# A Novel Framework for Modeling Weakly Lensing Shear Using Kinematics and
Imaging at Moderate Redshift

Brian DiGiorgio Department of Astronomy and Astrophysics, University of
California, 1156 High Street, Santa Cruz, CA 95064, USA  Kevin Bundy
Department of Astronomy and Astrophysics, University of California, 1156 High
Street, Santa Cruz, CA 95064, USA  University of California Observatories,
University of California, 1156 High Street, Santa Cruz, CA 95064, USA  Kyle B.
Westfall  University of California Observatories, University of California,
1156 High Street, Santa Cruz, CA 95064, USA  Alexie Leauthaud  Department of
Astronomy and Astrophysics, University of California, 1156 High Street, Santa
Cruz, CA 95064, USA  David Stark  Department of Physics and Astronomy,
Haverford College, 370 Lancaster Avenue, Haverford, PA 19041, USA

###### Abstract

Kinematic weak lensing describes the distortion of a galaxy’s projected
velocity field due to lensing shear, an effect recently reported for the first
time by Gurri et al. based on a sample of 18 galaxies at $z\sim 0.1$. In this
paper, we develop a new formalism that combines the shape information from
imaging surveys with the kinematic information from resolved spectroscopy to
better constrain the lensing distortion of source galaxies and to potentially
address systematic errors that affect conventional weak-lensing analyses.
Using a Bayesian forward model applied to mock galaxy observations, we model
distortions in the source galaxy’s velocity field simultaneously with the
apparent shear-induced offset between the kinematic and photometric major
axes. We show that this combination dramatically reduces the statistical
uncertainty on the inferred shear, yielding statistical error gains of a
factor of 2–6 compared to kinematics alone. While we have not accounted for
errors from intrinsic kinematic irregularities, our approach opens kinematic
lensing studies to higher redshifts where resolved spectroscopy is more
challenging. For example, we show that ground-based integral-field
spectroscopy of background galaxies at $z\sim 0.7$ can deliver gravitational
shear measurements with S/N $\sim 1$ per source galaxy at 1 arcminute
separations from a galaxy cluster at $z\sim 0.3$. This suggests that even
modest samples observed with existing instruments could deliver improved
galaxy cluster mass measurements and well-sampled probes of their halo mass
profiles to large radii.

##  1 Introduction

Weak gravitational lensing is a powerful and increasingly utilized tool for
measuring how mass is distributed throughout the Universe (e.g. Mandelbaum,
2018). Applications include use of cosmic shear to probe the mass density and
the growth of structure in the Universe (e.g. Troxel et al. 2018; Gatti et al.
2020; Hikage et al. 2019), galaxy-galaxy lensing to characterize halo masses
of galaxies in different stellar-mass regimes and perform $3\times 2$
correlation studies (e.g. Leauthaud et al. 2012; Krause et al. 2017), and
cluster lensing profiles to measure cluster-scale halo mass profiles and halo
shapes (e.g. Umetsu et al., 2018; Mandelbaum et al., 2006). These science
goals have motivated large-scale weak-lensing surveys with new instruments,
telescopes, and even satellite missions. Major projects include the Hyper
Suprime Cam (HSC, Aihara et al., 2018), the Kilo Degree Survey (KiDS, Kuijken
et al., 2015), the Legacy Survey of Space and Time (LSST, Ivezić et al.,
2019), Euclid (Laureijs et al., 2011), the Nancy Grace Roman Space Telescope
(Spergel et al., 2015), and the Dark Energy Survey (DES, Krause et al., 2017;
Troxel et al., 2018; Gatti et al., 2020).

Weak-lensing surveys seek to measure distortions in a galaxy’s shape caused by
lensing shear, a geometric effect caused by the gravitational lensing of a
background source by mass in the foreground. To first order, this distortion
results in background galaxies becoming elongated in the direction tangential
to the surface-density gradient of the foreground mass distribution.

A number of factors complicate lensing measurements in conventional surveys
that use imaging to measure galaxy shapes. Weak lensing distortions only
induce a ${<}1\%$ change in observed ellipticity. Measurements at this level
of precision are typically achieved by statistically stacking results from
multiple sources, but our uncertain knowledge of the intrinsic or “pre-lensed”
galaxy ellipticity distribution (“shape noise”) typically necessitates
thousands of stacked sources to detect a lensing signal. Potential biases
become a greater concern with added complications, such as intrinsic
alignments of source galaxies with near-foreground dark matter structure
(e.g., filaments) and systematic errors in shape measurements resulting from
instrumentation (e.g., Troxel & Ishak, 2015; Mandelbaum, 2018). Very large
samples are therefore required to drive down statistical errors and test for
systematics that can otherwise overwhelm the shear signal.

For instance, the CLASH survey (Umetsu et al., 2014) used ${\sim}10^{4}$
background galaxies around each of 20 foreground Abell clusters to recover
mass density profiles precise enough to constrain dark matter halo mass
profiles. Meanwhile, cosmic shear surveys like the DES (Troxel et al., 2018)
use ${\sim}10^{7}$ galaxies to sufficiently characterize their signal, but
even these large samples remain susceptible to systematic biases. All of these
efforts require deep imaging over wide areas, typically carried out on 2
meter- to 8 meter-class telescopes.g

The burgeoning subfield of kinematic weak lensing (KWL), also referred to as
“precision weak lensing” in the literature (Gurri et al., 2020, 2021),
provides an additional means of inferring lensing shear by measuring
distortions in the projected velocity field of source galaxies. Although KWL
requires more expensive spectroscopic observations, many fewer galaxies are
needed to detect a signal. This is because the velocity field encodes a trace
of the galaxy’s original, pre-lensed coordinate space. The induced lensing
distortions can be fit directly, eliminating or at least dramatically reducing
the shape noise. If future weak-lensing analyses could make use of KWL, they
could enable higher spatial resolution maps of foreground mass, more sensitive
mass measurements, and independent checks on results from conventional imaging
surveys (Huff et al., 2019).

The literature investigating applications of KWL has fallen into two regimes:
high-spatial-resolution kinematic measurements on small samples, and lower-
precision kinematic measurements derived from large samples. The idea of KWL
was first put forward by Blain (2002), who determined that weak lensing shear
would change the symmetry of a galaxy’s rotational velocity as measured in an
azimuthal ring of constant radius. The idea was further developed by Morales
(2006), who proposed measuring the angle between the kinematic major and minor
axes, which are no longer perpendicular in a lensed galaxy. A similar idea was
developed by de Burgh-Day et al. (2015), who suggested searching for lensing-
induced asymmetries in the reflection symmetries of galaxy kinematic data. All
these authors emphasize the potential of making a shear measurement with a
single galaxy and obtaining a measurement that would be independent of shape
noise. However, none of these studies reported a detection due to the small
size of lensing effects at nearby redshifts where the high-spatial-resolution
spectroscopic measurements required for precise measurements are most readily
available.

Huff et al. (2019) instead proposes implementing KWL on a survey scale,
considering large samples where individual measurements with less kinematic
information are statistically stacked. Building on the work of Huff et al.
(2013), they propose targeted measurements of lensing-induced differences in
the projected velocity along the major and minor axes. Wittman & Self (2021)
performs a Fisher Matrix analysis of this technique in a hypothetical DES-
scale survey to derive theoretical limits on the covariances of the derived
lensing parameters.

In the first reported detection of kinematic weak lensing, Gurri et al. (2020)
stake out a middle ground in sample size versus per-galaxy information
content. They collect and analyze 2D velocity fields of ${\sim}20$ galaxies at
$z<0.15$, and report a positive mean shear amplitude detected at $2.5\sigma$.
They forward-model each source galaxy with a rotating thin-disk model that
allows shear to vary and apply their technique to selected source galaxies
likely to be sheared by foreground halos. The detection reported by Gurri et
al. (2020) is an exciting development in the young field of KWL, but also
highlights upcoming observational challenges. For instance, they find
discrepancies between their theoretical and observed shear magnitudes, which
they say likely arise from a combination of intrinsic kinematic irregularities
in their sample galaxies (“dynamical shape noise”) and the scatter in the
stellar-halo mass relation (see Gurri et al., 2021). It will be difficult to
obtain the kinds of highly-sampled and high-S/N velocity fields in Gurri et
al. (2020) for source galaxies at $z\lesssim 1$, but it is at these redshifts
where probes of structure formation are most needed and the lensing kernel
more favorable (Weinberg et al., 2013).

With this challenge in mind, we build on the techniques of Gurri et al. (2020)
by introducing additional constraints from galaxy shape measurements into the
KWL formalism using a Bayesian forward model. We show that by including
measurements of the major-axis position angle of an ellipse fit to the surface
brightness profile and comparing it to the position angle derived for the
sheared velocity field, we can improve the per-galaxy shear S/N by several
times in many cases. This motivates us to explore the potential use case of
measuring the mass and shape of galaxy cluster halos at $z\sim 0.3$
(Bartelmann & Maturi, 2017) with IFU observations of 50–100 galaxies at $z\sim
0.7$. Immediately valuable for studies of individual clusters, a future survey
program could aid mass calibrations required for cluster cosmology (e.g.
Bocquet et al., 2019) and opens new possibilities for wide-scale KWL surveys
at $z\lesssim 1$ with a dedicated survey.

The paper is structured as follows: Section 2 gives background on relevant
weak-lensing theory and introduces our formalism for how lensing affects the
shape of a galaxy. Section 3 studies the impact of lensing shear on the shape
of a rotating galaxy and compares the merits of different techniques for
measuring kinematic shear observables. Section 4 develops our modeling
framework for including offsets between the kinematic axes and axes derived
from imaging for a given galaxy and characterizes the improvement of the
precision of lensing measurements and the important increase of S/N that
results. Section 5 gives a summary and looks forward to future applications of
the technique.

Throughout this paper, we assume a Planck 2018 cosmology (Planck Collaboration
et al., 2018) as implemented in the Colossus cosmology package from Diemer
(2018), with $H_{0}=67.36$ km s-1 Mpc-1 and $\Omega_{m}=0.3111$ at $z=0$.

##  2 Weak Lensing Effects on Imaging

To understand the benefits image position angle measurements can have for KWL,
we first review key aspects of how lensing geometry changes the shape of
galaxies. Imaging-based lensing surveys typically model background galaxies as
ellipses with a measured on-sky position angle and ellipticity that are used
to derive lensing amplitude. This serves both as context for the current state
of the field and as a basis for the techniques developed in Section 4. For a
more detailed treatment of the subject, see Bartelmann & Maturi (2017).

###  2.1 General Lensing Theory

Gravitational lensing is a well-established result of general relativity and
its effects have been well-characterized (Miralda-Escude, 1991). Because
photons travel along geodesics, when they pass through regions of spacetime
that have been distorted by a mass distribution, the geodesic is deflected,
curving their paths. So the observed image of a background galaxy will be
distorted by any foreground mass along the line of sight.

If we assume that the characteristic scale of the foreground mass distribution
is small compared to the other relevant distance scales in the system — the
angular diameter distance from the observer to the foreground mass, $D_{L}$,
the distance from the observer to the background source, $D_{S}$, and the
distance from the foreground mass to the background source, $D_{LS}$ — we may
adequately describe the system using the thin lens approximation. This
stipulates that the deflection of the photons in the system happens in an
infinitely thin plane located at the lens position and that the photon travels
in a straight line everywhere else. The magnitude of the angular deflection
relates to the projected surface density $\Sigma$ of the 3D foreground mass
distribution $\rho$, where

$\Sigma(\bm{\xi})=\int\rho(\bm{\xi,z})d\bm{z},$ (1)

for $\bm{\xi}=\sqrt{\bm{x}^{2}+\bm{y}^{2}}$ with $\bm{x}$ and $\bm{y}$ being
the physical coordinates of the mass distribution on-sky and $\bm{z}$ being
the line of sight vector from the observer. The angular deflection of a photon
is given by

$\bm{\alpha}(\bm{\xi})=\frac{4G}{c^{2}}\int\Sigma(\bm{\xi}^{\prime})\frac{\bm{\xi-\xi^{\prime}}}{|\bm{\xi-\xi^{\prime}}|^{2}}\,d^{2}\bm{\xi}^{\prime}.$
(2)

The deflection angle can also be given in terms of the lensing potential
$\psi$ as

$\bm{\alpha}(\theta)=\frac{D_{S}}{D_{LS}}\nabla_{\theta}\psi,$ (3)

where $\theta$ is the angular separation of the source as seen from the
perspective of the observer: $\theta=|\bm{\xi}|/D_{L}$. For a point mass lens
in the weak lensing regime, $\psi$ can be written as

$\psi=\frac{4GM}{c^{2}}\frac{D_{LS}}{D_{L}D_{S}}\ln|\bm{\xi}|.$ (4)

The magnitude of the lensing effect is often expressed in terms of the
critical surface mass density, which defines the characteristic angular scale
of a lensing system:

$\Sigma_{\textrm{crit}}=\frac{c}{4\pi G}\frac{D_{S}}{D_{L}D_{LS}}.$ (5)

The distortion manifests itself in two ways. The first is the convergence
$\kappa$, which has magnitude $\kappa=\Sigma/\Sigma_{\textrm{crit}}$ and
magnifies the background source. However, for the purposes of this paper, we
will be more interested in the lensing shear.

The shear, as its name implies, distorts the coordinate system of the source,
changing its observed shape. This distortion can be broken into two separate
components defined by their relation to the derivatives of the lensing
potential:

$\gamma_{+}=\frac{1}{2}\left(\frac{\partial^{2}\psi}{\partial\theta_{x}^{2}}-\frac{\partial^{2}\psi}{\partial\theta_{y}^{2}}\right),\
\ \
\gamma_{\times}=\frac{\partial^{2}\psi}{\partial\theta_{x}\partial\theta_{y}}.$
(6)

where $\theta_{x}$ and $\theta_{y}$ are perpendicular angular coordinates in
the arbitrarily defined coordinate system projected onto the lens plane.
Following the formalism of Huff et al. (2013), the shear aligned with the
coordinate axes is called $\gamma_{+}$, while the cross term $\gamma_{\times}$
is aligned with the diagonals of the coordinate system. For a simplified case
of an axisymmetric lens, the shear magnitude is given by

$|\gamma|=\frac{\Delta\Sigma}{\Sigma_{\textrm{crit}}},$ (7)

where $\Delta\Sigma$ is the differential projected surface mass density, the
difference between the value of $\Sigma$ at a given radius $r_{0}$ and the
mean $\Sigma$ within that radius:

$\Delta\Sigma=\bar{\Sigma}(r<r_{0})-\Sigma(r_{0}).$ (8)

In the simplified axisymmetric case, the two shear components can also be
expressed as

$\gamma_{+}=|\gamma|\cos 2\theta\ \ \ \gamma_{\times}=|\gamma|\sin 2\theta,$
(9)

with $\theta=\tan^{-1}(\theta_{y}/\theta_{x})$. Positive and negative values
for the shear result in distortions in opposite directions. We assume
$|\gamma|<0.1$ in this paper to remain in the weak-lensing regime where the
magnitude of the distortions are small and the small angle approximation is
still valid.

###  2.2 Image Distortions

Figure 1: Representation of how shear from gravitational lensing affects the
positions of major and minor axes derived from kinematics and
imaging/photometry. The original orientation of the galaxy is shown as a
dashed elliptical isophote, with the dashed lines marking the positions of the
unlensed major and minor axes (photometric and kinematic are the same). A lens
to the upper left of the galaxy applies a shear, distorting the galaxy’s
intrinsic coordinate system and velocity field contours according to the
transformation matrix $A$ (Equation 11). The solid ellipse represents how a
new elliptical isophote would be fit to the sheared galaxy, along with its new
imaged major and minor axes. The colored contours represent the observed
velocity field, with red and blue indicating receding and approaching sides of
the galaxy, respectively, and darker colors indicating higher velocity
magnitude. The point along the lensed isophote that is measured as the lensed
imaged major axis (filled black circle) was not on the major axis of the
unlensed galaxy (unfilled black circle), leading to a discrepancy between the
lensed axis observed in imaging and the kinematic axis (filled white circle).
A similar effect happens with the minor axis. The galaxy shown has an axis
ratio $q=0.8$ and the shear applied is $\gamma_{\times}=0.1$.

In the weak-lensing regime, the recognizable features of strong lensing, like
arcs and multiple images, give way to tiny distortions in background sources
that are usually only detectable through statistical methods applied to large
samples. Weak lensing effects are well-parameterized by a single
transformation matrix that can be applied to the coordinate system of the
background source:

$A=\begin{pmatrix}1-\kappa+\gamma_{+}&\gamma_{\times}\\\
\gamma_{\times}&1-\kappa-\gamma_{+}\end{pmatrix}$ (10)

If we do not care about the size and luminosity of the source, we may
disregard the convergence $\kappa$ since it is small in the weak-lensing
regime, so we may simplify $A$ to be:

$A\approx\begin{pmatrix}1+\gamma_{+}&\gamma_{\times}\\\
\gamma_{\times}&1-\gamma_{+}\end{pmatrix}.$ (11)

The effect of $\gamma_{+}$ is to symmetrically elongate the source along the
tangential axis111Note that this is distinct from $\kappa$, which elongates
the galaxy in all directions equally, causing a perceived magnification.,
meaning the on-sky orientation of the source is not substantially affected.
However, $\gamma_{\times}$ shears the source, causing an effective rotation.

We can apply this transformation to a toy model of a galaxy, a typical
background source in a weak-lensing survey. We may model the galaxy as a
circle inclined at some angle relative to the observer, so it is seen simply
as a series of concentric elliptical isophotes, each satisfying

$1=q^{2}x^{2}+y^{2},$ (12)

where $q$ is the axis ratio of the projected ellipse. Applying the shear
transformation matrix to this object, we find that, post-lensing, the
coordinate system has become $\bm{x}^{\prime}=A^{-1}\bm{x}$ for
$\bm{x}=(x,y)^{T}$, so

$1\approx q^{2}x^{\prime 2}(1-2\gamma_{+})+y^{\prime
2}(1+2\gamma_{\times})-2x^{\prime}y^{\prime}\gamma_{\times}(1+q^{2}),$ (13)

where we have dropped higher order terms in $\gamma_{+}$ and $\gamma_{\times}$
since we expect the magnitude of these terms to be small in the weak lensing
regime.

We then treat the deformation of the elliptical isophote as a rotation rather
than a shear. The general equation of an ellipse rotated by angle $\alpha$
relative to the origin is

$1=\left(\frac{\cos^{2}\alpha}{a^{2}}+\frac{\sin^{2}\alpha}{b^{2}}\right)x^{2}+\left(\frac{\sin^{2}\alpha}{a^{2}}+\frac{\cos^{2}\alpha}{b^{2}}\right)y^{2}\\\
-2\cos\alpha\sin\alpha\left(\frac{1}{b^{2}}-\frac{1}{a^{2}}\right),$ (14)

where $a$ and $b$ are the lengths of the major and minor axes, respectively.
We may greatly simplify this equation if we restrict ourselves to the weak-
lensing regime. By assuming $\alpha$ to be small, we can apply the small angle
approximation and ignore higher order terms in $\alpha$. If we then multiply
through by $b^{2}$ to put everything in terms of the axis ratio $q=b/a$ and
set $b=1$ in arbitrary units, we have

$1\approx q^{2}x^{2}+y^{2}-2\alpha xy\left(1-q^{2}\right).$ (15)

Equations 13 and 15 are identical except for the cross term. If we assume that
the shear can indeed be thought of as a rotation, we can set these two
equations equal to each other to find

$\alpha(1-q^{2})\approx\gamma_{\times}(1+q^{2})\ \ \ \Rightarrow\ \ \
\alpha\approx\gamma_{\times}\frac{1+q^{2}}{1-q^{2}}.$ (16)

So the effect of the shear on an isophote can be well-approximated in the
weak-lensing regime as a rotation, as seen in Figure 1. The rotation angle is
determined by $\gamma_{\times}$, which depends on the magnitude of the overall
shear, the relative positions of the foreground lens and the background
source, and the axis ratio of the background source, which are in turn
determined by Equation 9 and the inclination of the galaxy. The magnitude of
$\alpha$ is maximized when the background source has a low inclination, has a
position angle that is misaligned with the radial vector by $45^{\circ}$, and
when the foreground lens is more massive.

The axis ratio $q$ changes as a result of lensing as well, as seen by the
effect of the transformation matrix $A$ on a point at $(a,b)$:

$\begin{pmatrix}1+\gamma_{+}&\gamma_{\times}\\\
\gamma_{\times}&1-\gamma_{+}\end{pmatrix}\begin{pmatrix}a\\\ b\end{pmatrix}\\\
=\begin{pmatrix}(1+\gamma_{\times}+\gamma_{+})\,a\\\
(1+\gamma_{\times}-\gamma_{+})\,b\end{pmatrix}=\begin{pmatrix}a^{\prime}\\\
b^{\prime}\end{pmatrix},$ (17)

where we have defined $(a^{\prime},b^{\prime})$ as the major and minor axis
lengths of the lensed galaxy. Thus, the lensed axis ratio $q^{\prime}$ is

$q^{\prime}=\frac{b^{\prime}}{a^{\prime}}=\frac{(1+\gamma_{\times}-\gamma_{+})}{(1+\gamma_{\times}+\gamma_{+})}\frac{b}{a}\\\
\approx\left(1-\frac{2\gamma_{+}}{1+\gamma_{\times}}\right)q,$ (18)

where we have again exploited a Taylor series expansion to make a first order
approximation. What observers see is the post-lensing axis ratio $q^{\prime}$
rather than $q$, so we again approximate to write this as

$q\approx\frac{q^{\prime}}{1-2\gamma_{+}}.$ (19)

We see that for small $\gamma_{+}$, $q\approx q^{\prime}$, and because we will
be concerned mostly with cases where this is true, we will ignore the
distinction between pre- and post-lensing axis ratios for the remainder of the
paper.

However, neither the angular distortion nor the axis ratio distortion can be
directly measured in images of any single galaxy because nothing is known
about its original shape. Before the galaxy is lensed, it has some intrinsic
position angle and ellipticity that provide the baseline for any lensing
distortions. With conventional imaging surveys, the only way we can perceive
changes is by looking at the statistical properties of the sample in
aggregate. A large sample of galaxies under the same shear will show a net
alignment perpendicular to the induced shear that deviates from a random
distribution of position angles. Measuring this net alignment is a primary
goal of imaging-based weak lensing surveys.

##  3 Weak Lensing Effects on Kinematics

While it is impossible to tell the difference between an elliptical isophote
that has been rotated and elongated due to lensing shear and one that was
never sheared, the same is not true for projected kinematics of a rotating
disk. By taking spectroscopic measurements across the face of a rotating
galaxy, the relative velocity of different parts of the galaxy can be
determined from the Doppler shift of spectral features compared to the
systemic velocity of the galaxy.222In this paper, we assume an infinitely
thin, rotating disk. This assumption may cause problems with highly inclined
disks. The kinematic measurements that make up the velocity field are
associated with specific coordinates in the galaxy’s intrinsic plane, a
relationship that is not broken by gravitational lensing. This probe of the
galaxy’s intrinsic geometry enables much more precise per-galaxy lensing
measurements.

In what follows, our analysis is motivated by reducing the final statistical
errors on a galaxy’s shear measurement, as limited by the quality of the data.
We do not treat here the per-galaxy error that stems from intrinsic
irregularities in the physical structure of the galaxy. Features that deviate
from the model used in the fit will bias the results regardless of the quality
of the data. However, this dynamical shape noise (Gurri et al., 2020) is
expected to be randomly distributed across a sample and should beat down as
the sample size increases. We will return to the role of statistical and
systematic errors in Section 4.2.1.

###  3.1 Kinematic Axis Distortion

Unlike the photometric measurements made by fitting isophotes, the
transformation of a galaxy’s velocity field cannot be modeled simply as a
rotation of the projected field. In order to illustrate the effect of shear on
the velocity field, we examine the behavior of two reference locations: a
point on the major axis and one on the minor axis.

A point on the major axis of the unlensed galaxy (unfilled white circle on
Figure 1) can be described by the coordinates $(x,0)$ for a rectilinear
coordinate system aligned with the major and minor axes. We define the on-sky
angle of the major axis in a particular reference frame to be the kinematic
position angle (PA). Applying the transformation matrix $A$ to this coordinate
location, we find that this point gets moved to ${((1+\gamma_{+})\,x,\
\gamma_{\times}x)}$ (filled white circle), so the angular displacement of the
lensed major axis from the original major axis is

$\theta_{maj}=\tan^{-1}\left(\frac{\gamma_{\times}}{1+\gamma_{+}}\right)\approx\frac{\gamma_{\times}}{1+\gamma_{+}},$
(20)

where we have performed a Taylor series expansion of the inverse tangent and
taken just the first order since we expect the shear to be small in the weak-
lensing regime.

We can do a similar transformation to a point on the unlensed minor axis
$(0,y)$ to find

$\theta_{min}=\tan^{-1}\left(\frac{1-\gamma_{+}}{\gamma_{\times}}\right)\approx\frac{\pi}{2}-\frac{\gamma_{\times}}{1-\gamma_{+}},$
(21)

where we have again performed a first-order Taylor series expansion.

We can again exploit Taylor series to expand Equations 20 and 21 with respect
to $\gamma_{+}$:

$\theta_{maj}\approx\gamma_{\times}(1-\gamma_{+});\ \ \ \ \
\theta_{min}\approx\frac{\pi}{2}-\gamma_{\times}(1+\gamma_{+}).$ (22)

The dependence of the angular differences on $\gamma_{+}$ is on the order of
$|\gamma|^{2}$ (assuming that $\gamma_{+}\approx\gamma_{\times}$, which is
true for most galaxy-lens orientations). This means that we can neglect the
$\gamma_{+}$ term333The same is true for the $\kappa$ term that we have
already neglected. If we had kept it, it would have shown up in the
denominator here as well and could be neglected with the same logic. and just
write

$\theta_{maj}\approx\gamma_{\times};\ \ \ \ \
\theta_{min}\approx\frac{\pi}{2}-\gamma_{\times}.$ (23)

These simple relations allow us to estimate the shear by measuring how much
the axes differ from being orthogonal, as was proposed by Morales (2006). In
an ideal velocity field, the angle between the major and minor kinematic axes
should be

$\theta_{\perp}\approx\frac{\pi}{2}\pm 2\gamma_{\times}.$ (24)

If we could determine the position angles of the velocity maximum, minimum,
and zeros and see how they differ from perpendicularity using a technique like
the Radon transform (Stark et al., 2018) or kinemetry (Krajnović et al.,
2006), we could use the above result to gain an estimate of the induced shear.

###  3.2 Sheared Velocity Field Fitting

We can produce a more complete picture of the shear of a velocity field if we
model the distortion of the entire galaxy rather than just its major and minor
axes. We construct a Bayesian forward model that allows us to utilize all of
the velocity measurements to obtain a best-fit sheared velocity field.

For the purpose of comparing different KWL techniques in our regime of
interest, we create a simulated observation of a galaxy. We consider a source
galaxy at $z=0.7$ behind a $7\times 10^{14}\ M_{\odot}$ foreground cluster
(approximately equal in mass to the Coma cluster, Gavazzi et al. 2009) at
$z\sim 0.3$. We assume the mass profile of the cluster follows a Navarro-
Frenk-White profile (NFW, Navarro et al., 1997). Two-dimensional emission-line
kinematics are feasible at this redshift with 8m-class ground-based optical
telescopes (e.g., Contini et al. 2016). We assume an impact parameter of 0.3
Mpc in the plane of the lens corresponding to a 1 arcminute separation from
the cluster center, which is the field of view in the MUSE wide-field
configuration (Bacon et al., 2010). This system yields a shear magnitude of
$|\gamma|=0.0589$. We assume that the kinematic major axis of the background
galaxy is at a 45∘ angle to the radial vector to the lens, meaning
$\gamma_{\times}=0.0589$ and $\gamma_{+}=0$. With this shear magnitude in
mind, we explore what types of measurements are possible.

Figure 2:  Left: A mock intrinsic velocity field to test our KWL shear
extraction. This galaxy was generated using the model in Equation 25 with
parameters $v_{max}=220$ km/s , $i=45^{\circ}$, $\phi=2$ rad, $h=2$ kpc, and
$v_{sys}=0$ km/s. A shear of $\gamma_{\times}=0.0589$ has been applied to it.
The overlaid contours show the error on the velocity measurements in km/s,
which are normalized to a central value of 5 km/s. The inverse variance on the
velocity measurements is assumed to be proportional to surface brightness,
which is modeled as a Sérsic profile of the same galaxy with $n=1$ and
$R_{e}=2$ kpc. The velocity inverse variance has been normalized to 0.05
(km/s)-2, typical of a MaNGA galaxy. Right: The velocity field from the left
panel but smeared and sampled coarsely to emulate an observation at $z\sim
0.7$ with a FWHM of 0.7” on a MUSE-like instrument. See details in Section
3.2.

We assume that the inverse variances on the velocity measurements are
proportional to surface brightness, which we set as an $n=1$ Sérsic profile.
The scale, inclination, and position angle of the Sérsic profile are defined
by the same parameters that define the mock velocity field. normalized to a
peak value of 0.05 (km/s)-2, corresponding to a velocity error of about $4.5$
km/s at the center of the galaxy. This value was chosen because it was the
most common value for galaxies in MaNGA (Bundy et al., 2015) and should
capture some of the errors from approximating rotation as a thin disk as well.

For the velocity field, we first generate an idealized toy model of a rotating
galaxy using a simple model:

$v(r,\theta)=\frac{2}{\pi}v_{max}\arctan\left(\frac{r}{h}\right)\sin
i\,\cos(\theta-\phi)+v_{sys}.$ (25)

Here $r$ and $\theta$ are polar coordinates of the spectroscopic measurements,
$v_{max}$ is the asymptotic rotation speed, $h$ is the characteristic scale
radius, $i$ is the inclination, $\phi$ is the kinematic PA, and $v_{sys}$ is
the systemic velocity of the galaxy. The positions of the primary kinematic
axes, the part of the velocity field model that the shear measurement
primarily relies on, are not sensitive to the specific shape of the rotation
curve model. We compared results from this model with the more complex
empirically-derived Universal Rotation Curve from Persic et al. (1996) and
found negligible difference for the results of this paper, as did Wittman &
Self (2021) in their analysis. We then apply the transformation matrix
(Equation 11) to the data to shear the velocity field, an effect more easily
seen in Figure 1. For the remainder of this paper, we will use this model to
both simulate mock velocity fields and to fit our mock data, allowing the
model complete freedom to describe the mock galaxy.

To simulate a real ground-based observation, we first convolve the velocity
field with a Moffat point-spread function (PSF) with a full-width half maximum
(FWHM) of 0.7” and $\beta=2.9$ to the velocity field and inverse variance to
model atmospheric distortions, a very good night at a mountaintop observatory.
We weigh the PSF convolution using the observed surface brightness profile of
the galaxy. We then assume a spatial sampling of $0.2\times 0.2$ arcsec2, the
same as the wide field mode for MUSE, corresponding to ${\sim}1$ kpc spatial
resolution at redshift $z=0.7$. We generate a velocity field with a
characteristic scale radius $h=2$ kpc and apply a shear of
$\gamma_{\times}\approx 0.06$ to its spatial coordinates. We assume
measurements extending out to ${\sim}2R_{e}$ for an $n=1$ Sérsic profile also
with $R_{e}=2$ kpc444For all other mock galaxies generated in this paper, we
will always make the simplification that the characteristic scale radius of
the velocity field $h$ and the effective radius of its surface brightness
profile $R_{e}$ are equal., meaning we should have a grid of $8\times 8$
spatial samples across the field of the galaxy. We also apply this same PSF
convolution step when fitting this model to data in order to more accurately
recover the input parameters.

We also apply random Gaussian errors in the measured velocity according to the
assumed inverse variance profile, perturbing the results from the ideal
velocity field. The resulting measurements, shown in Figure 2, qualitatively
resemble actual data taken by MUSE (see Contini et al. 2016, Fig. 5). Because
we generate our mock data from ideal rotation curve models, we do not expect
per-galaxy systematic errors from dynamical shape noise to be present, leaving
only statistical errors. We do not apply this step when fitting the model to
the data, instead using the ideal unperturbed model.

To find the best velocity field parameters for each mock galaxy, we use the
Markov Chain Monte Carlo (MCMC) package emcee (Foreman-Mackey et al., 2013),
feeding in the velocity measurements across the face of a galaxy. We use the
same rotation curve model as was used to generate the velocity field, but we
add shear by simply applying the inverse of the transformation matrix given in
Equation 11 to the underlying coordinates of the measurements and letting
$\gamma_{\times}$ be a free parameter in the fit. Because our velocity field
model is sheared directly by the transformation matrix, it is not affected by
the approximations we made in Section 3.1. This, however, comes at the price
of analytic simplicity of the model.

Figure 3: Black: The resulting posteriors from applying the Bayesian velocity
field model from Equation 25 with a free shear parameter to the simulated
sheared galaxy described in Section 3.2. Most posteriors are centered near
their true values (shown as blue lines), but the uncertainty is inflated by
injected Gaussian errors, blurring from the PSF, and low spatial resolution.
This is especially apparent in inclination and shear. Since the magnitude of
the introduced errors greatly outweighs the signal from weak lensing, the
posterior for $\gamma_{\times}$ has very poor precision and the statistical
error dominates the measurement. Red: The performance of the model is greatly
improved when image PA is included as described in Section 4. With a 3 degree
error on the image PA, the posteriors are noticeably more constrained than the
kinematics-only model, especially the shear, which has a factor of 5 reduction
in spread, and the position angle $\phi$. The degeneracy between $\phi$ and
$\gamma_{\times}$ is also resolved.

We apply uniform priors to inclination and position angle to allow for random
variation in orientation. We use a Gaussian prior on asymptotic velocity with
mean 200 km/s and standard deviation 100 km/s and is truncated at 0,
parameters that are loosely based off of the MaNGA sample (Bundy et al.,
2015). We allow the rotation scale to vary uniformly up to 4 arcseconds. For
$\gamma_{\times}$, we apply a uniform prior bounded at $\pm 0.5$ to allow for
a reasonable amount of variation, but not so much as to allow the model to fit
any irregularities it sees in the data with unphysical amounts of shear. We
use a standard Gaussian likelihood function for comparing the model to the
mock data at each iteration. This model produces shear magnitudes and
statistical errors that are similar to those measured by Gurri et al. (2020)
when applied to the same data.

With these assumptions, we apply our Bayesian velocity-field model to the
simulated data and recover the posteriors shown in black in Figure 3. We can
see that the posteriors include the true values for most of the model
parameters, although only some appear near the median of the posterior
distributions. Two primary factors in the model precision are the effects of
the PSF and the added Gaussian noise. Inclination relies on the specific
shapes of isovelocity contours, but much of their variation with inclination
is masked by the PSF. In addition, the width of the shear posterior
$\sigma_{\gamma}$ is hampered by the relatively shallow velocity gradient,
limited spatial resolution, and a significant degeneracy with the kinematic
position angle. So while we can use this method to extract some of the
parameters of the velocity field, we cannot fit our key parameter of interest
with much precision. More information on the shear is needed if we want to
lower statistical errors enough to produce a successful fit for data of this
spatial resolution.

##  4 Combining Imaging and Kinematics

###  4.1 Kinematic and Photometric Position Angle Offset

To better constrain the lensing distortion of the velocity field, we
incorporate the image distortion we explored in Section 2.2. Comparing the
kinematic axis differences derived in Equations 22 and 23 to the photometric
angular difference from Equation 16, we can see (as in Figure 1) that there is
a difference between the angle measured by fitting an isophote and the angle
from the velocity field for the major and minor axes:

$\Delta\theta_{maj}=\gamma_{\times}\frac{1+q^{2}}{1-q^{2}}-\gamma_{\times}\\\
=\gamma_{\times}\left(\frac{1+q^{2}}{1-q^{2}}-1\right)=\frac{2\gamma_{\times}q^{2}}{1-q^{2}}$
(26)

$\Delta\theta_{min}=\left(\gamma_{\times}\frac{1+q^{2}}{1-q^{2}}+\frac{\pi}{2}\right)-\left(\frac{\pi}{2}-\gamma_{\times}\right)\\\
=\gamma_{\times}\left(\frac{1+q^{2}}{1-q^{2}}+1\right)=\frac{2\gamma_{\times}}{1-q^{2}}.$
(27)

The minor axis deviates more because the velocity field shearing effect goes
in the opposite direction of the apparent rotation of the image. A comparison
of the dependencies of $\Delta\theta$ on $\gamma_{\times}$ and $q$ for the
major and minor axes is seen in Figure 4. More face-on galaxies have larger
apparent rotation angles because their shape is easy to distort. A slight
elongation of an elliptical isophote in a given direction will have much more
effect on the orientation of the isophote if it starts off as relatively round
rather than relatively extended.

Figure 4: The angular difference between the imaged and kinematic axis for the
major axis (top) and minor axis (bottom) as a function of the shear
$\gamma_{\times}$ and the axis ratio $q=b/a$. The magnitude of the rotation of
the photometric axis increases more quickly for more face-on galaxies because
less elongated ellipses are more easily distorted in an arbitrary direction
than more elongated ellipses. The major axis shows slightly less angular
difference than the minor axis because the imaged and kinematic axes are being
shifted in the same direction, while for the minor axis, they are shifted in
opposite directions. This is also why the edge-on $q=0.2$ line for the minor
axis still shows some difference.

We can also see that these angular differences mainly depend on
$\gamma_{\times}$ and $q$. We can then solve for the shear $\gamma_{\times}$
in terms of the two observables: $\Delta\theta$ (the difference between
measured kinematic axis and imaged axis) and the axis ratio $q$. The
relationship between $\gamma_{\times}$ and the deflection of the major and
minor axis position angles differ by a factor of $q^{2}$:

$\gamma_{\times,maj}\approx\frac{\Delta\theta_{maj}(1-q^{2})}{2q^{2}}$ (28)

$\gamma_{\times,min}\approx\frac{\Delta\theta_{min}(1-q^{2})}{2}.$ (29)

The dependencies we recover here agree well with Wittman & Self (2021) and we
will use them in the next section to derive an improved lensing inference
based on a combination of imaging and kinematic measurements. By combining the
imaging measurements of position angle and axis ratio with velocity
measurements of the kinematic position angle, we can gain access to a
constraint on the gravitational shear induced on the galaxy.

###  4.2 KWL Models with Imaging Information

We rerun the previously described Bayesian velocity field fitting models but
with added constraints on the difference between the kinematic and image PAs.
We feed the model a mock image position angle, setting the value by perturbing
the expected value derived using Equation 26 with a Gaussian error based on
the assumed level of photometric uncertainty. Commonly-used photometric codes
like Galfit (Peng et al., 2002) often drastically under-report the magnitude
of their PA errors. Accurate error accounting must be done by comparing
intrinsic PAs in simulated galaxies with the values recovered by photometric
fitting. Häussler et al. (2007) suggests that photometric PA measurements have
average errors between 1 and 6 degrees depending on image depth, so we will
largely restrict our analyses to that range.

We then allow the model to fit the angular difference between the kinematic PA
and the perturbed image PA with its shear parameter, incorporating the result
as a new Gaussian term in the likelihood function. With this added
information, we see a significant reduction in the width of the posteriors
$\sigma_{\gamma}$, as shown by the red posteriors in Figure 3. Adding in the
image PA allows for the degeneracy between the kinematic PA and the shear
strength to be broken much more effectively, resulting in a factor of 5
reduction in $\sigma_{\gamma}$ for an image PA with an error of 3 degrees.
This vast reduction in statistical errors has large implications for regimes
where the measurement error would be dominated by statistical error.

These improvements persist even for larger PA errors, as shown in Figure 5.
Within the range of expected image PA errors (shown as the shaded blue
region), we see a reduction in $\sigma_{\gamma}$ by a factor of 2–6, meaning
that a KWL technique utilizing imaging shape information is more sensitive. We
also see that for large image PA errors, $\sigma_{\gamma}$ approaches its
value from before imaging information was added. As expected, as uncertainty
in the image PA measurement increases, the statistical error in the measured
shear tends towards what is obtained in a fit without image PA constraints.

Figure 5:  The gains in precision seen after adding information on image
position angle to our Bayesian KWL model for a mock galaxy like the one in
Figure 2, including the applied shear of $\gamma_{\times}\approx 0.06$. The
model that fits both the kinematic and imaging distortion (blue line) has an
error on the shear posterior $\sigma_{\gamma}$ that is 3-6 times smaller than
the model with only kinematics (dashed line) was able to achieve within the
region of expected errors on image PA in real data (blue shaded area). If
borne out in real observations, this could lead to significant decreases in
necessary sample sizes and exposure times to obtain a given lensing S/N. For
very large errors in image PA, $\sigma_{\gamma}$ tends back towards the value
obtained with no imaging information.

In Figure 6, we explore differing combinations of image PA error and velocity
error for a relatively face-on source galaxy (30 degree inclination). Larger
errors in image PA or velocity measurement may come from shallower exposures,
poor seeing or angular resolution, or irregularities in the galaxy. At this
low inclination, improvements in image PA precision improve the KWL shear
measurements as much or more than improvements to the velocity precision,
reducing the need for higher S/N or higher resolution spectroscopy. Even with
moderate image PA uncertainties on the order of several degrees, the precision
of KWL measurements can reach levels comparable to those of velocity-field-
only fits relying on kinematic measurements at a fraction of the velocity
error.

Figure 6: The change in the error on the shear measurement $\sigma_{\gamma}$
for a mock velocity field with simulated seeing with different amounts of
error on the image PA and different central velocity errors. The input shear
is $\gamma_{\times}\approx 0.06$ and the model velocity field is identical to
those used in previous figures except that it has a $30^{\circ}$ inclination
with respect to the observer. The blue shaded region represents the expected
range for image PA errors in real observations. Smaller errors on the image PA
lead to higher precision in shear, as is expected. For larger image PA errors,
the precision approaches the value derived from just the velocity field, but
the value it levels off at depends on the errors in the velocity field itself
and the inclination of the galaxy. For image PA errors expected in real data,
even poor-quality velocity fields can match the precision reached by very good
velocity field-only KWL measurements, presenting opportunities for exposure
time savings.

These gains are much larger for lower-inclination galaxies since their imaged
axes are much more distorted, allowing them to contribute more to the shear
measurement. Higher-inclination galaxies, as shown in Figure 7, gain less from
their photometric information because the magnitude of the angular deflection
is greatly reduced, forcing the shear measurement to rely more on kinematic
distortion. The distribution of inclinations within a population of randomly-
oriented disks is uniform over $\sin i$ (Law et al., 2009), so the inclination
distribution of a random sample from a survey will be weighted towards the
high end, with a mean inclination angle of 1 radian (about $57^{\circ}$) or a
mean axis ratio of $q\approx 0.54$. So unless a survey sample is deliberately
designed to sample low-inclination galaxies, most galaxies will see
improvements on par with Figure 7 rather than 6.

Still, adding imaging information to KWL measurements for these higher-
inclination galaxies that would make up the majority of a random sample would
cut required exposure times dramatically for spectroscopic observations. If we
assume that S/N scales with the square root of exposure time, then even a
factor of two improvement in S/N from including the image PA can cut exposure
time by a factor of four.

####  4.2.1 Comparison to Previous Work

As mentioned in Section 3.2, our kinematics-only model delivered similar
statistical errors as that of Gurri et al. (2020) when tested on their data,
so to quantify the benefits of our kinematics + imaging technique on
presently-available data sets, we again benchmark against their approach by
modifying our mock data to better match the Gurri et al. (2020) data. We move
the source galaxy to $z=0.15$ and the lens to $z=0.03$, increase the FWHM of
the PSF to 1.5”, and increase the spatial element size to 0.5” to roughly
mimic their data set. We add mock image PA measurements, as we have done for
the rest of our mock data, and we assume an error on the image PA of 3
degrees, which is in the middle of the expected error range.

Applying our kinematics + imaging algorithm to these simulated galaxies
results in significantly improved statistical uncertainties. Our estimate on
the resulting $\sigma_{\gamma}$ values are lower by a factor of ${\sim}6$ on a
per-galaxy basis, consistent with the results from higher redshift mock data
in Figure 5. This result demonstrates that even for existing data sets at
lower redshifts, this technique can increase lensing precision.

However, our inclusion of imaging data does not address the dynamical shape
noise term. In order to account for this error, Gurri et al. (2020) estimate
the magnitude of the error using a different sample of unsheared galaxies, and
find an amplitude that is similar to the statistical error on a per-galaxy
basis. This dynamical shape noise term is added in quadrature to determine the
final error for each galaxy in their sample. As a result, our estimated factor
of 6 improvement in the statistical error, when applied to the full sample in
Gurri et al. 2020, only increases their overall 2.5$\sigma$ shear detection to
3$\sigma$.

Figure 7: The same as Figure 6 but for a $45^{\circ}$ and $60^{\circ}$
galaxies. Higher-inclination galaxies have a much smaller image distortion
than lower-inclination galaxies, meaning the model cannot rely on this
information as much when constraining the shear. This creates a much larger
stratification in velocity error than for lower-inclination galaxies. Overall,
$\sigma_{\gamma}$ values are larger, particularly in the $60^{\circ}$ galaxy,
but improvements to the image PA observation have little effect.

###  4.3 Survey Design Considerations

While statistical error is small relative to systematic error for low redshift
samples, the same is not true at higher redshifts. Resolved spectroscopic
measurements become more difficult at higher redshifts due to lower spatial
resolution and surface brightness, resulting in higher statistical errors. So
the gains made by our imaging + kinematics model will be especially salient at
increased redshifts, where there are the added benefits of greater lensing
magnitudes (thanks to more favorable lensing kernels) and higher on-sky source
densities, allowing for easier collection of larger sample sizes.

If we observe a $1\times 1$ arcmin2 field near a galaxy cluster (e.g. using
MUSE), we will catch tens of galaxies with velocity fields that are well-
defined enough to perform our KWL analysis. A multiplexed fiber IFU instrument
like the proposed FOBOS (Bundy et al., 2019) would be able to patrol a
significantly wider field and collect velocity data from only the most
promising galaxies, enabling a larger sample in less overall exposure time. We
can stack the lensing information from the multiple galaxies to obtain higher
precision and reduce the random errors introduced by noise or galaxy
irregularities either by a weighted average of shear results (as was done in
Gurri et al. 2020) or a hierarchical Bayesian model that simultaneously fits
all galaxies in a given spatial bin. We expect these individual galaxy
“dynamical shape noise” errors to average out to zero in a large sample
because they will be randomly and symmetrically distributed (Gurri et al.,
2020).

Figures 6 and 7 show that, with attainable observational errors and low
inclinations, it is possible to obtain a $\sigma_{\gamma}$ value comparable in
magnitude to the applied shear, resulting in a ${\sim}1\sigma$ KWL measurement
per galaxy for the input shear of $\gamma_{\times}\approx 0.06$. However, if
we stack multiple such measurements within a radial bin defined by a selection
of foreground lenses and assume that the source measurements are statistically
independent samples of the ensemble mass density profile of the lenses, then
the error on this galaxy–galaxy shear measurement should scale as $n^{-1/2}$.
So for a sample of only $n=9$ galaxies in a spatial bin, $\sigma_{\gamma}$
should be lowered by a factor of 3, raising a 1$\sigma$ detection to
3$\sigma$. This scaling should apply to the systematic error as well since we
expect individual galaxy errors to average to zero over a large sample.

####  4.3.1 Source Redshift

Depending on the instrumentation, if we want to build a larger sample size,
higher source density on-sky may be beneficial if the number of independent
pointings can be minimized. Source density increases with redshift, as does
the magnitude of lensing effects according to Equations 5 and 7, but there is
a trade-off because surface brightness and physical size decrease with
redshift. To explore this trade-off, we run our KWL model that incorporates
imaging information on the same mock galaxy placed at a range of redshifts. We
ignore complications from galaxy evolution like mass/radius growth or
dynamical changes. We simulate the observations of different inclination
source galaxies at the varying distances as if they are behind a lens at
$z=0.3$. We hold the rest of the observational parameters constant, on-sky
spatial resolution, image PA error, central velocity error, and PSF width
(unless specified). Maintaining constant errors would require deeper exposures
for both imaging and spectroscopy, which we do not account for in this paper.

Figure 8: Top: The KWL S/N per galaxy for source galaxies at varying
inclinations with different redshifts behind a massive halo at $z=0.3$. We
maintain constant on-sky spatial resolution, imaging and velocity errors, and
PSF width, ignoring the effects of surface brightness dimming and galaxy
evolution. Error bars are based on the random variation from running 10
identical velocity fields with distinct randomized errors. Middle: The
$45^{\circ}$ inclination galaxy from the top panel but with FWHM of the PSF
varied to simulate different observing conditions, still holding on-sky
instrumental resolution constant. Realistic observing conditions provide
measurements that are a factor of 2–3 lower in S/N compared to an ideal
observation without PSF smearing. Improving the PSF by a factor of 2 with
something like a ground-layer adaptive optics system to 0.35” leads to S/N
gains of about 50%. Bottom: The overall magnitude of the lensing shear, shown
in the red dashed line, increases with redshift, making the effects of KWL
more noticeable. The fraction of a square FWHM covered by a square kpc in the
on-sky plane of the source galaxy, shown in blue solid line, indicates that
the PSF becomes very large compared to the spatial scales of the source
galaxy, worsening the quality of the fit. S/N largely tracks with shear
magnitude.

Although closer source galaxies have smaller $\sigma_{\gamma}$ uncertainties
because of their greater number of spatial resolution elements, Figure 8 shows
that they deliver a poorer S/N per galaxy because they are simply not sheared
as much as more distant galaxies. The top panel shows a high degree of S/N
stratification in inclination because the change in image PA is much greater
for lower inclinations. We find that there is modest gain in S/N for all
inclinations with redshift before leveling off around $z\sim 0.7-1$, the
redshift range assumed in this paper, before keeping a relatively constant
value at higher $z$. This mirrors the trend seen in the overall shear
magnitude seen as the red dashed line in the bottom panel of Figure 8.

Another factor that influences precision is the FWHM of the spatial PSF of the
resolved kinematic field. The blue line in the bottom panel of 8 shows that as
the source galaxy redshift increases, the size of the galaxy on-sky becomes
smaller relative to the size of the PSF, here represented as the ratio of the
square FWHM compared to the size of a square kpc in the plane of the galaxy.
Reductions to the FWHM result in higher surface brightness in many cases and
more independence of the spatial resolution elements of an IFU observation,
which would improve the precision of the velocity field fit even if the
spatial sampling is not changed. These types of gains could be obtained in the
real world using a ground-layer adaptive optics system (e.g. Chun et al.,
2018; Hartke et al., 2020), which would lead to an improvement in seeing by a
factor of 1.5 to 2 over a several arcminute-wide field. In the middle panel of
Figure 8, we vary the FWHM to simulate different observing conditions, from an
ideal observation without seeing to a FWHM of 2”. We find modest gains in S/N
with better seeing, especially at moderate and higher redshifts. However, even
fits with poor seeing are able to locate the kinematic axes with acceptable
precision, so the effects of better seeing are limited. Still, the increase in
surface brightness is valuable for driving down exposure time.

####  4.3.2 Lens Redshift

Figure 9: The same as Figure 8 but holding the source galaxy fixed at $z=1$
while moving the lens in front of it. The S/N again largely follows the shear
magnitude (bottom), which peaks at $z\sim 0.25-0.3$ because that is the
configuration of angular diameter distances that minimizes the critical radius
(Equation 5). FWHM2 per kpc2 is not shown because it is constant due to the
stationary source galaxy.

To test KWL’s ability to probe lenses at various redshifts, we also simulate
systems where the source is held at a constant $z=1$ for varying redshift
lenses (Figure 9). We find similar dependencies on inclination and FWHM as in
Figure 8. However, since shear magnitude scales with the angular diameter
distances to the source and the lens (Equation 5), shear magnitude peaks at
$z\sim 0.3$, falling off sharply from there. In fact, due to how angular
diameter distance is defined, for source galaxies further than $z\sim 0.7$,
shear magnitude will always be largest for lenses at $0.2<z<0.3$. So KWL, like
any other lensing technique, will be most sensitive to lenses at these
moderate redshifts.

####  4.3.3 General Strategy

An ideal strategy for a KWL survey would likely be to target low-inclination
galaxies that benefit more from the gains imaging can provide. They should be
regularly-rotating blue spiral galaxies that are likely to have a high
emission line flux for ease of kinematic measurements. The targets should have
regular, symmetric isophotes and imaging sufficient to measure image position
angles with 1–3 degree uncertainties. If the source galaxy population reaches
$z\sim 1$, the ideal lens sample would have a redshift range $z=0.2$–0.7 to
ensure shear magnitudes near the peak of the lensing kernel. Ideally, the
major-axis position angles of selected source galaxies should be misaligned
with respect to the on-sky direction to the lens center. A $45^{\circ}$ offset
maximizes the cross-term in the shear. A full optimization of a KWL survey
design would take into account the lens and source redshift distributions, the
intrinsic variation in galaxy shape, systematic errors in individual galaxies
due to dynamical shape noise, and observational constraints such as seeing and
exposure time as it relates to brightness dimming, distance, and size.

##  5 Conclusion

We have demonstrated a new formalism for combining imaging and kinematic
information that significantly improves the S/N of kinematic weak-lensing
observations. Our Bayesian models fit both the kinematic distortion and the
photometric offset caused by lensing shear in a given galaxy, leading to
decreases in per-galaxy statistical error by a factor of 2 to 6 compared to
kinematics alone. If borne out in real observations, this approach opens the
door for future studies to more effectively utilize kinematics-based lensing
observations of lens samples at moderate redshifts. Even with existing
instruments, the methodology appears promising for enabling probes of galaxy
cluster halo profiles with greater spatial resolution and S/N while mitigating
many of the systematics that affect conventional lensing measurements, like
shape noise and intrinsic alignments. Ultimately, this will allow for
individual, total mass measurements in the weak-lensing regime for a greater
number of clusters.

A purpose-designed survey on a new or existing instrument could provide the
observations necessary for a successful KWL measurement. Further study is
needed to determine the appropriate redshift range to balance shear magnitude
and source density with spatial resolution and surface brightness, while more
detailed galaxy rotation models are needed to quantify the systematic effects
of kinematic irregularities on KWL measurements.

The authors would like to acknowledge the anonymous reviewer for their helpful
comments in improving this paper. We woudl also like to acknowledge Eric Huff
and Tim Eifler for their early guidance on the theory of kinematic weak
lensing and Kevin McKinnon for early statistical guidance.

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