# Indirect Searches for Ultraheavy Dark Matter in the Time Domain

David E. Kaplan<EMAIL_ADDRESS>The William H. Miller III Department of
Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland,
21218, USA  Xuheng Luo<EMAIL_ADDRESS>The William H. Miller III Department of
Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland,
21218, USA  Ngan H. Nguyen<EMAIL_ADDRESS>The William H. Miller III
Department of Physics and Astronomy, The Johns Hopkins University, Baltimore,
Maryland, 21218, USA  Surjeet Rajendran<EMAIL_ADDRESS>The William H.
Miller III Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, Maryland, 21218, USA  Erwin H. Tanin<EMAIL_ADDRESS>Stanford
Institute for Theoretical Physics, Stanford University, Stanford, California,
94305, USA

###### Abstract

Dark matter may exist today in the form of ultraheavy composite bound states.
Collisions between such dark matter states can release intense bursts of
radiation that includes gamma-rays among the final products. Thus, indirect-
detection signals of dark matter may include unconventional gamma-ray bursts.
Such bursts may have been missed not necessarily because of their low arriving
gamma-ray fluxes, but rather their briefness and rareness. We point out that
intense bursts whose non-detection thus far are due to the latter can be
detected in the near future with existing and planned facilities. In
particular, we propose that, with slight experimental adjustments and suitable
data analyses, imaging atmospheric Cherenkov telescopes (IACTs) and Pulsed
All-sky Near-infrared and Optical Search for Extra-Terrestrial Intelligence
(PANOSETI) are promising tools for detecting such rare, brief, but intense
bursts. We also show that if we assume these bursts originate from collisions
of dark matter states, IACTs and PANOSETI can probe a large dark matter
parameter space beyond existing limits. Additionally, we present a concrete
model of dark matter that produces bursts potentially detectable in these
instruments.

###### Contents

  1. I Introduction
  2. II Detecting Rare Ultrashort Gamma-ray Bursts
    1. II.1 Sub-$10\,\mu\text{s}$ Blind Spot
    2. II.2 Wavefront Technique
    3. II.3 Expanding the Search for Rare Ultrashort Gamma-Ray Bursts
      1. II.3.1 Searches in the archival data of existing IACTs
      2. II.3.2 Piggyback trigger systems
      3. II.3.3 All-sky all-time searches with PANOSETI
    4. II.4 Projected Sensitivities
  3. III Electromagnetic Signals of Dark Blob Collisions
    1. III.1 Assumptions and Parameterization
    2. III.2 Galactic Gamma-Ray Transients
      1. III.2.1 Visibility depth
      2. III.2.2 Prospects for discovery
    3. III.3 Diffuse Extragalactic Gamma-Ray Background
    4. III.4 Cosmic Rays
  4. IV Dark Blob Models
    1. IV.1 Burst-Producing Model: Summary
    2. IV.2 Formation Mechanisms
  5. V Discussion
  6. VI Conclusion
  7. A Sensitivities of IACTs and PANOSETI to Ultrashort Gamma-Ray Bursts
    1. A.1 VERITAS
    2. A.2 CTA
    3. A.3 PANOSETI
  8. B Fireball
    1. B.1 Thermalization
    2. B.2 Output
  9. C Burst-Producing Model: Details
    1. C.1 Fermi Degenerate Blobs
    2. C.2 Sommerfeld Enhancement
    3. C.3 Weakly Dissipative Collision
      1. C.3.1 $\propto g_{S}^{0}$: Pauli exclusion
      2. C.3.2 $\propto g_{S}^{4}$: elastic $\chi$ scattering
      3. C.3.3 $\propto g_{S}^{6}$: bremsstrahlung of heavy scalar $S$
    4. C.4 Strongly Dissipative Collision
    5. C.5 Ultrashort Gamma-Ray Burst Signal
      1. C.5.1 Weakly dissipative collision: free-blob
      2. C.5.2 Strongly dissipative collision: free-blob
      3. C.5.3 Weakly dissipative collision: binary-blob
  10. D Rate Calculations for Burst-Producing Model
    1. D.1 Elastic Scattering $\chi\chi\rightarrow\chi\chi$
    2. D.2 Bremsstrahlung $\chi\chi\rightarrow\chi\chi S$

##  I Introduction

Numerous programs have been conducted to search for possible electromagnetic
signals of dark matter decay and annihilation Hooper (2019); Slatyer (2018);
Safdi (2024); Gaskins (2016). Such indirect dark matter searches have spanned
broadly across the electromagnetic spectrum, covering many orders of magnitude
in the frequency domain. Due to strong emphasis on minimal and popular dark
matter models such as weakly interacting massive particles Jungman  _et al._
(1996); Conrad (2014); Roszkowski  _et al._ (2018) and axion-like particles
Caputo and Raffelt (2024); O’Hare (2020), the majority of the searches have
been geared toward persistent signals. On the other hand, the Standard Model
universe hosts a large assortment of astrophysical objects that produce a
great diversity of transient signals with rich profiles in the time domain:
flares from pulsar wind nebulae, jets from microquasars, outbursts from
cataclysmic variable stars, etc Rieger (2019); 201 (2012); Van Putten and
Levinson (2012). This motivates us to expand the discovery space of indirect
dark matter detection by systematically covering not only across the
electromagnetic spectrum (frequency domain) but also over a broad range of
temporal structures (time domain).

As a start, we consider gamma-ray transient signals, which are well
parameterized by their arriving fluxes, occurrence rates, and time durations.
A thorough search for such signals should aim not only at achieving
sensitivity to the smallest fluxes but also at covering broad ranges of
occurrence rates and time durations. Even bursts that release a huge amount of
energy and arrive with very high fluxes can be missed if they are rare and
very brief. Catching signals that are off most of the time and appear at
unpredictable locations on the sky requires detectors with high exposure:
large field of view (FoV) and high duty cycle. Detecting short-duration events
poses its own challenges due to fundamental hardware limits on the sampling
rate of a detector. Additional limits may arise from practical implementation
details of the detectors such as their trigger algorithms and data
transmission speeds.

Fermi-LAT, the current-generation space-based gamma-ray detector, covers
$\mathcal{O}(1)$ of the sky with $\mathcal{O}(1)$ duty cycle, but it is
essentially blind to gamma-ray bursts with durations less than about ten
microseconds due to its electronic hardware limitations Atwood  _et al._
(2009); Schaefer  _et al._ (1993). Beside that, ground-based imaging
atmospheric Cherenkov telescopes (IACTs) Errando and Saito (2023); Prandini
_et al._ (2022); Bose  _et al._ (2022); Lacki (2011), while in principle have
nanosecond time resolutions, have not been utilized to its full potential when
it comes to searching for $\lesssim 10\,\mu\text{s}$ duration bursts, as we
will explain further in Section II. Moreover, IACTs have a typical FoV of
$\sim 10^{-3}\text{ sr}$ and a typical duty cycle of $10\%$, which means in a
decade-long observational program they are only sensitive to galactic bursts
happening frequently at the rate of $\gtrsim 10^{4}\text{ event/year}$ on the
entire sky, assuming any burst occurring within the FoV is detectable. We have
thus identified an interesting observational target to aim for, namely
ultrashort ($\lesssim 10\,\mu\text{s}$ duration) gamma-ray bursts. Since this
observational window has been underexplored thus far, there is a potential for
discovery when our detectors become sensitive to it. The goal of this paper is
to study the prospects for probing this observational space in the near future
using existing and planned experimental facilities.

To further motivate exploratory searches for such ultrashort burst signals,
let us recount how discoveries in time-domain astronomy occurred historically.
Many of these discoveries, including those of gamma-ray bursts and pulsars,
happened unexpectedly as new instruments turn on and enable us to access new
observational windows Fabian (2009). Notably, fast radio bursts occur
frequently over the whole sky at the rate of $\sim 10^{3}$ events/day Amiri
_et al._ (2021), and yet we missed them initially because historically radio
transient observations were only done in a targeted way on known variable
objects such as active galactic nuclei and X-ray binaries. Multiple fast radio
bursts signals were eventually found through new analyses of the archival data
of survey radio telescopes. These examples indicate that a large diversity of
electromagnetic signals may have been missed simply because we have not
optimized existing detectors to search for them or performed appropriate data
analyses to reveal their existence.

While primarily our motivation is to probe an unexplored regime in the space
of observables that is within technological reach, it is interesting to
speculate on possible sources of the class of signal we seek for. Hence, in
this paper we also construct and study a concrete model of dark matter that
produces ultrashort gamma-ray bursts. An extremely short timescale and a large
energy release imply a phenomenon involving a spatially-small object with
extraordinary density. Given that the light crossing time of a neutron star,
the most compact Standard Model object known, is $\sim 10\,\mu\text{s}$, a
sub-$10\,\mu\text{s}$ burst that is strong enough to be detectable may require
a beyond the Standard Model source Scargle and Babu . One possibility is that
the dark matter exists in the form of highly dense composite objects, dubbed
dark blobs in this paper, that produce gamma-ray bursts when two of them
collide Gresham  _et al._ (2018a); Bai  _et al._ (2019); Hardy  _et al._
(2015).111Another potential source of short gamma-ray burst is perhaps the
runaway evaporation of black holes with mass $\ll 10^{15}\text{ g}$ whose
lifetimes are much shorter than the age of the universe. Such black holes may
exist today if they form late Picker and Kusenko (2023); Boluna  _et al._
(2024). Accounting only for Standard Model particle spectrum, a
sub-$10\,\mu\text{s}$ duration burst is only achieved when the mass of the
black hole is tiny, resulting in an extremely dim signal. However, if black
hole evaporation proceeds differently from the standard Hawking evaporation,
the duration of the explosion may be shorter for a given black hole mass. See
also Crumpler (2023). As heavy blobs easily evade direct dark matter searches
due to their rare terrestrial transits Jacobs  _et al._ (2015); Grabowska  _et
al._ (2018); Ebadi  _et al._ (2021), currently viable dark blob parameter
space is vast and still allows for strong and distinctive indirect detection
signals.

The rest of the paper is organized as follows. We explore methods for
detecting ultrashort gamma ray bursts and project possible near-future
sensitivities to them in Section. II, provide simple parametrization for
ultrashort gamma-ray bursts from dark blob collisions and map the sensitivity
projections of Section. II onto the parameter space of dark blobs in Section.
III, briefly discuss a simple model of dark blobs that produce ultrashort
gamma-ray bursts in Section. IV.1 (detailed further in Appendix. C), review
existing dark blob formation mechanisms in Section. IV.2, identify interesting
directions for future exploration in Section. V, and conclude in Section VI.
For better readability, we collect some supplementary details in Appendices.
In Appendix. A, we provide simple estimates for the sensitivities of IACTs and
PANOSETI to ultrashort gamma-ray bursts based on the wavefront technique; in
Appendix. B we discuss the condition for and consequences of fireball
formation following a spatially and temporally concentrated injection of
Standard Model particles; in Appendix. C we describe the details of the model
summarized in Section. IV.1; in Appendix. D we provide calculational details
of the gamma-ray burst signal in the dark matter model of Section. IV.1 and
Appendix. C.

##  II Detecting Rare Ultrashort Gamma-ray Bursts

Many orders of magnitude in the time domain below the timescale of ten
microseconds remains a largely unexplored territory for gamma-ray transient
searches LeBohec  _et al._ (2002, 2005); Krennrich  _et al._ (2000). If the
dark matter has been emitting intermittent gamma-ray bursts with durations
less than about ten microseconds, indirect searches for dark matter thus far
would most certainly have missed them.

Consider as an example a scenario where the entire Galactic dark matter exists
in the form of blobs, i.e. large dark matter bound states, with mass
$10^{17}\text{ g}$ and radius $1\text{ cm}$. Assuming these blobs move with a
typical velocity of $v_{\rm DM}=10^{-3}$ and collide with the geometrical
cross section, blob collisions would occur all over the Milky Way at the rate
of $\sim 1\text{ collision/year}$. When two blobs collide, suppose that a
significant fraction of the blobs’ mass energy $10^{17}\text{ g}\sim
10^{38}\text{ erg}$ is released in the form of gamma-rays over a short
timescale, possibly set by the collision timescale $1\text{ cm}/v_{\rm DM}\sim
10\text{ ns}$. Within the brief burst duration, the luminosity of such an
event can be as high $10^{45}\text{ erg/s}$, comparable to the rate at which
energy is released in a supernova. If the burst originates from a collision
within the Galaxy at a distance of $1\text{ kpc}$, the resulting energy
fluence222Fluence here means the number of photons per unit ground area or,
equivalently, the time-integrated flux of the arriving photons. Energy fluence
is fluence multiplied with the (average) energy per photon. would be $\approx
5\mathinner{\mathrm{GeV}}/\text{m}^{2}$, which is in principle detectable
since this is higher than the sensitivities of existing gamma-ray detectors.

However, the briefness and rareness of the bursts in this example scenario
make them challenging to detect in practice. For such a short burst duration,
the detector’s pulse pile-up time and dead time become important limiting
factors to consider.333Pulse pile-up time is the time window within which
arriving photons are lumped into a single event. Dead time is the time after
each event during which the detector is unable to record another event.
Moreover, these burst events might be rejected by the standard trigger systems
employed in current and future gamma-ray detectors as these detectors are
optimized for conventional sources, e.g. conventional $\gtrsim 1\text{
s}$-long gamma-ray bursts. As such, specialized trigger systems may be
required to detect these events. Further, that the bursts occur infrequently
at random positions on the sky make them extra challenging to catch with
detectors that have small FoVs and low duty cycles. In this section, we
discuss strategies for overcoming such challenges in detecting gamma-ray burst
signals that are highly intense, but extremely brief in duration and rare in
occurrence.

###  II.1 Sub-$10\,\mu\text{s}$ Blind Spot

Space-based detectors, which detect transiting gamma-ray photons directly, are
limited by the finite processing time it takes for electronic devices in the
detectors to convert incident gamma-rays into electrical signals Atwood  _et
al._ (2009); Schaefer  _et al._ (1993); Scargle and Babu . This electronic
processing time is typically longer than a microsecond. For instance, Fermi-
LAT has a pulse pile-up time of $\sim 1\,\mu\text{s}$, which means a gamma-ray
burst shorter than $1\,\mu\text{s}$ will be recorded as a single pile-up event
with perceived energy given by the sum total of the energies deposited to the
calorimeter onboard by two or more photons arriving within the
$1\,\mu\text{s}$ window. Sub-microsecond gamma-ray bursts would appear in the
data as occasional fake excesses of high-energy events from the summation of
lower energy ones that do not reflect the incident gamma-ray spectrum, and it
would not be possible to infer if these events were due to bursts of photons.
LAT also has a dead time of $26\,\mu\text{s}$, and that means after each
triggered event the detector will not be able to record another event for a
period of $26\,\mu\text{s}$. Thus, LAT will register more than one photons
only if the burst lasts significantly longer than $26\,\mu\text{s}$.

IACTs such as HESS, MAGIC, and VERITAS avoid these issues by utilizing the
atmosphere to convert incoming gamma-ray photons into optical photons, which
are easier to process Errando and Saito (2023); Prandini  _et al._ (2022);
Bose  _et al._ (2022); Lacki (2011). This conversion occurs automatically as
gamma-ray photons that enter the Earth’s atmosphere initiate extensive showers
of secondary charged particles which emit photons in the optical range through
Cherenkov radiation. IACTs detect incoming gamma-rays indirectly by collecting
their Cherenkov light yields using a large aperture ($\sim 10\text{ m}$)
mirror and focusing them into a pixelized camera.444Charged particles produced
in gamma-ray initiated showers can also be detected directly in water
Cherenkov detectors, such as HAWC, LHAASO, ALPACA, ALTO, LATTES, and SGSO.
However, the poor ability of these detectors in rejecting cosmic ray
backgrounds likely undermine their sensitivity to ultrashort gamma-ray bursts.
By analyzing the parameters (centroid position, size, shape, orientation, etc)
of the digital images captured by the camera, the properties of the primary
gamma-rays that initiated the showers can be inferred. Since IACTs employ
fast, $\mathcal{O}(\text{ns})$ response time, photomultiplier tubes (PMTs) as
the default photosensors, they have the inherent capability to probe the time
domain at nanosecond or longer timescales. Moreover, compared to space-based
gamma-ray detectors, IACTs usually have orders of magnitude better effective
collection areas (and hence much better fluence sensitivities) because the
process of air shower greatly enlarges the area of influence of the incoming
gamma-ray photons to essentially the area of their showers.

While the nanosecond time resolution and large collection area of IACTs make
them suitable for detecting gamma-ray bursts with ultrashort durations, IACTs
unfortunately tend to have poor sky coverages Errando and Saito (2023). For
example, VERITAS covers only a $\theta_{\rm FoV}=3.5^{\circ}$ diameter FoV
corresponding to a solid angle of $\Omega_{\rm
FoV}=2\pi\left[1-\cos\left(\theta_{\rm FoV}/2\right)\right]=3\times
10^{-3}\text{ sr}$ at a given time, and so they would miss an event if they do
not happen to be pointing at the right part of the sky. Furthermore, since
IACTs must operate only during moonless nights and in good weather conditions,
their duty cycles are usually low. VERITAS, for example, has only $\sim
1000\text{ hr}$ of effective observation duration per year. This reduces
further the chance that a burst appears in the detector’s FoV while it is
collecting data.

To summarize, gamma-ray bursts in the sub-$10\,\mu\text{s}$ duration regime
remains, as yet, a poorly explored domain of observations due to mainly the
hardware limitations of space-based gamma-ray detectors and the tiny exposures
of ground-based gamma-ray detectors. It is thus possible that there are
classes of objects emitting such signals that have escaped detection. In other
words, there is an opportunity for discovery in this observational space.

###  II.2 Wavefront Technique

While searches for sub-$10\,\mu\text{s}$ gamma-ray bursts are still very
limited, there exists a technique suitable for detecting ultrashort bursts
devised by Porter and Weekes for a pair of non-imaging Cherenkov detectors
Porter and Weekes (1978), developed further for imaging telescopes Krennrich
_et al._ (2000); LeBohec  _et al._ (2002), and realized in SGARFACE Schroedter
_et al._ (2009); Lebohec  _et al._ (2003) and briefly in SGARFACE+VERITAS
Schroedter (2009, 2009); LeBohec  _et al._ (2005); Krennrich  _et al._ (2001).
These studies were motivated by the prospects of detecting the possible gamma-
ray counterparts to $\text{ns}-\mu\text{s}$ radio pulses from the Crab pulsar
Schroedter (2009); Hankins and Eilek (2007); Lyutikov (2007); Lyutikov  _et
al._ (2016); Bilous  _et al._ (2009) and explosive evaporation of primordial
black holes (PBHs) as predicted by the (outdated) Hagedorn model Schroedter
_et al._ (2009).555Within the Standard Model the explosive evaporation of a
PBH is expected to last for hundreds of seconds Picker and Kusenko (2023);
Boluna  _et al._ (2024). The Hagedorn model was proposed before the Standard
Model was experimentally confirmed. It predicts that as the Hawking
temperature of a black hole approaches the QCD scale of $\sim
200\mathinner{\mathrm{MeV}}$, the number of states available for Hawking
evaporation would increase exponentially, and this would accelerate the
evaporation process, resulting in a burst of gamma-ray photons with a much
shorter duration of hundreds of nanoseconds Halzen  _et al._ (1991). To keep
the distinction clear, we refer to this detection scheme as the gamma-ray
wavefront technique. We briefly highlight important aspects of this technique
here and detail it further in Appendix. A.666Note that the gamma-ray wavefront
technique is to be distinguished from the so-called Cherenkov wavefront
sampling technique, an outdated technique to detect individual gamma-rays that
was implemented in detectors such as ASGAT, Themistocle, CELESTE, and STACEE
de Naurois and Mazin (2015).

Gamma-ray photons from conventional sources arrive on Earth well-separated
spatially and are observed one at a time at IACTs through their Cherenkov
images Errando and Saito (2023); Prandini  _et al._ (2022); Bose  _et al._
(2022). By contrast, an ultrashort gamma-ray burst we have in mind would
arrive with a high fluence in the form of a thin, planar wavefront sweeping
through space, which creates a large number of overlapping showers when it
enters the Earth’s atmosphere. The rough condition for overlapping showers is
that the primary gamma-ray fluence $\mathcal{F}_{*}$ is much greater than
$10^{-5}\,\text{ ph}/\text{ m}^{-2}$, which corresponds to the inverse of the
typical ground area $A_{\rm pool}\sim 10^{5}\text{ m}^{2}$ of a gamma-ray
induced shower. Since the Cherenkov light from these showers are fully mixed
up in this case, the primary gamma-rays are not detected individually but
collectively Porter and Weekes (1978); Schroedter  _et al._ (2009); Schroedter
(2009); LeBohec  _et al._ (2005). This is the key difference between the
wavefront technique and the standard detection schemes of IACTs.

Extensive Monte Carlo simulations show that the Cherenkov images from the
superimposed showers initiated by a gamma-ray wavefront are distinct, both
morphologically and temporally, from that which result from a single gamma-ray
Krennrich  _et al._ (2000); LeBohec  _et al._ (2002); Schroedter  _et al._
(2009). Single gamma-ray induced images are elliptical, exhibit parallactic
variations among telescopes due to the different distances of the telescopes
to the shower core, and can only be detected within the $\sim 150\text{ m}$
pool radius of the shower. On the other hand, gamma-ray wavefront induced
showers do not have a well-defined core and instead extend more or less
uniformly on the ground. They create images that are much more circular and
would appear nearly identically in many telescopes spread over vast distances.
Further, while the typical Cherenkov flash of a single gamma-ray induced
shower is a few nanosecond long, the time profile of the Cherenkov light of
gamma-ray wavefront showers may last for $\gtrsim 10\text{ ns}$ Krennrich  _et
al._ (2000); LeBohec  _et al._ (2002); Schroedter  _et al._ (2009), reflecting
the intrinsic duration of the gamma-ray burst. The wavefront technique
utilizes these distinct properties of the Cherenkov images that result from
gamma-ray wavefronts entering the atmosphere to achieve essentially
background-free detection of ultrashort gamma-ray bursts.

A major challenge in detecting single or wavefront gamma-ray induced shower
signals with IACTs is achieving sufficient rejection of background events,
mainly due cosmic-ray induced showers and the light of the night sky. A
signal-background separation can in principle be made based on the recorded
camera images and their time development. In practice, current generation
IACTs reject backgrounds in two steps:

  1. 1.

Online Triggering
In order not to overwhelm the data acquisition system, the majority of the
backgrounds are already rejected online (in real time, before the data are
read out) by applying a three-level trigger system, usually dubbed Level 1, 2,
and 3. The single-pixel (Level 1) trigger checks if a threshold number of
counts is registered in each camera pixel within a certain time window,
enabling selection based on the temporal profile of the event. Next, the
multi-pixel (Level 2) trigger checks for coincident Level 1 triggers in
multiple adjacent pixels within a specified time window, enabling selection
based on the angular size of the event. Finally, the multi-telescope (Level 3)
trigger checks for coincident Level 2 triggers in at least two telescopes
within a time window, providing selection based on the spatial size of the
event on the ground.

  2. 2.

Offline Reconstruction
Events that pass the Level 3 trigger are read out by the data acquisition
system and stored in the memory after some delay, typically
$\mathcal{O}(10\,\mu\text{s})$. The latter contributes to the dead time of the
detector.777The dead time of IACTs is not an important limiting factor for
detecting ultrashort gamma-ray bursts with the wavefront technique because it
takes only one image, i.e. one event, to recognize a burst. Further background
rejections and reconstruction of the primary gamma-rays are done offline by
comparing the recorded images with expected signal images as found in Monte
Carlo simulations.

For more details on the trigger and data acquisition systems of IACTs, see
e.g. Paoletti  _et al._ (2004); Funk  _et al._ (2004); Weinstein (2007).

The standard Level 1 trigger employed in existing and planned IACTs typically
integrates over a time window of $\mathcal{O}(10\text{ ns})$. Such a short
trigger window is optimized for selecting single gamma-ray shower events whose
durations are $\mathcal{O}(10\text{ ns})$. To maximize the sensitivity to
ultrashort gamma-ray burst events that last $\gtrsim 10\text{ ns}$, different
Level 1 integration times are warranted. Ideally, the integration time should
match the duration of the signal Lebohec  _et al._ (2001).888With shorter
integration time, the sensitivity degrades because only a fraction of the
signal can fit into an integration window; with longer integration time, the
sensitivity suffers from both signal dilution and background exaggeration.
Since the burst duration is a priori unknown, it is useful to have a trigger
system that is sensitive to various burst durations at once. This is the idea
behind the multi-time scale discriminator (MTSD) of the SGARFACE Level 1
trigger, which integrates the signal on six integration windows: 60 ns, 180
ns, 540 ns, 1620 ns, 4860 ns, and 14580 ns Schroedter  _et al._ (2009);
Schroedter (2009). Employing an MTSD-like trigger not only achieves near-
optimal trigger sensitivity to a wide range of burst durations longer than
$10\text{ ns}$ but also enables better rejection of frequent 5-30 ns Cherenkov
flashes from cosmic-ray showers.

Even if the temporal profile of the gamma-ray burst is a delta function,
random air shower process inevitably introduces $\mathcal{O}(10\text{ ns})$
smearing of the arrival times of Cherenkov photons at the telescope Krennrich
_et al._ (2000); LeBohec  _et al._ (2002); Schroedter  _et al._ (2009).999The
time spread in the arrival times of the Cherenkov photons remains small,
despite the $\mathcal{O}(10\text{ km})$ uncertainty in the heights at which a
shower could start, because the showers develop at nearly the speed of light
and are highly beamed. Given that the typical angular spread of Cherenkov
light from a wavefront event is $\theta_{\rm Ch}\sim 1^{\circ}$, we can
estimate the time spread of the Cherenkov light to be $10\text{
km}\left(1-\cos\theta_{\rm Ch}\right)\sim 10\text{ ns}$, which explains the
time spread found in the Monte Carlo simulations of Krennrich  _et al._
(2000); LeBohec  _et al._ (2002); Schroedter  _et al._ (2009). Note also that
the geometry of a $\sim 10\text{ m}$ aperture telescope may also introduce a
time spread of similar order due to the different path lengths of rays
depending on where they hit the mirror. It follows that primary gamma-ray
bursts with any durations shorter than $\sim 10\text{ ns}$ would produce
virtually the same Cherenkov photon signals, independently of the burst
duration. Detecting such sub-10 ns bursts might be relatively challenging
compared to detecting longer duration ones because their online selection
necessitate short, $\mathcal{O}(10\text{ ns})$, integration times at Level 1
which result in poor temporal rejection of cosmic ray backgrounds at trigger
level based on arrival times of photons. While further simulations studies are
required, it may still be possible to identify $\lesssim 10\text{ ns}$ gamma-
ray burst signals among the cosmic-ray shower background without relying on
the temporal discrimination, i.e. based on mainly Level 2 and 3 triggers and
offline analysis of the Cherenkov images. This is more likely to work if a
large number of telescopes is used for stereoscopic rejection at Level
3.101010The past decade has seen the advent of deep learning techniques to
deal with data in the form of sequences of images. These techniques have been
demonstrated to improve event reconstruction Nieto  _et al._ (2022); Miener
_et al._ (2021) and background suppression Spencer  _et al._ (2021); Nieto
Castaño  _et al._ (2019); Parsons and Ohm (2020); Postnikov  _et al._ (2019);
Krause  _et al._ (2017) in the analysis of simulated IACT data. Similar
techniques may be applied to help achieve background-free search for gamma-ray
wavefront events.

###  II.3 Expanding the Search for Rare Ultrashort Gamma-Ray Bursts

Although the wavefront technique was developed more than two decades ago,
gamma-ray bursts at sub-$10\,\mu\text{s}$ timescales have not been searched
for with good sensitivity and exposure. SGARFACE has carried out a 1502 hr
search for ultrashort bursts but relying only on Level 1 and Level 2 triggers
in a single Whipple telescope Schroedter  _et al._ (2009). The lack of
stereoscopic (Level 3) rejection in this search leads to poor rejection of
cosmic ray backgrounds. SGARFACE and VERITAS have carried out a stereoscopic
search with multiple telescopes, achieving excellent Level 3 background
rejection, but only for 6.3 hr Schroedter (2009). On top of that, these
telescopes have poor FoVs.

We propose three (economical) efforts to extend the search for ultrashort
gamma-ray bursts in the near future: (1) by analyzing the archival data of
SGARFACE, VERITAS, and other IACTs (2) by installing piggyback trigger systems
in existing and planned IACTs (3) by using PANOSETI, an upcoming all-sky, all-
time survey for fast optical transients.

####  II.3.1 Searches in the archival data of existing IACTs

If the signals of our interest pass the trigger criteria of existing IACTs,
such signals might already be present in their archival data. In fact,
SGARFACE, which is optimized for $\sim 60\text{ ns}-15\,\mu\text{s}$ duration
bursts, has found, after a 1502-hour search, 14 events that passed its Level 2
trigger and cannot be ruled out as sub-$10\,\mu\text{s}$ gamma-ray wavefront
events Schroedter  _et al._ (2009). While stereoscopic observation with one or
more additional telescopes would most probably have rejected many of these
events, it is possible that our signals have been hiding among these events.

The standard trigger systems employed in current-generation IACTs, e.g.
VERITAS Adams  _et al._ (2022), have $\mathcal{O}(10\text{ ns})$ Level 1
integration windows. Such integration windows should at least be able to catch
bursts with durations not much longer than $10\text{ ns}$. Although some of
our signals seem to pass the standard triggers of IACTs, their subsequent
standard data analysis chains Daniel (2007) do not search for Cherenkov images
associated with gamma-ray wavefront events. Archival data searches for gamma-
ray bursts have been done in the past, but these searches focused on
conventional, longer-lasting sources whose gamma-rays are detected
individually and $\gtrsim 10^{3}\text{ s}$ integration time windows are needed
to achieve sufficient signal to noise ratios Krennrich  _et al._ (2000);
LeBohec  _et al._ (2002); Schroedter  _et al._ (2009); Tesic (2012); Skole
(2016); Archambault (2016, 2018); Doro  _et al._ (2021). It is thus worthwhile
to re-analyze the archival data of IACTs to search specifically for wavefront
events.

####  II.3.2 Piggyback trigger systems

The online trigger systems in existing and planned IACTs are optimized for
conventional sources. These standard trigger systems may not be optimal for
selecting putative ultrashort gamma-ray bursts. It is possible that our
signals are vetoed at the trigger level, and thus not recorded. In that case,
a specific trigger system that better selects these events should be installed
in IACTs. Building new telescopes that are entirely dedicated to exploratory
ultrashort gamma-ray burst searches may be expensive and not cost-effective.
We instead propose installing an MTSD-like trigger that integrates over
multiple time windows longer than $10\text{ ns}$ in existing and planned IACTs
in order to maximize their trigger sensitivities for a wide range of burst
durations.

As demonstrated by the SGARFACE collaboration, existing IACTs can be made more
sensitive to ultrashort gamma-ray bursts by simply having a new trigger system
installed in it. The SGARFACE collaboration installed a new trigger system for
the Whipple telescope in order to improve its sensitivity to ultrashort
bursts, without interfering with the standard operation of the telescope
Schroedter  _et al._ (2009); Schroedter (2009). This can be done by splitting
the PMT signals before they reach the standard trigger system and running the
copied signal through the new trigger system. Originally, the Whipple
telescope employs a trigger system that integrates over time windows of a few
seconds, which would completely wash away any sub-$10\,\mu\text{s}$ burst
signals Doro  _et al._ (2021). The SGARFACE trigger system, on the other hand,
integrates over multiple time windows, the shortest of which is 60 ns (three
consecutive intervals of 20 ns), thus greatly enhancing Whipple telescope’s
ability to probe ultrashort bursts.

In a way similar to how the SGARFACE system operates on the data collected by
the Whipple telescope, a new trigger system with improved sensitivity to
ultrashort bursts can piggyback on existing telescopes without affecting the
standard operation of the telescope. For example, the piggyback trigger can be
installed on VERITAS, MAGIC, HESS, and also CTA in the near future. The $\sim
3-5^{\circ}$ FoV diameter of an IACT is usually significantly bigger than both
its $\sim 0.1^{\circ}$ angular resolution and the angular extent of the object
being observed, which means most of the pixels of the IACT would be empty of
photoelectrons most of the time Cassanyes (2015). A Cherenkov flash from an
ultrashort gamma-ray burst could appear in the FoV of the telescope even when
it is performing a scheduled observations of another object. Hence, the full
$\sim 1000\text{ hr}/\text{yr}$ data-taking time of an IACT can in principle
be used to simultaneously search for ultrashort bursts, if a suitable trigger
system is running. In this way, we can carry out our proposed search in
parallel with the standard analysis routine of the telescope.

####  II.3.3 All-sky all-time searches with PANOSETI

Another challenge with detecting gamma-ray bursts of the type we have in mind
is to do with their possibly infrequent—perhaps once a year—rate of
occurrence. The signal would be absent most of the time and when a burst
occurs, it would appear for a short duration at a random point on the sky. An
effective search for such rare and very fast gamma-ray transients requires
detectors with not only short time-resolutions, but also with large
instantaneous field of coverage and operating continuously for a long
observation time.

The Pulsed All-sky Near-Infrared Optical SETI (PANOSETI) seems to have all
these qualities and is currently in its final design stage Maire  _et al._
(2022). Unlike most optical sky surveys (Pan STARRS, Zwicky Transient Factory,
Vera C. Rubin Observatory) which have integration times of several minutes or
longer Wright  _et al._ (2019), PANOSETI will probe the optical and near-
infrared band with 10 nanosecond resolution. While PANOSETI is developed to
search for fast-transient optical SETI targets, they can double duty as IACTs
Korzoun  _et al._ (2023). Two prototype PANOSETI telescopes were tested in
conjunction with VERITAS for about four nights. It was demonstrated that they
could detect showers induced by 15-50 TeV gamma-ray photons from the Crab
nebula. PANOSETI’s significantly larger FoV and nearly $100\%$ duty cycle sets
them apart from current gamma-ray detectors in their ability to survey the sky
for ultrashort gamma-ray transients that occur rarely.

###  II.4 Projected Sensitivities

Figure 1: Projected sensitivities of VERITAS, CTA, and PANOSETI to ultrashort
gamma-ray bursts, assuming a background-free search can be achieved with the
wavefront technique (see Appendix. A). The spectra of the bursts are assumed
monochromatic at an energy $E_{*}$ and the sensitivities are shown in terms of
the energy fluence of the burst $E_{*}\mathcal{F}_{*}$. Shown for comparison
in dashed red lines labeled VERITAS (S09) and SGARFACE (S09), respectively,
are the sensitivities of VERITAS and SGARFACE (separately) as reported in
Schroedter (2009). The $\mathcal{O}(1)$ difference between the VERITAS (S09)
line and our projected VERITAS sensitivity is due to the PMT upgrade of
VERITAS reported in Kieda (2011). Also shown for comparison in dot-dashed
lines are the sensitivity of Fermi-LAT, whose detection threshold is assumed
here to correspond to consecutive event triggers by at least five photons
crossing within its effective collection area of $0.95\text{ m}^{2}$ over a
time period longer than $100\,\mu\text{s}$ (roughly four times the dead time
of Fermi-LAT) Atwood  _et al._ (2009). The dotted lines correspond to the
gamma-ray fluence $\mathcal{F}_{*}$ satisfying $\mathcal{F_{*}}t_{\rm
int}/\text{max}\left[\Delta t_{*},10\text{ ns}\right]=10^{-5}\text{ ph }\text{
m}^{-2}$ which mark the rough boundaries above which the Cherenkov image
captured within an integration time $t_{\rm int}$ receives contributions from
numerous showers instead of just a single shower; below the dotted lines, the
wavefront technique does not apply, see Section. II.2 and Appendix. A. Top
Left:  photon energy $E_{*}$ dependence of the energy fluence sensitivities
$E_{*}\mathcal{F}_{\rm det}$. Top Right:  the burst occurrence rates at which
the energy fluence sensitivities shown in the other panels apply. While for
concreteness the threshold values of $E_{*}\mathcal{F}_{*}$ are shown for
photon energy $E_{*}=10\mathinner{\mathrm{GeV}}$ and burst duration $\Delta
t_{*}\lesssim 10\text{ ns}$, the threshold burst rates depend on neither
$E_{*}$ nor $\Delta t_{*}$. Bottom Left and Right: burst duration $\Delta
t_{*}$ dependence of the energy fluence sensitivities $E_{*}\mathcal{F}_{\rm
det}$ for a trigger system with a single integration time of $t_{\rm
int}=10\text{ ns}$ (similar to that employed by VERITAS Schroedter (2009)) and
for a trigger system with multiple integration times $t_{\rm int}$ ranging
from $10^{-8}\text{ s}$ to $10^{-3}\text{ s}$ [similar in spirit to the multi-
time-scale discriminator (MTSD) employed by SGARFACE Schroedter (2009)],
respectively. Because the randomness of air showers introduces
$\mathcal{O}(10\text{ ns})$ time spreads in the time-developments of the
Cherenkov images in these detectors, bursts with durations $\Delta
t_{*}\lesssim 10\text{ ns}$ result in virtually the same Cherenkov time
profiles, hence the flattening of the $E_{*}\mathcal{F}_{\rm det}$ at $\Delta
t_{*}\lesssim 10\text{ ns}$.

The search for coincident gamma-ray wavefront events in both VERITAS and
SGARFACE done in Schroedter (2009) is perhaps the only experiment performed
thus far that has the capability of discovering $\lesssim 10\,\mu\text{s}$
gamma-ray bursts. They found no coincident gamma-ray wavefront events during a
6.3 hour run Schroedter (2009), and thereby placed a limit on gamma-ray bursts
with rates greater than $10^{7}\text{ burst/year}$.

To illustrate our near-future ability to extend the search for ultrashort
gamma-ray bursts, we consider three representative detectors: VERITAS, CTA,
and PANOSETI. VERITAS is an existing current-generation IACT which has been
running since 2007 Errando and Saito (2023); Prandini  _et al._ (2022); Bose
_et al._ (2022) and is the only experiment apart from Whipple (with SGARFACE
trigger), its predecessor, that has been used specifically to search for
sub-$10\,\mu\text{s}$ gamma-ray bursts Schroedter (2009), albeit only for one
night. We estimate the fluence sensitivity of VERITAS to wavefront events (for
burst durations $\Delta t_{*}\lesssim 10\text{ ns}$) in Appendix A. Cherekov
Telescope Array (CTA), a next generation IACT that is currently being built,
is expected to reach about an order of magnitude of improvement in energy
fluence $E_{*}\mathcal{F}_{\rm det}(E_{*})$ as well as slightly better FoV and
duty cycle compared to current-generation IACTs. The significant increase in
the fluence sensitivity will enable us to better probe fainter bursts.
PANOSETI uses a large number of low-cost telescopes with relatively small
collecting area to cover a larger FoV. Essentially, it trades sensitivity for
better exposure, thus making it suitable for probing relatively strong bursts
that occur infrequently. We estimate the energy fluence sensitivities of CTA
and PANOSETI based on simple adaptation of the procedure for VERITAS, which we
also detail in Appendix A. The results are shown in Fig. 1.

Assuming the bursts occur at a constant and uniform rate $\Gamma_{\rm burst}$
over the whole sky, the probability of catching a burst within the field of
view $\Omega_{\rm FoV}$ and data-taking time of a detector $T_{\rm obs}$ is
$\Gamma_{\rm burst}T_{\rm obs}\left(\Omega_{\rm FoV}/4\pi\right)$. Requiring
this probability be greater than unity sets the minimum burst rate
$\Gamma_{\rm burst}$ that the detector is sensitive to. The rate sensitivities
of the three detectors we consider as per this criterion are displayed in Fig.
1. The assumed field of view solid angles $\Omega_{\rm FoV}$ of VERITAS, CTA,
and PANOSETI are $2.9\times 10^{-3}\text{ sr}$, $2\times 4.4\times
10^{-3}\text{ sr}$,111111The factor of 2 accounts for the northern and
southern arrays Hofmann and Zanin (2023). and $2.27\text{ sr}$, respectively.
Moreover, we assume that the detectors under consideration run for
approximately ten years (the typical lifetime of an experimental project, not
accounting for duty cycle). This amounts to $T_{\rm obs}=10\times 1000\text{
hr}$ for VERITAS Rajotte (2014) and $T_{\rm obs}=10\times 1300\text{ hr}$ for
CTA Hofmann and Zanin (2023) given the $\sim\mathcal{O}(10)\%$ duty cycle of
IACTs, and $T_{\rm obs}=10\times 8766\text{ hr}$ for PANOSETI which we assume
to have $100\%$ duty cycle. Further details of these detectors can be found in
Appendix. A.

For burst durations $\Delta t_{*}\gtrsim 10\text{ ns}$, the energy fluence
threshold of a detector varies with $\Delta t_{*}$ in a way that depends on
both the trigger algorithm (software) and electronic noise (hardware) of the
detector. For instance, the fluence sensitivity of VERITAS scales as
$E_{*}\mathcal{F}_{\rm det}\propto\Delta t_{*}$ whereas it scales as
$E_{*}\mathcal{F}_{\rm det}\propto\sqrt{\Delta t_{*}}$ for SGARFACE Schroedter
(2009); Schroedter  _et al._ (2009). VERITAS employs a trigger system with a
single integration time window of $t_{\rm int}=7\text{ ns}$, which means only
a fraction $t_{\rm int}/\Delta t_{*}$ of the fluence of a $\Delta t_{*}\gtrsim
t_{\rm int}$ gamma-ray burst would arrive within an integration interval
$t_{\rm int}$. In order to meet the same photoelectron threshold of the PMTs,
the gamma-ray fluence would need to be larger by a factor $\Delta t_{*}/t_{\rm
int}$, thus explaining the linear in $\Delta t_{*}$ scaling of the energy
fluence sensitivity of VERITAS. A linear in $\Delta t_{*}$ extrapolation of
the $E_{*}\mathcal{F}_{\rm det}$ obtained for $\Delta t_{*}\lesssim 10\text{
ns}$ in Appendix A to longer burst durations $\Delta t_{*}\gtrsim 10\text{
ns}$ can be seen in Fig. 1.

Given a burst duration $\Delta t_{*}\gtrsim 10\text{ ns}$, the optimal trigger
threshold is achieved when the trigger integration time matches the burst
duration, $t_{\rm int}\approx\Delta t_{*}$. While we do not a priori know the
$\Delta t_{*}$, such an optimal energy fluence sensitivity can be achieved
simultaneously for a broad range of burst durations $\Delta t_{*}$ with the
use of a trigger system that integrates over multiple integration times
$t_{\rm int}$ over a broad range of timescales, similar to the MTSD trigger of
SGARFACE. As explained in the preceding two subsections, such a trigger system
can work in parallel without interfering with the standard operation of the
telescopes. Now, suppose that an MTSD-like trigger is installed in the three
detectors under considerations: VERITAS, CTA, and PANOSETI. The improved
sensitivities of these detectors once $t_{\rm int}\approx\Delta t_{*}$ is
achieved will depend on their hardware details. For illustration purposes, we
also show in Fig. 1 the improved energy fluence sensitivities if they were to
scale as $E_{*}\mathcal{F}_{\rm det}\propto\sqrt{\Delta t_{*}}$, as is the
case for SGARFACE with its MTSD system.121212 Note that the shortest
integration time of the MTSD of SGARFACE is $t_{\rm int}=3\times 20\text{
ns}$, which explains why the $\propto\sqrt{\Delta t_{*}}$ scaling of its
energy fluence sensitivity pivots at $\Delta t_{*}\approx 60\text{ ns}$. In
our extrapolation of the sensitivities of VERITAS, CTA, and PANOSETI shown in
Fig. 1 we assume that the MTSD-like triggers for these telescopes have a
minimum integration time of $10\text{ ns}$.

We reiterate that the wavefront technique on which we base our analysis relies
on detecting a smooth, superimposed Cherenkov image from a large number of
contributing showers that are initiated by the same gamma-ray wavefront. The
applicability of this technique thus requires a sufficiently high gamma-ray
fluence: $\mathcal{F}\gg A_{\rm pool}^{-1}\sim 10^{-5}\text{ ph }\text{
m}^{-2}$; see Section II.2 and Appendix A. Now, the relevant fluence
$\mathcal{F}$ to satisfy this condition is the fluence that arrives on the
ground within the integration time $t_{\rm int}$ of the detector, which is not
necessarily the same as the full fluence of the burst $\mathcal{F}_{*}$. For
$t_{\rm int}\lesssim\text{max}\left[\Delta t_{*},10\text{ ns}\right]$, where
the $10\text{ ns}$ accounts for shower spreading, we have
$\mathcal{F}=\mathcal{F}_{*}$, and so the overlapping-shower condition amounts
to $\mathcal{F}_{*}\gg 10^{-5}\text{ ph }\text{ m}^{-2}$. However, for $t_{\rm
int}\gtrsim\text{max}\left[\Delta t_{*},10\text{ ns}\right]$, only a fraction
$t_{\rm int}/\text{max}\left[\Delta t_{*},10\text{ ns}\right]$ of the full
fluence $\mathcal{F}_{*}$ contributes to creating a Cherenkov image that is
captured within an integration time $t_{\rm int}$, which translates to a
stricter condition $\mathcal{F}_{*}t_{\rm int}/\text{max}\left[\Delta
t_{*},10\text{ ns}\right]\gg 10^{-5}\text{ ph }\text{ m}^{-2}$. The latter
implies a lower bound on $\mathcal{F_{*}}$ for the wavefront technique to
apply, that for $\Delta t_{*}\gtrsim 10\text{ ns}$ is linear in $\Delta t_{*}$
for a single integration time $t=10\text{ ns}$ trigger and independent of
$\Delta t_{*}$ for a multi-$t_{\rm int}$ (MTSD-like) trigger; see Fig. 1.

##  III Electromagnetic Signals of Dark Blob Collisions

An intense gamma ray burst with duration $\lesssim 10\,\mu\text{s}$ implies an
emitting blob of size $\lesssim 3\text{ km}$ (whose light travel time is
$\lesssim 10\,\mu\text{s}$) with very high energy density, possibly higher
than that of any known Standard Model object, that glows in gamma-ray for
$\lesssim 10\,\mu\text{s}$ when the time is ripe. It is plausible that the
dark matter comes in the form of such ultra-dense blobs which release an
ultrashort burst of gamma-rays when they collide.

Nearly all known mechanisms for dark blob formation, which we summarize in the
Section IV.2, require the existence of some particles that mediate
interactions between the fundamental dark matter particles constituting a
blob. When a pair of dark blobs collide in the present epoch, a large number
these mediators may be radiated as an energy loss channel. If the mediators
are coupled to the Standard Model in some way, the emitted mediators may
subsequently decay with some branching ratio into photons. Even if this
branching ratio to photon is small, non-photon decay products may thermalize
into a fireball which then releases most of its energy in photons when it
becomes optically-thin (see Appendix. B). It is thus reasonable to expect a
burst of photons to be produced in a blob collision. We present a model that
does exactly that in Section. IV.1 and Appendix. C. In this section, we
discuss in a model agnostic way the prospect for discovering gamma-ray
transients from dark blob collisions.

###  III.1 Assumptions and Parameterization

We assume the entire dark matter mass density comes in the form of identical
dark blobs with mass $M$. An upper bound on the blob mass is given by
microlensing constraints Smyth  _et al._ (2020); Jacobs  _et al._ (2015);
Croon  _et al._ (2020),

$\displaystyle M\lesssim 10^{23}\text{ g}$ (1)

For simplicity, we write the velocity-averaged blob collision cross-section in
the Milky Way as

$\displaystyle\left<\sigma v\right>_{\rm col}=v_{\rm DM}R^{2}$ (2)

where $v_{\rm DM}=10^{-3}$ is the typical virial velocity in the Milky Way and
$R$ is the effective radius of the blob as defined by the above
relation.131313What we refer to as the blob radius $R$ merely parameterizes
the square root of the blob collision cross section. If the blob collision
cross-section were Sommerfeld enhanced, for example, the actual blob radius
could be much smaller. The blob radius $R$ is bounded from above and below by
requiring, respectively, consistency with Bullet Cluster observations
($R^{2}/M\lesssim 1\text{ cm}^{2}/\text{g}$)141414Requiring the probability
that over the age of the universe each blob collides less than once (to
ensure, in some models, that the blobs are not depleted) yield a
parametrically similar constraint to the Bullet Cluster one. Jacobs  _et al._
(2015) and the blobs not be black holes ($R>2GM$), where $G$ is Newton’s
gravitational constant. Given these constraints, a wide range of blob radii
are still allowed

$\displaystyle 10^{-11}\text{ cm}\left(\frac{M}{10^{17}\text{
g}}\right)\lesssim R\lesssim 10^{8}\text{ cm}\left(\frac{M}{10^{17}\text{
g}}\right)^{1/2}$ (3)

Variations in the outcomes of a collision due to different impact parameters
are neglected in our analysis. We further assume that blob collisions occur
only after Galaxy formation. The blobs may have formed before Cosmic Microwave
Background (CMB) decoupling, however before structure formation we expect the
blobs to move extremely slowly and are thus unlikely to collide Mack  _et al._
(2007). Hence, there is no limits on blob parameters based on photon injection
in the early universe and the resulting CMB spectral distortions Deng  _et
al._ (2018).

To characterize a burst, we introduce three more parameters:

  * •

The fraction of $M$ released in gamma-ray $\epsilon_{*}$.
We assume that each blob collision produces a gamma-ray burst with total
energy $\epsilon_{*}M$. The $\epsilon_{*}$ parameter is in principle set by
the coupling strength of the blob constituents to the mediator as well as
other properties of the blob. Interesting values of $\epsilon_{*}$ include
those corresponding to scenarios where the burst energy saturates the blob’s
mass ($\epsilon_{*}\sim 1$), the blob’s virial kinetic energy
($\epsilon_{*}\sim v_{\rm DM}^{2}\sim 10^{-6}$), or the blob’s binding energy
($\epsilon_{*}\sim v_{\rm esc}^{2}$, where $v_{\rm esc}$ is the escape
velocity of a blob constituent from the confining potential of the blob). When
showing our results in plots, we set $\epsilon_{*}=1$ and
$\epsilon_{*}=10^{-6}$, with the understanding that the result for different
values of $\epsilon_{*}$ can be obtained by simple rescalings.

  * •

Single gamma-ray energy $E_{*}$.
We assume the bursts are monoenergetic. The individual photon’s energy $E_{*}$
may be set by the mass, translational kinetic energy, thermal energy, or Fermi
energy of a typical blob constituent. For simplicity, we will present our
results only for $E_{*}$ in the range $100\mathinner{\mathrm{MeV}}\leq
E_{*}\lesssim 100\mathinner{\mathrm{GeV}}$. This corresponds to the range of
$E_{*}$ for which simulations of gamma-ray wavefront induced showers are
currently available Krennrich  _et al._ (2000); LeBohec  _et al._ (2002);
Schroedter  _et al._ (2009); Lebohec  _et al._ (2003); Schroedter (2009);
LeBohec  _et al._ (2005). For $E_{*}$ above this range, we expect the
wavefront technique to still be applicable, though the gamma-ray fluence
$\mathcal{F}_{*}$ will be subject to a lower bound as per overlapping-shower
requirement that is higher than the sensitivity of the detectors under
consideration, complicating the analysis; see the last paragraph of Section.
II.4 and Fig. 1.

  * •

The duration of the burst $\Delta t_{*}$.
Depending on the nature of the collision, the burst duration $\Delta t_{*}$
may be set by the blob’s light crossing time $R$, the blob interpenetration
time $R/v_{\rm DM}$, the mediator’s decay lifetime, or a model-dependent
dynamical timescale (e.g. the blob collapse rate at the moment of photon
production). As explained in Section. II, Cherenkov light signals as seen by
ground-based gamma-ray detectors are independent of $\Delta t_{*}$ as long as
$\Delta t_{*}\lesssim 10\text{ ns}$, which we assume to be the case in our
blob parameter space plots.

An energy release in the form of Standard Model particles that is sufficiently
concentrated both spatially and temporally may lead to the formation of a
fireball, a thermalized, optically-thick plasma that expands under its own
radiation pressure. We consider this possibility in Appendix. B and find that
the above parametrization also applies to the gamma-ray burst that is released
from a fireball.

To further simplify our analysis, we confine ourselves to the regime where
burst events originating in the Galaxy can be treated independently. Very
crudely and regardless of the detector, this is the case if the $\Delta
t_{*}$-thick shells of photons created in blob collisions do not have
significant volume overlap in the Galaxy. In other words, the total volume
occupied by burst shells created in the typical time it takes to escape the
Milky Way, $\sim 10\text{ kpc}\sim 3\times 10^{4}\text{ yr}$, must be less
than the volume of the Milky way $\sim(10\text{ kpc})^{3}$, which boils down
to

$\displaystyle\Gamma_{\rm col}^{\rm MW}\Delta t_{*}\lesssim 1$ (4)

where $\Gamma_{\rm col}^{\rm MW}\sim\left(\rho_{\rm DM}/M\right)^{2}\times
R^{2}v_{\rm DM}\times(10\text{ kpc})^{3}$ is the average collision rate of
blobs in the Milky Way. This can be satisfied by choosing a sufficiently small
$\Delta t_{*}$.

###  III.2 Galactic Gamma-Ray Transients

####  III.2.1 Visibility depth

The arriving energy fluence of the burst emitted in a blob collision at a
distance $d$ from the Earth is151515Photons of a gamma-ray burst may hit other
photons in the interstellar medium (mainly the CMB and infrared radiation by
dust) and pair produce electron-positron pairs. Such gamma-ray attenuation
effect is negligible in the cases we consider. Gamma-rays with energies
$\lesssim 10\mathinner{\mathrm{TeV}}$ originating from essentially anywhere in
the Milky Way will reach the Earth with near-unity survival probabilities
Vernetto and Lipari (2016). At higher energies, gamma-rays may still survive
with $\mathcal{O}(1)$ probabilities.

$\displaystyle E_{*}\mathcal{F}_{*}=\frac{\epsilon_{*}M}{4\pi d^{2}}$ (5)

For a given detector, there is a minimum energy fluence above which a burst is
detectable, $E_{*}\mathcal{F}_{\rm det}(E_{*})$. Requiring
$E_{*}\mathcal{F}_{*}>E_{*}\mathcal{F}_{\rm det}(E_{*})$ translates to the
maximum distance at which a blob collision is detectable

$\displaystyle d_{\rm max}=\sqrt{\frac{\epsilon_{*}M}{4\pi
E_{*}\mathcal{F}_{\rm det}(E_{*})}}$ (6)

Fig. 2 shows the $d_{\rm max}$ as a function of blob mass $M$ for
$\epsilon_{*}=1$, $E_{*}=10\mathinner{\mathrm{GeV}}$, and the detectors
considered in Section II.4 whose energy fluence sensitivities
$E_{*}\mathcal{F}_{\rm det}$ are displayed in Fig. 1. As shown in Fig. 2, we
can roughly demarcate two cases based on whether $d_{\rm max}$ is larger than
10 kpc (the length scale associated to the size of the Milky Way halo):

  * •

Strong burst: If $d_{\rm max}\gtrsim 10\text{ kpc}$, then a blob collision
occurring anywhere in the Milky Way is detectable as long as it happens within
the search time and FoV of the detector.

  * •

Weak burst: If $d_{\rm max}\ll 10\text{ kpc}$, then blob collisions are
detectable if they happen sufficiently frequently in the Milky Way that at
least one of them takes place at a distance less than $d_{\rm max}$ within the
search time and FoV of the detector.

Figure 2: The characteristic visibility radii $d_{\rm max}$ of VERITAS, CTA,
and PANOSETI to individual gamma-ray transients from dark blob collisions in
the Milky Way as functions of the blob mass $M$. Each blob collision is
assumed to produce a burst of gamma-rays with a duration $\Delta t_{*}$, a
total energy $\epsilon_{*}M$, and a monochromatic energy per photon $E_{*}$,
where $\Delta t_{*}\lesssim 10\text{ ns}$, $\epsilon_{*}=1$ and
$E_{*}=10\mathinner{\mathrm{GeV}}$ as labeled in the figure. The $d_{\rm max}$
is given explicitly by Eq. (6) and is computed based on the energy fluence
sensitivities of the respective detectors as shown in Fig. 1. The dotted line
indicates the scale radius of the Milky Way halo and demarcates two distinct
regimes for burst detection. Individual bursts originating from anywhere in
the Galaxy are detectable in the strong burst regime ($d_{\rm max}\gtrsim
10\text{ kpc}$), whereas only bursts that occur sufficiently close can be
detected in the weak burst regime ($d_{\rm max}\ll 10\text{ kpc}$).

####  III.2.2 Prospects for discovery

The rate at which blob collisions occur is model dependent. It depends not
only on the mass and radius of the blob, but also on whether the blobs form
binaries and/or interact via long-range forces. Here, we simply assume that
the blob collision cross section is given by Eq. (2), where the role of the
blob radius $R$ is to parameterize the collision rate. While we will call $R$
as defined by Eq. (2) the blob radius, it does not necessarily reflect the
actual size of the blob. As such, the duration of the burst produced in a blob
collision $\Delta t_{*}$ is a priori independent of $R$.

The expected number of detectable blob collisions occurring within the field
of view (FoV) of the detector $\Omega_{\rm FoV}$ during the observation time
of the detector $T_{\rm obs}$ can be estimated as follows

$\displaystyle N_{\rm event}=$ $\displaystyle J_{\rm
col}\left(\frac{\rho_{s}}{M}\right)^{2}R^{2}v_{\rm DM}T_{\rm obs}\Omega_{\rm
FoV}$ $\displaystyle\times\text{min}\left[d_{\rm max}^{3},\left(10.6\text{
kpc}\right)^{3}\right]$ (7)

with the criterion for detecting bursts from blob collisions being $N_{\rm
event}>1$. Here

$\displaystyle J_{\rm col}=$ $\displaystyle\frac{\int_{\rm
FoV}d\Omega\,l^{2}dl\left(\rho_{\rm DM}^{\rm
NFW}/\rho_{s}\right)^{2}}{\Omega_{\rm FoV}\times\text{min}\left[d_{\rm
max}^{3},\left(10.6\text{ kpc}\right)^{3}\right]}$ (8)

is an $\mathcal{O}(1)$ numerical factor to account for the spatial
distribution of dark matter, $\Omega_{\rm FoV}$ is the FoV solid angle of the
detector being used, $l$ is the line of sight distance, and we assume that the
Galactic dark matter density $\rho_{\rm DM}^{\rm NFW}$ follows the NFW profile
Navarro  _et al._ (1996)

$\displaystyle\rho_{\rm DM}^{\rm
NFW}(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}}$ (9)

where $r$ is the Galactocentric radius, $\rho_{s}=9\times
10^{-3}M_{\odot}\text{pc}^{-3}$, and $r_{s}=16\text{ kpc}$ Navarro  _et al._
(1996); Bovy (2015). In calculating $J_{\rm col}$, we take the distance from
the Earth to the Galactic Center to be $r_{\oplus}=8\text{ kpc}$ and assume
that the detector’s FoV points perpendicularly to the Galactic plane. For
$\theta_{\rm FoV}\lesssim 100^{\circ}$ (corresponding to an FoV solid angle of
$\Omega_{\rm FoV}=2\pi\left[1-\cos\left(\theta_{\rm FoV}/2\right)\right]$), we
find that typically $J_{\rm col}\approx 0.28$, unless $d_{\rm max}\sim
10.6\text{ kpc}$ where $J_{\rm col}$ dips down by up to a factor of $\sim 3$
compared to the typical value. Nonetheless, in obtaining our results we always
compute the full integral in Eq. (8).

The condition $N_{\rm event}>1$ suggests the following figures of merit for
burst detectability in the weak bursts regime ($d_{\rm max}\ll 10\text{ kpc}$)

$\displaystyle\underbrace{\left[\epsilon_{*}^{3/2}M^{-1/2}R^{2}\right]}_{\text{blob
parameters}}\times\underbrace{\left[E_{*}\mathcal{F}_{\rm
det}(E_{*})\right]^{-3/2}}_{\text{sensitivity}}\times\underbrace{\left[\Omega_{\rm
FoV}T_{\rm obs}\right]}_{\text{exposure}}$ (10)

and in the strong bursts regime ($d_{\rm max}\gtrsim 10\text{ kpc}$)

$\displaystyle\underbrace{\left[M^{-2}R^{2}\right]}_{\text{blob
parameters}}\times\underbrace{\left[\Omega_{\rm FoV}T_{\rm
obs}\right]}_{\text{exposure}}$ (11)

These show that the fluence sensitivity $E_{*}\mathcal{F}_{\rm det}(E_{*})$ is
an important factor in the detectability of weak bursts. By contrast, the
discovery potential for strong bursts is mainly limited by the exposure of the
detector, namely the product $\Omega_{\rm FoV}T_{\rm obs}$. In that case, we
can afford to reduce the fluence sensitivity (increase $E_{*}\mathcal{F}_{\rm
det}$) and decrease the visibility depth $d_{\rm max}$ if that is what it
takes to improve exposure. This is reminiscent of an effective strategy for
directly detecting dark blobs: where we are looking for rare but spectacular
events, and the strategy is to maximize exposure at the expense of
sensitivity, see e.g. Grabowska  _et al._ (2018); Ebadi  _et al._ (2021).
Though we do not consider it here, single strong bursts from outside of the
Milky Way may also be detectable in some cases.

In Fig. 3, we show the projected sensitivities of VERITAS, CTA, and PANOSETI
to $\epsilon_{*}=1$ and $\epsilon_{*}=10^{-6}$ bursts in terms of the blob
parameters $(R,M)$. As we will discuss below, consistency with diffuse gamma-
ray and cosmic ray observations already sets some limits on the blob parameter
space. Nevertheless, a large parameter space that yields observable bursts in
VERITAS, CTA, and PANOSETI still remains. For a given blob mass $M$, the
constraints (13) on $R$ derived from limits on particle dark matter
annihilation cross-sections scale as $\epsilon_{*}^{-1/2}$, whereas the figure
of merits (10) and (11) imply that the threshold $R$ above which individual
Galactic bursts can be detected scale as $\epsilon_{*}^{-3/4}$ in the weak
burst regime and $\epsilon_{*}^{0}$ in the strong burst regime. It follows
that as we decrease $\epsilon_{*}$ from $\epsilon_{*}=1$ to $\epsilon_{*}\ll
1$, the parameter space with observable Galactic bursts will shrink and expand
in the weak and strong burst regime, respectively, until the
$\epsilon_{*}$-independent upper bound on $R$ from Bullet Cluster dominates.
Since $d_{\rm max}$ itself is proportional to $\epsilon_{*}^{1/2}$, the
boundary between weak burst and strong burst, $d_{\rm max}\sim 10\text{ kpc}$,
will also shift toward making the weak (strong) burst regime bigger (smaller)
in the $(R,M)$ space.

Figure 3: The radius $R$ vs mass $M$ parameter space of dark blobs. Shown in
gray is the combined limits from Bullet Cluster (slanted upper boundary),
microlensing (vertical boundary), and black hole avoidance (slanted lower
boundary), c.f. Eqs. (1) and (3). The blobs are assumed to collide with a
cross-section $\left<\sigma v\right>_{\rm col}=10^{-3}R^{2}$ and produce upon
a collision a burst of gamma-rays with a duration $\Delta t_{*}$, a total
energy $\epsilon_{*}M$, and a monochromatic energy $E_{*}$ as labeled in each
sub-figure. Along the dotted lines, blobs collide at the rate of once per year
in the Milky Way. The colored solid lines are the projected sensitivities of
VERITAS, CTA, and PANOSETI to the blob parameter space, obtained by requiring
the expected number of detectable burst events $N_{\rm event}$ found in Eq.
(7) be greater than 1. The change in the slope of the sensitivity line from
$R\propto M^{1/4}$ at relatively low $M$ to $R\propto M$ at relatively high
$M$ corresponds to switching from weak burst regime (only sufficiently close
bursts are detectable) to strong burst regime (bursts from any distance in the
Galaxy are detectable); see Eqs. (10) and (11). The colored regions show the
strongest existing indirect detection limits. For the case of
$E_{*}=10\mathinner{\mathrm{GeV}}$ shown here, these limits are from the
isotropic diffuse gamma-ray background measurement by Fermi (Fermi: IGRB)
Ackermann  _et al._ (2015a) and 6.3-hour-long stereoscopic ultrashort gamma-
ray burst search with VERITAS and SGARFACE (VERITAS+SGARFACE) Schroedter
(2009).

###  III.3 Diffuse Extragalactic Gamma-Ray Background

Figure 4: Existing indirect limits on the combination $R^{2}/M$ (left vertical
axis) of the blob radius $R$ and blob mass $M$ translated from limits on the
annihilation cross section to two photons per unit mass
$\left<\tilde{\sigma}v\right>_{\gamma\gamma}/\tilde{m}$ of particle dark
matter with mass $\tilde{m}=E_{*}$, as per Eq. (13). The limits shown are
obtained from positron spectrum measurement by AMS-02 Elor  _et al._ (2016),
isotropic diffuse gamma-ray background (IGRB) measurement by Fermi Ackermann
_et al._ (2015a), and galaxy cluster observations by Fermi Anderson  _et al._
(2016). The blobs are assumed to collide with a cross-section $\left<\sigma
v\right>_{\rm col}=10^{-3}R^{2}$ and produce upon a collision a burst of
gamma-rays with a total energy $\epsilon_{*}M$ with $\epsilon_{*}=1$ and a
monochromatic energy per photon $E_{*}$.

Some of the leading indirect detection constraints on annihilating particle
dark matter come from gamma-ray observations of the Galactic center and dwarf
spheroidal galaxies Hooper (2019); Slatyer (2018); Safdi (2024); Gaskins
(2016). Nevertheless, in the case of dark blobs, due to the infrequent nature
of their collisions (perhaps once per decade in the Milky Way, for example),
these constraints either do not apply or need to be re-analyzed. Since we
confine our analysis to the parameter space where dark blob collisions in the
Milky Way produce non-overlapping bursts, gamma-ray observations that probe
spatial volumes comparable to or smaller than the size of a Galaxy should
expect temporally discrete, non-diffuse signals from blob collisions.

However, at galaxy-cluster and cosmic scales we expect collisions of blobs to
likely lead to a diffuse gamma-ray signals, in which case the usual
constraints obtained in the case of particle dark matter annihilation should
also apply for dark blobs. By equating the predicted photon flux in the two
cases

$\displaystyle\left(\frac{\rho_{\rm DM}}{M}\right)^{2}\left(v_{\rm
DM}R^{2}\right)\left(\frac{\epsilon_{*}M}{E_{*}}\right)=\left[\left(\frac{\rho_{\rm
DM}}{\tilde{m}}\right)^{2}\left<\tilde{\sigma}v\right>_{\gamma\gamma}\right]_{\tilde{m}=E_{*}}$
(12)

where a tilde $\tilde{}$ denotes a particle dark matter quantity, we can
recast limits on particle dark matter annihilation cross section to a pair
photons, $\left<\tilde{\sigma}v\right>_{\gamma\gamma}^{\rm limit}$, to the
corresponding limits on the dark blob radius $R$ and mass $M$

$\displaystyle\frac{R^{2}}{M}\lesssim\frac{\left<\tilde{\sigma}v\right>_{\gamma\gamma}^{\rm
limit}}{\epsilon_{*}v_{\rm DM}E_{*}}$ (13)

Fermi measurement of the isotropic diffuse gamma-ray background (IGRB)
Ackermann  _et al._ (2015a) and Fermi observation of the 16 highest J-factor
galaxy clusters Anderson  _et al._ (2016) have set
$\left<\tilde{\sigma}v\right>_{\gamma\gamma}^{\rm limit}$ as function of the
dark matter mass $\tilde{m}=E_{*}$, which we translate into limits on
$R^{2}/M$ using Eq. (13). These limits on
$\left<\tilde{\sigma}v\right>_{\gamma\gamma}$ and the translated limits on
$R^{2}/M$ are shown in Fig. 4 and have been incorporated into Fig. 3.

While we assumed that the prompt products of a blob collision are
monoenergetic photons, the final photon spectrum to be compared with data
would generically have additional contributions from decaying pions, radiative
processes such as final state radiation and internal bremsstrahlung, and
subsequent interactions with the surrounding medium (gas, starlight, CMB,
cosmic rays, etc). We neglect the constraints on $R^{2}/M$ derived from those
secondary spectra as line searches usually give the strongest constraints.

We note that gamma rays from blob collisions are clustered in both their
arrival times (concentrated in intervals of $\Delta t_{*}$) and in their sky
locations. This means the statistics of the arriving gamma-ray photons are
inherently non-Poissonian, unlike in the case of particle dark matter which
predicts Poissonian statistics. This distinction can in principle be used to
infer the presence of burst signals in the diffuse gamma-ray background. For
instance, gamma-ray photons arriving from an angular patch in
sub-$10\,\mu\text{s}$ bunches of a few photons more often than the Poisson
expectation would indicate a non-Poissonian statistics. However, in practice
it might be difficult to detect the non-Poissonian nature of the photon
statistics as it requires detecting more than one photons within the burst
duration $\Delta t_{*}$. As discussed previously, Fermi-LAT, for instance,
cannot register more than one photons within $\sim 10\,\mu\text{s}$. A good
time resolution is thus essential for probing the non-Poissonian statistics.

###  III.4 Cosmic Rays

Generically, electrons and positrons should also be present among the final
states that include monoenergetic photons. Positrons with energy
$E\lesssim\mathinner{\mathrm{TeV}}$ would lose a significant fraction of their
energies in $\gtrsim 10^{5}\text{ yr}$ Joshi and Razzaque (2017). The current,
steady-state positron population in the Galaxy probes the accumulation of
$e^{\pm}$ injections from dark blob collisions over the long timescales set by
the energy loss or diffusion timescale of positrons. In order to have a
reasonable chance for detecting individual Galactic gamma-ray transients from
blob collisions, we need the blob collision rate to be greater than about once
per decade. Blob collisions at that rate would appear continuous over $\sim
10^{5}\text{ yr}$. Hence, an integration over such a long time scale washes
out the discrete and burst-like nature of $e^{\pm}$ injections from blob
collisions, the very features distinguishing them from that of particle dark
matter. While it is difficult to extract tell-tale signature of blob
collisions in cosmic ray data, constraints on the abundance of cosmic rays can
be used to limit the blob parameter space. As before, such limits can be
obtained by recasting cosmic-ray limits on the $\left<\sigma
v\right>_{\gamma\gamma}$ of particle dark matter annihilation into two photons
(which also produce cosmic rays automatically) into limits on $R^{2}/M$ using
Eq. (13). One such limits on $R^{2}/M$ can be obtained from the AMS-02
measurement of the cosmic-ray positron flux and the corresponding constraint
on $\left<\sigma v\right>_{\gamma\gamma}$, as found in Ref. Elor  _et al._
(2016). This cosmic-ray limit is shown in Fig. 4 and has been incorporated
into Fig. 3.

Dark matter injections of $e^{\pm}$ can also be probed through various other
considerations, including observations of the 511 keV line from $e^{\pm}$
annihilation Wilkinson  _et al._ (2016) as well as secondary photons Cirelli
_et al._ (2023); Crocker  _et al._ (2010); Ackermann  _et al._ (2015b);
Abazajian  _et al._ (2010) produced by synchrotron radiation of $e^{\pm}$ and
inverse Compton scattering of ambient photons (CMB, starlight, etc) off the
high energy $e^{\pm}$. The null result of these searches have been used to
bound DM annihilation into various channels except for the diphoton one
considered here. Studies on dark matter annihilation via the diphoton channel
which then create charged particles with higher order diagrams have been
lacking because dark matter is often assumed to be electrically neutral, which
leads to the standard expectation that its cross section to two photons must
be radiatively induced at one-loop and therefore suppressed. Obtaining these
other cosmic ray limits on $\left<\sigma v\right>_{\gamma\gamma}$ requires
further dedicated work and is beyond the scope of this paper. Given the very
large blob parameter space currently available for detection with VERITAS,
CTA, and PANOSETI as diplayed in Fig. 3 we do not expect the inclusion of
these other cosmic ray limits to affect our main results significantly.

##  IV Dark Blob Models

###  IV.1 Burst-Producing Model: Summary

We consider in this subsection an example model for dark matter blobs that
produce upon a collision a burst of mediator particles which subsequently
decay to photons, producing a gamma-ray burst. We briefly summarize the main
features of the model here. For the interested reader, a more complete
description of the model is given in Appendix. C.

The particle content of our model includes a fermion $\chi$ as the fundamental
dark matter particle and two mediators: a light scalar $\phi$ and a heavy
scalar $S$. We assume that the dark matter is completely asymmetric and comes
in the form of identical dark blobs: large composite bound states of fermions
$\chi$ confined by attractive forces mediated by the light scalar $\phi$. We
also assume that prior to a collision each blob has radiated away its internal
energy to the point of degeneracy, which effectively shuts off further
radiation. When two blobs collide, the relative motion of the blobs causes
them to appear excited with respect to each other’s Fermi sea. This enables a
fraction [up to $\mathcal{O}(1)$] of their translational kinetic energy at
impact to be released, in this case, via bremsstrahlung of heavy scalars $S$
which promptly decay to photons, thus producing a gamma-ray burst.

The nature of the blob collision in our model depends on the assumed coupling
strength $g_{S}$ of the fermion $\chi$ to the heavy scalar $S$, and can be
classified into two qualitatively different regimes:

  * •

Weakly dissipative collision
For a sufficiently weak $g_{S}$, the mean free path of $S$-mediated
scatterings between $\chi$ particles during a blob collision can be larger
than the blob radius. As such, the colliding blobs are for the most part
transparent to each other. A small fraction of $\chi$ particles may scatter
during the course of the blob collision. When they do there is a small
probability that a heavy scalar $S$ is emitted via bremsstrahlung. In this
case, the energy per unit blob mass $\epsilon_{*}$, peak photon energy
$E_{*}$, and duration $\Delta t_{*}$ of the gamma-ray burst from the decay of
$S$ are set by the bremsstrahlung rate, the Fermi energy of the $\chi$
particles, and the duration of the blob collision, respectively.

  * •

Strongly dissipative collision
We define this regime to be when the coupling strength $g_{S}$ is sufficiently
strong that the mean free path of $S$-mediated $\chi$ scatterings is shorter
than the blob radius. We expect the rapid $\chi$ scatterings to efficiently
dissipate all of the kinetic energy of the blob’s relative bulk motion into
random thermal motion, causing them to stop and merge into a single object.
Once the blobs’ bulk kinetic energy is converted to thermal energy, it is just
a matter of time for the thermal energy to be radiated away through $S$
bremsstrahlung. In this case, the resulting gamma-ray burst will have its
$\epsilon_{*}$ set by the entire bulk kinetic energy of the blobs at impact,
its $E_{*}$ set by the Fermi energy of the $\chi$ particles, and its $\Delta
t_{*}$ set by the longer of the blob collision timescale and the time it takes
for the merger product to radiate $\mathcal{O}(1)$ of its internal energy.

The gamma-ray burst signal in the weakly dissipative collision regime is
relatively weak but can be estimated robustly, whereas the signal in the
strongly dissipative collision regime is less tractable but can be many orders
of magnitude stronger. Accordingly, we find that the expected number
detectable of events in VERITAS, CTA, and PANOSETI is $N_{\rm
event}=\mathcal{O}\left(10^{-8}-10^{-7}\right)$ at the highest in the weakly
dissipative case, while it can go as high as $N_{\rm
event}=\mathcal{O}\left(10^{3}-10^{4}\right)$ and more in the strongly
dissipative case. If the blobs form binaries in the early universe Ali-Haïmoud
_et al._ (2017); Raidal  _et al._ (2019); Inman and Ali-Haïmoud (2019);
Jedamzik (2020); Raidal  _et al._ (2024); Diamond  _et al._ (2023a); Bai  _et
al._ (2023), we find that the significantly enhanced blob collision rate can
raise the expected number of detectable events in the three detectors to
$N_{\rm event}=\mathcal{O}\left(1-10\right)$ in the weakly dissipative case.

###  IV.2 Formation Mechanisms

The Standard Model sector has provided us with a wide variety of examples of
large fermion bound states with known formation mechanisms. The dark sector
should be able to form analogous bound states through similar mechanisms if it
possesses certain key features of the SM. We first discuss two classes of
dark-blob formation scenarios that are largely based on known SM processes,
namely nucleosynthesis and star formation, albeit with simplification and
tweaks.

In dark nucleosynthesis Hardy  _et al._ (2015); Gresham  _et al._ (2018b);
Krnjaic and Sigurdson (2015) production scenarios, the formation of dark blobs
proceeds in a way similar to how nuclei are synthesized during Big Bang
Nucleosynthesis (BBN): heavier dark bound states are produced sequentially
from lighter ones through a chain of exothermic reactions. Key ingredients in
this scenario are fermions with short-range attractive interactions analogous
to nucleons with nuclear forces that bind them together in a nucleus. Since
unlike in BBN dark nucleosynthesis may be free of bottlenecks and Coulomb
barriers, the synthesized fermion bound states may grow unthwarted to
macroscopic sizes.

Another class of dark blob formation scenarios draws inspiration from how
stars form. The formation mechanism begins with primordial dark-matter density
fluctuations growing gradually during matter domination until they become
nonlinear, followed by the nonlinear regions decoupling from the Hubble flow,
and collapsing into virialized halos. Note that the dark sector may evolve in
this way up to this point without any additional interactions beyond gravity.
With that said, long-range attractive self-interactions stronger than gravity
can enhance the tendency of the dark sector particles to clump and have an
instability similar to the gravitational collapse occur much earlier, during
radiation domination instead of matter domination Savastano  _et al._ (2019);
Domènech  _et al._ (2023). Finally, compact dark blobs would form if the
virialized halos can efficiently cool and further contract without
annihilating. The success of this scenario therefore requires the dark sector
be asymmetric and equipped with dissipative interactions Chang  _et al._
(2019); Bramante  _et al._ (2024); Flores and Kusenko (2021).

Large fermion bound states can also form in a class of scenarios involving a
cosmological first-order phase transition from a (higher-energy) false vacuum
to a (lower-energy) true vacuum Hong  _et al._ (2020); Witten (1984); Bai  _et
al._ (2019); Gross  _et al._ (2021); Asadi  _et al._ (2021a, b). While first-
order phase transition does not occur in the Standard Model Aoki  _et al._
(2006), it may occur in theories with a new higgs-like scalar Hong  _et al._
(2020) as well as in those with confining gauge theories, e.g. $SU(N)$, with
appropriate fermionic content Witten (1984); Bai  _et al._ (2019); Gross  _et
al._ (2021); Asadi  _et al._ (2021a, b). Independently of the first-order
phase transition, the theory must include pre-existing dark fermions (the
constituents of the eventual dark blobs) that are energetically favored to
remain in the false vacuum phase. These dark blob formation scenarios share
the following broad-brush chronology of events. Once the dark sector cools
below the critical temperature of the first-order phase transition, bubbles of
the true vacuum begin to nucleate sporadically and proceed to expand. As the
true-vacuum bubbles grow in both number and size, the dark fermions, unable to
enter the true vacuum bubbles, are collected in the shrinking false-vacuum
region between the true-vacuum bubbles. The true-vacuum region eventually
percolate and occupy most of the Hubble volume. By that point, the false-
vacuum region have been reduced to relatively small pockets containing trapped
dark fermions. These dark fermions pockets then cool down, become increasingly
compact, and gradually approach the compactness of the present epoch’s blobs.

There are also other dark blob formation mechanisms that do not fall into the
above categories Del Grosso  _et al._ (2024); Balkin  _et al._ (2023); Kusenko
and Shaposhnikov (1998); Amin  _et al._ (2010). Depending on the model and
parameter space, dark blobs may leave imprints on the CMB Dvorkin  _et al._
(2014); Caloni  _et al._ (2021), emit gravitational waves Diamond  _et al._
(2023a); Banks  _et al._ (2023); Bai  _et al._ (2023); Barrau  _et al._
(2024); Kuhnel  _et al._ (2020), produce cosmic-ray anti-nuclei Fedderke  _et
al._ (2024), cause damage tracks in various materials Ebadi  _et al._ (2021);
Bhoonah  _et al._ (2021); Acevedo  _et al._ (2023); Clark  _et al._ (2020),
cause dynamical heating of stars in ultrafaint dwarfs Graham and Ramani (2023,
2024), among others Jacobs  _et al._ (2015); Grabowska  _et al._ (2018); Baum
_et al._ (2022); Du  _et al._ (2023); Acevedo  _et al._ (2021); Bai and Berger
(2020); Croon and Sevillano Muñoz (2024); Mathur  _et al._ (2022); Xiao  _et
al._ (2024).

##  V Discussion

The success of the wavefront technique relies on being able to discriminate
gamma-ray burst signals from cosmic ray backgrounds based on the morphology
and time-development of their digitized Cherenkov images. To better identify
and reconstruct ultrashort gamma-ray burst events, it is useful to extend the
simulations of gamma-ray wavefront induced showers to a wider range burst
durations and individual gamma-ray energies, as well as to cases where the
incoming gamma-rays have non-monochromatic spectra and large zenith angles.
Beside that, to confirm our rough estimates of the burst signals in our
example dark matter model, it would be interesting to perform numerical
simulations of blob collisions based on the Boltzmann-Uehling-Uhlenbeck (BUU)
equation. We leave these studies as future works.

Space based gamma-ray detectors such as Fermi-LAT have the advantages of
having large FoVs and duty cycles, and are essentially free of astrophysical
backgrounds. If their pile-up time and dead time issues can be overcome, they
would be excellent detectors for ultrashort gamma-ray bursts. One approach to
effectively improve the time resolutions of space-based detectors is to use
two or more of them in conjunction to look for coincident signals. Proposals
for networks of space-based telescopes have been put forward (though not
intended for detecting ultrashort bursts) Inceoglu  _et al._ (2020); Greiner
_et al._ (2022). While each detector can register at most one photon within
its pile-up or dead time, simultaneous photon detection in multiple detectors
within a short time interval would indicate a gamma-ray burst signal with a
duration comparable or less than the time interval. A network of Fermi-LAT-
like detectors operating for a decade, for example, will allow us to extend
the search for rare gamma-ray bursts with occurrence rates $\gtrsim 0.1\text{
burst/year}$ to shorter burst durations $\Delta t_{*}\lesssim
10\,\mu\text{s}$. Moreover, a network of space-based detectors can also be
used to triangulate the distance of the burst source Ukwatta  _et al._ (2016).

In the dark matter model discussed in Section. IV.1 and Appendix. C, the
strongest burst energy per unit blob mass $\epsilon_{*}$ corresponds the
scenario where the entire kinetic energy of the colliding blobs at impact
(which may derive from the blobs’ mutual binding energy) is converted to
gamma-rays. Blob collisions can in principle create even stronger signals, up
to $\epsilon_{*}\approx 1$, in other models if we can tap into the mass energy
of the blobs, through either decay or annihilation of the blob constituents.
In these cases, one needs to explain why the decay/annihilation occurs only
when two blobs are colliding and not when they are freely floating in
isolation. Such a situation can be realized, for instance, if some particle-
antiparticle segregation mechanism in the early universe (see e.g. Goolsby-
Cole and Sorbo (2015); Shaposhnikov and Smirnov (2023)) leads to the formation
of blobs and anti-blobs which collide and annihilate into Standard Model
particles in the present epoch. Alternatively, one can consider dark matter
particles that are unstable when free but stable inside a blob, similar to how
neutrons behave inside and outside a neutron star. Along this line, a blob
collision may eject a large number of free, unstable dark matter particles
which subsequently decay to photons Bondorf  _et al._ (1980); Morawetz and
Lipavsky (2001). Yet another possibility is to have a blob collision trigger a
runaway collapse of the blob merger product that leads to many orders of
magnitude increase in its density Fedderke  _et al._ (2024). At some point,
the blob’s density become sufficiently high as to turn on some higher-
dimensional operators that convert the dark matter particles of the collapsing
blob merger product into Standard Model particles.

We have focused on ultrashort bursts of gamma-rays in this paper. It would be
interesting to consider ultrashort bursts of photons in different energy
ranges than that considered here. Sources of transient gamma-rays considered
here may also produce other types of transients, e.g. non-gamma-ray photons,
neutrinos, and gravitational waves. These possible counterparts to our gamma-
ray signals are also interesting exploratory targets. Since the burst
timescales we are interested in are too short for reorientation of detectors
for multi-messenger observations, large-FoV surveys instruments would be
needed to observe coincident events and identify these counterparts.

##  VI Conclusion

Motivated by the prospects of probing unexplored signals that are within
technological reach, we have explored methods for detecting gamma-ray
transients that may occur rarely, perhaps once a year, and last only briefly,
for less than ten microseconds. Because past and present gamma-ray detectors
have poor sensitivities to such infrequent and short bursts, there might exist
a class of astronomical, perhaps beyond the Standard Model, objects emitting
such signals that have escaped detection thus far. Hence, there are
opportunities for new discoveries once this observational window is opened.

We have proposed methods to search for ultrashort gamma-ray bursts based on an
existing detection scheme, devised by Porter and Weekes in 1978 and developed
further by the SGARFACE collaboration more than a decade ago, called the
wavefront technique. While gamma-rays from conventional transient sources
arrive sparsely enough to be detected individually through imaging their
showers with Cherenkov telescopes, bursts that are sufficiently intense and
brief can only be detected collectively since the thin wavefront of densely
arriving gamma-rays initiate air showers that are highly overlapping. The
wavefront technique seeks to identify ultrashort burst signals by matching the
unique Cherenkov images of their superimposed air showers with the results of
air shower simulations.

There are at least three ways to extend the search for ultrashort gamma-ray
bursts with current and near-future experiments. First, one can search for
wavefront signals in the archival data of existing IACTs, since such signals
are not already searched for in the standard analysis of IACT data. Similar
analyses can be done on the data that CTA will collect, with improved
sensitivity and reach compared to the existing IACTs. Beside IACTs, a near-
future facility called PANOSETI will survey large patches of the sky for
optical transients at ten-nanosecond time-resolutions and so will be sensitive
to Cherenkov lights from gamma-ray wavefronts. Further, dramatic improvements
in the sensitivities of these facilities to a wide range of burst durations
can be achieved by simply running a copy of the collected data through a new
trigger system. We have estimated the reaches of IACTs and PANOSETI if they
were to run for a decade. Their projected sensitivities to gamma-ray bursts
span many orders of magnitude in energy fluences, burst durations, and burst
rates that are thus far unprobed.

While ultrashort bursts can be searched for regardless of their sources, if we
assume the bursts arise from dark matter, several constraints specific to dark
matter become relevant. These include those arising from the more or less
known density distribution of dark matter, the Bullet Cluster, gravitational
microlensing, cosmic rays, gamma-ray observations of galaxy clusters, diffuse
gamma-ray background, and possibly more. We have examined the radius vs mass
plane of dark matter bound states (blobs) accounting for these limits and
assuming that when two blobs collide some fractions of the blobs’ mass energy
is released in a gamma-ray burst, but otherwise agnostic to the microphysics
of the blobs. We find that more than ten orders of magnitude in the masses and
radii of dark matter blobs can be probed with IACTs and PANOSETI.

We have also constructed a concrete model of large fermionic dark matter bound
states that produce an ultrashort gamma-ray burst upon a collision. In this
model, when two blobs collide the fermions of the blobs scatter and emit
copious amount of mediators via bremsstrahlung. These mediators then rapidly
decay to Standard Model particles, including gamma-rays, producing a gamma-ray
burst. While in general the burst strength is determined by the coupling
between the fermion and the mediator, among other considerations, up to
$\mathcal{O}(1)$ of the translational kinetic energy of the colliding blobs
can be radiated impulsively in this way. After accounting for model-specific
constraints, our estimates suggest that gamma-ray burst signals in this model
can be detected with the above mentioned techniques.

###### Acknowledgements.

We thank Savas Dimopoulos, Peter Graham, Simon Knapen, Alexander Kusenko,
Nicola Omodei for useful conversations. E.H.T. thanks Michael Fedderke and
Anubhav Mathur for useful discussions on a previous project. E.H.T.
acknowledges support by NSF Grant PHY-2310429 and Gordon and Betty Moore
Foundation Grant No. GBMF7946. This work was supported by the U.S. Department
of Energy (DOE), Office of Science, National Quantum Information Science
Research Centers, Superconducting Quantum Materials and Systems Center (SQMS)
under Contract No. DE-AC02-07CH11359. D.E.K. and S.R. are supported in part by
the U.S. National Science Foundation (NSF) under Grant No. PHY-1818899. S.R.
is also supported by the Simons Investigator Grant No. 827042, and by the DOE
under a QuantISED grant for MAGIS. D.E.K. is also supported by the Simons
Investigator Grant No. 144924.

##  Appendix A Sensitivities of IACTs and PANOSETI to Ultrashort Gamma-Ray
Bursts

Figure 5: The number of Cherenkov photons per unit ground area per unit solid
angle on the sky $d\mathcal{F}_{\rm Ch}/d\Omega$ produced in the showers
initiated by an infinitely-thin wavefront of gamma-rays arriving at zenith
with fluence $1\text{ ph m}^{-2}$ adopted from the simulation results of
LeBohec  _et al._ (2005); Schroedter  _et al._ (2009); Schroedter (2009),
plotted as a function of the energy $E_{*}$ of the primary gamma-rays. Also
shown for comparison is the $d\mathcal{F}_{\rm Ch}/d\Omega$ due to a single
gamma-ray photon at zenith, obtained from Ong (1998); Arbeletche and De Souza
(2021). The value of $d\mathcal{F}_{\rm Ch}/d\Omega$ shown here is what is
observed at $0.5^{\circ}$ zenith angle, which corresponds to the half width
half maximum of the angular profile of $d\mathcal{F}_{\rm Ch}/d\Omega$ due to
a gamma-ray wavefront. The dashed lines are the linear extrapolations of the
available simulation results.

A gamma-ray burst with a delta-function pulse profile originating from a
faraway source would strike the upper atmosphere in the form an infinitely-
thin, planar wavefront. Each of the arriving gamma-rays then produces an air
shower that is spread over an area of $A_{\rm pool}\sim 10^{5}\text{ m}^{2}$
at ground level, known as shower pool. For sufficiently high gamma-ray
fluence, $\mathcal{F}_{*}\gg 1\text{ ph}/A_{\rm pool}$, a large number of
showers overlap within an $A_{\rm pool}$, and the primary $\gamma$-ray photons
can thus be detected collectively through their cumulated Cherenkov yields.
This method of detecting the primary gamma-rays is called the wavefront
technique Porter and Weekes (1978); Krennrich  _et al._ (2000); LeBohec  _et
al._ (2002); Schroedter  _et al._ (2009); Lebohec  _et al._ (2003); Schroedter
(2009); LeBohec  _et al._ (2005) and is different from the typically used
technique for conventional sources that detects primary gamma-rays
individually Errando and Saito (2023); Prandini  _et al._ (2022); Bose  _et
al._ (2022). In this Appendix, we provide simple estimates for the
sensitivities of IACTs and PANOSETI, specifically to gamma-ray wavefront
events. The sensitivity will be expressed in terms of the minimum value of the
energy fluence $E_{*}\mathcal{F}_{*}$ of the incoming wavefront, i.e. the
minimum number of gamma-rays per unit ground area $\mathcal{F}_{*}$ multiplied
by the energy of individual gamma-rays $E_{*}$, for a detection.

The Cherenkov light from multi-photon-initiated showers was simulated and
shown to be detectable with IACTs Krennrich  _et al._ (2000); LeBohec  _et
al._ (2002); Schroedter  _et al._ (2009). According to the simulations, a
simultaneous wavefront of $E_{*}\gtrsim 10\mathinner{\mathrm{GeV}}$ gamma-ray
primaries with fluence $\mathcal{F}_{*}$ produces Cherenkov photons at ground
level with a typical fluence161616We choose to show the Cherenkov fluence per
solid angle of $(0.1^{\circ})^{2}$, which is significantly smaller than a
square degree, because the Cherenkov fluence varies considerably within and
does not extend beyond a square degree. Apart from that this choice is
arbitrary and is meant to reflect the typical FoV of a PMT used in IACTs.

$\displaystyle\frac{d\mathcal{F}_{\rm Ch}}{d\Omega}\approx 300\text{
ph}/\text{m}^{2}/\left(0.1^{\circ}\right)^{2}\left(\frac{E_{*}\mathcal{F}_{*}}{10\mathinner{\mathrm{GeV}}/\text{m}^{2}}\right)$
(14)

over a time spread of $\mathcal{O}(10\text{ ns})$ and an angular spread
$\theta_{\rm Ch}\sim 1^{\circ}$. Note that the above value of
$d\mathcal{F_{\rm Ch}}/d\Omega$ is half of the maximum value, which occurs at
$\approx 0.5^{\circ}$ zenith angle. The approximate linear dependence of
$d\mathcal{F}_{\rm Ch}/d\Omega$ on $E_{*}$ breaks at $E_{*}\lesssim
10\mathinner{\mathrm{GeV}}$. See Fig. 5 for the complete $E_{*}$ dependence.

Cherenkov photons from a gamma-ray burst would form a compact image of
triggered pixels on IACT or PANOSETI cameras. These images are then analyzed
in order to reconstruct the properties of the primary particles producing the
shower. The reconstruction is based on Monte Carlo simulations of multi-
photon-initiated air showers together with the responses of the detector. The
results of the simulations are compiled into a multi-dimensional table, called
look-up table, that maps the properties of the primary gamma-ray burst (photon
energy, fluence, incoming direction, and more) to the parameters of the
digitized shower images (centroid position, orientation, size, shape, and
more). Given the parameters of a shower image, one can invert the table to
determine the properties of the primary particles.

###  A.1 VERITAS

Detector properties vary considerably between telescopes. We take VERITAS as a
reference Kieda (2011); Schroedter (2009). VERITAS is an array of four optical
telescopes, where each telescope consists of a mirror with large collecting
area $A_{\rm mirror}=95\text{ m}^{2}$ to focus Cherenkov lights from secondary
shower particles toward $N_{\rm PMT}=499$ photomultiplier tubes (PMTs). Each
PMT employed by VERITAS records photons from a patch of the sky with a solid
angle of radius $\theta_{\rm PMT}=0.15^{\circ}$, corresponding to a solid
angle $\Omega_{\rm PMT}=2\pi\left(1-\cos(\theta_{\rm
PMT}/2)\right)=(0.13^{\circ})^{2}$, and converts them to electrons, with an
efficiency $\eta_{\gamma\rightarrow e}=0.34$ Kieda (2011) (this is the new
number after the PMTs were upgraded; the old value before the upgrade was
$\eta_{\gamma\rightarrow e}=0.14$ Schroedter (2009)). In our definition,
$\eta_{\gamma\rightarrow e}$ includes both the mirror reflectivity and the PMT
quantum efficiency. Assuming an integration time window just enough to catch
the entire shower duration and neglecting the angular dependence of the
collecting area of the mirror, the number of photoelectrons registered by a
PMT whose FoV is within the $\theta_{\rm shower}\sim 1^{\circ}$ shower spread
is given by

$\displaystyle N_{\text{ph}e}\sim\frac{d\mathcal{F}_{\rm
Ch}}{d\Omega}\Omega_{\rm PMT}A_{\rm mirror}\eta_{\gamma\rightarrow e}\ $

requiring this be greater than the trigger threshold of $N_{\text{ph}e}^{\rm
thr}=5$ Schroedter (2009) implies a minimum gamma-ray energy fluence for
triggering an event

$\displaystyle E_{*}\mathcal{F}_{\rm det}\approx$ $\displaystyle\,3\times
10^{-3}\mathinner{\mathrm{GeV}}/\text{m}^{2}\left(\frac{N_{\text{ph}e}^{\rm
thr}}{5}\right)^{-1}$ $\displaystyle\times\left(\frac{\Omega_{\rm
PMT}}{\left(0.13^{\circ}\right)^{2}}\right)^{-1}\left(\frac{A_{\rm
mirror}}{95\text{ m}^{2}}\right)^{-1}\left(\frac{\eta_{\gamma\rightarrow
e}}{0.34}\right)^{-1}$ (15)

for $E_{*}\gtrsim 10\mathinner{\mathrm{GeV}}$. The result for the full energy
range is shown in Fig. 1. When we set the photon to photoelectron conversion
efficiency to the old value $\eta_{\gamma\rightarrow e}=0.14$ used in
Schroedter (2009), we find that $\mathcal{F}_{\rm det}\approx 3.6\times
10^{-2}\text{ ph}/\text{m}^{2}$ at $E_{*}=1\mathinner{\mathrm{GeV}}$, which
roughly agrees with the fluence sensitivity of VERITAS to a 10 ns burst
reported in Schroedter (2009). We collect these assumed detector properties in
Table 1.

The range of primary gamma-ray energy for which IACTs are sensitive to is
often quoted to be $100\mathinner{\mathrm{GeV}}\lesssim E_{\gamma}\lesssim
10\mathinner{\mathrm{TeV}}$ Errando and Saito (2023); Prandini  _et al._
(2022); Bose  _et al._ (2022). The lower bound is set by the number of
Cherenkov photons produced in a gamma-ray induced shower that can be detected
with a $\sim 100\text{ m}^{2}$ IACT mirror, which drops rapidly below $\sim
100\mathinner{\mathrm{GeV}}$. The upper bound is set by the uncertainty in the
energy reconstruction from Cherenkov image analysis which becomes
$\mathcal{O}(1)$ or worse at $E_{\gamma}\gtrsim 10\mathinner{\mathrm{TeV}}$.
We note that these apply for single gamma-ray events but not necessarily for
gamma-ray wavefront events we are interested in. The wavefront technique has
been shown to be effective for $100\mathinner{\mathrm{MeV}}\lesssim
E_{*}\lesssim 100\mathinner{\mathrm{GeV}}$ Krennrich  _et al._ (2000); LeBohec
_et al._ (2002); Schroedter  _et al._ (2009); Lebohec  _et al._ (2003);
Schroedter (2009); LeBohec  _et al._ (2005). Further simulations will be
needed to determine the applicability of the wavefront technique at higher
$E_{*}$.

In the main text, we show results assuming energy fluence sensitivities that
could be achieved if all backgrounds can be rejected. For completeness, we
briefly discuss the most important backgrounds:

  * •

Night sky background (NSB).
The flux of NSB is approximately $d\Phi_{\rm NSB}/d\Omega\sim 10^{12}\text{
ph}/\text{m}^{2}/\text{s}/\text{sr}$ which, in terms of flux, can be a
significant background for the Cherenkov photon signals, even when considering
a 10 ns window. However, NSB occurs at random and so tends to trigger PMTs in
a sporadic way. False triggers from the NSB can be eliminated by requiring
several neighboring pixels to be triggered simultaneously.

  * •

Cosmic-ray induced showers.
The superimposed Cherenkov yields from a wavefront of gamma-rays have
distinctive angular and temporal distributions from that of cosmic ray
showers. Compared to most cosmic ray events, multi-photon-initiated showers
have images that are larger, more circular, and also smoother (superposing a
large number of gamma-ray initiated showers averages out shower fluctuations).
These differences can be exploited to reject cosmic ray backgrounds.
Additionally, the images from a wavefront of gamma-rays appear essentially the
same in different telescopes, whereas cosmic-ray induced showers have
different shapes and orientations in different detectors. By cross-correlating
events detected in different telescopes in search for coincident and similar
events, one can further reject background events. With enough telescopes, the
telescope array collectively can perhaps be made essentially background free.

  * •

Isotropic diffuse gamma-ray background (IGRB).
Fermi has measured the IGRB flux to be $\sim 0.3\text{
ph/m}^{2}/\text{s}/\text{sr}$ at $100\mathinner{\mathrm{MeV}}$ and lower at
higher photon energies Ackermann  _et al._ (2015a). The background showers
initiated by these individual gamma-rays will again yield Cherenkov images
different from that due to multi-photon-induced shower signals, allowing us to
reject them similarly to how cosmic ray events are rejected.

To reject these backgrounds, VERITAS adopts the following three-level trigger
algorithm. The Level 1 trigger operates at the single-pixel level, requiring
single PMT trigger threshold of 5 photo-electrons. The Level 2 trigger checks
for coincident triggers of multiple pixels: at least three adjacent PMTs need
to exceed the threshold within a $5\text{ ns}$ window. Finally, at Level 3 at
least two telescopes must be triggered within a 50 ns coincidence window. In
projecting the sensitivity of other telescopes, we ignore these complications
and simply assume that these backgrounds can be rejected completely using
similar trigger algorithms.

Table 1: Detector properties. Area stands for either the mirror area (VERITAS and CTA) or the Fresnel lens area (PANOSETI). | VERITAS | CTA | PANOSETI
---|---|---|---
| (1 telescope) | (1 LST) | (1 module)
$\theta_{\text{FoV}}[^{\circ}]$ | $3.5$ | $4.3$ | $9.92$
$\theta_{\rm PMT}[^{\circ}]$ | $0.15$ | $0.1$ | $0.31$
No. PMTs | 499 | 1855 | 1024
Area [m2] | 95 | 370 | 0.166
Peak $\eta$ | 0.34171717upgraded PMT from Kieda (2011) | 0.42 | 0.25

###  A.2 CTA

CTA is the next generation IACTs that will succeed the current generation
ones: VERITAS, MAGIC, and HESS Hofmann and Zanin (2023). CTA is planned to
have 3 sets of telescopes arrays: Large-Sized Telescopes (LSTs), Medium-Sized
Telescopes (MSTs), and Small-Sized Telescopes (SSTs). In projecting the
sensitivity of CTA, we adopt the properties of LSTs which have the largest
photon collection area among the three sets. LSTs consist of four telescopes,
each with a mirror of area $A_{\rm mirror}=370\text{ m}^{2}$ and $N_{\rm
PMT}=1855$ photomultiplier tubes (PMTs) at its focus. The PMTs used in the
LSTs have an FoV angular diameter of $\theta_{\rm PMT}=0.1^{\circ}$,
corresponding to a solid angle $\Omega_{\rm PMT}=2\pi\left(1-\cos(\theta_{\rm
PMT}/2)\right)=(0.089^{\circ})^{2}$, and a photon to photoelectron conversion
efficiency of $\eta_{\gamma\rightarrow e}=0.42$. These relevant properties of
CTA can be found in Table. 1

###  A.3 PANOSETI

Since its first proposal, the potential configurations of PANOSETI have been
updated frequently. For the purpose of deriving its projected sensitivity, we
adopt one of it latest proposed configurations as outlined in Maire  _et al._
(2021), specifically the one called Configuration A. In this configuration,
PANOSETI is a geodesic dome with 80 modules. These modules function as
independent telescopes pointing in different directions, which collectively
cover a large part of the sky. Unlike IACTs which use mirrors. PANOSETI
telescopes use Fresnel lenses to collect and focus light. Each PANOSETI module
consists of a $0.166\text{ m}^{2}$-area lens that focuses light onto a
$32\times 32$ pixel photon counting detector. Here, each pixel is a silicon
photomultiplier (SiPM) instead of a conventional PMT used in IACTs.181818While
SiPM suffers from dark current and noise from cross-communication between
cells, and requires more complex algorithm Ruch  _et al._ (2014), the trigger
criterion of an SiPM is similar to that of a conventional PMT Wright  _et al._
(2018). Each SiPM employed by PANOSETI records photons from a square-shaped
patch of the sky with a side length $\theta_{\rm SiPM}=0.31^{\circ}$,
corresponding to a solid angle $\Omega_{\rm SiPM}=(0.31^{\circ})^{2}$, and
converts them to photoelectrons, with an efficiency $\eta_{\gamma\rightarrow
e}=0.25$ Maire  _et al._ (2021); Korzoun  _et al._ (2023). Adding up all
$32\times 32=1024$ pixels in a telescope results in a field of view of
$\Omega_{\rm FoV}=(9.92^{\circ})^{2}$. Since there are 80 identical modules
within a dome, the total field of view of our chosen PANOSETI configuration is
$\Omega_{\rm SiPM}^{\rm tot}=2.27$ sr, excluding the redundant coverage Maire
_et al._ (2021).

We note that some of the modules in the PANOSETI dome are pointing at large
zenith angles. Simulations of the showers produced by gamma-ray wavefronts
from large zenith angles have not been reported. The shower profiles in those
cases may be qualitatively different from that at small zenith angles since
the amount of air traversed by shower particles is larger at higher zenith
angles. Dedicated simulation studies are thus needed to obtain an accurate
estimate for the sensitivity of large-FoV instruments like PANOSETI. In this
paper, we simply assume that Eq. (15) applies and substitute into it the
properties of PANOSETI as described above and outlined in Table 1.

##  Appendix B Fireball

An injection of a large amount of Standard Model particles in a small volume
of space over a short duration of time may lead to the formation of a fireball
Goodman (1986); Paczynski (1986); Piran (1999); Meszaros (2006), an optically-
thick plasma that self-accelerates to ultrarelativistic bulk radial velocities
under its own radiation pressure. In this appendix, we discuss the condition
for fireball formation and how fireball formation affects the properties of
the perceived gamma-ray bursts.

###  B.1 Thermalization

Consider a sudden and spatially-localized injection of Standard Model
particles with a total energy $\epsilon_{\rm SM}M$ and no net quantum number
of any kind. The condition for these particles to thermalize depends on the
details of the injection. Here, we discuss as an example the case where the
injection is entirely in the form of photons, such that the total energy
injected in the form of photons is $\epsilon_{\gamma,\rm inj.}M=\epsilon_{\rm
SM}M$. We further assume that the photon injection is concentrated in a
spherical volume of radius $R_{\rm inj.}$, lasts for a duration of $t_{\rm
inj.}$, and has a monochromatic spectrum with energy $E_{\rm inj.}$. These
input parameters $\epsilon_{\gamma,\rm inj.}$, $R_{\rm inj.}$, $t_{\rm inj.}$,
and $E_{\rm inj.}$ of the injection are to be distinguished from the output
parameters $\epsilon_{*}$, $\Delta t_{*}$, and $E_{*}$ of the burst used in
the main text.

In order for the injected particles to reach a chemical equilibrium, number-
changing reactions are required Mukaida and Yamada (2016); Harigaya and
Mukaida (2014); Davidson and Sarkar (2000); Diamond  _et al._ (2023b); Chang
_et al._ (2022); Garny  _et al._ (2019). As a rough estimate of the
thermalization condition, we consider only the number-changing process
$ee\rightarrow ee\gamma$, where $e$ without a superscript stands for either an
electron $e^{-}$ or a positron $e^{+}$. We assume that thermalization can be
achieved if the $ee\rightarrow ee\gamma$ process is efficient inside the
injection radius $R_{\rm inj.}$.191919Outside the injection radius, $r\gtrsim
R_{\rm inj}$, the efficiency of a process scales approximately as $n\sigma
v_{\rm rel}r\propto r^{-1}$ or faster, since the number density of the
relevant particles scales as $n\propto r^{-2}$ Fedderke  _et al._ (2024);
MacGibbon  _et al._ (2008). The cross-section $\sigma$ can be assumed
independent of $r$ for particles that have not interacted since their
production. The relative velocity of the particles participating in the
process $v_{\rm rel}$ may decrease with $r$, which is why $n\sigma v_{\rm
rel}r$ may decrease with increasing $r$ faster than $r^{-1}$. Since this
process involves electrons and/or positrons, we first need to estimate the
amount of $e^{\pm}$ produced by the injected photons. The efficiency of
$\gamma\gamma\rightarrow e^{+}e^{-}$ within the injection radius $r\lesssim
R_{\rm inj.}$ is

$\displaystyle\Gamma_{\gamma\gamma\rightarrow e^{+}e^{-}}R_{\rm inj.}\sim$
$\displaystyle~{}n_{\gamma,\rm inj.}\sigma_{\gamma\gamma\rightarrow
e^{+}e^{-}}R_{\rm inj.}$ $\displaystyle\sim$ $\displaystyle~{}4\times
10^{3}\times\text{ min}\left[1,\frac{R_{\rm inj.}}{\Delta t_{\rm
inj.}}\right]\left(\frac{\epsilon_{\rm SM}M}{10^{23}\text{ g}}\right)$
$\displaystyle~{}\times\left(\frac{E_{\rm inj.}}{1\text{
TeV}}\right)^{-3}\left(\frac{R_{\rm inj,}}{1\text{ ns}}\right)^{-2}$ (16)

In writing the second line, we assumed $\sigma_{\gamma\gamma\rightarrow
e^{+}e^{-}}\sim\pi\alpha^{2}/E_{\rm inj.}^{2}$ which applies for $E_{\rm
inj.}\gg m_{e}$ and

$\displaystyle n_{\gamma,\rm inj.}\sim\text{ min}\left[1,\frac{R_{\rm
inj.}}{\Delta t_{\rm inj.}}\right]\frac{\epsilon_{\rm SM}M}{R_{\rm
inj.}^{3}E_{\rm inj.}}$ (17)

where we included a factor $\text{min}\left[1,R_{\rm inj.}/\Delta t_{\rm
inj.}\right]$ to account for burst ($\Delta t_{\rm inj.}\lesssim R_{\rm
inj.}$) and wind ($\Delta t_{\rm inj.}\gtrsim R_{\rm inj.}$) branching of
cases Fedderke  _et al._ (2024). In the wind case, $n_{\gamma,\rm inj.}$ is
suppressed by a factor of $R_{\rm inj.}/\Delta t_{\rm inj.}\lesssim 1$ because
only a fraction $R_{\rm inj.}/\Delta t_{\rm inj.}$ of the total number of
injected photons is inside the injection region at a given time. If
$\Gamma_{\gamma\gamma\rightarrow e^{+}e^{-}}R_{\rm inj.}$ turns out to be
greater than unity then it means $e^{+}e^{-}\leftrightarrow\gamma\gamma$ will
reach a detailed balance, in which case $n_{e}\sim n_{\gamma,\rm inj.}$.
Accounting for both possibilities, namely $\Gamma_{\gamma\gamma\rightarrow
e^{+}e^{-}}R_{\rm inj.}\lesssim 1$ and $\Gamma_{\gamma\gamma\rightarrow
e^{+}e^{-}}R_{\rm inj.}\gtrsim 1$, the initial number density of electrons
produced from $\gamma\gamma\rightarrow e^{+}e^{-}$ can be written as

$\displaystyle n_{e}\sim n_{\gamma,\rm
inj.}\text{min}\left[1,\Gamma_{\gamma\gamma\rightarrow e^{+}e^{-}}R_{\rm
inj.}\right]$ (18)

These electrons and positrons can subsequently undergo $ee\rightarrow
ee\gamma$ with the rate Chang  _et al._ (2022); Garny  _et al._ (2019)

$\displaystyle\Gamma_{ee\rightarrow ee\gamma}\sim f_{\rm dead-
cone}\times\alpha\times\text{min}\left[\Gamma_{ee\rightarrow ee}^{\rm
soft},t_{\rm form}^{-1}\right]$ (19)

where $\Gamma_{ee\rightarrow ee}^{\rm soft}\sim\alpha^{2}n_{e}/\omega_{p}^{2}$
is the infrared-divergent electron collision rate regulated by the Debye
scale, $t_{\rm form}^{-1}\sim\text{max}\left[m_{e}^{2}/E_{\rm
inj.},\alpha^{1/2}\omega_{p}\right]$ is the inverse timescale for the produced
photons to separate from the electron that produces it (which accounts for the
Landau–Pomeranchuk–Migdal effect) Arnold  _et al._ (2001); Kurkela and Moore
(2011), $f_{\rm dead-cone}\sim\text{min}\left[1,(\omega_{p}/m_{e})^{2}\right]$
is a factor included to account for the dead-cone effect Dokshitzer and
Kharzeev (2001), and $\omega_{p}\sim\sqrt{\alpha n_{e}/E_{\rm inj.}}$ is the
order of magnitude of the plasma frequency of the initial, non-thermal
$e^{\pm}$ plasma Carrington  _et al._ (1999); Arnold  _et al._ (2003).

For example, for the set of parameters $M=10^{23}\text{ g}$, $R_{\rm inj.}\sim
t_{\rm inj.}\sim 1\text{ ns}$, and $E_{\rm inj.}=1\mathinner{\mathrm{TeV}}$,
we find $\Gamma_{ee\rightarrow ee\gamma}R_{\rm inj.}\sim 2\times 10^{8}$ if
$\epsilon_{\rm SM}=1$ and $\Gamma_{ee\rightarrow ee\gamma}R_{\rm inj.}\sim
5\times 10^{-5}$ if $\epsilon_{\rm SM}=10^{-6}$. These indicate that the rough
condition for fireball formation, $\Gamma_{ee\rightarrow ee\gamma}R_{\rm
inj.}\gtrsim 1$, can be satisfied in at least some part of the injection
parameter space ($\epsilon_{\gamma,\rm inj.}$, $R_{\rm inj.}$, $t_{\rm inj.}$,
$E_{\rm inj.}$). Nevertheless, as we will discuss below, the burst
parametrization and the results we obtained in the main text remain valid
whether or not a fireball forms following an injection from a blob collision.

###  B.2 Output

Once the fireball achieves thermalization and attains a semirelativistic bulk
radial velocity $v\sim 1$, its subsequent evolution is dictated by
hydrodynamics Chang  _et al._ (2022); Fiorillo  _et al._ (2024a, b) and can be
parameterized with its initial temperature $T_{0}$ and initial radius $R_{0}$.
The initial fireball temperature $T_{0}$ can be obtained from energy
conservation $\epsilon_{\rm SM}M\times\text{min}\left[1,R_{\rm inj.}/\Delta
t_{\rm inj.}\right]=g_{*,0}(\pi^{2}/15)T_{0}^{4}(4\pi R_{\rm inj.}^{3})$ while
the initial fireball radius $R_{0}$ is set by the injection radius $R_{\rm
inj.}$

$\displaystyle T_{0}\approx$
$\displaystyle~{}0.8\mathinner{\mathrm{GeV}}\times\text{min}\left[1,\frac{R_{\rm
inj.}}{\Delta t_{\rm inj.}}\right]^{1/4}\left(\frac{g_{*}}{12}\right)^{-1/4}$
$\displaystyle~{}\times\left(\frac{\epsilon_{\rm SM}M}{10^{23}\text{
g}}\right)^{1/4}\left(\frac{R_{\rm inj.}}{1\text{ ns}}\right)^{-3/4}$ (20)
$\displaystyle R_{0}\sim$ $\displaystyle~{}R_{\rm inj.}$ (21)

where $g_{*}=12$ corresponds to $\gamma,e^{\pm},\mu^{\pm},\pi^{0,\pm}$. As
long as the energy density of the fireball is dominated by radiation, the
comoving temperature $T^{\prime}$ and bulk Lorentz factor $\gamma$ of the
fireball evolves as

$\displaystyle T^{\prime}$ $\displaystyle\approx T_{0}\frac{R_{0}}{R}$ (22)
$\displaystyle\gamma$ $\displaystyle\approx\frac{R}{R_{0}}$ (23)

In what follows, primed quantities are defined in the comoving frame of the
fireball. Note that these scalings preserve the average energy per particle in
the center of mass frame of the fireball, $\gamma T^{\prime}\approx
T_{0}=\text{constant}$. Hence, when the photons are released from the
fireball, their energy spectrum will peak at $E_{*}\sim T_{0}$. The duration
of the burst will be set the thickness of the plasma shell when it becomes
optically-thin. In the absence of baryon loading Goodman (1986); Paczynski
(1986); Shemi and Piran (1990); Grimsrud and Wasserman (1998); Li and Sari
(2008), the fireball expansion preserves the plasma shell thickness $\Delta
R\approx R_{0}\sim R_{\rm inj.}$ in the burst case $\Delta t_{\rm
inj.}\lesssim R_{\rm inj.}$ or the wind length $\Delta R\approx\Delta t_{\rm
inj.}$ in the wind case $\Delta t_{\rm inj.}\gtrsim R_{\rm inj.}$. Thus, the
burst duration of the released gamma-rays will be given by $\Delta
t_{*}\sim\text{max}\left[\Delta t_{\rm inj.},R_{\rm inj.}\right]$.

We now estimate the comoving temperature at which the fireball plasma becomes
optically-thin. Consider the process $e^{+}e^{-}\leftrightarrow\gamma\gamma$
which initially keeps the $e^{\pm}$ number density at its chemical equilibrium
value

$\displaystyle\left(n_{e}^{\prime}\right)_{\rm
ch}\approx\left(\frac{m_{e}T^{\prime}}{2\pi}\right)^{3/2}e^{-m_{e}/T^{\prime}}$
(24)

where we have for simplicity assumed that the $e^{\pm}$ are non-relativistic
because the decoupling most likely occurs after Boltzmann suppression has
kicked in. The $e^{+}e^{-}$ annihilation ceases to be efficient when the
annihilation rate
$\Gamma_{e^{+}e^{-}\rightarrow\gamma\gamma}^{\prime}\sim\left(n_{e}^{\prime}\right)_{\rm
ch}(8\pi/3)(\alpha^{2}/m_{e}^{2})$ becomes slower than the comoving expansion
rate of the fireball $\gamma R^{-1}\sim R_{0}^{-1}$, i.e. when

$\displaystyle\left(\frac{m_{e}T^{\prime}}{2\pi}\right)^{3/2}e^{-m_{e}/T^{\prime}}\frac{8\pi\alpha}{3m_{e}^{2}}\sim
R_{0}^{-1}$ (25)

Solving this for $T^{\prime}$, we find that electron-positron annihilation
stops being efficient when the comoving temperature is

$\displaystyle
T^{\prime}_{e^{+}e^{-}\nrightarrow\gamma\gamma}=\mathcal{O}(10\mathinner{\mathrm{keV}})$
(26)

with some logarithmic dependence on $R_{0}$ Shemi and Piran (1990); Grimsrud
and Wasserman (1998); Li and Sari (2008).

Since $T^{\prime}_{e^{+}e^{-}\nrightarrow\gamma\gamma}\ll m_{e}$, by the time
the $n_{e}^{\prime}$ freezes out the electron-positron number density
$n_{e}^{\prime}$ is severely Boltzmann-suppressed compared to
$n_{\gamma}^{\prime}\sim T_{e^{+}e^{-}\nrightarrow\gamma\gamma}^{\prime 3}$ by
many orders of magnitude. It follows that the vast majority of the particles
released from the fireball are photons, which means
$\epsilon_{*}\approx\epsilon_{\rm SM}$. After the $e^{\pm}$ density has frozen
out, though most $e^{\pm}$ can scatter with the more abundant photons, most of
the photons do not scatter with electrons or positrons anymore since the
Thomson scattering rate from the point of view of the photons is of the same
order as $\Gamma_{e^{+}e^{-}\rightarrow\gamma\gamma}^{\prime}$. In other
words, the properties gamma-ray photons that escape the bulk of the fireball
are essentially the same as those at $e^{+}e^{-}\leftrightarrow\gamma\gamma$
decoupling.

We therefore find that in the case where the injected Standard Model particles
subsequently thermalize into a fireball, the properties of the resulting
gamma-ray burst are

$\displaystyle\epsilon_{*}$ $\displaystyle\approx\epsilon_{\rm SM}$ (27)
$\displaystyle E_{*}$ $\displaystyle\sim T_{0}$ (28) $\displaystyle\Delta
t_{*}$ $\displaystyle\sim\text{max}\left[\Delta t_{\rm inj.},R_{\rm
inj.}\right]$ (29)

with $T_{0}$ given by Eq. (20). The energy spectrum of the burst is that of a
boosted thermal distribution such that the peak energy is $E_{*}\sim T_{0}$.
Note that $E_{*}$ and $\Delta t_{*}$ can be arbitrarily large or small
depending on the details of the initial injections: $\epsilon_{\rm SM}M$,
$R_{\rm inj.}$ and $\Delta t_{\rm inj.}$.

To summarize, if a fireball forms following an injection of Standard Model
particles then the $\epsilon_{*}$ of the resulting gamma-ray burst will be
driven close to $\epsilon_{\rm SM}$, while the $E_{*}$ and $\Delta t_{*}$ can
still take up essentially any values depending on the input parameters
$\epsilon_{\rm SM}M$, $R_{\rm inj.}$, and $\Delta t_{\rm inj.}$. In other
words, fireball formation tends to increase $\epsilon_{*}$ and does not
significantly limit the possible values of $E_{*}$ and $\Delta t_{*}$. These
imply not only that fireball formation does not affect our main analysis
(which parameterizes bursts in terms of the output parameters $\epsilon_{*}$,
$E_{*}$, and $\Delta t_{*}$), but also that it provides further motivation to
assume most of the injected Standard Model energy ends up in gamma-rays.

##  Appendix C Burst-Producing Model: Details

The particle content of our model include a fermion $\chi$ with mass
$m_{\chi}$ which couples to a light scalar $\phi$ with mass $m_{\phi}$ and a
heavy scalar $S$ with mass $m_{S}\gg m_{\phi}$. The Lagrangian of the model
reads

$\displaystyle\mathcal{L}\supset$
$\displaystyle\bar{\chi}(i\not{\partial}+m_{\chi})\chi-
g_{\phi}\phi\bar{\chi}\chi+\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}$
$\displaystyle-g_{S}S\bar{\chi}\chi+\frac{1}{2}\left(\partial
S\right)^{2}-\frac{1}{2}m_{S}^{2}S^{2}-\frac{g_{S\gamma\gamma}}{4}SF^{2}$ (30)

where $F$ is the electromagnetic field strength tensor (with Lorentz indices
omitted), the fermions $\chi$ are the fundamental dark matter particles, the
light scalar field $\phi$ mediates long-range attractive forces that bind
$\chi$ particles together into blobs, and the heavy scalars $S$ are to be
emitted during a blob collision before subsequently decaying into photons (and
potentially other light Standard Model particles), producing a gamma-ray
burst.

###  C.1 Fermi Degenerate Blobs

We assume that the dark matter particles $\chi$ is completely asymmetric
Kaplan  _et al._ (2009) and exists in its entirety in the form of identical
blobs of mass $M$ and radius $R$. Each blob consists of $N_{\chi}=M/m_{\chi}$
fermions $\chi$ bound together by attractive forces mediated by the light
scalar $\phi$ whose range is longer than the size of the blob,
$m_{\phi}^{-1}\gg R$,202020Blobs bound together by short-range Yukawa forces
are also viable. The motivation behind the choice $m_{\phi}^{-1}\gtrsim R$ is
that it increases both the blob collision rate and the blob velocities prior
to an impact. As we will see, the latter enhances the amount of energy
available for radiation and ameliorates Pauli blocking as blob collision is
taking place. but short enough that constraints on dark matter self-
interactions at Galactic scales and beyond (e.g. Bogorad  _et al._ (2023)) do
not apply. The heavy scalar $S$ also mediates attractive forces between $\chi$
particles. In the parameter space we consider, where $m_{S}^{-1}\ll R$ and
$g_{S}\gg g_{\phi}$, the $S$-mediated attraction between $\chi$ particles
affect the structure of the blob negligibly but is the dominant force that
facilitates hard collisions between $\chi$ particles during a blob collisions.
Specifically, the condition that the pressure contribution from $S$-mediated
attraction is negligible compared to that from $\phi$-mediated attraction
boils down to $g_{S}^{2}\ll m_{S}^{2}/(m_{\chi}^{2}v_{F})$, which turns out to
be a subdominant constraint on $g_{S}$ in our analysis below.

We also assume that the fermions $\chi$ in each blob are degenerate, i.e.
their temperature is small compared to their Fermi energy. We expect the
internal evolution of a blob to naturally bring itself toward degeneracy.
Although the details of this process will depend on the formation mechanism of
the blob and so are beyond the scope of this work, we can make several
qualitative remarks. Depending on the temperature of the blob, the blob may
lose energy predominantly via either on-shell $S$ emissions or $S$-mediated
photon emissions. If the blob begins in a thermal-pressure supported
hydrostatic-equilibrium, which has a negative heat capacity, the energy loss
would initially heat the blob and at the same time cause it to shrink in
volume. In that case, the blob only begins to cool when it has shrunk
sufficiently that it switches from being thermal-pressure supported to
degeneracy-pressure supported, whereupon its heat capacity becomes positive.
The cooling then shuts off asymptotically as the momentum states available for
scattering are gradually restricted by Pauli exclusion to progressively
thinner vicinity of the Fermi sphere.

Furthermore, we assume for simplicity that the fermions $\chi$ are uniformly
distributed inside each blob with a mass density $\rho_{\chi}=M/(4\pi
R^{3}/3)$, a number density $n_{\chi}=\rho_{\chi}/m_{\chi}$, and a Fermi
momentum $p_{F}=(3\pi^{2}n_{\chi})^{1/3}$, which we assume to be non-
relativistic, $p_{F}\lesssim m_{\chi}$. Hydrostatic equilibrium between
$\phi$-mediated attraction and Fermi repulsion implies the following radius-
mass relation for the blobs Wise and Zhang (2014)

$\displaystyle R\approx\frac{60}{g_{\phi}^{2}M^{1/3}m_{\chi}^{2/3}}$ (31)

It follows that the Fermi velocity of the fermions $\chi$ in a blob is

$\displaystyle v_{F}=\frac{p_{F}}{m_{\chi}}\approx 5\times
10^{-2}g_{\phi}^{2}\left(\frac{M}{m_{\chi}}\right)^{2/3}$ (32)

which is a key quantity for our analysis below. Note that the escape velocity
of a $\chi$ particle out of the blob is comparable to its Fermi velocity,
$v_{\rm esc}\sim v_{F}$.

###  C.2 Sommerfeld Enhancement

Consider two blobs approaching each other with the typical virial velocity in
the Galaxy, $v_{\rm DM}\sim 10^{-3}$. Once their relative distance falls
within the range $m_{\phi}^{-1}$ of the $\phi$ mediator, $\phi$-mediated
attraction between the blobs turns on and accelerates the blobs toward each
other. At the moment of impact, the kinetic energy of the blobs, $Mv_{*}^{2}$,
is their kinetic energy at infinity, $Mv_{\rm DM}^{2}$, plus the binding
energy between the blobs

$\displaystyle Mv_{*}^{2}=Mv_{\rm
DM}^{2}+\frac{g_{\phi}^{2}N_{\chi}^{2}}{4\pi(2R)}$ (33)

The radius-mass relation (31) of the blob implies that the second term is
$\sim Mv_{F}^{2}$. Hence, the velocity of the blobs right before the impact is

$\displaystyle v_{*}\sim\text{max}\left[v_{\rm DM},v_{F}\right]$ (34)

In the interest of producing a strong burst signal, the case where $v_{F}\gg
v_{\rm DM}$ (but $v_{F}$ not ultrarelativistic) is more interesting,212121In
the opposite, $v_{F}\lesssim v_{\rm DM}$, case the scalar $S$ needs to be
sufficiently light ($m_{S}\lesssim m_{\chi}v_{\rm DM}^{2}$) in order to be
emitted, to the point that strong stellar cooling and BBN limits on $g_{S}$
become relevant. and this is what we will assume in our subsequent analysis.
Furthermore, the attractive force between the blobs leads to a Sommerfeld
enhancement of the velocity-averaged blob collision cross section

$\displaystyle\left<\sigma v\right>_{\rm col}\sim\left(\frac{v_{F}}{v_{\rm
DM}}\right)^{2}\times v_{\rm DM}R^{2}$ (35)

as long as $m_{\phi}^{-1}\gtrsim(v_{F}/v_{\rm DM})R$.

###  C.3 Weakly Dissipative Collision

We first consider in this subsection the simplest case where when two blobs
collide, the majority of the fermions $\chi$ do not participate in scatterings
and the blobs simply pass through each other to first approximation. We refer
to this case as weakly dissipative collision. The $S$ emission in this case is