# The Kinematic Sunyaev-Zel’dovich Effect with ACT, DES, and BOSS: a Novel
Hybrid Estimator

M. Mallaby-Kay  0000-0002-2018-3807 Department of Astronomy and Astrophysics,
University of Chicago, Chicago, IL 60637, USA  S. Amodeo  0000-0002-4200-9965
Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR
7550, F-67000 Strasbourg, France  J. C. Hill  0000-0002-9539-0835 Department
of Physics, Columbia University, New York, NY 10027, USA  M. Aguena
Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino
77, Rio de Janeiro, RJ - 20921-400, Brazil  S. Allam  Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA  O. Alves
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA  J.
Annis  Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL
60510, USA  N. Battaglia  0000-0001-5846-0411 Department of Astronomy, Cornell
University, Ithaca, NY 14853, USA  E. S. Battistelli  0000-0001-5210-7625
Physics Department, Sapienza University of Rome, Piazzale Aldo Moro 5, I-00185
Rome, Italy  E. J. Baxter  Institute for Astronomy, University of Hawai’i,
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d’Astrophysique de Paris, F-75014, Paris, France  Sorbonne Universités, UPMC
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University of Toronto, 60 St. George St., Toronto, ON M5S 3H4, Canada  D.
Brooks  Department of Physics & Astronomy, University College London, Gower
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Lokken  David A. Dunlap Department of Astronomy and Astrophysics, University
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Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
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000-0001-6606-7142 A. Astrophysics Research Centre, University of KwaZulu-
Natal, Westville Campus, Durban 4041, South Africa  B. School of Mathematics,
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###### Abstract

The kinematic and thermal Sunyaev-Zel’dovich (kSZ and tSZ) effects probe the
abundance and thermodynamics of ionized gas in galaxies and clusters. We
present a new hybrid estimator to measure the kSZ effect by combining cosmic
microwave background temperature anisotropy maps with photometric and
spectroscopic optical survey data. The method interpolates a velocity
reconstruction from a spectroscopic catalog at the positions of objects in a
photometric catalog, which makes it possible to leverage the high number
density of the photometric catalog and the precision of the spectroscopic
survey. Combining this hybrid kSZ estimator with a measurement of the tSZ
effect simultaneously constrains the density and temperature of free electrons
in the photometrically selected galaxies. Using the 1000 deg2 of overlap
between the Atacama Cosmology Telescope (ACT) Data Release 5, the first three
years of data from the Dark Energy Survey (DES), and the Baryon Oscillation
Spectroscopic Survey (BOSS) Data Release 12, we detect the kSZ signal at
4.8$\sigma$ and reject the null (no-kSZ) hypothesis at 5.1$\sigma$. This
corresponds to 2.0$\sigma$ per 100,000 photometric objects with a velocity
field based on a spectroscopic survey with 1/5${}^{\textrm{th}}$ the density
of the photometric catalog. For comparison, a recent ACT analysis using
exclusively spectroscopic data from BOSS measured the kSZ signal at
2.1$\sigma$ per 100,000 objects. Our derived constraints on the thermodynamic
properties of the galaxy halos are consistent with previous measurements. With
future surveys, such as the Dark Energy Spectroscopic Instrument and the Rubin
Observatory Legacy Survey of Space and Time, we expect that this hybrid
estimator could result in measurements with significantly better signal-to-
noise than those that rely on spectroscopic data alone.

††preprint: DES-2022-0684††preprint: FERMILAB-PUB-23-239-T

##  I Introduction

As photons from the cosmic microwave background (CMB) propagate towards Earth,
they interact with gas in the intervening space. These interactions leave
imprints on the CMB. The two interactions considered in this paper are the
thermal and kinematic Sunyaev-Zel’dovich effects (tSZ effect and kSZ effect,
respectively) [67, 68, 32]. The kSZ effect is caused when low-energy CMB
photons Compton scatter off of moving free electrons. The interaction results
in an energy shift that depends on the density and coherent motion of the free
electrons, scaling with the projected electron momentum along the line-of-
sight. The tSZ effect is also caused by the scattering of CMB photons, but is
dependent on the random thermal motion of the electrons as opposed to their
bulk motion, thus scaling with the product of the electron temperature and
density. Measurements of these effects can teach us about the distribution and
thermodynamic properties of the baryons in and between galaxies and clusters,
and the growth of cosmological structures.

Previous kSZ measurements [62, 35, 41] and recent fast radio burst
measurements (such as [43]), provide multiple low-z probes of the mean baryon
abundance that agree with the CMB and big-bang nucleosynthesis (BBN) cosmic
abundance. However, current measurements of the baryon densities within galaxy
halos only account for a portion of the overall cosmic abundance of baryons
[29, 15]. This suggests there must be additional baryons beyond the virial
radii of these halos. This uncertainty around the location of the baryons is
known as the missing baryon problem. Both SZ effects provide ways to trace
baryons in ionized gas at large halo-centric radii and potentially resolve
this problem (e.g., [46], and see [47] for a review).

Unlike X-ray surface brightness, which depends on the square of the gas
density, the surface brightness of the kSZ effect has a linear dependence on
density. This means that the kSZ effect can be used to trace low-density
regions of ionized gas that cannot be easily traced with X-ray observations;
consequently kSZ measurements are crucial for understanding galaxy evolution.
In particular, they test models of galaxy evolution and feedback that predict
different distributions of ionized gas in and around halos [9, 10].

Over the years the kSZ signal has been measured using a variety of techniques.
The pairwise-momentum estimator uses CMB temperature maps and galaxy catalogs
to measure the net difference in temperature between clusters that are moving
towards each other under the influence of gravity [32]. This estimator can
also be applied in fourier-space as demonstrated in [66]. The kSZ signal has
also been measured using projected fields [35, 25, 41, 13]. This method relies
on cross-correlating the square of a CMB temperature map with a projected
density map constructed from large-scale structure tracers. Velocity-
reconstruction stacking [61, 70, 69] is another approach that involves
stacking CMB temperature maps at the location of galaxies, weighted by the
line-of-sight velocities associated with those galaxies. This is the method
that we use for this paper.

The first measurement of the kSZ effect was presented in [32] in 2012 using
the pairwise-momentum estimator, cross-correlating Atacama Cosmology Telescope
(ACT) maps with spectroscopic galaxy measurements from the SDSS-III Baryon
Oscillation Spectroscopic Survey (BOSS) [24]. Since then, a number of
detections using the pairwise method have been made with ACT and BOSS data,
including [12] and [14]. This method was also applied to DESI imaging surveys
and Planck in [16] and to the DES redMaPPer cluster catalog and SPT data in
[63]. The first kSZ measurement with photometric data was presented in [35],
which used projected fields to measure the kSZ effect at 3.8-4.5$\sigma$
significance using Planck, _WMAP_ , and _WISE_ data.

Soon after, [65] presented the first pairwise kSZ measurement using
photometric data. This was done with maps from the South Pole Telescope and
galaxy catalogs from the Dark Energy Survey (DES). Doing so, they were able to
detect the kSZ signal at 4.2$\sigma$ and reject the null hypothesis at
2.4$\sigma$.

Recent velocity-reconstruction stacking measurements from ACT and BOSS [62]
[hereafter S21] present the highest-significance kSZ measurement to date at
7.9$\sigma$ and rule out the null hypothesis at 6.5$\sigma$.

These high-significance measurements have helped push the boundaries of what
we know about the kSZ effect, by leveraging large amounts of spectroscopic
data and high-resolution CMB maps. Except for [35, 65, 41], these measurements
have been primarily limited to spectroscopic catalogs. This is due to the fact
that measuring the kSZ effect generally depends on having very well-
constrained 3D locations of the objects in RA, declination, and redshift. For
stacking analyses, this is because it is necessary to first estimate the
radial velocities of the halos in question. These velocities are typically
inferred from the 3D density of galaxies in galaxy catalogs using the
continuity equation. Similarly, for pairwise estimators the estimator depends
on knowing the distances between pairs, and thus the 3D location of halos. In
both cases, these quantities are much more robustly inferred with high-
precision spectroscopic redshifts than with photometric redshifts. However, an
advantage of photometric surveys is that they typically have significantly
higher densities of objects than are found in spectroscopic surveys. It would
be advantageous to exploit the much denser photometric catalogs in order to
improve our ability to measure and understand the kSZ signal. This idea was
explored in [50], which used a forward modeling technique to combine
spectroscopic galaxy data from SDSS with photometric and spectroscopic
catalogs of clusters also from SDSS to measure the kSZ signal at 2$\sigma$.

In this work, we present a new hybrid estimator that involves interpolating
the velocity field constructed from spectroscopic galaxy catalogs to infer the
velocity field at the locations of galaxies in a photometric survey. This
alternative method allows us to leverage the more precise redshift
measurements from the spectroscopic catalogs and the higher density of
galaxies in the photometric catalogs. For the analysis presented here, the
hybrid estimator enables us to use 256023 photometric galaxies, whose
velocities we estimate using a significantly smaller sample of 51321
spectroscopic galaxies.

In addition to studying the kSZ effect, we include a measurement of the tSZ
effect [67], which depends on the temperature and density of the electrons. In
contrast, the kSZ effect traces the density of the electrons. By combining
these two effects, it is possible to tease out an estimate for the electron
temperature. We then go on to fit a physically motivated model to the measured
SZ signals, which can subsequently be compared to theoretical predictions from
hydrodynamical simulations. Throughout this work we compare to the results
from S21 and [5] [hereafter A21] which are companion papers that present a
similar, but independent, analysis of the kSZ and tSZ signals associated with
the BOSS CMASS objects. Overall, we find that the signals we measure, and the
models we fit to these signals, are consistent at the current level of
precision. Future, higher-precision data, may be able to probe differences in
the gas properties between different galaxy samples.

This paper is organized as follows. The data that we use for this measurement
are presented in §II. The hybrid estimator and our analysis pipeline are
presented in §III. We present our kSZ measurement in §IV and our tSZ
measurement in §V. The electron temperature profile is discussed in §VII and
modeling is discussed in §VIII. Finally, our discussion is in §IX and our
conclusion is in §X.

##  II Data

###  II.1 ACT CMB Maps

We study the profile of the gas in halos using CMB temperature maps from ACT
[28, 71]. ACT was a 6-meter telescope located in the Atacama Desert in Chile.
The telescope first started observing in 2007 and has had a series of upgrades
to its camera, the most recent of which is Advanced ACTPol [34, 36, 17, 42].

The data from these different instruments were combined as part of ACT’s Data
Release 5 (DR5) to produce coadded maps that cover nearly half the sky at
arcminute resolution [48]. The depth, resolution, and large sky coverage of
these maps are well-suited to our kSZ analyses. In particular, we focus on the
ACT DR5 2008-18 temperature maps in the f090 and f150 frequency channels.
These frequency channels are roughly centered at 98 and 150 GHz but see Figure
2 of [48] for an overview of the passbands. The ACT DR5 maps used here were
constructed by coadding multiple seasons worth of ACT data with _Planck_ data
[55]. The resulting maps have beams with full-width half-maxima of 2.1 and 1.3
arcminutes at f090 and f150, respectively.

For our core analysis, as in S21, we rely on maps containing both data
collected in the day and the night (referred to as daytime and nighttime
data); however, we also use maps that include just the nighttime data alone
for null tests. The inclusion of these extra data sets allows us to make
difference maps between the nighttime and day+night maps. These difference
maps should contain no signal and can thus be used for a null test. The sky
coverage shared by the BOSS, DES, and ACT maps mean that we primarily use data
in ACT’s Deep 56 (D56) area, which is the area of ACT’s deepest DR5 coverage,
with depths of 12–18 and 8–12 $\mu$K arcmin in f090 and f150, respectively. In
addition to these maps, we use the inverse variance maps (ivar maps), which
include an estimate of the non-atmospheric inverse variance in 1/$\mu{\rm
K}^{2}$ per pixel. These allow us to account for variations in survey depth
across the ACT sky area when estimating the kSZ signal.

For the tSZ measurement, we use the component-separated Compton-$y$ map
presented in [44], which was constructed from ACT DR4 and _Planck_ data. These
component-separated maps cover $\sim$ 2,100 deg2 at arcminute resolution and
overlap with the sky region used for this analysis. The $y$ maps are
constructed using an internal linear combination (ILC) of the ACT maps at f090
and f150, as well as Planck maps at 30, 44, 70, 100, 143, 217, 353, and 545
GHz, which fill in missing low-$\ell$ modes that ACT does not measure well and
also covers frequency channels that ACT does not observe in. We use the
Compton-$y$ maps with a fiducial cosmic infrared background (CIB) spectral
energy distribution (SED) deprojected to reduce contamination from
extragalactic dust. This choice was motivated by null tests in S21 that
demonstrated their sensitivity to CIB contamination in their tSZ measurements
(see fig 22 of S21). The CIB deprojection is described in detail in [44] but
to summarize, a modified blackbody spectrum is deprojected from the ILC map
via a constrained ILC technique [59], assuming an effective temperature of
$T_{CIB}=$ 24 K and spectral index $\beta=$ 1.2. The resulting map has a
Gaussian beam with FWHM = 2.4'.

###  II.2 BOSS CMASS Catalog

As in S21, we use the CMASS (“constant stellar mass”) sample from BOSS DR12
[21, 57]; this is a galaxy catalog from the Sloan Digital Sky Survey (SDSS).
The catalog is selected such that the stellar mass limit is constant over
redshift, and it includes galaxies with spectroscopic redshift measurements
ranging from 0.4 to 0.7 [21].

For this analysis, we limit the sample to those objects that overlap with the
DES redMaGiC and ACT DR5 sky area. This results in a sample of 51321 objects.
The redshift distribution is presented in Figure 1 and Figure 2 presents the
sky region, which has $\sim$ 6000 deg2 overlapping between BOSS and ACT but
just 1000 deg2 between all three surveys.

Figure 1: Redshift distribution of the trimmed DES redMaGiC and BOSS CMASS
catalogs. This includes only the 51321 BOSS and 256023 redMaGiC objects that
are used in our final analysis.

We use the reconstructed radial velocity measurements used in S21 and
described in [73]. The measurements use the 3D number density of galaxies,
smoothed with a fixed smoothing scale of 5$h^{-1}$ Mpc, to infer their line-
of-sight velocity. This is done by solving the continuity equation ([53],
S21):

$\bm{\nabla}\cdot\bm{v}+f\bm{\nabla}\cdot\left[\left(\bm{v}\cdot\bm{\hat{n}}\right)\bm{\hat{n}}\right]=-\frac{aHf\delta_{g}}{b}\
.$ (1)

Here the galaxy over-density is $\delta_{g}$, the logarithmic linear growth
rate is $f$, $\bm{v}$ is the velocity vector, $\bm{\hat{n}}$ is the line-of-
sight direction, $H$ is the Hubble parameter, and $b$ is the linear galaxy
bias. This method works by placing the galaxies in a 3D box that is
significantly larger than the sky area covered by the sample to map out the
density field. It then assumes linear theory such that the velocity field is a
gradient of the scalar field. Initially, this assumption is valid because the
vector component of the initial velocity field decays with the inverse of the
scale factor. At late times this assumption remains reasonable because,
although nonlinear evolution under gravity sources a vector component, this
component is small on the large scales that we probe, and so we can neglect it
here.

We interpolate the reconstructed velocity field from [73] to estimate the
velocity field at the locations traced by our photometric sample, described in
§II.3; this process is described in §III.1.

###  II.3 DES redMaGiC Catalog

DES [2] is a photometric survey in the $grizY$ bands that covers $\sim$ 5000
deg2 using the Dark Energy Camera [26], which is mounted on the Blanco 4-meter
Telescope at the Cerro Tololo Inter-American Observatory (CTIO) in the Chilean
Andes. From the full survey, DES selects red galaxies with their red-sequence
Matched-filter Galaxy Catalog algorithm (redMaGiC, [60]). This results in a
catalog of objects with optimal photometric redshift data that is well-suited
for SZ measurements.

Here, we use the DES-Year-3 (DES-Y3) high-density redMaGiC galaxy catalog,
which includes the first three years of DES data. This data has well-
calibrated photo-$z$ measurements with a mean redshift error of approximately
0.02 [64]. After limiting the catalog to the region that overlaps with the
BOSS CMASS footprint and the ACT DR5 map area we are left with 256,023 objects
(approximately five times more than in the CMASS sample) as shown in Figure 1.
This sample has a mean redshift of 0.57 and the overlapping area between the
three surveys is approximately 1000 deg2, as shown in Figure 2. In Figure 2 we
also give an outline of the sky area covered by other surveys including LSST
and DELVE (DECam Local Volume Exploration Survey) [22], which could be used to
expand upon this work in the future. Finally we estimate an average halo mass
for this sample of $M_{H}\approx 10^{13.4\pm 0.1}\,M_{\odot}$ based on [77],
which models the weak gravitational lensing signal of halos hosting redMaGiC
galaxies with a halo occupation distribution framework (see Section VIII.1 for
more details).

Figure 2: Overlap of the ACT DR5 maps (blue), the DES redMaGiC catalog (red), and the BOSS CMASS catalog (yellow), plotted in equatorial coordinates. The overlapping region for all three surveys is shown in black and constitutes 1000 deg2. We also show the outline of the upcoming LSST survey (below the white line), which covers a similar region as DELVE . For future studies we note that the overlap of ACT and BOSS constitutes 5737 deg2 and ACT, BOSS, and LSST will cover 3773 deg2. Data Set |  Total number of galaxies |  Galaxies in overlapping sky area |  Total sky area
---|---|---|---
ACT DR5 frequency maps | - | - | 18,000
ACT DR4 $y$ maps | - | - | 2,100
BOSS CMASS | 777,202 | 51,321 | 10,000
DES redMaGiC | 2,766,815 | 256,023 | 5,000
Table 1: The number of galaxies and the sky area available for each survey.
All sky areas are listed for the unmasked maps and are in deg2. The total
overlapping sky area is 1000 deg2.

##  III Analysis

We measure the kSZ effect using spectroscopic data to estimate the 3D velocity
field, photometric data to trace the location of galaxies, and CMB data to
measure the kSZ signal. This process is summarized in Figure 3 and described
in detail in the subsections that follow.

Figure 3: The hybrid estimator presented in this work relies on combining
three data sets in order to measure the kSZ effect. Here we outline the
process that goes into this measurement.

###  III.1 The Hybrid Velocity Estimator

One of the challenges associated with kSZ measurements is obtaining accurate
velocity estimates for the objects in question. Typical methods for velocity
estimation use the 3D density of objects to infer their velocities [73]. These
methods are typically limited to spectroscopic data because they rely heavily
on the accuracy of the redshift measurements. However, in many cases
photometric catalogs contain significantly more objects, so it would be
beneficial to make use of these larger catalogs.

We use the velocity reconstruction from the BOSS catalog to estimate the 3D
velocity field. We then evaluate the velocity of this field at the 3D location
of DES redMaGiC galaxies. As such, the photometric redshifts of the redMaGiC
objects are only used to determine the three-dimensional position of the
objects, which means that the relatively large photometric redshift errors (in
comparison to those from spectroscopy) only lead to small errors in the
redMaGiC velocity estimates. For comparison, directly estimating the velocity
field from photometry could result in much larger errors since it requires
taking the derivative of noisy data111Refs. [38] and [65] both studied the
effect of using photometric data in kSZ measurements using the pairwise
estimator and found that the photo-$z$ errors play a significant role in
suppressing the kSZ amplitude. We note that while the hybrid estimator
presented here provides one way of combining spectroscopic and photometric
data, a better alternative could be to model the velocity field using both
data sets and one overall forward modeling pipeline. Such an approach could
build upon the work presented in [11] and [56].. We find that, at the mean
redshift of our sample, the average photo-$z$ error corresponds to an error in
the 3D localization of less than 50 Mpc. In comparison, the velocity fields
are coherent on scales of $\sim$100 Mpc, a factor of two larger. This makes it
possible to use our hybrid estimator, which relies on the long coherence
lengths of the velocity field and well-calibrated photo-$z$, to estimate
radial velocity values for the redMaGiC galaxies from the BOSS velocity field.

We implement this method using nbodykit [31] to convert the coordinates of
each object in RA, DEC, and redshift to Cartesian positions. To do so, we
assume a fiducial $\Lambda$CDM cosmology with parameters from the Planck 2015
CMB analysis [4]222This is the same cosmology used in the BOSS velocity
reconstruction.. Next, we project the BOSS objects onto a 3D grid of voxels
that measure 10 Mpc in each dimension. We then interpolate over this regular
grid to reconstruct the velocities of the redMaGiC objects.

This interpolation process will inherently lead to some bias in the resulting
velocities. Traditionally, these biases are analyzed using simulations such as
in [73]. However, in order to perform that type of test, it would be necessary
to have a simulation that included both a CMASS-like and redMaGiC-like sample
in the same simulation. In addition to the overlapping sample, the simulation
would need to have a similar number density of objects, given that these
reconstructions rely heavily on understanding the 3D density field. Currently,
such a simulation is not available, meaning a direct analysis of the bias is
difficult, and we instead perform partial tests of the reconstruction where
possible.

One test we include is to examine the efficacy of the interpolation method we
use in our reconstruction. We do this using exclusively the BOSS CMASS
objects, for which we already have velocity estimates. We take the sample of
CMASS objects and split it into a test sample, which contains 10% of the data
and base sample, which contains the remaining 90% of the data. We then
interpolate over the base sample to reconstruct the velocities of the test
sample. We refer to the original reconstructed velocities from [73] as the
“Original Velocities” and the velocities calculated through our interpolation
method as the “Interpolated Velocities”.

Figure 4: We verify the efficacy of our interpolation method by testing our
ability to recover the velocities for the objects in the BOSS CMASS catalog.
We find that the correlation coefficient for this test is 0.95 and the bias
(defined as bias = interpolated velocities/original velocities - 1) is -0.07.

Once we have the interpolated velocities, we can measure the correlation
between the original velocities and the interpolated velocities. This process
is then repeated 100 times, randomly splitting the sample each time, and the
results are shown in Figure 4. We find that the process produces a sample with
a correlation coefficient of 0.95. We also look at the bias from this test,
defined as bias = interpolated velocities/original velocities - 1, and find
that it is -0.07. We test propagating this bias through to our kSZ measurement
below and find that the effect is an order of magnitude smaller than the
errors on our measurement.

###  III.2 Aperture Photometry and the SZ signals

We measure the stacked radial kSZ and tSZ profiles of the redMaGiC galaxies in
the ACT maps by using aperture photometry (AP) filters following the procedure
in S21. To do so, we apply AP filters of varying radii to each object, which
yield the integrated radial profiles of the signals.

To apply a filter, we integrate over a disk of angular radius $\Theta$
centered on the halo in question. We then subtract off an annulus with an
outer radius of $\sqrt{2}\Theta$ that surrounds the disk. The resulting radial
profile is given by:

$T_{AP}(\Theta)=\int_{0}^{\Theta}\delta
T(\theta)d\theta-\int_{\Theta}^{\sqrt{2}\Theta}\delta T(\theta)d\theta\,,$ (2)

where $\delta T$ is the temperature map. AP filters are particularly useful
for kSZ measurements because they reduce the noise from CMB fluctuations,
which cannot be separated from kSZ fluctuations using multifrequency
information, as both signals have the same blackbody spectrum. The AP filter
mitigates noise from primary CMB fluctuations with wavelengths longer than the
size of the filter, as these modes are effectively removed when we subtract
off the annulus.

We apply these filters by first selecting 20$\times$20 arcminute cut-outs of
the CMB maps centered on each of the DES galaxies. For the tSZ measurements,
we extract similar cut-outs from Compton-$y$ maps with the CIB deprojected.
For the kSZ measurements, we use ACT DR5 CMB maps. In both cases, we take care
to reproject our CMB maps such that the projection is centered on each galaxy.
This mitigates distortions caused by taking cut-outs of the maps from
different declinations in the Plate Carreé-projected ACT maps. We then apply
the AP filters to each of the cut-outs as well as to the inverse-variance
maps. These profiles are then used to calculate the kSZ signal, discussed in
Section IV, and the tSZ signal, discussed in Section V.

###  III.3 Covariance Matrix

For both the tSZ and kSZ effects, we calculate covariance matrices by
bootstrap resampling our data as was done in S21. To do so we generate
catalogs by selecting galaxies, with repetition, from our sample such that our
bootstrap catalog has the same number of objects as our original one. Next we
calculate the associated SZ signals by repeating the stacking and weighting
process outlined in Sections IV and V and then repeat the process 10,000
times333We found that the $\chi^{2}$ measurements associated with the
covariance matrices stabilized at 8,000 bootstrap iterations, thus we allowed
for 10,000 iterations to ensure these numbers had converged.. We then
calculate the covariance of the bootstrapped samples in order to get the
relevant covariance matrices. For the kSZ signal, we calculate the covariance
matrix of the combined f090 and f150 profiles, which we can then use to
calculate the combined signal-to-noise of our results.

Figure 5 shows the correlation matrix calculated from this covariance. The
bins at larger radii are highly correlated due to the aperture photometry
filtering. In Appendix A we test whether these high correlations affect the
expected number of degrees of freedom and conclude that the expected number of
degrees of freedom is consistent with the number of spatial degrees of
freedom.

Figure 5: The f090 and f150 correlation matrix calculated by bootstrapping the
kSZ data. This is equivalent to Figure 7 in S21 and shows similarly high
correlations at large radii.

We calculate the signal-to-noise ratio (SNR) by first calculating the
$\chi^{2}$ as:

$\chi^{2}=(d-m)^{t}C^{-1}(d-m)\,,$

where $C$ is the covariance matrix, $d$ represents the data, and $m$
represents the given model. The model is either null, when considering
rejection of the null hypothesis, or our best-fit model. To convert our
$\chi^{2}_{\rm null}$ into SNR, we first calculate the probability-to-exceed
(PTE), which we then express in terms of Gaussian standard deviations,

$(1-\textrm{PTE})=\textrm{erf}(\textrm{SNR}/\sqrt{2})\,,$ (3)

where erf refers to the error function.

For the best-fit models, we calculate the SNR using the same approach as in
S21, which is:

$\textrm{SNR}\equiv\sqrt{\Delta\chi^{2}_{\textrm{null}-\textrm{best-
fit}}}=\sqrt{\chi^{2}_{\textrm{null}}-\chi^{2}_{\textrm{best-fit}}}\,.$ (4)

This quantity corresponds to the significance of the preference for the best-
fit model over the null hypothesis.

##  IV The kSZ Signal

Since the kSZ effect is dependent on velocity, the $T_{AP}(\Theta)$ needs to
be weighted by velocity in order to avoid canceling the signal. The
oscillating signs of the line-of-sight velocity also means that by weighting
our stack by the velocity we make the estimator highly robust against
foregrounds, which are velocity independent. We use the following estimator,
which was proposed in S21:

$T_{\rm{kSZ}}(\Theta)=\frac{v^{\rm{rms}}_{\rm{rec}}}{c}\frac{\sum_{i}T_{AP_{i}}(\Theta)(v_{{\rm
rec},i}/c)w_{i}}{\sum_{i}(v_{{\rm rec},i}/c)^{2}w_{i}}\,,$ (5)

where $v_{\rm{rec},\textit{i}}$ is the mean-subtracted line-of-sight velocity
for object $i$ reconstructed with the hybrid technique and
$v^{\rm{rms}}_{\rm{rec}}$ is the RMS of this velocity. We also define inverse-
variance weights using $w_{i}=1/(\sigma_{z,i}^{2}\sigma_{m,i}^{2}$). Here we
account for the noise in the CMB maps with $\sigma_{m,i}$, which is calculated
by applying the AP filters to the ivar maps, such that each individual halo
has its own value for $\sigma_{m,i}$. This allows us to account for the large
distribution in noise across the ACT DR5 maps. The $\sigma_{z,i}$ is the
redshift error in comoving coordinates associated with each object, and it
allows us to preferentially weight objects that have low redshift errors when
stacking. We choose to combine the two weights in order to optimize for both
the CMB-map-based noise and the noise associated with the location of each
redMaGiC object.

This equation would ordinarily include a scaling factor of $\frac{1}{r_{v}}$,
which is the correlation between the true and reconstructed velocities and is
used to account for bias caused by the velocity reconstruction method. While
we know the bias for the reconstruction of the BOSS catalog, as discussed in
Section II we do not have measurements for the full bias of the velocity
reconstruction of the photometric DES data. For this reason, we include this
as a free parameter (defined as the scale factor $S$) in our model instead of
trying to estimate it and include it in our kSZ calculation. Future analyses
using this method could look at measuring this number using simulations. We
note that characterizing this factor more precisely is necessary to accurately
determine the amplitude of our profile; however, it should not affect the
profile shape or the SNR of our measurement, but instead just suppresses the
amplitude of the signal.

Excepting the inclusion of $\sigma_{z,i}$ used to account for the redshift
errors and the exclusion of the velocity bias, Equation 5 is equivalent to the
one used to calculate the kSZ signal in S21.

By applying our pipeline to the f090 and f150 maps separately, we obtain the
results shown in Figure 6.

Figure 6: Stacked profile of the kSZ signal of DES redMaGiC galaxies in units
of $\mu$K arcmin2. The scaling on the y-axis includes bias from the velocity
reconstruction that is later included as a free parameter in the model shown
in Figures 10 and 12, but not accounted for here. The solid gray line is an
NFW profile, which assumes the gas follows the dark matter. The two dashed
gray lines near the NFW profile are Gaussian profiles with FWHM of 1.3' and
2.1', respectively. The gray line that most closely follows the data
represents a Gaussian profile that is much more extended with FWHM = 6', this
means that the data prefer a gas profile that is more extended than the NFW.
These are normalized to the signal in the largest aperture for f150. We also
plot the virial radius (1.4' at $z=0.57$), calculated based on the mean mass
of the sample, added in quadrature with the beam, as a vertical blue line for
the f090 beam and as an orange line for the f150 beam.

These profiles allow us to rule out the null hypothesis at 5.1$\sigma$ for the
combined data, and 4.9$\sigma$ and 3.7$\sigma$ for the f090 and f150 bands,
respectively. For both frequencies, this data is shown with the beam
unconvolved. The presence of the beams should lead to a small difference
between the profiles at f090 and f150, despite the kSZ signal having the same
SED at both. We would expect the larger f090 beam to result in a slight
suppression of the f090 compared to the f150 data. Instead, what we see is a
slight positive fluctuation for the f090 data. This fluctuation is explored
later in this paper.

In addition to ruling out the null hypothesis, we follow S21 in comparing the
data to a simple Navarro-Frenk-White profile (NFW) [49], which assumes that
the gas follows the dark matter. To do so we use the mass-concentration
relation laid out in [23] to simulate a 3D NFW profile at the mean halo mass
of the sample, which we estimate as $10^{13.4}\,M_{\odot}$ as discussed in
Section II.3. This profile is then converted to a number density of free
electrons; to do so, we assume the gas is fully ionized with primordial helium
abundance and cosmological baryon abundance. The 3D profile is truncated at
one virial radius and then projected to a 2D plane at the mean redshift of the
sample. We then convolve it with the ACT f150 beam and apply the AP filtering.
We note that, as shown in Figure 6, the data disfavor this NFW model
$(\sqrt{\chi^{2}_{\textrm{NFW}}-\chi^{2}_{\textrm{best-fit}}}=58)$; a similar
result was shown in S21.

We also show that the data indicate the presence of an extended gas profile
compared to the NFW. This extended profile can also be seen by comparing the
data to the three Gaussian profiles shown in gray dashed lines in Figure 6.
The profiles, from highest to lowest, represent Gaussians with FWHM of 1.3',
2.1', and 6' that have been passed through the AP filters and then normalized
to the last radial bin of the f150 profile. The first two profiles approximate
the ACT beam sizes.

It is evident that the data prefer a gas profile that is more extended than
the NFW, which is consistent with predictions that the kSZ traces gas in the
warm hot intergalactic medium (WHIM) in an extended halo around galaxies [20].
It is worth noting that some of the signal we measure at large radii could
also be due to contributions from neighboring halos or nearby satellites. We
consider these neighboring halos in more detail when we model the kSZ signal
in §VIII, and discuss comparisons with the results of S21 and A21.

##  V The tSZ Signal

For the tSZ measurements, we use the same sky area and DES objects used for
the kSZ measurement in order to maintain consistency. The measurement is
simpler since it is not necessary to take into account radial velocities;
instead we just account for the noise as given by:

$T_{\rm{tSZ}}(\Theta)=\frac{\sum_{i}T_{AP_{i}}(\Theta)/\sigma_{m,i}^{2}}{\sum_{i}1/\sigma_{m,i}^{2}}\,.$
(6)

For the sake of interpretation we can convert the above $T_{\rm{tSZ}}(\Theta)$
from $y$ units to CMB temperature units by multiplying by the following:

$f_{\rm{tSZ}}(\nu)=x\textrm{coth}(x/2)-4\,,$ (7)

where $x=\frac{h\nu}{k_{B}T_{\rm{CMB}}}$. Using the above relation at f150, a
value $y=1$ corresponds to $-2.59\times 10^{6}\,\mu$K.

We note that the Compton-$y$ maps do not include inverse variance maps,
meaning that in this case $\sigma_{m,i}$ is estimated by calculating the
variance from the mean of the map for each AP filter radius and for each
object $i$.

By applying the tSZ pipeline to the Compton-$y$ map, we measure the tSZ signal
as shown in Figure 7. The tSZ signal is generally an order of magnitude
stronger than the kSZ signal, and so is much easier to detect. For our data,
we are able to rule out the null hypothesis at 15$\sigma$.

Figure 7: The tSZ profile of DES redMaGiC galaxies, measured using Compton-$y$
maps with a fiducial CIB SED deprojected. We plot the profile in dimensionless
Compton-$y$ units on the left-hand side and on the right-hand side we give the
signal in temperature units at f150. We convert from $y$ to $T_{\rm{tSZ}}$
using Equation 7. We also plot the virial radius, added in quadrature with the
beam, as a vertical blue line.

##  VI  Null Tests

We test the results of our hybrid estimator pipeline by performing a suite of
null tests in both the f090 and f150 channels. The results of these tests are
shown in Figure 8.

For our first null test (Type 1 in Figure 8) we produce 100 simulated CMB-only
maps based on a fiducial power spectrum using HEALPIX’s synfast function [30].
We then pass these maps through our pipeline (i.e., apply the AP filters,
weight with the velocities, and calculate the signal).

The second null test (Type 2) is based on shuffling the velocities associated
with each object. We perform this test on our data by repeatedly scrambling
the velocities associated with each object and then calculating the kSZ
profile with these scrambled velocities. We repeat this test 250 times for
each frequency channel and average over the results.

Finally, we perform a series of null tests based on taking difference maps,
referred to as Type 3. ACT DR5 includes maps using both day time and nighttime
data, as well as nighttime data only in both f090 and f150. We take the
nighttime and the day+night versions of the maps and reconvolve them to the
same beam in order to account for the broader day+night beam. We then take the
difference of the two maps. We also take the difference between the f090 and
f150 maps, again reconvolved to the same beam. We pass these two difference
maps through our pipeline and measure the resulting signal.

Figure 8: This figure shows the three types of kSZ null tests we perform. Type
1 (shown in blue): Gaussian map tests performed by simulating CMB maps,
passing those maps through the analysis pipeline. Type 2 (shown in orange):
shuffled velocities associated with each object. Type 3 (shown in green): we
pass difference maps, produced by taking the difference between the day +
night maps and the night only maps as well as the difference between f090 and
f150 maps, through the pipeline. Table 2 presents the PTE for each null test.

For the simulated CMB maps and the velocity shuffle null test we base the
error bars on the covariance between multiple iterations of each test. For the
difference maps, we instead rely on bootstrapping to measure the covariance
matrix, similar to what is done for the fiducial analysis. For all of the
tests, we compute the PTE (shown in Table 2), convert that to equivalent
Gaussian standard deviations, and then accept null tests if they pass at
3$\sigma$ (PTE $>$ 0.00275). We find that using this benchmark all of our null
tests pass.

We note that the f150 - f090 test passes our null test, but has a slightly low
PTE of 0.09 and visually seems to deviate the furthest from zero in Figure 8.
This seems to suggest that we may not be fully nulling out the signal when we
difference the two maps. This small difference could potentially be attributed
to small-scale issues in the DR5 maps, possibly related to contamination from
the source subtraction algorithm. Alternatively, it could indicate an issue
with the beam profiles we used, or possibly some type of contamination from a
frequency-dependent signal. Interestingly, the same null test performed in S21
showed a similar trend. We measure our lowest PTE of 0.03 for the daynight-
night null test and we see a small indication of a residual for this test in
Figure 8. This could indicate a slight calibration error, which would be due
to the fact that the DR5 maps are not fully calibrated as discussed in [48].
Additionally, as stated in [48], these DR5 maps include data products that
were not fully characterized and as such gain/beam errors on the order of a
few percent are expected in the night maps and slightly larger errors on the
order of 10% may be present for the day+night maps.

Null Test | PTE
---|---
Difference map: f150 - f090 daynight | 0.09
Difference map: f150 daynight - night | 0.03
Velocity shuffle - f090 | 0.95
Velocity shuffle - f150 | 0.86
Simulated CMB maps | 0.97
Table 2: The PTE values for the null tests shown in Figure 8. Both sets of
difference maps have slightly low PTEs. These low values could be indicative
of some inconsistencies at the map or beam level between these datasets, and
are discussed in more detail in Section VI and in Section VIII.3.

##  VII Electron Temperature Profile

The kSZ signal is dependent upon $n_{e}$, the electron number density. In
comparison, the tSZ signal depends on both $n_{e}$ and $T_{e}$, the electron
temperature. By leveraging this relationship, we can estimate the electron
temperature as a function of radius for these filters.

The electron temperature is effectively given by the ratio between the two
profiles. Or more specifically, we can use:

$T_{e}=\left(\frac{m_{e}c^{2}}{k_{B}}\right)\left(\frac{y_{AP}}{\tau_{AP}}\right)\,.$
(8)

In this equation $y_{AP}$ represents the signal from our AP filtered
Compton-$y$ maps and
$\tau_{AP}=(c/v_{\rm{rms}}^{\rm{true}})T_{\rm{kSZ}}/T_{\rm{CMB}}$, where
$v_{\rm{rms}}=314$ km/s is the RMS of the peculiar velocities along the line-
of-sight for the mean redshift of our sample ($z=0.57$), assuming a linear
relation between velocity and density (as in A21). Because the measurements of
$y_{AP}$ and $\tau_{AP}$ are done using different maps, the beam differences
between the maps needs to be accounted for. To do so, we reconvolve the f150
temperature map to the wider beam of the $y$ map before applying the filters
and calculating $\tau_{AP}$. The resulting profile has the same AP
dependencies as our kSZ and tSZ profiles, meaning that it should not be
interpreted as a radial differential profile, but instead as an integrated
mean profile.

Because the amplitude of the kSZ profile depends on the correlation
coefficient of the velocity reconstruction, this measurement of the
temperature depends on that coefficient. For that reason, we assume the
profile has some overall scaling factor ($S$, see §IV for details) that would
affect the amplitude but not the shape of the profile. If this factor was less
than 1, which is what we expect, then this would result in a reduction in the
amplitude of the temperature profile.

The temperature profile, shown in Figure 9, decreases with radius. This shape
indicates that while the virial temperature may be a reasonable estimate for
the central temperature, there is evidence that the temperature decreases
towards the edge of the halos beyond the virial radius. A similar profile was
seen in A21 and S21 when they measured the electron temperature profile for
their sample; however their measurement showed evidence for a more rapid drop
in temperature.

Figure 9: The electron temperature profile given by the ratio of the kSZ and
tSZ measurements. We also show the temperature derived from the tSZ and kSZ
models, which is based on the spread of the posteriors for those models (and
described in SectionVIII.5). For comparison, we plot the virial temperature as
a horizontal dashed line with a 1$\sigma$ error band as estimated from the
error on the virial mass. The vertical line gives the virial radius, added in
quadrature with the beam.

##  VIII  Modeling the SZ Signals

###  VIII.1 The GNFW Profile

We model the SZ signals presented in §IV and §V using a three-dimensional
generalized Navarro-Frenk-White (GNFW) model following A21 and using the
convention in [78] 444This is implemented using the Mop-c GT code
(https://github.com/samodeo/Mop-c-GT) presented in A21.. This model describes
the electron number density ($n_{e}$), which is related to the kSZ signal by
the following equation:

$\frac{\Delta
T_{\rm{kSZ}}}{T_{\rm{CMB}}}=\frac{\sigma_{T}}{c}\int_{los}e^{-\tau}n_{e}v_{p}dl\
,$ (9)

where $los$ refers to the line-of-sight, $v_{p}$ is the line-of-sight peculiar
velocity, and $\tau$ is the optical depth to Thomson scattering.

The model also describes the electron pressure profile, which is related to
the tSZ signal by:

$\frac{\Delta T_{\rm{tSZ}}}{T_{\rm{CMB}}}=f_{\rm{tSZ}}(\nu)y\ ,$ (10)

where $f_{\rm{tSZ}}(\nu)$ describes the frequency dependence of the tSZ signal
and $y$ is given by:

$y(\theta)=\frac{\sigma_{T}}{m_{e}c^{2}}\int_{los}P_{e}(\sqrt{l^{2}+d_{A}(z)^{2}|\theta|^{2}})dl\
.$ (11)

Here $m_{e}$ is the electron mass and $\sigma_{T}$ is the Thomson cross-
section.

For both profiles, we use the same formalism given in A21. Thus, the GNFW
density profile is given by:

$\displaystyle\begin{split}\rho(x)&=\rho_{0}(x/x_{c,k})^{\gamma_{k}}[1+(x/x_{c,k})^{\alpha_{k}}]^{-\frac{\beta_{k}-\gamma_{k}}{\alpha_{k}}}\,.\\\
\rho_{gas}(x)&=\rho_{cr}(z)f_{b}\rho(x)\ .\end{split}$ (12)

Here $\rho_{0}$ is the central density, $\rho_{cr}(z)$ is the critical density
of the Universe at redshift z and $f_{b}$ is the baryon fraction. We define
$x=r/R_{200}$ where $M_{200}$ is the mass within $R_{200}$, within which the
halo density is 200 times $\rho_{cr}(z)$. $x_{c,k}$ is the core scale;
$\alpha_{k}$ is the slope at $x\sim$ 1 while $\beta_{k}$ and $\gamma_{k}$ give
the slopes at $x>>1$ and $x<<1$, respectively.

The thermal pressure profile is given by:

$\displaystyle\begin{split}P(x)&=P_{0}(x/x_{c,t})^{\gamma_{t}}[1+(x/x_{c,t})^{\alpha_{t}}]^{-\beta_{t}}\,.\\\
P_{th}(x)&=P(x)P_{200}\,.\\\
P_{200}&=GM_{200}\frac{200\rho_{cr}(z)f_{b}}{2R_{200}}.\end{split}$ (13)

For these relations $P_{th}$ is the thermal pressure, the core scale is given
by $x_{c,t}$ and, just as for the density, $\alpha_{t},\beta_{t}$ and
$\gamma_{t}$ give the slopes at $x\sim 1$, $x>>1$ and $x<<1$, respectively.

We also include a two-halo term, which is used to account for excess signal
from correlated neighboring halos. We implement the method used in [74] and
A21. This method constructs the two-halo term by considering an average
neighboring halo and computing its contribution to the halo-density cross-
power spectrum555See Appendix A and the related Figure 7 of A21 for details..
This results in a two-halo term that is dependent on the mean redshift and
halo mass of our sample, which is then included in the model with an overall
amplitude factor, A2H for the density profile and AP2H for the pressure
profile. We assume an average halo mass of $M_{H}=10^{13.4}\,M_{\odot}$. This
is based on the results of [77], which quotes an average halo mass per
redshift bin. We take those averages and weight them by the number of objects
in each redshift bin from our sample to estimate the overall average halo
mass. Ideally this two-halo term would be estimated based on the mass and
redshift of each individual object in the catalog instead of the mean redshift
and mass; however, for this dataset we have a limited understanding of the
mass. This limitation means that there could be a certain amount of noise,
both in the actual mass measurement and in the redshift binning used to find
the mean mass, that could affect our estimate of the two-halo term and affect
our estimate for the amplitude of the term later on. For a full discussion of
the two-halo term, see [74] and A21.

We then combine the GNFW model with the two-halo term to arrive at our model
for the density profile,

$\rho(x)=\rho_{1H}(x)+\textrm{A}_{2H}\rho_{2H}(x)\,,$ (14)

and for the pressure profile,

$P(x)=P_{1H}(x)+\textrm{A}_{P2H}P_{2H}(x)\,.$ (15)

These models generate three-dimensional profiles that we project into a two-
dimensional plane, convolve with the ACT beam profiles, and then apply AP
filters, following the procedure used to process the data. When fitting the
models, we use Markov Chain Monte Carlo calculations (MCMC) [45] and the emcee
algorithm [27]. We assume convergence has been reached when the chain length
reaches at least 20 times the auto-correlation time, as recommended in [27].

In the following sections, we present a few different approaches to fitting
this model and in all cases we measure the SNR of the model by comparing the
preference for the model to the preference for a null signal, as written in
equation 4.

###  VIII.2 kSZ Individual Frequency Fits

We begin by fitting the f090 and f150 data independently of each other. In
order to do it we follow the procedure in A21, for the kSZ profiles, and we
fix $\gamma_{k}=-0.2$ and $\alpha_{k}=1$. From there we assume the same
uniform priors used in A21 for $x_{c,k}$ and $\beta$ but slightly wider
uniform priors for $\rho_{0}$ and $\textrm{A}_{2H}$: 1 $<$
log${}_{10}\rho_{0}<$ 10, 0.1 $<x_{c,k}$ $<$ 1.0, 1 $<\beta_{k}<10$, and 0
$<\textrm{A}_{2H}<$ 15\. The priors for $x_{c,k}$ and $\beta$ are set by
physical constraints; $x_{c,k}$ is a ratio of the core and halo radii and as
such must be a fraction and $\beta$ is the outer slope which must be larger
than the intermediate slope set by $\alpha_{k}$ and small enough to be
physically reasonable. For log${}_{10}\rho_{0}$ we set the lower bound to
avoid negative densities and then allow for a high enough upper bound for the
parameter to be constrained. The two-halo amplitude is set such that it cannot
be negative and the upper limit is set higher than that of S21 to accommodate
the data’s preference for a larger two-halo term.

To account for the potential bias in the amplitude of the kSZ profile
discussed in Section IV we include a scale parameter, $S$, with uniform prior
of 0 $<S\leq$ 1.

In Figures 10(a) and 10(b) we present the fit to the f090 and f150 data,
respectively. The best-fit models are preferred over the no-signal hypothesis
at 6.2$\sigma$ and 4.6$\sigma$, respectively. For these fits, we show the
posteriors in Figure 11. We find that our fits are mainly prior dominated,
particularly for the core radius $x_{c}$ and the outer slope $\beta_{k}$.

(a) The f090-only GNFW profile fit for the kSZ profile. (b) The f150-only GNFW
profile fit for the kSZ profile.

Figure 10: For both figures, the band gives the 1$\sigma$ spread of the
posterior, while the dashed line gives the median and the solid gives the
maximum-likelihood solution. We also show the virial radius, added in
quadrature with the beam, as a vertical blue line. At f090 and f150
respectively, the maximum-likelihood solution is preferred over the null
hypothesis at 6.2$\sigma$ and 4.6$\sigma$ as defined by Equation 4. Figure 11:
The independent f090 (red) and f150 (blue) posteriors for the kSZ models in
Figure 10. We note that, although the parameters are not well-constrained
overall, the parameters for the two frequencies are consistent.

The posteriors of the individual fits show good agreement between the f090 and
f150 data, and all the parameters are in statistical agreement between the two
fits.

###  VIII.3 kSZ Combined Frequency Fits

Given that we sample the same population of objects at both f090 and f150, it
is also possible to fit the model for the combined data. The kSZ signal is
frequency-independent in CMB thermodynamic temperature units, such that the
expected signal should be consistent between the two frequency channels. The
only difference between the two channels is the beam profiles. For this
reason, when simultaneously fitting the model to both frequency channels, we
fit one density profile to both channels convolved with the f090 and f150
beams.

When fitting both the f090 and f150 data simultaneously, we take into account
the off-diagonal blocks of the covariance matrix, which account for the
covariance between the f090 and f150 data.

GNFW |  |
---|---|---
Density | Priors | Constraints
Parameters |  | f090 | f150 | Combined
$\log_{10}\rho_{0}$ | [1, 10] | $3\pm 1$ | $3\pm 1$ | 2.1
$x_{c}$ | [0.1, 1.0] | $0.6\pm 0.3$ | $0.6\pm 0.3$ | 1.0
$\beta_{k}$ | [1, 10] | $5^{+3}_{-2}$ | $5\pm 3$ | 3.9
$\textrm{A}_{2H}$ | [0, 15] | $8\pm 4$ | $7^{+5}_{-3}$ | 4.1
$S$ | [0, 1] | $0.6^{+0.3}_{-0.2}$ | $0.5^{+0.3}_{-0.2}$ | 0.7 $\pm$ 0.1
Table 3: The marginalized constraints for the individual and combined
frequency kSZ fits. For the combined fit we fixed the four shape parameters to
the best-fit values for the minimum $\chi^{2}$ solution found using scipy
optimize.

We find that jointly fitting the two frequencies does not improve our overall
detection SNR for the best-fit model (4.8$\sigma$ compared to 6.2$\sigma$ and
4.6$\sigma$ for the f090 and f150 fits, respectively). Because this fit is
less well constrained, we elect to fix our shape parameters
($\log_{10}\rho_{0}$666We note that $\log_{10}\rho_{0}$ represents the
amplitude of the radial density profile, because this profile is used to model
a 3D gas density, projected onto a 2D surface and then filtered with AP
filters it is not the same as the amplitude factor $s$ which is used to fit
the amplitude of the final kSZ profile., $x_{c}$, $\beta_{k}$ and
$\textrm{A}_{2H}$) to the best-fit values for the minimum $\chi^{2}$ solution
found using scipy optimize and then fit an overall amplitude using emcee.
Doing so gives us a 4.8$\sigma$ detection and tightens the constraint on the
amplitude to $S=0.7\pm 0.1$. The best-fit values from the individual frequency
and combined fits are given in Table 3 and the results of the combined model
are shown in Figure 12.

In A21, the authors also fit the same GNFW profile to their measurement of the
kSZ signal; we find that our results agree with theirs at the 2$\sigma$ level.
They measured $\textrm{log}_{10}\rho_{0}=2.6^{+0.4}_{-0.3}$, $x_{c}=0.6\pm
0.3$, $\beta_{k}=2.6^{+1.0}_{-0.6}$ and $\textrm{A}_{2H}=1.1^{+0.8}_{-0.7}$.
In comparison, we find that our model for the combined frequency fit favors a
higher best-fit value for $\textrm{A}_{2H}$, a marginally higher best-fit
value for $x_{c}$, and a slightly lower best-fit value for
$\textrm{log}_{10}\rho_{0}$. For the individual frequency fits, we find that
our model again favors a higher best-fit value for $\textrm{A}_{2H}$. The
values in A21 were calculated for a portion of the BOSS CMASS halos; that
sample has a lower average redshift and a larger average mass than the DES
redMaGiC halos studied here. In general, halos of different mass and redshift
are likely to have different gas profiles, due to differing impacts of
feedback and structure growth; while these two samples’ properties are
currently consistent with one another, that may not be the case in future,
higher-precision measurements.

Finally, we consider the reduced statistical power of the joint fit compared
to the single band fits, which appears to arise from the differing central
values of the single-frequency fits. Figure 12 shows that the f090 data points
tend to lie above the f150 data, while the model expects these to be identical
except for beam convolution, which would lead to a small difference in the
opposite direction. The observed difference in signal between the two
frequency channels makes it impossible for the GNFW profile to perfectly
describe our measurements and reduces the overall detection significance in
the combined fit.

The error bars on the f090 and f150 data are such that this difference could
easily be caused by a statistical fluctuation and not be an indication of any
true tension beyond noise. Alternatively, this tension could point to a
systematic error that we have not been able to identify. Such a systematic
error could also be the reason for the marginally passing null test between
the f090 and f150 data described in Section VI.

Figure 12: The joint fit to the f090 and f150 data. The band represents the
1$\sigma$ posterior spread of the fits. The best-fit model given by the solid
lines is detected at the 4.8$\sigma$ level. We find that the model expects a
stronger signal at f150, while the data has a stronger signal at f090. This
tension is interesting to consider and is discussed more throughout the paper
(see Sec VIII.3 and VI), however, because the f090 and f150 error bars overlap
for all but one bin, the significance of the tension is minimal.

###  VIII.4 Modeling the tSZ signal

For the tSZ profile, we fix $\gamma_{t}=-0.3$ and we set $x_{c,t}$ using the
results from [8].

We then assume uniform priors of: 0.1 $<P_{0}<$ 30, 0.1 $<\alpha_{t}<$2, 1
$<\beta_{t}<$ 10, and 0 $<\textrm{A}_{P2H}<$ 5.

The results for the tSZ parameter constraints are given in Table 4 and Figure
13. The model is then plotted in Figure 14.

We find that our model is favored over null at 16.2$\sigma$. This higher
detection is due mainly to the fact that the tSZ signal is much stronger than
the kSZ signal, which in turn makes it much easier to fit our model. While the
model provides a good fit to the data we do see some evidence for an increase
in the signal (in units of $\mu$K arcmin2) at large radii. This increase is
not yet statistically significant due to the large error bars at these radii,
but would be interesting to study in more detail in the future.

As has been seen in previous tSZ studies, we note the degeneracy between
$\beta_{t}$ and $P_{0}$. There is also a degeneracy between $\beta_{t}$ and
$\alpha_{t}$ that is observable in the measurements from A21. We find that our
fits are completely consistent with A21. The only deviation we see is for our
measurement of the two-halo amplitude, for which they found a 1.8$\sigma$
preference for a non-zero two-halo term, but for which we find no evidence.
However, they measured the tSZ signal using both the ILC maps (as done in this
analysis) and single-frequency CMB maps combined with a dust model. Their
fiducial analysis was based on the CMB maps + dust model and when compared to
that analysis we see the difference mentioned above. However, if we compare to
their measurements done using the same ILC maps used for this analysis, we
find that this slight tension in the two-halo amplitude disappears.

Figure 13: The GNFW pressure profile posteriors. Figure 14: GNFW model of the tSZ profile. The gray band gives the $1\sigma$ spread of the posterior, while the dashed blue line represents the 50th percentile and the solid blue curve gives the maximum-likelihood solution. We plot the profile in Compton-$y$ units on the left-hand side, and on the right-hand side we give the signal in temperature units at 150 GHz. We also plot the virial radius, added in quadrature with the beam, as a vertical blue line. We measure the SNR of the best-fit model to be $\sqrt{\chi^{2}_{\rm null}-\chi^{2}_{\rm best-fit}}=16.2$. GNFW Pressure Parameters | Priors | Constraints
---|---|---
$P_{0}$ | [0.1, 30] | $2.0^{+11.0}_{-1.0}$
$\alpha_{t}$ | [0.1, 2] | $0.9^{+0.6}_{-0.4}$
$\beta_{t}$ | [1, 10] | $4.0^{+3.0}_{-2.0}$
$\textrm{A}_{P2H}$ | [0, 5] | $0.1\pm 0.1$
Table 4: The marginalized constraints from our MCMC for the tSZ profiles.

We leave a more extensive analysis of the SZ signals to future work. These
analyses could include comparing the results of these models to simulated
galaxies. Such comparisons could result in improving our understanding of
feedback mechanisms and galaxy evolution.

###  VIII.5 Modeled Electron Temperature Profile

Given a model for the kSZ profile as well as one for the tSZ profile, it is
also possible to extract an electron temperature profile from the two models
as was done in A21. We use the same process, as outlined in Section VII. The
temperature profile is shown in Figure 9.

We find that the temperature profile generated by the models agrees well with
the data, and all the data points lie within 1$\sigma$ of the model. We do
notice a slightly higher temperature is preferred by the data over the model
for the central bins with radii between 3' and 4.5'. This effect is also seen
in the Compton-$y$ data, which prefers a slightly higher value for Compton-$y$
than the model for these same bins.

Both the model and the data point to a temperature profile that decreases at
higher radii, such that the temperature drops below the virial temperature
beyond the virial radius. We caution that this profile is still an aperture
photometry filtered profile, meaning it should not be interpreted as a radial
temperature profile but instead can be thought of as an integrated profile.
This suggests that the underlying radial temperature profile should drop off
more steeply than the aperture photometry filtered profile shown here.

##  IX  Discussion

In this paper, we present a new hybrid estimator that results in a 4.8$\sigma$
detection of the kSZ effect. We expect our SNR to scale with the square root
of the sky area and for it to increase with more objects, although simulations
would be needed to determine a more exact scaling. For this reason, we compare
the SNR per object to a recent measurement done using exclusively
spectroscopic data in S21. The comparison, shown in Table 5, demonstrates
that, while our signal-to-noise is lower than that of the spectroscopic
measurement, our method still performs well. This is an indication that this
estimator will be an important tool for future high statistical significance
kSZ studies.

Paper |  Sky area deg2 |  Number of Spectroscopic Objects |  Number of Photometric Objects | SNR |  SNR per 100,000 objects
---|---|---|---|---|---
This work | 1000 | 51,000 | 256,000 | 5.1$\sigma$ | 2.0$\sigma$
S21 | 4000 | 311,000 | 0 | 6.5$\sigma$ | 2.1$\sigma$
Table 5: Comparison of these results to those of S21, which used the same
stacking method but with exclusively spectroscopic data. In both cases, we
take the SNR for rejecting the null hypothesis (S21 also quotes an SNR for
their best-fit model of 7.9$\sigma$). To calculate the SNR per 100,000 objects
we consider just the objects that are used for the stacking analysis, for this
work that is the 256,000 photometric objects and for S21 it is the 311,000
spectroscopic objects. We find that our SNR per object is lower than in S21,
but still competitive when taking into account potential gains in the number
of objects available for future photometric surveys.

In addition to the new hybrid kSZ estimator and the kSZ detection, we also
measure the tSZ signal associated with these galaxies, the electron
temperature profile and fit a GNFW profile to the SZ profiles. These
measurements complement our main kSZ measurement and allow us to further probe
the distribution of gas in these halos.

We find evidence for an extended gas profile with mean temperature close to
the virial temperature expected for halos of this size, in agreement with S21
and A21. Understanding these density profiles is a key ingredient to modeling
galaxy feedback mechanisms, and the density profiles can be compared to
simulations to determine whether simulations predict similar gas profiles to
those detected via SZ measurements (see e.g. [39] for an example). This type
of study was done in A21 where they found that at larger radii the measured
pressure profiles are significantly higher than those predicted by
simulations. Here, we expanded the work done by A21 by measuring and modeling
density and pressure profiles for a different sample of galaxies. Because a
comparison to simulations would depend on selecting a sample that is
representative of the catalog used for this work, a precise, direct comparison
between A21 and this work is not possible. A qualitative comparison indicates
that the density profiles measured in this work are similar to those of A21.

Future studies will continue to improve upon these measurements and our
ability to measure the gas profiles of high-redshift galaxies. Going forward,
we expect the signal-to-noise of our kSZ measurements to scale as the square
root of the sky area, or the square root of the number of objects times the
mass of the objects for surveys with different object densities and mass
distributions. Additionally, at small scales, we can expect to see some
improvements as the depth of our CMB maps increase.

We can also consider the improvements we might see with future photometric and
spectroscopic catalogs. Using LSST [19] and DESI [18] in the future would mean
an increase in both depth and sky fraction. DESI projects they will observe 30
million galaxies, while LSST forecasts 20 billion. In terms of density, this
means that DESI would observe $\sim$ 2000 objects per deg2 while LSST will
observe $\sim$ 650,000 objects per deg2. Because photometric surveys often
observe less massive objects, and the kSZ signal scales with mass, a direct
comparison based on the number of objects is not accurate. However, due to the
significantly higher density of the photometric catalogs, the inclusion of
this data should still result in gains over using exclusively the
spectroscopic data. In addition to the added depth, these surveys should have
of order 6,000 deg2 of overlap with Simons Observatory (SO) [3] and 7,000 deg2
with CMB-S4 [1, 51]. The hybrid estimator presented here is one method that
could be used to take advantage of this data in the future.

The signal-to-noise of this measurement is also dependent on the depth of the
CMB maps available to us. For this study, we have used ACT DR5 maps, which
include data through 2018 from ACT. With subsequent seasons of data from ACT
and future measurements from SO and CMB-S4 we expect to see improvements to
the CMB maps that will translate to improvements for these kSZ measurements.

We note that the methodology behind the velocity reconstruction for the
spectroscopic data from BOSS is constantly evolving and improving (see e.g.,
[40]). These improved velocity reconstructions will also lead to improvements
with this technique.

##  X  Conclusion

High signal-to-noise kSZ measurements depend on having both high density
galaxy catalogs and high-quality velocity reconstruction. Using our new hybrid
kSZ estimator, we demonstrate that those elements can come from different
datasets without having a substantially negative impact on the measurement. We
also present a complementary measurement of the tSZ signal and constraints on
the thermodynamics of the DES redMaGiC halos. Finally, we use the two SZ
signals to extract an electron temperature profile.

Our hybrid kSZ estimator uses the spectroscopic catalog to estimate the
underlying velocity field and then interpolates over that field to reconstruct
the line-of-sight velocity data for the photometric catalog. From there, we
apply AP filters to the ACT DR5 CMB maps at the location of the objects and
weight the profiles by the reconstructed velocity. This pipeline leverages the
more precise spectroscopic catalog while still taking advantage of the greater
depth of the photometric catalog. While this technique does not yet result in
the highest signal-to-noise, it does establish a new method that, when
expanded to larger regions, should prove to be a valuable technique.

Going forward, this method will make it possible to leverage the incredible
depth of photometric surveys to probe the gas profiles of galaxies in
increasing detail, and will shed new light on the evolution of galaxies and
galaxy clusters.

##  XI  Acknowledgments

ACT operated in the Parque Astronómico Atacama in northern Chile under the
auspices of the Agencia Nacional de Investigación y Desarrollo (ANID). Support
for ACT was through the U.S. National Science Foundation through awards
AST-0408698, AST-0965625, and AST-1440226 for the ACT project, as well as
awards PHY-0355328, PHY-0855887 and PHY-1214379. Funding was also provided by
Princeton University, the University of Pennsylvania, and a Canada Foundation
for Innovation (CFI) award to UBC. The development of multichroic detectors
and lenses was supported by NASA grants NNX13AE56G and NNX14AB58G. Detector
research at NIST was supported by the NIST Innovations in Measurement Science
program. Computing for ACT was performed using the Princeton Research
Computing resources at Princeton University, the National Energy Research
Scientific Computing Center (NERSC), and the Niagara supercomputer at the
SciNet HPC Consortium.

MMK is supported by the NSF Graduate Research Fellowship under Grant No.
DGE-1256260. JCH acknowledges support from NSF grant AST-2108536, NASA grants
21-ATP21-0129 and 22-ADAP22-0145, DOE grant DE-SC00233966, the Sloan
Foundation, and the Simons Foundation. CS acknowledges support from the
Agencia Nacional de Investigación y Desarrollo (ANID) through FONDECYT grant
no. 11191125 and BASAL project FB210003. This work was supported by a grant
from the Simons Foundation (CCA 918271, PBL). KM acknowledges support from the
National Research Foundation of South Africa. EC acknowledges support from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 849169).

We would also like to gratefully acknowledge the many publicly available
software packages that made this work possible. We relied heavily on emcee
[27], Mop-c-GT 777https://github.com/samodeo/Mop-c-GT A21, pixell
888https://github.com/simonsobs/pixell, and Astropy,999http://www.astropy.org
a community-developed core Python package for Astronomy [6, 7]. We also used
HEALPix [30] and the python wrapper for HEALPix which is healpy [79]. In
addition to these we made use of libsharp [58], the matplotlib [37] package,
numpy [52, 72, 33], pandas [54, 76] and scipy [75].

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##  Appendix A Correlation and Degrees of Freedom

In order to convert the $\chi^{2}$ measurements in this paper to signal-to-
noise measurements, we assume our kSZ measurement has 18 degrees of freedom,
consistent with the number of spatial bins used. However, large correlations
in the covariance matrix could mean that the true number of degrees of freedom
is not consistent with the spatial degrees of freedom. This would then bias
the PTEs for both the measurements and the null tests. Here, we examine this
possibility by simulating the $\chi^{2}$ distribution associated with the
covariance matrix. To do so, we repeatedly draw random samples from the
multivariate normal distribution described by the covariance matrix. We repeat
this process 10,000 times and calculate the $\chi^{2}$ for each sample. The
result is shown in Figure 15. We find that the mean $\chi^{2}=18$, which is
consistent with the number of spatial degrees of freedom.

Figure 15: The probability density function of the simulated $\chi^{2}$
associated with the kSZ covariance matrix. The mean of these simulations is 18
and is consistent with the number of spatial degrees of freedom.