# Inferring the Energy and Distance Distributions of Fast Radio Bursts using
the First CHIME/FRB Catalog

Kaitlyn Shin MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, USA  Department of Physics, Massachusetts Institute of Technology, 77
Massachusetts Ave, Cambridge, MA 02139, USA  Kiyoshi W. Masui MIT Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA  Department of
Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave,
Cambridge, MA 02139, USA  Mohit Bhardwaj McGill Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Department of
Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
Tomas Cassanelli Department of Electrical Engineering, Universidad de Chile,
Av. Tupper 2007, Santiago 8370451, Chile  Pragya Chawla Anton Pannekoek
Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  Matt Dobbs Department of Physics, McGill
University, 3600 rue University, Montréal, QC H3A 2T8, Canada  McGill Space
Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7,
Canada  Fengqiu Adam Dong Department of Physics and Astronomy, University of
British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1 Canada
Emmanuel Fonseca Department of Physics and Astronomy, West Virginia
University, PO Box 6315, Morgantown, WV 26506, USA  Center for Gravitational
Waves and Cosmology, West Virginia University, Chestnut Ridge Research
Building, Morgantown, WV 26505, USA  B. M. Gaensler Dunlap Institute for
Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada  David A. Dunlap Department of Astronomy &
Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S
3H4, Canada  Antonio Herrera-Martín David A. Dunlap Department of Astronomy &
Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S
3H4, Canada  Department of Statistical Sciences, University of Toronto, 700
University Ave., Toronto, ON M5G 1Z5, Canada  Jane Kaczmarek Dominion Radio
Astrophysical Observatory, Herzberg Research Centre for Astronomy and
Astrophysics, National Research Council Canada, PO Box 248, Penticton, BC V2A
6J9, Canada  Victoria Kaspi Department of Physics, McGill University, 3600 rue
University, Montréal, QC H3A 2T8, Canada  McGill Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Calvin Leung
MIT Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts
Ave, Cambridge, MA 02139, USA  Marcus Merryfield Department of Physics, McGill
University, 3600 rue University, Montréal, QC H3A 2T8, Canada  McGill Space
Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7,
Canada  Moritz Münchmeyer Department of Physics, University of Wisconsin-
Madison, 1150 University Ave, Madison, WI 53706, USA  Aaron B. Pearlman
Department of Physics, McGill University, 3600 rue University, Montréal, QC
H3A 2T8, Canada  McGill Space Institute, McGill University, 3550 rue
University, Montréal, QC H3A 2A7, Canada  Masoud Rafiei-Ravandi Department of
Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
McGill Space Institute, McGill University, 3550 rue University, Montréal, QC
H3A 2A7, Canada  Kendrick Smith Perimeter Institute of Theoretical Physics, 31
Caroline Street North, Waterloo, ON N2L 2Y5, Canada  Shriharsh P. Tendulkar
Department of Astronomy and Astrophysics, Tata Institute of Fundamental
Research, Mumbai, 400005, India  National Centre for Radio Astrophysics, Post
Bag 3, Ganeshkhind, Pune, 411007, India Kaitlyn Shin<EMAIL_ADDRESS>
###### Abstract

Fast radio bursts (FRBs) are brief, energetic, extragalactic flashes of radio
emission whose progenitors are largely unknown. Although studying the FRB
population is essential for understanding how these astrophysical phenomena
occur, such studies have been difficult to conduct without large numbers of
FRBs and characterizable observational biases. Using the recently released
catalog of 536 FRBs published by the Canadian Hydrogen Intensity Mapping
Experiment/Fast Radio Burst (CHIME/FRB) collaboration, we present a study of
the FRB population that also calibrates for selection effects. Assuming a
Schechter luminosity function, we infer a characteristic energy cut-off of
$E_{\mathrm{char}}=$ $2.38^{+5.35}_{-1.64}\times 10^{41}$ erg and a
differential power-law index of $\gamma=$ $-1.3^{+0.7}_{-0.4}$.
Simultaneously, we infer a volumetric rate of
[$7.3^{+8.8}_{-3.8}$(stat.)${}^{+2.0}_{-1.8}$(sys.)]$\times 10^{4}$ bursts
Gpc-3 year-1 above a pivot energy of 1039 erg and below a scattering timescale
of 10 ms at 600 MHz, and find we cannot significantly constrain the cosmic
evolution of the FRB population with star formation rate. Modeling the host
dispersion measure (DM) contribution as a log-normal distribution and assuming
a total Galactic contribution of 80 pc cm-3, we find a median value of
$\mathrm{DM}_{\mathrm{host}}=$ $84^{+69}_{-49}$ pc cm-3, comparable with
values typically used in the literature. Proposed models for FRB progenitors
should be consistent with the energetics and abundances of the full FRB
population predicted by our results. Finally, we infer the redshift
distribution of FRBs detected with CHIME, which will be tested with the
localizations and redshifts enabled by the upcoming CHIME/FRB Outriggers
project.

##  1 Introduction

Ever since fast radio bursts (FRBs) were first discovered (Lorimer et al.,
2007; Thornton et al., 2013), their mysterious origin has been an open
question with no simple resolution. Studying FRBs as a population can provide
valuable insight into their nature, as well as their utility for cosmological
applications (e.g., Masui & Sigurdson, 2015; Madhavacheril et al., 2019). A
natural way to study FRBs is through fitting their luminosity function, which
can tell us about observables such as volumetric rate and maximum FRB energy
(Luo et al., 2020). The fitted parameters can be compared with those of other
known astrophysical events to better contextualize the question of FRB
origins. However, populations studies of FRBs are challenging — as a
population, FRBs have been observed in a variety of observational contexts; it
is yet unknown how many distinct populations there may be, and whether there
is a dominant progenitor channel (James et al., 2022a). For example, some FRBs
have been observed to repeat, while many more have only been observed as
apparent one-off bursts. Indeed, recent studies have put forth the
possibility, on the basis of observational properties such as pulse widths and
observed frequency bandwidths, that repeaters and non-repeaters may comprise
two distinct populations of FRBs (e.g., Cui et al., 2021; Pleunis et al.,
2021).

Furthermore, FRBs have been found to be associated with a variety of types of
galaxies, and in some cases, localized to Galactic regions with
$\lesssim$arcsecond precision. Repeaters have been shown to originate both
within star-forming regions (e.g., Marcote et al., 2020; Nimmo et al., 2022)
and offset by $\sim$250 pc from the closest knot of local star formation
(e.g., Tendulkar et al., 2021). By contrast, the repeating FRB 20200120E was
localized to a globular cluster associated with M81 (Bhardwaj et al., 2021;
Kirsten et al., 2022), surprising given that globular clusters host a much
older stellar population than do star-forming regions. FRB-like bursts have
also been observed from the Galactic magnetar SGR 1935+2154 (CHIME/FRB
Collaboration et al., 2020a; Bochenek et al., 2020; Kirsten et al., 2021). It
is possible that more such FRB-like events occur in the Milky Way, but remain
undetected due to interstellar medium (ISM) scattering (Gohar & Flynn, 2022).
FRB localizations are not limited to repeating bursts; apparent one-offs have
also been localized to host galaxies, and in some cases, the localizations are
precise enough to be associated with regions of low star formation rate as
well (e.g., Bannister et al., 2019; Heintz et al., 2020; Bhandari et al.,
2020; Li & Zhang, 2020; Mannings et al., 2021). A comprehensive overview of
the breadth of such localization results can be found in Petroff et al.
(2022).

Nonetheless, there are many more FRBs that have been detected than just the
$\sim$20 FRBs robustly associated with host galaxies. By 2020, there were over
100 verified FRBs111https://www.frbcat.org (Petroff et al., 2016), and as of
writing, the number of verified FRB events is $\sim$800; a full catalog of
FRBs is maintained at the Transient Name Server (TNS).222https://wis-
tns.weizmann.ac.il Based on these numbers alone, it is clear that studying
FRBs as a population based only on the sub-sample of FRBs with precise
localizations has strong limitations — that is, optical follow-up is time-
intensive and expensive, and cannot currently be feasibly pursued for the
hundreds of detected FRBs. There is also valuable information possessed by the
many FRBs that do not have host galaxy associations, e.g., FRB energetics and
abundances. Using a larger sample of FRBs adds valuable information, and
should be very beneficial for statistically constraining how FRB energies are
distributed. However, the central challenge to measuring the FRB luminosity
function is, again, the lack of precise distance (localization) information
for the majority of FRBs. Without a good handle on distance, the FRB
energetics, distances, and volumetric abundances are all degenerate.

For FRBs, the dispersion measure ($\mathrm{DM}$) is the (imperfect) distance
proxy that allows us to break this modeling degeneracy. Since we expect
$\mathrm{DM}$ to correlate with distance, while brightness will anti-
correlate, any correlation/anti-correlation in modeling between $\mathrm{DM}$
and brightness should be a distance effect, thus providing a distance scale.
As the uncertainties when modeling FRB energetics, distances, and abundances
all effect each other, it is important to work with these properties
simultaneously to minimize bias. Luo et al. (2020) and James et al. (2022b)
have recently implemented this approach to studying the FRB population.

Previously, in order to conduct large-number statistical studies of the FRB
population, it was necessary to combine burst detections from multiple surveys
or telescopes such as the Australian Square Kilometre Array Pathfinder (ASKAP)
or the Parkes radio telescope (e.g., Luo et al., 2020; James et al., 2022b).
Using a heterogeneously observed sample of FRBs introduces differing, non-
uniform selection effects that are difficult to calibrate. This motivates the
use of FRBs observed from a single observing campaign to study the FRB
population as a whole. Recently, the Canadian Hydrogen Intensity Mapping
Experiment/Fast Radio Burst (CHIME/FRB) collaboration released a catalog of
536 FRBs, the largest sample of bursts detected thus far in a single survey,
and hereafter referred to as “Catalog 1” (CHIME/FRB Collaboration et al.,
2021). With Catalog 1, we now possess the ability to conduct a statistical
study with uniform selection effects and probe the intrinsic FRB population
with greater numbers and logistical simplicity. Crucially, the selection
effects are carefully characterizable with the use of the CHIME/FRB injections
system (Merryfield et al., 2022). With 536 bursts, the Catalog 1 sample
dominates the currently available sample of observed FRBs.

In this paper, we present a study of FRB population parameters, including the
intrinsic luminosity function, derived from the sample of Catalog 1 bursts. We
fit a joint brightness–$\mathrm{DM}$ model developed by James et al. (2022b),
and use formalism that accounts for selection effects detailed by CHIME/FRB
Collaboration et al. (2021). The outline of this paper is as follows. In
Section 2, we describe the datasets used for this analysis. Section 3 outlines
key components of the model we are fitting to the Catalog 1 dataset, using the
methodology described in Section 4. Results are presented in Section 5 and
discussed in Section 6. Implications of our results for the CHIME/FRB
Outriggers project are presented in Section 7, and we conclude in Section 8.
We adopt Planck cosmological parameters (Planck Collaboration et al., 2020)
throughout this analysis.

##  2 Dataset

The data used in this work consist of observed bursts from the CHIME/FRB
Catalog 1 sample, corrected for selection effects in order for our analyses to
accurately reflect properties of the intrinsic population of FRBs as much as
possible. The following subsections will detail the datasets used for this
work.

###  2.1 CHIME/FRB Catalog 1 observations

CHIME/FRB is a collaboration that uses the Canadian Hydrogen Intensity Mapping
Experiment (CHIME; The CHIME Collaboration et al., 2022), located at the
Dominion Radio Astronomical Observatory (DRAO). The telescope consists of four
20-m $\times$ 100-m cylindrical paraboloid reflectors, each oriented N-S and
populated with 256 equispaced dual-linear-polarization antennae. This
telescope is sensitive to bursts in the frequency range 400–800 MHz. Further
technical details of the CHIME/FRB system can be found by CHIME/FRB
Collaboration et al. (2018).

The Catalog 1 sample released by CHIME/FRB contains 536 bursts observed
between from 2018 July 25 to 2019 July 1. In the work presented by CHIME/FRB
Collaboration et al. (2021), the population analysis of Catalog 1 bursts
considered six observed FRB properties measured for each burst: fluence ($F$),
dispersion measure ($\mathrm{DM}$), scattering timescale ($\tau$), pulse width
($w$), spectral index (denoted with $\gamma$ in CHIME/FRB Collaboration et al.
(2021), but denoted with $\alpha$ in this paper), and spectral running ($r$).
These observed properties were modeled by a least-squares fitting routine
called fitburst, which processes the intensity data of FRBs offline and fits
the two-dimensional dynamic spectrum with a 2D analytic model. For bursts with
one or more components $i$, fitburst models $\mathrm{DM}$, burst time of
arrival ($t_{\mathrm{arr},i}$), signal amplitude ($A_{i}$), temporal width
($w_{i}$), power-law spectral index ($\alpha_{i}$), spectral running
($r_{i}$), and scattering timescale ($\tau$). The same $\mathrm{DM}$ and
$\tau$ are assumed for all sub-bursts, and we also assume
$\tau\propto\nu^{-4}$ (Lang, 1971; Lorimer & Kramer, 2012), with 600 MHz as
the reference frequency for scattering. For each FRB, fitburst was run twice:
once with $\tau$ fixed to 0, and once allowing $\tau$ to vary. In order to
choose the preferred model, an $F$-test333$F$-tests are good for comparing the
statistics of nested models, and the model for bursts with scattering can be
interpreted as nested from the model for bursts without scattering. The code
for fitburst is not yet public. was used to compare the $\chi^{2}$ values from
both models, adopting a threshold of $p<0.001$ to determine the significance
of scattering. Electron density models of the Milky Way $\mathrm{DM}$, NE2001
(Cordes & Lazio, 2002) and YMW2016 (Yao et al., 2017), were used in
combination with the $\mathrm{DM}$ value determined by fitburst to validate
the extragalactic nature of each FRB. Further details about fitburst can be
found in Section 3.3 of CHIME/FRB Collaboration et al. (2021).

One further thing to note about CHIME/FRB observations is that fluence
measurements are uncertain estimates of the true fluence, limited primarily by
burst localization uncertainty and, to a lesser extent, by beam model
uncertainty. While these fluence measurements are biased low (Section 3.4 in
CHIME/FRB Collaboration et al., 2021), our analysis is not affected by this
systematic because we use $\mathrm{S/N}$ as a more reliable observable proxy
to study fluence. Further discussion of our treatment of fluence values is in
Sections 2.3 and 3.1.

###  2.2 CHIME/FRB Catalog 1 injections

In order to properly characterize and account for instrumental effects when
inferring the intrinsic distributions of FRB properties, a synthetic signal
injection system was designed and implemented (Merryfield et al., 2022).
Synthetically generated pulses were injected into the software detection
system in order to allow for the accounting of real-time detection effects
such as the RFI environment. These pulses will be hereafter referred to as
“injections.”

There are some important properties to note for the population of synthetic
bursts injected into the real-time detection pipeline, which are briefly noted
here but elaborated on by CHIME/FRB Collaboration et al. (2021). Of the
$N_{\mathrm{inj}}=5\times 10^{6}$ total injected FRBs assigned random
locations on the sky, some locations were automatically discarded: locations
below the horizon, where the band-averaged primary beam response was
$<10^{-2}$, and where the response was $<10^{-3}$ for all 1024 synthesized
beams. The fraction of surviving sky locations is $f_{\mathrm{s}ky}=0.0277$.
This “forward-modeling” method, using the beam models to simulate
$f_{\mathrm{sky}}$, was employed to account for the beam response of the
telescope on the sky. Each of these FRBs had properties drawn from initial
probability density functions $P_{\mathrm{init}}(F)$,
$P_{\mathrm{init}}(\mathrm{DM})$, $P_{\mathrm{init}}(\tau)$,
$P_{\mathrm{init}}(w)$, and $P_{\mathrm{init}}(\alpha,r)$, designed to fully
and densely sample phase space of observed properties in the catalog. Of these
$5\times 10^{6}$ injected FRBs, a cut was applied to events with little chance
of being detected, and 96,942 events were ultimately scheduled for injection.
Due to system errors that affected an effectively random subset of injections,
84,697 bursts were successfully injected with an injection efficiency
$\epsilon_{\mathrm{inj}}=0.874$ in 2020 August, with 39,638 events detected
with a signal-to-noise ratio ($\mathrm{S/N}$) assigned by the CHIME/FRB
detection pipeline, where detected events had $\mathrm{S/N}$ $>12$. Full
technical details on how the injections are detected by the real-time pipeline
can be found in CHIME/FRB Collaboration et al. (2021) and Merryfield et al.
(2022).

###  2.3 Determining the fiducial property distributions

A primary goal of the initial Catalog 1 populations analysis was to fit a
fiducial model to the intrinsic FRB property distributions after correcting
for selection effects. Such a model would be an imperfect, but still
reasonable, match to the data. However, although the Catalog 1 sample makes up
the first large sample of FRBs observed in a single survey with uniform
selection effects, a number of bursts are affected by certain selection
effects that are unquantifiable (e.g., non-nominal telescope operation). If we
want to characterize the population while measuring and compensating for
selection effects, we must only use the bursts with the most robustly known
statistics. Therefore, bursts affected by the following criteria were excluded
from both the Catalog 1 sample as well as from the corresponding injections
campaign sample:

  1. 1.

Events detected during pre-commissioning, during epochs of low sensitivity, or
on days with software upgrades. These events were correspondingly excluded
from the Catalog 1 survey duration time, $\Delta t$.

  2. 2.

Far-sidelobe events, for which the primary beam response is poorly understood.

  3. 3.

Events with a detected $\mathrm{S/N}$ $<$ 12, to avoid human inspection error
affecting the completeness of identifying low $\mathrm{S/N}$ FRBs.

  4. 4.

Events with $\mathrm{DM}<1.5\;\mathrm{max(DM_{NE2001},DM_{YMW16})}$, to avoid
mistakenly classifying extragalactic bursts as Galactic or vice versa.

  5. 5.

Events with $\mathrm{DM}<100$ pc cm-3.

  6. 6.

Highly scattered events ($\tau>10$ms at 600 MHz), which are poorly
observationally constrained due to a strong selection bias against them from
the CHIME instrument, which would dominate uncertainties.

  7. 7.

Repeat bursts after the first-detected burst from any known repeating FRBs in
the sample.

The first exclusion criterion (requiring $\mathrm{S/N}$ $\geq$ 12) dominates
the 271 total excluded bursts in the populations analysis conducted by
CHIME/FRB Collaboration et al. (2021). All of the above exclusion criteria are
used in this analysis as well; for more information about these cuts, refer to
Sections 5 and 6.1 in CHIME/FRB Collaboration et al. (2021). One further
exclusion criterion that was used by CHIME/FRB Collaboration et al. (2021) is
used in this analysis: Events with $\mathrm{DM}_{\mathrm{NE2001}}>100$ pc
cm-3, in order to minimize the variance in our results involved with the Milky
Way contribution to the $\mathrm{DM}$. This was done because in our model, we
take a fixed value of $\overline{\mathrm{DM}}_{\mathrm{MW}}$ to be 50 pc cm-3;
further discussion on this can be found in Section 5.3.1. It is worth noting
that this extra exclusion criteria brings the total Catalog 1 sample to 225
bursts, down from the sample of 265 bursts used by CHIME/FRB Collaboration et
al. (2021). Thus, hereafter, when referring to the Catalog 1 sample used for
populations studies in this analysis, we are referring to the 225 bursts that
survived all of these cuts.

To elaborate on exclusion criterion #7, simply excising all known repeating
bursts would not work as there is no way to know if an apparent one-off burst
will later turn out to be a repeater. Choosing the highest $\mathrm{S/N}$
burst will bias the sample to higher fluences, and choosing “random” bursts of
the repeat bursts also introduce a subtle bias as the detection trigger
threshold is set to be more sensitive to possible repeat bursts, i.e., bursts
detected in the same directions and DMs as previously detected FRBs. Thus, to
minimize bias (especially with respect to trigger thresholds), we include only
the first-detected burst from any known FRB source that otherwise passes other
cuts (i.e., $\mathrm{S/N}>12$). As detailed by Pleunis et al. (2021), there
are differences in burst widths and bandwidths for observed repeaters and
apparent non-repeaters, raising the question of whether it is fair to assume
identical detection biases for these bursts. However, applying all the
exclusion criteria except for the exclusion criterion for repeaters excludes
306 bursts instead of 311. Thus, even though the selection biases may differ
for repeaters and apparent non-repeaters, the difference is unlikely to
strongly affect this analysis. As the sample of observed repeaters grows,
further investigation into the differing selection biases for repeaters and
apparent non-repeaters will be crucial; further discussion is in Section 6.6.

Using the response of the real-time detection pipeline to the injected bursts,
in CHIME/FRB Collaboration et al. (2021) we were able to determine the
selection functions for fluence, $\mathrm{DM}$, scattering, and pulse width.
Also, as detailed by CHIME/FRB Collaboration et al. (2021), the injections
system does not have the ability to forward model fluence measurement
processes in a robust manner, and measurements of the observed fluence are
currently limited in their certainty. Thus, the intrinsic fluence distribution
was studied by using the detection $\mathrm{S/N}$ as a proxy. The key
assumptions that go into using $\mathrm{S/N}$ as a proxy for fluence are (1)
detection $\mathrm{S/N}$ is strongly correlated with intrinsic fluence, and
(2) we can statistically accurately forward model how fluence maps to
detection $\mathrm{S/N}$ with the injections system. As detailed in Section
3.1, the same methodology is employed in this analysis. Additionally, although
spectral index and spectral running were two properties also assigned to each
injected pulse and observed in each real burst, due to the correlated nature
of those properties, a functional form was not fit to the distribution
$P(\alpha,r)$.

With the selection functions, the following selection-corrected fiducial
distributions were determined: $P_{\mathrm{fid}}(F)$,
$P_{\mathrm{fid}}(\mathrm{DM})$, $P_{\mathrm{fid}}(\tau)$, and
$P_{\mathrm{fid}}(w)$. For details on their derivation, refer to Section 6.1
of CHIME/FRB Collaboration et al. (2021). Here we note that a number of
simplifying assumptions were made in the Catalog 1 analysis, notably that the
$F$–$\mathrm{DM}$ distribution factorizes into a power law in $F$ and a free
function of $\mathrm{DM}$. In this paper, we improve upon this
$F$–$\mathrm{DM}$ relation and use a more realistic one following the model
detailed by James et al. (2022b). Thus, we hold the distributions of $\tau$
and $w$ to the fiducial models $P_{\mathrm{fid}}(\tau)$ and
$P_{\mathrm{fid}}(w)$ found in Catalog 1 while fitting for a new joint
distribution of $F$ and $\mathrm{DM}$ here. For simplicity of notation, we
define the parameter set $\mathbf{\Theta}=\\{\tau,w\\}$ such that
$P(\mathbf{\Theta})=P(\tau)P(w)$, and $\int
d\mathbf{\Theta}P(\mathbf{\Theta})=1$.

##  3 Fluence–DM distribution model

The model detailed by James et al. (2022b) contains components that detail how
both fluence (i.e., brightness) and $\mathrm{DM}$ depend on redshift (i.e.,
distance). Contained within this populations model is information about FRB
energetics, how the FRB population may evolve over cosmic time, as well as the
distance distribution as inferred from $\mathrm{DM}$ observations.

We define the joint rate distribution of $F$ and $\mathrm{DM}$ as

$R(F,\mathrm{DM}|\bm{\lambda})=\int
dzR(F,z|\bm{\lambda_{1}})P(\mathrm{DM}|z,\bm{\lambda_{2}}),$ (1)

where the parameters $\bm{\lambda}$ for the fluence$\mathrm{DM}$ distribution
considered will be detailed shortly. Inside the integrand, we can see two
specific components: the joint rate distribution of fluence and redshift, and
the probability distribution of $\mathrm{DM}$ given redshift. Appendix A
details how to derive $R(F,\mathrm{DM}|\bm{\lambda})$, closely following James
et al. (2022b). The key components of the model are as follows.

We model the FRB energy distribution with a Schechter (1976) function

$P(E)dE\propto\frac{1}{E_{\mathrm{char}}}\left(\frac{E}{E_{\mathrm{char}}}\right)^{\gamma}\exp{\left[-\frac{E}{E_{\mathrm{char}}}\right]}dE,$
(2)

where $E_{\mathrm{char}}$ is the characteristic exponential cutoff energy and
$\gamma$ is the differential power-law index. Note that we are assuming $P(E)$
does not evolve with redshift. We also model the evolution of the FRB
population by smoothly scaling the star formation rate (SFR) with a power-law
index $n$,

$\Phi(z)=\frac{\Phi_{0}}{1+z}\left(\frac{\mathrm{SFR}(z)}{\mathrm{SFR}(0)}\right)^{n},$
(3)

where $\Phi(z)$ represents the rate of FRBs per comoving volume above a fixed
pivot energy $E_{\mathrm{pivot}}$, and SFR$(z)$ (Equation A9) is from Madau &
Dickinson (2014). A pivot energy is required for us to be able to quote a rate
of FRBs (Appendix A.1). Thus, we choose a pivot energy of $10^{39}$ erg, above
the threshold of ruled out minimum FRB energies (i.e., James et al. 2022b
rules out $E_{\mathrm{min}}>10^{38.5}$ erg at 90% C.L.). The parameter
$\Phi_{0}$ is the volumetric rate in units bursts Gpc-3 year-1 at $z=0$. The
population redshift evolution with the star formation history (SFH) is
parameterized by $n$, where $n=0$ would imply no evolution with cosmic SFH and
$n=1$ would imply evolution that perfectly traces the cosmic SFH. This model
also allows for the possibility of FRB progenitors that evolve faster than the
SFH of the universe (e.g., $n>1$). Thus, the free parameters in
$\bm{\lambda_{1}}$ are $\Phi_{0}$, $E_{\mathrm{char}}$, $\gamma$, the spectral
index $\alpha$, and $n$. It is worth noting that $\alpha$ comes into the model
in the conversion from $E$ to $F$ in Equation A4, such that this model assumes
the intrinsic $\alpha$ is the same for every FRB. The “rate interpretation” of
$\alpha$, further discussed in Section 5.4, Appendix B, and by James et al.
(2022b), takes the other extreme where every FRB has a unique spectral index.
As FRBs have been observed to have a wide range of spectral properties, the
treatment of $\alpha$ is a systematic uncertainty that is model-dependent.

The $\mathrm{DM}$ distribution accounts for contributions from the disk and
halo of the Milky Way, cosmological contributions through the intergalactic
medium (IGM; refer to Equation A12 in Section A.2), and contributions from the
host galaxy of the FRB. The Milky Way contribution to the $\mathrm{DM}$ is
fixed to be $\overline{\mathrm{DM}}_{\mathrm{MW}}$ = 80 pc cm-3 (discussed in
Section 5.3.1), and the host $\mathrm{DM}$ contribution is modeled with a log-
normal distribution

$P(\mathrm{DM}_{\mathrm{host}}^{\prime})=\frac{1}{\mathrm{DM}_{\mathrm{host}}^{\prime}}\frac{1}{\sigma_{\mathrm{host}}\sqrt{2\pi}}e^{-\frac{(\log\mathrm{DM}_{\mathrm{host}}^{\prime}-\mu_{\mathrm{host}})^{2}}{2\sigma^{2}_{\mathrm{host}}}},$
(4)

where the host $\mathrm{DM}$ contribution is parameterized by
$\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ and corrected for redshift
with $\mathrm{DM}_{\mathrm{host}}=\mathrm{DM}_{\mathrm{host}}^{\prime}/(1+z)$.
(Note, the parameters as presented in the exponential of Equation 4 are in
natural log space.) Thus, the free parameters in $\bm{\lambda_{2}}$ are
$\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$.

Note, $\mathrm{DM}_{\mathrm{host}}^{\prime}$ is the log-normal variate
parameterized by $\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$. Thus,
$\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ are the expected value and
standard deviation of log($\mathrm{DM}_{\mathrm{host}}^{\prime}$), not of
$\mathrm{DM}_{\mathrm{host}}^{\prime}$ itself. The median and standard
deviation of $\mathrm{DM}_{\mathrm{host}}^{\prime}$ can be obtained with the
parameters $\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ as
follows:444https://sites.google.com/site/probonto (Swat et al., 2016)

$\displaystyle\mathrm{Median}\;$ $\displaystyle
P(\mathrm{DM}_{\mathrm{host}}^{\prime})=\exp({\mu_{\mathrm{host}}})$ (5)
$\displaystyle\mathrm{Std.dev.}\;$ $\displaystyle
P(\mathrm{DM}_{\mathrm{host}}^{\prime})=$ (6)
$\displaystyle\sqrt{(\exp(\sigma_{\mathrm{host}}^{2})-1)\exp(2\mu_{\mathrm{host}}+\sigma_{\mathrm{host}}^{2})}.$

Throughout this work, to avoid confusion, we consider the log-normal variate
$\mathrm{DM}_{\mathrm{host}}^{\prime}$ so as to keep all values in physical
units of pc cm-3. We also quote the median of the log-normal distribution
rather than the mean, as the median is more robust to outliers present in a
skewed distribution.

###  3.1 Comparing the model to an observed distribution

The model we aim to fit to the Catalog 1 data is a joint rate distribution of
$F$ and $\mathrm{DM}$, $R(F,\mathrm{DM}|\bm{\lambda})$. However, as noted in
Sections 2.1 and 2.3, observed fluence measurements are uncertain estimates of
the true fluence. While the synthetic bursts are assigned a “true” fluence
value before injection, there is no good way to forward-model how their
fluence values would be “measured” by the real-time detection pipeline. Thus,
directly comparing the “true” fluence values of the injected bursts with the
“observed” fluence values of the Catalog 1 bursts would be subject to
systematics not yet fully characterized. The better alternative is to use
$\mathrm{S/N}$ as a proxy for fluence when fitting for
$R(F,\mathrm{DM}|\bm{\lambda})$, as fluence is strongly correlated with
$\mathrm{S/N}$ with known calibration factors (Merryfield et al., 2022).

Thus, we wish to make a prediction for the observed (as opposed to intrinsic)
CHIME/FRB Catalog 1 DM–$\mathrm{S/N}$ distribution using a given
fluence–$\mathrm{DM}$ model. The next section will detail how we fit for the
$\mathrm{S/N}$$\mathrm{DM}$ distribution, and by extension, obtain the
parameters $\bm{\lambda}$ in $R(F,\mathrm{DM})$.

##  4 Methodology

The observed CHIME/FRB Catalog 1 $\mathrm{S/N}$–$\mathrm{DM}$ distribution is
denoted with $n_{ij}$, where $i$ indexes the $\mathrm{S/N}$ bin and $j$
indexes the $\mathrm{DM}$ bin. We want to compare the observable $n_{ij}$ to
our model prediction $\zeta_{ij}$, which depends on parameters
$\bm{\lambda}=\\{\Phi_{0},E_{\mathrm{char}},\gamma,\alpha,n,\mu_{\mathrm{host}},\sigma_{\mathrm{host}}\\}$.

The candidate population model $\zeta_{ij}$ is parameter-dependent — and thus
model-dependent — because we construct weights to convert the original single
injected population model to any other population model. The alternative
approach would have been to conduct an injections campaign for every candidate
population model we wished to test, which is logistically infeasible (the full
duration of the injections campaign used in this analysis was over a month).
Whereas $n_{ij}$ is an unweighted histogram of the observed Catalog 1 data in
$\mathrm{S/N}$ and $\mathrm{DM}$ bins, $\zeta_{ij}$ is a weighted histogram of
the synthetic data “detected” by the real-time pipeline. Each synthetic burst
$m$ in a given bin $n_{ij}$ has a weight
$W_{m}(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$ assigned to it, i.e.,

$\zeta_{ij}=\sum_{m=1}^{n_{ij}}W_{m}(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda}).$
(7)

The construction of the weights
$W_{m}(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$ is elaborated on in
Section 4.1. Note also that the units of $\zeta_{ij}$ is counts per $n_{ij}$
bin. If we divide each bin in $\zeta_{ij}$ by the bin area
($\Delta\mathrm{S/N}\;\Delta\mathrm{DM}$), then we would obtain an observed
rate of bursts per $\mathrm{S/N}$ per $\mathrm{DM}$, e.g.,
$R_{\mathrm{obs}}(\mathrm{S/N},\mathrm{DM}|\lambda)$. This quantity is the
prediction for the observed (as opposed to intrinsic) CHIME/FRB Catalog 1
DM–$\mathrm{S/N}$ distribution.

We employ Markov chain Monte Carlo (MCMC) techniques to sample the likelihood
while simultaneously fitting for all seven parameters in
$R(F,\mathrm{DM}|\bm{\lambda})$. We use the logarithmic binned Poisson
likelihood (Zyla et al., 2020):

$\log\mathcal{L}(\bm{\lambda})\propto\sum_{ij}n_{ij}\log(\zeta_{ij})-\zeta_{ij}.$
(8)

The likelihood was evaluated with 15 logarithmically spaced bins in
$\mathrm{S/N}$, ranging from 12–200, and 20 logarithmically spaced bins in
$\mathrm{DM}$, ranging from 100–3500 pc cm-3.

###  4.1 Weighting formalism

Each injected event $m$ has its own parameter-dependent weight
$W_{m}(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$, which means each weight
depends on the properties of each synthetic burst. These weights are
constructed to be the ratio of how many bursts were injected to how many
occurred on the sky during the survey for a given set of model parameters
$\bm{\lambda}$. Thus, assuming the parameters $\bm{\lambda}$ are true, then
for every injected burst with properties $F$ and $\mathrm{DM}$, we should have
instead injected a number equal to
$W_{m}(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$ in order to exactly match
the observed sky population.

To obtain the property-dependent weights
$W_{m}(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$, for each synthetic burst
$m$ with properties $F$ and $\mathrm{DM}$, using the model parameters
$\mathbf{\Theta}$ and $\bm{\lambda}$, we divide the full injected population
of bursts on the sky by the initial injected population of bursts:

$\displaystyle W(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$ $\displaystyle=$
$\displaystyle\frac{R(F,\mathrm{DM},\mathbf{\Theta}|\bm{\lambda})\Delta
t}{R_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta})}$ (9) $\displaystyle=$
$\displaystyle\frac{\Delta
tf_{\mathrm{sky}}}{N_{\mathrm{inj}}\epsilon_{\mathrm{inj}}}\frac{R(F,\mathrm{DM},\mathbf{\Theta}|\bm{\lambda})}{P_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta})}.$
(10)

The quantity $R_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta})$ is the full
initial population of injected bursts, with units of bursts per $F$ per
$\mathrm{DM}$ per injections campaign. It is equivalent to the normalized
initial population distribution of the injected synthetic bursts,
$P_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta})$, multiplied by the total
number of FRBs injected ($N_{\mathrm{inj}}$) corrected for the efficiency of
injections ($\epsilon_{\mathrm{inj}}$) and the sky fraction
($f_{\mathrm{sky}}$). That is,

$R_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta})=\frac{N_{\mathrm{inj}}\epsilon_{\mathrm{inj}}}{f_{\mathrm{sky}}}P_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta}).$
(11)

The model $R(F,\mathrm{DM},\mathbf{\Theta}|\bm{\lambda})$ gives a rate in
bursts per year. When multiplied by $\Delta t$, the survey duration in days
($\Delta t=214.8$ days), we can then get a rate of observed bursts per survey,
which can be directly compared against
$R_{\mathrm{init}}(F,\mathrm{DM},\mathbf{\Theta})$.

Also, as mentioned at the end of Section 2.3, the distributions of
$\mathbf{\Theta}$ were held to the fiducial models found in Catalog 1.
Therefore,

$\displaystyle W(F,\mathrm{DM},\mathbf{\Theta},\bm{\lambda})$ $\displaystyle=$
$\displaystyle\frac{\Delta
tf_{\mathrm{sky}}}{N_{\mathrm{inj}}\epsilon_{\mathrm{inj}}}\frac{R(F,\mathrm{DM}|\bm{\lambda})}{P_{\mathrm{init}}(F,\mathrm{DM})}\frac{P_{\mathrm{fid}}(\mathbf{\Theta})}{P_{\mathrm{init}}(\mathbf{\Theta})},$
(12)

where the fiducial distributions and initial distributions are all
factorizable.

Each synthetic burst can have an associated weight because each burst has
“intrinsic” $F$ and $\mathrm{DM}$ information associated with it. Because each
burst also has “observed” $\mathrm{S/N}$ information, each weighted burst can
be assigned a bin in $\zeta_{ij}$, thus allowing for $\zeta_{ij}$ to be model-
dependent (i.e., dependent on $\bm{\lambda}$) and directly comparable to
$n_{ij}$ in the likelihood function (Equation 8).

Note that the equations in this section are simply a more explicit definition
of the weights also defined in CHIME/FRB Collaboration et al. (2021), wherein
the weights were used only for obtaining the model parameters for
$P_{\mathrm{fid}}(F)$, $P_{\mathrm{fid}}(\mathrm{DM})$, and
$P_{\mathrm{fid}}(\mathbf{\Theta})$. For these distributions, the pre-factors
used to accurately determine the rate ($\Delta
tf_{\mathrm{sky}}/N_{\mathrm{inj}}\epsilon_{\mathrm{inj}}$) do not matter.

##  5 Results

Figure 1:  Corner plot of the results of the MCMC run, thinned by a factor of
20 for visualization purposes. Overlaid solid blue lines denote the median of
the posterior distributions, with the dashed blue lines enclosing the central
68% of the samples.

Using the methodology described by CHIME/FRB Collaboration et al. (2021) and
elaborated on in Section 4, we use the injections system to calibrate out
selection effects while fitting this full population model,
$R(F,\mathrm{DM}|\bm{\lambda})$, to the Catalog 1 data. We use the emcee
package (Foreman-Mackey et al., 2013) to generate MCMC samples of the
posterior, using scipy.optimize.curve_fit to obtain the initial parameters for
the MCMC chains. In order to ensure the posterior distribution is proportional
to the likelihood in Equation 8, all of our parameters are drawn from uniform
priors. We then run the MCMC sampler with 40 walkers, 2000 burn-in steps, and
then 25000 steps after that. While there is no simple way to ensure
convergence, the first and second halves of the chains show similar posterior
distribution morphology to each other, heuristically indicating confidence our
chains have converged (Hogg & Foreman-Mackey, 2018). Additionally, the chains
are longer than 75 times the integrated autocorrelation time for all
parameters, with an acceptance fraction of $\approx$0.26 — within the range of
recommended values (Foreman-Mackey et al., 2013). The results can be seen in
Figure 1, generated using the corner package (Foreman-Mackey, 2016). We quote
the median of the posterior distributions, along with the 16% quantile as the
lower 1$\sigma$ error bar and the 84% quantile as the upper 1$\sigma$ error
bar. These values are tabulated in Table 1.

Table 1:  Table of results of the parameters fit in $R(F,\mathrm{DM})$ (see
Equation 1). Best-fit results quote the median values of the posterior
distributions, with the error bars containing the central 68% of the samples.
Where illustrative, fits are paired with corresponding derived physically
meaningful quantities.

Parameter | Uniform prior range | Best-fit result
---|---|---
$\log_{10}\;\Phi_{0}\>^{a}$ | [$-0.96$, $6.43$] | $4.86^{+0.34}_{-0.33}$
$\Phi_{0}$ | $\ldots$ |  $7.3^{+8.8}_{-3.8}\times 10^{4}$ bursts Gpc-3 yr${-1}$
$\gamma$ | [$-2.50$, $2.00$] | $-1.3^{+0.7}_{-0.4}$
$\log_{10}\;E_{\mathrm{char}}\>^{a}$ | [$38.00$, $49.00$] | $41.38^{+0.51}_{-0.50}$
$E_{\mathrm{char}}$ | $\ldots$ |  $2.38^{+5.35}_{-1.64}\times 10^{41}$ erg
$n$ | [$-2.00$, $8.00$] | $0.96^{+0.81}_{-0.67}$
$\alpha$ | [$-5.00$, $5.00$] | $-1.39^{+0.86}_{-1.19}$
$\mu_{\mathrm{host}}\;\times\log_{10}(e)\>^{a}$ | [$1.30$, $2.70$] | $1.93^{+0.26}_{-0.38}$
$\mathrm{Median}\;P(\mathrm{DM}^{\prime})$ | $\ldots$ |  $84^{+69}_{-49}$ pc cm-3
$\sigma_{\mathrm{host}}\;\times\log_{10}(e)\>^{a}$ | [$0.04$, $1.74$] | $0.41^{+0.21}_{-0.20}$
$\mathrm{Std.dev.}\;P(\mathrm{DM}^{\prime})$ | $\ldots$ |  $174^{+319}_{-128}$ pc cm-3

aafootnotetext: These parameters were fit in natural $\log$ space but are
presented in $\log_{10}$ space for ease of interpretation.

Figure 2 shows slices through the $\mathrm{S/N}$–$\mathrm{DM}$ distribution of
the observed Catalog 1 data $n_{ij}$ compared against the best-fit model
prediction from the MCMC run, $\zeta_{ij}$, in each parameter space.

Figure 2:  Observed distributions, denoted with solid lines, compared with
best-fit model predictions, denoted with dashed lines. Each sample range is
determined by the logarithmic binning described in Section 4, and not by the
number of events that fall in each bin. Left panel: The $\mathrm{S/N}$
distribution is shown for a low-DM sample and a high-DM sample. Each shown
$\mathrm{DM}$ range spans ten bins. Right panel: The $\mathrm{DM}$
distribution is shown in three $\mathrm{S/N}$ samples. Each shown
$\mathrm{S/N}$ range spans five bins. The best-fit model predictions use the
median of the posterior distributions of each parameter after the MCMC run. In
every viewed slice, the model appears consistent with the observed data.

In both the low and high $\mathrm{DM}$ slices, the observed $\mathrm{S/N}$
distribution peaks at lower values closer to the $\mathrm{S/N}$ cutoff
threshold, with negligibly few sources at the high-end of the $\mathrm{S/N}$
distribution. Additionally, when viewing the data in $\mathrm{S/N}$ slices,
there are more sources in the lower $\mathrm{S/N}$ ranges than in the higher
ones, with the $\mathrm{DM}$ distribution appearing to peak at $\sim$500 pc
cm-3 for the slice most representative of the Catalog 1 data (the lowest
$\mathrm{S/N}$ sample). Note the rarity of detecting exceptionally high
$\mathrm{S/N}$ events at any $\mathrm{DM}$ range. In each view of the slices
through the data, the best-fit model appears to be consistent with the
observed data.

In Figure 3, one can see a side-by-side comparison of the 2D view of the
observed $\mathrm{S/N}$–$\mathrm{DM}$ distribution and model prediction.
Qualitatively, it is clear that the data appear to be consistent with the
model and added Poissonian noise. Both the observed data and the model
prediction show that the majority of FRBs are detected at lower $\mathrm{S/N}$
ranges, peaking around $\mathrm{DM}$ values of hundreds of pc cm-3.

###  5.1 Goodness of fit

One advantage of using maximum likelihood methods with binned data, as was
mentioned in Section 4, is that a goodness-of-fit statistic can be
simultaneously obtained (Zyla et al., 2020). Maximizing the likelihood
$\mathcal{L}(\bm{\lambda})$ is equivalent to maximizing the likelihood ratio
$\xi(\bm{\lambda})$, and thus is also equivalent to minimizing the quantity
$-2\log\xi(\bm{\lambda})$. We define the quantity $T=-2\log\xi(\bm{\lambda})$.
For Poissonian data, this expression takes the form

$T=2\sum_{ij}\left[\zeta_{ij}-n_{ij}+n_{ij}\log\left(\frac{n_{ij}}{\zeta_{ij}}\right)\right].$
(13)

The smaller the value of $T$, the better the data $n_{ij}$ agree with the
model $\zeta_{ij}$. If Wilks’ theorem holds, then the minimum of $T$ follows a
$\chi^{2}$ distribution (Wilks, 1938), and the value can be interpreted as a
$p$-value. However, one of the conditions of Wilks’ theorem is that the sample
size must be large, which our binned data do not satisfy. Thus, to quantify
the goodness of fit, we aim to estimate the null distribution of $T$ by Monte
Carlo simulations. We generate 10,000 Monte Carlo samples of Poissonian-
distributed $n_{ij}$ from our $\zeta_{ij}$ and calculate the test statistic
$T$. The results of the Monte Carlo simulations are in Figure 4. Based on this
statistic, over 100 runs, $\approx$25% of the simulations have a worse fit
than our best-fit model prediction. In other words, we have estimated a
$p$-value of 0.25 where in 10,000 different realizations of the data, the
model $R(F,\mathrm{DM}|\bm{\lambda})$ was a worse fit. This estimated
$p$-value is consistent with the conclusion that our model matches the data
reasonably well, albeit with room for improvement in some combination of the
model details, the collected data, and our treatment of observed selection
effects.

###  5.2 $R(F,z)$ parameters

The parameters in $\bm{\lambda_{1}}$ are $\Phi_{0}$, $E_{\mathrm{char}}$,
$\gamma$, $\alpha$, and $n$. Contained within this component of the fit model
are predictions about the energetics and abundances of FRBs based on the
CHIME/FRB Catalog 1 sample.

Figure 3:  Comparison of the $\mathrm{S/N}$–DM distribution between the
observed data (left) and the best-fit model prediction (right). The best-fit
model predictions use the median of the posterior distributions of each
parameter after the MCMC run. The two appear to behave qualitatively
similarly, in-line with expectations from Figure 2.  Figure 4:  The estimated
probability distribution $\hat{P}(\hat{T})$ of the test statistic $T$, defined
in Equation 13, from Monte Carlo sampling. The value of this test statistic
computed for our best-fit model prediction is overlaid in a vertical dashed
line. Approximately 25% of $\hat{P}(\hat{T})$ lies to the right of the dashed
line, giving an estimated $p$-value of 0.25.

We find a volumetric rate at $z=0$ above $E_{\mathrm{pivot}}=10^{39}$ erg,
with our statistical uncertainties given by the MCMC. In the work presented by
CHIME/FRB Collaboration et al. (2021), the net systematic uncertainty for the
sky rate was ${}^{+27\%}_{-25\%}$; we assume the systematics propagate forward
in a similar manner for the volumetric rate derived here. Thus, the volumetric
rate is $\Phi_{0}=$ [$7.3^{+8.8}_{-3.8}$(stat.)${}^{+2.0}_{-1.8}$(sys.)]
$\times 10^{4}$ bursts Gpc-3 year-1. In CHIME/FRB Collaboration et al. (2021),
the systematic errors are by far the largest for factors that affect the
overall rate. Given that in this analysis, the systematic errors are
subdominant to the statistical errors for the rate calculation, we assume that
the remainder of the parameters are dominated by statistical errors as well.
Thus, for all other parameters we quote only statistical errors on our values.

The energy distribution within $R(F,z)$ is modeled with a Schechter function
(Equation 2), and we find best-fitting values of $E_{\mathrm{char}}=$
$2.38^{+5.35}_{-1.64}\times 10^{41}$ erg and $\gamma=$ $-1.3^{+0.7}_{-0.4}$,
where $\gamma$ is the differential power-law index.

Our best-fitting value for the spectral index in $F\propto\nu^{\alpha}$ is
$\alpha=$ $-1.39^{+0.86}_{-1.19}$. However, one can also interpret $\alpha$ as
a frequency-dependent rate; see Section 5.4 for details of this
interpretation. For the parameter $n$ which parameterizes evolution with the
cosmic SFH, we find a best-fitting value of $n=$ $0.96^{+0.81}_{-0.67}$. As
can be seen in Figure 1, the parameters $\alpha$ and $n$ are strongly anti-
correlated — intuitively, the more strongly FRBs appear to evolve with (or
faster than) the cosmic SFH (higher $n$), the more sources would originate
from further away and appear fainter, thus leading to a steeper observed “true
spectral index.” This degeneracy is stronger under the “rate interpretation”
of $\alpha$. The parameter for the cutoff energy $E_{\mathrm{char}}$ is also
correlated (anti-correlated) with $n$ ($\alpha$), as the more energetic we
expect FRBs can be, the more FRBs we should be able to observe further away
(fainter).

###  5.3 $P(\mathrm{DM}|z)$ parameters

The parameters in $\bm{\lambda_{2}}$ are $\mu_{\mathrm{host}}$ and
$\sigma_{\mathrm{host}}$. Modeling the host galaxy contribution of the FRB to
its $\mathrm{DM}$ as a log-normal distribution, our best-fit values are
$\mu_{\mathrm{host}}\times\log_{10}(e)=$ $1.93^{+0.26}_{-0.38}$ and
$\sigma_{\mathrm{host}}\times\log_{10}(e)=$ $0.41^{+0.21}_{-0.20}$. These
values correspond to a median $\mathrm{DM}_{\mathrm{host}}^{\prime}$ value of
$84^{+69}_{-49}$ pc cm-3, with a standard deviation of $174^{+319}_{-128}$ pc
cm-3.

As can be seen in Figure 1, there is somewhat of a correlation between
$\mu_{\mathrm{host}}$ and the volumetric rate $\Phi_{0}$. Intuitively,
$\mu_{\mathrm{host}}$ sets the overall distance scale for FRBs – the higher
$\mu_{\mathrm{host}}$ is, the more we expect the $\mathrm{DM}$ contribution to
be from the host galaxy rather than the IGM, and thus FRBs would be closer,
less energetic, and more volumetrically abundant.

####  5.3.1 Varying assumed $\overline{\mathrm{DM}}_{\mathrm{MW}}$

When fitting for $P(\mathrm{DM}|z)$, we adopted
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ = 80 pc cm-3 as our fiducial value.
However, the Galactic contribution to the $\mathrm{DM}$ of an FRB has non-
negligible uncertainties, especially when considering the Galactic halo. We
tried to minimize this uncertainty by excluding all bursts with a
$\mathrm{DM}_{\mathrm{NE2001}}>100$ pc cm-3 (Section 2.3), but wish to further
test how sensitive our analysis may be to different values of
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ = 80 pc cm-3. Thus, we performed
multiple more iterations of the MCMC analysis, varying
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ to between be 50 pc cm-3 and 100 pc
cm-3 in increments of 10 pc cm-3. We do these discrete iterations because
otherwise, $\overline{\mathrm{DM}}_{\mathrm{MW}}$ as a free parameter would be
degenerate with $\mu_{\mathrm{host}}$ (see, e.g., Macquart et al., 2020).
Having $\overline{\mathrm{DM}}_{\mathrm{MW}}$ as a free parameter, and also
having it be independent of $\mu_{\mathrm{host}}$, would require significant
changes to the model formalism, and is beyond the scope of this analysis.

Predictably, the varying of this assumption most affects the posterior
distributions of the parameters that go into the log-normal distribution model
of DMhost, e.g., $\bm{\lambda_{2}}$ = {$\mu_{\mathrm{host}}$,
$\sigma_{\mathrm{host}}$}. None of the five other posterior distributions from
the MCMC run appreciably changed. Shown in Figure 5 are the posterior
distributions for the derived physically meaningful quantities corresponding
to $\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ — i.e., the median and
the standard distribution of $P(\mathrm{DM}^{\prime}_{\mathrm{host}})$, which
is parameterized by $\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ — from
the runs where $\overline{\mathrm{DM}}_{\mathrm{MW}}$ was varied.

Figure 5:  Overlaid posterior distributions from the MCMC runs run with
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ = 50 pc cm-3 (blue dotted-dashed
lines), 80 pc cm-3 (gray shaded region), and 100 pc cm-3 (orange dashed
lines). On the left is the normalized posterior distribution for
$\mu_{\mathrm{host}}$ and on the right is the normalized posterior
distribution for $\sigma_{\mathrm{host}}$. When
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ is varied away from 80 pc cm-3, the
resulting posterior distributions of $\mu_{\mathrm{host}}$ and of
$\sigma_{\mathrm{host}}$ appear insignificantly changed. Thus the parameters
we find assuming a log-normal distribution for $\mathrm{DM}_{\mathrm{host}}$
are relatively robust to reasonable assumptions of
$\overline{\mathrm{DM}}_{\mathrm{MW}}$.

Comparing Figure 5 to the results presented in Table 1, the variation of where
the posterior distributions peak appears to be within 1$\sigma$ statistical
uncertainties. Thus, while we know our assumption of
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ is imperfect, its effect on the
modeling of $\mathrm{DM}_{\mathrm{host}}$ is sub-dominant to statistical
uncertainties, and it also does not appreciably affect the other fit
parameters in $R(F,\mathrm{DM}|\bm{\lambda})$. As such, we continue with our
fiducial value of $\overline{\mathrm{DM}}_{\mathrm{MW}}$ = 80 pc cm-3.

###  5.4 “Rate interpretation” of $\alpha$

Throughout this work, we have treated $\alpha$ as a true FRB spectral index,
i.e., a spectral index for an individual FRB (observed as a broad-band burst)
that also applies to the full population of studied FRBs. The flux behaves as
$F\sim\nu^{\alpha}$. However, as James et al. (2022b) noted, $\alpha$ can also
be interpreted as a frequency-dependent rate, particularly when considering
experiments with a similar bandwidth to that of the detected FRB. As FRBs can
be band-limited (e.g., Law et al., 2017; Pleunis et al., 2021), this means
that there are either a larger number of low-frequency FRBs, or that FRBs are
stronger at low frequencies. Under such an interpretation, the rate now
behaves as $\Phi\sim\Phi(z)\nu^{\alpha}$, and through the changed
$k$-correction, the factor of $(1+z)^{\alpha}$ is directly added to the source
evolution (Equation A8), thus most directly affecting the parameter $n$. It is
likely that the most realistic treatment of $\alpha$ is more complicated than
either single interpretation — for example, it is reasonable to think FRBs may
be both broad-band (as per the “true spectral index” interpretation) and have
unique spectral indices (as per the “rate interpretation”). Thus the treatment
of $\alpha$ remains a model-dependent systematic uncertainty.

For completeness, we redo the analysis, this time treating $\alpha$ as a
frequency-dependent rate. The complete results of the parameters can be found
in Figure 11 and Table 2. Of the parameters, $\alpha$ and $n$ had the most
significant changes; they become far more correlated under the “rate
interpretation” of $\alpha$. Under the “true spectral index interpretation” of
$\alpha$, we obtain $\alpha=$ $-1.39^{+0.86}_{-1.19}$ and $n=$
$0.96^{+0.81}_{-0.67}$; under the “rate interpretation” of $\alpha$, we obtain
$\alpha^{\prime}=$ $-1.10^{+0.67}_{-0.99}$ and $n^{\prime}=$
$1.72^{+1.48}_{-1.10}$.

##  6 Discussion

Using the population model developed by James et al. (2022b), this analysis
simultaneously constrains the redshift, energy, and host $\mathrm{DM}$
distributions of FRBs using CHIME/FRB Catalog 1 data. There are a total of
seven free parameters fit in the model; when interpreting $\alpha$ as a
frequency-dependent rate, another set of fit parameters can be obtained. In
this section, we discuss our results and the property distributions they
describe. Notably, in Section 6.6, we discuss the implications for considering
repeating and non-repeating FRBs for this fitted one-population model.

###  6.1 FRB volumetric rate

We find a rate of bursts above $10^{39}$ erg and below a scattering of 10 ms
at 600 MHz of $7.3^{+8.8}_{-3.8}\times 10^{4}$ bursts Gpc-3 year-1. This
volumetric rate appears consistent with the rate presented by James et al.
(2022a), who found a rate of bursts above $10^{39}$ erg of
$8.7^{+1.7}_{-3.9}\times 10^{4}$ bursts Gpc-3 year-1, and is also broadly
consistent with other studies as well (e.g., Lu & Piro, 2019; Ravi, 2019; Luo
et al., 2020). On the surface, the consistent volumetric rates is interesting
to note when considering the different frequency bands of the respective
samples of FRBs. However, great caution should be employed when directly
comparing these values, as the rates presented by other studies do not employ
a scattering cut. As noted by CHIME/FRB Collaboration et al. (2021), there is
evidence that there is a notable population of FRBs with a scattering
timescale at 600 MHz larger than 10 ms that is missed by CHIME/FRB
observations. Although highly scattered bursts were excluded from this
analysis due to the poorly constrained nature of their population, including
such highly scattered bursts may greatly increase our volumetric rate,
possibly making the rate found in this analysis less comparable with other
derived rates.

This volumetric rate of $7.3^{+8.8}_{-3.8}\times 10^{4}$ bursts Gpc-3 year-1
exceeds that of known populations of cataclysmic phenomena. As mentioned in
Section 2.3, consistent with the methodology by CHIME/FRB Collaboration et al.
(2021), only the first burst of every known source in Catalog 1 was kept.
Conceptually, this sample selection methodology would indicate that we are
studying FRB source distributions rather than burst distributions. Of course,
as further discussed in Section 6.6, our treatment of repeaters is far from
perfect. The question then is whether to interpret this volumetric rate as
that of burst activity or activity from sources.

Going back to the sample selection methodology employed by CHIME/FRB
Collaboration et al. (2021), most of the FRBs removed from the populations
analyses were rejected at the $\mathrm{S/N}<12$ cut. Without the cut on
repeaters, the Catalog 1 sample for populations analysis would have been 230
bursts instead of 225. However, given the total net systemic uncertainty on
the sky rate found by CHIME/FRB Collaboration et al. (2021) was
${}^{+27\%}_{-25\%}$, this systemic uncertainty of treatment of repeaters is
almost certainly a sub-dominant effect. Thus, from a practical point of view,
this volumetric rate can be interpreted as either a rate of bursts or of
sources. This reasoning is undeniably complicated by the fact that the
observed rates are also strong functions of telescope sensitivity and the FRB
repetition luminosity function.

Nonetheless, since it is possible that one-off bursts in our sample can
actually be bursts from repeating sources, and the cut on repeaters is a sub-
dominant effect in this analysis, if one looks purely at the obtained
volumetric rate above 1039 erg, it may be physically sensible to interpret it
as a rate of bursts. Extrapolating below the pivot energy introduces a large
amount of uncertainty, and our model does not fit for a minimum energy, but it
is reasonable to expect that the volumetric rate of FRBs is larger when
including energies below $E_{\mathrm{pivot}}$. If our observed volumetric rate
is interpreted as a rate of bursts, as this rate exceeds that of known one-off
event populations (e.g., Nicholl et al., 2017), such a scenario could suggest
that FRBs are predominantly from repeaters, most of which instruments such as
CHIME/FRB may missed. This in turn could imply that many apparent one-off FRBs
are in fact repeat bursts from progenitors with long stretches of burst
quiescence (Caleb & Keane, 2021), though the notion that all — or even most —
FRBs are repeaters is one that has fallen out of relative favor (e.g., Ravi,
2019; James et al., 2020; Ai et al., 2021). There is also the possibility that
repeaters and one-off FRBs are distinct populations with distinct populations
parameter values; further discussion on this consideration can be found in
Section 6.6. Interpreting our observed volumetric rate as a rate of bursts can
also suggest that FRBs predominantly come from a yet-unknown population of
one-off events. Nonetheless, it is important to emphasize that this discussion
is driven purely by consideration of the numerical volumetric rate result; our
model $R(F,\mathrm{DM}|\bm{\lambda})$ does not directly make any claims as to
which progenitor origin scenario is preferred. (The model does allow for
source evolution, which could more directly argue for preferred progenitor
scenarios based on evolution with star formation rate, but as discussed in
Section 6.4, our results there are inconclusive.)

###  6.2 FRB energy function

Constraining the energy function of FRBs is particularly illuminating for
better understanding FRB origins. Although we model the energy distribution as
a Schechter function for all energies, we only quote a volumetric rate above a
pivot energy $E_{\mathrm{pivot}}=10^{39}$ erg. We do not test for a minimum
energy. An allowable range of energy distributions is plotted in Figure 6. We
see that the shape of the function is quite well-constrained — the
characteristic energy after which the distribution exponentially falls off
appears to visually consistently be on the order of $\approx$1041 erg. We find
$\log_{10}E_{\mathrm{char}}$ (erg) to be $41.26^{+0.42}_{-0.41}$, in fairly
good agreement with the James et al. (2022a) result of
$\log_{10}E_{\mathrm{max}}$ (erg) of $41.70^{+0.53}_{-0.06}$. (Note James et
al. (2022a) preferred to model the energy distribution with a power law with a
maximum energy cutoff rather than as a Schechter function, although both
models were tested with comparable results.)

Figure 6:  Plotted in gray are the results of the allowable Schechter function
values scaled by energy and the volumetric rate from the MCMC run, thinned by
a factor of 500 for visualization purposes. The best-fit Schechter function
from the MCMC run is overlaid in solid blue. Abundances at a given energy can
be directly read off the y-axis.

We also find a differential power-law energy distribution index of $\gamma=$
$-1.3^{+0.7}_{-0.4}$. This result is in broad agreement with the results of
Hashimoto et al. (2022), who also used Catalog 1 data to explore the energy
function of FRBs, albeit employing a different methodology (e.g., they do not
fit for a distance distribution simultaneously) and using a different subset
of bursts released by CHIME/FRB Collaboration et al. (2021). Within error
bars, this power-law index is also in agreement with what Luo et al. (2020)
found ($-1.79^{+0.31}_{-0.35}$) using a heterogenous sample of FRBs, including
bursts from UTMOST-SS (Caleb et al., 2017) and from CRAFT (Shannon et al.,
2018). Interesting to note is that James et al. (2022a)
($-2.09^{+0.14}_{-0.10}$), who developed the model
$R(F,\mathrm{DM}|\bm{\lambda})$ we use in this analysis, find a steeper power-
law index value using a sample of ASKAP and Parkes FRBs, though it is still
consistent with our results within 2$\sigma$ statistical uncertainties.
Further data and larger numbers of FRB observations are needed to explore the
consistency of FRB energy distributions across different frequency bands.

###  6.3 FRB spectral dependence with $\alpha$

When interpreting $\alpha$ as a true spectral index, every FRB is observed as
a broad-band burst with the same spectral index. Using a sample of 23 FRBs
detected with ASKAP, Macquart et al. (2019) find $\alpha=-1.5$, a default
value adopted in other studies (e.g., Lu & Piro, 2019). Using a Gaussian prior
on $\alpha=-1.5\pm 0.3$, James et al. (2022b) find a consistent
$\alpha=-1.55^{+0.18}_{-0.18}$. Our result of $\alpha=$
$-1.39^{+0.86}_{-1.19}$ is consistent with $\alpha\approx-1.5$, although our
statistical uncertainties are larger than those of the aforementioned quoted
values.

The treatment of $\alpha$ is far from standardized in the literature. Multiple
studies fix $\alpha=0$ (e.g., Caleb et al., 2016; Luo et al., 2020; Chawla et
al., 2022) when studying their sample. In particular, Chawla et al. (2022) set
$\alpha=0$ for the Catalog 1 sample in order to be consistent with the
implicit assumptions made when measuring fluences for Catalog 1 bursts
(CHIME/FRB Collaboration et al., 2021). Also, as has been detailed by James et
al. (2022b) and mentioned by Law et al. (2017), there is little meaning of a
“spectral index” for an individual FRB with a limited band occupancy. Instead,
when interpreting $\alpha$ as a frequency-dependent rate, every FRB is
observed as a narrow-band burst, depending on the frequency-dependent survey
sensitivity. Our rate interpretation result is $\alpha^{\prime}=$
$-1.10^{+0.67}_{-0.99}$, which is slightly flatter than our “spectral index”
interpretation of $\alpha=$ $-1.39^{+0.86}_{-1.19}$ but again, not
significantly so — especially when considering the uncertainties. Certainly
more work remains to better constrain the spectral index of FRBs, and how best
to treat the systemic uncertainty of how to interpret $\alpha$.

###  6.4 FRB evolution with star formation history

The progenitors of FRBs are a widely debated topic, and one way to probe their
origins as a population is to track their redshift distribution with respect
to star formation history. If FRBs arise predominantly from magnetars, as the
association of the FRB-like burst with the Galactic magnetar SGR 1935+2154
might suggest (CHIME/FRB Collaboration et al., 2020a; Bochenek et al., 2020),
then they would closely track the star formation history (SFH) of the
universe. On the other hand, if FRBs arise predominantly from older stellar
populations, as the association of FRB 20200120E with a globular cluster in
M81 might suggest (Bhardwaj et al., 2021; Kirsten et al., 2022), then they
would not appear to trace the SFH. There is also the possibility that FRBs
arise from younger and older stellar populations in comparable proportions
(e.g., Guo & Wei, 2022), but given the limited information regarding FRB host
environments and the risk of over-interpreting available data, we do not
consider this scenario in this work.

Instead, the formalism of this population model simply aims to quantify how
closely the FRB population evolves with SFR. James et al. (2022a) claim that
the FRB population does evolve with SFR, a result that is not incompatible
with our results of $n=$ $0.96^{+0.81}_{-0.67}$ when considering the different
samples of FRBs used. The frequency coverage differs between Parkes (1157 –
1546 MHz) and ASKAP (1129 – 1465 MHz) compared to CHIME (400 – 800 MHz),
raising the possibility that there may be frequency-dependent evolution
behavior. However, there is no concrete evidence in our analysis to support
this claim, and again, any related investigation is outside the scope of this
work. Other studies have investigated the cosmic evolution of FRBs (e.g.,
Hashimoto et al., 2022; Zhang & Zhang, 2022; Qiang et al., 2022) using Catalog
1 data, and have concluded that FRBs from the CHIME/FRB Catalog 1 sample do
not evolve with SFR. Our result of $n=$ $0.96^{+0.81}_{-0.67}$ is much less
conclusive; the large uncertainties on our results are also visualized in
Figure 7. Of our $10^{6}$ chains, $\approx$7.2% give $n\leq 0$ (the rest of
the parameter results vary across chains; there is no real physical
interpretation for $n<0$, but we allowed the range for completeness). Our
result under the “rate interpretation” of $\alpha$ also gives a poorly
constrained $n^{\prime}=$ $1.72^{+1.48}_{-1.10}$; of those chains,
$\approx$4.5% give $n\leq 0$. We note that under the “rate interpretation” of
$\alpha$, our results tend towards stronger evolution with SFR — consistent
with what James et al. (2022b) found. However, the statistical uncertainties
on both parameters, in both interpretations, are too large to draw any
significant conclusions about how the interpretation of $\alpha$ may affect
the modeling of FRB evolution with SFR.

Figure 7:  Results of how the Catalog 1 FRBs evolve with SFR, $\Phi(z)$. Gray
lines denote allowable behavior of $\Phi(z)$ using parameters from the MCMC
run, thinned by a factor of 500 for visualization purposes. The best-fit
$\Phi(z)$ from the MCMC is overlaid in solid blue. The lower dashed orange
curve represents no evolution with SFR, and the upper dashed yellow curve
represents perfect evolution with SFR.

The difference between the results of this study and of previous studies using
Catalog 1 data can be explained in a few ways. Using the full injections
sample, this analysis fits a population model which simultaneously models both
the energy distributions and the redshift distributions of FRBs. In the work
presented by Hashimoto et al. (2022), redshifts are directly inferred from
$\mathrm{DM}$ measurements rather than treating them as separate parameters,
and they thus explored the redshift evolution of the derived FRB energy
functions. Both Qiang et al. (2022) and Zhang & Zhang (2022) generate Monte
Carlo distributions from varied redshift and energy distributions, and then
test them against the publicly available CHIME/FRB Catalog 1 data. While this
approach allows for explicit testing of different evolutionary models (i.e.,
including time delays), it does not fit specific parameters to the data — it
can only test how consistent a dataset may be with a tested distribution. We
do test one more toy model, elaborated on in Appendix C, to explore the
potential constraining power of our data for FRBs from older stellar
populations. Ultimately, though, because our result of $n=$
$0.96^{+0.81}_{-0.67}$ has such large uncertainties, we conclude that our data
cannot meaningfully constrain how the FRB population evolves with the star
formation history of the universe. However, localizations to host galaxies and
direct redshift measurements for the FRBs may enable better constraints in the
future. They would enable determination of information such as stellar masses,
which one can then use to also test whether FRB host galaxies track the
stellar mass function (e.g., as Bhandari et al. 2022 did with ASKAP FRBs).

###  6.5 Host galaxy DM contribution

The rest-frame host galaxy contribution to the $\mathrm{DM}$ of an FRB was
modeled with a log-normal distribution, parameterized by $\mu_{\mathrm{host}}$
and $\sigma_{\mathrm{host}}$. However, the values of $\mu_{\mathrm{host}}$ and
$\sigma_{\mathrm{host}}$ do not directly correspond to physical values.
Instead, the median and standard deviation of the log-normal rest-frame
$\mathrm{DM}_{\mathrm{host}}$ distribution described by those parameters are
$84^{+69}_{-49}$ pc cm-3 and $174^{+319}_{-128}$ pc cm-3, respectively. As
illustrated by the statistical uncertainties, these parameters are not well-
constrained; how poorly constrained these parameters are can also be
visualized with the allowable $P(\mathrm{DM}_{\mathrm{host}}^{\prime})$
distributions plotted in Figure 8. As such, caution must be employed when
comparing these values against those found in the literature. Nonetheless, we
highlight a few observations.

Figure 8:  Plotted in gray are the results of the allowable distributions of
the rest-frame contribution to the $\mathrm{DM}$ from the host galaxy of the
FRB, $P(\mathrm{DM}_{\mathrm{host}}^{\prime})$, from the MCMC run, thinned by
a factor of 500 for visualization purposes. The best-fit distribution from the
MCMC run is overlaid in solid blue. Due to the extended tail nature of this
distribution, only values up to $\mathrm{DM}_{\mathrm{host}}^{\prime}=200$ pc
cm-3 are plotted. However, higher $\mathrm{DM}_{\mathrm{host}}^{\prime}$
values are certainly allowed.

Macquart et al. (2020) and James et al. (2022a, b) also generically modeled
the intrinsic host galaxy contribution to the $\mathrm{DM}$ with a log-normal
distribution, with only the latter study explicitly quoting derived values. In
the work presented by James et al. (2022a), they quote a log-mean host
$\mathrm{DM}$ contribution of $129^{+66}_{-48}$ pc cm-3 on top of a typical
Galactic contribution of
$\mathrm{DM}_{\mathrm{disk}}+\mathrm{DM}_{\mathrm{halo}}=50+35=$ 85 pc cm-3.
It is important to note that James et al. (2022a) used a different sample of
FRBs from different surveys (at different observing frequencies), did not
apply a scattering cut for $\tau>10$ms to their model as we did, and also
quote a typical Galactic contribution of 85 pc cm-3 instead of $=$80 pc cm-3,
although this latter difference is likely to be minimal (Section 5.3.1).
However, in Figure 1, we can see that our MCMC chains also strongly disfavor
low $\sigma_{\mathrm{host}}$ and high $\mu_{\mathrm{host}}$ values, a finding
consistent with James et al. (2022a).

Both James et al. (2022a) and our analysis find “central” values of
$\mathrm{DM}_{\mathrm{host}}^{\prime}$ well within statistical errors of each
other, as well as within “typical” host galaxy contributions which are often
quoted as $\sim$50 pc cm-3 (e.g., Shannon et al., 2018) or $\sim$100 pc cm-3
(e.g., Cordes & Chatterjee, 2019). One should also note that the log-normal
nature of the distribution allows for the occasional extraordinarily high host
galaxy $\mathrm{DM}$ such as the estimated $\mathrm{DM}_{\mathrm{host}}\approx
903^{+72}_{-111}$ pc cm-3 found for FRB 20190520B by Niu et al. (2021). This
is also in agreement with Rafiei-Ravandi et al. (2021) finding statistical
evidence for some FRBs having a large host $\mathrm{DM}$ (defined therein as
$\sim$400 pc cm-3).

It is worth noting that the assumptions built into the $P(\mathrm{DM}|z)$
distribution are not perfect. Notably, we assume an average constant Milky Way
$\mathrm{DM}$ contribution, despite knowing that the disk contribution to the
$\mathrm{DM}$ varies greatly as a function of Galactic latitude and that there
is currently great uncertainty associated with the halo contribution to the
$\mathrm{DM}$. Although we currently aim to minimize the uncertainty
associated with the halo contribution by excluding sources with
$\mathrm{DM}_{\mathrm{NE2001}}>100$ pc cm-3, a more sophisticated analysis
would take into account updated models of the Milky Way halo $\mathrm{DM}$
(e.g., Cook et al. in prep). Modifying the formalism to include
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ as a free parameter would likely have
constraining power only after the sample of FRBs grows to reduce the already
large statistical uncertainties already associated with $\mu_{\mathrm{host}}$
and $\sigma_{\mathrm{host}}$.

Overall, we note that the poor constraints we obtain with the Catalog 1 data
are perhaps unsurprising when one considers that no redshift information was
used for any FRBs in this study. However, the commissioning of the forthcoming
CHIME/FRB Outriggers project (e.g., Leung et al., 2021; Cary et al., 2021;
Mena-Parra et al., 2022; Cassanelli et al., 2022) should provide localizations
for hundreds of FRBs and thus enable redshift determinations for a much larger
sample of FRBs, which in turn would allow for much more sensitive constraints
on the intrinsic $\mathrm{DM}_{\mathrm{host}}$ contribution.

###  6.6 Considering repeaters vs non-repeaters

A growing number of FRBs has been observed to repeat (e.g., Spitler et al.,
2016; Scholz et al., 2016; CHIME/FRB Collaboration et al., 2019, 2020b;
Fonseca et al., 2020; Bhardwaj et al., 2021; Lanman et al., 2021). There is
currently no reliable method to identify which FRBs are from repeaters, but
the growing sample of observed repeat bursts has led to some promising
methods. Pleunis et al. (2021) found that repeaters could be a distinct
population from apparent one-offs based on the temporal and spectral studies
of the Catalog 1 sample. Also using the Catalog 1 sample, Chen et al. (2022)
have recently used unsupervised machine learning to create a catalog of
repeater source candidates. Of course, all this is assuming that there are two
distinct populations of FRB sources: those that repeat, and those that do not.
Although observationally such a classification scheme is intuitive, it is also
possible that all FRBs repeat, or all FRBs represent distinct phases of the
same class of progenitors (e.g., Caleb et al., 2019; Pleunis et al., 2021; Cui
et al., 2021).

If most FRBs are from repeaters, the results presented here are likely to be
affected to a stronger degree than if a lower fraction of FRBs is from
repeaters. As mentioned in Section 2.3, to minimize unquantifiable bias, we
include only the first-detected burst from any observed FRB source. However,
one should expect the repetition rate of FRBs to affect distributions of
observed parameters. Naively, the presence of increasing numbers of repeating
FRBs would cause the burst rate within an observed volume of the universe to
be increasingly greater than expected. Likewise, Gardenier et al. (2021) noted
that repeating FRBs should have a lower mean $\mathrm{DM}$ than apparent one-
off FRBs, although CHIME/FRB Collaboration et al. (2021) found that the
distributions of repeating FRBs and apparent one-off FRBs are consistent with
being drawn from the same underlying $\mathrm{DM}$ distribution. Pleunis et
al. (2021) show that there are observable differences in the spectral
parameters (e.g., $\alpha$) of repeaters and non-repeaters, and of course the
differences could extend to physically different values of
$\mathrm{DM}_{\mathrm{host}}$ contributions (e.g., $\mu_{\mathrm{host}}$ and
$\sigma_{\mathrm{host}}$), evolution with SFH (e.g., $n$), and more.

Although it is tempting to conduct a populations study considering only
apparent non-repeaters from the Catalog 1 sample, it is likely that some
apparent one-off bursts are actually from repeater sources, with both Pleunis
et al. (2021) and Chen et al. (2022) proposing candidate apparent one-off
bursts that may actually be from repeater sources. In fact, FRB20190304A was
identified as an apparent non-repeater source in Catalog 1, but since then
another burst has been observed from this source (CHIME/FRB Collaboration et
al., private communication), establishing its nature as a repeater instead.
There is also the fact that CHIME/FRB has only observed one burst from the
first repeating source FRB 121102 (Josephy et al., 2019), despite there many
more additional observed bursts present in the literature (Rajwade et al.,
2020; Li et al., 2021). The presence of such as-yet-unidentified repeat bursts
could represent a not insignificant “contamination” of the nominal non-
repeating sample of FRBs. Chen et al. (2022) suggest a large potential
“contamination” of repeaters, but Pleunis et al. (2021) warn that instrumental
effects, such as chromatic sensitivities away from the detection beam centers,
make it difficult to isolate the true repeaters. This is because such
instrumental effects would result in observed bursts with biased low
bandwidths that also cause mixing of bursts with spectra that are
intrinsically broad and narrow. Further complicating this unquantifiable
effect of as-yet unidentified repeaters in a sample of nominally one-off
bursts is that the injections framework of CHIME/FRB currently only injects
bursts from a single population distribution of FRB parameters. Nonetheless,
as larger samples of well-defined repeaters are identified, we expect our
ability to do comparative populations analyses between observed repeat bursts
and one-off bursts will improve.

##  7 CHIME/FRB Outrigger Predictions

For the majority of FRBs detected by CHIME/FRB, the localization precision is
insufficient for identifying host galaxies for these bursts. Host galaxies,
and thus redshifts, for FRBs are highly desirable for both understanding the
nature of FRB origins and using them as cosmological probes (Bhandari et al.,
2022; Masui & Sigurdson, 2015). Enabling larger-scale localization abilities
is a high-priority science goal for FRBs, and one that will be addressed with
the CHIME/FRB Outriggers project. Through CHIME/FRB Outriggers, CHIME-like
outrigger telescopes will be deployed at continental baselines to form a very-
long-baseline interferometry (VLBI) network. While CHIME/FRB Outriggers will
trigger on a subset of sources observed by CHIME/FRB (e.g., not on
low–$\mathrm{S/N}$ bursts), the VLBI network will still obtain near real-time,
sub-arcsecond localization capability for hundreds of FRBs a year (Mena-Parra
et al., 2022). Although it is unrealistic to expect that every FRB will have
its VLBI position followed-up with optical observations to obtain
spectroscopic redshifts, it would certainly be possible for spectroscopic
redshifts to be obtained for a much more significant fraction of FRBs than is
currently possible. Additionally, a larger fraction of FRBs may have
reasonably reliable photometric redshifts obtained by cross-matching the VLBI
localizations with photometric galaxy surveys such as the Dark Energy
Spectroscopic Instrument (DESI) Legacy Imaging Surveys (Dey et al., 2019).

With the methodology used in this paper, we can obtain a prediction for the
observed FRB redshift distribution that CHIME/FRB Outriggers may see. Recall
Equation 1:

$R(F,\mathrm{DM}|\bm{\lambda})=\int
dzR(F,z|\bm{\lambda_{1}})P(\mathrm{DM}|z,\bm{\lambda_{2}}).$

It is here that the model $R(F,\mathrm{DM}|\bm{\lambda})$ is marginalized over
$z$. As described in Section 4, $R(F,\mathrm{DM}|\bm{\lambda})$ is used to
construct parameter-dependent weights, which are then used to create a model
prediction $\zeta_{ij}$, a weighted histogram of the synthetic data in
$\mathrm{S/N}$ and $\mathrm{DM}$ (Equation 7). Properly normalized, this
earlier obtained $\zeta_{ij}$ can be translated to an observed rate
distribution of $\mathrm{S/N}$ and $\mathrm{DM}$, given parameters
$\bm{\lambda}$, over the redshift range $z=1\times 10^{-3}-4$.

Instead of integrating over $z$ and ending up with a 2D model prediction
(histogram) in $\mathrm{S/N}$ and $\mathrm{DM}$, we can construct a 3D model
prediction that depends on $\mathrm{S/N}$, $\mathrm{DM}$, and $z$:

$\displaystyle\zeta_{ij}{\Bigr{|}}_{z_{\mathrm{val}}}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{n_{ij}}W_{m}(F,\mathrm{DM},z_{\mathrm{val}},\mathbf{\Theta},\bm{\lambda})$
(14) $\displaystyle=$
$\displaystyle\sum_{m=1}^{n_{ij}}\frac{R(F,\mathrm{DM},z_{\mathrm{val}}|\bm{\lambda})\Delta
t}{R_{\mathrm{init}}(F,\mathrm{DM})}$ (15)

where

$R(F,\mathrm{DM},z_{\mathrm{val}}|\bm{\lambda})=\Delta
z\;R(F,z_{\mathrm{val}}|\bm{\lambda_{1}})P(\mathrm{DM}|z_{\mathrm{val}},\bm{\lambda_{2}}),$
(16)

and $\Delta z$ is estimated by the midpoint spacing between the evaluated
$z_{\mathrm{val}}$ points across the full redshift range $z=1\times
10^{-3}-4$. (Note that for simplicity of notation, $\mathbf{\Theta}$ was
dropped as their distributions were held to the fiducial models found in
Catalog 1 for all evaluated models, for each burst.)

This 3D histogram, after dividing by the bin widths in $z$, gives the
prediction for the observed CHIME/FRB Catalog 1
$\mathrm{DM}$–$\mathrm{S/N}$–$z$ distribution per $z$ for a given model
$R(F,\mathrm{DM}|\bm{\lambda})$. We can thus sum over the $\mathrm{S/N}$ bins
(which span the values 12–200) and the $\mathrm{DM}$ bins (which span the
values 100–3500 pc cm-3) to obtain the prediction for the observed redshift
distribution, $P_{\mathrm{obs}}(z)$. The range of predicted redshift
distributions our MCMC run allows appears reasonably well-constrained, and is
shown in Figure 9.

Figure 9:  Plotted in gray are the results of the allowable observed FRB
redshift distributions $R_{\mathrm{obs}}(z)$ from the MCMC run, thinned by a
factor of 2000 for visualization purposes. The $R_{\mathrm{obs}}(z)$
prediction using the best-fit distribution from the MCMC run is overlaid in
solid blue. Although the model was fit up to $z=4$, only redshifts up to
$z=2.5$ are plotted.

The $P_{\mathrm{obs}}(z)$ distribution predicted by the best-fit model peaks
at around a redshift of $z\approx 0.36$. This result is consistent with the
$\mathrm{DM}$ distribution observed by CHIME/FRB shown in CHIME/FRB
Collaboration et al. (2021), where the peak of that distribution is around
$\mathrm{DM}\approx 500$ pc cm-3. That the redshift peak is below $z\sim 0.5$
suggests that there is a noticeable fraction of low to intermediate-$z$ FRBs
that have a sizable $\mathrm{DM}$ contribution from astrophysical environments
other than the cosmological intergalactic medium (e.g., local environments,
host galaxies, or intervening galaxies). This $P_{\mathrm{obs}}(z)$ prediction
also contains $\approx$99.4% of its area within $z\leq 2$, suggesting that
CHIME/FRB Outriggers will see a negligible number FRBs with $z>2$. Such a
result implies that the sample of FRBs observed by CHIME/FRB Outriggers will
be limited in its ability to be used as high-$z$ cosmological probes (e.g.,
Linder, 2020). Obtained values for the $P_{\mathrm{obs}}(z)$ distribution
predicted by the best-fit model is provided in Table 3. It is worth noting
this prediction is based on the sample that survived the statistical cuts
detailed in Section 2.3; therefore, it is likely that the true
$R_{\mathrm{obs}}(z)$ distribution will differ when including observed bursts
that do not pass these cuts (such as bursts with $\mathrm{DM}<1.5$
max($\mathrm{DM}_{\mathrm{NE2001}},\mathrm{DM}_{\mathrm{YMW16}}$) and
$\mathrm{S/N}<12$). Nevertheless, this prediction supports the notion that
low-$z$ cosmological studies such as lensing from foreground structures will
likely be a promising scientific application of FRBs in the near future (e.g.,
Kader et al., 2022; Leung et al., 2022).

Figure 10:  Left panel: The median $P_{\mathrm{obs}}(z|\mathrm{DM})$
distribution for the results of the MCMC run, thinned by a factor of 2000 for
computational speed purposes. Evidence for a “turnover” of the DM–$z$ relation
can be seen at higher DMs and lower redshifts, although most of the FRBs
appear to occur at $\mathrm{DM}<2000$ pc cm-3 and $z<2$. Right panel: The
standard deviation of the of the $P_{\mathrm{obs}}(\mathrm{DM},z)$
distributions at each point divided by the median
$P_{\mathrm{obs}}(\mathrm{DM},z)$ (shown on the left) to visualize the
fractional uncertainty of $P_{\mathrm{obs}}(z|\mathrm{DM})$ values across MCMC
realizations.

We can also take the 3D histogram that gives the prediction for the observed
CHIME/FRB Catalog 1 $\mathrm{DM}$-$\mathrm{S/N}$-$z$ distribution, given by
Equations 14 and 16, and divide by the bin widths in $z$ and $\mathrm{DM}$.
Then, we can sum over the $\mathrm{S/N}$ bins and normalize the distribution
such that we obtain a prediction for the observed joint redshift and
$\mathrm{DM}$ distribution, per $z$ and per $\mathrm{DM}$:
$P_{\mathrm{obs}}(z|\mathrm{DM})$. We show a visualization of what
$P_{\mathrm{obs}}(z|\mathrm{DM})$ predictions are allowed by the MCMC run in
Figure 10.

In the left-hand panel of Figure 10, the $P_{\mathrm{obs}}(z|\mathrm{DM})$
result shows that, although the relation between the redshift and
$\mathrm{DM}$ of FRBs is a reasonably tight, single-modal distribution (e.g.,
Shannon et al., 2018; Macquart et al., 2020), for a survey sample as large as
Catalog 1, there is a “turnover” at high $\mathrm{DM}$ where the true redshift
of the FRB is much lower than would be expected by applying a naive
$\sim$linear DM–$z$ relation (first shown by James et al., 2022b). This can be
seen the apparent excess of probability in the overlapping region of
$0.5\lesssim z\lesssim 1.0$ and $\mathrm{DM}\gtrsim 2000$ pc cm-3. There, the
plot is saying that, for an FRB observed by CHIME/FRB Outriggers with a
measured $\mathrm{DM}\gtrsim 2000$ pc cm-3, it is more likely for the source
to have a redshift $z<1$ with a high $\mathrm{DM}_{\mathrm{host}}$
contribution than a redshift $z>2$ with a large $\mathrm{DM}_{\mathrm{IGM}}$
contribution. This finding is consistent with Rafiei-Ravandi et al. (2021),
who find statistical evidence for a nonzero correlation between
high-$\mathrm{DM}$ FRBs ($\geq 785$ pc cm-3) and galaxies at $z\sim 0.4$, with
a $\mathrm{DM}_{\mathrm{host}}$ contribution $\sim$greater than the
$\mathrm{DM}_{\mathrm{IGM}}$ contribution. This result once again emphasizes
the importance of actually obtaining redshifts for FRBs rather than relying on
inferred redshift values when doing cosmological studies, especially for high-
DM FRBs.

The right-hand panel of this figure shows fractional uncertainties associated
with our prediction for $P_{\mathrm{obs}}(z|\mathrm{DM})$ across the chains in
our MCMC run. This plot illustrates that, along the relation between $z$ and
$\mathrm{DM}$, the predicted $P_{\mathrm{obs}}(z|\mathrm{DM})$ is quite
consistent across realizations. The further the deviation from that relation,
the more “scatter” there is between predictions in the run. For example, it
seems that between different MCMC realizations, the prediction for FRBs with
$z\lesssim 0.5$ varies greatly; some chains exhibit a strong “turnover” in the
$\mathrm{DM}$–$z$ relation, while others (a fewer number) do not. Though as
the left-hand panel of this figure shows, the median obtained
$P_{\mathrm{obs}}(z|\mathrm{DM})$ does exhibit evidence for a “turnover” of
the $\mathrm{DM}$–$z$ relation at higher DMs and lower redshifts. It is also
worth noting that “fractional uncertainty” can also mean “scatter surrounding
how close this value is to zero across runs,” as appears to be the case in the
region with $\mathrm{DM}<500$ pc cm-3 and $0.5<z<1.0$ — a region that appears
to have negligible probability in the left-hand panel of this figure, but can
have scatter due to variations in large scale structure noise.

##  8 Conclusion

Using the CHIME/FRB Catalog 1 data, we improve upon the initial Catalog 1
populations analysis by fitting a more physical model to the fluence and DMs
of the FRBs, a model initially developed in previous works (e.g., Luo et al.,
2020; James et al., 2022b). We use the weighting formalism developed in
CHIME/FRB Collaboration et al. (2021), which makes use of the injected
synthetic pulses to calibrate out selection function effects when fitting the
model to the observed FRB data. The model fits for 7 parameters: a volumetric
rate $\Phi_{0}$, a spectral index $\alpha$, a differential power-law energy
index $\gamma$, a characteristic exponential cutoff energy
$E_{\mathrm{char}}$, a power-law index $n$ for smooth-scaling evolution with
SFR, and parameters $\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ that
parameterize the log-normal distribution of the host $\mathrm{DM}$
contribution. We also explore the “rate interpretation” of $\alpha$.

We find a volumetric rate of FRBs above $10^{39}$ erg of $\Phi_{0}=$
[$7.3^{+8.8}_{-3.8}$(stat.)${}^{+2.0}_{-1.8}$(sys.)]$\times 10^{4}$ bursts
Gpc-3 year-1. We also find tight constraints on the energy distribution
observed by CHIME/FRB, modeled as a Schechter function, with
$E_{\mathrm{char}}=$ $2.38^{+5.35}_{-1.64}\times 10^{41}$ erg and $\gamma=$
$-1.3^{+0.7}_{-0.4}$. However, the constraints on the remainder of the
parameters are much weaker. In particular, the constraints on $n$,
$\mu_{\mathrm{host}}$, and $\sigma_{\mathrm{host}}$ may significantly improve
with the use of redshift information for many FRBs. Such a sample of redshift
information would be enabled by the upcoming CHIME/FRB Outriggers project, for
which we predict the observed FRB redshift distribution. The distribution
shows that the application of CHIME-detected FRBs to cosmological studies may
be limited to the $z<2$ regime, which limits the possibility of using these
FRBs to probe events such as Helium reionization, but does allow for FRB
lensing studies.

As the amount of data seems to be the limiting factor for this FRB populations
study, it is promising that CHIME/FRB has continued to detect a large number
of FRBs since Catalog 1 (which only included bursts up to 2019 July 1). These
observed FRBs, combined with knowledge about survey selection effects provided
by the injections system, and possibly redshift information with the
commissioning of CHIME/FRB Outriggers, suggest much more powerful populations-
level constraints in the near future. With more data comes more responsibility
to handle subtle effects — namely, how best to treat repeating sources versus
apparent one-off bursts when modeling the FRB distribution. Such a question
would be of particular importance when exploring population behavior in a
parameter space where repeater bursts would dominate, such as events with
large pulse widths. Further in the future, care must be taken to properly
incorporate redshift information into populations studies while accounting for
the selection biases (e.g., repeater or high-$\mathrm{S/N}$) that lead to
some, but not all, FRBs having associated redshifts.

## acknowledgments

We acknowledge that CHIME is located on the traditional, ancestral, and
unceded territory of the Syilx/Okanagan people. We are grateful to the staff
of the Dominion Radio Astrophysical Observatory, which is operated by the
National Research Council of Canada. CHIME is funded by a grant from the
Canada Foundation for Innovation (CFI) 2012 Leading Edge Fund (Project 31170)
and by contributions from the provinces of British Columbia, Québec and
Ontario. The CHIME/FRB Project is funded by a grant from the CFI 2015
Innovation Fund (Project 33213) and by contributions from the provinces of
British Columbia and Québec, and by the Dunlap Institute for Astronomy and
Astrophysics at the University of Toronto. Additional support was provided by
the Canadian Institute for Advanced Research (CIFAR), McGill University and
the McGill Space Institute thanks to the Trottier Family Foundation, and the
University of British Columbia. A.B.P. is a McGill Space Institute (MSI)
Fellow and a Fonds de Recherche du Quebec - Nature et Technologies (FRQNT)
postdoctoral fellow. A.B.P. is a McGill Space Institute (MSI) Fellow and a
Fonds de Recherche du Quebec - Nature et Technologies (FRQNT) postdoctoral
fellow. A.H.M is supported by the CANSSI CRT program. B.M.G. is supported by
an NSERC Discovery Grant (RGPIN-2022-03163), and by the Canada Research Chairs
(CRC) program. C.L. was supported by the U.S. Department of Defense (DoD)
through the National Defense Science & Engineering Graduate Fellowship (NDSEG)
Program. F.A.D is supported by the U.B.C Four Year Fellowship. K.S. is
supported by the NSF Graduate Research Fellowship Program. K.W.M. is supported
by NSF grants 2008031 and 2018490. M.B. is supported by an FRQNT Doctoral
Research Award. M.D. is supported by a Canada Research Chair, NSERC Discovery
Grant, CIFAR, and by the FRQNT Centre de Recherche en Astrophysique du Québec
(CRAQ). M.Me. is supported by an NSERC PGS-D award. MM is supported by DOE
grant DE-SC0022342. SPT is a CIFAR Azrieli Global Scholar in the Gravity and
Extreme Universe Program. V.M.K. holds the Lorne Trottier Chair in
Astrophysics & Cosmology, a Distinguished James McGill Professorship, and
receives support from an NSERC Discovery grant (RGPIN 228738-13), from an R.
Howard Webster Foundation Fellowship from CIFAR, and from the FRQNT CRAQ.

##  Appendix A Details of the Fluence–DM Distribution Model

###  A.1 Joint fluence and redshift distribution

In this section, we focus on how to obtain $R(F,z|\bm{\lambda_{1}})$. The
parameter vector $\bm{\lambda_{1}}$ contains information related to FRB
energetics.

First, we define the energy function $P(E)$. We model $P(E)$ with a Schechter
function with a characteristic exponential cutoff energy $E_{\mathrm{char}}$
and power-law index $\gamma$, e.g.,

$P(E)dE\propto\frac{1}{E_{\mathrm{char}}}\left(\frac{E}{E_{\mathrm{char}}}\right)^{\gamma}\exp{\left[-\frac{E}{E_{\mathrm{char}}}\right]}dE.$
(A1)

In order for $\int E\>P(E)\>dE$ in Equation A1 to not diverge, we must have
$\gamma<-2$. However, this function is not truly normalizable, i.e., we cannot
guarantee $\int_{0}^{\infty}P(E)\>dE=1$ for all redshifts. Thus, we shall
choose to normalize the function such that
$\int_{E_{\mathrm{pivot}}}^{\infty}P(E)\>dE=1$ while assuming the energy
itself extrapolates to follow the Schechter function. This allows us to assume
this energy distribution is valid for all FRB energies down to a minimum (but
unknown) energy $E_{\mathrm{min}}$, while quoting a rate of FRBs above
$E_{\mathrm{pivot}}$. For a reasonable choice of $E_{\mathrm{pivot}}$,
$\Phi_{0}$ should be weakly correlated with $\gamma$. We chose a pivot energy
of $10^{39}$ erg as it is above the threshold of ruled out minimum FRB
energies (e.g., James et al., 2022b); in our results in Figure 1, we can see
that $\Phi_{0}$ and $\gamma$ are clearly correlated, but because their
correlation isn’t strong, we did not fine tune $E_{\mathrm{pivot}}$ further so
as to have $\Phi_{0}$ and $\gamma$ be more weakly correlated.

To normalize this energy distribution, recall that the integral of a Schechter
function is an upper incomplete gamma function, defined as

$\Gamma\left(\gamma+1,\frac{E_{\mathrm{pivot}}}{E_{\mathrm{char}}}\right)=\int_{E_{\mathrm{pivot}}}^{\infty}\frac{1}{E_{\mathrm{char}}}\left(\frac{E}{E_{\mathrm{char}}}\right)^{\gamma}\exp{\left[-\frac{E}{E_{\mathrm{char}}}\right]}dE.$
(A2)

Thus, the “normalized” Schechter function is

$P(E)dE=\frac{1}{\Gamma(\gamma+1,E_{\mathrm{pivot}}/E_{\mathrm{char}})}\frac{1}{E_{\mathrm{char}}}\left(\frac{E}{E_{\mathrm{char}}}\right)^{\gamma}\exp{\left[-\frac{E}{E_{\mathrm{char}}}\right]}dE.$
(A3)

With this normalization, all rates should be interpreted as a rate above the
pivot energy $E_{\mathrm{pivot}}$.

We can convert between the FRB energy and fluence with

$E=\frac{4\pi D^{2}_{L}(z)}{(1+z)^{2+\alpha}}\Delta\nu F,$ (A4)

where $E$ is the FRB energy, $D_{L}(z)$ is the luminosity distance, $\alpha$
is the spectral index ($F\propto\nu^{\alpha}$), and $\Delta\nu$ is the
frequency bandwidth (taken as 1 GHz).

$R(F,z)$ and $P(E)$ are related by

$\displaystyle R(F,z)$ $\displaystyle=$ $\displaystyle
P(F|z)R(z)=R(E,z)\frac{dE}{dF}$ (A5) $\displaystyle=$ $\displaystyle
P(E)R(z)\frac{dE}{dF}.$ (A6)

$R(z)$ is a rate per unit redshift, and contains the comoving volume element
scaled by the star formation rate (SFR) at that given redshift:

$R(z)=\chi(z)^{2}\frac{d\chi}{dz}\Phi(z)$ (A7)

Since $\chi(z)$ is the comoving distance, the component
$\chi(z)^{2}\frac{d\chi}{dz}$ is equivalent to a volume element
$\frac{dV}{dz}$. $\Phi(z)$ is the rate of FRBs per comoving volume, which can
then be generically modeled as

$\Phi(z)=\frac{\Phi_{0}}{1+z}\left(\frac{\mathrm{SFR}(z)}{\mathrm{SFR}(0)}\right)^{n},$
(A8)

with $\Phi_{0}$ representing the rate of FRBs per comoving volume at $z=0$ and
$E>E_{\mathrm{pivot}}$, taking the units of bursts per proper time per
comoving volume, i.e., bursts year-1 Gpc-1.

The star formation rate $\mathrm{SFR}(z)$ is given by Madau & Dickinson (2014)
as

$\mathrm{SFR}(z)=1.0025738\frac{(1+z)^{2.7}}{1+\left(\frac{1+z}{2.9}\right)^{5.6}}.$
(A9)

This means the distribution of FRB energy as a function of distance is
proportional to the distribution of energies at a given volume element and
scaled by the star formation rate SFR.

We can obtain $\frac{dE}{dF}$ from Equation A4, which we can then combine with
Equations A1 and A7 to get

$R(F,z)=P(E)\chi(z)^{2}\frac{d\chi}{dz}\Phi(z)\frac{4\pi
D^{2}_{L}(z)}{(1+z)^{2+\alpha}}\Delta\nu.$ (A10)

Thus, the parameters in $\bm{\lambda_{1}}$ are $\alpha$, the spectral index;
$n$, the power-law scaling with SFR; and $\gamma$ and $E_{\mathrm{char}}$, the
power-law index and exponential cutoff energy which together characterize the
Schechter energy function.

This aspect of the model was evaluated on an intrinsic grid in $F$ and $z$
space. There were 200 $F$ points spaced logarithmically from $F=1$–$5\times
10^{5}$ Jy ms, encompassing the range of burst fluences from the Catalog 1
sample. There were 100 points in $z$ space, spaced logarithmically from
$z=1\times 10^{-3}$–1 and then linearly from $z=1$–4. This sampling was done
in order to densely sample and reduce numerical error for the relatively
low-$z$ region, where we would expect most FRBs to be.

###  A.2 DM distribution

We now have the first component of the integrand needed to find
$R(F,\mathrm{DM}|\bm{\lambda})$; the second component of the integrand is
given by

$P(\mathrm{DM}|z,\bm{\lambda_{2}})=P(\mathrm{DM}=\mathrm{DM}_{\mathrm{EG}}+\overline{\mathrm{DM}}_{\mathrm{MW}}|z,\bm{\lambda_{2}})$
(A11)

where the observed $\mathrm{DM}$ has an extragalactic contribution
$\mathrm{DM}_{\mathrm{EG}}$ and an average Galactic contribution
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ consisting of contributions from the
disk and halo of the Milky Way. To obtain $\mathrm{DM}_{\mathrm{disk}}$, the
interstellar medium contribution to the $\mathrm{DM}$ in the Milky Way disk,
we consider the NE2001 model of electron density (Cordes & Lazio, 2002) and
the YMW2016 model of electron density (Yao et al., 2017) for the observed
Catalog 1 bursts. The former gives a median value of
$\mathrm{DM}_{\mathrm{disk}}\approx 50$ pc cm-3, while the latter gives a
median value of $\mathrm{DM}_{\mathrm{disk}}\approx 45$ pc cm-3. The halo
contribution is much more uncertain, with estimates ranging from
$\approx$10–80 pc cm-3 (e.g., Dolag et al., 2015; Prochaska & Zheng, 2019;
Keating & Pen, 2020). As done by Chawla et al. (2022), we start with the
NE2001 model for estimating $\mathrm{DM}_{\mathrm{disk}}$, and based on Dolag
et al. (2015), adopt $\mathrm{DM}_{\mathrm{halo}}$ = 30 pc cm-3. Thus,
$\overline{\mathrm{DM}}_{\mathrm{MW}}$ is originally fixed to be 80 pc cm-3 in
our model, where deviations from the assumption will be absorbed into
contributions from the host galaxy of the FRB. We test the sensitivity of our
results to the adopted $\overline{\mathrm{DM}}_{\mathrm{MW}}$ in Section
5.3.1. Since $\overline{\mathrm{DM}}_{\mathrm{MW}}$ is fixed, obtaining
$P(\mathrm{DM}_{\mathrm{EG}}|z,\bm{\lambda_{2}})$ will allow us to then obtain
$P(\mathrm{DM}|z,\bm{\lambda_{2}})$.

$\mathrm{DM}_{\mathrm{EG}}$ has contributions from the cosmological ionized
gas distribution and host galaxy of the FRB, e.g.,
$\mathrm{DM}_{\mathrm{EG}}=\mathrm{DM}_{\mathrm{cosmic}}+\mathrm{DM}_{\mathrm{host}}$.
Using the mean density of ions $\bar{n}_{e}$ and relevant cosmological
parameters, the expected cosmological contribution to the $\mathrm{DM}$ can be
written as

$\left<\mathrm{DM}_{\mathrm{cosmic}}\right>(z)=\int_{0}^{z}\frac{c\bar{n}_{e}(z^{\prime})dz^{\prime}}{H_{0}(1+z^{\prime})^{2}\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}.$
(A12)

Then

$\mathrm{DM}_{\mathrm{cosmic}}=\Delta_{\mathrm{DM}}\left<\mathrm{DM}_{\mathrm{cosmic}}\right>(z),$
(A13)

where $\Delta_{\mathrm{DM}}$ is the deviation from the mean. Thus,

$\displaystyle P(\mathrm{DM}_{\mathrm{cosmic}}|z)$ $\displaystyle=$
$\displaystyle
P(\Delta_{\mathrm{DM}})\frac{d\Delta_{\mathrm{DM}}}{d\mathrm{DM}_{\mathrm{cosmic}}}$
(A14) $\displaystyle=$ $\displaystyle
P(\Delta_{\mathrm{DM}})\frac{1}{\left<\mathrm{DM}_{\mathrm{cosmic}}\right>(z)}.$
(A15)

The expression $P(\Delta_{\mathrm{DM}})$ is given by

$P(\Delta_{\mathrm{DM}}|z)=\kappa\Delta_{\mathrm{DM}}^{-B}\exp{\left[-\frac{(\Delta_{\mathrm{DM}}^{-A}-C_{0})^{2}}{2A^{2}\sigma^{2}_{\mathrm{DM}}}\right]}$
(A16)

with $A=3$, $B=3$, $\kappa$ the normalization factor, $C_{0}$ set by the
requirement of $\langle\Delta_{\mathrm{DM}}\rangle=1$, and
$\sigma_{\mathrm{DM}}$ parameterized by feedback with $Fz^{-0.5}=0.32z^{-0.5}$
(James et al., 2022b). Note that these parameters are fixed in all the fits.

The host galaxy contribution can be generically modeled as a log-normal
distribution

$P(\mathrm{DM}_{\mathrm{host}}^{\prime})=\frac{1}{\mathrm{DM}_{\mathrm{host}}^{\prime}}\frac{1}{\sigma_{\mathrm{host}}\sqrt{2\pi}}\exp{\left[-\frac{(\log\mathrm{DM}_{\mathrm{host}}^{\prime}-\mu_{\mathrm{host}})^{2}}{2\sigma^{2}_{\mathrm{host}}}\right]},$
(A17)

where $\mu_{\mathrm{host}}$ and $\sigma_{\mathrm{host}}$ are free parameters
that make up the parameter vector $\bm{\lambda_{2}}$. The median of this
distribution is $\exp(\mu_{\mathrm{host}})$, and the variance is
$[\exp(\sigma_{\mathrm{host}}^{2})-1]\exp(2\mu_{\mathrm{host}}+\sigma_{\mathrm{host}}^{2})$.
Thus, $\bm{\lambda_{2}}$ contains information related to the host galaxy
contribution to the extragalactic $\mathrm{DM}$.

The host contribution should also be corrected for redshift via

$\mathrm{DM}_{\mathrm{host}}=\frac{\mathrm{DM}_{\mathrm{host}}^{\prime}}{1+z}.$
(A18)

Similarly to the cosmological $\mathrm{DM}$,

$\displaystyle P(\mathrm{DM}_{\mathrm{host}}|z)$ $\displaystyle=$
$\displaystyle
P(\mathrm{DM}_{\mathrm{host}}^{\prime})\frac{d\mathrm{DM}_{\mathrm{host}}^{\prime}}{\mathrm{DM}_{\mathrm{host}}}$
(A19) $\displaystyle=$
$\displaystyle\frac{1}{\mathrm{DM}_{\mathrm{host}}^{\prime}}\frac{1}{\sigma_{\mathrm{host}}\sqrt{2\pi}}\exp{\left[-\frac{(\log\mathrm{DM}_{\mathrm{host}}^{\prime}-\mu_{\mathrm{host}})^{2}}{2\sigma^{2}_{\mathrm{host}}}\right]}(1+z).$
(A20)

With expressions for $P(\mathrm{DM}_{\mathrm{cosmic}}|z)$ and
$P(\mathrm{DM}_{\mathrm{host}}|z)$, the expression for the extragalactic
contribution to the $\mathrm{DM}$ becomes a convolution:

$P(\mathrm{DM}_{\mathrm{EG}}|z)=\int
d\mathrm{DM}_{\mathrm{host}}P(\mathrm{DM}_{\mathrm{host}}|z)P(\mathrm{DM}_{\mathrm{cosmic}}=\mathrm{DM}_{\mathrm{EG}}-\mathrm{DM}_{\mathrm{host}}|z).$
(A21)

However, our expression for $P(\mathrm{DM}_{\mathrm{host}}|z)$ is analytical,
while our expression for $P(\mathrm{DM}_{\mathrm{cosmic}}|z)$ is numerical.
Thus, a change of variables in the convolution is better for ease of
evaluation:

$P(\mathrm{DM}_{\mathrm{EG}}|z)=\int
d\mathrm{DM}_{\mathrm{cosmic}}P(\mathrm{DM}_{\mathrm{host}}=\mathrm{DM}_{\mathrm{EG}}-\mathrm{DM}_{\mathrm{cosmic}}|z)P(\mathrm{DM}_{\mathrm{cosmic}}|z).$
(A22)

Note that while Chawla et al. (2022) modeled host galaxy contributions to the
$\mathrm{DM}$ using the Catalog 1 sample, the results of the simulations were
predictions with noise. While it was the more complete way to model host
galaxy contribution compared to this model, this methodology needs a more
precise analytical form.

By Equation A11, we can obtain $P(\mathrm{DM}|z)$. Combining that with the
earlier expression for $R(F,z)$ gives us the brightness distribution
$R(F,\mathrm{DM})$.

This aspect of the model was evaluated on an intrinsic grid in $z$ and
$\mathrm{DM}$ space. The points in $z$ space are the same as outlined in
Section A.1, whereas there are 150 points in $\mathrm{DM}$ space linearly
spaced from $\mathrm{DM}=$ 100–6500 pc cm-3. When evaluating on the
$\mathrm{DM}_{\mathrm{EG}}$ grid, $\overline{\mathrm{DM}}_{\mathrm{MW}}$ is
subtracted from the $\mathrm{DM}$.

##  Appendix B Supplemental results for the “rate interpretation” of $\alpha$

This section contains the equivalent results of Figure 1 and Table 1, except
with the analysis done under the “rate interpretation” of $\alpha$. They are
presented in Figure 11 and Table 2. The logic in following few paragraphs was
originally presented in Macquart & Ekers (2018), and are presented again here
to emphasize how the equations change between the “rate interpretation” of
$\alpha$ and the “true spectral index interpretation”.

Figure 11:  Corner plot of the results of the MCMC run under the “rate
interpretation” of $\alpha$, thinned by a factor of 20 for visualization
purposes. Overlaid solid blue lines denote the median of the posterior
distributions, with the dashed blue lines enclosing the central 68% of the
samples.  Table 2: Table of results of the parameters fit in
$R(F,\mathrm{DM})$ assuming the “rate” interpretation of $\alpha$. Best-fit
results quote the median values of the posterior distributions, with the error
bars containing the central 68% of the samples. Where illustrative, fits are
paired with corresponding physical values.

Parameter | Uniform prior range | Best-fit result
---|---|---
$\log_{10}\;\Phi_{0}^{\prime}\>{}^{a}$ | [$-0.96$, $6.43$] | $4.83^{+0.35}_{-0.33}$
$\Phi_{0}^{\prime}$ | $\ldots$ |  $6.8^{+8.3}_{-3.6}\times 10^{4}$ bursts Gpc-3 yr${-1}$
$\gamma^{\prime}$ | [$-2.50$, $2.00$] | $-1.3^{+0.8}_{-0.5}$
$\log_{10}\;E_{\mathrm{char}}^{\prime}\>{}^{a}$ | [$38.00$, $49.00$] | $41.29^{+0.47}_{-0.46}$
$E_{\mathrm{char}}^{\prime}$ | $\ldots$ |  $1.95^{+3.75}_{-1.28}\times 10^{41}$ erg
$n^{\prime}$ | [$-2.00$, $8.00$] | $1.72^{+1.48}_{-1.10}$
$\alpha^{\prime}$ | [$-5.00$, $5.00$] | $-1.10^{+0.67}_{-0.99}$
$\mu_{\mathrm{host}}^{\prime}\;\times\log_{10}(e)\>^{a}$ | [$1.30$, $2.70$] | $1.91^{+0.27}_{-0.38}$
$\mathrm{Median}^{\prime}\;P(\mathrm{DM}^{\prime})$ | $\ldots$ |  $81^{+69}_{-48}$ pc cm-3
$\sigma_{\mathrm{host}}^{\prime}\;\times\log_{10}(e)\>^{a}$ | [$0.04$, $1.74$] | $0.41^{+0.22}_{-0.21}$
$\mathrm{Std.dev.}^{\prime}\;P(\mathrm{DM}^{\prime})$ | $\ldots$ |  $174^{+347}_{-130}$ pc cm-3

aafootnotetext: These parameters were fit in natural $\log$ space but are
presented in $\log_{10}$ space for ease of interpretation.

The flux density in the frame of the observer is related to the luminosity in
the frame of the emitting source by the following relation:

$S_{\nu}=(1+z)\frac{L_{(1+z)\nu}}{L_{\nu}}\frac{L_{\nu}}{4\pi D_{L}^{2}},$
(B1)

where $D_{L}$ is the luminosity distance, and the ratio $L_{(1+z)\nu}/L_{\nu}$
represents the $k$-correction, necessary because the radiation was emitted in
a different band than it was observed in. Since $S_{\nu}\sim\nu^{\alpha}$,
this $k$-correction factor is $(1+z)^{\alpha}$.

Observed flux and emitted luminosity are related via the following relation
(Marani & Nemiroff, 1996):

$F_{\nu}=\Delta t_{e}(1+z)^{2}\frac{L_{(1+z)\nu}}{L_{\nu}}\frac{L_{\nu}}{4\pi
D_{L}^{2}},$ (B2)

where $\Delta t_{e}$ represents the burst duration in the emission frame.
Given that $E_{\nu}=\int L_{\nu}(t)dt$, and we are only considering $E$ and
$F$ within a frequency bandwidth $\Delta\nu$, we can rewrite the above
relation as

$F=(1+z)^{2}(1+z)^{\alpha}\frac{E\Delta\nu}{4\pi D_{L}^{2}}.$ (B3)

With some rearranging, this equation is equivalent to Equation A4. This factor
of $\alpha$ is explicitly due to the $k$-correction from the “true spectral
index interpretation” of $\alpha$, i.e., $F\sim\nu^{\alpha}$.

Under the “rate interpretation” of $\alpha$, the $k$-correction instead comes
into play by directly modifying the rate. As mentioned in Section 5.4,
Equation A8 is modified such that it becomes

$\Phi(z)=\frac{\Phi_{0}}{1+z}\left(\frac{\mathrm{SFR}(z)}{\mathrm{SFR}(0)}\right)^{n}(1+z)^{\alpha}.$
(B4)

Additionally, Equation A4 is now

$E=\frac{4\pi D^{2}_{L}(z)}{(1+z)^{2}}\Delta\nu F.$ (B5)

It is worth noting that although the parameter values of $n$ and $\alpha$
changed under this interpretation, the constraints on both values were not
very powerful to begin with. We present these results for completeness
regarding systematic modeling uncertainties.

##  Appendix C Testing a FRB evolution model delayed with respect to SFR

In our fiducial $R(F,\mathrm{DM})$ model, the evolution of the FRB population
smoothly scales with the SFR as a power-law parameterized by $n$:

$\Phi(z)=\frac{\Phi_{0}}{1+z}\left(\frac{\mathrm{SFR}(z)}{\mathrm{SFR}(0)}\right)^{n}.$
(C1)

Despite the poor constraints on $n$, we decide to explore whether our data can
constrain FRB population in a toy model where the rate of FRBs strictly lags
behind the SFR by a delay parameter $\tau$ Gyr. Such a model is represented by

$\Phi(z)=\frac{\Phi_{0}}{1+z}\frac{\mathrm{SFR}(z_{a}(t_{\mathrm{MW}}(z)-\tau))}{\mathrm{SFR}(0)},$
(C2)

where $z_{a}$ is the function that obtains the redshift at a given lookback
time. Of course, it is not physical to expect that for each star that forms,
an FRB occurs $\tau$ Gyr later. However, if this toy model shows no
constraining power, then a more complex model where there is a distribution of
delay times would almost certainly have no constraining power as well. Thus,
we run this toy model first with the same procedure as in Sections 4 and 5,
with uniform priors on $\tau$ from 0 – 14 Gyr. The results of the MCMC are
plotted in Figure 12.

Figure 12:  Corner plot of the results of the MCMC run using a time-delay toy
model, thinned by a factor of 20 for visualization purposes. Overlaid solid
blue lines denote the median of the posterior distributions, with the dashed
blue lines enclosing the central 68% of the samples.

We immediately see that $\tau$ is even more poorly constrained than $n$; in
other words, with this toy model, our data still hold no constraining power
for the FRB evolution with SFR. The posterior distribution for $\tau$ appears
bimodal, with peaks at both negligible ($\sim$0 Gyr) time delay and
significant ($\sim$12 Gyr) time delay. Of our $10^{6}$ chains, $\approx$25.8%
have $\tau<2$ Gyr and $\approx$13.3% have $12$ Gyr $<14$ Gyr (the rest of the
parameter results vary across chains). Although it is tempting to ascribe
significance to this bimodality, we caution against over-interpretation for a
number of reasons. First, the middle of the posterior distribution does not
dip down to negligible probabilities, mitigating the bimodal appearance of
this distribution. Second, and more importantly, this is an unphysical toy
model whose primary purpose was to explore whether the Catalog 1 data could
constrain a model where FRBs as a population are delayed with respect to the
SFH of the universe. We conclude that the Catalog 1 data cannot constrain such
a model.

##  Appendix D Tabulated values for predicted observed redshift distribution
with CHIME/FRB Outriggers

Here, we present tabulated values for the $P_{\mathrm{obs}}(z)$ distribution
predicted by the best-fit model from our results.

Table 3:  Table containing the value of the 100 data points for the $P_{\mathrm{obs}}(z)$ distribution predicted by the best-fit model. This distribution is visualized in blue in Figure 9. The redshift array $z$ is spaced logarithmically from $10^{-3}$ to 1, and linearly afterwards.  $z$ |  $P_{\mathrm{obs}}(z)$ [$z^{-1}$] |  |  $z$ (cont.) |  $P_{\mathrm{obs}}(z)$ [$z^{-1}$] (cont.)
---|---|---|---|---
$1.00\times 10^{-3}$ | $1.90\times 10^{-3}$ |  | $\vdots$ | $\vdots$
$1.11\times 10^{-3}$ | $2.19\times 10^{-3}$ |  | $1.73\times 10^{-1}$ | $8.54\times 10^{-1}$
$1.23\times 10^{-3}$ | $2.51\times 10^{-3}$ |  | $1.92\times 10^{-1}$ | $9.17\times 10^{-1}$
$1.36\times 10^{-3}$ | $2.89\times 10^{-3}$ |  | $2.13\times 10^{-1}$ | $9.91\times 10^{-1}$
$1.51\times 10^{-3}$ | $3.31\times 10^{-3}$ |  | $2.36\times 10^{-1}$ | $1.08$
$1.67\times 10^{-3}$ | $3.80\times 10^{-3}$ |  | $2.62\times 10^{-1}$ | $1.14$
$1.86\times 10^{-3}$ | $4.36\times 10^{-3}$ |  | $2.90\times 10^{-1}$ | $1.19$
$2.06\times 10^{-3}$ | $4.99\times 10^{-3}$ |  | $3.22\times 10^{-1}$ | $1.22$
$2.28\times 10^{-3}$ | $5.71\times 10^{-3}$ |  | $3.57\times 10^{-1}$ | $1.24$
$2.53\times 10^{-3}$ | $6.53\times 10^{-3}$ |  | $3.95\times 10^{-1}$ | $1.24$
$2.80\times 10^{-3}$ | $7.46\times 10^{-3}$ |  | $4.38\times 10^{-1}$ | $1.21$
$3.11\times 10^{-3}$ | $8.52\times 10^{-3}$ |  | $4.86\times 10^{-1}$ | $1.16$
$3.45\times 10^{-3}$ | $9.70\times 10^{-3}$ |  | $5.39\times 10^{-1}$ | $1.09$
$3.82\times 10^{-3}$ | $1.10\times 10^{-2}$ |  | $5.97\times 10^{-1}$ | $1.01$
$4.24\times 10^{-3}$ | $1.25\times 10^{-2}$ |  | $6.62\times 10^{-1}$ | $9.11\times 10^{-1}$
$4.70\times 10^{-3}$ | $1.42\times 10^{-2}$ |  | $7.34\times 10^{-1}$ | $8.02\times 10^{-1}$
$5.20\times 10^{-3}$ | $1.61\times 10^{-2}$ |  | $8.14\times 10^{-1}$ | $6.83\times 10^{-1}$
$5.77\times 10^{-3}$ | $1.82\times 10^{-2}$ |  | $9.02\times 10^{-1}$ | $5.65\times 10^{-1}$
$6.40\times 10^{-3}$ | $2.05\times 10^{-2}$ |  | $1.000$ | $4.56\times 10^{-1}$
$7.09\times 10^{-3}$ | $2.32\times 10^{-2}$ |  | $1.098$ | $3.64\times 10^{-1}$
$7.86\times 10^{-3}$ | $2.62\times 10^{-2}$ |  | $1.196$ | $2.77\times 10^{-1}$
$8.72\times 10^{-3}$ | $2.96\times 10^{-2}$ |  | $1.294$ | $2.02\times 10^{-1}$
$9.66\times 10^{-3}$ | $3.36\times 10^{-2}$ |  | $1.392$ | $1.47\times 10^{-1}$
$1.07\times 10^{-2}$ | $3.82\times 10^{-2}$ |  | $1.490$ | $1.08\times 10^{-1}$
$1.19\times 10^{-2}$ | $4.35\times 10^{-2}$ |  | $1.588$ | $7.85\times 10^{-2}$
$1.32\times 10^{-2}$ | $4.97\times 10^{-2}$ |  | $1.686$ | $5.63\times 10^{-2}$
$1.46\times 10^{-2}$ | $5.68\times 10^{-2}$ |  | $1.784$ | $3.96\times 10^{-2}$
$1.62\times 10^{-2}$ | $6.49\times 10^{-2}$ |  | $1.882$ | $2.72\times 10^{-2}$
$1.79\times 10^{-2}$ | $7.41\times 10^{-2}$ |  | $1.980$ | $1.80\times 10^{-2}$
$1.99\times 10^{-2}$ | $8.46\times 10^{-2}$ |  | $2.078$ | $1.16\times 10^{-2}$
$2.20\times 10^{-2}$ | $9.65\times 10^{-2}$ |  | $2.176$ | $7.43\times 10^{-3}$
$2.44\times 10^{-2}$ | $1.10\times 10^{-1}$ |  | $2.274$ | $4.77\times 10^{-3}$
$2.71\times 10^{-2}$ | $1.25\times 10^{-1}$ |  | $2.371$ | $3.09\times 10^{-3}$
$3.00\times 10^{-2}$ | $1.42\times 10^{-1}$ |  | $2.469$ | $2.03\times 10^{-3}$
$3.33\times 10^{-2}$ | $1.61\times 10^{-1}$ |  | $2.567$ | $1.34\times 10^{-3}$
$3.69\times 10^{-2}$ | $1.82\times 10^{-1}$ |  | $2.665$ | $8.78\times 10^{-4}$
$4.09\times 10^{-2}$ | $2.06\times 10^{-1}$ |  | $2.763$ | $5.63\times 10^{-4}$
$4.54\times 10^{-2}$ | $2.32\times 10^{-1}$ |  | $2.861$ | $3.50\times 10^{-4}$
$5.03\times 10^{-2}$ | $2.61\times 10^{-1}$ |  | $2.959$ | $2.12\times 10^{-4}$
$5.58\times 10^{-2}$ | $2.93\times 10^{-1}$ |  | $3.057$ | $1.28\times 10^{-4}$
$6.18\times 10^{-2}$ | $3.28\times 10^{-1}$ |  | $3.155$ | $7.87\times 10^{-5}$
$6.85\times 10^{-2}$ | $3.66\times 10^{-1}$ |  | $3.253$ | $4.93\times 10^{-5}$
$7.60\times 10^{-2}$ | $4.08\times 10^{-1}$ |  | $3.351$ | $3.16\times 10^{-5}$
$8.42\times 10^{-2}$ | $4.53\times 10^{-1}$ |  | $3.449$ | $2.05\times 10^{-5}$
$9.34\times 10^{-2}$ | $5.01\times 10^{-1}$ |  | $3.547$ | $1.32\times 10^{-5}$
$1.03\times 10^{-1}$ | $5.53\times 10^{-1}$ |  | $3.645$ | $8.43\times 10^{-6}$
$1.15\times 10^{-1}$ | $6.07\times 10^{-1}$ |  | $3.743$ | $5.30\times 10^{-6}$
$1.27\times 10^{-1}$ | $6.65\times 10^{-1}$ |  | $3.841$ | $3.27\times 10^{-6}$
$1.41\times 10^{-1}$ | $7.28\times 10^{-1}$ |  | $3.939$ | $1.97\times 10^{-6}$
$1.56\times 10^{-1}$ | $7.94\times 10^{-1}$ |  | $4.037$ | $1.14\times 10^{-6}$
. . . |  . . . |  | $4.135$ | $6.31\times 10^{-7}$

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