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# Testing Velocity Field Lensing on Realistic Galaxy Models

Jean Donet David Wittman Department of Physics & Astronomy, University of
California, Davis, CA 95616 David Wittman<EMAIL_ADDRESS>
###### Abstract

Weak gravitational lensing shear could be measured far more precisely if
information about unlensed attributes of source galaxies were available. Disk
galaxy velocity fields supply such information, at least in principle, with
idealized models predicting orders of magnitude more Fisher information when
velocity field observations are used to complement images. To test the level
at which realistic features of disk galaxies (warps, bars, spiral arms, and
other substructure) inject noise or bias into such shear measurements, we fit
an idealized disk model, including shear, to unsheared galaxies in the
Illustris TNG simulation. The inferred shear thus indicates the extent to
which unmodeled galaxy features inject noise and bias. We find that
$\gamma_{+}$, the component of shear parallel to the galaxy’s first principal
axis, is highly biased and noisy because disks display a range of intrinsic
axis ratios from 0.8–1. The other shear component, $\gamma_{\times}$, shows
little bias and is well-described by a double Gaussian distribution with
central core scatter $\sigma_{\text{core}}\approx$ 0.03, with low-amplitude,
broad wings. This is the first measurement of the natural noise floor in the
proposed velocity field lensing technique. We conclude that the technique will
achieve impressive precision gains for measurements of $\gamma_{\times}$, but
little gain for measurements of $\gamma_{+}$.

Gravitational lensing (670), Weak gravitational lensing (1797), Gravitational
lensing shear (671), Galaxy kinematics (602), Magnetohydrodynamical
simulations (1966)

††journal: ApJ

##  1 Introduction

Weak gravitational lensing is a key tool in modern astrophysics because it
enables mapping of gravitational potentials (see Bartelmann & Maturi, 2017 for
a recent review). The physical basis for this mapping is the gravitational
deflection of light from a distant source (assumed here to be a galaxy) as it
passes the potential on its way to the observer. The deflection itself is not
measurable in the absence of constraints on the true source position, but
variations in deflection angle across the face of a source galaxy manifest as
anisotropic stretching (shearing) of the galaxy image. Measuring shear thus
requires some knowledge of the unlensed shape of the source. Rather than
attempt to model unlensed shapes on a source-by-source basis, weak lensing has
proceeded on the assumption that unlensed sources on average have no preferred
direction. Hence shear measurements are averaged over many source galaxies in
a patch of sky. In this approach, an individual source carries little
information because its unlensed shape is poorly constrained—a fundamental
source of uncertainty called shape noise. This approach is well suited to
modern wide-field imagers that collect data on many sources simultaneously,
but limits the angular resolution of shear measurements.

Information about the unlensed condition of a source galaxy could enable shear
constraints along the specific line of sight to that galaxy, hence enabling
shear maps with better angular resolution. Blain (2002) suggested a method
based on the fact that a circular orbit inclined to the line of sight produces
an observed velocity pattern that is antisymmetric across its apparent minor
axis and symmetric across its apparent major axis. This symmetry is disturbed
by lensing, hence any observed asymmetry could be attributed to lensing.
Morales (2006) extended this idea to disks, deriving a relation between shear
and the angle between the principal axes of the observed velocity and
intensity fields. de Burgh-Day et al. (2015) analyzed velocity maps of nearby
(hence unlensed) disk galaxies as well as simulated velocity maps, and found
${\sim}0.01$ shear precision per source, compared to ${\sim}0.2$ for
traditional weak lensing.

However, these approaches constrain only one of two shear components. The
component of shear directed along the unlensed major axis, which we define as
$\gamma_{+}$, preserves the symmetry of the velocity field but changes the
apparent axis ratio. Huff et al. (2013) proposed to constrain $\gamma_{+}$ as
follows: the Tully-Fisher relation (TFR; Tully & Fisher, 1977) predicts the
(3-D) rotation speed; this supports an inference of the inclination angle $i$
using the measured line-of-sight speed; constraining $i$ in turn constrains
the unlensed axis ratio of a circular disk viewed at this inclination. Hence
the observer can compare lensed and unlensed axis ratios to deduce
$\gamma_{+}$. Wittman & Self (2021 , hereafter WS21) presented a Fisher
Information Matrix (FIM) analysis of this method using an idealized disk
model. They concluded that the shear precision is a function of inclination
angle $i$, with face-on ($i{=}0$) galaxies offering more precision. They found
that $\gamma_{+}$, as well as the symmetry-violating shear component
$\gamma_{\times}$, could be inferred to a precision of 0.01 or better (at
least at $i\leq 25^{\circ}$) in this idealized model. However, they warned
that their model accounted only for random measurement errors in the velocity
and intensity fields. Nature must introduce additional variations, such as
intrinsically noncircular disks upsetting the axis-ratio calculation, or
warped disks that violate the assumed velocity symmetry. These variations
would constitute the equivalent of shape noise—a fundamental limit on per-
galaxy shear precision due to variance in unlensed galaxy properties.

The goal of this paper is to quantify these natural sources of noise using
simulated galaxies, which enable high-fidelity mock observations from a
variety of viewing angles. Although simulated galaxies may omit some details
found in nature, modern simulations incorporate a variety of relevant physical
processes and produce realistic distributions of satellites and neighbors,
bulges, warps, and so on. Hence they offer a useful starting point free of the
complications inherent in using noisy data obtained from a fixed, unknown
viewing angle. Nevertheless, later in the paper we will discuss how our
results compare to observational work on closely related issues.

The remainder of this paper is organized as follows: §2 describes our methods;
§3, the results; and §4 provides discussion and interpretation.

##  2 Methods

Overview: we fit the idealized disk model of WS21, including the shear
parameters $\gamma_{+}$ and $\gamma_{\times}$, to hundreds of unlensed
simulated disk galaxies. The distribution of inferred shear parameters
directly indicates the noise and bias due to unmodeled galactic structures,
which is likely an irreducible source of error for the velocity field method.
In this section we describe the model briefly (§2.1), then the galaxy
selection (§2.2), Tully-Fisher calibration (§2.3), mock data preparation
(§2.4), and parameter estimation (§2.5).

###  2.1 Idealized Model

We briefly recap the WS21 model and refer readers to that paper for more
details. The model assumes an infinitesimally thin disk with no bulge, viewed
at an angle $i$ between the disk axis and the line of sight. The intensity
field is described by $I=I_{0}\exp(\frac{-2.99R}{r_{80}})$, where $R$ is a
radius in polar coordinates attached to the disk, $r_{80}$ is the radius that
encircles 80% of the light, and $I_{0}$ is the central intensity. The rotation
curve is described by $v=v_{\rm max}\arctan\frac{R}{r_{0}}$ where $r_{0}$ is a
scale length independent of $r_{80}$, and $v_{\rm max}$ is prescribed by the
TFR. The parameter $A_{\rm TF}$ describes $v_{\rm max}$ as a fraction of the
TFR prediction, with a prior allowing for scatter at fixed luminosity. WS21
prescribed 4% scatter, but we adjusted this for Illustris as described below.
Lensing parameters in WS21 include $\gamma_{+}$, $\gamma_{\times}$, and the
magnification $\mu$; we fix $\mu$ at unity to focus on the ability of galaxy
features to mimic shear. The galaxy location on the sky and in velocity space
is described by three nuisance parameters $x_{0}$, $y_{0}$, and $v_{0}$. Table
1 summarizes the parameters.

Table 1: Model parameters

Symbol | Typical value | Bounds | Unit | Description
---|---|---|---|---
$A_{\rm TF}$ | 1 | [0,$\infty$) | - |  $v_{\rm max}$ as a fraction of the Tully-Fisher prediction
$I_{0}$ | 90 | [0,$\infty$) | - | intensity S/N at center
$i$ | varies | [0,90] | deg | inclination angle
$\phi_{\rm sky}$ | 90 | [0,180) | deg | sky position angle of unlensed major axis
$r_{0}$ | 4 | [0,$\frac{L}{2}$)$*$$*$footnotemark:  | pixel | rotation curve scale length
$r_{80}$ | 12.5 | [0,$\frac{L}{2}$) | pixel | radius of 80% encircled light
$x_{0}$ | 0 | (-$\frac{L}{2}$,$\frac{L}{2}$) | pixel | center of galaxy in $x$ coordinate
$y_{0}$ | 0 | ($-\frac{L}{2}$,$\frac{L}{2}$) | pixel | center of galaxy in $y$ coordinate
$v_{0}$ | 0 | - | km/s | galaxy systemic radial velocity
$\gamma_{+}$ | 0 | (-1,1) | - | shear parallel to unlensed major axis
$\gamma_{\times}$ | 0 | (-1,1) | - | shear at 45∘ to unlensed major axis
$\mu$ | 1 | fixed | - | magnification

**footnotetext: $L$ refers to the image side length in pixels (typically
$\approx 50$ pixels)

###  2.2 Simulated Galaxies from Illustris TNG100

We select galaxies from the $z{=}0$ snapshot of Illustris TNG100-1 (Nelson et
al., 2019), a magnetohydrodynamical simulation with a volume of $110.7^{3}$
Mpc3 in a Planck2015 cosmology. The dark matter and baryon particle masses are
$7.5\times 10^{6}$ and $1.4\times 10^{6}$ respectively; the softening lengths
are 0.74 kpc for dark matter and a minimum of 0.125 kpc for gas; and the
median radius of star-forming gas cells is 355 pc, with a minimum gas cell
size of 14 pc. Illustris TNG includes a wealth of physical effects such as
radiative cooling, stochastic star formation, stellar feedback, and stellar
evolution. Figure 1 shows a face-on view of a disk galaxy in our analysis,
demonstrating realistic features that go far beyond the idealized model of a
bulgeless exponential disk.

Figure 1: Stellar composite mock-up of subhalo 421835. This visualization was
generated using Illustris’ visualization tool. We analyze this subhalo in
later figures.

To access TNG100 data we use Illustris’ web-based Application Programming
Interface (API) framework which allows for a search of all TNG runs and
subhalo metadata. This is done via a specified object id search using the
requests Python module. Our goal is to study massive, rotation-supported
galaxies as these are the galaxies most likely to be targeted in observations
if this shear inference method proves practical. Furthermore, because
Illustris TNG has many galaxies, we begin with tight selection criteria aimed
at producing a fairly uniform sample of subhalos:

  * •

the total mass is required to be within 10% of 1.4 $\times 10^{12}$ M⨀

  * •

$\geq$40% of the stellar mass is required to be in the disk and $\leq$30% of
the stellar mass is allowed to be in the bulge

Disk star particles are defined as those with specific angular momenta nearly
as large as the maximum angular momentum of the 100 particles with the most
similar binding energy; in other words with a circularity parameter (Abadi et
al., 2003) $\epsilon{>}0.7$. The fraction of bulge star particles is
quantified as twice the fraction with $\epsilon{<}0$.

These criteria yield 386 subhalos for the fitting described below.

###  2.3 TFR Calibration

We calibrate the TFR on the selected galaxies by using two metadata attributes
precomputed by Illustris: the K-band luminosity and the maximum circular
speed. Figure 2 shows these two quantities for each galaxy as well as the
best-fit relation $L\propto v_{\rm max}^{4.3}$. The residual scatter about
this relation is 22 km/s or about10%. This is larger than assumed by WS21, but
in keeping with the fact that we are using K-band luminosity rather than a
baryonic TFR that would require more mock photometry. According to WS21,
scatter in the TFR at this level is not a limiting factor. With this scatter,
the prior on $A_{\rm TF}$ in our fitting procedure becomes $P(A_{\rm
TF})\propto\exp-\frac{1}{2}(\frac{A_{\rm TF}-1}{\sigma_{A_{\rm TF}}})^{2}$
where $\sigma_{A_{\rm TF}}=0.1$.

Figure 2: Tully-Fisher relation for the selected halos. The best-fit relation
is $L\propto v_{\rm max}^{4.3}$ with a scatter of 22 km/s or about 10%.

###  2.4 Mock Data Preparation

For each selected subhalo, we calculate the moment of inertia tensor of the
star particles, and diagonalize the tensor to yield three principal axes
$A\geq B\geq C$ (we use upper-case letters for intrinsic quantities, reserving
lower-case $a$ and $b$ for the apparent semimajor and semiminor axes). The
first principal axis is chosen to coincide with the $x$ axis of a mock
detector, and the second principal axis with the $y$ axis. For views other
than face-on, the second principal axis is foreshortened by a factor of $\cos
i$. This forces any intrinsic noncircularity to manifest as an inference of
$\gamma_{+}>0$, which stretches the $x$ axis and shrinks the $y$ axis.

Next we compute the $K$-band intensity field and the line-of-sight stellar
velocity field. We choose a grid with 0.25 kpc/pixel, extending to
$\frac{2}{3}$ of the SubhaloStellarPhotometricsRad attribute, which is the
radius defined by a surface brightness of 20.7 mag arcsec-2 in the $K$ band.
This yields grids with ${\approx}50\times 50$ pixels. We assign uncertainties
to each pixel using the WS21 recipe, which is meant to emulate achievably high
signal-to-noise data as follows. The intensity field uncertainty is the same
in each pixel, and is set to $I_{0}/90$. The velocity field uncertainty is set
to 10 km/s at the center and scales inversely with the square root of the
local intensity.

###  2.5 Least-Squares Optimization

We construct a data vector consisting of a concatenated list of intensity and
velocity pixel values, and use an off-the shelf optimizer to find the best-fit
model. Many of the parameters in our model are bounded by physical or
geometric arguments. For example $i$ is defined on [0,90∘], $\phi_{\rm sky}$
is defined on [0,180∘), and the shear components are defined on (-1,1); see
Table 1 for a full list of bounds. To enforce these bounds while efficiently
searching for the best-fit model we use the Python function
scipy.optimize.least_squares which implements the “Trust Reflective Region”
method (Branch et al., 1999) for dealing with boundaries. Within the
boundaries, scipy.optimize.least_squares does not directly support the use of
priors. Therefore, we implement the TFR prior indirectly as follows.
scipy.optimize.least_squares requires a user-defined function that returns a
vector of residuals, which it then internally squares and sums to obtain a
$\chi^{2}$ value. Because $\chi^{2}=-2\ln\mathcal{L}$ where $\mathcal{L}$ is
the likelihood, the $A_{\rm TF}$ prior in §2.3 becomes simply $(\frac{A_{\rm
TF}-1}{\sigma_{A_{\rm TF}}})^{2}$ in $\chi^{2}$ units. Hence we concatenate
one element to the data array with the value $\frac{A_{\rm
TF}-1}{\sigma_{A_{\rm TF}}}$ which is then, internally to
scipy.optimize.least_squares, squared to obtain the $\Delta\chi^{2}$
corresponding to the prior.

As a test of the code, we fitted mock data generated by the idealized model.
The fitted parameters matched the input parameters to within one part in
$10^{6}$.

##  3 Results

Figure 3: An illustrative TNG100 subhalo (421835, at $z=0$) viewed at
$i{=}60^{\circ}$ from face-on. The top (bottom) row shows the velocity
(intensity) field, while the left, middle, and right columns show the mock
data, model, and residuals (in units of the uncertainty on each pixel)
respectively. The idealized model does not include a bar nor spiral arms,
hence these features appear in the residuals.

We present one example in detail in Figure 3. The left column contains the
data from TNG100, the middle column contains the least-squares optimized
model, and the residual of the difference is displayed in the right column.
The rows contain the velocity and intensity data and model fields as labelled.
The “observed” intensity field has substructure near its center, which
violates the idealized model of an axisymmetric disk, but there is little
corresponding substructure in the velocity field. The extent to which these
nonaxisymmetric features result in spuriously inferred shear may vary with
viewing angle as well as from galaxy to galaxy. Hence, we repeat for each of
the 386 galaxies viewed at a series of inclination angles (5, 20, 40, and 60
degrees).

###  3.1 Inferred shear

Figure 4 displays histograms of the inferred shear, broken down by component
and by true inclination angle. Table 2 lists the mean and standard deviation
of each of these distributions. Some of the distributions are non-Gaussian, so
for a more complete description we tested Gaussian mixture models (GMM) with a
range of Gaussian components and used the Bayesian Information Criterion to
determine the preferred number of components in each case. The result is that
$\gamma_{+}$ is best fit at each inclination by a single Gaussian, while
$\gamma_{\times}$ is best fit at each inclination by a double Gaussian. The
resulting descriptive curves are overplotted in Figure 4. We discuss each
component in more detail below.

Figure 4: Histograms for inferred $\gamma_{+}$ and $\gamma_{\times}$ with
overlaid best fit functions. We use a Gaussian mixture model to determine the
statistics of the distribution. A Bayesian Information Criterion analysis was
conducted to determine the optimal number of model components. $\gamma_{+}$ is
best fit in all cases by a single Gaussian, while $\gamma_{\times}$ is best
fit by a double Gaussian. Table 2: Distribution of inferred shear as function
of inclination.

| $\gamma_{+}$ | $\gamma_{\times}$
---|---|---
$i$ (deg) | mean | $\sigma$ | mean | $\sigma$ | $\sigma_{\rm core}$
5 | 0.213 | 0.179 | -0.016 | 0.103 | 0.033
20 | 0.176 | 0.138 | -0.018 | 0.089 | 0.036
40 | 0.096 | 0.143 | -0.015 | 0.076 | 0.029
60 | 0.084 | 0.214 | -0.014 | 0.062 | 0.025

###  3.2 $\gamma_{+}$

According to Figure 4, $\gamma_{+}$ is biased toward positive values, and
shows large scatter (${\approx}0.1-0.2$) at all inclinations. The sign of the
bias suggests that real galaxies depart substantially from the assumption of
face-on circularity. We arranged all mock observations such that the true
first principal axis $A$ was aligned with the detector’s $x$ axis; hence if
the second principal axis $B$ is smaller, the galaxy appears as an ellipse
with apparent major axis aligned with $x$, even when face-on. The model has no
way to account for this other than introducing $\gamma_{+}{>}0$, which
stretches the apparent $x$ axis of the assumed circular disk. To confirm this
hypothesis we plot inferred $\gamma_{+}$ against the intrinsic axis ratio
$\frac{B}{A}\equiv q_{0}$ in Figure 5. The inverse relation between
$\gamma_{+}$ and $q_{0}$ is clear. We also plot the predicted relationship
$\gamma_{+}=\frac{1-q_{0}}{1+q_{0}}$ which follows from inverting the well-
known relationship $\frac{b}{a}=\frac{1-\gamma}{1+\gamma}$ where $\frac{b}{a}$
is the apparent axis ratio of a sheared circular disk.

Figure 5: Scatter plots of inferred $\gamma_{+}$ v. $q_{0}$ as a function of
$i$. We overlay the function $\gamma_{+}(q_{0})=\frac{1-q_{0}}{1+q_{0}}$ which
describes the expected shear along the major axis arising from the intrinsic
axis ratio. We include a colorbar to illustrate the difference between the
inferred $i$ and the preset value, indicated at the top right of each panel.

The variation in face-on axis ratio explains much but not all of the
spuriously inferred $\gamma_{+}$: the inferred $\gamma_{+}$ is generally
larger than predicted by the $q_{0}$ argument, particularly when viewed near
face-on. One complicating factor is the anticorrelation with $i$:
underestimating $i$ lowers the apparent axis ratio prediction, which can
compensate for overestimating $\gamma_{+}$. This trend is revealed by the
color gradient in the $i=40^{\circ}$ and $i=60^{\circ}$ panels of Figure 5. It
is not clear why $\gamma_{+}$ remains overestimated in other panels of this
figure where the trend with $i$ is not strong.

Figure 5 also demonstrates that constraints on $\gamma_{+}$ degrade as targets
become more edge-on. This was predicted by the Fisher forecasts of WS21, and
can be understood in terms of a uniform (face-on) velocity field allowing very
little wiggle room in $i$.

###  3.3 $\gamma_{\times}$

As illustrated in Figure 5, $\gamma_{\times}$ performs significantly better
than $\gamma_{+}$ with a narrow peak at values close to the expected value of
$\gamma_{\times}{=}0$. At each inclination the distribution of
$\gamma_{\times}$ is best described by a GMM of degree $n{=}2$, which captures
a sharp central peak plus low-amplitude broad wings. To compare with the
Fisher forecast of WS21 we examine the width of the central peaks, listed in
Table 1. The $\gamma_{\times}$ precision is several times worse than forecast,
and the least favorable viewing angle is near face-on rather than near edge-on
as forecast. There is no correlation between $\gamma_{\times}$ and other
parameters such as $q_{0}$, $i$, or $\gamma_{+}$, so we conclude that the
effects discussed in the preceding subsection are not responsible for the
greater than expected scatter.

The bias in $\gamma_{\times}$, although small, is statistically significant
(the typical standard error in the mean in each row in Table 1 is 0.005). This
suggests that spiral arms could be responsible for the bias and scatter.
Spiral arms will have a greater effect near face-on orientation, and they can
cause a bias as follows. We consistently orient the mock data such that the
disk is rotating clockwise. Spiral arms that trail (which are the vast
majority; Iye et al., 2019) will then trail in an Z-wise rather than S-wise
direction on the sky. It is plausible that this could systematically affect
the inference of the $\gamma_{\times}$ component. Warping of disks could also
play a role, but it is more difficult to see why warps would become more
important for face-on orientations, or create a systematic effect.

The systematic effect would not be seen in a straightforward average of real
observations because any given galaxy is, immutably, seen as S-wise or Z-wise.
Z-wise galaxies would have a small bias toward negative $\gamma_{\times}$ and
S-wise galaxies would have a small bias toward positive $\gamma_{\times}$.

##  4 Discussion & Summary

The scatter and bias in $\gamma_{+}$ will be serious obstacles in attempting
to make the velocity-field method competitive with traditional image-based
weak lensing, at least for this component of shear. Image-based weak lensing
yields a slightly larger shear uncertainty per galaxy (${\sim}0.2$ vs.
0.1–0.2), but involves far less expensive observations that collect many more
source galaxies. Much of the $\gamma_{+}$ scatter and bias in this work is due
to scatter in the face-on axis ratio $q_{0}$, which causes a bias because
galaxies can only scatter in one direction from the assumed value of
$q_{0}=1$. Hence, our conclusion rests on the assumption that Illustris
reproduces the $q_{0}$ distribution in the observed universe, at least
qualitatively. The bulk of this section examines the literature on that issue
and confirms the conclusion.

We briefly recap two methods of inferring $q_{0}$ from observations. The first
(Ryden, 2004, 2006) finds the distribution of $q_{0}$ that, when viewed from
random angles, best matches the observed distribution of apparent axis ratios.
In 12,764 SDSS galaxies Ryden (2004) found a median axis ratio of 0.85 and a
scatter ${\approx}0.1$ in $i$ band. Figure 6 shows that her best-fit
distribution (blue curve) is quite similar to the distribution we extracted
from Illustris (black curve). Ryden (2006) extended this type of analysis to
include $K$-band data from 2MASS, as well as a breakdown into early and late
spirals. She found a strong trend in median $q_{0}$ with Hubble type: from
${\approx}0.98$ for late spirals to ${\approx}0.7$ for early spirals. This is
again comparable to the range in Illustris. If $q_{0}$ is indeed a strong
function of Hubble type, a possible workaround for velocity-field shear
inference is to adopt a prior on $q_{0}$ for each observed galaxy based on its
Hubble type. However, there remains substantial scatter in $q_{0}$ at fixed
Hubble type so this prior would remove only some of the scatter in inferred
$\gamma_{+}$.

The second method (Andersen et al., 2001; Andersen & Bershady, 2002) is
capable of inferring $q_{0}$ on a per-galaxy basis. Andersen et al. (2001)
used intensity and velocity field data to compare the sky position angle (PA)
of the photometric and kinematic major axes. Such a misalignment can be caused
by lensing, but not in the nearby sample studied by Andersen et al. (2001).
They attributed any misalignment to the disk having circular orbits but an
intrinsically photometric ellipticity $\epsilon\equiv 1-q_{0}$. They examined
seven galaxies and found $\epsilon$ ranging from 0.02–0.20. Andersen &
Bershady (2002) examined 28 galaxies and found
$\epsilon=0.06^{+0.064}_{-0.031}$; Ryden (2004) corrected this for selection
effects and derived a distribution shown as the red curve in Figure 6. Our
Illustris distribution differs from the two Ryden results by less than those
two results differ from each other. We conclude that Illustris is accurately
reproducing the intrinsic noncircularity of disk galaxies.

Figure 6: Understanding the scatter in $\gamma_{+}$: Left: Distribution
comparison for our sample of 386 subhalos in the $K$ band, Ryden’s correction
of Andersen & Bershady (2002) in the $I$ band, and Ryden (2004) in the $i$
band. Our sample is best described by a lognormal distribution peaking at
$\ln(\epsilon){=}-2.14\pm 0.83$, we include the equivalent axis ratio values
at the top for clarity. Right: Corner plot showing that $q_{0}$ is
anticorrelated with $\gamma_{+}$ but has no correlation with
$\gamma_{\times}$. Contours show 68.27%, 95.45%, and 99.73% confidence
intervals.

We note that velocity fields are typically measured for interstellar gas—e.g.
Andersen et al. (2001) and Andersen & Bershady (2002) used the H$\alpha$
emission line—while intensity fields are typically measured for stars, e.g.
these references used broadband $R$ and $I$ images. Any misalignment between
gas and stellar velocity fields would contribute to the ellipticity inferred
by these authors, but would not manifest in our results because we used the
stellar velocity field, nor in Ryden’s results which are based entirely on
photometry. The consistency of these three results suggests that gas-star
misalignments are a subdominant effect.

Although the details may vary with galaxy selection and wavelength, this body
of work consistently implies that disks are not intrinsically circular, which
leads to a bias of $\approx 0.1$ on the $\gamma_{+}$ inference. Even if the
mean effect of intrinsic ellipticity could be removed from the shear
inference—an open question, considering the unknown PA of the intrinsic
ellipticity—there remains a scatter of the same order, possibly reducible only
through careful selection of very late types if the Ryden (2006) results are
correct. This casts serious doubt on the viability of the kinematic method for
inferring $\gamma_{+}$.

This issue does not affect $\gamma_{\times}$, where we see promising precision
gains compared to traditional weak lensing. The $\gamma_{\times}$ scatter has
a tight central core with $\sigma_{\text{core}}{\approx}0.03$, dramatically
smaller than the ${\sim}0.2$ from image-based weak lensing. The scatter is
larger than predicted by the idealized model of WS21, as expected from
galaxies with realistic features such as spiral arms and warps; this work
documents for the first time how much scatter is contributed by such features.

It is instructive to compare our $\gamma_{\times}$ results with those of de
Burgh-Day et al. (2015), whose inference method involves a nonparametric disk
model created by reflecting the data itself. In their method, the inferred
value of $\gamma_{\times}$ is that which minimizes the asymmetry when
unlensed; $\gamma_{+}$ is not constrained. They tested this method on
idealized disks similar to those created by our parametric model, and found
uncertainties ${\approx}0.005$ with mock data similar in quality to ours. This
is in line with the WS21 Fisher prediction, and suggests that the
${\gtrsim}0.03$ scatter we see is largely due to galaxy features not captured
by our parametric model. Surprisingly, when de Burgh-Day et al. (2015) applied
this method to two actual (nearby, hence unlensed) galaxies, the
$\gamma_{\times}$ precision was much tighter, ${\approx}0.001$. It is unclear
how higher precision was obtained from less idealized data, and perhaps no
conclusion can be drawn from a sample of two. Still, it is entirely possible
that, for $\gamma_{\times}$ inference, nonparametric galaxy models are more
robust against features such as bars and warps. The inability to infer
$\gamma_{+}$ was a weakness of the nonparametric method, but the current work
has shown that $\gamma_{+}$ inference is confounded by other factors in any
case. Application of a nonparametric disk model to mock Illustris data is a
possible step for future work. Nevertheless, this work has shown that the
parametric model already provides substantial precision gains over traditional
image-based weak lensing.

We thank Dylan Nelson for patiently answering many questions about Illustris-
TNG. This work was supported by NSF grant AST-1911138.

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