega}),$

where we drop the superscripts for each field. We will now consider the
integer and half-integer terms in turn, since these have distinct impacts on
the power spectrum. This also serves as a pedagogical exploration of the
impact of these various different biases. In Section 2 we present how integer
quadratic terms can impact shape measurement and the cosmic shear power
spectrum; both of which can be understood analytically. In Section 3 we add
half-integer quadratic terms and show how then can impact impact shape
measurement and the cosmic shear power spectrum. In Section 4 and 5 we discuss
the results and present conclusions.

##  2 The Impact of Quadratic-Order Integer Terms on Shape Measurement

We first consider how biases of integer powers of shear are related to a
canonical Cartesian parameterisation of the shear as $\gamma=\gamma_{1}+{\rm
i}\gamma_{2}$ (where $\gamma_{1}$ are local distortions parallel to a local
axes, and $\gamma_{2}$ are at $45$ degrees). In this case we find that

$\displaystyle\Delta\gamma_{1}=\widetilde{\gamma}_{1}-\gamma_{1}$
$\displaystyle=$
$\displaystyle(m^{R}_{0}+m^{R}_{4})\gamma_{1}+(m^{R}_{2}+m^{R}_{-2}+m^{R}_{6})\gamma^{2}_{1}+(m^{R}_{2}-m^{R}_{-2}-m^{R}_{6})\gamma^{2}_{2}$
$\displaystyle-$
$\displaystyle(m^{I}_{0}-m^{I}_{4})\gamma_{2}-2(m^{I}_{-2}-m^{I}_{6})\gamma_{1}\gamma_{2}$
$\displaystyle+$ $\displaystyle c_{1}$
$\displaystyle\Delta\gamma_{2}=\widetilde{\gamma}_{2}-\gamma_{2}$
$\displaystyle=$
$\displaystyle(m^{R}_{0}-m^{R}_{4})\gamma_{2}+2(m^{R}_{-2}-m^{R}_{6})\gamma_{1}\gamma_{2}$
(5) $\displaystyle+$
$\displaystyle(m^{I}_{0}+m^{I}_{4})\gamma_{1}+(m^{I}_{2}+m^{I}_{-2}+m^{I}_{6})\gamma^{2}_{1}+(m^{I}_{2}-m^{I}_{-2}-m^{I}_{6})\gamma^{2}_{2}$
$\displaystyle+$ $\displaystyle c_{2}$

where we have introduced the notation $\Delta\gamma_{i}$ to mean the
difference between the measured and true shear for component $i$. Each
spin-$s$ multiplicative bias field is split into its real and imaginary parts
$m_{s}=m^{R}_{s}+{\rm i}m^{I}_{s}$ or $m_{s}=|m_{s}|{\rm exp}(s\theta_{m})$
where $\theta_{m}$ is the complex angle in the bias’ coordinate frame. Note
that we consider $m_{s}$ and $m_{-s}$ to be independent multiplicative bias
fields; one could also assume these are related via $m_{-s}=m^{*}_{s}$ (and
such a model could be tested against the calibration data). The inclusion of
imaginary parts in the $m_{s}$ fields allows for a rotation of the observed
shear with respect to the true shear222In Kitching et al. (2008) a different
approach was taken where a rotation operator was applied to the shear where
$m=|m|{\rm exp}({\rm i}\phi)$, where $\phi$ is an arbitrary rotation angle
between the bias coordinate frame and the galaxy image’s coordinate frame, so
that $\widetilde{\gamma}=|m||\gamma|{\rm exp}(2{\rm i}(\theta+\phi/2))$, where
$\theta={\rm atan}(\gamma_{2}/\gamma_{1})$. Using similar notation, in general
the multiplicative terms from equation 2 could be rewritten like
$\displaystyle m^{i,j}_{s}\gamma^{i/2}\gamma^{*,{j/2}}{\rm e}^{{\rm
i}\phi}=|m^{i,j}_{s}||\gamma|^{i/2-j/2}{\rm e}^{2{\rm
i}([i/2-j/2]\theta+s\theta/2}){\rm e}^{{\rm
i}\phi}=|m^{i,j}_{s}||\gamma|^{2}{\rm e}^{2{\rm i}\theta}{\rm e}^{{\rm
i}\phi},$ (6) here $\phi$ is rotation angle between coordinate frames,
$\theta$ is the angle within a coordinate frame, and $s=2(1-i/2+j/2)$. In this
case the angle $\phi$ can be incorporated into the multiplicative bias field
like  $\displaystyle m^{i,j}_{s}=|m^{i,j}_{s}|{\rm e}^{s{\rm
i}(\theta+2\phi/s)}.$ (7) In this case $\theta$ is the angle within the
coordinate of the bias field, and $\psi=2\phi/s$ is a rotation angle. It can
then be seen that $m^{R}_{s}=|m^{i,j}_{s}|\cos(s[\theta+\psi])$ and
$m^{I}_{s}=|m^{i,j}_{s}|\sin(s[\theta+\psi])$. For $s\not=0$ a rotation can
therefore be incorporated into the multiplicative bias. In this approach we
also see that, for $s=0$ one can combine a rotation with the $m_{0}$ term so
that $m^{R}_{0}=|m^{i,j}_{0}|\cos(\psi)$ and
$m^{I}_{0}=|m^{i,j}_{0}|\sin(\psi)$ i.e. an ‘effective’ imaginary term that
results from combining the scalar and rotational biases, even though $m_{0}$
is itself a scalar..

One can also make a transformation of variables to
$u=(1/2^{1/2})(\gamma_{1}+\gamma_{2})$ and
$v=(1/2^{1/2})(\gamma_{1}-\gamma_{2})$, so that equation (2) can be rewritten

$\displaystyle\Delta\gamma_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{1/2}}(m^{R}_{0}+m^{R}_{4})(u+v)+\frac{1}{2}(m^{R}_{2}+m^{R}_{-2}+m^{R}_{6})(u^{2}+v^{2}+2uv)+\frac{1}{2}(m^{R}_{2}-m^{R}_{-2}-m^{R}_{6})(u^{2}+v^{2}-2uv)$
$\displaystyle-$
$\displaystyle\frac{1}{2^{1/2}}(m^{I}_{0}-m^{I}_{4})(u-v)-(m^{I}_{-2}-m^{I}_{6})(u^{2}-v^{2})$
$\displaystyle+$ $\displaystyle c_{1},$ $\displaystyle\Delta\gamma_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{1/2}}(m^{R}_{0}-m^{R}_{4})(u-v)+(m^{R}_{-2}-m^{R}_{6})(u^{2}-v^{2})$
(8) $\displaystyle+$
$\displaystyle\frac{1}{2^{1/2}}(m^{I}_{0}+m^{I}_{4})(u+v)+\frac{1}{2}(m^{I}_{2}+m^{I}_{-2}+m^{I}_{6})(u^{2}+v^{2}+2uv)+\frac{1}{2}(m^{I}_{2}-m^{I}_{-2}-m^{I}_{6})(u^{2}+v^{2}-2uv)$
$\displaystyle+$ $\displaystyle c_{2}.$

One may then perform the following regression: $\Delta\gamma_{1}$ vs.
$\gamma_{1}$; $\Delta\gamma_{1}$ vs. $\gamma_{2}$; $\Delta\gamma_{2}$ vs.
$\gamma_{1}$; $\Delta\gamma_{2}$ vs. $\gamma_{2}$; and also $\Delta\gamma_{2}$
vs. $u$ (or $v$). By taking derivatives of equation (2) we can predict the
behaviour of these relationships and how they depend on the multiplicative
bias fields. The non-zero derivatives to second order include (we do not list
all of these, but they are indicative):

$\displaystyle\frac{\partial\Delta\gamma_{1}}{\partial\gamma_{1}}$
$\displaystyle=$ $\displaystyle
g_{11}=(m^{R}_{0}+m^{R}_{4})+2(m^{R}_{2}+m^{R}_{-2}+m^{R}_{6})\gamma_{1}-2(m^{I}_{-2}-m^{I}_{6})\gamma_{2};$
$\displaystyle\frac{\partial\Delta\gamma_{1}}{\partial\gamma_{2}}$
$\displaystyle=$ $\displaystyle
g_{12}=2(m_{2}-m_{-2}-m_{6})\gamma_{2}-(m^{I}_{0}-m^{I}_{4})-2(m^{I}_{-2}-m^{I}_{6})\gamma_{1};$
$\displaystyle\frac{\partial^{2}\Delta\gamma_{1}}{\partial\gamma^{2}_{1}}$
$\displaystyle=$ $\displaystyle q_{111}=2(m^{R}_{2}+m^{R}_{-2}+m^{R}_{6});$
$\displaystyle\frac{\partial^{2}\Delta\gamma_{1}}{\partial\gamma^{2}_{2}}$
$\displaystyle=$ $\displaystyle q_{122}=2(m^{R}_{2}-m^{R}_{-2}-m^{R}_{6});$
$\displaystyle\frac{\partial^{2}\Delta\gamma_{1}}{\partial\gamma_{1}\partial\gamma_{2}}$
$\displaystyle=$ $\displaystyle q_{112}=-2(m^{I}_{-2}-m^{I}_{6});$ (9)

for example. We use the notation $g_{ij}$ to mean a first derivative of
$\Delta\gamma_{i}$ with respect to $\gamma_{j}$ i.e. the gradient
$\partial\gamma_{i}/\partial\gamma_{j}$, and $q_{ijk}$ to mean a second
derivative i.e. the quadratic term
$\partial^{2}\gamma_{i}/\partial\gamma_{j}\partial\gamma_{k}$ (we also will
label $q_{iuu}=\partial^{2}\gamma_{i}/\partial u^{2}$, and similar for $v$).
By performing regression between all the combinations of $\Delta\gamma_{i}$
vs. $\gamma_{j}$ and $u$ (or $v$), one can fit a quadratic function to each of
these, and thereby determine the value of the multiplicative bias terms

$\displaystyle m^{R}_{0}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}[g_{11}+g_{22}];\,\,\,\,\,\,\,m^{R}_{4}\simeq\frac{1}{2}[g_{11}-g_{22}];$
$\displaystyle m^{R}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[q_{111}+q_{122}];\,\,\,\,\,\,\,m^{R}_{-2}=\frac{1}{8}[q_{111}-q_{122}+2q_{221}];$
$\displaystyle m^{R}_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{8}[q_{111}-q_{122}-2q_{221}]$ $\displaystyle m^{I}_{0}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{2}[g_{21}-g_{12}];\,\,\,\,\,\,\,m^{I}_{4}\simeq\frac{1}{2}[g_{21}+g_{12}];$
$\displaystyle m^{I}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[q_{211}+q_{222}];\,\,\,\,\,\,\,m^{I}_{-2}=\frac{1}{8}[q_{211}-q_{222}+2q_{112}];$
$\displaystyle m^{I}_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{8}[q_{211}-q_{222}-2q_{112}],$ (10)

where $q_{221}=q_{2uu}-(1/4)(q_{211}+q_{222})$, and
$q_{112}=q_{2vv}-(1/4)(q_{111}+q_{122})$. $m^{R}_{0}$, $m^{R}_{4}$, and
$m^{I}_{4}$ can only be approximately determined, but can be solved exactly by
subtracting off the quadratic terms from equation (2) once determined and
performing a linear regression. Alternatively a regression can be performed
directly on equation (2), i.e. not performing regression in the individual
sub-spaces, by creating a loss-function function that is

$\displaystyle\chi^{2}=\frac{1}{2}\sum_{g}\left\\{\frac{[\widetilde{\Delta}\gamma_{1,g}-\Delta\gamma_{1}(m_{s})]^{2}}{\sigma^{2}(\widetilde{\Delta}\gamma_{1,g})}+\frac{[\widetilde{\Delta}\gamma_{2,g}-\Delta\gamma_{2}(m_{s})]^{2}}{\sigma^{2}(\widetilde{\Delta}\gamma_{1,g})}\right\\},$
(11)

where $\widetilde{\Delta}\gamma_{i,g}$ are the measured deviations from
calibration data where each simulated galaxy is labelled $g$, and
$\Delta\gamma_{i}(m_{s})$ are given in equation (2) over the whole
($\gamma_{1}$, $\gamma_{2}$) plane.
$\sigma^{2}(\widetilde{\Delta}\gamma_{i,g})$ are possible uncertainties on the
measured deviations. This is simply the $\chi^{2}$ fit to the data, in future
work this could be generalised to more sophisticated and Bayesian approaches.
One then can minimise this function with respect to the values of $m_{s}$. In
practice this is the approach that we take in Section 2.2.

This analysis differs from standard approaches that typically regress
$\Delta\gamma_{1}$ vs. $\gamma_{1}$ and $\Delta\gamma_{2}$ vs. $\gamma_{2}$
and fit a linear relation where $g_{11}=m^{\rm lin}_{1}$ and $g_{22}=m^{\rm
lin}_{2}$, we will compare our more general analysis with this approach. If
one fits a linear relation one should expect that $m^{\rm lin}_{1}\simeq
m_{0}+m_{4}$ and $m^{\rm lin}_{2}\simeq m_{0}-m_{4}$. The quadratic parameter
$q$ fit to methods in Heymans et al. (2006) to $\gamma_{1}$ is equivalent to
$(m^{R}_{0}+m^{R}_{-2}+m^{R}_{6})$. The $q$ in Kitching et al. (2012)
corresponds to a $\gamma|\gamma|$ term, which can be written as
$\gamma(\gamma\gamma^{*})^{1/2}=\gamma^{3/2}(\gamma^{*})^{1/2}$, which in
reference to equation (2) corresponds to a $m_{0}$ term.

Finally, we note that equation (2) can be written in a compact form like

$\displaystyle\Delta\gamma_{i}$ $\displaystyle=$ $\displaystyle
m^{R}_{0}\gamma_{i}+(-1)^{i-1}m^{R}_{4}\gamma_{i}+\left[\sum^{2}_{j=1}\sum^{2}_{k=1}[m^{R}_{2}|2-i|\sigma_{0,jk}+m^{R}_{-2}\sigma_{5-2i,jk}+m^{R}_{6}(-1)^{i-1}\sigma_{5-2i,jk}]\gamma_{j}\gamma_{k}\right]$
(12) $\displaystyle+$
$\displaystyle(-1)^{i}\sum^{2}_{j=1}m^{I}_{0}\sigma_{1,ij}\gamma_{j}+\sum^{2}_{j=1}m^{I}_{4}\sigma_{1,ij}\gamma_{j}+\left[\sum^{2}_{j=1}\sum^{2}_{k=1}[m^{I}_{2}|1-i|\sigma_{0,jk}+m^{I}_{-2}\sigma_{2i-1,jk}+m^{I}_{6}(-1)^{i}\sigma_{2i-1,jk}]\gamma_{j}\gamma_{k}\right]$
$\displaystyle+$ $\displaystyle c_{i},$

where $i={1,2}$. We use Pauli matrices as a convenient indexing scheme, where
$\sigma_{x,jk}$ are Pauli matrices with index $jk$, where the sums pick up
$\sigma_{0}=$$\tiny{\left({\begin{matrix}1&0\\\ 0&1\end{matrix}}\right)}$ (the
identity matrix), $\sigma_{1}=$$\tiny{\left({\begin{matrix}0&1\\\
1&0\end{matrix}}\right)}$ and $\sigma_{3}=$$\tiny{\left({\begin{matrix}1&0\\\
0&-1\end{matrix}}\right)}$ terms.

It is important to note that in order to determine if a shape measurement
method has quadratic dependence it is not sufficient to fit a linear relation
over a very small range of $\gamma_{i}$ (e.g. $\ll 0.1$); where quadratic
terms may not be measurable in a calibration procedure, but nonetheless may be
present. In such a case $m_{2}$, $m_{-2}$, and $m_{6}$ may still be
significantly non-zero (and therefore lead to a misestimation of the power
spectrum, see Section 2.3), but simply not be well-determined by the
calibration approach. In such cases these would be “hidden” biases that were
unaccounted for. Throughout we use the range of the distribution from Massey
et al. (2007) of $|\gamma_{i}|\leq 0.06$.

###  2.1 Test on Gaussian simulations

To investigate the impact of assuming a linear relation between
$\Delta\gamma_{1}$ vs. $\gamma_{1}$ we generate $10$,$000$ samples from a
Gaussian distribution $\gamma_{1}={\mathcal{N}}(0,\sigma)$ with mean zero and
width $\sigma=0.1$. We then construct
$\widetilde{\gamma}_{1}=\gamma_{1}+(m_{0}+m_{4})\gamma_{1}+(m_{2}+m_{-2}+m_{6})\gamma^{2}_{1}+(m_{2}-m_{-2}-m_{6})\gamma^{2}_{2}+c_{1}$
with $m_{0}=2\times 10^{-3}$, $m_{4}=0$, $m_{2}=1\times 10^{-2}$,
$m_{-2}+m_{6}=m_{2}$ (in order to remove any $\gamma_{2}$ dependency for this
simple test) and $c_{1}=0$. We then generate $10$,$000$ realisations where
$\gamma^{R}_{1}=\gamma_{1}+\mathcal{N}(0,\sigma/\sqrt{N_{\rm sim}})$ this
mimic the process of measuring $\Delta\gamma_{1}$ from $N_{\rm sim}$
simulations, and we generate $10$,$000$ sets of these; we take $N_{\rm
sim}=1\times 10^{5}$. To each realisation we fit functions
$\Delta\gamma_{1}=g_{11}\gamma_{1}+q_{111}\gamma^{2}_{1}+c_{1}$ and
$\Delta\gamma_{1}=g_{11}\gamma_{1}+c_{1}$.

In Figure 1 we show the result of these tests. As expected if the correct
model is fitted (a quadratic form) then the estimated values for $m_{0}$,
$m_{2}+m_{-2}+m_{6}$ and $c_{1}$ are unbiased, however if a linear function is
fitted the estimates for $m_{0}$ and $c_{1}$ are biased. In particular $c_{1}$
is biased by $\delta c_{1}\simeq 2\times 10^{-4}$. In Figure 1 we also show an
example realisation of the $\Delta\gamma_{1}$ vs. $\gamma_{1}$ plane in this
test setup. In Figure 2 we vary $m_{2}$ by up to $1.5\times 10^{-2}$, keeping
$m_{-2}+m_{6}=m_{2}$, and show how the bias in $c_{1}$ and $m_{0}$ vary. We
compare these to requirements on $m_{0}$ and $c_{1}$ for a Stage-IV (Albrecht
et al., 2006) dark energy experiment (Cropper et al., 2013). We find that
(under the assumptions of this simple test) if one assumes a linear fit then
this introduces a spurious $\delta c_{1}\simeq 0.02m_{2}$ whereas the $m_{0}$
term is relatively unaffected.

Figure 1: A test on randomly generated $\gamma_{1}$ values ($10$,$000$
realisations of $10$,$000$ galaxies) with
$\widetilde{\gamma}_{1}=\gamma_{1}+(m_{0}+m_{4})\gamma_{1}+(m_{2}+m_{-2}+m_{6})\gamma^{2}_{1}+(m_{2}-m_{-2}-m_{6})\gamma^{2}_{2}+c_{1}$
with $\smash{m_{0}=2\times 10^{-3}}$, $m_{4}=0$, $\smash{m_{2}=1\times
10^{-2}}$, $m_{-2}+m_{6}=m_{2}$ (in order to remove any $\gamma_{2}$
dependency for this simple test) and $c_{1}=0$. Shown are the distribution in
the measured values of $g_{11}$, $c_{11}$ and $q_{11}$ assuming a quadratic
fit to values
$\smash{\Delta\gamma_{1}=g_{11}\gamma_{1}+q_{111}\gamma^{2}_{1}+c_{1}}$ (top
row), and a linear fit to the values $\Delta\gamma_{1}=g_{11}\gamma_{1}+c_{1}$
(bottom row). The bottom right plot shows an example realisation.

Figure 2: The biased caused in the inferred $c_{1}$ and $m_{0}$ caused by a
non-zero $m_{2}$ term, keeping $m_{-2}+m_{6}=m_{2}$ and $m_{4}=0$. The left
plot shows the bias in $c_{1}$ (blue line) and a shaded region $\pm 1\times
10^{-4}$, the right plot shows the bias in $m_{0}$ (red line) and a shaded
region $\pm 2\times 10^{-3}$.

###  2.2 Test on a real method

To test whether such quadratic terms may be present in a real shape
measurement algorithm we use the GalSim (Rowe et al., 2015) demo2 which uses a
sheared, exponential profile for the galaxy; convolves it by a circular Moffat
PSF; and adds Poisson noise to the image. The HSM (Hirata & Seljak, 2003;
Mandelbaum et al., 2005) method is then used to determine the shear values,
and we investigate the PSF correction method ‘KSB’ (Kaiser et al., 1995). We
generate $N_{\rm gal}=100$,$000$ realisations where for each we sample the
true $\gamma_{1}$ and $\gamma_{2}$ from a uniform distribution between
$[-0.06,0.06]$ with $|\gamma|<0.06$. We then minimise the function in equation
(11) over the set of $m_{s}$ values333This is done using the
scipy.optimize.minimize function; throughout we use
$\sigma(\widetilde{\Delta}\gamma_{i,g})$ equal to the standard deviation of
the distribution of measured $\widetilde{\Delta}\gamma_{i,g}$ point estimates
which is $\sigma(\widetilde{\Delta}\gamma_{i,g})\simeq 0.01$. The validity of
this approach was tested by creating data with known $m_{s}$ values and
checking they were recovered for the noise level matching those in the
calibration sample used..

We show the results in Figure 3 in the $\Delta\gamma_{1}$ vs. $\gamma_{1}$;
$\Delta\gamma_{1}$ vs. $\gamma_{2}$; $\Delta\gamma_{2}$ vs. $\gamma_{1}$;
$\Delta\gamma_{2}$ vs. $\gamma_{2}$; and $\Delta\gamma_{2}$ vs. planes. It is
important to appreciate that these planes are projections of the 3D
$(\Delta\gamma_{i},\gamma_{1},\gamma_{2})$ fit into 2D planes, therefore in
each plot we show the projection of the fitted surface into these planes. We
also show the full planes at the bottom of the Figure. We find a reduced
$\chi^{2}_{R}-1={\rm min}[\chi^{2}/(N_{\rm gal}-N_{\rm par})]-1=-6\times
10^{-5}$ where $N_{\rm par}=11$.

The individual biases, as well as those found assuming a linear relation are
shown in Table 1. The level of spurious additive bias introduced is similar to
that found using the simple Gaussian tests. We find that in particular $m_{2}$
is significantly non-zero. The key feature in these results that indicates
non-zero quadratic terms is a non-zero $\Delta\gamma_{1}$ vs. $\gamma_{2}$
dependency, which coupled with a near-zero $\Delta\gamma_{2}$ vs. $\gamma_{1}$
indicates a significant non-zero $m_{2}$ term (see equations 2). These tests
are not meant to be a definitive statement on the amplitude of these biases
for the HSM (KSB) method, but are meant to demonstrate that in a simple real
context such biases are non-negligible.

Figure 3: Regression of differences between measured and true shear values in the the $\Delta\gamma_{1}$ vs. $\gamma_{1}$; $\Delta\gamma_{1}$ vs. $\gamma_{2}$; $\Delta\gamma_{2}$ vs. $\gamma_{2}$; and $\Delta\gamma_{2}$ vs. $\gamma_{1}$; for the HSM (Hirata & Seljak, 2003; Mandelbaum et al., 2005) method using the ‘KSB’ PSF correction method; using GalSim (Rowe et al., 2015) demo2. In each plot we show the quadratic fit projected into each plane (blue regions lines), and in all cases a linear fit is shown (red lines). The bottom two plots show the $(\gamma_{1},\gamma_{2})$ plane with colour showing the amplitude of $\Delta\gamma_{1}$ (left) and $\Delta\gamma_{2}$ (right). Parameter | HSM (KSB)
---|---
$m^{R}_{0}$ | $-0.00011\pm 0.0007$
$m^{I}_{0}$ | $0.00003\pm 0.0007$
$m^{R}_{4}$ | $-0.00078\pm 0.0007$
$m^{R}_{2}$ | $0.031\pm 0.024$
$m^{R}_{-2}$ | $-0.0052\pm 0.012$
$m^{R}_{6}$ | $-0.00092\pm 0.0012$
$m^{I}_{4}$ | $-0.00078\pm 0.0007$
$m^{I}_{2}$ | $-0.022\pm 0.024$
$m^{I}_{-2}$ | $0.031\pm 0.012$
$m^{I}_{6}$ | $-0.0036\pm 0.0012$
$c_{1}$ | $-0.00011\pm 0.00004$
$c_{2}$ | $0.00013\pm 0.00004$
$m^{\rm lin}_{1}$ | $-0.00096\pm 0.0007$
$m^{\rm lin}_{2}$ | $0.00074\pm 0.0007$
$c^{\rm lin}_{1}$ | $0.00005\pm 0.00004$
$c^{\rm lin}_{2}$ | $0.00004\pm 0.00004$
$\delta c_{1}$ | $-0.00005\pm 0.00004$
$\delta c_{2}$ | $0.00009\pm 0.00004$
Table 1: Measured biases from GalSim demo 2, using the HSM method with PSF
correction ‘KSB’. The top seven rows use a quadratic fit over all the
$\Delta\gamma_{1}$ vs. $\gamma_{1}$; $\Delta\gamma_{1}$ vs. $\gamma_{2}$;
$\Delta\gamma_{2}$ vs. $\gamma_{1}$; $\Delta\gamma_{2}$ vs. $\gamma_{2}$ and
$\Delta\gamma_{2}$ vs. $(\gamma_{1}+\gamma_{2})/2^{1/2}$ planes. The middle
four rows assume a linear fit only to the $\Delta\gamma_{1}$ vs. $\gamma_{1}$;
and $\Delta\gamma_{2}$ vs. $\gamma_{2}$ planes. The bottom two rows show the
different in the additive biases caused by assuming a linear fit relative to
the quadratic case.

###  2.3 The Impact of Quadratic-Order Integer Terms on Cosmic Shear Power
Spectra

In this Section we investigate how the presence of quadratic-order integer
biases may impact the power spectrum of the shear field. In this derivation we
ignore the effect of masks and spatially varying multiplicative bias fields
(both of which should be negligible for propagation of such biases into power
spectra, see Kitching et al., 2020, 2021). The two shear components can be
combined to extract the E-mode component of the shear field via

$\displaystyle\widetilde{E}_{\alpha}(\boldsymbol{\ell})=\sum_{i}T_{i}(\boldsymbol{\ell})\widetilde{\gamma}_{i,\alpha}(\boldsymbol{\ell}).$
(13)

Throughout we use Greek subscripts to label tomographic bins $\alpha$ and
$\beta$, and Roman subscripts to denote shear components $i$ and $j$. The
trigonometric weighting functions, $T_{i}(\boldsymbol{\ell})$ are defined as:
$T_{1}(\boldsymbol{\ell})=\cos(2\phi_{\ell})$ and
$T_{2}(\boldsymbol{\ell})=\sin(2\phi_{\ell})$, where $\phi_{l}$ is the angular
component of vector $\boldsymbol{\ell}$ which has magnitude $\ell$. The
$E$-mode auto-correlation and cross-correlation spectra,
$C_{\ell;\alpha\beta}^{\gamma\gamma}$ can now be defined as:

$\widetilde{E}_{\alpha}(\boldsymbol{\ell})\widetilde{E}_{\beta}(\boldsymbol{\ell^{\prime}})=(2\pi)^{2}\delta^{2}(\boldsymbol{\ell}+\boldsymbol{\ell^{\prime}})C_{\ell;\alpha\beta}^{\gamma\gamma},$
(14)

where $\delta^{2}$ is the two-dimensional Dirac delta. Assuming the Limber
(Limber, 1953; Kitching et al., 2017; Lemos et al., 2017), flat-Universe
(Taylor et al., 2018a), reduced shear (Deshpande & Kitching, 2020), and flat-
sky (Kamionkowski et al., 1998) approximations, the
$C_{\ell;\alpha\beta}^{\gamma\gamma}$ can be related to the three-dimensional
matter over-density power spectrum $P_{\delta\delta}(\boldsymbol{k},\chi)$ by:

$\displaystyle
C_{\ell;\alpha\beta}^{\gamma\gamma}=\frac{(\ell+2)!}{(\ell-2)!}\frac{1}{\ell^{4}}\int_{0}^{\chi_{\rm
lim}}{\rm
d}\chi\frac{W_{\alpha}(\chi)W_{\beta}(\chi)}{\chi^{2}}P_{\delta\delta}(\boldsymbol{k},\chi);$
(15)

where

$\displaystyle W_{\alpha}(\chi)=\frac{3}{2}\Omega_{\rm
m}\frac{H_{0}^{2}}{c^{2}}\frac{d_{A}(\chi)}{a(\chi)}\int_{\chi}^{\chi_{\rm
lim}}{\rm
d}\chi^{\prime}n_{\alpha}(\chi^{\prime})\frac{d_{A}(\chi^{\prime}-\chi)}{d_{A}(\chi^{\prime})},$
(16)

and $\chi$ is comoving distance. W${}_{\alpha}(\chi)$ is the lensing kernel
for sources in bin $\alpha$, where $\Omega_{\rm m}$ is the dimensionless
present-day matter density of the Universe, $a(\chi)$ is the scale factor of
the Universe, $H_{0}$ is the Hubble constant, $n_{\alpha}(\chi)$ is the
probability distribution of galaxies within bin $\alpha$, and $c$ is the speed
of light in a vacuum.

To construct the change in the power spectrum caused by the presence of non-
zero $m_{2}$, $m_{-2}$ and $m_{6}$ multiplicative biases we start with
equation (12). We will follow the derivation of Deshpande et al. (2020b) (that
is for an analogous problem of reduced shear) and throughout we will assume
that there is no strong spatial dependence of the multiplicative bias terms
(which should be accurate to a good approximation, see Kitching et al., 2020);
in the following equations $m_{s}$ should be read as the mean of the real part
of the multiplicative bias fields (the imaginary parts will not contribute to
the EE power spectrum). We will also only consider terms of cubic order or
lower in the lens potential (i.e. that depend on matter over-density power
spectrum or bispectrum). As an example of how bispectrum terms arise we
consider just $m^{R}_{-2}$ contribution where from equation (12) we have

$\widetilde{\gamma}_{i}(\mathbf{\Omega})=\gamma_{i}(\mathbf{\Omega})+\sum^{2}_{j=1}\sum^{2}_{k=1}\sigma_{5-2i,jk}m^{R}_{-2}\gamma_{j}(\mathbf{\Omega})\gamma_{k}(\mathbf{\Omega}),$
(17)

where $i={1,2}$, and we include the $\mathbf{\Omega}$ explicitly. By computing
the power spectrum of this quantity we find the uncorrected power spectrum
plus an additional term

$(2\pi)^{2}\delta^{2}(\boldsymbol{\ell}+\boldsymbol{\ell^{\prime}})\delta
C^{m_{-2}}_{\ell;\alpha\beta}=m^{R}_{-2}\sum_{i}\sum_{j}\sum_{k}\sum_{l}T_{i}(\boldsymbol{\ell})T_{j}(\boldsymbol{\ell^{\prime}})\sigma_{5-2i,kl}\langle(\gamma_{k}\gamma_{l})_{\alpha}(\boldsymbol{\ell})\gamma_{j,\beta}(\boldsymbol{\ell^{\prime}})\rangle+T_{i}(\boldsymbol{\ell^{\prime}})T_{j}(\boldsymbol{\ell})\sigma_{5-2i,kl}\langle(\gamma_{k}\gamma_{l})_{\beta}(\boldsymbol{\ell^{\prime}})\gamma_{j,\alpha}(\boldsymbol{\ell})\rangle,$
(18)

where $\delta C^{m_{-2}}_{\ell;\alpha\beta}$ are the resulting corrections to
the angular auto and cross-correlation spectra. Applying the Limber
approximation, and summing over all terms we obtain an expression:

$\delta
C^{m^{R}_{-2}}_{\ell;\alpha\beta}=2m_{-2}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}}),$
(19)

where the factor $2$ comes from $\sum_{ijk}\sigma_{5-2i,jk}$. The factors of
$[(\ell+2)!/(\ell-2)!]^{3/2}$ and $1/\ell^{6}$ arise from foregoing the flat-
sky approximation. $B_{\alpha\beta}^{\kappa\kappa\kappa}$, is the two-redshift
convergence bispectrum, given by

$\displaystyle
B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}},\boldsymbol{\ell_{3}})=\frac{1}{2}\int_{0}^{\chi_{\rm
lim}}\frac{{\rm
d}\chi}{\chi^{4}}W_{\alpha}(\chi)W_{\beta}(\chi)[W_{\alpha}(\chi)+W_{\beta}(\chi)]B_{\delta\delta\delta}(\boldsymbol{k_{1}},\boldsymbol{k_{2}},\boldsymbol{k_{3}},\chi),$
(20)

where $\boldsymbol{k_{x}}=(\boldsymbol{\ell_{x}}+1/2)/\chi$ (for $x=1,2,3$).
$B_{\delta\delta\delta}(\boldsymbol{k_{1}},\boldsymbol{k_{2}},\boldsymbol{k_{3}},\chi)$
is the three-dimensional matter over-density bispectrum.

Going beyond this demonstrative case, by taking the full power spectrum of
equation (4), and focussing the real parts of the multiplicative bias fields,
we find the uncorrected standard terms that only depend on $m^{R}_{0}$ and
$m^{R}_{4}$ plus the following correction

$\displaystyle\delta C_{\ell;\alpha\beta}$ $\displaystyle=$ $\displaystyle
2(1+m^{R}_{0})m^{R}_{-2}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}})$
(21) $\displaystyle-$ $\displaystyle
2(1+m^{R}_{0})m^{R}_{6}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(-\boldsymbol{\ell},-\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}})$
$\displaystyle+$ $\displaystyle
2(1+m^{R}_{0})m^{R}_{2}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},-\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}})$
$\displaystyle-$ $\displaystyle
2m^{R}_{4}m^{R}_{-2}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},\boldsymbol{\ell^{\prime}},\boldsymbol{\ell}+\boldsymbol{\ell^{\prime}})$
$\displaystyle+$ $\displaystyle
2m^{R}_{4}m^{R}_{6}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(-\boldsymbol{\ell},-\boldsymbol{\ell^{\prime}},\boldsymbol{\ell}+\boldsymbol{\ell^{\prime}})$
$\displaystyle-$ $\displaystyle
2m^{R}_{4}m^{R}_{2}\left[\frac{(\ell+2)!}{(\ell-2)!}\right]^{3/2}\frac{1}{\ell^{6}}\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},-\boldsymbol{\ell^{\prime}},\boldsymbol{\ell}+\boldsymbol{\ell^{\prime}}).$

The varying prefactors come from the sums over the prefectors associated in
equation (12), and the differing signs in convergence bispectrum occur due to
the various combinations of complex conjugation that result from the two-point
correlation of equation (4), making use of the fact that
$\kappa^{*}(\boldsymbol{\ell})=\kappa(-\boldsymbol{\ell})$ since $\kappa$ is a
real field (see e.g. Schneider et al., 2005). Note that the imaginary terms do
not contribute to the EE power spectrum.

To first order in multiplicative bias terms, and assuming the $\ell$-mode
dependent prefactor is unity, this reduces to

$\displaystyle\delta C_{\ell;\alpha\beta}$ $\displaystyle\simeq$
$\displaystyle 2(m^{R}_{-2})\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}})$
(22) $\displaystyle-$ $\displaystyle 2(m^{R}_{6})\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(-\boldsymbol{\ell},-\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}})$
$\displaystyle+$ $\displaystyle 2(m^{R}_{2})\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},-\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}}).$

It can also be shown the integral over the second and third terms in equation
(22) is in fact equal to the first term; because the only angular dependence
comes from the cosine in the outermost integral, and since
$\cos(2(\phi_{\ell}-[\phi_{\ell^{\prime}}+\pi]))=\cos(2([\phi_{\ell}+\pi]-[\phi_{\ell^{\prime}}+\pi]))=\cos(2(\phi_{\ell}-\phi_{\ell^{\prime}}))$
(there is no angular dependence in the matter bispectrum because there is only
one valid closed triangle for any given combination of $\ell$ sides, so the
angles between the sides are always the same; see e.g. Takahashi et al., 2020,
for details). Therefore the final expression, to linear order in $m_{s}$ is

$\displaystyle\delta C_{\ell;\alpha\beta}\simeq
2[m^{R}_{2}+m^{R}_{-2}-m^{R}_{6}]\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}}).$
(23)

This is equal to the reduced shear approximation (Deshpande et al., 2020b),
except with a prefactor $2(m^{R}_{2}+m^{R}_{-2}-m^{R}_{6})$. Therefore the
expected bias on cosmological parameters caused by ignoring quadratic terms in
multiplicative biases should be
$\delta\theta_{i}=2(m^{R}_{2}+m^{R}_{-2}-m^{R}_{6})\delta\theta^{\rm RS}_{i}$
where $\delta\theta_{i}$ is a bias in cosmological parameter $\theta_{i}$, and
$\delta\theta^{\rm RS}_{i}$ are the biases caused by the reduced shear
approximation, that are given in (Deshpande et al., 2020b) for a _Euclid_
-like Stage-IV dark energy experiment.

The biases on cosmological parameters derived in (Deshpande et al., 2020b),
relative to their uncertanties (denoted $\sigma$) are $\delta\Omega_{\rm
m}=-0.10\sigma$, $\delta\Omega_{\rm b}=0.023\sigma$, $\delta h=0.072\sigma$,
$\delta n_{\rm s}=-0.10\sigma$, $\delta\sigma_{8}=0.055\sigma$; all of which
are relatively small. However, $\delta\Omega_{\rm DE}=0.31\sigma$, $\delta
w_{0}=-0.32\sigma$, and $\delta w_{a}=0.40\sigma$, all of which are
significant biases. We refer to Deshpande et al. (2020b) for the full details
of these calculations.

A reasonable requirement is $\delta\theta_{i}/\sigma\simeq 0.25$, to ensure
90% overlap between biased and unbiased confidence regions (see e.g. Massey et
al., 2013). Therefore if one wishes the contribution from quadratic terms to
be sub-dominant to other sources of potential bias, and only account for $Q\%$
of the total budget then a requirement can be set on the sum of the quadratic
multiplicative biases of $|m_{2}+m_{-2}-m_{6}|<Q(0.25/0.4)/2=0.3125Q$ (taking
the largest of the potential biases, $\delta w_{a}=0.40\sigma$). If one
chooses a reasonable $Q=0.1$, this leads to a requirement that
$|m_{2}+m_{-2}-m_{6}|<0.031$. If one wishes $Q$ be much smaller this would
lead to a proportionately smaller requirement.

###  2.4 Discussion of Integer Terms

By combining the result from the previous Sections we therefore see that there
are two effects of ignoring quadratic biases of integer power in shear.
Firstly that spurious additive biases may be present where $\delta c_{1}\simeq
0.02m_{2}$ for example. Because the requirement for Stage-IV surveys is on the
error on the additive bias where $\sigma(\delta c_{i})\leq 10^{-4}$ (Massey et
al., 2013; Cropper et al., 2013; Kitching et al., 2021) this implies that
$\sigma(m_{s})\simeq 5\times 10^{-3}$. The exact impact of such a spurious
additive term will depend on the spatial variation of the spurious additive
bias field (as shown in Kitching et al., 2021) with $\delta
C_{\ell;ij}=2C^{\gamma\delta c}_{\ell;ij}+C^{\delta c\delta c}_{\ell;ij}$,
where the autocorrelation of the additive bias and the cross-correlation with
the shear field contribute to the change in the power spectrum. Secondly,
ignoring quadratic dependence can bias cosmological parameters by up to
$2(m^{R}_{2}+m^{R}_{-2}-m^{R}_{6})0.4\sigma$. A reasonable requirement on the
amplitude of these terms is $|m^{R}_{2}+m^{R}_{-2}-m^{R}_{6}|\leq 0.031$.
However, if $m_{2}$, $m_{-2}$ and $m_{6}$ are significantly non-zero, and
cannot be reduced, but are known accurately then the bispectrum could be
included in any inference and the values of $m^{R}_{2}$ $m^{R}_{-2}$ and
$m^{R}_{6}$ marginalised over with some associated priors.

To mitigate the impact of these biases it should be noted that methods to
remove small-scale sensitivity from the cosmic shear power spectrum, e.g.
k-cut (or x-cut) cosmic shear (Taylor et al., 2018b; Taylor et al., 2020), can
be used to remove susceptibility to reduced shear corrections that have the
same scale-dependent form as quadratic bias corrections. Using k-cut cosmic
shear Deshpande et al. (2020a) find that reduced shear corrections can be
removed at the expense of only a 10% increase in $1\sigma$ error bars on
cosmological parameters. If k-cut cosmic shear is applied this increase in
error would be the total for both effects (i.e. 10% total, not 20%). This
worst-case (removal of problematic scales) should not adversely impact the
objective for Stage-IV experiments to constrain dark energy parameters to have
a Figure-of-Merit of more than $400$ (Amendola et al., 2013; Euclid
Collaboration et al., 2020).

Combining both effects the overall change in the power spectrum caused by
ignoring quadratic terms will be

$\displaystyle\delta C_{\alpha\beta}\simeq 2C^{\gamma\delta
c}_{\ell;\alpha\beta}+C^{\delta c\delta
c}_{\ell;\alpha\beta}-2[m^{R}_{2}+m^{R}_{-2}-m^{R}_{6}]\int_{0}^{\infty}\frac{{\rm
d}^{2}\boldsymbol{\ell^{\prime}}}{(2\pi)^{2}}\cos(2\phi_{\ell^{\prime}}-2\phi_{\ell})B_{\alpha\beta}^{\kappa\kappa\kappa}(\boldsymbol{\ell},\boldsymbol{\ell^{\prime}},-\boldsymbol{\ell}-\boldsymbol{\ell^{\prime}}),$
(24)

where any additional spurious additive bias causes the first two terms (the
first of which may be negative), and the third is caused by misestimation of
the power assuming $m_{2}=m_{-2}=m_{6}=0$.

##  3 Including the Impact of Quadratic-Order Half-Integer Terms on Shape
Measurement

We next turn to the half-integer terms in equation (4). In general these are
more complex to understand because they do not propagate into shape
measurement calibration statistics or the power spectrum in a straightforward
manner. If we consider the $m_{1}\gamma^{1/2}$ term and express this in
$\gamma_{1}$ and $\gamma_{2}$ we find for example that

$\displaystyle
m_{1}\gamma^{1/2}=m_{1}\left[\frac{(\gamma^{2}_{1}+\gamma^{2}_{2})^{1/4}}{[(\gamma_{1}+(\gamma^{2}_{1}+\gamma^{2}_{2})^{1/2})^{2}+\gamma^{2}_{2}]^{1/2}}\right](\gamma_{1}+(\gamma^{2}_{1}+\gamma^{2}_{2})^{1/2}+{\rm
i}\gamma_{2}).$ (25)

Hence, unlike the integer terms the analytic propagation of such biases
$\Delta\gamma_{1}$ and $\Delta\gamma_{2}$ is not trivial. However, the
prefactor
$(\gamma^{2}_{1}+\gamma^{2}_{2})^{1/4}/[(\gamma_{1}+(\gamma^{2}_{1}+\gamma^{2}_{2})^{1/2})^{2}+\gamma^{2}_{2}]^{1/2}$
is approximately constant over the range $|\gamma|<0.06$, with median value of
$f\simeq 5.4$. Therefore, a rule-of-thumb is that $m_{1}\gamma^{1/2}\approx
fm_{1}\gamma_{1}+{\rm i}fm_{1}\gamma_{2}+f|\gamma|m_{1}$, hence $m_{1}$
introduces both a $m_{0}$-like term and an additive bias.

We can write the change in $\gamma_{1}$ and $\gamma_{2}$ including such terms
as

$\displaystyle\Delta\gamma_{1}=\widetilde{\gamma}_{1}-\gamma_{1}$
$\displaystyle=$
$\displaystyle(m^{R}_{0}+m^{R}_{4})\gamma_{1}+(m^{R}_{2}+m^{R}_{-2}+m^{R}_{6})\gamma^{2}_{1}+(m^{R}_{2}-m^{R}_{-2}-m^{R}_{6})\gamma^{2}_{2}$
$\displaystyle-$
$\displaystyle(m^{I}_{0}-m^{I}_{4})\gamma_{2}-2(m^{I}_{-2}-m^{I}_{6})\gamma_{1}\gamma_{2}$
$\displaystyle+$ $\displaystyle c_{1}$ $\displaystyle+$
$\displaystyle(m^{R}_{1}+m^{R}_{3})\mathbb{R}(\gamma^{1/2})+m^{R}_{2}\mathbb{R}(\gamma^{1/2}\gamma^{*,1/2})+m^{R}_{0}\mathbb{R}(\gamma^{3/2}\gamma^{*,1/2})+m^{R}_{4}\mathbb{R}(\gamma^{1/2}\gamma^{*,3/2})$
$\displaystyle+$ $\displaystyle
m^{R}_{1}\mathbb{R}(\gamma\gamma^{*,1/2})+m^{R}_{3}\mathbb{R}(\gamma^{1/2}\gamma^{*})+m^{R}_{-1}\mathbb{R}(\gamma^{3/2})+m^{R}_{5}\mathbb{R}(\gamma^{*,3/2})$
$\displaystyle-$
$\displaystyle(m^{I}_{1}-m^{I}_{3})\mathbb{I}(\gamma^{1/2})-m^{I}_{2}\mathbb{I}(\gamma^{1/2}\gamma^{*,1/2})-m^{I}_{4}\mathbb{I}(\gamma^{1/2}\gamma^{*,3/2})$
$\displaystyle-$ $\displaystyle
m^{I}_{1}\mathbb{I}(\gamma\gamma^{*,1/2})-m^{I}_{3}\mathbb{I}(\gamma^{1/2}\gamma^{*})+-m^{I}_{-1}\mathbb{I}(\gamma^{3/2})-m^{I}_{5}\mathbb{I}(\gamma^{*,3/2})$
$\displaystyle\Delta\gamma_{2}=\widetilde{\gamma}_{2}-\gamma_{2}$
$\displaystyle=$
$\displaystyle(m^{R}_{0}-m^{R}_{4})\gamma_{2}+2(m^{R}_{-2}-m^{R}_{6})\gamma_{1}\gamma_{2}$
(26) $\displaystyle+$
$\displaystyle(m^{I}_{0}+m^{I}_{4})\gamma_{1}+(m^{I}_{2}+m^{I}_{-2}+m^{I}_{6})\gamma^{2}_{1}+(m^{I}_{2}-m^{I}_{-2}-m^{I}_{6})\gamma^{2}_{2}$
$\displaystyle+$ $\displaystyle c_{2}$ $\displaystyle+$
$\displaystyle(m^{R}_{1}-m^{R}_{3})\mathbb{I}(\gamma^{1/2})+m^{R}_{2}\mathbb{I}(\gamma^{1/2}\gamma^{*,1/2})+m^{R}_{0}\mathbb{I}(\gamma^{3/2}\gamma^{*,1/2})+m^{R}_{4}\mathbb{I}(\gamma^{1/2}\gamma^{*,3/2})$
$\displaystyle+$ $\displaystyle
m^{R}_{1}\mathbb{I}(\gamma\gamma^{*,1/2})+m^{R}_{3}\mathbb{I}(\gamma^{1/2}\gamma^{*})+m^{R}_{-1}\mathbb{I}(\gamma^{3/2})+m^{R}_{5}\mathbb{I}(\gamma^{*,3/2})$
$\displaystyle+$
$\displaystyle(m^{I}_{1}+m^{I}_{3})\mathbb{R}(\gamma^{1/2})+m^{I}_{2}\mathbb{R}(\gamma^{1/2}\gamma^{*,1/2})+m^{I}_{4}\mathbb{R}(\gamma^{1/2}\gamma^{*,3/2})$
$\displaystyle+$ $\displaystyle
m^{I}_{1}\mathbb{R}(\gamma\gamma^{*,1/2})+m^{I}_{3}\mathbb{R}(\gamma^{1/2}\gamma^{*})+m^{I}_{-1}\mathbb{R}(\gamma^{3/2})+m^{I}_{5}\mathbb{R}(\gamma^{*,3/2})$

where $\mathbb{R}$ and $\mathbb{I}$ refer to the real and imaginary parts of
complex numbers respectively.

In total, to second order in shear, this results in $20$ free parameters
consisting of the real and imaginary parts of the sequence of multiplicative
biases $m_{s}$ with $s=-2$–$5$, plus the real and imaginary parts of $c$.

In Figure 4 we show this full model fit to the calibration data for the HSB
‘KSB’ method described in Section 2.2. It can be seen that the surface fit to
the $(\Delta\gamma_{1},\Delta\gamma_{2})$ plane is significantly more complex
than that described by the fit only including integer terms, but that it is
also a better fit to the data, capturing several features that were not well-
fit by the integer terms’ behaviour only. We find a reduced
$\chi^{2}_{R}-1={\rm min}[\chi^{2}/(N_{\rm gal}-N_{\rm par})]-1=-4\times
10^{-5}$ where $N_{\rm par}=20$; which represents an improvement over the
integer fit although both are good fits to the data. We find that the best-fit
values of the full set of parameters are $m^{R}_{0}=0.012$, $m^{R}_{4}=0.006$,
$m^{R}_{2}=0.009$, $m^{R}_{-2}=0.089$, $m^{R}_{6}=0.043$, $c_{1}=-0.034$,
$c_{2}=-0.019$, $m^{R}_{1}=-0.003$, $m^{R}_{3}=-0.002$, $m^{R}_{-1}=-0.044$,
$m_{5}=-0.023$, $m^{I}_{4}=-0.026$, $m^{I}_{2}=0.011$, $m^{I}_{-2}=0.064$,
$m^{I}_{6}=0.017$, $m^{I}_{1}=0.001$, $m^{I}_{3}=0.002$,$m^{I}_{-1}=-0.011$,
$m^{I}_{5}=0.006$; errors on each parameter are similar in order of magnitude
to those in Table 1 (we exclude them here for clarity).

It can be seen that by allowing flexibility in the model to include all terms
up to second order that the best fit values can change significantly compared
to the case of fitting only integer terms (see Table 1), in particular the
additive bias values. This is a more general, higher-dimensional, feature that
was observed in Section 2.1 i.e. that if a simplified model is fit to
calibration data that the inferred values of the biases can be biased.

Figure 4: Regression of differences between measured and true shear values in
the the $\Delta\gamma_{1}$ vs. $\gamma_{1}$; $\Delta\gamma_{1}$ vs.
$\gamma_{2}$; $\Delta\gamma_{2}$ vs. $\gamma_{2}$; and $\Delta\gamma_{2}$ vs.
$\gamma_{1}$; for the HSM (Hirata & Seljak, 2003; Mandelbaum et al., 2005)
method using the ‘KSB’ PSF correction method; using GalSim (Rowe et al., 2015)
demo2 using all terms up to second order (equation 3). In each plot we show
the fit projected into each plane (blue regions lines), and in all cases a
linear fit is shown (red lines). The bottom two plots show the
$(\gamma_{1},\gamma_{2})$ plane with colour showing the amplitude of
$\Delta\gamma_{1}$ (left) and $\Delta\gamma_{2}$ (right).

###  3.1 The Impact of Quadratic-Order Half-Integer Terms on Cosmic Shear
Power Spectra

Taking all terms to linear order in bias the impact of half-integer terms on
the cosmic shear power spectrum is to introduce the following

$\displaystyle\delta C_{\ell;\alpha\beta}$ $\displaystyle=$ $\displaystyle
2[m^{R}_{-1}C^{\gamma(\gamma^{3/2})}_{\ell;\alpha\beta}+m^{R}_{1}C^{\gamma(\gamma^{1/2})}_{\ell;\alpha\beta}+m^{R}_{3}C^{\gamma(\gamma^{*,1/2})}_{\ell;\alpha\beta}+m^{R}_{5}C^{\gamma(\gamma^{*,3/2})}_{\ell;\alpha\beta}$
(27) $\displaystyle+$ $\displaystyle
m^{R}_{0}C^{\gamma(\gamma^{3/2}\gamma^{*,1/2})}_{\ell;\alpha\beta}+m^{R}_{1}C^{\gamma(\gamma\gamma^{*,1/2})}_{\ell;\alpha\beta}+m^{R}_{2}C^{\gamma(\gamma^{1/2}\gamma^{*,1/2})}_{\ell;\alpha\beta}+m^{R}_{3}C^{\gamma(\gamma^{1/2}\gamma^{*})}_{\ell;\alpha\beta}+m^{R}_{4}C^{\gamma(\gamma^{1/2}\gamma^{*,3/2})}_{\ell;\alpha\beta}]+{\mathcal{O}}(m^{2})$

where $C^{X(Y)}_{\ell;\alpha\beta}$ corresponds to the correlation $\langle
XY^{*}\rangle$.

The impact of half-integer terms on the cosmic shear power spectrum is
complicated and not simply a matter of taking a suitable exponent of the power
spectrum or the bispectrum. In general, for a tomographic redshift bin
$\alpha$, the spherical harmonic representation of the shear can be written:

$\displaystyle\widetilde{\gamma}_{\alpha;\ell m}$
$\displaystyle=\frac{1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}4\pi
i^{\ell}\int_{0}^{\chi_{\rm lim}}{\rm d}\chi
W_{\alpha}(\chi)\int_{0}^{\infty}\frac{{\rm
d}^{3}k}{(2\pi)^{3}}j_{\ell}(k\chi){}_{2}Y^{*}_{\ell
m}(\boldsymbol{\hat{k}})\widetilde{\delta}(\boldsymbol{k},\chi),$ (28)

where $\widetilde{\delta}$ is the matter overdensity of the Universe, and the
spatial momentum vector $\boldsymbol{k}$ has magnitude $k=|\boldsymbol{k}|$;
$W_{\alpha}$ is given in equation (16).

To construct the power spectrum corresponding to the cross-correlation between
shear and a half-integer power $\langle\gamma\gamma^{*,1/2}\rangle$ for
example would therefore require a calculation that involved the following
correlation in $\widetilde{\gamma}_{i;\ell m}$

$\displaystyle\delta C^{\gamma\gamma^{1/2}}_{\ell;\alpha\beta}\propto$
$\displaystyle\frac{1}{2\ell+1}\sum_{m}\left[\int_{0}^{\chi_{\rm lim}}{\rm
d}\chi W_{\alpha}(\chi)\int_{0}^{\infty}\frac{{\rm
d}^{3}k}{(2\pi)^{3}}j_{\ell}(k\chi){}_{2}Y^{*}_{\ell
m}(\boldsymbol{\hat{k}})\widetilde{\delta}(\boldsymbol{k},\chi)\right]$ (29)
$\displaystyle\left[\int_{0}^{\chi_{\rm lim}}{\rm d}\chi
W_{\beta}(\chi)\int_{0}^{\infty}\frac{{\rm
d}^{3}k}{(2\pi)^{3}}j_{\ell}(k\chi){}_{2}Y^{*}_{\ell
m}(\boldsymbol{\hat{k}})\widetilde{\delta}(\boldsymbol{k},\chi)\right]^{*,1/2},$

a similar expression would exist to third order for the bispectrum. It can be
seen that the standard factoring of the power spectrum into kernels multiplied
by a matter overdensity power spectrum is not possible in this case.

To understand the magnitude of such effects, rather than calculate such terms
explicitly we instead create Gaussian random fields consistent with a true
shear field with a power spectrum described in equation (14), for a single
tomographic bin, with cosmological parameters set to _Planck_ maximum
likelihood values (Planck Collaboration et al., 2018) and a Flaugher et al.
(2015) number density (see Deshpande & Kitching, 2020, for details). We then
take half-integer powers of the generated shear field and compute
$C^{\gamma\gamma^{1/2}}_{\ell}$ and $C^{\gamma\gamma^{3/2}}_{\ell}$. We show
the result in Figure 5. We make an ansatz that the functional form of such
terms will be a simple amplitude scaling of the shear power spectrum
$AC^{\gamma\gamma}_{\ell}$. We find that this ansatz does accurately fit the
power spectra with $A=15$ and $A=0.1$ for $C^{\gamma\gamma^{1/2}}_{\ell}$ and
$C^{\gamma\gamma^{3/2}}_{\ell}$ respectively. We also find that
$C^{\gamma\gamma^{1/2}}_{\ell}=C^{\gamma\gamma^{*,1/2}}_{\ell}$ and
$C^{\gamma\gamma^{3/2}}_{\ell}=C^{\gamma\gamma^{*,3/2}}_{\ell}$ to a good
approximation.

Similarly we find that the second order terms are approximately fit with
amplitude scaling of the contribution to the power spectrum from the integer-
shear bispectrum $AC^{\gamma(\gamma\gamma)}_{\ell}$ with $A=10$ and $1$ for
$C^{\gamma(\gamma\gamma^{1/2})}_{\ell}$ and
$C^{\gamma(\gamma\gamma^{3/2})}_{\ell}$ respectively. We emphasise that these
tests are only using Gaussian random fields and future work should test if
such anstaz apply for more realistic simulations.

Figure 5: Left: the power spectrum of a Gaussian realisation of a true shear
field with $C^{\gamma\gamma}_{\ell}$ (red line) compared to the input model
(black solid line), and the cross-correlation power spectra
$C^{\gamma\gamma^{1/2}}_{\ell}$ (blue) and $C^{\gamma\gamma^{3/2}}_{\ell}$
(green) compared to a simple amplitude scaling of $AC^{\gamma\gamma}_{\ell}$
with $A=6$ and $A=0.035$ (dashed and dotted lines respectively). Right: the
impact on the power spectrum of the observed bispectrum of Gaussian
realisation of a true shear field with $C^{\gamma\gamma}_{\ell}$. The red line
shows $C^{\gamma(\gamma\gamma)}_{\ell}$, the blue line
$C^{\gamma(\gamma\gamma^{1/2})}_{\ell}$ and the green line
$C^{\gamma(\gamma\gamma^{3/2})}_{\ell}$. The solid, dashed and dot-dashed
lines show $AC^{\gamma(\gamma\gamma)}_{\ell}$ with $A=1,10$ and $1$
respectively.

###  3.2 Discussion of Half-Integer Terms

We find in general that the inclusion of half-integer terms is much more
complex than integer terms for several reasons. First that the amplitude of
such effects may be large – because shear is less than unity, the square root
is larger than the original quantity; secondly the impact on the change in
$\gamma_{1}$ and $\gamma_{2}$ is highly non-trivial and not open to analytic
derivation, as was the case with the integer terms; thirdly the power spectrum
and bi-spectrum terms are also difficult to compute analytically because they
do not factor in the way that integer terms do.

We find for the HSB method that the fit including half-integer terms is in
general a better fit to the projected $\Delta\gamma_{i}$ vs. $\gamma_{j}$
planes, but that the shape is complex. Furthermore all $20$ free parameters
corresponding to the real and imaginary parts of $m_{s}$ (with $s=-2$–$5$)
plus the additive terms are non-zero.

Regarding the power spectrum, we find that a simple amplitude scaling of the
shear power spectrum is sufficient to capture the behaviour of linear order
half-integer terms, and that a simple scaling of bi-spectrum is sufficient to
capture quadratic order half-integer terms. If the ansatz for the bispectrum
terms is correct then applying $k$-cut cosmic shear should, in a similar
manner to the quadratic integer terms, remove sensitivity to such terms.

However, the half-integer terms also introduce new half-integer order bias
parameters that lead to an overall change in power spectrum of

$\displaystyle\delta C_{\alpha\beta}\simeq
2[15(m^{R}_{1}+m^{R}_{3})C^{\gamma\gamma}_{\ell;\alpha\beta}+0.1(m^{R}_{-1}+m^{R}_{5})C^{\gamma\gamma}_{\ell;\alpha\beta}],$
(30)

assuming the amplitude scaling ansatz is correct.

##  4 Discussion

In this section we address the overall impact of quadratic terms including
both integer and half-integer terms. We have found that quadratic terms
introduce additional bispectra to the cosmic shear power spectrum. For integer
terms these are multiplied by $m^{R}_{2}$, $m^{R}_{-2}$ and $m^{R}_{6}$, in
the half-integer case by $m^{R}_{0}$, $m^{R}_{1}$, $m^{R}_{2}$, $m^{R}_{3}$
and $m^{R}_{4}$. If no mitigation approach is taken cosmic shear power spectra
will depend on bispectra that include contributions from all these fields.
However, as discussed in Section 4 bispectrum terms can be removed by making
scale-dependent cuts to the data vectors. Therefore the important
contributions to the power spectrum, up to second-order in shear, are

$\displaystyle\delta C_{\ell;\alpha\beta}\simeq
2[m^{R}_{0}+15(m^{R}_{1}+m^{R}_{3})+0.1(m^{R}_{-1}+m^{R}_{5})]C^{\gamma\gamma}_{\ell;\alpha\beta},$
(31)

this differs from the usual expression that only includes the $m^{R}_{0}$
term. For Stage-IV experiments requirements on $m^{R}_{0}$ have been set such
that if $\sigma(m^{R}_{0})\leq 0.01$ cosmological constraints should be
unbiased (see Kitching et al., 2020; Cragg et al., 2022). We find that this
requirement should be on
$\sigma[m^{R}_{0}+15(m^{R}_{1}+m^{R}_{3})+0.1(m^{R}_{-1}+m^{R}_{5})]$ i.e.
that $m^{R}_{0}$, $m^{R}_{1}$, $m^{R}_{3}$, $m^{R}_{-1}$ and $m^{R}_{5}$ need
to be determined with an overall summed uncertainty of $0.01$. Assuming
equipartition between the five terms would suggest that each term would need
to be constrained to $4\times 10^{-3}$. Of the five terms the large prefactor
multiplying $(m^{R}_{1}+m^{R}_{3})$ suggested that this is the most important
to constraint. Only $m^{R}_{0}$ has been determined in experiments to date. We
note that whilst the the metacalibration method (see e.g. Huff & Mandelbaum,
2017; Yamamoto et al., 2022) currently does not correct for non-scalar
multiplicative bias fields that is could be generalised to do so.

Finally, we note that the $20$ dimensional parameter fit is very large, and
there is considerable flexibility in the model and degeneracy. Future work
should investigate how to incorporate any uncertainties into cosmological
inference, and how to place appropriate priors on each multiplicative bias.

##  5 Conclusions

In this paper we revisit potential biases in cosmic shear power spectra caused
by quadratic multiplicative bias terms. We find that such biases can be
measured by performing general regression on $\Delta\gamma_{i}$ as a function
of $\gamma_{1}$ and $\gamma_{2}$.

We find that terms of integer power in shear and first order in bias impact on
the power spectrum is the same as that caused by the reduced shear
approximation, an additional bispectrum dependency, except multiplied by
$2(m_{2}+m_{-2}-m_{6})$. Ignoring quadratic terms can lead to biases in
cosmological of up to $2(m_{2}+m_{-2}-m_{6})0.4\sigma$. A reasonable
requirement on the amplitude of these terms is $|m_{2}+m_{-2}-m_{6}|\leq
0.031$. If one ignores quadratic terms entirely and instead fits only a linear
dependence to calibration data then this can in addition cause spurious
multiplicative and additive biases.

Investigating terms of half-integer power in shear we find new linear order
terms, and new quadratic terms. The linear order terms introduce power
spectrum changes proportional to the shear power spectrum, and the quadratic
order terms introduce power spectrum changes proportional to the bispectrum.
We find that a requirement of
$\sigma[m^{R}_{0}+15(m^{R}_{1}+m^{R}_{3})+0.1(m^{R}_{-1}+m^{R}_{5})]\leq 0.01$
is needed for cosmological constraints from Stage-IV experiments to be
unbiased.

In future therefore Stage-IV dark energy experiment should therefore seek to
measure and minimise quadratic bias terms, and investigation should be done to
apply Bayesian parameter estimation and model comparison to find the best
model and parameter combination to apply to calibration data.

###### Acknowledgements.

_Acknowledgements:_ TDK acknowledges funding from the EU’s Horizon 2020
programme, grant agreement No 776247. ACD acknowledges funding from the Royal
Society. We thank Henk Hoekstra for comments on an early draft. We thank Andy
Taylor, Alex Hall and Giuseppe Congedo for discussion and comments.

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# Domain wall motion at low current density in a synthetic antiferromagnet
nanowire

Christopher E. A. Barker  School of Physics and Astronomy, University of
Leeds, Leeds LS2 9JT, United Kingdom  Simone Finizio  Paul Scherrer Institut,
5232 Villigen PSI, Switzerland  Eloi Haltz  Present address: Université
Sorbonne Paris Nord, 99 Av. Jean Baptiste Clément, 93430 Villetaneuse, France
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom  Sina Mayr  Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  Philippa M. Shepley  School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom  Bragg Centre for Materials
Research, University of Leeds, Leeds LS2 9JT, United Kingdom  Thomas A. Moore
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom  Gavin Burnell  School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom  Jörg Raabe  Paul Scherrer Institut, 5232
Villigen PSI, Switzerland  Christopher H. Marrows<EMAIL_ADDRESS>School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom  Bragg Centre for Materials Research, University of Leeds, Leeds LS2
9JT, United Kingdom

###### Abstract

The current-driven motion of magnetic domain walls (DWs) is the working
principle of magnetic racetrack memories. In this type of spintronic
technology, high current densities are used to propel DW motion in magnetic
nanowires, causing significant wire heating. Synthetic antiferromagnets are
known to show very fast DW motion at high current densities, but lower current
densities around onset of motion have received less attention. Here we use
scanning transmission x-ray microscopy to study the response of DWs in a SAF
multilayer to currents. We observe that the DWs depin at $\sim 3\times
10^{11}$ A/m2 and move more quickly in response to 5 ns duration current
pulses than in comparable conventional multilayers. The results suggest that
DWs in SAF structures are superior to conventional Néel DWs for low energy
consumption racetrack technologies.

Magnetic domain walls (DWs) separate uniformly magnetized domains in a
ferromagnet. They are narrow regions where the magnetization rotates between
the directions in the domains and both influence and respond to spin-polarised
currents [1]. The use of a spin-polarized electron flow and its resulting
torques on magnetic textures has been demonstrated for driving DWs in magnetic
wires. New generations of devices have been proposed based on these effects,
such as the so-called magnetic racetracks that can be used as storage-class
memories [2, 3] or for novel forms of information processing [4, 5, 6]. In
this type of technology, a stream of bits is encoded as a series of domains
separated by domain walls in a magnetic nanowire, which can be shifted along
the wire using electrical current pulses by means of the spin-torque
mechanism. The DW velocity influences the speed of operation whilst the power
dissipated by the current pulse influences the energy consumption of the
device.

Initial versions of the magnetic racetrack were developed using a simple strip
of an in-plane magnetised soft magnetic material such as Permalloy in which
the domain wall motion was actuated by the volume spin-transfer-torque [2].
Further generations have incorporated developments such as multilayer wires
with interface-induced effects such as perpendicular magnetic anisotropy (PMA)
and Dzyaloshinskii-Moriya interactions (DMI) to enforce wall chirality [3],
with domain wall motion now being driven by spin-orbit torques [7, 8].

Most recently, synthetic antiferromagnets (SAFs) [9] have been introduced as
race track materials [10]. These systems are composed of two ferromagnetic
layers coupled to each other antiferromagnetically through a non-magnetic
spacer layer. The magnetic moments of the two layers compensates for each
other leading to fast magnetization dynamics. In particular, very fast domain
wall motion at several 100 m/s for high current drives of a few TA/m2 have
been observed in these systems using a Ru spacer [11]. It has subsequently
been shown that replacing the spacer material with Rh reduces the DW velocity
[12], and that DW velocity in a SAF can be controlled by means of iontronic
gating [13].

Power consumption is a key constraint for such applications. In particular,
fast motion at low current densities, just above the critical current density
$J_{\mathrm{c}}$ at which DWs depin and can be set in motion, is critical.
Here we study, using scanning transmission x-ray microscopy (STXM), the low
current density dynamics of DWs around the onset of motion. We show that
$J_{\mathrm{c}}\sim 3\times 10^{11}$ A/m2 and that DWs move more quickly at
lower current densities than in comparable conventional multilayers. The
multilayers that we study contain ten magnetic layers, showing that the
advantages of the SAF structure extend beyond a pair of magnetic layers
separated by a single spacer layer.

The SAF multilayers that we studied were deposited by magnetron sputtering in
a chamber with a base pressure of $1.0\times 10^{-9}$ mbar under a working
pressure of $4.5\times 10^{-3}$ mBar of Ar. Typical deposition rates,
calibrated by x-ray reflectometry on test films, were 0.5 Å/s. Multilayers
were grown simultaneously on x-ray transparent Si3N4 membranes for
transmission microscopy and solid thermally oxidised Si substrates for
magnetometry measurements.

Figure 1:  (a) SAF multiayer stack structure, with layer thicknesses given in
Å; (b) SEM image of device.

The SAF multilayer stack that we studied is shown in Fig. 1(a). It features
alternating layers of Co68B32 and Co40Fe40B20, which form oppositely
magnetised sublattices in the SAF ground state, owing to the indirect
antiferromagnetic exchange through the Ru/Pt spacers, whose thicknesses were
chosen to ensure this form of coupling[14]. There are five pairs of such
layers in the completed stack. The ferromagnetic (FM) layer thicknesses were
selected so that they have equal and opposite magnetic moments. PMA and DMI
are induced in the layers at their interfaces, predominantly by the heavy
metal Pt.

Multilayers on the membranes were patterned into 2 $\upmu$m wide wires by
electron beam lithography and lift off, with current contacts at either end
made from Cu, which is a light enough element to remain transparent to x-rays
at the Co and Fe $L_{3}$ absorption edges. The design is similar to that used
for skyrmion injection in Ref. 15, with a wide drain contact at the right hand
end and a finger-shaped source contact on the left.

The multilayer on the solid substrate was characterized magnetically using
superconducting quantum interference device vibrating sample magnetometry
(SQUID-VSM). A magnetic hysteresis loop, acquired at room temperature and
normalized to the saturation magnetization, is shown in fig. 2. For applied
fields smaller than about 45 mT the magnetisation is constant and very small,
less than 10 % of the saturated value, showing that stack does indeed have a
SAF ground state, and that the relative FM layer thicknesses are indeed close
to the balance point.

Figure 2:  (a) SQUID-VSM hysteresis loop; (b),(c) STXM images showing opposite
contrast at the Co and Fe $L_{3}$ edges, respectively.

The samples on membranes were imaged by means of STXM at the PolLux beamline
at the Swiss Light Source. Images of the region of the microwire close to the
source contact are shown Fig. 2(b) and (c), acquired using x-ray photons with
energies at the Co and Fe $L_{3}$ absorption edges, respectively. Since
magnetic contrast in this technique arises owing to the x-ray magnetic
circular dichroism effect (XMCD), we can separately probe the magnetism of
these two elements by this means. The x-ray beam passed through the multilayer
at normal incidence, and so magnetic contrast arises that is proportional to
the out-of-plane component of the magnetization, $M_{z}$.

In each image a domain that spans the width of the wire can be observed under
the tip of the source contact, appearing as a band of dark contrast in Fig.
2(b) and a band of light contrast in Fig. 2(c). In this case we are imaging
the as-patterned state, prior to the application of any fields or current
pulses. Since only one sublattice of FM layers, made from CoFeB, contains Fe,
only those layers contribute to the contrast in Fig. 2(c). Meanwhile, the Co
content of the CoB layers is higher than that of the CoFeB layers, so that
although the contrast in Fig. 2(b) arises from both sublattices, it will be
predominantly from the CoB sublattice. The fact that the contrast is of
opposite sign in this pair of images confirm the SAF state, with CoB and CoFeB
sublattices being oppositely magnetized.

We have studied the response of such DWs to nominally 5 ns duration pulses of
electrical current injected into the magnetic wire. The pulses were generated
with a Keysight M8195A arbitrary waveform generator (64 GSa/s sampling rate)
combined with a SHF-826H amplifier. An oscilloscope (50 $\Omega$-terminated)
trace of such a pulse after it has passed through the wire is shown in Fig.
3(a). The pulse retains a high degree of squareness with modest ringing after
the pulse, which peaks at a current density $J\sim 3.0\times 10^{11}$ A/m2.

Fig. 3(b) and (c) show STXM images acquired before and after the application
of five such pulses, separated by a few seconds to allow for a complete
cooling of the magnetic wire. The positions of the DWs are marked with dashed
red lines. The left-hand DW is unaffected by the current pulses, as expected
since the current is shunted by the overlying Cu finger electrode. On the
other hand, the right hand wall moves away from the finger electrode, in the
direction of the conventional current flow, indicating spin-orbit torques as
the most likely mechanism to drive the motion. The movement is a distance of a
few hundred nm, and is not completely uniform across the wire, with the DW
moving further along the edge of the wire in the lower part of the image than
the edge in the upper part in this case.

Figure 3:  (a)pulse from oscilloscope. (b), (c) STXM images showing before and
after images of DW motion, (d),(e) difference images showing DW motion and
fitting.

This sort of DW tilting under spin-orbit torques has been previously observed
in Co/Ni/Co multilayers with DMI [16], subsequently shown to be an intrinsic
effect of the dynamics of DWs under DMI [17]. The model of Ref. 17 predicts
that the tilt angle should be $\propto M_{\mathrm{s}}$ and so ought to be zero
for a SAF. Whilst it is quite common for us to see some tilt in our images
like the example in Fig. 3(c), the tilt angle is quite variable and can be of
either sign. Based on these considerations, we think it is likely that the
tilts we observe are extrinsic and arise stochastically from pinning of the
DWs at defects, particularly on lithographic imperfections at the edges of our
microwires.

In order to study this DW motion in a more systematic way, we acquired a
series of images using positive helicity photons at the Co $L_{3}$ edge before
and after series of current pulses of different $J$ but with the same
duration, 5 ns. Subtracting pairs of these images yields difference images:
examples are shown in Fig. 3(d) and (e). These difference images show the area
swept out by the DW during motion driven by sequences of 10 positive polarity
(d), and then 10 negative polarity (e) pulses of $6.35\times 10^{11}$ A/m2.
Bright contrast in the difference corresponds to rightward motion, and dark
contrast to leftward motion.

To determine the distance traveled by the DW, the contrast values in a column
of pixels across the wire were binned together. The red lines in Fig. 3(c) and
(d) show how this binned contrast varies along the length of the wire, with
clear excursions away from the background level in the region where the DW
motion has occurred. The distance traveled by the DW is then taken to be the
full-width at half maximum value of this excursion, giving the average value
for distance traveled in the case of non-uniform wall motion. We determined
the velocity as this distance divided by the total nominal duration of the
pulses in the pulse sequence, and so use this term to refer to the average
velocity of the domain wall during its motion.

The velocities derived from this analysis are plotted as a function of current
density in Fig. 4. There is a region of low $J$ where there is no observable
DW motion. Once $|J|$ exceeds a critical value of $J_{\mathrm{c}}\sim 3\times
10^{11}$ A/m2 then DW motion in the direction of conventional current flow
takes place. We only measured up to a values of $J=6\times 10^{11}$ A/m2 owing
to concerns about the thermal stability of our nanostructure on the poorly
heat-sunk membrane substrate. Nevertheless, the DW velocity rises as the
current becomes stronger, and reaches a value just exceeding 40 m/s at that
maximum current density.

Figure 4:  DW velocity vs current density. Red data points are derived from
prior work on comparable CoB-based samples: data point [1] is derived from
data reported in Ref. 18, data points [2] and [4] from Ref. 19, and data point
[3] from Ref. 20.

To compare the results we have obtained here on DWs in SAF multilayers we have
performed a meta-analysis on prior work on STXM imaging of current-driven DW
motion in conventional multilayers based around a Pt/CoB/Ir repeat unit that
have a FM-aligned ground state [18, 20, 19]. Data points obtained from those
previous works are also plotted in Fig. 4. We can see that comparable
velocities were reached in those experiments, but substantially higher current
densities were required to do so. Our results mirror a reduction in
$J_{\mathrm{c}}$ seen previously in in-plane magnetized SAFs when compared to
simple Permalloy wires [21].

The depinning current observed here is also about half of that reported by
Yang et al. in Ref. 11 for a two-layer SAF. In that case, only one of the two
layers had an interface with a heavy metal–Pt–to supply both DMI and spin-
orbit torque. An important difference is that we have more magnetic layers in
our SAF and every one of them is in contact with Pt to supply these two
critical effects.

To summarise, we have studied DW motion in a SAF multilayer using STXM. We
find motion at comparable velocities to conventional perpendicularly
magnetised multilayers, but at reduced current densities that imply power
dissipation reduced by up to 50 %. We also find a critical current density
$J_{\mathrm{c}}$ that is substantially less than that in a two-layer SAF,
indicating the advantages of a stack design where every magnetic layer is
subject to a spin-orbit torque. Our results offer promise for low power
consumption DW racetrack technologies as well as suggesting that the motion of
skyrmions in such SAFs [22, 23, 24] is also likely to only require modest
current densities.

###### Acknowledgements.

This work was supported by the UK EPSRC (grant numbers EP/T006803/1 and
EP/R00661X/1). We acknowledge support from the Sir Henry Royce Institute for
access to the the Royce Deposition System for sample growth. C.E.A.B.
acknowledges support from the National Physical Laboratory. Part of this work
was performed at the PolLux (X07DA) beamline of the Swiss Light Source, Paul
Scherrer Institut, Villigen PSI, Switzerland. The research leading to these
results received funding from the European Community’s Seventh Framework
Programme (No. FP7/2007-2013) under Grant Agreement No. 290605 (PSI-
FELLOW/COFUND), the Swiss National Science Foundation under Grant Agreement
No. 172517.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to declare.

## Data Availability

Data pertaining to this work is available from the Research Data Leeds
repository at https://doi.org/TBD.

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