# EXTRAGALACTIC MILLIMETER TRANSIENTS IN THE ERA OF NEXT GENERATION CMB
SURVEYS

T. Eftekhari Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA) and Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA  Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA  E. Berger Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA  B. D. Metzger Department of Physics and Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA  Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  T. Laskar Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK  Department of Astrophysics, Radboud University, 6525 AJ Nijmegen, The Netherlands  V. A. Villar Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA  Institute for Computational & Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA  K. D. Alexander Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA  G. P. Holder Department of Physics, University of Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL, 61801, USA  J. D. Vieira Department of Physics, University of Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL, 61801, USA  Astronomy Department, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA  Center for AstroPhysical Surveys, National Center for Supercomputing Applications, Urbana, IL, 61801, USA  N. Whitehorn Department of Physics and Astronomy, Michigan State University, 567 Wilson Road, East Lansing, MI 48824  Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA  P. K. G. Williams Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA  American Astronomical Society, 1667 K Street NW, Suite 800 Washington, DC 20006 USA

###### Abstract

The next generation of wide-field cosmic microwave background (CMB) surveys
are uniquely poised to open a new window for time-domain astronomy in the
millimeter band. Here we explore the discovery phase space for extragalactic
transients with near-term and future CMB experiments to characterize the
expected population. We use existing millimeter-band light curves of known
transients (gamma-ray bursts, tidal disruption events, fast blue optical
transients, neutron star mergers) and theoretical models, in conjunction with
known and estimated volumetric rates. Using Monte Carlo simulations of various
CMB survey designs (area, cadence, depth, duration) we estimate the detection
rates and the resulting light curve characteristics. We find that existing and
near-term surveys will find tens to hundreds of long-duration gamma-ray bursts
(LGRBs), driven primarily by detections of the reverse shock emission, and
including off-axis LGRBs. Next-generation experiments (CMB-S4, CMB-HD) will
find tens of fast blue optical transients (FBOTs) in the nearby universe and
will detect a few tidal disruption events. CMB-HD will additionally detect a
small number of short gamma-ray bursts, where these will be discovered within
the detection volume of next generation gravitational wave experiments like
Cosmic Explorer.

radio continuum: transients

††software: Astropy (Astropy Collaboration et al., 2018), Boxfit (van Eerten
et al., 2012)

##  1 Introduction

Our view of the transient sky has been revolutionized in the past few decades
by the advent of untargeted wide-field optical surveys and all-sky
X-ray/$\gamma$-ray satellites. More recently, wide-field centimeter-band radio
searches have been successful at finding millisecond-duration transients (fast
radio bursts; Shannon et al. 2018; CHIME/FRB Collaboration et al. 2019, 2021),
as well as a few long-duration ($\sim$years) transients (Law et al., 2019;
Anderson et al., 2020; Dong et al., 2021; Ravi et al., 2021; Stroh et al.,
2021). Outside of the electromagnetic (EM) band, gravitational wave (GW)
observatories have detected various types of compact object mergers (Abbott et
al., 2017, 2019, 2021), and neutrino detectors have found some evidence for
flaring neutrino emission from active galactic nuclei (AGN) (IceCube
Collaboration et al., 2018). These various approaches have significantly
increased the discovery rate of known events, enabled the discovery of new
phenomena, and have led to multi-messenger signals from some events. Upcoming
powerful facilities such as the Vera Rubin Observatory (VRO; LSST Science
Collaboration et al. 2009) and the Roman Space Telescope (Foley et al., 2019)
will continue to reshape the landscape of time-domain astronomy.

At centimeter wavelengths, wide-field searches are in their infancy, with
surveys such as the Australian Square Kilometer Array Pathfinder (ASKAP)
Survey for Variables and Slow Transients (VAST; Murphy et al. 2013) and the
Very Large Array Sky Survey (VLASS; Lacy et al. 2020) only just beginning to
provide the depth and sky coverage required for the discovery of extragalactic
transients. In the coming decade, the improved sensitivity and wide areal
coverage of facilities such as the Square Kilometer Array (SKA) will enable a
broader range of time-domain discoveries (Metzger et al., 2015; Dobie et al.,
2021; Leung et al., 2021).

At millimeter wavelengths, the study of transients has largely been limited to
targeted follow-up of events discovered at other wavelengths. These include
relativistic sources like gamma-ray bursts (GRBs; e.g., Berger et al. 2003)
and some tidal disruption events (TDEs; e.g., Berger et al. 2012; Cendes et
al. 2021b), as well as a few supernovae in the nearby universe (e.g.,
Soderberg et al. 2008; Horesh et al. 2013b). The direct discovery of
transients in the millimeter band may be enabled by the next generation of
cosmic microwave background (CMB) experiments, as these surveys will cover a
large fraction of the sky at high cadence and with unprecedented sensitivity,
complementing centimeter-band and optical surveys (Metzger et al., 2015).
Indeed, the discovery of millimeter transients at early times in CMB surveys
will uniquely facilitate the follow-up of sources across the electromagnetic
spectrum, lending to improved localizations and robust source
characterizations.

There are already possible hints of an interesting discovery space from on-
going CMB surveys. The first search for transients in a CMB survey led to the
detection of a single event with the South Pole Telescope polarization-
sensitive SPTpol survey (Whitehorn et al., 2016); the lack of a clear host
galaxy and the low detection significance, however, prevented a conclusive
determination of the nature of the event. A similar search using the third
generation camera on the South Pole Telescope, SPT-3G, revealed two
extragalactic transient events with timescales of a few weeks, some evidence
of light curve variability, and potential host galaxies in the localization
regions (Guns et al., 2021). The nature of these sources remains unclear,
although the observed variability may be indicative of an AGN origin. Finally,
using archival Planck data, Yuan et al. (2016) discovered a bright millimeter
source coincident with the TDE IGR J12580+0134, suggesting the event may
represent one of two known TDEs with off-axis jetted emission (see also
Mattila et al. 2018). An increasing number of Galactic stellar flares have
also been serendipitously detected in CMB surveys, but with much shorter
durations and obvious stellar counterparts (Guns et al., 2021; Naess et al.,
2021).

Given the potential of CMB surveys to discover transients, it is essential to
explore their actual discovery space and the survey strategies that would
maximize the discovery rate. This has been previously discussed in Metzger et
al. (2015), although with a focus on centimeter-band transients and surveys.
Here, we follow the approach of Metzger et al. (2015) and present for the
first time a comprehensive exploration of the discovery of millimeter-band
transients in CMB surveys. We consider a wide range of phenomena that span a
broad range of energies and ambient densities (and hence timescales and
luminosities). We use known and estimated volumetric rates and their
cosmological evolution, and define a robust detection criterion to explore how
detection rates vary as a function of survey depth, cadence, and sky coverage.
The wide range of known transients considered here, along with the basic
physical constraints of synchrotron emission, provide a robust
characterization of the parameter space.

We note that we do not consider Galactic transients such as stellar flares,
and we further distinguish between bona-fide transients and variable sources
like AGN. Although in principle wide-field experiments may lead to the
serendipitous discovery of new populations of previously unknown events, as we
show, the physical constraints of synchrotron emission limit the allowed phase
space for such sources. The broad range of energies and environment densities
probed by the models we consider here thus provide an adequate
characterization of the allowed parameter space.

The paper is structured as follows. In §2 we review the landscape of known
transients with detected millimeter emission, and present basic arguments that
govern the timescales and luminosities of synchrotron sources (§2.1). In §2.2,
we discuss the properties and rates of various classes of transients
considered in our analysis. We provide a brief overview of relevant CMB
surveys in §3, and describe our simulations in §4. Our results are presented
in §5, and we summarize our main findings in §6. Throughout the paper, we use
the following Planck cosmological parameters: $H_{0}=67.8\ \rm km\ s^{-1}\
Mpc^{-1}$, $\Omega_{m}=0.308$, and $\Omega_{\Lambda}=0.692$ (Planck
Collaboration et al., 2016).

Figure 1: Millimeter light curves of extragalactic transients, including tidal
disruption events (AT2019dsg: Cendes et al. 2021a, Sw J1644+57: Berger et al.
2012, IGR J1258+0134: Yuan et al. 2016, AT2022cmc: Alexander et al. 2022;
Perley et al. 2022; Smith et al. 2022), the fast blue optical transients
AT2018cow (Ho et al., 2019) and AT20202xnd (Ho et al., 2021a), core-collapse
supernovae (stars), and long GRBs (gray circles; see Table 1). Limits
correspond to $3\sigma$.

##  2 Extragalactic Transients

We consider a wide variety of extragalactic synchrotron transients that span a
range of timescales and luminosities in the millimeter band, corresponding to
a fiducial range of $\sim 30-300$ GHz. These include detected sources, which
span over seven orders of magnitude in luminosity ($L\sim 10^{37}-10^{44}\ \rm
erg\ s^{-1}$) and three orders of magnitude in time ($t\sim 10^{-1}-10^{2}$
d); see Figure 1. We also consider events that are theoretically predicted
despite no unambiguous direct detections to date (e.g., off-axis “orphan” GRB
afterglows), as well as more speculative transients (neutron star mergers
leaving stable magnetar remnants).

In Table 1 and Figure 1 we provide a complete list of extragalactic transients
with millimeter detections from the literature through March 2022 using the
NASA Astrophysics Data System (ADS). Long GRBs, which dominate the sample,
span luminosities of $L\sim 10^{41}-10^{44}\ \rm erg\ s^{-1}$ (see de Ugarte
Postigo et al. (2012) for a review). These LGRB detections have also revealed
reverse shock emission at early times with ALMA (Laskar et al., 2013, 2018c,
2019a).

A small, but growing, sample of TDEs have also been detected in the
millimeter. The most luminous examples are the relativistic TDEs Swift
J164449.3+573451 (hereafter, Sw J1644+57; Bloom et al. 2011; Burrows et al.
2011; Levan et al. 2011; Zauderer et al. 2011) and AT2022cmc (Andreoni et al.,
2022) with a luminosity comparable to typical LGRBs detected over a timescale
$\sim 200$ days (Berger et al. 2012; Alexander et al. 2022; Perley et al.
2022; Smith et al. 2022). More recently, Cendes et al. (2021b) used ALMA to
reveal the first millimeter detection of a non-relativistic TDE, AT2019dsg,
with a luminosity of $\sim{\rm few}\times 10^{38}$ erg s-1.

Finally, in the nearby universe, a small number of supernovae have also been
detected in the millimeter (e.g., Weiler et al. 2007; Soderberg et al. 2008;
Horesh et al. 2013b), with luminosities of $\sim 10^{37}-10^{38}$ erg s-1. At
higher luminosities ($\sim 10^{41}$ erg s-1), a growing number of luminous,
rapidly evolving blue transients (see §2.2.6) have also been discovered (e.g.,
Ho et al. 2019; Margutti et al. 2019; Coppejans et al. 2020; Ho et al. 2021a).

Despite the wide diversity of physical phenomena, the production of
synchrotron emission in these various transients is a generic feature of shock
mediated outflows. Synchrotron emission is generated at the interface between
the outflowing ejecta and a dense ambient environment. The wide diversity in
luminosities and timescales reflects the range of outflow velocities, kinetic
energies, and geometries, as well as a range of ambient environments. For
example, energetic relativistic outflows viewed on-axis (e.g., LGRBs) produce
luminous millimeter emission peaking on $\sim$day timescales, while non-
relativistic outflows with modest kinetic energies (e.g., supernovae) result
in low luminosities and longer durations.

###  2.1 Synchrotron Emission: Timescales and Luminosities

We begin with a general investigation of the luminosities and timescales
expected for millimeter-band synchrotron emitting transients. The discussion
below is meant to provide a broad overview of the landscape of transients, and
not a detailed derivation for any particular class of events. We consider an
outflow with kinetic energy $E_{K}$ and initial Lorentz factor
$\Gamma\equiv(1-\beta^{2})^{-1/2}$ where $\beta=v/c$ for an ejecta velocity
$v$ expanding into an ambient medium with constant density $n$. The outflow
sweeps up a mass comparable to its own, and transfers its energy to the
surrounding medium, at the deceleration radius $R_{\rm dec}$ given by (Nakar &
Piran, 2011):

$\begin{split}R_{\rm dec}=&\bigg{(}\frac{3E_{K}}{4\pi
nm_{p}c^{2}\beta^{2}\Gamma^{2}}\bigg{)}^{1/3}\\\ &\approx 10^{18}\ {\rm cm}\
E_{K,52}^{1/3}n^{-1/3}\beta^{-2/3}\Gamma^{-2/3},\end{split}$ (1)

where we have scaled the kinetic energy to $E_{K}=10^{52}$ erg and density to
$n=1\rm\ cm^{-3}$. The deceleration time is given by:

$t_{\rm dec}=\frac{R_{\rm dec}}{2c\beta^{2}\Gamma^{2}}\approx 225\ {\rm days}\
E_{K,52}^{1/3}n^{-1/3}\beta^{-5/3}\Gamma^{-8/3}.$ (2)

Before $t_{\rm dec}$ the light curve will rapidly brighten, before
subsequently declining following the Sedov-Taylor self-similar solution. This
timescale therefore defines a minimum peak time for millimeter transients; we
note that relativistic events will exhibit much shorter timescales (i.e.,
$t_{\rm dec}\propto\Gamma^{-8/3}$). For an observing frequency (here,
$\nu_{\rm obs}=100$ GHz) above the characteristic synchrotron frequency,
$\nu_{m}$, and the self-absorption frequency, $\nu_{a}$, the peak flux at
$t_{\rm dec}$ is given by (Nakar & Piran, 2011):

$F_{\rm\nu,dec}\approx 3\ {\rm mJy}\ E_{K,52}n^{0.83}\epsilon_{\rm
e,-1}^{1.3}\epsilon_{\rm B,-2}^{0.83}\beta^{2.3}d_{\rm L,27}^{-2}\nu_{\rm 100\
GHz}^{-0.65},$ (3)

where $\epsilon_{e}$ and $\epsilon_{B}$ are the fractions of post-shock energy
in the radiating electrons and magnetic field, respectively, $d_{L}=10^{27}\
d_{\rm L,27}$ is the luminosity distance, and we adopt a power-law index
$p=2.3$ for the electron energy distribution given by
$N(\gamma)\propto\gamma^{-p}$ for $\gamma$ above a minimal Lorentz factor
$\gamma_{m}=\Gamma\epsilon_{\rm e}m_{p}(p-2)/m_{e}(p-1)\approx
40\epsilon_{e,-1}\Gamma$ (Sari et al., 1998), where $m_{p}/m_{e}\approx 1800$
is the ratio of proton and electron masses.

For non-relativistic transients, or initially off-axis relativistic events
that reach sub-relativistic velocities at late times, the flux density
therefore scales linearly with energy, while the timescale scales as
$E_{K}^{1/3}$. High density environments similarly lead to brighter events
($F_{\rm\nu,dec}\propto n^{0.83}$) and overall shorter timescales ($t_{\rm
dec}\propto n^{-1/3}$).

In the case where the density of the ambient medium is shaped by the
progenitor’s stellar wind (e.g., $\rho=Ar^{-2}$, where $A\equiv\dot{M}/4\pi
v_{w}=5\times 10^{11}A_{\star}$ for a mass-loss rate $\dot{M}=10^{-5}\
M_{\odot}\ \rm yr^{-1}$ and a wind velocity $v_{w}=10^{3}\ \rm km\ s^{-1}$),
the deceleration time increases and is given by

$t_{\rm dec}\approx 1025\ {\rm days}\ A_{\star}E_{K,52}\beta^{-3}\Gamma^{-4}.$
(4)

For on-axis relativistic sources, the relevant timescale to reach peak
brightness is not $t_{\rm dec}$ but the timescale when $\nu_{\rm
obs}=\nu_{m}$, where the value of $\nu_{m}$ is determined by the minimum
Lorentz factor $\gamma_{m}$, and is given by (Granot & Sari, 2002):

$\nu_{m}\approx 100\ {\rm GHz}\ E_{\rm
K,iso,54}^{1/2}\epsilon_{e,\rm-1}^{2}\epsilon_{B,\rm-2}^{1/2}t_{1}^{-3/2},$
(5)

where we have scaled to $t=10$ days, and re-scaled to an isotropic-equivalent
kinetic energy $E_{\rm K,iso}=10^{54}$ erg assuming a beaming correction of
$0.01$. Equation 5 makes clear that relativistic sources reach peak brightness
in the millimeter-band on a timescale of $\sim 10$ d, as indeed is observed in
LGRBs (Figure 1). This is in contrast to the centimeter band where events
evolve on much longer ($\sim 100$ days) timescales. While the longer
timescales at GHz frequencies present a challenge for transient detection in
radio surveys (Metzger et al., 2015), the short durations at $\sim 100$ GHz
suggest that relativistic transients can be more readily identified in CMB
surveys.

In addition to the above arguments, the brightness temperature $T_{B}$ of a
synchrotron source sets a limit on the minimum observed variability timescale.
The inverse Compton catastrophe requires that synchrotron sources obey
$T_{B}<10^{12}$ K; above this temperature, electrons rapidly lose energy as
inverse Compton scattering of synchrotron photons dominates the energy output
(Kellermann & Pauliny-Toth, 1969). From Kulkarni et al. (1998), the brightness
temperature of an expanding synchrotron source in the plasma frame is given
by:

$T_{B}^{\prime}=\frac{L_{\nu}}{8\pi^{2}k}\frac{\Gamma^{2}}{\mathcal{D}\beta^{2}\nu^{2}t^{2}}\lesssim
10^{12}{\rm K}$ (6)

where $\mathcal{D}=[\Gamma(1-\beta\ {\rm cos}\ \theta)]^{-1}$ is the Doppler
factor and is equal to $\Gamma$ for the fastest moving material. This
constraint implies that a source with a given emission radius cannot vary on a
timescale shorter than the light travel time, where $\delta t_{\rm min}\approx
t\beta/2\Gamma^{2}$ for relativistic expansion, corresponding to a minimum
timescale for a source of luminosity $L_{\nu}$ given by (Metzger et al.,
2015):

$\delta t_{\rm
min}\approx\bigg{[}\frac{L_{\nu}}{16\pi^{2}kT_{B}^{\prime}\nu^{2}\Gamma^{3}}\bigg{]}^{1/2}\approx
0.8\ {\rm days}\ \Gamma^{-3/2}L_{\nu,\rm 30}^{1/2}\nu_{\rm 100\ GHz}^{-1}.$
(7)

The minimum variability timescale for sources above $L_{\nu}\gtrsim 10^{30}\
\rm erg\ s^{-1}\ Hz^{-1}$ at 100 GHz is therefore of order days, while the
characteristic timescales at frequencies of a few GHz are $\sim 100$ days
(Metzger et al., 2015). Figure 2 shows the boundary corresponding to
$T_{B}<10^{12}$ K for $\Gamma\beta=1$ compared to the timescales and
luminosities for a range of millimeter transients. The shorter variability
timescales at millimeter wavelengths further emphasizes the need for faster
cadence in CMB surveys.

Figure 2: Duration-luminosity phase space for extragalactic synchrotron
transients at 100 GHz. Timescales for individual sources are defined as the
time interval over which the flux is within a factor of 2 of the peak flux
(Table 2). Off-axis events correspond to $\theta_{\rm obs}=0.8$. The black
line indicates the brightness temperature constraint $T_{B}<10^{12}$ K, above
which inverse Compton scattering dominates the luminosity. With the exception
of relativistic events (e.g., on-axis LGRBs), synchrotron sources obey
$T_{B}<10^{12}$ K. We also show this limit at 1.3 GHz (gray line) which
restricts sources to longer timescales in the centimeter band. Horizontal
dotted lines correspond to the minimum volumetric event rate leading to one
transient detection across the entire sky for a seven year survey and a
limiting sensitivity of 60 mJy ($10\sigma$). Sources with lower luminosities
(and hence shorter timescales) require large intrinsic rates to compensate for
the decreased detection volume.

Finally, we calculate the limiting luminosity ($L_{\nu}$) leading to a single
detection over the course of a given survey by estimating the instantaneous
number of events across the sky above a flux density threshold
($F_{\rm\nu,lim}$):

$N=\frac{4\pi}{3}\bigg{(}\frac{L_{\nu}}{4\pi
F_{\rm\nu,lim}}\bigg{)}^{3/2}\mathcal{R}t_{\rm dur}$ (8)

where $\mathcal{R}$ is the volumetric event rate, and we adopt $t_{\rm dur}=7$
yr, motivated by the typical survey duration of planned and upcoming CMB
surveys. In Figure 2, we plot the contours corresponding to $N=1$ and unique
values of $\mathcal{R}$ assuming $F_{\nu,\rm lim}=60$ mJy (i.e., a $10\sigma$
detection in CMB-S4; see §3.4). The results suggest that transients varying on
timescales of $\lesssim 1$ day require large volumetric rates of
$\mathcal{R}\gtrsim 10\ \rm Gpc^{-3}\ yr^{-1}$ in order to result in a single
detection for nominal CMB survey designs. Conversely, for a volumetric rate of
$\mathcal{R}=0.1\ \rm Gpc^{-3}\ yr^{-1}$, comparable to the rate of
relativistic transients (Table 2), events with timescales as short as $\sim 2$
days are detectable. Thus while the combination of event rates and minimum
variability timescales precludes the detection of short-duration ($\lesssim
100$ d) centimeter wavelength transients (Metzger et al., 2015), millimeter
surveys will be sensitive to sources varying on timescales as short as a few
days.

Figure 3: Light curves of the extragalactic synchrotron transients considered
in this work (Table 2) at an observer frequency of 100 GHz. Sources include
on- and off-axis long and short GRBs, on-axis relativistic TDEs (Sw1644+57),
off-axis TDEs, FBOTs (AT2018cow), and neutron star binary mergers leaving
behind a stable magnetar remnant. We also show the 100 GHz flux density
corresponding to a typical star forming galaxy with a star formation rate of
$1\ \rm M_{\odot}\ yr^{-1}$ (horizontal red bar). Flux densities are
normalized to $D_{L}=10^{27}$ cm ($z\approx 0.07$).

###  2.2 Models and Volumetric Rates

In Figure 3 we show the model 100 GHz light curves for the extragalactic
transients we consider in this work, scaled to a common distance of $10^{27}$
cm ($z\approx 0.07$). This includes both on- and off-axis long and short GRBs,
on- and off-axis TDEs, binary neutron star mergers with stable magnetar
remnants, and fast blue optical transients (FBOTs). We also show lightcurves
for the reverse shock models which peak at much earlier times (§2.2.1) in
Figure 4. While a handful of SNe have been detected in the millimeter, the low
luminosities of these events ($L\sim 10^{37}-10^{38}\rm erg\ s^{-1}$; Figure
1) precludes their detection in CMB surveys. Indeed, given the sensitivity of
CMB experiments (Table 3), except in the most extreme cases, SNe would only be
detected to a distance of a few Mpc, thus we do not consider them further here
(Yadlapalli et al., 2022).

To determine the detection rates we also consider the volumetric rates and
their redshift evolution for each class of transients; see Table 2. For
transients involving massive stars or compact objects, we assume that the
volumetric rates track the cosmic star formation rate density, given by (Madau
& Dickinson, 2014):

${\rm SFH}(z)=0.015\dfrac{(1+z)^{2.7}}{1+[(1+z)/2.9]^{5.6}}\ \rm M_{\odot}\
Mpc^{-3}\ yr^{-1}.$ (9)

For these systems, the volumetric rate for a given transient class is
normalized to the cosmic star formation history and assumed to evolve with
redshift in the same manner. For TDEs, where the event rate depends on the
number density of supermassive black holes (SMBH), we use the black hole
accretion rate density from Sijacki et al. (2015) (their Figure 2) for SMBHs
in the mass range $10^{6}-10^{7}\ \rm M_{\odot}$.

Below we provide a brief overview of the different classes and models
considered in our simulations.

####  2.2.1 Long Gamma-ray Bursts

LGRBs mark the energetic explosions following the core-collapse of stripped
massive stars (Woosley & Bloom, 2006). These events produce highly collimated
outflows or relativistic jets that power afterglow emission across the
electromagnetic spectrum as the jets decelerate and expand into the
surrounding circumstellar medium (CSM) (Mészáros & Rees, 1997).

LGRBs have beaming-corrected kinetic energies of $\sim 10^{51}-10^{52}$ erg
with jet opening angles of $\theta_{j}\sim 10^{\circ}$ (Frail et al., 2001;
Berger et al., 2003). The environments around LGRBs span densities of $\sim
0.1-100$ cm-3 (e.g., Panaitescu & Kumar 2002). Here we consider both the
forward shock (FS) and reverse shock (RS) components of the afterglow
emission. The RS propagates into the ejecta as the jet decelerates, providing
a unique probe of the baryon content and magnetization of the ejecta (Nakar &
Piran, 2004; Granot & Taylor, 2005). Unlike the FS component which peaks on
timescales of days to months depending on the observer viewing angle, the RS
component is expected to peak in the millimeter band on timescales of
$\lesssim$ 1 day (Laskar et al., 2013, 2018c, 2019a).

For the FS emission, we generate LGRB millimeter-band light curves using the
two-dimensional relativistic hydrodynamical code Boxfit v2 (van Eerten et al.,
2012). We consider two cases: $E_{K}=10^{51}$ erg and $n=1\ \rm cm^{-3}$
(“typical”) and $E_{K}=10^{52}$ erg and $n=10\ \rm cm^{-3}$ (“high energy”).
In both cases, we assume typical parameters for the jet opening angle
$\theta_{j}=0.2$, the synchrotron power law index $p=2.5$, and the
microphysical parameters describing the fraction of energy in electric and
magnetic fields $\epsilon_{e}=0.1$ and $\epsilon_{B}=0.01$, respectively. We
further consider events viewed both on- and off-axis, noting that off-axis
events comprise the vast majority of the intrinsic LGRB population, despite a
paucity of detections to date (owing to their lack of prompt $\gamma$-ray
emission) (Ghirlanda et al., 2014). For the off-axis case, we generate models
with observer viewing angles of $\theta_{\rm obs}=0.4$ and $0.8$.

We compute RS light curves at 100 GHz for on-axis top-hat jets in an ISM
environment for both typical and high energy models, each with two sets of jet
Lorentz factors at the shell crossing time, $\Gamma(t_{\rm dec})=200$ and 50.
We assume a “thin shell” model with a non-relativistic RS, such that $t_{\rm
dec}$ corresponds to the RS shell crossing time (Kobayashi, 2000). In this
model, the shell expands adiabatically at the sound speed after $t_{\rm dec}$
and the Lorentz factor of the shocked shell evolves as $\Gamma\propto R^{-g}$.
We set $g=3.5$, corresponding to the minimum value expected in an ISM
environment as the shocked shell must lag behind the FS (Kobayashi & Sari,
2000). We scale the RS cooling frequency, and characteristic synchrotron
frequency, and RS peak flux to that of the FS at $t_{\rm dec}$ using the
relations in Kobayashi & Zhang (2003), using $p=2.5$ and fiducial values for
the microphysical parameters, $\epsilon_{\rm e}=0.1$ and $\epsilon_{\rm
B}=0.01$, as for the FS. For simplicity, we assume equal electron acceleration
efficiency in both RS and FS regions and no additional fireball magnetization
($\epsilon_{\rm e,RS}/\epsilon_{\rm e,FS}=1$ and $\epsilon_{\rm
B,RS}/\epsilon_{\rm B,FS}=1$) or magnetic field decay (which may suppress the
RS; e.g. Salafia et al. 2021). RS synchrotron self-absorption is not included
as it does not have a strong effect on the millimeter-band light curves for
these parameters.

For the local volumetric rate of on-axis LGRBs, we use $\mathcal{R_{\rm
LGRB}}=0.2^{+0.03}_{-0.02}\ \rm Gpc^{-3}\ yr^{-1}$ for events with $L_{\rm
iso}\gtrsim 5\times 10^{51}\ \rm erg\ s^{-1}$ (corresponding to the “typical”
model) and $\mathcal{R_{\rm LGRB}}=0.1^{+0.01}_{-0.01}\ \rm Gpc^{-3}\ yr^{-1}$
for our “high energy” events with $L_{\rm iso}\gtrsim 5\times 10^{52}\ \rm
erg\ s^{-1}$ (Sun et al., 2015). For off-axis events, we quantify the fraction
of events, $f_{\rm obs}=\int_{\theta_{1}}^{\theta_{2}}sin\theta\,d\theta$,
contributing to the total event rate as a function of viewing angle
$\theta_{\rm obs}$. Here the integration is carried out from
$\theta_{1}=\theta_{\rm obs}/2$ to $\theta_{2}=\theta_{\rm obs}$,
corresponding to the range of solid angles subtended for an observer with a
viewing angle $\theta\approx\theta_{\rm obs}$. The off-axis rate is then given
by $\mathcal{R_{\rm LGRB}}(\theta_{\rm obs})=\mathcal{R_{\rm LGRB}}f_{\rm
obs}^{-1}$. Finally, we scale the rates to the cosmic star formation rate
density using Equation 9.

####  2.2.2 Short Gamma-ray Bursts

SGRBs are produced by binary neutron star mergers (Berger, 2014), but
otherwise their afterglow emission physics is similar to that of LGRBs (Sari &
Piran, 1995), albeit with lower energy jets ($E_{K}\approx 10^{49}-10^{50}$
erg) and generally lower density environments ($n\approx 10^{-3}-10^{-2}\ \rm
cm^{-3}$; Berger 2014; Fong et al. 2015). Such sources are expected to peak in
the millimeter at early times when the synchrotron peak frequency is located
in the millimeter band. Moreover, for environments with $n\gtrsim 10^{-2}\ \rm
cm^{-3}$, self-absorption will initially suppress the centimeter band
emission.

We generate the afterglow models using Boxfit with a beaming-corrected energy
$E_{K}=10^{50}$ erg, $\theta_{j}=0.2$, and $n=10^{-2}\ \rm cm^{-3}$ (and
microphysical parameters identical to the LGRB case). We also consider off-
axis events with $\theta_{\rm obs}=0.4$.

We adopt a local volumetric rate of on-axis SGRBs of $\mathcal{R_{\rm
SGRB}}=1.3^{+0.4}_{-0.3}\ \rm Gpc^{-3}\ yr^{-1}$ for events above $L_{\rm
iso}\gtrsim 10^{50}\ \rm erg\ s^{-1}$ (Sun et al., 2015), and scale it with
redshift using Equation 9. We note that the event rate for SGRBs is depdendent
on the assumed merger delay model. Here we use the rate as derived for a
lognormal delay model, consistent with existing data (Wanderman & Piran,
2015). For the off-axis rates, we scale the on-axis rate as in §2.2.1.

Figure 4: Reverse shock models at 100 GHz for the LGRB jet energies and
densities considered in this work. Solid and dashed lines correspond to
initial Lorentz factors of $\Gamma=200$ and $50$, respectively. Thin lines
correspond to the forward shock component which peaks at later times.

####  2.2.3 Neutron Star Mergers with a Stable Magnetar Remnant

The merger of two neutron stars may lead to the formation of a long-lived,
rapidly spinning magnetar (e.g., Metzger et al. 2008). For a strong dipole
magnetic field $B\gtrsim 10^{13}$ G and millisecond spin period, the neutron
star will transfer its rotational energy, $E_{\rm rot}\sim 10^{52}-10^{53}$
erg, to the merger ejecta via magnetic spin-down, accelerating it to mildly
relativistic velocities and leading to bright radio and millimeter emission
(e.g., Metzger & Bower 2014).

We model the millimeter-band emission using the spherical blastwave model of
Nakar & Piran (2011), with $E_{K}=3\times 10^{52}$ erg, $n=0.1\ \rm cm^{-3}$,
$p=2.3$, $\epsilon_{e}=0.1$, $\epsilon_{B}=0.01$, and $\beta_{i}=1$, but note
that such systems may instead produce less energetic, sub-relativistic
outflows which would produce much fainter millimeter emission. Furthermore,
losses due to gravitational waves may limit the total available kinetic energy
for synchrotron radiation (Corsi & Mészáros, 2009).

For the volumetric rate of neutron star mergers leaving behind a stable
remnant, we assume that the extragalactic population of binary neutron star
mergers tracks the rate of Galactic double neutron star systems, and consider
that recent constraints on the neutron star equation of state imply that at
most a few percent of mergers leave stable remnants (Margalit & Metzger,
2019), consistent with late-time radio upper limits of SGRBs (Metzger & Bower,
2014; Horesh et al., 2016; Schroeder et al., 2020). We therefore adopt a total
volumetric rate of $R_{\rm NSM}=2.5\ \rm Gpc^{-3}\ yr^{-1}$, corresponding to
$\sim 0.5\%$ the total neutron star merger (NSM) rate, as in Metzger et al.
(2015). The actual rate of NSMs leaving stable remnants depends on the highly
uncertain binary neutron star mass distribution and neutron star equation of
state. We scale the event rate with redshift using Equation 9.

####  2.2.4 On-axis TDEs

The discovery of the $\gamma$-ray/X-ray transient Sw J1644+57 provided the
first observational evidence for relativistic jet formation from a TDE (Bloom
et al., 2011; Burrows et al., 2011; Levan et al., 2011; Zauderer et al.,
2011). Radio and millimeter observations were initiated within $0.3$ d of the
Swift trigger, revealing a peak in the millimeter band and a steep spectral
slope at lower frequencies (Zauderer et al., 2011). The millimeter emission
remained detectable for about a year, while the centimeter-band emission is
still detectable a decade later (Berger et al., 2012; Zauderer et al., 2013b;
Eftekhari et al., 2018; Cendes et al., 2021b). Since the discovery of Sw
J1644+57, three additional relativistic TDEs have been discovered (Cenko et
al., 2012; Pasham et al., 2015; Brown et al., 2015, 2017; Andreoni et al.,
2022).

Here we use the observed light curve of Sw J1644+57 at 87 GHz from Berger et
al. (2012) as a template. We assume an on-axis rate of TDEs based on the rate
of Swift detected events, with $\mathcal{R_{\rm Sw,TDE}}=0.03^{+0.04}_{-0.02}\
\rm Gpc^{-3}\ yr^{-1}$ for events above $L_{\rm iso}\gtrsim 10^{48}\ \rm erg\
s^{-1}$ (Sun et al., 2015) and note that the rate of radio non-detections of
thermally-detected TDEs sets an approximate upper limit of $\mathcal{R_{\rm
on-axis\ TDE}}=1\ \rm Gpc^{-3}\ yr^{-1}$ (assuming a beaming fraction of $100$
and that $\lesssim 10\%$ of TDEs produce powerful jets; Bower et al. 2013). We
scale the TDE rate with redshift using the black hole accretion rate density
of Sijacki et al. (2015) for black holes in the mass range $10^{6}-10^{7}\rm
M_{\odot}$.

####  2.2.5 Off-axis TDEs

The vast majority of TDEs are radio-quiet, with luminosities $\nu
L_{\nu}\lesssim 10^{40}\ \rm erg\ s^{-1}$ (Alexander et al., 2020). In such
cases, the radio emission may arise from a non-relativistic quasi-spherical
outflow or an initially off-axis relativistic jet that will decelerate and
expand into the observer’s line of sight at late times (Giannios & Metzger,
2011). Indeed, to date, there have been two examples of radio TDEs interpreted
as harboring off-axis jets (Lei et al., 2016; Mattila et al., 2018).
Nevertheless, deep radio limits of TDEs suggest that off-axis jets are not
ubiquitous (Bower et al., 2013; van Velzen et al., 2013; Generozov et al.,
2017).

We model the emission from off-axis TDEs using the formalism of Nakar & Piran
(2011) for mildly relativistic, quasi-spherical emission, which provides a
good approximation for the late-time emission from jetted TDEs viewed off-
axis. Following Metzger et al. (2015), we select model parameters
$E_{K}=10^{52}$ erg, $p=2.3$, $\epsilon_{e}=0.2$, $\epsilon_{B}=0.01$, and
$n=0.1\ \rm cm^{-3}$, based on the late-time emission from Sw J1644+57
(Metzger et al., 2012).

For the local volumetric rate, we adopt $\mathcal{R_{\rm TDE,off-
axis}}=3^{+4}_{-2}\ \rm Gpc^{-3}\ yr^{-1}$, a factor $f_{b}^{-1}\sim 100$
times the on-axis rate. We scale the event rate according to the SMBH
accretion rate density (Sijacki et al., 2015).

####  2.2.6 Fast Blue Optical Transients

Rapid cadence surveys have led to the discovery of a subset of luminous
transients that evolve on short timescales and exhibit strong blue continuum
(Drout et al., 2014). Commonly referred to as “fast blue optical transients”
(FBOTs), these events reach typical luminosities of $-15>M>-20$ on timescales
of $\lesssim 10$ days (Pursiainen et al., 2018). The overall sample of FBOTs
is largely heterogeneous, and exhibits wide diversity in the observed spectral
features (e.g., Arcavi et al. 2016; Whitesides et al. 2017). A number of
theories have been put forward to reconcile the range of luminosities,
observed spectral diversity, and short timescales, including shock breakout in
a dense stellar envelope (Drout et al., 2014), CSM interaction (Ofek et al.,
2010b), white dwarf detonation (Arcavi et al., 2016), and energy deposition by
a central engine (Margutti et al., 2019).

A small but growing sample of FBOTs have been detected at radio and millimeter
wavelengths (Coppejans et al., 2020; Ho et al., 2020; Perley et al., 2021).
The discovery of the nearby ($\sim 60$ Mpc) fast-rising transient AT2018cow
(Prentice et al., 2018; Smartt et al., 2018) provided a unique opportunity to
study one of these events in detail (Ho et al., 2019; Margutti et al., 2019;
Perley et al., 2019). A follow-up campaign initiated across the
electromagnetic spectrum revealed long-lasting ($\sim 100$ d) millimeter
emission with an optically thick spectrum below $\nu\lesssim 100$ GHz,
pointing to the presence of a sub-relativistic outflow in a dense medium (Ho
et al., 2019). Coupled with highly variable X-ray emission, the properties of
the event implicate a central engine, in the form of a compact object or
embedded internal shock (Margutti et al., 2019). While the majority of known
FBOTs have not been detected in the radio, this may in part be due to the fact
that most FBOTs have been discovered archivally in optical surveys and thus
have not been observed in the millimeter bands at early times when the
emission is expected to peak (Ho et al., 2021b).

To model the millimeter emission from FBOTs, we use the empirical light curve
of AT2018cow (Ho et al., 2019). We scale the light curve at 215 - 243 GHz to
100 GHz assuming $F_{\nu}\propto\nu^{-0.7}$ and a center frequency of 230 GHz,
designed to match the 100 GHz data at peak ($\delta t\sim 22$ d). The
volumetric rate for FBOTs is constrained to $4-7\%$ the core-collapse
supernovae rate (Drout et al., 2014), while the rate of AT2018cow-like events
is shown to be $0.01-0.1\%$ the core-collapse supernovae rate (Ho et al.,
2021b) (see also Coppejans et al. 2020). We therefore adopt $\mathcal{R}_{\rm
FBOT}=70\ \rm Gpc^{-3}\ yr^{-1}$ ($0.1\%$ the core-collapse rate). Finally, we
scale the event rate to the cosmic star formation history.

##  3 CMB Surveys

Here we give a brief overview of upcoming and long-term CMB surveys and their
technical capabilities as they pertain to transient searches: point source
sensitivity, cadence, survey area, survey duration, and angular resolution.
The key survey parameters for various CMB experiments are summarized in Table
3. While most of these surveys operate at multiple frequencies, we use the
parameters corresponding to the 90 or 100 GHz bands for our simulations. At
lower frequencies, the sensitivity of CMB surveys typically degrades, while
frequencies above 100 GHz will provide largely comparable sensitivity and
higher spatial resolution. Thus while we focus here on the 100 GHz band for
uniformity, we note that CMB transient searches will benefit from the use of
higher frequencies for cross-checking any detections at 100 GHz.

###  3.1 Atacama Cosmology Telescope

The Atacama Cosmology Telescope (ACT) in Chile first began operations in 2007
(Fowler et al., 2007). Since 2016, the instrument has been surveying roughly
$40\%$ of the sky between 98 and 225 GHz with a variable cadence of $2-8$
days. The point source sensitivity at 90 GHz is approximately $17$ mJy
(assuming 2 arrays and a 4 minute exposure; Sigurd Naess, private
communication),111This is comparable to the per-visit noise levels reported in
Naess et al. (2021). with an average beam FWHM of $\theta_{\rm res}\approx
2^{\prime}$. For the purpose of our simulations, we adopt a 5 day cadence and
a 7 year survey duration.

###  3.2 South Pole Telescope

The South Pole Telescope (SPT) in Antarctica is designed to operate in three
frequency bands at 95, 150 and 220 GHz, with an angular resolution of about
$1.5^{\prime}$ and $1^{\prime}$ at 95 and 220 GHz, respectively (Bender et
al., 2018). The third generation SPT-3G survey covers an area of $\sim 1500\
\rm deg^{2}$, with an average cadence of 12 hours; for simplicity we assume
here a 0.5 day cadence. The integration time for a given point on the sky is
20 minutes, achieving an rms noise of about 6 mJy per visit at 95 GHz (Guns et
al., 2021). The total survey duration is 5.5 years.

###  3.3 Simons Observatory

The Simons Observatory (SO), located on Cerro Toco in Chile, is under
construction and expected to achieve first light in 2023 (Ade et al., 2019).
The experiment will consist of both small and large aperture telescopes (LAT)
to optimally probe the CMB across a wide range of angular scales. The
fractional sky coverage for the LAT is projected to be $40\%$ and designed to
maximally overlap with LSST and the Dark Energy Spectroscopic Instrument
(DESI) survey. The observatory will operate in 6 frequency bands between 27
and 280 GHz, and an angular resolution of about $2.2^{\prime}$. As a rough
approximation for the per-visit sensitivity, we scale the ACT rms by the
inverse square root of the ratio of the number of arrays for ACT and SO
(Sigurd Naess, private communication), leading to an rms of about 10 mJy. The
total survey duration is 5 years.

###  3.4 CMB-S4

The Stage-4 CMB experiment (CMB-S4) represents the next generation of CMB
surveys (Abazajian et al., 2019). With planning underway, construction is
slated for completion in 2029. CMB-S4 observatories will be located in both
Chile and the South Pole, and will consist of an ultra deep survey, covering
$3\%$ of the sky, and a deep and wide, high-resolution legacy survey covering
over half the sky with a daily cadence; we use the latter here. The frequency
range will be $30-280$ GHz, with a projected resolution of $7.4^{\prime}$ and
$0.9^{\prime}$ at 30 and 280 GHz, respectively. For our simulations, we adopt
a resolution of 2.2’ at 93 GHz and a daily rms of $6$ mJy, corresponding to an
anticipated total noise depth of 1.97 $\mu$K-arcmin in this frequency band.
The total survey duration is expected to be 7 years.

###  3.5 CMB-HD

CMB-HD is a proposed CMB experiment submitted to the Astro2020 Decadal Survey
(Sehgal et al., 2019). The instrument resolution will improve markedly on
previous experiments with $0.4^{\prime}$ at 90 GHz. The project is forecast to
cover over half the sky with a daily cadence and a per-visit rms of $2$ mJy at
90 GHz for a total survey duration of 7.5 years.

##  4 Simulated CMB Surveys

Here we describe our methodology for determining the rate of transient
detections in CMB surveys. We define a detection criterion and combine the
theoretical and empirical light curves from §2.2 with the various survey
parameters in §3 to determine the number of transients detected in each
survey, as well as their basic light curve properties.

###  4.1 Detection Criteria

Our sole criterion for detecting and identifying a source as a transient event
is at least one $\geq 10\sigma$ detection in a single visit. The large
detection significance is motivated by the large number of independent
measurements (beams, or resolution elements) in wide-field CMB surveys, and
the likelihood for spurious detections as evidenced by past radio surveys
(Bower et al., 2007; Ofek et al., 2010a; Frail et al., 2012). In Table 3, we
estimate the total number of independent beams searched over the course of
each survey, $\rm N_{\rm beam}$, and use this to calculate the expected number
of false detections assuming a $5\sigma$ detection threshold, $N_{5\sigma}$;
we find that for most CMB surveys $N_{5\sigma}\gtrsim 10^{4}$. We also
calculate the minimum source detection threshold leading to one false
detection per year, $\sigma_{\rm 1\,f.d.}$, which we find to be $\approx
6.4-7.1$. In addition to Gausssian noise, data errors and mis-calibration may
further increase the number of false detections. We therefore adopt $10\sigma$
as a conservative threshold to mitigate false detections, especially in the
presence of systematic uncertainties.

Figure 5: Total number of predicted event detections (column $N$ in Tables
4-5) for each survey and event class, including results for the stacked
surveys. We note that the stacked analyses do not include the reverse shocks
as these are generally too fast to be detectable.

Previous work focused on the centimeter band additionally required a factor of
two change in brightness for identifying events as being transient (Metzger et
al., 2015). Given that the majority of transients evolve more rapidly in the
millimeter band (days to weeks) relative to the CMB survey durations, we do
not include this condition here. Furthermore, we note that once a source is
detected at high significance, it may be recovered in multiple epochs at lower
significance, leading to source classifications and inferences of their
physical properties.

Given that the cadence of most CMB surveys is about a day, shorter than the
duration of most transients (Figure 1, 3), we also consider searches in
“stacked” images. Specifically, we explore the impact of averaging the data to
7 day and 30 day cadences, thus increasing the overall depth of the searches
by about a factor of 2.5 and 5.5, respectively. This effectively accounts for
the fact that a source may be detected at $<10\sigma$ significance in multiple
epochs and thus be recovered. Indeed, such an approach has been adopted by the
SPT-3G collaboration (Guns et al., 2021). In principle, the limiting timescale
for producing stacked surveys is set by the transient duration and the
timescale for which there is at least one detection above a given detection
threshold. Thus we note that a number of transients that we consider in this
work may benefit from generating stacked images on even longer timescales.

Finally, we consider the effect of contamination due to emission from host
galaxies. This effect is less prevalent at 100 GHz given that the non-thermal
power-law spectrum of star-forming galaxies, which peaks at a few GHz, follows
$F_{\nu}\propto\nu^{-0.7}$. At 100 GHz thermal dust emission is also minimal,
although it may become problematic at a few hundred GHz (Carilli & Yun, 1999).
For a star formation rate of $1\ \rm M_{\odot}\ yr^{-1}$, we estimate a
typical 100 GHz luminosity of $L_{\nu}\approx 10^{38}\rm\ erg\ s^{-1}$. This
is roughly two orders of magnitude lower than the luminosity of the faintest
transients considered here, and thus we conclude that host galaxy
contamination is not likely to be an issue for the majority of transients at
100 GHz. We note that while present day CMB surveys like ACT and SPT-3G have
demonstrated that stellar flares will dominate the total transient detection
rates (when considering both Galactic and extragalactic events), these sources
will have obvious stellar associations, and therefore can be easily identified
and removed in extragalactic transient detection pipelines (Guns et al., 2021;
Naess et al., 2021). We discuss the issue of AGN contamination, and how this
might be alleviated, in §5.4.

###  4.2 Monte Carlo Simulations

We run a series of Monte Carlo simulations to estimate the detection rates and
measured properties for various transients as a function of CMB survey
strategy. For each transient and survey, we inject $\gtrsim 10^{4}$ mock light
curves using the models described above with a random start time relative to
the survey duration and a random distance weighted by the comoving
cosmological volume. We scale the volumetric event rate by the cosmic star
formation history or SMBH density (see Section 2.2). The light curves are
calculated at each redshift, taking into account time-dilation and a ($1+z$)
bandpass correction. We ignore the effects of $K$-corrections, which are
largely negligible for the moderate range of redshifts accessible to CMB
surveys. An absolute detection rate is calculated for each transient class and
CMB survey by weighting the mock detections by the volumetric event rate and
survey duration.

##  5 Results

###  5.1 Detection Rates

The results of the simulations are summarized in Tables 4-8 and in Figures 5
and 6. For each survey and transient event class, we report the total number
of detected events over the course of the survey ($N$), the mean redshift of
detected events ($\bar{z}$), and the mean number of $3\sigma$ and $10\sigma$
detections ($\bar{n}_{3\sigma}$ and $\bar{n}_{10\sigma}$, respectively). We
additionally quantify the fraction of events with a measured rise ($f_{\rm
rise}$) and decline ($f_{\rm fall}$) times, defined as those with at least a
factor of $2$ change in brightness. Finally, we quantify the fraction of
events for which the peak of the light curve is captured during the survey
duration ($f_{\rm peak}$).

###  5.2 Detection Rates and Redshift Distributions

We find that the overall detection rates are dominated by LGRBs. In daily
cadence searches, LGRBs are detected most readily by the RS emission, while
the FS emission dominates the detection rates for the weekly and monthly
stacked cadences. Indeed, the high luminosity associated with the RS (a factor
of $\sim 10^{2}$ times brighter than the FS; Figure 4) lends to a large number
of detections in next generation experiments like CMB-S4 and CMB-HD, where the
detection rates reach values of $\sim 160-400$ for the most luminous “high-
energy” RS events ($\Gamma=50$). Even for the fiducial LGRB energy RS models,
current and upcoming surveys like ACT, SPT-3G, and SO are expected to detect a
few events, and as high as $\sim 15-60$ events in the case of the “high-
energy” models. Given that the timescale for RS emission is $\lesssim 1$ day,
after which point emission from the FS dominates, we do not explore RS
detections in the stacked survey cadences, but we note that a few such RS
detections are still possible with the ACT nominal $5$-day cadence.

We caution that our analysis assumes that all on-axis LGRBs produce a
detectable RS. Although there are not yet robust estimates constraining the
fraction of LGRBs with detectable RS emission, empirical estimates currently
indicate a lower limit of $\gtrsim 30\%$ based on targeted observations (e.g.,
Laskar et al. 2013; Perley et al. 2014; Laskar et al. 2016, 2018b, 2018c,
2019a, 2019b). If we scale our RS detection rates by $\sim 1/3$, we instead
find that the number of LGRB detections based on FS emission and RS detections
are comparable.

The average redshifts of detected on-axis LGRBs (based on their FS emission)
span $z\sim 0.1-0.4$, while surveys like CMB-S4 and CMB-HD will detect the
most energetic LGRBs to $z\sim 1$; see Figure 6. For events that are detected
based on their bright RS emission, the detection range increases to $z\sim 4$.
In Figure 6, we plot the normalized redshift distributions for detected on-
axis “high-energy” LGRBs for both the FS and RS emission for CMB-S4. Although
the redshift distribution for RS detections extend to higher redshifts, it
eventually drops off following the rate of cosmic star formation at high
redshift. We note that in the case of the FS component, searches in stacked
data increase the average redshift of detected events across all surveys, and
thus the total detection volume (Figure 6).

The light curves of detected on-axis LGRBs are well-characterized, with
$\bar{n}_{3\sigma}\sim 10$ epochs, given the rapid daily cadence of most CMB
surveys (Table 3). Although ACT may detect a small number of LGRBs, the
irregular cadence suggests that such events will not have rise times that are
well-characterized. Conversely, the $0.5$ day cadence of SPT-3G implies that
the light curves of all detected LGRBs will be well-sampled. While the stacked
surveys result in significantly fewer detections per event, in practice, once
a source is identified in the stacked images, additional epochs before and
after the stacked detection may be recovered by relaxing the $10\sigma$
detection threshold. In the case of RS detections, the short timescales of
these events relative to the typical CMB survey cadence implies only a single
detection, demonstrating the importance of rapid ($\sim$ hour timescale)
alerts to facilitate multi-wavelength follow-up observations.

For less energetic LGRBs, CMB-S4 and CMB-HD will detect $5-24$ events with
average redshifts of $\bar{z}\sim 0.2-0.3$, while the sensitivities of ACT and
SPT-3G and the combination of reduced sensitivity and smaller areal coverage
for SO relative to CMB-S4, largely preclude such detections. On the other
hand, weekly and monthly stacks of SPT-3G and SO data may recover a few such
sources during the entire survey duration.

Of particular interest are the detection rates for off-axis LGRBs, which will
not be recovered in $\gamma$-ray searches. We find that CMB-S4 and CMB-HD will
detect $\sim 2-15$ such events in the daily cadence data, or $\sim 10-56$
events using weekly stacks and as high as $\sim 20-100$ in the monthly stacks.
The lower luminosities of off-axis events lowers their detection ranges to
$z\lesssim 0.2$ and $z\lesssim 0.4$ for the daily and stacked surveys,
respectively. The longer timescales associated with off-axis events on the
other hand lends to a larger number of detection epochs ($\bar{n_{\rm
3\sigma}}\approx 60-200$).

Figure 6: Normalized redshift distributions for "high-energy" LGRBs (including
the RS emission) and FBOTs for CMB-S4. Also shown in grey are the
distributions for the stacked weekly cadences which push the total detection
volume out to higher redshifts.

We further find that CMB surveys will detect a small number of FBOTs similar
to AT2018cow, and that such detections will typically be made in the stacked
survey data given the low luminosity of these events relative to the survey
sensitivities. We find that surveys like CMB-S4 and CMB-HD will detect a few
to $\approx 20$ events in the weekly and monthly stacked cadences.

We show the distance distributions for FBOTs in the daily and weekly stacked
searches in Figure 6. Unlike LGRBs, these events will be exclusively detected
at nearby distances of $\lesssim 50$ Mpc, comparable to the distance of
AT2018cow (60 Mpc).

While we find that such sources will not be detected in surveys like ACT,
SPT-3G, and SO, we emphasize that the rate of FBOTs with millimeter emission
is highly uncertain. If millimeter emission is more ubiquitous in the overall
population of FBOTs, then the enhanced volumetric event rate will lead to
detections across most CMB surveys.

Beyond LGRBs and FBOTs, we do not expect other transients to be detected in
CMB surveys in appreciable numbers. The exception is CMB-HD, which may detect
a few TDEs, SGRBs, and NSMs at redshifts $z\sim 0.04-0.3$, while CMB-S4 may
similarly recover $\sim 1$ TDE and $\sim 1$ SGRB during the survey duration by
searching the stacked images (30 day cadence). We note that we consider the
number of TDE detections across all surveys a lower limit based on the
uncertain event rate; if the rate of on-axis TDEs is closer to the upper limit
of $\mathcal{R_{\rm on-axis\ TDE}}=1\ \rm Gpc^{-3}\ yr^{-1}$, then we expect
roughly $30\times$ the number of detections. We note that the extended
timescales ($\gtrsim 100$ days) associated with on-axis TDEs (and both SGRBs
and NSMs) in particular suggests that CMB surveys may benefit from stacking
the data on longer ($\gtrsim 30$ days) timescales. Indeed, as demonstrated by
Tables 7 and 8, on-axis TDEs that are detected will be detected in $\sim 20$
epochs even in the 30 day stacks.

Finally, we note that we can quantify the typical post-explosion time of
detection by considering both the transient duration ($\delta t_{100}$) and
the survey cadence ($\Delta T$). Namely, if the transient duration is
comparable to (e.g., on-axis LGRBs) or shorter than (e.g., LGRB RS emission)
the survey cadence, then the typical post-explosion time is set by the survey
cadence. Conversely, if the transient duration is much longer than the survey
cadence, the typical post-explosion time is set by the transient duration. We
discuss optimal survey strategies and implications for multi-wavelength
follow-up in light of these results.

###  5.3 Optimal Survey Strategies

Based on our simulations and in light of the brightness temperature
constraints discussed in §2, we find that the optimal survey design for
millimeter transient detection prioritizes a rapid ($\sim 1$ day timescale)
cadence. Such a rapid cadence lends to the detection of LGRBs (which dominate
the detection rates due to their intrinsic brightness), as well as less
luminous, longer duration events (such as TDEs and off-axis LGRBs) that can be
recovered in the stacked data streams. We emphasize that this is in contrast
to the centimeter-band where characteristic timescales of $\gtrsim 100$ days
(Metzger et al., 2015) generally preclude the detection of extragalactic
transients at early times, further highlighting the utility of rapid-timescale
millimeter surveys to detect these sources early in their evolution.

Furthermore, we note that observations on an even faster cadence (e.g.,
2$\times$ per night) would allow for multiple detections of the RS emission
from LGRBs. The typical short duration ($\lesssim 1$ day) associated with the
RS implies that follow-up millimeter observations (e.g., with ALMA) on the
relevant timescales will be challenging. Thus, rapid cadence CMB surveys
provide the potential to characterize the early time millimeter light curves
from LGRBs, while also reaching the requisite sensitivity to detect less
luminous, longer-lived transients using stacked images.

###  5.4 Multi-wavelength Follow-up and Detections

Beyond the question of detectability we address here the question of
localization and source classification. Obtaining host galaxy identifications
and hence a distance will be key for constraining the energy scale of detected
millimeter transients and elucidating their origin (Figure 1). The next
generation of CMB experiments will drastically improve in angular resolution,
leading to more precise localizations that will enable host identifications.
Given our $10\sigma$ threshold for source detection and a fiducial beam size
of $\sim 120^{\prime\prime}$ in CMB surveys, typical localizations will be
$\sim 10^{\prime\prime}$ ($\sim\theta_{\rm res}/$SNR). Within such a region,
there are nominally a few galaxies brighter than $m_{r}\approx 25$ mag (Driver
et al., 2016). Absent any additional observations, identifying host galaxies
purely via chance coincidence arguments may therefore be feasible, allowing
for a measurement of the redshift and thus source luminosity (Eftekhari &
Berger, 2017). In general, given a detection threshold in CMB surveys, sources
will be readily detected with ALMA follow-up using short integration times,
leading to sub-arcsecond localizations and definitive host galaxy
associations.

We further consider the latency requirements for multi-wavelength follow-up to
facilitate source classifications, taking into account the typical post-
explosion time of detection for each source class (§5.2). With early-time
millimeter detections of LGRBs ($\sim$ day) and FBOTs ($\sim$ weeks), these
sources will be ideally suited for multi-wavelength follow-up and transient
classification. Given the small distances of any detected FBOTs and their
typical post-explosion time of detection ($\sim$ weeks), these events are
likely to be detected (and potentially classified) first in optical surveys.
Nevertheless, CMB survey detections will lead to an unbiased sample of
millimeter FBOTs for the first time, as we discuss below, and it will also
draw attention to specific events based on the presence of quasi-relativistic
outflows given that most optical transients are likely to remain unclassified.

While both on- and off-axis LGRBs are expected to pass the detection threshold
for transient alerts in wide-field optical surveys like LSST, given an LSST
survey cadence of $\sim 3-5$ days, many of these events will be detected
initially in CMB surveys. In general, the rapid daily cadence of CMB surveys
will constrain the explosion time to within $\lesssim 1$ day, which will be
sufficient for triggering both optical, cm, and mm-band observations to
confirm an afterglow origin. While rapid response ALMA observations to detect
the LGRB RS emission may be challenging, the sensitivity of ALMA will enable
detections of the afterglow forward shock associated with any RS detections in
CMB surveys. We note that the short timescales associated with the LGRB RS
underlines the need for rapid alerts on $\sim$hour timescales. Indeed, among
the sources we consider here, LGRB reverse shocks impose the strictest latency
requirement for follow-up, requiring immediate ALMA observations on $\sim$ day
timescales (Laskar et al., 2013, 2016, 2019a, 2019b). Finally, accounting for
the sky coverage of Swift, we expect $\sim 15\%$ of on-axis LGRBs detected
with CMB surveys to be detected initially with gamma-ray satellites.
Millimeter detections of on-axis GRBs within the Swift field of view that are
unaccompanied by high energy gamma-ray emission will constrain the presence of
“dirty fireballs” for the first time.

For SGRBs, the small number of events that may be detected with CMB-HD will be
well within the detection volume of next generation GW experiments like Cosmic
Explorer and Einstein Telescope and are therefore likely to have associated GW
emission (Hall et al., 2020) and thus can be readily classified. In the
absence of dedicated target-of-opportunity (TOO) programs at optical
wavelengths, CMB surveys may provide a useful tool for identifying and
localizing EM counterparts to future GW events. Even with dedicated optical
follow-up programs, the reduced number of contaminants in millimeter surveys
may provide an edge over optical searches. In Chen et al. (2021), the authors
estimate that an LSST TOO program using 30 s integration times will lead to
$\sim 40\%$ of GW sources with detected EM counterparts within $700$ Mpc
(roughly the detection volume for CMB-HD). Thus, a small fraction of GW events
that go undetected in optical searches may be uniquely localized with CMB
experiments.

Although on-axis TDEs will be detected with typical post-explosion times of
$\sim$ months, the rarity of on-axis events suggests that they may be
unambiguously associated with gamma-ray triggers in archival searches. High-
cadence wide-field optical surveys may also identify the jet synchrotron
afterglow as a rapidly fading, red transient (Andreoni et al., 2022). Absent
such gamma-ray or optical counterparts, such sources will be difficult to
distinguish from AGN with limited multi-wavelength data, as we discuss below.
This further emphasizes the utility of community alerts to enable a wide range
of multi-wavelength follow-up, noting that in the case of TDEs, the prolonged
timescales relaxes the latency requirement for alerts.

Finally, we highlight a few key multi-wavelength diagnostics that will be
particularly useful for distinguishing between transients and variable AGN.
Initial results from SPT-3G suggest that AGN flares may dominate over
extragalactic transients (Guns et al., 2021). To first order, any sources
detected with CMB surveys should be cross-matched against all-sky radio,
millimeter, and submillimeter catalogs to constrain the presence of pre-
existing AGN. Similarly, cross-matching against existing WISE data in CMB
transient detection pipelines can further help to distinguish between
transients and AGN using color-based selections. Following these initial
checks, a number of follow-up observations can be leveraged to definitively
constrain or rule out an AGN origin. Given the low redshifts of detected
events, high resolution ALMA observations can provide source positions within
the host galaxy and determine whether they are offset from the galaxy nucleus.
This will be feasible for most events below $z\lesssim 0.4$ but will become
challenging at higher redshifts and for smaller galaxy sizes. Similarly,
ground-based optical spectroscopy can be used to probe high-ionization
emission lines from AGN.

###  5.5 Science Questions Addressed by Discoveries

The detection of transient sources in CMB surveys will facilitate a wide range
of scientific discovery, providing answers to key questions governing the
nature and phenomenology of transient sources. Here we highlight a number of
scientific questions that can be uniquely addressed with millimeter detections
in CMB surveys.

####  5.5.1 How common is reverse shock emission in LGRBs?

RS detections in LGRBs by CMB surveys will lead to an unbiased probe of such
emission for the first time, providing a constraint on the ubiquity of RS
emission among the population of LGRBs. Targeted observations to search for RS
emission from known LGRBs are limited in nature, relying on initial detections
by gamma-ray satellites, which cover a small fraction of the sky, and are
subject to limited availability and response times with millimeter telescopes
such as ALMA. Thus, while a growing sample of LGRBs with detectable RS
emission are now being detected in targeted millimeter surveys (Laskar et al.,
2013, 2019b, 2019a), the simulations presented here demonstrate the utility of
CMB surveys to drastically improve the existing landscape of RS detections.

Obtaining a large sample of RS detections will facilitate detailed studies of
the mechanisms that drive jet launching and collimation (Zhang & Kobayashi,
2005). While the afterglow FS emission from LGRBs is insensitive to the
properties of GRB jets, RS detections provide critical insight into the jet
magnetization and initial Lorentz factor.

Finally, we note that in addition to blind transient searches afforded by CMB
surveys, targeted searches for millimeter emission from GRBs that are
initially detected with gamma-ray satellites will provide an alternate means
of enabling high impact science. By searching the CMB data streams for a
millimeter source each time there is a GRB alert, the coincidence criterion
allows for a lower detection threshold. This will not only help to facilitate
the identification of strong candidates for continued multi-wavelength follow-
up, but also provide automatic constraints for the jet energies and ambient
densities of gamma-ray detected GRBs.

####  5.5.2 What fraction of TDEs produce relativistic jets?

Radio follow-up observations have been conducted for several tens of TDEs,
revealing a diversity in terms of luminosity and outflow properties (e.g.,
Alexander et al. 2020). These observations indicate that TDEs span over six
orders of magnitude in radio luminosity, and generally bifurcate into radio-
loud (relativistic) and radio-quiet (non-relativistic) events.

Key open questions include the fraction of TDEs that produce relativistic
jets, as well as the fraction that drive non-relativistic outflows. While on-
axis events may produce bright gamma-ray emission at early times (e.g., Sw
J1644+57), in the case of off-axis TDEs gamma-ray emission will not be
detectable. In contrast, the millimeter emission is expected to persist on
timescales of months, highlighting the utility of wide-field millimeter
surveys for TDE detections. In particular, CMB surveys will facilitate the
first non-optical or X-ray selected sample of TDEs. Searches for radio and
millimeter emission from TDEs routinely rely on initial detections at optical
and/or gamma-ray surveys. However, the discovery of a potential TDE in the
Caltech-NRAO Stripe 82 Survey (CNSS) suggests that wide-field radio surveys
can uniquely discover TDEs that are otherwise missed at other wavelengths
(Anderson et al., 2020).

####  5.5.3 The Radio Diversity of FBOTs

Targeted radio and millimeter observations of the most luminous FBOTs indicate
that at least a subset of these sources produce associated radio and
millimeter emission (Ho et al., 2019; Coppejans et al., 2020; Ho et al.,
2020). Indeed, the present day sample of four radio-detected FBOTs trace out a
region of parameter space corresponding to the shortest duration and highest
luminosity events (Ho et al., 2021b). To date, however, the vast majority of
FBOTs have been detected archivally in optical survey data, and thus radio
observations on the relevant timescales and with adequate sensitivity have
been limited (Ho et al., 2021b). This in turn has hindered progress in our
understanding of what drives the observed radio diversity in the FBOT
population.

The wide areal coverage of CMB surveys will provide an unprecedented
opportunity to characterize the millimeter emission from FBOTs in an unbiased
sample for the first time. While our work here demonstrates that a few such
detections are expected in the stacked data streams from CMB surveys, in
principle, these surveys may detect additional events if radio and millimeter
emission is prevalent in less optically luminous FBOTs which dominate the
observed event rates (Drout et al., 2014). Even if radio/millimeter loud FBOTs
are strictly correlated with the most luminous events, the daily cadence,
improved sensitivity, and large fractional sky coverage of CMB surveys will
ensure coverage for nearly all detected events, allowing us to determine
whether all FBOTs generate trans-relativistic outflows.

##  6 Conclusions

We characterized the expected extragalactic transient detection rates in wide-
field current and future CMB surveys for a diverse set of transient classes
spanning a wide range of timescales and luminosities. Our key results can be
summarized as follows:

  * •

The basic physical arguments for synchrotron emission imply that relativistic
millimeter transients will peak on short characteristic timescales of
$\lesssim{\rm few}$ days. This is in contrast to the centimeter band, where
events will reach peak brightness on much longer timescales of $\sim 100$ d.
Events with initial off-axis orientations will peak at later times and evolve
more slowly, while high densities will lead to more luminous and short-lived
events. Conversely, non-relativistic events exhibit much longer rise times of
hundreds of days.

  * •

CMB experiments will detect a large number of on-axis LGRBs, ranging from
$\sim$ a few to tens of events in surveys like SPT-3G, ACT, and Simons
Observatory, to $\sim 20-400$ in upcoming next generation experiments like
CMB-S4 and CMB-HD. The detection rates on daily cadences will be driven
primarily by the reverse shock emission, whereas the forward shock component
dominates the detection rates for the time-averaged stacked data streams. We
note however that our RS detection rates assume that all on-axis LGRBs produce
detectable RS emission. Events detected via their forward shock emission will
be limited to redshifts $z\lesssim 1$, but with well-sampled lightcurves given
the daily cadence afforded by most surveys. Conversely, events detected on the
basis of their reverse shock emission will be detected out to $z\sim 4$.
CMB-S4 and CMB-HD will further detect a large number of off-axis LGRBs. The
stacked survey data will lead to $N\sim 8-44$ events at $z\lesssim 0.3$.

  * •

In addition to LGRBs, CMB surveys will detect a small number of FBOTs,
assuming a fiducial FBOT rate of $\sim 0.1\%$ the core-collapse supernova
rate. Indeed, CMB-S4 and CMB-HD will detect between $\sim 1-2$ FBOTs in the
daily cadence data, while these numbers increase to a few to $\sim 20$ in the
time-averaged stacked data streams. We consider these numbers lower limits
given the highly uncertain rate of FBOTs with associated millimeter
counterparts and that less luminous FBOTs may also produce millimeter
emission.

  * •

The redshift distributions of detected LGRBs and FBOTs correspond to unique
detection volumes. While LGRBs will be detected out to $z\sim 1$ with CMB-HD
(and $z\sim 4$ in the case of RS detections), FBOTs will be detected
exclusively in the local universe ($\lesssim 50$ Mpc). Thus FBOTs are likely
to be detected initially in wide-field optical surveys, whereas for LGRBs,
while the limited $3-5$ day cadence of wide-field optical surveys like LSST
implies that they may be detected in larger numbers in CMB surveys.

  * •

On-axis TDEs may be detected in sizeable numbers in CMB surveys depending on
the highly uncertain event rate. We find at most a few TDE detections with
surveys like CMB-S4 and CMB-HD, but note that these numbers may be $\approx
30$ times larger depending on whether the event rate more closely tracks the
lower ($0.03\ \rm Gpc^{-3}\ yr^{-1}$) or upper ($1\ \rm Gpc^{-3}\ yr^{-1}$)
limit based on existing observations. Detections in CMB surveys will allow for
constraints on an unbiased rate of jetted TDEs.

  * •

CMB-HD may detect a few SGRBs out to $\sim 500$ Mpc, well within the detection
volume of next generation gravitational wave experiments. Such sources are
therefore likely to have gravitational wave counterparts. While this is also
within the detection volume of proposed targeted programs to search for
kilonovae with the VRO, the efficiency for optical detections is $\sim 0.4$.
Coupled with the fact that there are fewer contaminants in millimeter surveys,
CMB experiments may uniquely localize a handful of GW events.

  * •

Typical transient localizations will be of order $\sim 10^{\prime\prime}$,
which may be sufficient for identifying a host galaxy and hence redshift.
However, multi-wavelength follow-up will play a key role in classifying events
and distinguishing variable AGN from true transients. ALMA observations will
be ideally suited for resolving source locations, given the nearby distances,
while ground-based spectroscopy, coupled with information from existing radio,
millimeter, and submillimeter catalogs, will help to identify AGN.

  * •

Inherently unique to millimeter surveys is the potential to answer key
questions relating to transient phenomena, including how common is reverse
shock emission in LGRBs, what fraction of TDEs produce relativistic jets, and
what is the radio diversity in an unbiased sample of FBOTs?

While our results broadly characterize the detection landscape for millimeter
transients in CMB surveys, it is worth noting that there are a number of
parameters that imply large uncertainties on the absolute detection rates. For
example, in our models we make the simplifying assumption that a given
transient class can be characterized by a single energy and density. Moreover,
the volumetric event rates for certain classes of events, particularly those
with few (on-axis TDEs) or no (neutron star mergers) known examples, are ill-
constrained, and subject to large uncertainties. Nevertheless, our results
indicate the utility of CMB surveys for transient science in the millimeter
band.

We thank the referee for their helpful comments. We thank Sam Guns and Sigurd
Naess for helpful discussions regarding the point source sensitivity of
SPT-3G, ACT, and SO. K.D.A. acknowledges support from NASA through NASA Hubble
Fellowship grant #HST-HF2-51403.001-A awarded by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., for NASA, under contract NAS5-26555.

Table 1: Millimeter detections of extragalactic transients from the literature Source | Type | $z$ | Frequencies [GHz] | Reference
---|---|---|---|---
AT2018cow | FBOT | 0.013 | 90 - 360 | Ho et al. 2019
AT2020xnd | FBOT | 0.2433 | 100 - 200 | Ho et al. 2021a
SN2008D | SN Ib | 0.007 | 95 | Soderberg et al. 2008; Gorosabel et al. 2010
PTF11qcj | SN Ic-BL | 0.029 | 93 | Corsi et al. 2014
SN1993J | SN IIb | 0.0008 | 87, 99 | Weiler et al. 2007
SN1998bw | SN Ic-BL | 0.0085 | 150 | Kulkarni et al. 1998
SN2011DH | SN IIb | 0.002 | 93, 107 | Horesh et al. 2013b
iPTF13bvn | SN Ib/c | 0.0059 | 95 | Horesh et al. 2013a
SN2013ak | SN IIb | 0.0037 | 85, 270 | Chakraborti et al. 2013; Zauderer et al. 2013a
SN2014C | SN Ib | 0.0027 | 85 | Zauderer et al. 2014
SN2020oi | SN Ic | 0.0052 | 100 | Maeda et al. 2021
GRB 970508 | GRB | 0.835 | 86 | Bremer et al. 1998
GRB 980329 | GRB | $\lesssim 5$ | 350 | Smith et al. 1999
GRB 991208 | GRB | 0.707 | 100, 250 | Galama et al. 2000
GRB 000301C | GRB | 2.329 | 100, 250 | Berger et al. 2000
GRB 021004 | GRB | 2.329 | 90, 230 | de Ugarte Postigo et al. 2005
GRB 030329 | GRB | 0.169 | 86, 100, 235, 350 | Sheth et al. 2003; Kuno et al. 2004; Kohno et al. 2005; Resmi et al. 2005
GRB 041219A | GRB | 0.31 | 99 | de Ugarte Postigo et al. 2012
GRB 050904 | GRB | 6.29 | 89 | de Ugarte Postigo et al. 2012
GRB 051022 | GRB | 0.809 | 90 | Castro-Tirado et al. 2007
GRB 060904B | GRB | 0.70 | 92 | de Ugarte Postigo et al. 2012
GRB 070125 | GRB | 1.55 | 95, 250 | Chandra et al. 2008; de Ugarte Postigo et al. 2012
GRB 070306 | GRB | 1.5 | 86 | de Ugarte Postigo et al. 2012
GRB 071003 | GRB | 1.6 | 86 | de Ugarte Postigo et al. 2012
GRB 080129 | GRB | 4.35 | 250 | Greiner et al. 2009
GRB 080319B | GRB | 0.937 | 97 | Pandey et al. 2009
GRB 090313 | GRB | 3.38 | 87, 92, 105 | Bock et al. 2009; Melandri et al. 2010
GRB 090404 | GRB | – | 87, 108 | Castro-Tirado et al. 2009a
GRB 090423 | GRB | 8.2 | 97 | Castro-Tirado et al. 2009b
GRB 100418A | GRB | 0.62 | 100, 345 | Martin et al. 2010; de Ugarte Postigo et al. 2018a
GRB 100621A | GRB | 0.54 | 345 | Greiner et al. 2013
GRB 110715A | GRB | 0.825 | 345 | Sánchez-Ramírez et al. 2017
GRB 120326A | GRB | 1.798 | 92.5, 230 | Urata et al. 2012, 2014; Laskar et al. 2015
GRB 130215A | GRB | 0.597 | 93 | Perley & Keating 2013
GRB 130418A | GRB | 1.218 | 93 | Perley 2013a
GRB 130427A | GRB | 0.34 | 90 | Perley et al. 2014
GRB 130702A | GRB | 0.145 | 93 | Singer et al. 2013
GRB 131108A | GRB | 2.4 | 93 | Perley 2013b
GRB 131231A | GRB | 0.642 | 93 | Perley 2014
GRB 140304A | GRB | 5.283 | 86 | Laskar et al. 2018b
GRB 140311A | GRB | 4.954 | 86 | Laskar et al. 2018a
GRB 160623A | GRB | 0.367 | 230 | Chen et al. 2020
GRB 161023A | GRB | 2.71 | 100, 345 | de Ugarte Postigo et al. 2018b
GRB 161219B | GRB | 0.148 | 91, 103 | Laskar et al. 2018c
GRB 171205A | GRB | 0.037 | 100, 345 | Perley et al. 2017; Smith & Tanvir 2017; Laskar et al. 2020
GRB 181201A | GRB | 0.450 | 97 | Laskar et al. 2019a
GRB 190114C | GRB | 0.435 | 97.5 | Laskar et al. 2019b
GRB 190829A | GRB | 0.0785 | 100, 343.5 | Dichiara et al. 2022
GRB 191221B | GRB | 1.148 | 90.5 | Laskar & a larger Collaboration 2019
GRB 210610B | GRB | 1.135 | 90.5 | Laskar et al. 2021c
GRB 210619B | GRB | 1.937 | 97.5 | Laskar et al. 2021a
GRB 210702A | GRB | 1.1757 | 97.5 | Laskar & Perley 2021; Laskar 2021
GRB 210905A | GRB | 6.318 | 97.5 | Laskar et al. 2021b
GRB 220101A | GRB | 4.618 | 97.5 | Laskar 2022
AT2019dsg | TDE | 0.051 | 97.5 | Cendes et al. 2021a
Sw J1644+57 | TDE | 0.354 | 87, 200, 230, 345 | Berger et al. 2012
IGR J1258+0134 | TDE | 0.004 | 100 | Yuan et al. 2016
AT2022cmc | TDE | 1.193 | 90, 230, 350 | Alexander et al. 2022; Perley et al. 2022; Smith et al. 2022
Table 2: Extragalactic Transient Models

Transient | $\mathcal{R}\ (z=0)$ | $E_{K}$ | $n$ | $\beta_{i}$ | $\delta t_{100}^{\rm c}$ | Model Type | Reference
---|---|---|---|---|---|---|---
|  (Gpc-3 yr-1) | (erg) |  (cm-3) |  | (days) |  |
LGRB, On-Axis |  $0.2^{+0.02}_{-0.03}$a | $10^{51}$ | 1 | 1 | 5 | Theoretical | van Eerten et al. 2012
LGRB, $\theta_{\rm obs}=0.4$ |  $0.8^{+0.1}_{-0.1}$a | $10^{51}$ | 1 | 1 | 23 | Theoretical | van Eerten et al. 2012
LGRB, $\theta_{\rm obs}=0.8$ |  $3.0^{+0.5}_{-0.3}$a | $10^{51}$ | 1 | 1 | 96 | Theoretical | van Eerten et al. 2012
LGRB high energy, On-axis |  $0.1^{+0.01}_{-0.01}$a | $3\times 10^{51}$ | 10 | 1 | 4 | Theoretical | van Eerten et al. 2012
LGRB high energy, $\theta_{\rm obs}=0.4$ |  $0.4^{+0.04}_{-0.04}$a | $3\times 10^{51}$ | 10 | 1 | 21 | Theoretical | van Eerten et al. 2012
LGRB high energy, $\theta_{\rm obs}=0.8$ |  $1.5^{+0.2}_{-0.2}$a | $3\times 10^{51}$ | 10 | 1 | 72 | Theoretical | van Eerten et al. 2012
SGRB, On-Axis |  1.3${}^{+0.4}_{-0.3}$a | $10^{50}$ | 0.01 | 1 | 16 | Theoretical | van Eerten & MacFadyen 2011
SGRB, $\theta_{\rm obs}=0.4$ |  $5.0^{+1.7}_{-1.0}$a | $10^{50}$ | 0.01 | 1 | 78 | Theoretical | van Eerten & MacFadyen 2011
TDE, On-Axis |  0.03${}^{+0.04}_{-0.02}$b | $10^{52}$ | 0.1 | 1 | 135 | Empirical | Berger et al. 2012
TDE, Off-Axis (spherical) |  $3^{+4}_{-2}$b | $10^{52}$ | 0.1 | 0.8 | 900 | Theoretical | Giannios & Metzger 2011
NSM: stable remnant |  2.5a | $3\times 10^{52}$ | 0.1 | 1 | 1300 | Theoretical | Metzger & Bower 2014
FBOTs | 70 | $4\times 10^{48}$ | $3\times 10^{5}$ | 0.13 | 28 | Empirical | Ho et al. 2019

Note. —

a Scaled with redshift according to the cosmic star formation history (Madau &
Dickinson, 2014).

b Scaled with redshift according to the SMBH accretion rate density (Sijacki
et al., 2015).

c Light curve duration at observer frequency $\nu=100$ GHz.

Table 3: CMB Survey Parameters

Name | $\theta_{\rm res}$ | $\nu$ | $\sigma^{\rm a}$ | Area | $T_{\rm dur}^{\rm b}$ | $\Delta T^{\rm c}$ | $\rm N_{obs}^{\rm d}$ | $\rm N_{beam}^{\rm e}$ | $\sigma_{\rm 1f.d.}^{\rm f}$ | $\rm N_{5\sigma}^{\rm g}$
---|---|---|---|---|---|---|---|---|---|---
| (arcsec) | (GHz) | (mJy) |  (deg2) | (yr) | (days) |  |  |  ($\sigma$) |
ACT | 123 | 98, 150, 225 | 17 | $1.7\times 10^{4}$ | 7 | 5 | 511 | $6.6\times 10^{9}$ | 6.4 | 3785
SPT-3G | 92.4 | 95, 150, 220 | 6 | 1500 | 5.5 | 0.5 | 4020 | $8.1\times 10^{9}$ | 6.4 | 4645
Simons Observatory | 132 | 27, 39, 93, 145, 225, 280 | 10 | $1.7\times 10^{4}$ | 5 | 1 | 1825 | $2\times 10^{10}$ | 6.6 | 11500
CMB-S4 | 132 | 27, 39, 93, 145, 225, 278 | 6 | $2\times 10^{4}$ | 7 | 1 | 2555 | $3.4\times 10^{10}$ | 6.4 | 19500
CMB-HD | 25 | 30, 40, 90, 150, 220, 280 | 2 | $2\times 10^{4}$ | 7.5 | 1 | 2740 | $1.0\times 10^{12}$ | 7.13 | $5.7\times 10^{5}$

Note. — We list all frequency bands for each experiment for completeness but
note that the angular resolution and sensitivity are given at the frequency
band nearest 100 GHz.

a 1$\sigma$ rms sensitivity.

b Survey duration.

c Survey cadence.

d Number of independent observations.

e Number of independent synthesized beams searched over the course of the
survey.

f Source detection threshold leading to one false detection per year due to
thermal fluctuations.

g Number of false detections over the course of the survey if the detection
threshold is $5\sigma$.

Table 4: ACT

Transient | Na | $\bar{z}^{\rm b}$ | $\bar{n}_{3\sigma}^{\rm c}$ | $\bar{n}_{10\sigma}^{\rm d}$ | $f_{\rm peak}^{\rm e}$ | $f_{\rm rise}^{\rm f}$ | $f_{\rm fall}^{\rm g}$ | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | 5 Day Cadence | 30 Day Stack
——————————————————————————————————– | ——————————————————–
LGRBs (on-axis) | 0.61 | 0.10 | 2.2 | 1.4 | 0.12 | 0.0 | 0.66 | 0.17 | 0.09 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0
LGRBs ($\theta_{\rm obs}=0.4$) | 0.02 | 0.02 | 11.9 | 5.5 | 0.19 | 0.34 | 0.99 | 0.03 | 0.03 | 1.0 | 1.0 | 0.0 | 0.0 | 0.01
LGRBs ($\theta_{\rm obs}=0.8$) | 0.0 | 0.004 | 45.3 | 19.4 | 0.22 | 0.97 | 0.91 | 0.0 | 0.01 | 2.1 | 1.4 | 0.0 | 0.0 | 0.27
LGRBs RS, $\Gamma=200$ | 0.8 | 0.75 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs RS, $\Gamma=50$ | 1.9 | 0.58 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs high energy, (on-axis) | 6.5 | 0.3 | 2.0 | 1.3 | 0.12 | 0.0 | 0.62 | 1.3 | 0.2 | 1.0 | 1.0 | 0.01 | 0.0 | 0.0
LGRBs high energy, $\theta_{\rm obs}=0.4$ | 0.46 | 0.07 | 9.7 | 5.1 | 0.21 | 0.33 | 0.99 | 0.65 | 0.08 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0
LGRBs high energy, $\theta_{\rm obs}=0.8$ | 0.02 | 0.02 | 33.6 | 15.6 | 0.26 | 0.99 | 0.98 | 0.10 | 0.03 | 1.6 | 1.1 | 0.0 | 0.0 | 0.61
LGRBs high energy RS, $\Gamma=200$ | 2.1 | 1.2 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs high energy RS, $\Gamma=50$ | 10.4 | 1.1 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
SGRB, On-axis | 0.02 | 0.02 | 7.6 | 4.0 | 0.21 | 0.40 | 0.99 | 0.02 | 0.02 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0
SGRB ($\theta_{\rm obs}=0.4$) | 0.0 | 0.004 | 33.8 | 18.3 | 0.23 | 0.98 | 0.95 | 0.01 | 0.01 | 1.4 | 1.1 | 0.0 | 0.0 | 0.36
TDE, On-axis | 0.01 | 0.05 | 126.1 | 43.6 | 0.19 | 0.18 | 0.94 | 0.14 | 0.10 | 5.0 | 1.9 | 0.03 | 0.0 | 0.91
TDE, Off-Axis (spherical) | 0.0 | 0.005 | 437.6 | 178.4 | 0.11 | 0.78 | 0.74 | 0.03 | 0.01 | 15.2 | 8.3 | 0.59 | 0.81 | 0.78
NSM: magnetar | 0.0 | 0.006 | 451.6 | 210.2 | 0.10 | 0.79 | 0.55 | 0.03 | 0.01 | 16.0 | 7.7 | 0.71 | 0.76 | 0.59
FBOTs | 0.1 | 0.01 | 11.3 | 6.0 | 0.19 | 0.24 | 0.99 | 0.14 | 0.01 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0

Note. —

a Number of detected events.

b Average redshift of detected events.

c Average number of $3\sigma$ detections per event.

d Average number of $10\sigma$ detections per event.

e Fraction of events with a measured peak.

f Fraction of events with measured rise times.

g Fraction of events with measured fall times.

Table 5: SPT-3G

Transient | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$ | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | 0.5 Day Cadence | 7 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) | 0.53 | 0.17 | 18.9 | 9.5 | 0.99 | 0.96 | 0.99 | 1.9 | 0.26 | 3.3 | 1.7 | 0.28 | 0.0 | 0.99
LGRBs ($\theta_{\rm obs}=0.4$) | 0.01 | 0.03 | 93.5 | 42.6 | 0.99 | 0.99 | 0.99 | 0.06 | 0.05 | 13.7 | 6.0 | 0.95 | 0.99 | 0.99
LGRBs ($\theta_{\rm obs}=0.8$) | 0.0 | 0.01 | 361.2 | 173.2 | 0.97 | 0.98 | 0.97 | 0.0 | 0.01 | 52.4 | 25.4 | 0.98 | 0.98 | 0.98
LGRBs RS, $\Gamma=200$ | 3.8 | 1.2 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs RS, $\Gamma=50$ | 10.0 | 0.9 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs high energy, (on-axis) | 5.4 | 0.48 | 22.1 | 10.5 | 0.99 | 0.99 | 0.99 | 18.9 | 0.74 | 4.1 | 1.9 | 0.58 | 0.60 | 0.99
LGRBs high energy, $\theta_{\rm obs}=0.4$ | 0.32 | 0.12 | 79.8 | 39.7 | 0.99 | 0.99 | 0.99 | 1.3 | 0.19 | 12.4 | 6.3 | 0.92 | 0.81 | 0.99
LGRBs high energy, $\theta_{\rm obs}=0.8$ | 0.01 | 0.03 | 271.7 | 130.5 | 0.98 | 0.98 | 0.99 | 0.05 | 0.04 | 40.8 | 19.4 | 0.97 | 0.97 | 0.99
LGRBs high energy RS, $\Gamma=200$ | 8.5 | 1.8 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs high energy RS, $\Gamma=50$ | 39.1 | 1.6 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
SGRB, On-axis | 0.02 | 0.03 | 59.3 | 30.3 | 0.99 | 0.99 | 0.99 | 0.06 | 0.05 | 8.9 | 4.5 | 0.90 | 0.76 | 0.99
SGRB ($\theta_{\rm obs}=0.4$) | 0.0 | 0.01 | 37.4 | 19.6 | 0.96 | 0.98 | 0.98 | 0.0 | 0.01 | 35.8 | 18.9 | 0.96 | 0.95 | 0.95
TDE, On-axis | 0.01 | 0.08 | 1044.9 | 313.1 | 0.95 | 0.95 | 0.98 | 0.04 | 0.13 | 169.6 | 53.5 | 0.56 | 0.21 | 0.95
TDE, Off-Axis (spherical) | 0.0 | 0.01 | 4293.7 | 1500.7 | 0.71 | 0.87 | 0.82 | 0.01 | 0.01 | 600.7 | 215.0 | 0.89 | 0.85 | 0.80
NSM: magnetar | 0.04 | 0.01 | 5096.5 | 2053.1 | 0.59 | 0.82 | 0.76 | 0.01 | 0.02 | 708.1 | 1282.8 | 0.90 | 0.83 | 0.74
FBOTs | 0.06 | 0.01 | 85.4 | 41.5 | 0.99 | 0.99 | 0.99 | 0.26 | 0.02 | 12.7 | 6.3 | 0.91 | 0.61 | 0.99
——————————————————————————————————– | ————————————————————-
|  |  | 30 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 2.0 | 0.27 | 1.0 | 1.0 | 0.0 | 0.0 | 0.03
LGRBs ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.11 | 0.07 | 3.9 | 2.2 | 0.16 | 0.0 | 0.99
LGRBs ($\theta_{\rm obs}=0.8$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.0 | 0.01 | 13.5 | 6.6 | 0.95 | 0.66 | 0.98
LGRBs high energy, (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 28.9 | 0.84 | 1.0 | 1.0 | 0.0 | 0.0 | 0.05
LGRBs high energy, $\theta_{\rm obs}=0.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.2 | 0.25 | 3.4 | 1.9 | 0.44 | 0.0 | 0.99
LGRBs high energy, $\theta_{\rm obs}=0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.13 | 0.06 | 9.5 | 4.7 | 0.91 | 0.32 | 0.98
SGRB, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.12 | 0.06 | 2.4 | 1.6 | 0.15 | 0.0 | 0.29
SGRB ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.01 | 0.02 | 8.7 | 4.7 | 0.94 | 0.27 | 0.98
TDE, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.11 | 0.19 | 43.6 | 13.9 | 0.72 | 0.0 | 0.95
TDE, Off-Axis (spherical) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.02 | 0.02 | 146.0 | 53.6 | 0.89 | 0.84 | 0.82
NSM: magnetar |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.02 | 0.02 | 167.5 | 67.9 | 0.91 | 0.84 | 0.72
FBOTs |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.66 | 0.03 | 3.4 | 1.6 | 0.63 | 0.99 | 0.99

Note. — Columns are as in Table 4.

Table 6: Simons Observatory

Transient | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$ | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Daily Cadence | 7 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) | 1.3 | 0.14 | 10.5 | 5.4 | 0.99 | 0.99 | 0.99 | 1.8 | 0.17 | 2.1 | 1.2 | 0.0 | 0.0 | 1.0 |
LGRBs ($\theta_{\rm obs}=0.4$) | 0.03 | 0.03 | 59.9 | 28.4 | 0.98 | 0.99 | 0.98 | 0.14 | 0.04 | 8.7 | 4.1 | 0.64 | 0.50 | 0.99 |
LGRBs ($\theta_{\rm obs}=0.8$) | 0.0 | 0.004 | 241.0 | 111.8 | 0.94 | 0.97 | 0.95 | 0.002 | 0.007 | 34.0 | 15.9 | 0.95 | 0.95 | 0.95 |
LGRBs RS, $\Gamma=200$ | 7.6 | 0.7 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
LGRBs RS, $\Gamma=50$ | 12.8 | 0.7 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
LGRBs high energy, (on-axis) | 12.6 | 0.37 | 9.9 | 5.1 | 0.99 | 0.99 | 0.99 | 34.3 | 0.5 | 1.6 | 1.1 | 0.0 | 0.0 | 0.55 |
LGRBs high energy, $\theta_{\rm obs}=0.4$ | 0.8 | 0.09 | 46.9 | 23.9 | 0.98 | 0.99 | 0.99 | 3.0 | 0.15 | 7.1 | 3.5 | 0.58 | 0.99 | 0.98 |
LGRBs high energy, $\theta_{\rm obs}=0.8$ | 0.03 | 0.02 | 169.0 | 79.7 | 0.94 | 0.96 | 0.96 | 0.11 | 0.03 | 24.2 | 11.9 | 0.96 | 0.97 | 0.97 |
LGRBs high energy RS, $\Gamma=200$ | 13.6 | 1.5 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
LGRBs high energy RS, $\Gamma=50$ | 63.3 | 1.4 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
SGRB, On-axis | 0.04 | 0.02 | 38.0 | 20.0 | 0.99 | 0.99 | 0.99 | 0.1 | 0.04 | 5.7 | 3.1 | 0.48 | 0.0 | 0.94 |
SGRB ($\theta_{\rm obs}=0.4$) | 0.0 | 0.006 | 162.8 | 85.1 | 0.96 | 0.96 | 0.96 | 0.01 | 0.01 | 23.0 | 12.1 | 0.96 | 0.95 | 0.95 |
TDE, On-axis | 0.03 | 0.07 | 564.1 | 195.1 | 0.91 | 0.70 | 0.92 | 0.1 | 0.1 | 90.4 | 33.7 | 0.34 | 0.0 | 0.93 |
TDE, Off-Axis (spherical) | 0.0 | 0.01 | 1754.7 | 790.8 | 0.40 | 0.68 | 0.67 | 0.02 | 0.01 | 247.1 | 116.4 | 0.74 | 0.70 | 0.65 |
NSM: magnetar | 0.0 | 0.008 | 1828.3 | 948.3 | 0.29 | 0.71 | 0.49 | 0.02 | 0.01 | 259.6 | 134.3 | 0.79 | 0.71 | 0.49 |
FBOTs | 0.1 | 0.01 | 55.1 | 27.8 | 0.99 | 0.99 | 0.98 | 0.59 | 0.02 | 8.1 | 4.4 | 0.45 | 0.11 | 0.99 |
——————————————————————————————————– | ————————————————————-
|  |  | 30 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.9 | 0.16 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0 |
LGRBs ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.25 | 0.05 | 2.4 | 1.4 | 0.0 | 0.0 | 0.3 |
LGRBs ($\theta_{\rm obs}=0.8$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.0 | 0.01 | 8.1 | 3.7 | 0.89 | 0.53 | 0.96 |
LGRBs high energy, (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 16.6 | 0.41 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0 |
LGRBs high energy, $\theta_{\rm obs}=0.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.9 | 0.18 | 2.1 | 1.2 | 0.0 | 0.0 | 0.98 |
LGRBs high energy, $\theta_{\rm obs}=0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.32 | 0.04 | 5.9 | 2.9 | 0.87 | 0.02 | 0.96 |
SGRB, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.27 | 0.04 | 1.4 | 1.1 | 0.0 | 0.0 | 3.9 |
SGRB ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.03 | 0.01 | 5.9 | 3.0 | 0.78 | 0.96 | 0.96 |
TDE, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.3 | 0.15 | 20.3 | 7.5 | 0.54 | 0.0 | 0.93 |
TDE, Off-Axis (spherical) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.05 | 0.02 | 57.6 | 26.9 | 0.78 | 0.71 | 0.64 |
NSM: magnetar |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.05 | 0.02 | 61.1 | 32.5 | 0.82 | 0.69 | 0.49 |
FBOTs |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.9 | 0.02 | 2.3 | 1.8 | 0.0 | 0.0 | 0.31 |

Note. — Columns are as in Table 4.

Table 7: CMB-S4

Transient | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$ | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Daily Cadence | 7 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) | 4.5 | 0.17 | 10.4 | 5.4 | 0.99 | 0.99 | 0.99 | 12.4 | 0.25 | 1.7 | 1.1 | 0.08 | 0.0 | 0.66 |
LGRBs ($\theta_{\rm obs}=0.4$) | 0.12 | 0.03 | 58.6 | 26.6 | 0.99 | 0.99 | 0.99 | 0.46 | 0.05 | 8.6 | 4.0 | 0.73 | 0.50 | 0.99 |
LGRBs ($\theta_{\rm obs}=0.8$) | 0.0 | 0.006 | 243.6 | 112.6 | 0.95 | 0.97 | 0.96 | 0.0 | 0.01 | 34.5 | 16.0 | 0.99 | 0.97 | 0.96 |
LGRBs RS, $\Gamma=200$ | 16.9 | 1.3 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
LGRBs RS, $\Gamma=50$ | 40.9 | 0.9 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
LGRBs high energy, (on-axis) | 43.7 | 0.47 | 9.8 | 5.0 | 0.99 | 0.99 | 0.99 | 114.3 | 0.65 | 1.6 | 1.1 | 0.04 | 0.0 | 0.5 |
LGRBs high energy, $\theta_{\rm obs}=0.4$ | 2.4 | 0.12 | 46.7 | 23.7 | 0.99 | 0.99 | 0.99 | 10.4 | 0.19 | 7.3 | 3.7 | 0.68 | 0.99 | 0.99 |
LGRBs high energy, $\theta_{\rm obs}=0.8$ | 0.1 | 0.03 | 169.3 | 81.9 | 0.96 | 0.97 | 0.97 | 0.4 | 0.04 | 24.8 | 12.0 | 0.97 | 0.97 | 0.97 |
LGRBs high energy RS, $\Gamma=200$ | 35.2 | 1.8 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
LGRBs high energy RS, $\Gamma=50$ | 162.5 | 1.7 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
SGRB, On-axis | 0.1 | 0.03 | 37.7 | 19.5 | 0.99 | 0.99 | 0.99 | 0.49 | 0.05 | 5.7 | 3.1 | 0.63 | 0.0 | 0.95 |
SGRB ($\theta_{\rm obs}=0.4$) | 0.0 | 0.01 | 166.3 | 87.2 | 0.96 | 0.96 | 0.98 | 0.04 | 0.01 | 23.5 | 12.5 | 0.97 | 0.96 | 0.97 |
TDE, On-axis | 0.09 | 0.09 | 617.7 | 198.4 | 0.94 | 0.70 | 0.94 | 0.34 | 0.13 | 93.0 | 32.6 | 0.50 | 0.0 | 0.94 |
TDE, Off-Axis (spherical) | 0.01 | 0.01 | 2164.7 | 885.7 | 0.55 | 0.78 | 0.72 | 0.05 | 0.01 | 314.7 | 136.0 | 0.85 | 0.78 | 0.73 |
NSM: magnetar | 0.01 | 0.01 | 2302.5 | 1071.3 | 0.45 | 0.75 | 0.57 | 0.06 | 0.02 | 324.7 | 141.0 | 0.86 | 0.78 | 0.58 |
FBOTs | 0.5 | 0.01 | 54.9 | 27.3 | 0.99 | 0.99 | 0.99 | 2.1 | 0.02 | 8.0 | 4.4 | 0.59 | 0.12 | 0.99 |
——————————————————————————————————– | ————————————————————-
|  |  | 30 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 6.3 | 0.2 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0 |
LGRBs ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.89 | 0.07 | 2.4 | 1.4 | 0.12 | 0.0 | 0.31 |
LGRBs ($\theta_{\rm obs}=0.8$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.03 | 0.01 | 8.3 | 3.9 | 0.93 | 0.57 | 0.97 |
LGRBs high energy, (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 57.7 | 0.51 | 1.0 | 1.0 | 0.0 | 0.0 | 0.0 |
LGRBs high energy, $\theta_{\rm obs}=0.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 20.5 | 0.23 | 2.1 | 1.2 | 0.16 | 0.0 | 0.98 |
LGRBs high energy, $\theta_{\rm obs}=0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.16 | 0.06 | 6.0 | 2.9 | 0.90 | 0.03 | 0.97 |
SGRB, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.92 | 0.06 | 1.5 | 1.1 | 0.12 | 0.0 | 0.4 |
SGRB ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.11 | 0.02 | 6.0 | 3.0 | 0.87 | 0.97 | 0.97 |
TDE, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.0 | 0.19 | 22.1 | 7.9 | 0.66 | 0.0 | 0.93 |
TDE, Off-Axis (spherical) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.16 | 0.02 | 73.0 | 30.8 | 0.85 | 0.77 | 0.72 |
NSM: magnetar |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.2 | 0.02 | 78.5 | 36.9 | 0.88 | 0.76 | 0.60 |
FBOTs |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.4 | 0.02 | 2.3 | 1.8 | 0.02 | 0.0 | 0.31 |

Note. — Columns are as in Table 4.

Table 8: CMB-HD

Transient | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$ | N | $\bar{z}$ | $\bar{n}_{3\sigma}$ | $\bar{n}_{10\sigma}$ | $f_{\rm peak}$ | $f_{\rm rise}$ | $f_{\rm fall}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Daily Cadence | 7 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) | 23.5 | 0.30 | 10.5 | 5.5 | 0.99 | 0.99 | 0.99 | 63.1 | 0.41 | 1.8 | 1.1 | 0.58 | 0.0 | 0.68
LGRBs ($\theta_{\rm obs}=0.4$) | 0.67 | 0.06 | 58.2 | 26.4 | 0.99 | 0.99 | 0.99 | 2.6 | 0.09 | 8.6 | 4.0 | 0.91 | 0.48 | 0.99
LGRBs ($\theta_{\rm obs}=0.8$) | 0.01 | 0.01 | 250.8 | 122.8 | 0.93 | 0.96 | 0.99 | 0.05 | 0.02 | 36.4 | 16.8 | 0.98 | 0.97 | 0.98
LGRBs RS, $\Gamma=200$ | 53.1 | 1.7 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs RS, $\Gamma=50$ | 197.8 | 1.4 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs high energy, (on-axis) | 182.4 | 0.74 | 10.4 | 5.9 | 0.99 | 0.99 | 0.99 | 515.5 | 1.1 | 1.7 | 1.1 | 0.47 | 0.0 | 0.6
LGRBs high energy, $\theta_{\rm obs}=0.4$ | 15.0 | 0.20 | 47.1 | 24.1 | 0.99 | 0.99 | 0.98 | 55.7 | 0.31 | 7.4 | 3.7 | 0.89 | 0.99 | 0.99
LGRBs high energy, $\theta_{\rm obs}=0.8$ | 0.52 | 0.04 | 171.8 | 82.2 | 0.96 | 0.97 | 0.97 | 2.2 | 0.07 | 24.3 | 11.6 | 0.99 | 0.97 | 0.97
LGRBs high energy RS, $\Gamma=200$ | 71.2 | 1.9 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
LGRBs high energy RS, $\Gamma=50$ | 412.6 | 1.9 | 1.0 | 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
SGRB, On-axis | 0.7 | 0.05 | 37.6 | 19.3 | 0.99 | 0.99 | 0.99 | 2.7 | 0.08 | 5.8 | 3.2 | 0.88 | 0.0 | 0.95
SGRB ($\theta_{\rm obs}=0.4$) | 0.05 | 0.01 | 162.5 | 84.1 | 0.97 | 0.98 | 0.97 | 0.22 | 0.02 | 23.6 | 12.3 | 0.98 | 0.97 | 0.97
TDE, On-axis | 0.4 | 0.15 | 631.5 | 209.4 | 0.93 | 0.70 | 0.96 | 1.9 | 0.22 | 91.9 | 32.4 | 0.83 | 0.0 | 0.94
TDE, Off-Axis (spherical) | 0.07 | 0.02 | 2282.1 | 927.6 | 0.56 | 0.78 | 0.75 | 0.3 | 0.02 | 325.1 | 134.3 | 0.89 | 0.78 | 0.74
NSM: magnetar | 0.07 | 0.02 | 2451.5 | 1091.2 | 0.46 | 0.79 | 0.62 | 0.31 | 0.03 | 345.3 | 156.2 | 0.92 | 0.78 | 0.61
FBOTs | 2.0 | 0.02 | 55.4 | 27.7 | 0.99 | 0.99 | 0.99 | 11.5 | 0.03 | 8.0 | 4.3 | 0.89 | 0.11 | 0.99
——————————————————————————————————– | ————————————————————-
|  |  | 30 Day Stack
——————————————————————————————————– | ————————————————————-
LGRBs (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 33.3 | 0.34 | 1.0 | 1.0 | 0.02 | 0.0 | 0.01
LGRBs ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4.8 | 0.11 | 2.4 | 1.4 | 0.44 | 0.0 | 0.31
LGRBs ($\theta_{\rm obs}=0.8$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.15 | 0.02 | 8.2 | 3.8 | 0.97 | 0.55 | 0.96
LGRBs high energy, (on-axis) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 282.6 | 0.89 | 1.0 | 1.0 | 0.02 | 0.0 | 0.0
LGRBs high energy, $\theta_{\rm obs}=0.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 117.2 | 0.40 | 2.1 | 1.2 | 0.45 | 0.0 | 0.98
LGRBs high energy, $\theta_{\rm obs}=0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 6.5 | 0.10 | 6.0 | 2.9 | 0.95 | 0.03 | 0.97
SGRB, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.1 | 0.10 | 1.4 | 1.1 | 0.45 | 0.0 | 0.41
SGRB ($\theta_{\rm obs}=0.4$) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.60 | 0.03 | 6.0 | 3.1 | 0.93 | 0.98 | 0.97
TDE, On-axis |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.6 | 0.3 | 22.4 | 7.9 | 0.79 | 0.0 | 0.94
TDE, Off-Axis (spherical) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.84 | 0.04 | 75.4 | 31.4 | 0.90 | 0.78 | 0.73
NSM: magnetar |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.9 | 0.04 | 81.8 | 37.7 | 0.93 | 0.78 | 0.62
FBOTs |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 18.2 | 0.04 | 2.3 | 1.8 | 0.38 | 0.0 | 0.30

Note. — Columns are as in Table 4.

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