# A measurement of circumgalactic gas around nearby galaxies using fast radio
bursts

Xiaohan Wu,1,2 Matthew McQuinn,3
1Canadian Institute for Theoretical Astrophysics, University of Toronto,
Toronto, ON, Canada M5S 3H8
2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge
02138, MA, USA
3Astronomy Department, University of Washington, Seattle, WA 98195, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The distribution of gas in the circumgalactic medium (CGM) of galaxies of all
types is poorly constrained. Foreground CGMs contribute an extra amount to the
dispersion measure (DM) of fast radio bursts (FRB). We measure this DM excess
for the CGMs of $10^{11}-10^{13}\ M_{\odot}$ halos using the CHIME/FRB first
data release, a halo mass range that is challenging to probe in any other way.
Because of the uncertainty in the FRBs’ angular coordinates, only for nearby
galaxies is the localization sufficient to confidently associate them with
intersecting any foreground halo. Thus we stack on galaxies within $80$ Mpc,
optimizing the stacking scheme to approximately minimize the stack’s variance
and marginalize over uncertainties in FRB locations. The sample has 20-30 FRBs
intersecting halos with masses of $10^{11}-10^{12}\ M_{\odot}$ and also of
$10^{12}-10^{13}\ M_{\odot}$, and these intersections allow a marginal
$1-2\,\sigma$ detection of the DM excess in both mass bins. The
$10^{11}-10^{12}\ M_{\odot}$ halos bin also shows a DM excess at 1-2 virial
radii. By comparing data with different models for the CGM gas profile, we
find that all models are favored by the data up to 2-$\sigma$ level compared
to the null hypothesis of no DM excess. With 2000-3000 more bursts from a
future CHIME data release, we project a 4-$\sigma$ detection of the CGM.
Distinguishing between viable CGM models by stacking FRBs with CHIME-like
localization would require tens of thousands of bursts.

###### keywords:

keyword1 – keyword2 – keyword3

††pubyear: 2015††pagerange: A measurement of circumgalactic gas around nearby
galaxies using fast radio bursts–A

##  1 Introduction

The circumgalactic medium (CGM) – the diffuse gas that sits outside of galaxy
disks and inside the halo virial radii – is a crucial component in the
baryonic processes in the universe. It receives inflow of the intergalactic
gas, fuels star formation in galaxies, and bears the impact of galactic
feedback (Naab & Ostriker, 2017; Tumlinson et al., 2017). An understanding of
the CGM would resolve an aspect of the “missing baryon problem”, that a
substantial fraction of the baryons associated with halos has not been
detected (McGaugh et al., 2010; Dai et al., 2010; McQuinn, 2016).

Questions remain about how the CGM is distributed around galaxies and how much
mass there is in the CGM (e.g. Tumlinson et al., 2017). Fast radio bursts
(FRB) are, however, starting to provide an unprecedented probe of the CGM with
the advent of many radio telescopes geared for this science coming online
across the globe (Cordes & Chatterjee, 2019; Petroff et al., 2019, 2022;
Prochaska & Zheng, 2019; Keating & Pen, 2020). FRBs are bright (up to a few
hundred Jy) transient radio pulses (typically up to a few milliseconds) of
mostly extragalactic origin, and have been detected over a wide range of
frequencies (400 MHz-8 GHz). As the radio waves from FRBs travel through the
intervening plasma, the interaction with free electrons causes a frequency-
dependent delay of the arrival time, which is proportional to the integral of
the electron number density along the line of sight – the dispersion measure
(DM). Since most of the CGM gas is ionized, they contribute an extra amount to
the total DM along a sightline, which can be used to constrain the CGM gas
distribution (McQuinn, 2014; Ravi, 2019).

There are many challenges to precisely measuring the excess DM from the CGM,
especially in a sample of FRBs that are not well localized. First of all, the
host galaxy and local environment around the FRB can contribute from a few
tens to a few hundreds of pc cm-3 (Kulkarni et al., 2015; Connor et al., 2016;
Tendulkar et al., 2017; Yang et al., 2017; Niu et al., 2022; Cordes et al.,
2022). Cosmic structures can also contribute scatter at the hundreds of pc
cm-3 level (McQuinn, 2014; Macquart et al., 2020). If the redshifts of the FRB
host and intervening galaxies can be measured, requiring arcsecond
localizations, one can subtract off the average cosmological contribution to
the total DM, which significantly reduces the variance in the measurement.
Previous theoretical works predict that $\mathcal{O}(100)$ FRBs are required
to put constraints on the CGM when stacking arcsecond-localized FRBs (McQuinn,
2014; Ravi, 2019). However, there are approximately 20 FRBs that are
sufficiently well localized to date (Petroff et al., 2022), while thousands of
predominantly unlocalized FRBs have been observed with more expected in the
next few years. Without arcsecond localizations, many more FRBs are required
to detect an excess DM from foreground CGMs (but see Platts et al., 2020, for
constraining the Mikly Way DM using a small number of transients). This paper
considers this case.

We expand upon the recent work of Connor & Ravi (2021) and present a
measurement of the CGM of nearby galaxies using the first FRB catalog
published by the Canadian Hydrogen Intensity Mapping Experiment (CHIME) FRB
project (Amiri et al., 2021). CHIME is a drift scan radio telescope operating
across 400 MHz to 800 MHz. The first CHIME/FRB catalog contains 535 FRBs
detected between 2018 July 25 and 2019 July 1, including 18 repeating ones.
While the typical localization error of $0.2^{\circ}$ is too large for any
sophisticated analysis on the CGM, nearby ($\lesssim 100$ Mpc) halos that
cover even larger areas on the sky ($>0.2^{2}$ deg2) makes it possible to
detect the DM excess from these halo CGMs. We thus stack FRBs that intersect
$10^{11}-10^{13}\ M_{\odot}$ halos within two virial radii and measure the DM
excess, with a flexible weighting function that considerably reduces the
stack’s variance and, importantly, down-weights high-DM FRBs. We show that
with our weighting scheme, having 2000-3000 more FRBs from CHIME data release
would lead to a $>3-4\,\sigma$ detection of the CGM of $10^{11}-10^{13}\
M_{\odot}$ halos, a number that is easily achievable with CHIME in the next
few years. This will open up a novel window for understanding the baryon
physics in the universe, since other methods of studying the CGM such as the
thermal and kinetic Sunyaev-Zeldovich effect (Schaan et al., 2021) and halo
X-ray emission (Chadayammuri et al., 2022) usually probes higher mass
($>10^{12}-10^{13}\ M_{\odot}$) halos (but see Bregman et al., 2021, for a
measurement of nearby Milky Way size halos).

This paper is organized as follows. Section 2.1 presents our galaxy catalog
and FRB sample selection. Section 2.2 describes our weighting method and
measurement of the DM excess from CGM. Section 2.3 compares our measurements
with CGM models. Section 2.4 makes forecasts for future CHIME data release.

##  2 Methods and results

###  2.1 FRB selection

We use the non-repeaters in the CHIME FRB catalog. Following Connor & Ravi
(2021), we exclude FRBs of Galactic latitudes within 5 degrees of the Galactic
plane to avoid regions where the Milky Way DM is higher such that modeling
errors for this contribution can be larger. This gives a total number of 453
FRBs. We have verified that our results are not sensitive to this Galactic
latitude threshold. For the FRB DM we use DM values provided by the CHIME
catalog that have the Milky Way disk contribution removed using the Cordes &
Lazio (2002) model. We have checked that using the Yao et al. (2017) model for
the Milky Way disk DM does not change our conclusions. We do not attempt to
subtract the Milky Way halo DM since stacking should average out the
fluctuations in the Milky Way halo DM, and the Milky Way halo DM is
anticipated to be similar for all sightlines. The Milky Way DM is also poorly
constrained (Prochaska & Zheng, 2019; Keating & Pen, 2020; Platts et al.,
2020).

As in Connor & Ravi (2021), we use the galaxies in the Gravitational Wave
Galaxy Catalogue (GWGC) catalog (White et al., 2011). At 40 Mpc, the angular
sizes of the virial radii of $10^{11},10^{12},10^{13}\ M_{\odot}$ halos are
$0.15^{\circ},0.33^{\circ},0.70^{\circ}$ respectively, assuming $R_{\rm
vir}=250(M/1.3\times 10^{12}\ M_{\odot})^{1/3}$ kpc. We use this expression
for $R_{\rm vir}$ throughout the paper. Since the typical CHIME localization
1-$\sigma$ error is 0.2 degree, we use $10^{11}-10^{12}\ M_{\odot}$ halos at
$0.5$-40 Mpc and $10^{12}-10^{13}\ M_{\odot}$ halos at $0.5$-80 Mpc for our
analysis. Later on, we develop a method to account for location uncertainties
when comparing CGM models. We calculate the halo masses of galaxies by
converting from their stellar masses using the relation in Moster et al.
(2010).111Different models predict similar stellar mass-halo mass relations at
$z=0$, e.g. Figure 34 of Behroozi et al. (2019). To estimate the galaxy
stellar mass, we adopt any of three methods depending on what photometric data
is available (higher priority comes first): cross-matching with NASA-Sloan
Atlas catalog222Provided by Matt Wilde. (NSA, 4200 galaxies), JHK band SED
fitting333We performed SED fitting using CIGALE (Boquien et al., 2019,
https://cigale.lam.fr/) and find that using JHK bands alone provides more
reliable stellar masses than using optical bands. A lot of galaxies do not
have optical photometry, as we queried the NASA/IPAC Extragalactic Database
web service with astropy. (3400 galaxies) or K-band mass-to-light ratio (100
galaxies), SDSS g-r color (400 galaxies). For the remaining 1900 galaxies in
GWGC for which we were unsuccessful at obtaining optical or infrared
photometric data, we convert their B band luminosities to stellar masses by
calibrating a conversion relation using the other galaxies. Among the galaxies
in the NSA catalog, 1800 also have JHK photometric data, for which we find
that our stellar mass estimates using JHK bands mostly lie within a factor of
3 from the NSA stellar masses. We have verified that our main conclusions do
not change if we use the JHK stellar masses for these galaxies instead.

For the candidate $>10^{11}\ M_{\odot}$ halos, we identify galaxy groups and
remove satellite galaxies. For a given galaxy, we define it as a satellite if
it lies within $1.2\ R_{\rm vir}$ of a more massive galaxy nearby in terms of
their 3D distance. We do not consider M33 as a satellite of M31. Since the
distances listed in GWGC have a typical error of 20%, we also test determining
a satellite using the 2D projected distance and the difference in the radial
velocities. We find that our results change very little if a satellite is
identified by requiring that the radial velocities differ within 3 times the
halo circular velocity instead of using the 3D distances. Our group finding
leaves 4600 $10^{11}-10^{12}\ M_{\odot}$ galaxies at $0.5$-40 Mpc and 4000
$10^{12}-10^{13}\ M_{\odot}$ galaxies at $0.5$-80 Mpc. We also remove 5 FRBs
that we identify to intersect $>10^{13}\ M_{\odot}$ halos at $<1\ R_{\rm
vir}$.

Figure 1: Locations of the CHIME catalog FRBs used in our analysis as well as of galaxies having FRB intersection with an impact parameter of $b<1\ R_{\rm vir}$. Dots show the position of FRBs, with colors denoting their DM. Blue and red crosses illustrate the centers of $10^{11}-10^{12}\ M_{\odot}$ and $10^{12}-10^{13}\ M_{\odot}$ halos, respectively. Ellipses around these halos map a circle of radius $1R_{\rm vir}$ into these angular coordiantes. Table 1: Number of FRBs and halos in each impact parameter ($b$) bin and the measured DM excess in units of parsec cm-3.. From top to bottom: $b\in[0,1],[1,1.5],[1.5,2]\ R_{\rm vir}$. Second to fourth column: number of FRBs intersecting $10^{11}-10^{12}\ M_{\odot}$ halos, number of FRBs intersecting M33, and the number of $10^{11}-10^{12}\ M_{\odot}$ halos with FRB intersections. Numbers in parentheses indicate the 1-$\sigma$ range (16th and 84th percentiles) of the number of FRBs owing to localization uncertainties, calculated by perturbing FRB locations assuming Gaussian errors. Fifth column: the measured DM excess and 1-$\sigma$ error, with numbers in brackets denoting the results excluding M33. Sixth to ninth column: the same but for $10^{12}-10^{13}\ M_{\odot}$ halos. $b$ range | $N_{\rm FRB}$ | $N_{\rm FRB}$ | $N_{\rm halo}$ | DM excess | $N_{\rm FRB}$ | $N_{\rm FRB}$ | $N_{\rm halo}$ | DM excess
---|---|---|---|---|---|---|---|---
| $10^{11}-10^{12}$ | M33 | $10^{11}-10^{12}$ |  | $10^{12}-10^{13}$ | M31 | $10^{12}-10^{13}$ |
$[0,1]\ R_{\rm vir}$ | 26 (26-31) | 6 | 24 | $71\pm 53(18\pm 61)$ | 30 (28-33) | 16 | 18 | $69\pm 49(96\pm 72)$
$[1,1.5]\ R_{\rm vir}$ | 31 (28-34) | 5 | 26 | $87\pm 48(78\pm 53)$ | 38 (36-42) | 22 | 18 | $-4\pm 43(6\pm 61)$
$[1.5,2]R_{\rm vir}$ | 39 (32-40) | 9 | 32 | $41\pm 43(35\pm 46)$ | 57 (49-56) | 28 | 31 | $37\pm 35(-11\pm 46)$

We select FRBs that intersect nearby halos out to $2\ R_{\rm vir}$ and bin
then according to their $b/R_{\rm vir}$. Figure 1 shows the map of FRBs, with
colors indicating the DM values, and locations of galaxies that have FRB
intersections within $1\ R_{\rm vir}$. Blue and red crosses represent the
centers of $10^{11}-10^{12}\ M_{\odot}$ and $10^{12}-10^{13}\ M_{\odot}$ halos
in this sample, respectively. Ellipses of corresponding colors map a circle of
radius $1\ R_{\rm vir}$ into RA-Dec coordinates. The two biggest circles at RA
$<50$ deg are M31 and M33. With our selection choices, there are 26 FRBs
intersecting 24 halos with masses of $10^{11}-10^{12}\ M_{\odot}$ within $1\
R_{\rm vir}$, and 30 FRBs intersecting 18 halos with $10^{12}-10^{13}\
M_{\odot}$, although half of these intersections are with M31. Among the 53
FRBs selected, 6 pass through two halos. Table 1 lists the number of FRBs and
halos falling in impact parameter bins of $[0,1],[1,1.5],[1.5,2]\ R_{\rm
vir}$. The numbers in brackets in this table represent the 1-$\sigma$ error on
these numbers owing to localization uncertainties, obtained by perturbing FRB
locations assuming Gaussian errors.444We note that the error-bars on the
number of FRBs in a bin are likely asymmetric around the number calculated
using the maximum likelihood localization. This occurs because some FRBs fall
on the edge of halos. If a FRB falls into multiple radial bins owing to
intersections with different halos, we only count this FRB once, grouping it
with the halo it intersects at the smallest impact parameter. We have tested
that counting the halo twice does not change our results.

###  2.2 Measuring DM excess

We aim to examine whether the FRBs with impact parameter $b<2\ R_{\rm vir}$
exhibit a statistically significant excess DM over the full sample of FRBs. A
simple estimator for the DM excess is the sample mean DM minus the mean of all
CHIME FRBs. However, since the number of FRBs with halo intersections is
small, one or two FRBs with high DM can significantly affect the sample mean
DM. For a sample size of 20, one FRB with DM $=2000$ pc cm-3 contributes 100
pc cm-3 to the sample mean, and the CHIME FRBs have a high-DM tail of DM
$\gtrsim 1500$ (see the black histogram in Figure 2; hereafter we omit the
unit of DM pc cm-3). An unweighted mean is thus not an ideal estimator for the
DM excess as high DM bursts add to the variance without contributing much
signal. Therefore, instead of calculating the sample mean DM, we compute a
weighted-average DM where we down-weight contributions from high-DM FRBs using
a flexible weighting function

$w({\rm DM})\propto\exp\left(-({\rm DM}/\alpha)^{\beta}\right),$ (1)

where $\alpha$ and $\beta$ shape how quickly the weight cuts off. If a sub-
sample of FRBs ($\mathcal{I}$) have a constant DM excess $\delta{\rm DM}$, an
estimate for $\delta{\rm DM}$ is

$\widetilde{\delta{\rm DM}}=\frac{1}{\sum_{i\in\mathcal{I}}w({\rm
DM}_{i})}\sum_{i\in\mathcal{I}}w({\rm DM}_{i})({\rm DM}_{i})-\overline{{\rm
DM}},$ (2)

where $i$ denotes the indices of FRBs and

$\overline{{\rm DM}}=\frac{1}{\sum_{i}w({\rm DM}_{i})}\sum_{i}w({\rm
DM}_{i}){\rm DM}_{i}$ (3)

is the weighted mean DM of the whole CHIME sample. This down-weighting is a
major difference of our work from Connor & Ravi (2021), who used uniform
weighting. Optimizing our weighting function allows us to detect a DM excess
with added precision.

One worry with such weighting is that sightlines with excess DM will receive
smaller weights, such that the weighted average will return a smaller excess.
Specifically, since our weighting function is a non-linear function of the DM
excess, if we input an array of DM values drawn from some the CHIME DM
distribution and artificially put in an excess $\delta$DM, the estimator
becomes

$\displaystyle\widetilde{\delta{\rm DM}}$
$\displaystyle=\frac{1}{\sum_{i}w({\rm DM}_{i})+\delta{\rm DM}}\sum_{i}w({\rm
DM}_{i}+\delta{\rm DM})({\rm DM}_{i}+\delta{\rm DM})$
$\displaystyle-\;\frac{1}{\sum_{i}w({\rm DM}_{i})}\sum_{i}w({\rm DM}_{i}){\rm
DM}_{i}.$

This does not return $\delta{\rm DM}$ but rather is biased by a factor that we
denote by

$f=\widetilde{\delta{\rm DM}}/\delta{\rm DM},$ (4)

which conveniently we find is nearly independent of the DM excess for
$\delta{\rm DM}<200$, a limit that is easily satisfied in models for the
foreground CGMs of galactic halos. This independence of $f$ is convenient as
it means that the correction does not depend on the true excess. For a sub-
sample $\mathcal{I}$ that has a constant DM excess $\delta$DM, an estimator
for $\delta$DM is given by iteratively solving

$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}=\frac{1}{\sum_{i\in\mathcal{I}}w\left({\rm
DM}_{i}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}\right)}\sum_{i\in\mathcal{I}}w\left({\rm
DM}_{i}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}\right)\ {\rm DM}_{i}-\overline{{\rm DM}}.$ (5)

This iterative estimator takes out the bias by removing the weightings’
dependence on $\delta DM$. For our DM excess of interest ($\lesssim 100$), the
solution roughly converges at

$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}\approx\widetilde{\delta{\rm DM}}/f.$ (6)

In the limit of a large sample, the above equation gives the unbiased excess,
$\delta{\rm DM}$. We use this estimator,
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$, to measure the DM excess of FRBs with CGM intersections. We
explore the best choices for $\alpha,\beta$ in what follows.

Figure 2: Top panel: contours of the standard deviation (std) of the weighted-
average DM as a function of the parameters of our weight function $\alpha$ and
$\beta$ (c.f. eqn 1), obtained by randomly sampling 20 FRBs from the CHIME DM
distribution. The gray shaded region represents $\alpha,\beta$ values that can
suppress all sampled DM values. The line at $\beta=0$ and red crosses indicate
the weighting functions we examine in the bottom panels: uniform weighting,
$(\alpha,\beta)=(600,4)$, $(\alpha,\beta)=(400,1)$. Left column of the
following rows: the black lines show the DM histogram of the CHIME FRBs, and
red lines represent the corresponding weighing functions in arbitrary units.
Right column: histogram of the excess weighted-average DM (equation 6), for
different values of $\alpha,\beta$, and uniform weighting (blue lines). Orange
lines represent Gaussian fits to the histograms. Also quoted are the standard
deviations of the weighted-mean DM. All DM units are in parsec cm-3.

We examine how much reduction in the variance of the weighted-average DM that
our weighting function can give. To this end, we created a mock FRB catalogue
of size $10^{5}$ that has the same DM distribution as the CHIME catalog. We
then randomly sample $N_{\rm DM}$ FRBs from this mock catalogue and for each
sample calculate
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$ using equation (6) for a range of $\alpha,\beta$ values. We
repeat this step 10,000 times and compute the standard deviation (std) of
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$ for each $\alpha,\beta$.

The top panel of Figure 2 shows contours of the std of
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$ as a function of $\alpha,\beta$, obtained by randomly sampling
$N_{\rm DM}=20$ FRBs from the CHIME DM distribution. Twenty is chosen to
roughly match the number of CHIME FRBs with halo intersections, although we
find that the contours of the std of
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$ remain mostly unchanged when assuming different $N_{\rm DM}$
such as $N_{\rm DM}=100$. The gray shaded region represents $\alpha,\beta$
values that can lead to all zero weights to machine precision, for at least
one time out of the 10,000 times of random sampling. The best estimator for
mean DM when sampling 20 FRBs seems to be given by values near
$(\alpha,\beta)=(100,1)$, but such a weighting falls off exponentially above
the low value of ${\rm DM}\sim 100$, selecting the few sightlines with the
smallest DM. Therefore, the minimum variance estimator is close to selecting
the minimum DM – for many distributions the minimum value in the DM array
converge faster than the mean. Such an estimator is problematic because the
minimum is more affected by problems such as some FRBs being at very low-
redshift in the foreground of our halos (e.g. the low-DM FRBs in Bhardwaj et
al., 2021a, b). We investigate this possibility in Appendix A, finding that it
is likely that $\sim 5$ of our FRBs out of 20-30 are likely to originate from
the galaxy in the halo that they are identified to intersect or from a
foreground galaxy; this contamination is problematic for the minimum variance
estimator. We thus explore $(\alpha,\beta)=(600,4)$ and $(400,1)$, which –
while giving substantial weight to about half of the FRBs – still give a
factor of $1.6$ and 2 reduction of the variance relative to an unweighted
estimator, respectively, and just modestly larger variances than the minimum
variance $(\alpha,\beta)=(100,1)$.555The bias factor $f$ of the estimator with
$(\alpha,\beta)=(600,4)$ is $0.65$, while $(400,1)$ is unbiased.

The black lines in the left column of Figure 2 illustrate the DM histogram of
the CHIME FRBs, and red lines represent these weighing functions in arbitrary
units. From top to bottom we use uniform weighting, $(\alpha,\beta)=(600,4)$,
and $(\alpha,\beta)=(400,1)$. Using $(\alpha,\beta)=(600,4)$ has a much
stronger cut-off at high DM values, leading to 20-30% of the CHIME DM
distribution contributing negligibly. The right panels of Figure 2 show the
histograms of
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$ by randomly sampling 20 FRBs from mocks. Blue lines illustrate
the histograms, and the orange lines represent Gaussian fits to the
histograms. Also quoted are the standard deviations of
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\delta{\rm
DM}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\delta{\rm
DM}}{\tmpbox}$. Since $(\alpha,\beta)=(400,1)$ gives a slightly more skewed
distribution of the weighted-mean DM with a high-value tail, by default we use
$(\alpha,\beta)=(600,4)$, but we have verified that $(\alpha,\beta)=(400,1)$
yields very similar conclusions.

Figure 3: Statistics of CHIME FRBs that intersect nearby $10^{11}-10^{12}\
M_{\odot}$ (blue) and $10^{12}-10^{13}\ M_{\odot}$ (red) halos within $1\
R_{\rm vir}$. Left panel: the blue and red histograms show the DM
distributions of the 26 and 30 FRBs intersecting the smaller and larger halo
mass bins, respectively. The black line illustrates the DM distribution of the
full CHIME sample. The gray dashed line represents our weighting function in
arbitrary units, which to calculate the stacked DM is effectively multiplied
by the DM distribution. Middle panel: The blue and red lines represent the
likelihood of our estimator $\widehat{\delta{\rm DM}}$ with
$(\alpha,\beta)=(600,4)$ for the whole CHIME sample, where the central value
is our measurement – the estimator applied to the observational data – and the
Gaussian PDF is calculated by random sampling of the CHIME DM distribution
with 26 and 30 FRBs respectively. Shaded regions show the 1-$\sigma$ and
2-$\sigma$ bounds of the distributions. Right panel: The same as the middle
panel, except that we have excluded FRBs that intersect M31 and M33, which
results in 20 FRBs intersecting $10^{11}-10^{12}\ M_{\odot}$ halos and 15
intersecting $10^{12}-10^{13}\ M_{\odot}$ ones.

Using the FRBs that intersect halos, we measure their excess DM by using the
estimator given by equation (6). The left panel of Figure 3 shows the DM
distributions of the 26 FRBs that intersect $10^{11}-10^{12}\ M_{\odot}$ halos
(blue) within $1\ R_{\rm vir}$, and the 30 FRBs intersecting $10^{12}-10^{13}\
M_{\odot}$ halos (red). The black line represents the DM distribution of the
whole CHIME sample. The gray dashed line represents our weighting function.
The middle panel shows the probability distributions of the measured DM
excess, centered on the solution of equation (6) with the std determined by
randomly sampling 26 (blue line) and 30 (red line) FRBs from the CHIME DM
distribution respectively. The shaded regions indicate the 1-$\sigma$ and
2-$\sigma$ bounds of the distributions. The excess DM of both halo groups
seems to be detected at $1.5$-$\sigma$ level. Owing to the large extent of M31
and M33 on the sky, we also performed our analysis without the FRBs
intersecting these galaxies. This gives 20 FRBs intersecting $10^{11}-10^{12}\
M_{\odot}$ halos and 14 intersecting $10^{12}-10^{13}\ M_{\odot}$ ones. The
right panel shows the distributions of the excess DM when excluding M31 and
M33. While the DM excess from $10^{12}-10^{13}\ M_{\odot}$ halos can still be
detected at $1.3$-$\sigma$ level, an insignificant excess in the
$10^{11}-10^{12}\ M_{\odot}$ mass bin is statistically preferred.

Our 1-$1.5$ $\sigma$ level detection of the foreground CGM is less significant
than the $>2$ $\sigma$ level of detection in Connor & Ravi (2021), even though
the std of our estimator is a factor of two smaller. However, there are two
FRBs with ${\rm DM}>2000$ intersecting $10^{11}-10^{12}\ M_{\odot}$ halos
(left panel of Figure 3). Without a weighting function, these FRBs raise the
sample mean DM by over 100, for a sample size of 30. This likely leads to the
t-test returning $p<0.05$ in Connor & Ravi (2021).

We performed the same analysis on the DM excess of FRBs intersecting halos
within impact parameters of $1-1.5\ R_{\rm vir}$ and $1.5-2\ R_{\rm vir}$. We
find that the FRBs intersecting $10^{11}-10^{12}\ M_{\odot}$ halos continue to
show a DM excess at $\gtrsim 1$-$\sigma$ level in these bins, while the
$10^{12}-10^{13}\ M_{\odot}$ halos no longer get a detection of DM excess.
Figure 4 summarizes our results. Dots with error-bars show the measured DM
excess as a function of $b/R_{\rm vir}$ assuming constant DM excess in each
radial bin. Left and right panels illustrate results for the $10^{11}-10^{12}\
M_{\odot}$ and $10^{12}-10^{13}\ M_{\odot}$ halos, respectively. The black and
gray colors indicate including all halos and excluding M31 and M33,
respectively. The $b$ values are calculated as the weighted-average $b$ of all
FRBs, with the weights given by our weighting function and the DM. As a
reference for the expected DM excess, this figure also shows are the radial DM
profiles for a model in which the CGM gas traces the NFW profile and another
where it is distributed as a top hat with radius of $2\,R_{\rm vir}$ (STH2)
model. Our DM excess measurements are largely consistent with the model
predictions. Both models are discussed in what follows.

Figure 4: The measured and model-predicted DM excess as a function of impact
parameter at different impact parameters $b$ to the foreground halo. Left and
right panels show our results for the $10^{11}-10^{12}\ M_{\odot}$ and
$10^{12}-10^{13}\ M_{\odot}$ halos, respectively. Black points with errorbars
illustrate the observed DM excess including all halos, and the gray points
excluding M33 (left panel) or M31 (right panel). As a reference for the
expected signal amplitude, the blue and orange lines show the radial DM
profiles in the NFW-tracing and the $2\,R_{\rm vir}$ spherical top hat (STH2)
models for the CGM gas, respectively (see § 2.3). These models are calculated
assuming halo masses of $4\times 10^{11}\ M_{\odot}$ (left) and $4\times
10^{12}\ M_{\odot}$ (right).

###  2.3 Comparison with CGM models

Since our constraints are in the ballpark of the expected CGM DM excess, it is
possible that they already rule out some CGM models. Suppose we are in the
limit of a very large number of FRBs, although with imprecise localization. We
expect the following relation to hold for the sub-sample ($\mathcal{I}$) of
FRBs that have CGM intersections:

$\frac{1}{\sum_{i}w_{i\in\mathcal{I}}}\sum_{i\in\mathcal{I}}\left[{\rm
DM}_{i}-\delta{\rm DM}_{i}(b,M)\right]w_{i}=\overline{{\rm DM}},$ (7)

where $i$ is the index of a FRB, ${\rm DM}$ is its measured DM, $\delta{\rm
DM}_{i}$ is its excess DM from foreground CGMs, and $w_{i}$ is the weight
calculated from the no-excess DM (i.e. from ${\rm DM}_{i}-\delta{\rm
DM}_{i}$). $\overline{{\rm DM}}$ is the weighted-average DM of the whole CHIME
sample. We note that unlike equation (5), $\delta{\rm DM}_{i}(b,M)$ models the
excess DM in terms of the impact parameter $b$ and halo mass $M$ of each
foreground galaxy (and not necessarily just one halo despite our notation).

We aim to find the model for $\delta{\rm DM}_{i}$ that best describes the
data. We therefore define a measure for the goodness of fit

$\xi^{2}=\left(\left\\{\frac{1}{\sum_{i\in\mathcal{I}}w_{i}}\sum_{i\in\mathcal{I}}\left[{\rm
DM}_{i}-\delta{\rm DM}_{i}(b,M)\right]w_{i}\right\\}-\overline{{\rm
DM}}\right)^{2}.$ (8)

We can also take the expectation value of $\xi^{2}$ by random sampling the
same number of intersections from the full DM distribution:

$\langle\xi^{2}\rangle=\Bigg{\langle}\left(\frac{1}{\sum_{i}w({\rm
DM}_{i})}\sum_{i}w({\rm DM}_{i}){\rm DM}_{i}-\overline{{\rm
DM}}\right)^{2}\Bigg{\rangle},$ (9)

where the summation goes over the desired number of FRBs $N_{\rm DM}$. The
expectation scales with $N_{\rm DM}$ as $\langle\xi^{2}\rangle\propto N_{\rm
DM}^{-1}$.

Our formalism still holds when dividing the FRBs into impact parameter bins
based on their $b/R_{\rm vir}$ values, as long as there are enough FRBs in a
bin. For each radial bin $j$, we can calculate the $\chi^{2}$ statistic by
dividing $\xi^{2}$ by $\langle\xi^{2}\rangle$ in that bin:

$\chi^{2}_{j}=\frac{\xi_{j}^{2}}{\langle\xi^{2}\rangle_{j}}\left(\mathbf{b}\right),$
(10)

which is a function of the set of impact parameters $\mathbf{b}$. Summing
these over radial bins gives the total:

$\chi^{2}=\sum_{j}\chi^{2}_{j}.$ (11)

The above is the procedure without accounting for localization error. However,
for some of our intersections, the localization is comparable to the halo
viral radius. To take into account of the localization uncertainties, we
calculate the marginalized $\chi^{2}$ value of bin $j$:

$\exp\left(-\chi^{2}_{j}/2\right)=\int\exp\left(-\frac{1}{2}\frac{\xi^{2}_{j}}{\langle\xi^{2}\rangle_{j}}\left(\mathbf{b}\right)\right)\
P(\mathbf{b})\ \mathrm{d}\mathbf{b}.$ (12)

The total $\chi^{2}$ is

$\exp\left(-\chi^{2}/2\right)=\int\exp\left(-\frac{1}{2}\sum_{j}\frac{\xi^{2}_{j}}{\langle\xi^{2}\rangle_{j}}\left(\mathbf{b}\right)\right)\
P(\mathbf{b})\ \mathrm{d}\mathbf{b}.$ (13)

Given a model of the CGM, the radial profile of DM around a sample of halos
has a one-halo term owing to the gas surrounding these halos themselves, and
also a rather flat two-halo term coming from the overlapping of gas from other
halos. We use the DM radial profiles calculated using the CGMBrush algorithm
(Williams et al., 2022) for our $\chi^{2}$ calculation. This algorithm
subtracts off the dark matter associated with each halo in an N-body
simulation and then pastes on different models for the distribution of the gas
associated with that halo. It assumes that gas outside of halos traces the
dark matter.666To compute the total DM excess for a FRB for the few sightlines
that have multiple intersections, we first sum up the one-halo terms from all
the nearby $10^{11}-10^{13}\ M_{\odot}$ halos that this FRB can intersect
along its sightline. We compute an average two-halo term using these halos and
add to the total excess DM, since an FRB may intersect multiple halos at $>1\
R_{\rm vir}$. It perhaps makes the most sense to take the maximum value of the
two-halo terms if the intersecting halos are within several impact parameters,
and to sum the two halo the two-halo terms if halos are further apart than
several impact parameters. However, the exact algorithm does not affect our
results by $\Delta\chi^{2}>1$. Using these models for the DM excess, we
calculate $\chi^{2}$ values using equations (8) and (12) for FRBs in each of
the three radial bins $[0,1],[1,1.5],[1.5,2]$, in units of $R_{\rm vir}$. To
speed up the calculation, we restrict to using FRBs that lie within $(2\
R_{\rm vir}+3\times 0.2^{\circ})$ of galaxies, where $0.2^{\circ}$ is the
typical localization error of CHIME. As mentioned in Section 2.1, we count
each FRB only once if any intersects multiple halos. Counting galaxy-FRB pairs
instead leads to somewhat larger $\Delta\chi^{2}$, but with differences
smaller than 1-2. To evaluate the high-dimension integral on the right hand
side of equation (12) or equation (13), we draw from a 2D Gaussian for the
location of each FRB on the sky plane so that the draws trace $P(\mathbf{b})$
and sum up $10^{4}$ realizations for a Monte Carlo evaluation.

We perform this $\chi^{2}$ calculation for the $2\,R_{\rm vir}$ spherical top
hat (STH2) model of the CGM and the NFW model. These extreme models roughly
bound the DM excess predicted by the more realistic models in Williams et al.
(2022). The STH2 model has only an eighth of the halo-associated baryons
within $1\,R_{\rm vir}$, whereas the NFW contains all of them within
$1\,R_{\rm vir}$. Indeed, owing to its diffuseness, we find that the STH2 DM
excess is shaped at most radii by the two-halo term and not the halo profile.
Figure 4 illustrates DM excess as a function of $b/R_{\rm vir}$ predicted by
the NFW (blue lines) and the STH2 (orange lines) models. These illustrative
curves in the left and right panels assume halo masses of $4\times 10^{11}\
M_{\odot}$ and $4\times 10^{12}\ M_{\odot}$ respectively, roughly the mean of
our two halo samples. Despite these two profile making much different
assumptions for the halo associated baryons, these models give similar
predictions outside $0.5\ R_{\rm vir}$, with DM differences smaller than 50
because of the large two-halo excess that depends weakly on the halo gas
profiles. Our sample contains only 7 and 4 FRBs intersecting with $b<0.5\
R_{\rm vir}$ the small and large halo mass bins respectively. The STH2 model
predicts a flat DM excess of 30-50 at $0.5-1\ R_{\rm vir}$ for the
$10^{11}-10^{12}\ M_{\odot}$ halos, and 50-80 for the $10^{12}-10^{13}\
M_{\odot}$ halos. The NFW model produces 30-50 higher DM excess. The two-halo
term is about 30-50 for both halo mass bins outside $1\ R_{\rm vir}$. Because
the DM differences among models are only $\gtrsim 50$, we expect the
$\chi^{2}$ values of these two models (and other models in Williams et al.
2022) to differ only at $<1$-$\sigma$ level.

Table 2: $\chi^{2}$ and $\Delta\chi^{2}$ values of different models in each impact parameter ($b$) bin, for the $10^{11}-10^{12}\ M_{\odot}$ (upper) and $10^{11}-10^{12}\ M_{\odot}$ (lower) halos. From top to bottom: $b\in[0,1],[1,1.5],[1.5,2]\ R_{\rm vir}$, and the total $\chi^{2}$ considering all bins. For each $\chi^{2}$ value we list the $\chi^{2}$ values by marginalizing over localization uncertainties, and the number in brackets shows the $\chi^{2}$ obtained by using the maximum likelihood localization. Second column: $\chi^{2}$ of the no DM excess model. Third and fourth columns: $\Delta\chi^{2}$ of no DM excess minus the $2R_{\rm vir}$ spherical top hat model (STH2), and the NFW model. Fifth column: $\Delta\chi^{2}$ of no DM excess minus the NFW model, excluding M33 or M31. $10^{11}-10^{12}\ M_{\odot}$ | $\chi^{2}$ | $\Delta\chi^{2}$ | $\Delta\chi^{2}$ |  $\Delta\chi^{2}$ no M33
---|---|---|---|---
| no DM exc | no DM exc - STH2 | no DM exc - NFW | no DM exc - NFW
$[0,1]\ R_{\rm vir}$ | 2.3 (1.1) | 1.6 (1.0) | 1.6 (0.7) | 0.5 (0.0)
$[1,1.5]\ R_{\rm vir}$ | 1.7 (4.1) | 1.2 (3.2) | 1.2 (3.0) | 0.8 (2.3)
$[1.5,2]\ R_{\rm vir}$ | 1.3 (0.8) | 0.8 (0.8) | 0.9 (0.8) | 0.7 (0.5)
total | 6.3 (6.0) | 4.5 (5.0) | 4.6 (4.5) | 2.6 (2.8)
$10^{12}-10^{13}\ M_{\odot}$ | $\chi^{2}$ | $\Delta\chi^{2}$ | $\Delta\chi^{2}$ |  $\Delta\chi^{2}$ no M31
| no DM exc | no DM exc - STH2 | no DM exc - NFW | no DM exc - NFW
$[0,1]\ R_{\rm vir}$ | 1.7 (2.0) | 1.5 (1.9) | 1.4 (1.9) | 1.6 (1.9)
$[1,1.5]\ R_{\rm vir}$ | 0.3 (0.0) | -0.7 (-1.3) | -0.5 (-1.0) | -0.6 (-0.5)
$[1.5,2]\ R_{\rm vir}$ | 1.0 (1.2) | 0.8 (1.2) | 0.8 (1.2) | -1.1 (-1.2)
total | 3.1 (3.2) | 1.8 (1.8) | 1.7 (2.2) | -0.3 (0.1)

Table 2 lists the $\chi^{2}$ values as well as their differences,
$\Delta\chi^{2}$, in each radial bin. It also lists the total $\chi^{2}$
summing over all bins. By default, we calculate $\chi^{2}$ by marginalizing
localization errors, but we show the $\chi^{2}$ computed using maximum
likelihood localization in brackets (i.e. assuming $P(\mathbf{b})$ is a
$\delta$-function at the best fit value).777We note that technically
$\chi^{2}\geq\sum_{j}\chi^{2}_{j}$, with equality achieved when
$P(\mathbf{b})$ is a delta function. This is why the total $\chi^{2}$
differences are larger than summing up the $\Delta\chi^{2}$ values in
individual bins. Not marginalizing over localization errors generally changes
the results at the $<1\sigma$ level, but in some cases the differences are
somewhat larger showing that accounting for positional errors is important.
The $\Delta\chi^{2}$ values are calculated by subtracting the $\chi^{2}$ of
the no DM excess model with that of the STH2 or NFW model. We have also
verified that further dividing the radial bins (but keeping the number of FRBs
in a bin at least 15-20) do not affect the $\Delta\chi^{2}$ values. With 3
radial bins we would expect total $\chi^{2}\sim 3$, and the models produce a
slight over-fit of total $\chi^{2}\sim 1.5$.

The $\chi^{2}$ in all radial bins of the $10^{11}-10^{12}\ M_{\odot}$ halos
disfavors the no DM excess model at $\sqrt{\Delta\chi^{2}}\approx 1$-$\sigma$
level, although the significance level drops when excluding M33. The total
$\chi^{2}$ of NFW and STH2 models considering all bins is smaller than that of
the no DM excess case by 2-5, regardless of marginalizing over the
localization uncertainties or using maximum likelihood localization. Thus the
significance that a DM excess is preferred over no excess ranges from 1.4-2.2
$\sigma$. The $10^{12}-10^{13}\ M_{\odot}$ halos, on the other hand, favor a
DM excess only in the $[0,1]\ R_{\rm vir}$ bin and cannot distinguish any
models for the larger impact-parameter bins. The $\Delta\chi^{2}$ values are
consistent with our results in Figure 3. Finally, the $\chi^{2}$ of the STH2
and NFW models are very similar, as anticipated, and so not distinguishable by
the data.

###  2.4 How many FRBs are needed to get a significant detection of DM excess

Thus, we have found a marginal detection of an excess DM when stacking on
foreground halos, but the data set of 453 FRBs is insufficient to distinguish
between realistic CGM models. Larger samples are of course required to get a
more significant detection of a given DM excess of 50-100 – the anticipated DM
excess for the CGM of $10^{11}-10^{13}\ M_{\odot}$ halos and this is also the
difference between viable models for $b<0.5\ R_{\rm vir}$. Since we currently
get an error bar in DM of 50 with $\approx 30$ intersections, we anticipate
that the error bar scales with the number of intersections as $\sigma_{\rm
DM}=50\,\sqrt{30/N_{\rm DM}}$. With 200 intersections then $\sigma_{\rm
DM}=20$; this many intersections would require about 3000 FRBs in the next
CHIME data release.

Currently the number of sightlines passing through halos at $<0.5\ R_{\rm
vir}$ is too small, but this is the region where viable models for the CGM gas
profile differ the most. With about $10^{4}$ more FRBs from CHIME, the number
of intersections with $10^{11}-10^{13}\ M_{\odot}$ halos at $b<0.5\ R_{\rm
vir}$ will reach $\sim 200$. Such a sample would be able to put novel
constraints on CGM models.

##  3 Conclusions

We have measured the DM excess owing to the CGM of $10^{11}-10^{13}\
M_{\odot}$ halos at $<80$ Mpc using the CHIME/FRB first data release. To this
end, we have developed a weighted stacking scheme to reduce the variance of
the observed DM distribution and to lower the bias of high-DM FRBs on the
sample mean DM. With 20-30 FRBs intersecting $10^{11}-10^{12}$ and
$10^{12}-10^{13}\ M_{\odot}$ halos at $<1\ R_{\rm vir}$, we find that the DM
excess of these halo groups can be detected at 1-2 $\sigma$. We also
tentatively detect a DM excess at impact parameters of 1-2 $R_{\rm vir}$ for
$10^{11}-10^{12}\ M_{\odot}$ halos, but not for the $10^{12}-10^{13}\
M_{\odot}$ halos. With more FRB data from CHIME coming in the near future,
each stack’s signal-to-noise ratio will continue to improve with an error of
$\sigma_{\rm DM}=50\,\sqrt{30/N_{\rm DM}}$ pc cm-3, where $N_{\rm DM}$ is the
number of intersections (which was approximately $30$ in our stacks).

We have also calculated the likelihood of different CGM models given the CHIME
FRB data. All models are favored by the data at 1.4-2.2 $\sigma$ over a model
with no DM excess, consistent with our measurements of a DM excess. We find
that viable models for the CGM gas distribution produce DM differences smaller
than 50 outside $0.5\ R_{\rm vir}$ because of the importance of the two-halo
term around $10^{11}-10^{13}\ M_{\odot}$ halos. Owing to the paucity of
intersections at these impact parameters, a large increase in the number of
FRBs to at least ten thousand would be needed to discriminate between viable
CGM model with our method that stacks on nearby galaxies.

In the next few years, our weighted stacking method will continue to be a
valuable tool to detect and measure the CGM of nearby halos as more data from
CHIME comes along. Especially for the relatively low-mass ones as we
considered here, whose CGM is hard to probe in any other way. With new surveys
launching that aim to observe FRBs with arcsecond localizations, other
methodologies may provide better constraints on the CGM gas profile. For
instance, the Canadian Hydrogen Observatory and Radio-transient Detector
(CHORD Vanderlinde et al., 2019) will receive $>20$ bursts per day, with the
goal to provide milli-arcsecond localization accuracy of FRBs with VLBI.
DSA-2000 will also start observing $\sim 75\%$ of the full sky with arcsecond
spatial resolution (Hallinan et al., 2019). Once localized to a galaxy, the
mean cosmic dispersion to a redshift can be removed from each FRB, which will
dramatically reduce the noise in the stack allowing constraints with just
hundreds of bursts (McQuinn, 2014; Williams et al., 2022). It might even be
possible to forward model each component (intergalactic, Milky Way, host
galaxy) of the FRB DM instead of performing weighted stacking. Especially, the
intergalactic contribution might be modeled with a reconstruction of the
underlying density field (e.g. Burchett et al., 2020). The improved angular
resolution of these surveys will also allow stacking on galaxies that are
further away than in this study. Another interesting related direction is to
constrain the fraction of cool ionized gas via scattering and lensing of FRBs
(Vedantham & Phinney, 2019; Prochaska & Zheng, 2019).

## Acknowledgements

We thank Ian Williams for providing the CGMBrush profiles, Matt Wilde for
offering the NSA galaxy catalog, and Bryan Gaensler, Yakov Faerman, Ue-Li Pen,
Sandro Tacchella for useful discussions. We acknowledge support from NSF award
AST-2007012.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Possibility of FRBs originating from nearby halos

While we have implicitly assumed that all the FRBs used in this work should
originate from halos further away than 40-80 Mpc, it is likely that some FRBs
may come from nearby halos, especially low-DM (DM $\lesssim 100$) ones
(Bhardwaj et al., 2021a, b). Even for FRBs with DM of several hundred, it is
not unlikely that they are from nearby galaxies because the host DM may well
be a few hundred (Table 2 of Cordes et al., 2022).

To address this issue, we ran the Probabilistic Association of Transients to
Hosts (PATH) code888https://github.com/FRBs/astropath to determine which FRBs
might originate from $<80$ Mpc galaxies (Aggarwal et al., 2021). PATH
calculates the probability that an extragalactic transient source is
associated with a candidate host galaxy using the Bayes’ rule. For each FRB,
we take all $<80$ Mpc galaxies in GWGC that lie within 3 times the FRB RA Dec
error-bars to be the candidates which the FRB can be associated with. Instead
of using the “inverse” prior that assumes brighter galaxies have higher
probabilities, we adopt a uniform prior. We assume that the distribution of
transients around galaxies follows an exponential profile, where the size of
the exponential function is given by the tabulated major and minor diameters
of galaxies in GWGC. PATH then integrates over the FRB localization ellipse,
and finds the galaxy with a $>0.95$ posterior to be the most likely host of
the FRB. It does not take into account the FRB DM, however.

For the 26 (30) FRBs that we find to pass through nearby $10^{11}-10^{12}\
M_{\odot}(10^{12}-10^{13}\ M_{\odot})$ halos at $<1\ R_{\rm vir}$, PATH shows
that 6 (1) of them might originate from these halos themselves. We find 7 more
FRBs likely associated with the intersected galaxies when focusing on $b<2\
R_{\rm vir}$, and 3 FRBs may arise from halos closer to the ones they are
found to intersect with. We thus performed our measurement of DM excess again
without these FRBs identified, and find that our main conclusions remain
unchanged. The PATH code, however, is only a rough way of estimating the
probability of FRBs being associated with galaxies and much more careful
visual inspection on the locations of galaxies versus the localization
contours of FRBs should be done before drawing conclusions.