$\displaystyle\begin{split}\langle
E_{rr^{\prime}}(\mathcal{D})\rangle_{\mathcal{D},\bm{w}^{*}}(\rho=1)=&\frac{\gamma_{rr^{\prime}}}{1-\gamma_{rr^{\prime}}}\zeta^{2}+\frac{1}{M(1-\gamma_{rr^{\prime}})}\left(\bm{1}_{M}^{\top}\bm{\Sigma}_{s}\bm{1}_{M}\right)\\\
&-\frac{1}{M(1-\gamma_{rr^{\prime}})}\bm{1}_{M}^{\top}\bm{\Sigma}_{s}\left(\frac{1}{\nu_{rr}}\hat{q}_{r}\bm{A}_{r}^{\top}\bm{G}_{r}^{-1}\bm{A}_{r}+\frac{1}{\nu_{r^{\prime}r^{\prime}}}\hat{q}_{r^{\prime}}\bm{A}_{r^{\prime}}^{\top}\bm{G}_{r^{\prime}}^{-1}\bm{A}_{r^{\prime}}\right)\bm{\Sigma}_{s}\bm{1}_{M}^{\top}\\\
&+\frac{\hat{q}_{r}\hat{q}_{r^{\prime}}}{M(1-\gamma_{rr^{\prime}})}\frac{1}{\sqrt{\nu_{rr}\nu_{r^{\prime}r^{\prime}}}}\bm{1}_{M}^{\top}\bm{\Sigma}_{s}\bm{A}_{r}^{\top}\bm{G}_{r}^{-1}\tilde{\bm{\Sigma}}_{rr^{\prime}}\bm{G}_{r^{\prime}}^{-1}\bm{A}_{r^{\prime}}\bm{\Sigma}_{s}\bm{1}_{M}\end{split}$
(189)

Calculating these contractions and traces and expanding to leading order in
$M$, we get:

$\begin{split}\langle
E_{rr^{\prime}}(\mathcal{D})\rangle_{\mathcal{D},\bm{w}^{*}}(\rho=0)=\frac{s(1-c)}{1-\gamma_{rr^{\prime}}}&\left(1-\frac{s(1-c)\nu_{rr}\hat{q}_{r}}{\nu_{rr}+(s(1-c)+\omega^{2})\hat{q}_{r}}-\frac{s(1-c)\nu_{r^{\prime}r^{\prime}}\hat{q}_{r^{\prime}}}{\nu_{r^{\prime}r^{\prime}}+(s(1-c)+\omega^{2})\hat{q}_{r^{\prime}}}\right.\\\
&\left.+\frac{s(1-c)(s(1-c)+\omega^{2})\nu_{rr^{\prime}}\hat{q}_{r}\hat{q}_{r^{\prime}}}{(\nu_{rr}+(s(1-c)+\omega^{2})\hat{q}_{r})(\nu_{r^{\prime}r^{\prime}}+(s(1-c)+\omega^{2})\hat{q}_{r^{\prime}})}\right)\\\
&+\frac{\gamma_{rr^{\prime}}}{1-\gamma_{rr^{\prime}}}\zeta^{2}\end{split}$
(190)

$\begin{split}\langle
E_{rr^{\prime}}(\mathcal{D})\rangle_{\mathcal{D},\bm{w}^{*}}(\rho=1)=\frac{1}{1-\gamma_{rr^{\prime}}}&\left(\frac{s(1-c)(\nu_{rr^{\prime}}-\nu_{rr}\nu_{r^{\prime}r^{\prime}})+\omega^{2}\nu_{rr^{\prime}}}{\nu_{rr}\nu_{r^{\prime}r^{\prime}}}\right)+\frac{\gamma_{rr^{\prime}}}{1-\gamma_{rr^{\prime}}}\zeta^{2}\end{split}$
(191)

Taking $\lambda\rightarrow 0$ we get the ridgeless limit:

$\gamma_{rr^{\prime}}\to\frac{4\alpha\nu_{rr^{\prime}}}{(\alpha+\nu_{rr}+|\alpha-\nu_{rr}|)(\alpha+\nu_{r^{\prime}r^{\prime}}+|\alpha-\nu_{r^{\prime}r^{\prime}}|)}\qquad(\lambda\to
0)$ (192)

$\langle
E_{rr^{\prime}}(\mathcal{D})\rangle_{\mathcal{D},\bm{w}^{*}}(\rho=0)=\frac{1}{1-\gamma_{rr^{\prime}}}F_{0}(\alpha)+\frac{\gamma_{rr^{\prime}}}{1-\gamma_{rr^{\prime}}}\zeta^{2}\quad(r\neq
r^{\prime})$ (193)

where

$F_{0}(\alpha)\equiv\left\\{\begin{array}[]{lr}\frac{(c-1)s\left(\nu_{r}\nu_{r^{\prime}}((2\alpha-1)(c-1)s+\omega^{2})-\alpha^{2}(c-1)s\nu_{rr^{\prime}}\right)}{\nu_{r}((c-1)s-\omega^{2})\nu_{r^{\prime}}},&\text{if
}\alpha\leq\nu_{rr}\leq\nu_{r^{\prime}r^{\prime}}\\\
\frac{(c-1)s\left(\nu_{r^{\prime}}\left((c-1)s\nu_{r}+(\alpha-1)(c-1)s+\omega^{2}\right)-\alpha(c-1)s\nu_{rr^{\prime}}\right)}{((c-1)s-\omega^{2})\nu_{r^{\prime}}},&\text{if
}\nu_{rr}\leq\alpha\leq\nu_{r^{\prime}r^{\prime}}\\\
\frac{(c-1)s\left((c-1)s\nu_{r^{\prime}}-cs\nu_{rr^{\prime}}+(c-1)s\nu_{r}-cs+s\nu_{rr^{\prime}}+s+\omega^{2}\right)}{(c-1)s-\omega^{2}},&\text{if
}\nu_{rr}\leq\nu_{r^{\prime}r^{\prime}}\leq\alpha\end{array}\right\\}$ (194)

$\langle
E_{rr^{\prime}}(\mathcal{D})\rangle_{\mathcal{D},\bm{w}^{*}}(\rho=1)=\frac{1}{1-\gamma_{rr^{\prime}}}\left(\frac{s(1-c)(\nu_{rr^{\prime}}-\nu_{rr}\nu_{r^{\prime}r^{\prime}})+\omega^{2}\nu_{rr^{\prime}}}{\nu_{rr}\nu_{r^{\prime}r^{\prime}}}\right)+\frac{\gamma_{rr^{\prime}}}{1-\gamma_{rr^{\prime}}}\zeta^{2}\qquad(\lambda\to
0)$ (195)

###  H.3 Optimal Regularization

Here, we derive the “locally” optimal regularization which minimizes the
prediction error of the ensemble members independently. Equivalently we derive
the optimal regularization for a single readout ($k=1$) or the regularization
which minimizes the diagonal terms $E_{rr}$ of the generalization errror. By
differentiating $E_{rr}$ with respect to the regularization $\lambda$, one can
show that for the optimal regularization, we will have

$\displaystyle\frac{1}{(1-\gamma_{rr})}\frac{dS_{r}}{d\lambda}\left[\frac{a^{2}\nu_{rr}}{\alpha(1-\gamma_{rr})}S_{r}\left((1-\rho^{2})I_{rr}^{0}+\rho^{2}I_{rr}^{1}+\zeta^{2}+\eta_{r}^{2}\right)+(1-\rho^{2})s^{2}(1-c^{2})\nu_{rr}(aS_{r}-1)\right]=0$
(196)

It is easy to show that $\frac{dS_{r}}{d\lambda}$ cannot be equal to $0$.
Setting the term in brackets to zero and solving for $\lambda$, we find with
the aid of the computer algebra system Mathematica [54]:

$\lambda^{\star}=a\left(\frac{a\left(\nu_{rr}(\zeta^{2}+\eta^{2})+a\rho^{2}+s(1-c)\nu_{rr}(1-2\rho^{2})\right)}{(1-c)^{2}\nu_{rr}^{2}\left(1-\rho^{2}\right)s^{2}}-1\right),\qquad(0<c\leq
1)$ (197)

In the limiting case of isotropic data ($c=0$), the optimal regularization
reduces to:

$\lambda^{\star}=\frac{1}{\nu_{rr}}(\zeta^{2}+\eta^{2})+s\left(\frac{1}{\nu_{rr}}-1\right),\qquad(c=0)$
(198)

We note that while generalization error is discontinuous at $c=0$, this result
can be quickly obtained from eq. 197 by noticing that the formula for the
generalization error with $0<c\leq 1$ reduces to the formula for
generalization error with $c=0$ when we set $\rho=c=0$ (when $c=0$, the
generalization error does not depend on $\rho$ because there is no special
direction in the data).

###  H.4 Homogeneous Ensembling with Resource Constraints

We make further simplifications in the special case where
$\lambda_{r}=\lambda$, $\eta_{r}=\eta$, $\nu_{rr}=\frac{1}{k}$ for all
$r=1,\dots k$. For simplicity, we also set $\nu_{rr^{\prime}}=0$ for all
$r\neq r^{\prime}$ so that ensemble members sample mutually exclusive subsets
of the features. We will consider both the ridgeless limit $\lambda\to 0$ and
the case of “locally optimal” regularization $\lambda=\lambda^{*}$ (see eq.
197). Concretely, we show here that under these special cases, generalization
error has the forms given in eqs. 25, 26.

We start by analyzing the saddle-point equations 111, 112. Note that in this
special case the saddle point equations will be identical for all
$r=1,\dots,k$. We will therefore suppress the $r$ index. Solving this
quadratic system of equations explicitly, we encounter the following radical,
which we assign variable name $x$:

$x\equiv\sqrt{(a\alpha-a\nu+\lambda\nu)^{2}+4a\lambda\nu^{2}}$ (199)

In order to simplify this radical to begin extracting the factor of $s(1-c)$
that appears in eqs. 25, 26, we define a reduced regularization:

$\Lambda\equiv\frac{\lambda}{s(1-c)}$ (200)

And substitute $\Lambda$, $H$, $W$, and $Z$ (recall eq. 24) for $\lambda$,
$\eta$, $\omega$, and $\zeta$, giving

$\displaystyle x$ $\displaystyle=s(1-c)\mathcal{X}(\alpha,\nu,W,\Lambda)$
(201) $\displaystyle\text{ where }\mathcal{X}(\alpha,\nu,W,\Lambda)$
$\displaystyle\equiv\sqrt{(\alpha(W+1)+\nu(\Lambda-W-1))^{2}+4\Lambda\nu^{2}(W+1)}$
(202)

making the same substitutions, we can then write

$\displaystyle\hat{q}$ $\displaystyle=\left[s(1-c)\right]^{-1}\mathcal{Q}$
(203) $\displaystyle\text{where}\quad\mathcal{Q}$
$\displaystyle\equiv\frac{2\alpha\nu}{(-\alpha(W+1)+\nu(\Lambda+W+1)+X)}$
(204)

Continuing to make substitutions in this manner, we arrve at:

$\displaystyle S$ $\displaystyle=[s(1-c)]^{-1}\mathcal{S}$ (205)
$\displaystyle\text{where}\quad\mathcal{S}$
$\displaystyle\equiv\frac{\mathcal{Q}}{\nu+\mathcal{Q}(1+W)}$ (206)

Finally, we may express the generalization error in a reasonably compact form.
In the special case at hand, the generalization error is written:

$E_{k}=\frac{1}{k}E_{rr}(\nu_{rr}=\frac{1}{k})+\frac{k-1}{k}E_{rr^{\prime}}(\nu_{rr^{\prime}}=\frac{1}{k}\delta_{rr^{\prime}})$
(207)

Substituting 205 for S in eq. 16, and making further substitutions as
necessary, we arrive at:

$\displaystyle
E_{rr}(\nu_{rr}=\frac{1}{k})=s(1-c)\mathcal{E}_{rr}(k,\alpha,\rho,\Lambda,H,W,Z)$
(208)

where

$\displaystyle\mathcal{E}_{rr}$ $\displaystyle=\frac{N_{1}+N_{2}}{D}$ (209)
$\displaystyle N_{1}$
$\displaystyle=\alpha(H+1)k+\mathcal{S}(\mathcal{S}(W+1)(\alpha+WZ+Z)-2\alpha)$
(210) $\displaystyle N_{2}$
$\displaystyle=\alpha\rho^{2}(k-\mathcal{S})(W(k+\mathcal{S})+k+\mathcal{S}-2)$
(211) $\displaystyle D$ $\displaystyle=\alpha k-\mathcal{S}^{2}(W+1)^{2}$
(212)

We also obtain

$E_{rr^{\prime}}(\nu_{rr^{\prime}}=\frac{1}{k}\delta_{rr^{\prime}})=s(1-c)\mathcal{E}_{rr^{\prime}}(k,\alpha,\rho,\Lambda,H,W,Z)$
(213)

where

$\displaystyle\mathcal{E}_{rr^{\prime}}$
$\displaystyle=\frac{2\left(\rho^{2}-1\right)\mathcal{S}}{k}-2\rho^{2}+1$
(214)

Combining these, we obtain the error of the ensemble as:

$\displaystyle E_{k}$
$\displaystyle=s(1-c)\mathcal{E}(k,\alpha,\rho,\Lambda,H,W,Z)$ (215)

where:

$\displaystyle\mathcal{E}=\frac{1}{k}\mathcal{E}_{rr}+\frac{k-1}{k}\mathcal{E}_{rr^{\prime}}$
(216)

These result are derived in the Mathematica notebook titled
“EquiCorrParameterReduction.nb” included with the available code. It follows
from eq. 215 that when $\lambda=0$ error can be written in the form of eq. 22.
It follows from equations 215, 213 that at “locally optimal” regularization
$\lambda^{*}$, error can be written in the form of eq. 23. To see this, note
that the reduced regularization $\Lambda^{*}$ which minimizes $E_{rr}$ will
only depend on the other arguments of $\mathcal{E}_{rr}$. Full expresions for
the generalization error in the case $\lambda\to 0$ and $\lambda=\lambda^{*}$
can be found in the mathematica notebooks “EquiCorrPhaseDiagram_ZeroReg.nb”
and “EquiCorrPhaseDiagram_LocalReg.nb” included with the available code. These
equations are long and difficult to interperet – nor are they directly used in
our code – and so are omitted here.

####  H.4.1 The Intermediate to Noise-Dominated Transition

The transition between the intermediate regime where $1<k^{*}<\infty$ and the
noise-dominated regime where $k^{*}=\infty$ can be studied analytically
relatively painlessly in the ridgeless limit $\lambda\to 0$. The strategy we
employ to determine this phase boundary is to examine the large-$k$ asymptotic
expansion of $E_{k}$ to determine whether the error approaches its asymptotic
value from below or above. If $E_{k}$ approaches $E_{\infty}$ from below, then
$k^{*}$ must be finite. If $E_{k}$ approaches $E_{\infty}$ from above, then
$k^{*}$ may be infinite – however, there is still the possibility of
$k^{*}<\infty$ if $E_{k}$ is non-monotonic in $k$. In practice, we check the
values of $E_{k}$ for $k=1,2,\dots,100$ and $k=\infty$.

Setting $\Lambda=0$ and expanding $E_{k}$ around $k=\infty$, we find:

$\frac{E_{k}}{s(1-c)}=1-\rho^{2}+\rho^{2}W+\frac{\left((1+W)^{2}\rho^{2}+\alpha(-2+H+HW+2\rho^{2})\right)}{(1+W)\alpha
k}+\mathcal{O}\left(\frac{1}{k^{2}}\right)$ (217)

Setting the coefficient of $k^{-1}$ to zero, we find the phase boundary as:

$\alpha=\frac{(1+W)^{2}\rho^{2}}{2(1-\rho^{2})-H(1+W)}$ (218)

This equation explains the shapes of the boundaries between the intermediate
and noise-dominated regions in the phase diagrams of fig. 4 with $\lambda=0$
(see black dotted lines in panels b, c, d).

An analytical formula for the boundary between the intermediate and noise-
dominated regimes at locally optimal regularization $\lambda=\lambda^{*}$
cannot be easily obtained. To understand why, we can asses the large-$k$
asymptotic expansion of the generalization error at locally optimal
regularization:

$E_{k}(\lambda=\lambda^{*})=s(1-c)\left[1-\rho^{2}+\rho^{2}W+\frac{H}{k}\right]+\mathcal{O}\left(\frac{1}{k^{2}}\right)$
(219)

This shows that at large $k$, when $H>0$, error always approaches its
asymptotic value from above (recall that when $H=0$ we always have $k^{*}=1$,
so that there is no phase boundary unless $H>0$). Thus, determining the phase
boundary requires determining the value of $k$ which minimizes $E_{k}$, which
is not analytically tractable.

###  H.5 Infinite Data Limit

In this section we consider the behavior of generalization error in the
equicorrelated data model as $\alpha\rightarrow\infty$ while keeping the
$\lambda\sim\mathcal{O}(1)$. For simplicity, we assume $\nu_{rr^{\prime}}=0$
for $r\neq r^{\prime}$, isotropic features ($c=0$), no feature noise
($\omega=0$) and uniform readout noise $\eta_{r}=\eta$ as in main text Fig. 3.
This limit corresponds to data-rich learning, where the number of training
examples is large relative to the number of model parameters. In this case,
the saddle point equations reduce to:

$\displaystyle\hat{q}_{r}$ $\displaystyle\to\frac{\alpha}{\lambda}$ (220)
$\displaystyle q_{r}$ $\displaystyle\to\frac{\nu_{rr}\lambda}{\alpha}$ (221)

In this limit, we find that $\gamma_{rr^{\prime}}\to 0$. Using this, we can
simplify the generalization error as follows:

$E_{g}=\frac{1}{k^{2}}\sum_{rr^{\prime}=1}^{k}E_{rr^{\prime}}=s\left[1-\left(2-\frac{1}{k}\right)\left(\frac{1}{k}\sum_{r=1}^{k}\nu_{rr}\right)\right]+\frac{\eta^{2}}{k}$
(222)

Interestingly, we find that the readout error in this case depends on the
subsampling fractions $\nu_{rr}$ only through their mean. Therefore, with
infinite data, there will be no distinction between homogeneous and
heterogeneous subsampling.

##  Appendix I Theoretical Learning Curves and Optimal Subsampling Phase
Diagrams

Here, we provide additional learning curves and phase diagrams of $k^{*}$ such
as those in Fig. 4a,b,c,d, exploring more parameter values for the task-model
alignment $\rho$ and the Reduced noises $H$, $W$, and $Z$. Generalization
errors are calculated for a homogeneous ensemble of $k$ linear regressors, as
described in sections 5, H.4. We also show diagrams of generalization error
$E_{k^{*}}$.

Figure S6:  (a) Optimal ensemble size $k^{*}$ (eqs. 25, 26) in the parameter
space of sample size $\alpha$ and reduced readout noise scale $H$ setting
$W=Z=0$. Grey indicates $k^{*}=1$ and white indicates $k^{*}\to\infty$, with
intermediate values given by the colorbar. Appended vertical bars show
$\alpha\to\infty$. $\rho$ and $\lambda$ indicated in panel titles.
$\lambda=\lambda^{*}$ denotes the ‘locally optimal” regularization (see
section H.3) (b) Optimal generalization error $E_{k^{*}}$ for the same
parameter values in (a). Figure S7:  (a) Optimal ensemble size $k^{*}$ (eqs.
25, 26) in the parameter space of sample size $\alpha$ and reduced feature
noise scale $W$ setting $H=Z=0$. Grey indicates $k^{*}=1$ and white indicates
$k^{*}\to\infty$, with intermediate values given by the colorbar. Appended
vertical bars show $\alpha\to\infty$. $\rho$ and $\lambda$ indicated in panel
titles. $\lambda=\lambda^{*}$ denotes the ‘locally optimal” regularization
(see section H.3) (b) Optimal generalization error $E_{k^{*}}$ for the same
parameter values in (a). Figure S8:  (a) Optimal ensemble size $k^{*}$ (eqs.
25, 26) in the parameter space of sample size $\alpha$ and reduced label noise
scale $Z$ setting $H=W=0$. Grey indicates $k^{*}=1$ and white indicates
$k^{*}\to\infty$, with intermediate values given by the colorbar. Appended
vertical bars show $\alpha\to\infty$. $\rho$ and $\lambda$ indicated in panel
titles. $\lambda=\lambda^{*}$ denotes the ‘locally optimal” regularization
(see section H.3) (b) Optimal generalization error $E_{k^{*}}$ for the same
parameter values in (a). Figure S9: Reduced generalization errors
$E_{g}/s(1-c)$ with $\lambda=0$ and $W=Z=0$ (given by eq. 22) for linear ridge
ensembles of varying size $k$. $\rho$ and $H$ values indicated above plots.
Grey lines indicate $k=1$, dashed black lines $k\to\infty$, and intermediate
$k$ values by the colorbar. Figure S10:  Reduced generalization errors
$E_{g}/s(1-c)$ with $\lambda=\lambda^{*}$ and $W=Z=0$ (given by eq. 23) for
linear ridge ensembles of varying size $k$. $\rho$ and $H$ values indicated
above plots. Grey lines indicate $k=1$, dashed black lines $k\to\infty$, and
intermediate $k$ values by the colorbar. Figure S11:  Reduced generalization
errors $E_{g}/s(1-c)$ with $\lambda=0$ and $H=Z=0$ (given by eq. 22) for
linear ridge ensembles of varying size $k$. $\rho$ and $W$ values indicated
above plots. Grey lines indicate $k=1$, dashed black lines $k\to\infty$, and
intermediate $k$ values by the colorbar. Figure S12: Reduced generalization
errors $E_{g}/s(1-c)$ with $\lambda=\lambda^{*}$ and $H=Z=0$ (given by eq. 23)
for linear ridge ensembles of varying size $k$. $\rho$ and $W$ values
indicated above plots. Grey lines indicate $k=1$, dashed black lines
$k\to\infty$, and intermediate $k$ values by the colorbar. Figure S13: Reduced
generalization errors $E_{g}/s(1-c)$ with $\lambda=0$ and $H=W=0$ (given by
eq. 22) for linear ridge ensembles of varying size $k$. $\rho$ and $Z$ values
indicated above plots. Grey lines indicate $k=1$, dashed black lines
$k\to\infty$, and intermediate $k$ values by the colorbar Figure S14: Reduced
generalization errors $E_{g}/s(1-c)$ with $\lambda=\lambda^{*}$ and $H=W=0$
(given by eq. 23) for linear ridge ensembles of varying size $k$. $\rho$ and
$Z$ values indicated above plots. Grey lines indicate $k=1$, dashed black
lines $k\to\infty$, and intermediate $k$ values by the colorbar.


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# Observational Constraints on Sunyaev-Zeldovich Effect Halos Around High-z
Quasars

Kyle Massingill  School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287, USA<EMAIL_ADDRESS>Brian Mason
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA
22903, USA  Mark Lacy  National Radio Astronomy Observatory, 520 Edgemont
Road, Charlottesville, VA 22903, USA  Bjorn H. C. Emonts  National Radio
Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
Ilsang Yoon  National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903, USA  Jianrui Li  Department of Astronomy, Tsinghua
University, Beijing 100084, People’s Republic of China  Craig Sarazin
Department of Astronomy, University of Virginia, P. O. Box 400325,
Charlottesville, VA 22904-4325, USA

###### Abstract

We present continuum observations from the Atacama Large
Millimeter/submillimeter Array (ALMA) of 10 high-redshift ($2.2\leq z\leq
2.7$) ultraluminous quasars (QSOs) and constrain the presence of hot, ionized,
circum-galactic gas in a stacking analysis. We measure a Compton-y parameter
profile with a peak value of $(1.7\pm 1.1)\times 10^{-6}$ at a radius of $\sim
50$ kpc. We compare our stacked observations to active galactic nucleus (AGN)
feedback wind models and generalized Navarro-Frenk-White (gNFW) pressure
profile models to constrain the wind luminosity and halo mass of the stacked
QSOs. Our observations constrain the observed stack’s halo mass to $<1\times
10^{13}M_{\odot}$ and the stack’s feedback wind power $<1\times
10^{12}L_{\odot}$, which is $<1$% of the bolometric luminosity of the quasar.

##  1 Introduction

The lack of super massive galaxies at low redshift (Drory & Alvarez 2008; Treu
et al. 2005) and the “downsizing” (Cowie et al., 1996) of star-forming
galaxies below $z\sim 2$ indicates some process regulating the growth of
galaxies on long time scales. Active galactic nucleus (AGN) feedback has been
postulated as the mechanism regulating galaxy formation (Scannapieco & Oh
2004; Bower et al. 2006) and remains the best explanation for the observed
galaxy sizes. An AGN has enough energy to heat gas in the circumgalactic
medium (CGM) and even possibly expel gas out of the galaxy completely (Silk &
Rees 1998; Bower et al. 2006). It is not yet well understood what role
different feedback mechanisms play in stifling galaxy growth (Ostriker et al.,
2010; Fabian, 2012). The relative importance of feedback mechanisms, such as
jets, winds, and radiation, is still being studied.

In order to better understand AGN feedback, observations must be made to
constrain the energy of the the different feedback mechanisms. Quasar winds or
outflows have been detected directly in X-ray for two low-z cases (Greene et
al. 2014; Harrison et al. 2014). X-ray detections of outflows from high-$z$
systems where AGN are most powerful are more difficult, as the faint, diffuse
X-ray emission from the winds is hard to detect against the strong point
source emission from the quasar (e.g. Kar Chowdhury et al., 2022). Instead,
detections can be obtained using the Sunyaev-Zel’dovich effect (SZE) (Sunyaev
& Zeldovich, 1972) around QSOs. The thermal SZE (tSZE) is the process where
cosmic microwave background (CMB) emission is distorted by traveling through
hot gas, inducing an inverse Compton scattering. Bulk motion of electrons in
the gas also produce a kinetic SZE (kSZE). In the case of AGN feedback, most
studies focus on detecting the tSZE from the superposition of multiple wind
events in the CGM of the quasar, when the tSZE is dominant (Scannapieco et
al., 2008). By stacking large numbers of quasars in tSZE maps from mm-wave
telescopes a number of studies have claimed to detect a significant tSZE
signal from quasar winds (e.g. Chatterjee et al. 2010; Ruan et al. 2015;
Crichton et al. 2016; Verdier et al. 2016; Hall et al. 2019; Meinke et al.
2021). With the large beams of the mm-wave telescopes used and uncertainties
regarding the contamination of the SZ signal by dust emission from the quasar
host galaxies, however, it can be hard to tell if the detections correspond
simply to a combination of dust emission and the tSZE from virialized gas in
the massive dark matter haloes in which the quasars reside, or whether there
is an extra component of the tSZE due to quasar feedback. tSZE from hot
intracluster medium has been detected in the Spiderweb radio galaxy (Mascolo
et al. 2023).

Typical AGN exist in relatively rich environments, even at high redshift.
Quasar-quasar clustering analyses (e.g. White et al., 2012; Timlin et al.,
2018; Eftekharzadeh et al., 2019), lensing of the Cosmic Microwave Background
by quasar hosts (Geach et al., 2019), the kinematics of the warm ionized gas
from quasar absorption lines (Lau et al., 2018) and Ly-$\alpha$ haloes
(Fossati et al., 2021) all suggest that quasars at $z\sim 1-4$ lie in dark
matter haloes of masses $\approx 2-10\times 10^{12}h^{-1}M_{\odot}$,
consistent with galaxy groups in the local Universe. Most estimates for radio-
quiet quasars are towards the lower end of that range, whereas radio-loud AGN
cluster more strongly, consistent with halo masses at the upper end of that
range (Miley & Breuck, 2008; Retana-Montenegro & Röttgering, 2017). These
relatively high mass haloes led Cen & Safarzadeh (2015) and Soergel et al.
(2017) to argue that the tSZE signal from stacking studies is dominated by
that from the virialized haloes of the quasars. By using a halo occupation
distribution model for quasar clustering Chowdhury & Chatterjee (2017)
suggested that the signal from feedback in the stacking studies made prior to
2017 is significantly higher than that expected from the virialized gas in the
quasar halo only at high redshifts ($z>2.5$).

Some of the ambiguities of stacking studies can be overcome using high
resolution images from mm-wave interferometers, such as ALMA. These studies
enable the subtraction of emission by the quasar host and any companion or
foreground/background galaxies. The morphology of the SZE signal can also be
compared to, for example, that of emission line gas to determine whether the
signal is associated with an outflow, or more generally with the halo in which
the quasar resides. However, these observations are challenging, requiring
$\stackrel{{\scriptstyle>}}{{{}_{\sim}}}10$ hours of integration time on ALMA
to detect a signal around even the most luminous quasars. It is also not
possible to separate the tSZE from any kSZE contribution without further very
deep observations at other frequencies. To date, there is only one example of
an SZE detection around a single QSO, the hyperluminous (bolometric luminosity
$\sim 10^{15}L_{\odot}$) quasar HE 0515-4414 (Lacy et al., 2019).

Target | Position | SNR | Redshift | log${}_{10}(L_{\rm bol}/L_{\odot}$) | RMS
---|---|---|---|---|---
|  | ($\sigma$) | ($z$) |  | (Jy/beam)
Q0050+0051 | ICRS 00:50:21.2200, 00:51:35.000 | 14 | 2.22 | 13.79 | 2.14e-5
Q0052+0140 | ICRS 00:52:33.6700, 01:40:40.800 | 11 | 2.30 | 13.99 | 2.00e-5
Q0101+0201 | ICRS 01:01:16.5400, 02:01:57.400 | 13 | 2.46 | 13.91 | 1.91e-5
Q1227+2848 | ICRS 12:27:27.4800, 28:48:47.900 | 7 | 2.26 | 13.76 | 1.64e-5
Q1228+3128 | ICRS 12:28:24.9700, 31:28:37.700 | 307 | 2.22 | 14.55 | 2.80e-5
Q1230+3320 | ICRS 12:30:35.4700, 33:20:00.500 | 13 | 2.32 | 13.66 | 1.99e-5
Q1416+2649 | ICRS 14:16:17.3800, 26:49:06.200 | 21 | 2.29 | 13.51 | 1.86e-5
Q2121+0052 | ICRS 21:21:59.0400, 00:52:24.100 | 3 | 2.37 | 13.69 | 2.17e-5
Q2123$-$0050 | ICRS 21:23:29.4600, $-$00:50:52.900 | 7 | 2.65 | 14.52 | 2.21e-5
Q0107+0314 | ICRS 01:07:36.9000, 03:14:59.200 | 9 | 2.28 | 13.76 | 2.18e-5
Table 1: Quasars observed in this study, their position and the signal to
noise ratio (SNR) achieved in the $6^{{}^{\prime\prime}}$ tapered continuum
ALMA 12m observations. Here SNR is defined as the peak brightness of the QSO
divided by overall RMS of the image. We have estimated tapered RMS from the
median of the absolute deviations from the median (MAD) of the source
subtracted images. We rescaled the MAD as $RMS=k\times MAD$, where $k\approx
1.4826$, so as to provide the same results as the RMS for a Gaussian
distribution. $L_{bol}$ is the bolometric luminosity of the Quasar.

We therefore decided to try a different approach to constrain the tSZE signal
from a sample of more typically-luminous quasars, also by using ALMA, but
stacking the signal from a moderate number ($\approx 10$) of ultraluminous
quasars (with bolometric luminosities $\sim 10^{14}L_{\odot}$). These quasars
were selected to have extended Ly$\alpha$ nebulae around them (Cai et al.
2019), which allow a rough estimate of the typical halo mass in the sample to
be made based on the gas dynamics (uncertain due to radiative transfer effects
and possible non-gravitational motions in the Ly$\alpha$-emitting gas). Using
ALMA allows us to take advantage of the capability to subtract emission from
discrete sources, both in the field and associated with the quasar, whilst
still building up enough signal-to-noise to make a detection or place a
meaningful limit on the tSZE signal from feedback.

In this paper we present observations of ten ultraluminous QSOs at $2.2\leq
z\leq 2.7$ observed by the ALMA 12m array. We constrain the Compton parameter
of the observed population. We compare our measured Compton parameters to
generalized Navarro–Frenk–White (gNFW) (Arnaud et al. 2010) profiles and
spherical feedback models.

##  2 Observations

Our ALMA QSO targets were observed by project 2019.1.01251.S. This was a
commensal program between our continuum study of the SZE and emission-line
observations of CO(4-3) and [CI] as part of the SUPERCOLD-CGM survey (Li et
al., 2023), which both benefited from sensitive low-surface-brightness
observations with ALMA Band 4. Ten pointings were used to observe ten QSO’s on
the ALMA 12m array band 4 ($\sim 145\,{\rm GHz}$) with an angular resolution
of $\sim$$2^{{}^{\prime\prime}}$. Targets were observed by the 12m array for
$1-2$ hours in order to achieve an root mean square RMS of $\approx 11\mu\rm
Jy$ in the continuum. Observations of these sources were also made with the
ALMA 7m array but at a much higher noise level, so we do not utilize them in
our analysis here. See Table 1 for a full list of targets and the achieved
signal to noise ratio of the observations. Observations were made with four
$2\,\rm GHz$ spectral windows (SPWs), two containing line emission (from CO
and CI) and two only containing continuum.

###  2.1 Data Reduction and Calibration

The standard pipeline calibrations of the ALMA data (Hunter et al., 2023) were
utilized for the QSO observations. Data reduction scripts supplied with the
ALMA archival data were used together with the Common Astronomy Software
Application (CASA; The CASA Team et al. 2022) version 5.6.1-8, which includes
the ALMA pipeline (Masters et al. 2020) version r42866. We reduced the data by
first splitting off the calibrated on source observations from the full data
set then flagging spectral line in each SPW. For flagging spectral lines, an
image cube with a channel width of $\sim 8$ MHz (base bands have a full
bandwidth of $1875$ MHz) was produced for each base band. Then channels with
localized emission $4\sigma$ or more above the continuum of the channel were
flagged. Flagged channels were then ignored in subsequent cleaning/imaging.
All four SPWs were then combined to make a single, aggregate continuum image.

Imaging and cleaning was done using CASA task “tclean”. We imaged an area of
60$\times$60 arcseconds centered on each quasar using a pixel scale of
$0.4^{{}^{\prime\prime}}$/pixel. We started the imaging process by first
cleaning the ALMA 12m images at full resolution. We cleaned using a robustness
of $0.5$. For each of the ten targets, the sources (the QSO and any nearby
serendipitous sources) were masked manually; we then cleaned the masked region
to the threshold of the overall image RMS. All subsequent analysis was
performed upon the clean residual images, which have effectively had the
central QSO (and any strong, positive emission) subtracted. We applied
Gaussian tapers to the (u,v)-data during imaging to improve surface brightness
measurements of extended signals. We made tapered images for each source at
$\sim 6^{{}^{\prime\prime}}$ and $\sim 3^{{}^{\prime\prime}}$ resolutions.

The object Q1228+3128 (previously identified as a radio loud QSO in Li et al.
2021) was detected with very high ($\sim 307\sigma$) signal-to-noise ratio
(SNR) in the ALMA continuum observations such that the image was dynamic range
limited and benefited from self-calibration. To self-calibrate we use the
central source to correct for changes in phase. We solved for phase in
intervals of scan length following ALMA NAASC recommended self-calibration
procedure (Brogan et al. 2018). However, even with self-calibration we
determined the observation to not have enough dynamic range to detect the
$\mu$Jy level signal of the tSZE. Q1228+3128 was therefore excluded from the
analyses. Our second highest QSO SNR detection in the continuum observations
was Q1416+2649 with a much lower significance of $\sim 21\sigma$ (we do not
utilize self-calibration for this Q1416+264). Continuum emission from the
central QSO was clearly detected in 9 of the 10 targets, and weakly ($\sim
3\sigma$) detected in the other.

Figure 1: Stacked QSO environments expressed as a Compton-y map. A
$6^{{}^{\prime\prime}}$ Gaussian taper has been applied to the ALMA 12m
observations. Regions with source flux have been masked. Bins of differing
radius have been made, with the width of each bin being about $\sim
6.7^{{}^{\prime\prime}}$ wide.

###  2.2 Stacked SZE Signal

We stacked the source subtracted ALMA images by first measuring the median
absolute deviation (MAD) of each residual image and applying weighting to each
residual image based on the MAD. The masks from the cleaning stage (see
section 2.1 Data Reduction and Calibration) were applied to the residuals and
masked regions were not included in the stack. This was done to ensure any
leftover source emission was not added to the final stacked image. All the
residuals were then stacked by summation in units of Jy/beam. We produced a
$\sim 6^{{}^{\prime\prime}}$ and $\sim 3^{{}^{\prime\prime}}$ resolution
stack.

We interpret the stacked continuum flux observations as SZE signal by
converting to Compton parameter ($y$). Observed decrement in the continuum
flux will correspond to a positive Compton parameter. Our stacking analysis
will not be sensitive to the non-spherically symmetric kSZE. For stacked QSO
environments we use a pure thermal SZE model to determine the Compton
parameters. We therefore assume $v_{r}/c<<(k_{B}T_{e})/(m_{e}c^{2})f_{1}(x)$.
Following Sazonov & Sunyaev 1998, the change in intensity of the CMB due to
SZE is given by:

$\Delta
I(x)=\frac{2k_{B}T_{CMB}}{\lambda^{2}}\frac{x^{2}e^{x}}{(e^{x}-1)^{2}}\tau\frac{k_{B}T_{e}}{m_{e}c^{2}}f_{1}(x),$
(1)

where $x\equiv h\nu/k_{B}T_{CMB}$, and $f_{1}(x)=x\coth(x/2)-4$. $\tau$ is the
Thompson scattering optical depth along the line of sight and $f_{1}(x)$ is
the frequency dependence of thermal SZE. We therefore describe the spectral
distortion along a line of sight as the Compton parameter (Sunyaev & Zeldovich
1972):

$y=\frac{\tau k_{B}T_{e}}{m_{e}c^{2}}.$ (2)

We convert observed flux into Compton parameter for the purposes of comparing
observations to theoretical models.

Fig. 1 shows the Compton parameter of the stacked QSO environments tapered to
an effective beam of $\sim 6^{\prime\prime}$. A region of high $y$ can be seen
in the east side of the image, at a radius of $\sim 5^{\prime\prime}$ from the
center, peaking at $y=2.4e-5$ or $\sim 2.6\sigma$. Regions of null pixels
exist (prominently seen as white pixels at the center of Fig. 1) due to
masking as described earlier in this section.

##  3 Analysis

###  3.1 gNFW Models

In order to compare our observations to theoretical models of different halo
masses, we use a generalized Navarro-Frenk-White (gNFW) pressure profile:

$\mathbb{P}(x)=\frac{P_{0}}{(c_{500}x)^{\gamma}[1+(c_{500}x)^{\alpha}]^{(\beta-\gamma)/\alpha}}$
(3)

as described in Arnaud et al. 2010 and Nagai et al. 2007. $x\equiv r/r_{s}$
and $r_{s}=r_{500}/c_{500}$, where $r_{500}$ is the radius containing matter
at 500 times the ambient density. We treat the halo as a cube divided into
cells; the cube is $300^{3}$ cells. Each is cell is $0.4^{{}^{\prime\prime}}$
per side and the total volume of the cube, within which we are integrating the
line-of-sight pressure, is $\sim 1\times 10^{10}$ kpc3at our average redshift
of $z=2.34$. We then calculate the pressure of each cell:

$P(r)=P_{500}\left[\frac{M_{500}}{3\times
10^{14}h_{70}^{-1}M_{\odot}}\right]^{\alpha_{P}+\alpha^{\prime}_{P}(x)}\mathbb{P}(x)$
(4)

with parameters:

$\displaystyle[P_{0},c_{500},\gamma,\alpha,\beta]=$
$\displaystyle\left[8.403h^{-3/2}_{70},1.177,0.3081,1.0510,5.4905\right]$ (5)

where $P_{500}$ and $M_{500}$ are the corresponding pressure and mass
respectively. To determine $y$ we integrate the pressure through the cube. We
generate halo simulation images (see section 3.3) for halo masses of $1\times
10^{12}M_{\odot}$ , $3\times 10^{12}M_{\odot}$, $1\times 10^{13}M_{\odot}$,
and $3\times 10^{13}M_{\odot}$. For the nominal mass case of $3\times
10^{12}M_{\odot}$, $y\sim 10^{-6}$ out to a radius of $\sim 100$ kpc (or
$\approx 12^{{}^{\prime\prime}}$ at $z=2.34$) from the center of the profile.

###  3.2 Feedback Models

The gNFW models do not include the effects of feedback. We therefore
constructed a set of simple spherical wind models of AGN feedback using the
prescription of Rowe & Silk (2011) as adapted for AGN winds by Lacy et al.
(2019). These give the radius of the bubble, $R$:

$R=\beta\left(\frac{L_{W}T^{3}}{\Omega_{b}\rho_{c}(1+z)^{3}(1+b_{Q}\delta)}\right)^{1/5}$
(6)

where $\beta=0.8828$, $L_{W}$ is the wind power, $T$ is the age of the
outflow, $\Omega_{b}$ is the fraction of the critical density of the Universe
$\rho_{c}$ in baryons, $b_{Q}\approx 13$ is the cosmological bias factor for
quasars and $\delta=180$ is the assumed overdensity corresponding to collapsed
structures. The pressure inside the bubble is approximately constant and is:

$P_{\rm bub}=\frac{2}{5}\frac{3}{4\pi}\frac{L_{W}T}{R^{3}}$ (7)

and the peak Compton-$y$ signal (through the center of the bubble):

$y_{\rm max}=1.08\frac{P_{e}\sigma_{T}}{m_{e}c^{2}}2R$ (8)

where the electron pressure, $P_{e}=P_{\rm bub}/1.92$, $\sigma_{T}$ is the
Thompson cross-section and $m_{e}$ the electron mass. We constructed 2-D
models of the SZE signal to then run through the simulator, as described
below.

Figure 2: This is an example of a feedback model ($T=1\times 10^{8}$years and
$L_{W}=1\times 10^{12}L_{\odot}$). LEFT: The wind model calculated as
described in section 3.2. RIGHT: The wind model convolved with the ALMA beam
using the simulator described in section 3.3. In the right hand image there
are primary beam and sidelobe effects such as the negative Compton-y ring at a
radius of $\sim 0.16^{{}^{\prime}}$. Figure 3: Radial profile analysis
comparing the stacked QSO images to gNFW profiles of four different halo
masses. Both the real and theoretical observations contain primary beam
effects from the telescope configuration. Uncertainty in the bin mean is
$\sigma=RMS/\sqrt{beams-1}$, where “beams” is the number of ALMA beams in the
bin.

###  3.3 Simulating Observations

The gNFW pressure profiles and wind models are simulations of sky images. In
order to compare them to our observations in the image plane, we had to add
the effects of the ALMA beam to our simulated images. We do this by utilizing
the ALMA simulator, a facility within CASA (The CASA Team et al. 2022). The
ALMA simulator takes a sky image and a given set of observational conditions
and produces the expected ALMA visibilities. This is done by convolving the
telescope configuration with the theoretical sky image. We used simulator task
“simpredict” from CASA version 5.6.1-8. For each of calculated models, we
produce a set of theoretical visibilities based on the QSO observational
conditions. These simulated visibities are then imaged using the same
parameters as the QSOs (see Fig. 2 for an example). These imaged models now
have the same primary beam and side lobe effects as the observations.

###  3.4 Radial Profile Analysis

To compare the gNFW models to our observations, the Compton parameter of the
stack was analyzed as a radial profile in bins slightly wider then the width
of the tapered beam. We utilized the lower resolution stack for this analysis,
as the pressure profiles are relatively extended ( $>10^{{}^{\prime\prime}}$)
sources. The beam of the smoothed ALMA image is $\sim 6^{{}^{\prime\prime}}$
while we used $\sim 6.7^{{}^{\prime\prime}}$ wide bins. The central pixels of
the image (out to $r\sim 3^{{}^{\prime\prime}}$) were not utilized in the
radial profile analysis as that region is mostly masked out. We measured the
mean Compton parameter and RMS in each bin of the stack. Figure 3 shows our
radial profile analysis comparing the stacked QSO images to gNFW profiles of
different halo masses.

The feedback models were prepared using the same procedure as the gNFW models
(see section 3.3). We compared the feedback models to the higher resolution
stack ($\sim 3^{{}^{\prime\prime}}$ beam) in order to be sensitive to wind
bubbles with radii on order of a few arcseconds predicted by recent
simulations (Chakraborty et al., 2023). We analyzed this stack in Compton-$y$
as a radial profile using bins with a width of $\sim 5.3^{{}^{\prime\prime}}$.
As on the lower resolution image, the central pixels of the image (out to
$r\sim 3^{{}^{\prime\prime}}$) were not utilized in the radial profile
analysis. We compared this profile to the feedback models at a variety of wind
powers ($L_{W}$) and two outflow ages ($T$). The chosen outflow ages are based
on the estimated age of a previously detected feedback wind bubble in Lacy et
al. 2019 and are of the order of the Salpeter timescale for quasar growth
(e.g. Shen, 2013). Figures 4 and 5 show our radial profile analysis.

##  4 Results and Discussion

Model Halo Mass | $\chi^{2}$ | p-value
---|---|---
$1\times 10^{12}M_{\odot}$ | $4.03$ | $0.4$
$3\times 10^{12}M_{\odot}$ | $4.38$ | $0.36$
$1\times 10^{13}M_{\odot}$ | $8.02$ | $0.09$
$3\times 10^{13}M_{\odot}$ | $31.43$ | $2.51\times 10^{-6}$
Table 2: Results of image plane analysis, comparing stacked observations of
QSO environments to gNFW halo profiles at four different halo masses as a
goodness of fit $\chi^{2}$ and associated statistical p-value with four
degrees of freedom. All models use a concentration of 1. Figure 4: Radial
profile analysis comparing the stacked QSO images to feedback models with an
outflow age of $T=3\times 10^{7}$ years. Both the real and theoretical
observations contain primary beam effects from the telescope configuration.
Uncertainty in the bin mean is $\sigma=RMS/\sqrt{\text{beams}-1}$, where
“beams” is the number of ALMA synthesized beams in the bin. Figure 5: Radial
profile analysis comparing the stacked QSO images to feedback models with an
outflow age of $T=1\times 10^{8}$ years. Both the real and theoretical
observations contain primary beam effects from the telescope configuration.
Uncertainty in the bin mean is $\sigma=RMS/\sqrt{\text{beams}-1}$, where
“beams” is the number of ALMA synthesized beams in the bin.

###  4.1 Halo Mass

Signal from a gNFW halo (CMB decrement) is strongest at the center of the
profile, which would not be detectable by these observations as they are
centered on continuum bright QSOs. On the image plane we instead try to fit
the regions around the QSO to the profile the same distance from the center.
In our stacked data we do see a Compoton signal in the inner radial bin (see
figure 3), but the error is large due to the small radius; this bin only
contains $\sim 10$ beams. The other notable features of the gNFW halo when
observed through the ALMA simulator are the dark sidelobes. Larger mass halo
models will have brighter (in Compton-$y$) centered decrement and darker
sidelobes than lower mass models. Our observations lack dark sidelobes that
would indicate a higher mass halo. We quantify the relationship between the
stacked QSO observations and the simulated gNFW models by calculating the
$\chi^{2}$ and p-value (where “p” is the probability that the difference
between model and observations are due to chance) for each halo mass model
compared to the stacked observations. $\chi^{2}$ is calculated from 4 bins in
the case of halo mass. Looking at Table 2 we can see that as we go to higher
halo mass models, it becomes statistically less likely that they are
consistent with our observations. The gNFW model with halo mass $1\times
10^{13}M_{\odot}$ has a p-value $0.09$. We can therefore constrain our
observed stack of QSOs to have a halo mass $<1\times 10^{13}M_{\odot}$ with
$90\%$ confidence.

$T=3\times 10^{7}$years
---
$L_{W}$ | $\chi^{2}$ | p-value
$1\times 10^{11}L_{\odot}$ | $1.49$ | $0.19$
$3\times 10^{11}L_{\odot}$ | $1.6$ | $0.16$
$1\times 10^{12}L_{\odot}$ | $2.08$ | $0.06$
$3\times 10^{12}L_{\odot}$ | $4.93$ | $1.6\times 10^{-4}$
$1\times 10^{13}L_{\odot}$ | $30.56$ | $<1\times 10^{-16}$
Table 3: Results of image plane analysis, comparing stacked observations of
QSO environments to outflow age $3\times 10^{7}$ years feedback models at five
different wind powers as a goodness of fit $\chi^{2}$ and associated
statistical p-value with five degrees of freedom. $T=1\times 10^{8}$years
---
$L_{W}$ | $\chi^{2}$ | p-value
$1\times 10^{11}L_{\odot}$ | $1.55$ | $0.17$
$3\times 10^{11}L_{\odot}$ | $1.88$ | $0.09$
$1\times 10^{12}L_{\odot}$ | $3.32$ | $0.01$
$3\times 10^{12}L_{\odot}$ | $7.78$ | $2.51\times 10^{-7}$
$1\times 10^{13}L_{\odot}$ | $13.68$ | $2.18\times 10^{-13}$
Table 4: Results of image plane analysis, comparing stacked observations of
QSO environments to outflow age $1\times 10^{8}$ years feedback models at five
different wind powers as a goodness of fit $\chi^{2}$ and associated
statistical p-value with five degrees of freedom.

###  4.2 Feedback

We have modeled feedback as a spherical wind bubble that will cause a positive
Compton-$y$ signal within the radius of the bubble. Sidelobe effects from the
ALMA observatory are accounted for by the simulator and cause negative
Compton-$y$ signal around the feedback wind bubble. On the image plane we fit
the regions around the QSO to the feedback models at the same distance from
image center. We quantify the relationship between the stacked QSO
observations and the simulated feedback models by calculating the $\chi^{2}$
and p-value for each wind power and outflow age model compared to the stacked
observations. Looking at Tables 3 and 4 we see that in both outflow age cases,
as we go to higher power models, it becomes statistically less likely that
they are consistent with our observations. At a wind power of $1\times
10^{12}L_{\odot}$ and an outflow age $1\times 10^{8}$ years the p-value is
$0.01$ We can therefore constrain our observed stack of QSOs to have a
feedback wind power $<1\times 10^{12}L_{\odot}$ , or
$\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$% of the bolometric luminosity of
the quasar.

##  5 Conclusions

We use the constraints on the tSZE from our stack of quasar observations to
show that (1) if the quasars are in quiescent halos of virialized gas, the
halo mass is $<1\times 10^{13}M_{\odot}$ (consistent with estimates $\sim
3\times 10^{12}M_{\odot}$ from clustering analyses), and (2), if feedback from
thermal winds extends to spatial scales $\sim 100$ kpc, these winds carry
$<1\%$ of the bolometric luminosity of the quasars. This finding is consistent
with the only direct detection of SZE around a single QSO, Lacy et al. 2019,
which found the hyperluminous quasar HE 0515-4414 to have a wind luminosity of
$\sim 0.01\%$ of the bolometric luminosity of the quasar. The $\chi^{2}$ of
our best fitting models is similar to zero signal. Further fitting of halo and
feedback models is therefore not merited by the data available in this survey.
We do not see evidence of a strongly peaked central decrement that would be
indicative of a gNFW profile or feedback wind bubble. We note that the
simulations of Chakraborty et al. (2023) show that strong jet-mode feedback is
effective at suppressing the SZE from both the halo and thermal winds, thus
jet feedback may be occurring in these objects. Only one of the quasars
(Q1228+3128, which was not included in our stacking analysis) is currently
radio loud, however there could be intermittent jet activity (e.g. Nyland et
al., 2020) and it has been show that non radio-loud quasars can still have
radio mode feedback that effect the QSO environment on scales of $\sim$10-100
kpc (Villar-Martín et al. 2021).

##  6 Acknowledgments

This paper makes use of the ALMA data: 2019.1.01251.S. ALMA is a partnership
of ESO (representing its member states), NSF (USA) and NINS (Japan), together
with NRC (Canada), MOST and ASIAA (Tai-wan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. We thank the North American ALMA
Science Center faculty and staff for their support. We thank the Arizona State
University SZ science group, including Seth Cohen, Phil Mauskopf, Sean Bryan,
Jenna Moore, Emily Lunde, Jeremey Meinke, Skylar Grayson and Evan Scannapieco,
for their advice and contributions.

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