# Coordinated Followup Could Have Enabled the Discovery of the GW190425
Kilonova

Ido Keinan The School of Physics and Astronomy, Tel Aviv University, Tel Aviv
69978, Israel  Iair Arcavi The School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 69978, Israel Ido Keinan<EMAIL_ADDRESS>
###### Abstract

The discovery of a kilonova associated with the GW170817 binary neutron star
merger had far-reaching implications for our understanding of several open
questions in physics and astrophysics. Unfortunately, since then, only one
robust binary neutron star merger was detected through gravitational waves,
GW190425, and no electromagnetic counterpart was identified for it. We analyze
all reported electromagnetic followup observations of GW190425 and find that
while the gravitational-wave localization uncertainty was large, most of the
90% probability region could have been covered within hours had the search
been coordinated. Instead, more than 5 days after the merger, the
uncoordinated search covered only 50% of the probability, with some areas
observed over 100 times, and some never observed. We further show that,
according to some models, it would have been possible to detect the GW190425
kilonova, despite the larger distance and higher component masses compared to
GW170817. These results emphasize the importance of coordinating followup of
gravitational-wave events, not only to avoid missing future kilonovae, but
also to discover them early. Such coordination, which is especially important
given the rarity of these events, can be achieved with the Treasure Map, a
tool developed specifically for this purpose.

††software: astropy (Astropy Collaboration et al., 2022), healpy (Górski et
al., 2005), matplotlib (Hunter, 2007), MOSFiT, numpy (van der Walt et al.,
2011), pandas, pyphot.

##  1 Introduction

Binary neutron star (BNS) mergers were predicted to produce both gravitational
waves (GWs; Clark & Eardley, 1977) and electromagnetic (EM) radiation. The
latter was theorized to be emitted in the form of two types of transients: a
short gamma-ray burst (GRB; Eichler et al., 1989), and longer-duration
emission from the radioactive decay of heavy elements synthesized via the
$r$-process in the ejecta (Li & Paczyński, 1998). This transient has been
nicknamed “macronova” (Kulkarni, 2005) or “kilonova” (Metzger et al.,
2010)111Hereafter we use the term “kilonova” rather than “macronova” because
it has been more widely adopted by the community..

All three predictions were confirmed with the first discovery of a GW signal
from a BNS merger, GW170817, detected and localized by the Advanced Large
Interferometer Gravitational-wave Observatory (LIGO; LIGO Scientific
Collaboration et al., 2015), and the Advanced Virgo detector (Acernese et al.,
2014), during their second observation run (O2), on August 17, 2017 (Abbott et
al., 2017a). The GW source was localized to a region of $\sim 30\ deg^{2}$ at
a distance of $\sim 40\ Mpc$ and had a total binary mass of
$2.74^{+0.04}_{-0.01}\ M_{\odot}$ (Abbott et al., 2017a). A short GRB,
GRB170817A, consistent with the GW localization, was detected $\sim$2 seconds
after the GW-determined merger time (Abbott et al., 2017b) by the Fermi Gamma-
ray Burst Monitor (Fermi-GBM; Meegan et al., 2009), and the International
Gamma-Ray Astrophysics Laboratory (INTEGRAL; Winkler et al., 2003). An optical
transient, AT 2017gfo, also consistent with the GW localization, was detected
11 hours later (Coulter et al., 2017a). Both EM transients were later
confirmed to be physically associated with the BNS merger (Abbott et al.,
2017).

Specifically, AT 2017gfo was consistent with kilonova predictions whereby at
the coalescence of a BNS system, $10^{-4}-10^{-2}M_{\odot}$ of neutron-rich
material are ejected at velocities of $0.1-0.3c$ in the equatorial plane due
to tidal effects (e.g. Rosswog et al., 1999; Hotokezaka et al., 2013).
Additional mass was predicted to be ejected in the polar direction from the
contact region at the time of the merger (e.g. Bauswein et al., 2013;
Hotokezaka et al., 2013). Lanthanides formed in low electron-fraction
material, such as the equatorial tidal ejecta, have a high opacity (Kasen et
al., 2013; Tanaka & Hotokezaka, 2013), making the light curve redder, fainter,
and longer lived compared to low lanthanide ejecta (Barnes & Kasen, 2013;
Grossman et al., 2014), such as that from to polar regions. The result, seen
in AT 2017gfo, is a kilonova of at least two components: one blue and short-
lived from the lanthanide-poor ejecta, and one red and longer-lived from the
lanthanide-rich ejecta. For a review of this event, see for e.g. Nakar (2020)
and Margutti & Chornock (2021).

GW170817 and its EM counterparts confirmed that BNS mergers are sites of
$r$-process nucleosynthesis (e.g. Kasen et al., 2017), and confirmed the
connection between short GRBs and BNS mergers. This event also demonstrated a
novel method to constrain the Hubble constant (Abbott et al., 2017). However,
many open questions regarding the properties of the kilonova emission remain,
such as whether the source of the early blue emission is indeed a distinct
lanthanide-poor ejecta component (other physical mechanisms were proposed,
e.g. Kasliwal et al., 2017; Piro & Kollmeier, 2018; Waxman et al., 2018). More
BNS detections and multi-wavelength observations of their EM counterparts,
especially during the first few hours (e.g. Arcavi, 2018) are needed to answer
these questions.

The second (and so far, last) robust GW detection of a BNS merger occurred at
the start of the third GW-detector observing run (O3). GW190425 was discovered
on April 25, 2019, at 08:18:05 UTC, at a distance of $159^{+69}_{-72}\ Mpc$
and with a total binary mass of $3.4^{+0.3}_{-0.1}\ M_{\odot}$ (Abbott et al.,
2020). The event was detected only by a single LIGO detector, making its
localization poorly constrained, with a $10183\ deg^{2}$ 90% uncertainty
region initially (Ligo Scientific Collaboration & VIRGO Collaboration, 2019a).
This region shrank about one day later to $7461\ deg^{2}$ (Ligo Scientific
Collaboration & VIRGO Collaboration, 2019b), but grew back to $9881\ deg^{2}$
in the final localization published in the second GW Transient Catalog
(GWTC-2; Abbott et al., 2021). The localization parameters are summarized in
Table 1, and the localization maps are shown in Figure 1.

No coincident GRB was detected by the Fermi Large Area Telescope (Fermi-LAT;
Atwood et al., 2009; Axelsson et al., 2019), the Swift Burst Alert Telescope
(Swift/BAT; Gehrels et al., 2004; Sakamoto et al., 2019) nor by the Fermi-GBM
(Fletcher et al., 2019). The INTErnational Gamma-Ray Astrophysics Laboratory
(INTEGRAL; Winkler et al., 2003) reported a detection of a low significance
signal, associated with a BNS merger (Pozanenko et al., 2020), but with no
localization information to robustly tie it to GW190425 (Savchenko et al.,
2019).

Table 1: GW190425 localization areas and distances.

Epoch | Date | 90% Area | 50% Area | Distance
---|---|---|---|---
| [UTC] | [$deg^{2}$] | [$deg^{2}$] | [Mpc]
Initial | 2019-04-25 | 10183 | 2806 | $154\pm 45$
Alert | 08:19:27 |  |  |
Update | 2019-04-26 | 7461 | 1378 | $156\pm 41$
Alert | 10:48:17 |  |  |
GWTC-2 | 2020-07-27 | 9881 | 2400 | $157\pm 43$
| 16:41:49 |  |  |

Note. — “90% (50%) Area” refers to the sky area that contains 90% (50%) of the
probability for the location of the source.

Figure 1: GW190425 localizations released in the initial (top) and update
(middle) alerts, and in the final GWTC-2 publication (bottom; Abbott et al.,
2021). White and grey contours denote the regions containing 50% and 90% of
the probability of the source location, respectively. The Swift/Bat and Fermi-
LAT footprints at the time of the merger are presented in red solid and dashed
lines, covering areas of $4596\ deg^{2}$ and $8602\ deg^{2}$, respectively.
The position of the Sun and Moon at the time of the merger are denoted in
their respective symbols.

An extensive EM followup effort to find the kilonova associated with GW190425
began shortly after the initial alert. However, no significant kilonova
candidate was identified. This was attributed to the larger distance, higher
component masses, more equal mass ratio, and to the larger localization
compared to GW170817 (e.g. Hosseinzadeh et al., 2019a; Lundquist et al., 2019;
Abbott et al., 2020; Foley et al., 2020; Gompertz et al., 2020; Kyutoku et
al., 2020; Sagués Carracedo et al., 2021; Camilletti et al., 2022; Dudi et
al., 2022; Radice et al., 2024). However, here we show that, according to some
models, the emission from the GW190425 kilonova could have been detected by
existing facilities, and that most of the localization could have been covered
within hours, had the EM followup effort been coordinated.

Coordinating EM followup of GW events is challenging. Reports of observations
and possible kilonova candidates during O2 and O3 were primarily communicated
through GRB Coordinates Network222https://gcn.nasa.gov/ (GCN; now the General
Coordinates Network) circulars. GCN circulars were designed to be human
readable, with no unified text format. As such they are not ideal when
reporting lists of coordinates, bands, and sensitivities observed. The non
machine-readable format makes it very difficult to parse, store and interpret
such information in real time.

One tool designed to address the challenge of coordinating EM followup of GW
events is the Treasure Map333https://treasuremap.space/ (Wyatt et al., 2020).
Treasure Map is a web application that collects, stores and distributes
followup reports through a machine-friendly Application Programming Interface
(API). The API allows observers to report their planned and executed
observations and to retrieve those of other facilities in order to inform
their own counterpart search plan in real time. In addition, an interactive
visualization allows observers to see their observations and those of other
facilities on a map, together with GW localization in an intuitive way. All of
the reports are also available to download at a later date for analyzing
followup efforts and non-detection statistics. Unfortunately, the Treasure Map
did not exist during GW190425, leading to very inefficient followup, as we
will show here. Our goal is to encourage the community to use tools such as
the Treasure Map during the next BNS merger, in order to increase the chances
of finding the EM counterpart, and to do so as quickly as possible. We do so
by demonstrating the impact of uncoordinated observations on missing the
counterpart to GW190425.

In this work, we analyze all reported infrared, optical and ultraviolet
followup efforts of GW190425. In Section 2 we describe our data sources and
collection process; in Section 3, we analyze the sky coverage and depth
achieved in the context of the expected kilonova emission from this event;
and, finally, in Section 4, we discuss our conclusions.

##  2 Data

In order to analyze the efficiency of the search for the GW190425 EM
counterpart we collected pointing and photometric data of infrared, optical
and ultraviolet observations obtained as followup of GW190425 up to 2019 April
30, 23:57:21 UTC, which were reported in one of the following sources
(removing duplicate reports of the same observations): The Treasure Map (1419
observations reported retroactively), 122 GCN circulars (1339 observations),
the MASTER global robotic net (MASTER-Net; Lipunov et al., 2010)
website444http://observ.pereplet.ru/ (4491 observations), the Coulter et al.
(2024) report on the One-meter Two-hemispheres (1M2H) followup (345
observations), and the Smartt et al. (2024) report of the Panoramic Survey
Telescope and Rapid Response System (PAN-STARRS) and the Asteroid Terrestrial-
impact Last Alert System (ATLAS) followup (6631 observations). In total, 14255
unique observations were collected, including time stamps, limiting
magnitudes, and bands used. Instrument footprints and fields of view (FOVs)
were taken from the Treasure Map when available, from GCN reports when
provided there, and from instrument-specific websites and publications
otherwise (all references are provided in Table 2). Some galaxy-targeted
searches provided target galaxy names rather than exact coordinates pointed
to. In such cases, we obtained the galaxy coordinates from the Galaxy List for
the Advanced Detector Era (GLADE) catalog (Dálya et al., 2018) or the SIMBAD
Astronomical Database (Wenger et al., 2000) and assumed the target galaxies
were positioned at the center of the observed footprint. A summary of the
observations and instruments used in this analysis is presented in Table 2,
and the full list of all pointings gathered is presented in Table 3.

Table 2: GW190425 EM followup observations reported in circulars, the Treasure
Map, by MASTER-Net on their website, and in Coulter et al. (2024) and Smartt
et al. (2024), out to 5.65 days after merger.

Group/Facility/Instrument | No. of | Bands | Field of View | Median $5\sigma$ Depth | Pointings Reference
---|---|---|---|---|---
| Pointings |  | [$deg^{2}$] | [AB Mag] |
Pan-STARRSa | 6209 | w,i,z | 7.07 | 21.58 | 1
MASTER-Netb | 4491 | Clear | 8.00 | 19.10 | 2
ZTFc | 596 | g,r,i | 46.73 | 21.05 | 3
ATLASd | 422 | o,c | 28.89 | 19.26 | 1
Swift/UVOTe | 392 | u | 8.0e-02 | 19.40 | 3
GOTOf | 303 | V,g | 18.85 | 19.85 | 3
J-GEM/Subarug | 154 | r | 2.0e-03 | 24.00 | 4
1M2H/Swopeh | 151 | B,V,u,g,r,i | 0.25 | 21.55 | 5
COATLIi | 128 | w | 3.3e-02 | n/a | 6
KMTNet 1.6mj | 120 | R | 4.00 | n/a | 7
KAITk | 101 | Clear | 1.2e-02 | 19.00 | 8
Harold Johnson Telescope/RATIRl | 99 | J,H,Y,Z,g,r,i | 8.1e-03 | 20.74 | 9, 10
GROWTH/LOTm | 93 | R,g,r,i | 3.6e-02 | 19.79 | 11, 12, 13
1M2H/Nickeln | 93 | r,i | 1.1e-02 | 20.24 | 5
MMTO/MMTCamo | 81 | g,i | 2.0e-03 | 21.48 | 3, 14
GWAC-F60Ap | 80 | Clear | 9.0e-02 | 18.97 | 15
CNEOSTq | 75 | R | 9.00 | 19.86 | 16
BOOTES-5r | 63 | Clear | 2.78 | 20.50 | 17
SAGUARO/CSS 1.5ms | 61 | Clear | 5.00 | 21.30 | 3
LCO 1mt | 58 | g,r,i | 0.20 | 21.30 | 18, 19, 20, 21
1M2H/Thacheru | 53 | g,r,z | 0.12 | 19.55 | 5
GRANDMA/LesMakes T60v | 52 | Clear | 17.64 | 19.20 | 22
GRANDMA/AZT$-$8w | 52 | B,R | 0.25 | 19.00 | 22
GRANDMA/Abastumani T70x | 40 | R | 0.25 | 16.29 | 22
Xinglong 60/90cmy | 35 | Clear | 2.25 | 18.00 | 23
McDonald Observatory 2.1m/CQUEANz | 30 | i | 6.1e-03 | 20.44 | 24
1M2H/ANDICAM-CCDaa | 27 | I | 1.1e-02 | 20.70 | 5
GROWTH/Kitt Peak 2.1m/KPEDab | 25 | I,g,r | 7.0e-03 | 20.40 | 25, 26, 27
YAHPTac | 23 | R | 3.4e-02 | 18.18 | 28
1M2H/ANDICAM-IRad | 21 | J,H,K | 1.5e-03 | 13.45 | 5
Liverpool Telescope/IO:Oae | 19 | g,r,i,z | 2.8e-02 | 22.24 | 29, 30
LOAO 1maf | 13 | R | 0.22 | 19.23 | 31
GROWTH-Indiaag | 10 | g,r,i | 0.49 | 18.41 | 32, 33
Konkoly 0.8mah | 10 | g,r | 9.0e-02 | 20.40 | 34
IRSF 1.4m/SIRIUSai | 7 | J,H,K | 1.6e-02 | 15.31 | 35, 36
TAROT TREaj | 7 | R | 17.64 | 17.23 | 37
Konkoly 0.6/0.9mak | 5 | Clear | 1.36 | 21.50 | 34
Lijiang 2.4mal | 5 | g,r | 2.56 | 19.04 | 38
GRANDMA/CAHA 2.2mam | 5 | U,B,V,R,I | 4.5e-02 | 22.30 | 39
DECaman | 4 | g,r,i,z | 3.35 | 23.85 | 40
MPG/GRONDao | 3 | r,i,z | 8.1e-03 | 20.00 | 41
NOT/ALFOSCaq | 3 | g,r,i | 1.1e-02 | 20.39 | 42
OSN 1.5mar | 2 | R | 6.8e-02 | 21.30 | 43, 44
ANU/SkyMapperas | 2 | i | 5.62 | 20.00 | 45
GTC/OSIRISat | 1 | r | 1.8e-02 | 22.80 | 44
VISTA/VIRCAMau | 1 | K | 2.14 | 19.55 | 46

††footnotetext: Instrument references: aPanoramic Survey Telescope and Rapid
Response System (Chambers et al., 2016) bMobile Astronomical System of
TElescope Robots (Lipunov et al., 2010); cZwicky Transient Facility (Masci et
al., 2018); dAsteroid Terrestrial-impact Last Alert System Tonry et al. (2018)
eSwift Ultraviolet/Optical Telescope (Roming et al., 2005); fGravitational-
wave Optical Transient Observer (Dyer et al., 2022); gJapanese Collaboration
for Gravitational-wave ElectroMagnetic followup (Yoshida et al., 2000;
Kashikawa et al., 2002); hOne-Meter Two-Hemispheres/Swope (Bowen & Vaughan,
1973); iCOATLI (Watson et al., 2016); jKorea Microlensing Telescope Network
(Kim et al., 2016); kKatzman Automatic Imaging Telescope (Filippenko et al.,
2001); lReionization and Transients Infrared Camera (Farah et al., 2010);
mGlobal Relay of Observatories Watching Transients Happen/Lulin One-meter
Telescope (Kasliwal et al., 2019); nNickel Direct Imaging Camera
(https://mthamilton.ucolick.org/techdocs/instruments/nickel_direct/intro/);
oMMT Observatory (Fabricant et al., 2019); pGround Wide Angle Cameras Array
(Han et al., 2021); qChinese Near Earth Object Survey Telescope; rBurst
Observer and Optical Transient Exploring System 5; sSearches After
Gravitational-waves Using ARizona’s Observatories/Catalina Sky Survey (Drake
et al., 2009); tLas Cumbres Observatory (Brown et al., 2013); uThacher ACP
Camera Swift et al. (2022); vGlobal Rapid Advanced Network Devoted to the
Multi-messenger Addicts (Antier et al., 2020); wAstronomical Reflecting
Telescope 8 (Antier et al., 2020); xAbastumani(Antier et al., 2020); yXinglong
60/90cm, Chinese Academy of Sciences; zCamera for QUasars in EArly uNiverse
(Park et al., 2012); aaA Novel Dual Imaging CAMera (ANDICAM) CCD (DePoy et
al., 2003); abKitt Peak EMCCD Demonstrator; acYAoan High Precision Telescope
(Guo et al., 2022); adANDICAM-IR (DePoy et al., 2003); aeInfrared-
Optical:Optical (Steele et al., 2004); afLemmonsan Optical Astronomy
Observatory (Han et al., 2005); agKasliwal et al. (2019); ahKonkoly 0.8m
Telescope, Research Centre for Astronomy and Earth Sciences; aiInfraRed Survey
Facility/Simultaneous InfraRed Imager for Unbiased Survey (Nagayama et al.,
2003); ajTélescope à Action Rapide pour les Objets Transitoires (Boër et al.,
1999); ak Konkoly 0.6/0.9m (https://old.konkoly.hu/konkoly/telescopes.html);
al(Wang et al., 2019); amCentro Astronómico Hispano en Andalucía (Antier et
al., 2020); anDark Energy Camera (Flaugher et al., 2015); aoMax Planck
Gesellschaft/Gamma-Ray burst Optical and Near-infrared Detector (Greiner et
al., 2008); aqNordic Optical Telescope/ALhambra Faint Object Spectrograph and
Camera (Djupvik & Andersen, 2010); arObservatorio de Sierra Nevada
(https://www.osn.iaa.csic.es/en/page/15-m-telescope); asAustralian National
University (Keller et al., 2007); atGran Telescopio Canarias/Optical System
for Imaging and low-intermediate Resolution Integrated Spectroscopy (Cobos et
al., 2002); auVisible and Infrared Survey Telescope for Astronomy/VISTA
InfraRed CAMera (Emerson et al., 2006; Dalton et al., 2006).

††footnotetext: Pointings references: 1Smartt et al. (2024)
2https://observ.pereplet.ru/ (Lipunov et al., 2010);
3https://treasuremap.space/ (Wyatt et al., 2020); 4Sasada et al. (2019);
5Coulter et al. (2024); 6Watson et al. (2019); 7Kim et al. (2019); 8Zheng et
al. (2019); 9Butler et al. (2019); 10Troja et al. (2019); 11Tan et al.
(2019a); 12Tan et al. (2019b); 13Kong (2019); 14Hosseinzadeh et al. (2019b);
15Xin et al. (2019); 16Li et al. (2019a); 17Hu et al. (2019a); 18Burke et al.
(2019); 19Hiramatsu et al. (2019a); 20Hiramatsu et al. (2019b); 21Arcavi et
al. (2019); 22Howell et al. (2019); 23Xu et al. (2019); 24Im et al. (2019);
25Ahumada et al. (2019b); 26Ahumada et al. (2019a); 27Ahumada et al. (2019c);
28Sun et al. (2019); 29Perley & Copperwheat (2019); 30Perley et al. (2019);
31Paek et al. (2019); 32Bhalerao et al. (2019); 33Waratkar et al. (2019);
34Vinko et al. (2019); 35Morihana et al. (2019a); 36Morihana et al. (2019b);
37Blazek et al. (2019); 38Li et al. (2019b); 39Kann et al. (2019); 40Bloom et
al. (2019); 41Schady et al. (2019); 42Izzo et al. (2019); 43Castro-Tirado et
al. (2019); 44Hu et al. (2019b); 45Chang et al. (2019); 46Tanvir et al.
(2019).

Note. — Some sources did not provide their observation depths. These are
marked with “n/a” in the depth column.

Table 3: List of the observations used in this work.

Facility/Instrument | MJD | R.A. [deg] | Dec. [deg] | Band |  $5\sigma$ Limiting Mag. | Source
---|---|---|---|---|---|---
ZTF | 58598.3460 | 180.0000 | 62.1500 | r | 20.46 | Treasure Map
ZTF | 58598.3465 | -168.6100 | 54.9500 | r | 20.95 | Treasure Map
ZTF | 58598.3470 | -165.0502 | 47.7500 | r | 21.05 | Treasure Map
ZTF | 58598.3474 | -167.7500 | 40.5500 | r | 20.80 | Treasure Map

Note. — This table is published in its entirety in the machine readable
format. A portion is shown here for guidance regarding its form and content.

##  3 Analysis and Results

In order to measure the efficiency of the search for the GW190425 EM
counterpart, we first analyze the observability of the GW localization region
at the time of the merger (Section 3.1), followed by the amount of the GW
localization that was covered as a function of time, in terms of the total
probability, area, and galaxy luminosity (Sections 3.2 and 3.3). We then
examine the reported non-detection limiting magnitudes and compare them to
light curves of the GW170817 kilonova and of kilonova models tuned to the
parameters of GW190425 (Section 3.4).

GW localizations are provided as Hierarchical Equal Area isoLatitude
Pixelation555https://HEALPix.sourceforge.io/ (HEALPix; Górski et al., 2005)
maps. These maps divide the sky into pixels of equal area and assign a
probability value for the location of the GW source to each pixel. Each pixel
is also assigned a distance to the source (assuming it is at that position)
and a distance error estimate. The number of pixels $N_{pixels}$ in the map is
$12\cdot(nside)^{2}$ with $nside=2^{n}$ for some integer $n\geq 0$. The
localization maps provided for GW190425 have a resolution of $nside=256$, or
$N_{pixels}=786432$. This translates into pixels with an area of $0.052\
deg^{2}$ each. We perform all of our analysis using the maps in this
resolution, as they were provided, except for the observability analysis
(Section 3.1), for which we downsample the maps to $nside=32$ (i.e.
$N_{pixels}=12288$ and a pixel area of $3.36\ deg^{2}$) for computational
ease.

###  3.1 Observability

We define that a region on the sky is “visible” at a point in time from a
certain location on Earth when the Sun is a least $12^{\circ}$ below the
horizon, the Moon separation is at least $20^{\circ}$ (the moon illumination
was 66% at the time of the GW190425 merger) and the airmass is less than 2.5.
According to this definition, of the $9881\ deg^{2}$ ($2400\ deg^{2}$) of the
90% (50%) final localization, $8182\ deg^{2}$ ($2182\ deg^{2}$), or roughly
83% (90%), were visible during the 24 hours following the merger from the
combined locations of all ground-based observatories listed in Table 2.

Next, we define the “accessibility” of each area of the sky to each instrument
as the amount of time per day that area is visible to that instrument,
$t_{visible}$, weighted by the instrument FOV (to take into account that
larger FOV instruments can cover more area simultaneously) and divided by the
pixel area of the map ($3.36\ deg^{2}$ in this case):

$t_{accessibility}=t_{visible}\left(\frac{FOV}{3.36deg^{2}}\right)$ (1)

We define an inaccessible area as one with total accessibility (i.e. summed
over all ground-based instruments considered here) $t_{accessibility}<5\ min$
(corresponding to a typical single exposure time). Using this definition, 6284
$deg^{2}$ of the sky were inaccessible due to sun and moon constraints, of
which 2098 $deg^{2}$ were in the GWTC-2 90% localization region. The rest of
the localization region, encompassing 75% of the probability, was visible with
a median total accessibility time of 41.5 hours. The total accessibility of
the 90% localization region during the first day since merger is presented in
Figures 2 and 3, and in Table 4.

Figure 2: Accessible hours during the first day after the merger summed over all ground-based observatories listed in Table 2. An accessible hour is one that is over an airmass of 2.5 and not sun or moon-constrained, normalized to the FOV of each instrument (see text for details). Grey regions indicate inaccessible areas $(t_{accessibility}<5\ min)$. Black contours are of the GWTC-2 localization. The sun and moon locations in the figure at the time of the merger. Table 4: GW190425 accessible localization areas and probabilities, and median accessibility. Epoch | 90% Area | Accessible | Accessible | Median
---|---|---|---|---
|  | 90% Area | Probability | Accessibility
| [$deg^{2}$] | [$deg^{2}$] |  | [Hr]
Initial | 10183 | 8216 | 72% | 80.0
Update | 7461 | 5833 | 76% | 32.0
GWTC-2 | 9881 | 7783 | 75% | 41.5
Figure 3: Accessible hours of the GWTC-2 90% localization during the first day
after merger. Out of the 90% localization area, 2098 $deg^{2}$ were
inaccessible ($t_{accessibility}<5\ min$, leftmost bin), leaving 75% of the
probability accessible, with a median total accessibility of 41.5 hours.

###  3.2 Area and Probability Coverage

Figure 4: Total number of infrared, optical and ultraviolet followup
observations up to 5.65 days post merger for GW190425. Grey areas are where no
observations were reported. Lines mark the 50% and 90% probability regions of
the final GWTC-2 localization. While some parts of the localization were
observed over 100 times, some were not observed at all, even though they were
accessible (Figure 2).

We count the sky area and the probability observed up to 5.65 days after the
time of the merger, both with and without overlap between observations, in
order to measure how much of the localization was covered and how much of it
was observed multiple times. Measuring total area and probability on a HEALPix
map requires counting the pixels that overlap with an observed footprint. Due
to the resolution of the map, the minimal area that can be counted per
footprint is $0.052\ deg^{2}$. However, there are footprints with a smaller
FOV (as seen in Table 2). Here we use the
healpy666https://healpy.readthedocs.io/ python package, which includes two
methods for counting (“querying”) pixels inside an observed footprint. The
“inclusive” method counts every pixel that overlaps with the footprint, while
the “exclusive” method only counts pixels that have their center fall inside
the footprint. An illustration of these two methods is presented in Figure 8
in Appendix A. We choose the exclusive method for our analysis, as it is a
more conservative accounting of the actual area and probability covered, while
the inclusive method over counts pixels, causing an overestimation of the area
and probability covered.

Figure 4 shows the total coverage of all reported followup observations up to
5.65 days post-merger outlined in Section 2. Some regions, in the northern
part of the localization, were observed more than 100 times while most of the
accessible southern part of the localization was not observed at all.

The GW localization was updated 26 hours after the initial localization was
released. At the time of the update, earlier footprints had their probability
coverage changed. We take that into account by retroactively recounting the
probability covered by all of the observations using the updated localization
from the time it was released. The area covered does not depend on the
localization map and thus, no recounting of the area covered is needed at the
time of the update. The sum of area and probability covered is performed in
two different ways, first without regard to whether observations overlap or
not, and second with overlapping regions counted only once. We compare the
probability covered to the 100% probability of the entire sky and the area
covered to the initial and updated 90% probability areas.

The probability and area covered as a function of time are shown in Figure 5.
Within 5.65 days following the merger, 11486 $deg^{2}$ were covered (once or
more). This is more than the area of the GWTC-2 90% probability region, but
not all of the area covered is in that region (Fig. 4). In fact, at 5.65 days
post-merger, only 54% of the probability (of the GWTC-2 localization) was
covered, while enough observations were made to cover over 120000 $deg^{2}$ by
that time. The 90% probability region could have theoretically been covered
within three hours from merger777Here we provide the theoretical time period
it would have taken ideally coordinated observations to cover the
localization, without simulating a detailed coordinated observing strategy in
order to provide a simple order-of-magnitude estimate of the time it would
have taken to cover the region. Instrument-specific observability constraints
could prolong this time estimate..

Figure 5: Cumulative area (top) and probability (bottom) observed during the
5.65 days following the GW190425 merger. Colored dashed lines are for all
observations (including overlapping ones) while solid lines are for unique
positions on the sky (without overlap). Horizontal grey dotted lines denote
the total area inside the 90% localization region and 100% probability
reported at the time, for the top and bottom plots, respectively, while the
grey dash-dotted lines are for the accessible area and probability (see
Section 3.1). The vertical grey dotted line denotes the time of the updated
localization.

###  3.3 Galaxy Luminosity Coverage

We define the galaxy luminosity coverage as the fraction of $B$-band
luminosity of galaxies observed out of the total $B$-band luminosity of all
galaxies with $L>L_{B}^{*}$ in the 90% localization volume within 3$\sigma$ of
the mean localization distance. Here, $L_{B}^{*}$ is the $B$-band luminosity
corresponding to $L^{*}$, the charactesitic luminosity of the Schechter
Function (Schechter, 1976). We adopt a corresponding absolute magnitude in the
$B$ band of $M_{B}^{*}=-20.7$, following Arcavi et al. (2017a). The GLADE
catalog is less complete to galaxies fainter than $L^{*}$ at large distances
(see Dálya et al., 2018, for more details).

We count only galaxies that have a $B$-band apparent magnitude and a distance
estimate in Version 2.3 of the GLADE catalog. There are a total of 47664
(34302) such galaxies in the catalog inside the initial (updated) 90%
localization.

We count galaxies that are inside observed footprints and the localization
volume, once with recounting galaxies and once without. Figure 6 shows the
galaxy luminosity coverage as a function of time. After 5.65 days from the
merger, only 52% of the $B$-band luminosity in galaxies inside the updated
localization region was observed, while 11.88 times the localization
luminosity was observed in total due to overlapping observations.

Figure 6: $B$-band Galaxy luminosity fraction observed out of all galaxies
with luminosity larger than $L_{B}^{*}$ from the GLADE 2.3 catalog that are
inside the 90% localization region and up to a distance of 3$\sigma$ of the
mean estimated distance. The initial and updated localizations are separated
by the vertical dotted grey line. The dashed line shows the total luminosity
observed including overlapping galaxies, while the solid line shows the total
luminosity observed in unique galaxies.

###  3.4 Kilonova Detection

Table 5: Parameters and distributions used for generating the ensemble of
light curves from the Nicholl et al. (2021) BNS merger kilonova model.

Parameter | Description | Type | Values
---|---|---|---
$\mathcal{M}^{a}$ | Chirp mass | Gaussian |  $\mu=1.44$, $\sigma=0.02$
$q^{a}$ | Mass ratio | uniform |  $\min=0.4$, $\max=1.0$
s | Shock density profile power law index | fixed | 1.0
$M_{TOV}$ | NS maximum theoretical mass | fixed | 2.2
$\alpha^{b}$ | Enhancement of blue ejecta | uniform |  $\min=0.1$, $\max=1.0$
$disk_{frac}^{b}$ | Fraction of disk ejected | uniform |  $\min=0.05$, $\max=0.5$
$\cos\theta^{b}$ | Cosine of viewing angle | uniform |  $\min=0.0$, $\max=1.0$
$\cos\theta_{cocoon}^{b}$ | Cosine of cocoon opening angle | uniform |  $\min=0.707$, $\max=1.0$
$\cos\theta_{open}^{b}$ | Cosine of squeezed ejecta opening angle | fixed | 0.707
$\log N_{H}^{b}$ | Hydrogen column density | uniform |  $\min=19$, $\max=23$

aafootnotetext: Based on the inferred values in Abbott et al. (2020).

bbfootnotetext: Based on the priors used in Nicholl et al. (2021).

Even if the localization would have been covered as efficiently as possible,
there is still a possibility that the EM counterpart was too faint to have
been detected. In order to check this, we compare the non-detection limiting
magnitudes reported for each of the followup observations to the light curve
of GW170817 and to light curves generated using the Kasen et al. (2017) and
Nicholl et al. (2021) kilonova models.

We use the InfraRed Survey Archive (IRSA) dust extinction service
queries888https://astroquery.readthedocs.io/en/latest/ipac/irsa/irsa_dust/irsa_dust.html
to correct all reported magnitude limits for Milky Way dust extinction at the
center of their footprint by using the Schlafly & Finkbeiner (2011) galactic
dust extinction estimates. For unfiltered observations, we assume $r$-band
extinction. We then translate all limits to absolute magnitudes using the mean
distance of the pixel that overlaps with the footprint center.

The GW170817 data is taken from the Arcavi (2018) compilation of observations
by Andreoni et al. (2017); Arcavi et al. (2017b); Coulter et al. (2017b);
Cowperthwaite et al. (2017); Díaz et al. (2017); Drout et al. (2017); Evans et
al. (2017); Hu et al. (2017); Kasliwal et al. (2017); Lipunov et al. (2017);
Pian et al. (2017); Shappee et al. (2017); Smartt et al. (2017); Tanvir et al.
(2017); Troja et al. (2017); Utsumi et al. (2017); Valenti et al. (2017) and
Pozanenko et al. (2018).

The Kasen et al. (2017) model is based on a 1D Monte Carlo simulation of the
radiative transfer of photons from the radioactive decay of $r$-process
elements through kilonova ejecta. The ejecta is modeled using three
parameters: ejecta mass $M_{ej}$, characteristic expansion velocity $v_{k}$,
and lanthanide fraction power index $X_{lan}$ (where the lanthanide fraction
is $10^{X_{lan}}$). The calculation produces a spectral energy distribution of
the emission. We use pyphot999https://mfouesneau.github.io/pyphot/index.html
to create synthetic light curves in the relevant bands from these model
spectra. We use the ejecta parameters assumed by Foley et al. (2020) for the
GW190415 kilonova: $M_{ej}=0.04M_{\odot}$, $v_{k}=0.15c$ and $X_{lan}=-2$,
which they propose assuming a BNS system with masses $1.4M_{\odot}$ and
$2.0M_{\odot}$. This lanthanide fraction results in a high opacity of
${\kappa_{ej}}\approx 10\ cm^{2}/g$ and a transient that is observable mostly
in the infrared. This led Foley et al. (2020) to conclude that the GW190425
kilonova would not have likely been observable to most facilities that
participated in the search.

Nicholl et al. (2021) provide an analytic 2D kilonova model that assumes a
three-component ejecta: one component is the “tidal ejecta” in the equatorial
region and a second is the “polar ejecta” concentrated in the polar region.
For both components, a blackbody is assumed. The third component is from GRB-
shocked material in a “cocoon” around the polar region (Nakar & Piran, 2016).
The model parameters relate directly to the GW-measured binary system chirp
mass $\mathcal{M}$ and mass ratio $q$, and to neutron star (NS) equation of
state parameters, such as the NS maximum theoretical mass $M_{TOV}$. These are
used to calculate the mass $M_{ej}$, velocity $v_{k}$, and the opacity
${\kappa_{ej}}$ of each ejecta component corresponding to three opacity
regimes: the “red” high opacity (low $Y_{e}$) tidal ejecta, the “blue” low
opacity (high $Y_{e}$) polar ejecta, and the intermediate-opacity “purple”
ejecta. The blue and red opacities are assumed to be $\kappa_{ej,blue}=0.5\
cm^{2}/g$ and $\kappa_{ej,red}=10\ cm^{2}/g$, respectively. The asymmetry of
the ejecta is parameterized by the opening angle around the poles
${\theta_{open}}$, which separates the polar ejecta and the tidal ejecta, and
the opening angle of the shocked cocoon ${\theta_{cocoon}}$. Finally, the
observer’s viewing angle ${\theta}$ and the host hydrogen column density
$N_{H}$ are taken as parameters as well. The model is implemented in the
Modular Open Source Fitter for Transients (MOSFiT; Guillochon et al. 2018),
which can be used to generate light curves in various bands. We generate an
ensemble of models, using the binary parameters $\mathcal{M}$ and $q$ from
Abbott et al. (2020). For the rest of the parameters, we sample 100
realizations of the ensemble from the prior distributions that Nicholl et al.
(2021) used for their GW170817 kilonova fit (which included three fixed
parameters, as we repeat here). The full set of parameters and the values used
are described in Table 5. These parameters produce a blue kilonova ejecta
component with $M_{ej,blue}=0.00154\pm 0.00036M_{\odot}$, and
$v_{k,blue}=0.1513\pm 0.0024c$, a red ejecta component with
$M_{ej,red}=0.0426\pm 0.0047M_{\odot}$, and $v_{k,red}=0.2586\pm 0.0024c$, and
a purple ejecta component with $M_{ej,purple}=0.0235\pm 0.00036M_{\odot}$,
$v_{k,purple}=0.0485\pm 0.0011c$, and $\kappa_{ej,purple}\simeq 5.6\
cm^{2}/g$. The total mass ejected of $M_{ej}=0.0676\pm 0.0045M_{\odot}$ has a
mean opacity of $\kappa_{ej}=7.95\pm 0.21\ cm^{2}/g$, making the light curves
bluer and more luminous overall than the Kasen et al. (2017) ones.

The comparison of reported non-detection limiting magnitudes to the GW170817
kilonova and the model light curves described above can be seen in Figure 7.

Figure 7: 5$\sigma$ non-detection limits of the GW190425 kilonova (triangles)
compared to a single kilonova model from Kasen et al. (2017) (dashed lines),
an ensemble of models from Nicholl et al. (2021) (semi-transparent solid grey
lines; see text for details), and the GW170817 kilonova (circles; see text for
data sources). The upper limit triangles are displayed fully opaque for
observations within the 50% GWTC-2 localization region, partially opaque for
observations outside the 50% GWTC-2 localization region but within the 90%
GWTC-2 localization region, and almost transparent for observations outside
the 90% GWTC-2 localization region. The limits are converted to absolute
magnitude after Milky Way extinction correction and using the distance
estimate at the HEALPix closest to the center of the pointing for each limit.
Data in the ‘clear’ band is unfiltered and is compared to models in the
$r$-band. Limits with magnitudes brighter than -20 are not shown.

We find that, had the GW190425 kilonova been similar to that of GW170817, it
should have been easily detected if the entire localization region had been
observed, even given the larger distance of GW190425 compared to that of
GW170817. In fact, limits in the $g$\- and $r$-bands are 3–4 magnitudes deeper
than what would have been required to observe a GW170817-like kilonova at
peak. However, given the different merger parameters, it is not clear whether
the GW190425 kilonova would have been as luminous as that of GW170817. Indeed,
the Kasen et al. (2017) models predict much fainter optical emission.
According to that model, the kilonova would just have barely been detected in
a few of the bands used in the search. The Nicholl et al. (2021) models, on
the other hand, predict a kilonova that would have been detected easily in the
optical bands as well. The two model predictions differ by as much as $6$
magnitudes in the optical bands.

##  4 Summary and Conclusions

We analyzed all reported ultraviolet, optical and infrared followup
observations of GW190425, the only robust BNS merger detected since GW170817,
and showed that:

  1. 1.

Other than some sun and moon constrained regions, most of the localization
region of GW190425 was observable at the time of the trigger.

  2. 2.

Enough observational resources were invested in the followup of GW190425 to
allow the accessible part of the 90% probability region to have been covered
potentially in a few hours. Instead, several regions were observed over 100
times, while others were never observed, and more than 5 days after the
merger, only 50% of the probability was covered.

  3. 3.

Even if the GW190425 kilonova were 3–4 magnitudes fainter than the GW170817
kilonova in its optical peak, it could still have been detected. According to
more conservative models without a blue emission component, the kilonova might
have been only marginally detected around peak.

These results only take into account observations undertaken by facilities
which decided to followup GW190425. It is not possible to know which
additional facilities prefered not to trigger given the large localization
region, but which might have participated in the search, had they known that
roughly half of the probability was already being covered by others.

We conclude that even for relatively distant BNS mergers with large
localizations, coordinated followup can significantly increase the chances of
discovering an EM counterpart. The lack of coordination during the followup of
GW190425 strongly contributed to the failure to identify its EM counterpart.
In addition, lack of coordination weakens our ability to constrain models of
kilonova emission, given the amount of localization probability that was not
covered, and for which no limits are available. Even for smaller localiztions,
coordination can help find the kilonova sooner. Observations on few-hour time
scales are critical for constraining emission models (e.g. Arcavi, 2018).

Real-time coordination in such a competitive field, requiring very rapid
response, is challenging. Tools like the Treasure Map have been built exactly
to overcome this challenge. We encourage the community to report their
pointings to the Treasure Map, and to use the information there to guide their
search, even if it means searching lower probability areas (instead of
contributing the 100th observation to a higher probability region).

With the lack of additional significant BNS merger discoveries since GW190425,
it is possible that the BNS merger rate is on the lower end of its large
uncertainty range, and events as nearby and well localized as GW170817 are
likely very rare. If we wish to unleash the potential of GW-EM multi-messenger
astronomy for nuclear physics, astrophysics, and cosmology, we must better
coordinate our GW followup searches.

We thank A. Tohuvavohu for assisting with querying Treasure Map. We also thank
M. Nicholl for help using the kilonova model implemented in MOSFiT. I.A.
acknowledges support from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant agreement
number 852097), and from the Pazy foundation (grant number 216312). I.K. and
I.A. acknowledge support from the Israel Science Foundation (grant number
2752/19), from the United States - Israel Binational Science Foundation (BSF;
grant number 2018166). This paper has made use of version 2.3 of the GLADE
galaxy catalog (Dálya et al., 2018), the NASA/IPAC Infrared Science Archive,
which is funded by the National Aeronautics and Space Administration and
operated by the California Institute of Technology (Schlafly & Finkbeiner,
2011) and the SIMBAD database, operated at CDS, Strasbourg, France.

## References

  * Abbott et al. (2017a) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, Phys. Rev. Lett., 119, 161101, doi: 10.1103/PhysRevLett.119.161101
  * Abbott et al. (2017b) —. 2017b, ApJ, 848, L13, doi: 10.3847/2041-8213/aa920c
  * Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, The Astrophysical Journal Letters, 848, L12, doi: 10.3847/2041-8213/aa91c9
  * Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, Nature, 551, 85, doi: 10.1038/nature24471
  * Abbott et al. (2020) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2020, The Astrophysical Journal Letters, 892, L3, doi: 10.3847/2041-8213/ab75f5
  * Abbott et al. (2021) Abbott, R., Abbott, T. D., Abraham, S., & et al. 2021, Physical Review X, 11, 021053, doi: 10.1103/PhysRevX.11.021053
  * Acernese et al. (2014) Acernese, F., Agathos, M., Agatsuma, K., et al. 2014, Classical and Quantum Gravity, 32, 024001, doi: 10.1088/0264-9381/32/2/024001
  * Ahumada et al. (2019a) Ahumada, T., Coughlin, M. W., Staats, K., Burdge, K., & Dekany, R. G. 2019a, GRB Coordinates Network, 24320, 1
  * Ahumada et al. (2019b) Ahumada, T., Coughlin, M. W., Staats, K., & Dekany, R. G. 2019b, GRB Coordinates Network, 24198, 1
  * Ahumada et al. (2019c) Ahumada, T., Coughlin, M. W., Staats, K., et al. 2019c, GRB Coordinates Network, 24343, 1
  * Andreoni et al. (2017) Andreoni, I., Ackley, K., Cooke, J., et al. 2017, PASA, 34, e069, doi: 10.1017/pasa.2017.65
  * Antier et al. (2020) Antier, S., Agayeva, S., Aivazyan, V., et al. 2020, MNRAS, 492, 3904, doi: 10.1093/mnras/stz3142
  * Arcavi (2018) Arcavi, I. 2018, The Astrophysical Journal Letters, 855, L23, doi: 10.3847/2041-8213/aab267
  * Arcavi et al. (2019) Arcavi, I., Howell, D. A., McCully, C., et al. 2019, GRB Coordinates Network, 24307, 1
  * Arcavi et al. (2017a) Arcavi, I., McCully, C., Hosseinzadeh, G., et al. 2017a, ApJ, 848, L33, doi: 10.3847/2041-8213/aa910f
  * Arcavi et al. (2017b) Arcavi, I., Hosseinzadeh, G., Howell, D. A., et al. 2017b, Nature, 551, 64, doi: 10.1038/nature24291
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071, doi: 10.1088/0004-637X/697/2/1071
  * Axelsson et al. (2019) Axelsson, M., Omodei, N., Kocevski, D., & Longo, F. 2019, GRB Coordinates Network, 24174, 1
  * Barnes & Kasen (2013) Barnes, J., & Kasen, D. 2013, The Astrophysical Journal, 775, 18, doi: 10.1088/0004-637X/775/1/18
  * Bauswein et al. (2013) Bauswein, A., Goriely, S., & Janka, H. T. 2013, ApJ, 773, 78, doi: 10.1088/0004-637X/773/1/78
  * Bhalerao et al. (2019) Bhalerao, V., Kumar, H., Karambelkar, V., et al. 2019, GRB Coordinates Network, 24201, 1
  * Blazek et al. (2019) Blazek, M., Christensen, N., Howell, E., et al. 2019, GRB Coordinates Network, 24227, 1
  * Bloom et al. (2019) Bloom, J. S., Zucker, C., Schlafly, E., et al. 2019, GRB Coordinates Network, 24337, 1
  * Boër et al. (1999) Boër, M., Bringer, M., Klotz, A., et al. 1999, A&AS, 138, 579, doi: 10.1051/aas:1999356
  * Bowen & Vaughan (1973) Bowen, I. S., & Vaughan, A. H., J. 1973, Appl. Opt., 12, 1430, doi: 10.1364/AO.12.001430
  * Brown et al. (2013) Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, PASP, 125, 1031, doi: 10.1086/673168
  * Burke et al. (2019) Burke, J., Hiramatsu, D., Arcavi, I., et al. 2019, GRB Coordinates Network, 24206, 1
  * Butler et al. (2019) Butler, N., Watson, A. M., Troja, E., et al. 2019, GRB Coordinates Network, 24238, 1
  * Camilletti et al. (2022) Camilletti, A., Chiesa, L., Ricigliano, G., et al. 2022, Monthly Notices of the Royal Astronomical Society, 516, 4760, doi: 10.1093/mnras/stac2333
  * Castro-Tirado et al. (2019) Castro-Tirado, A. J., Hu, Y. D., Li, X. Y., et al. 2019, GRB Coordinates Network, 24214, 1
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560, doi: 10.48550/arXiv.1612.05560
  * Chang et al. (2019) Chang, S. W., Wolf, C., Onken, C. A., Luvaul, L., & Scott, S. 2019, GRB Coordinates Network, 24325, 1
  * Clark & Eardley (1977) Clark, J. P. A., & Eardley, D. M. 1977, ApJ, 215, 311, doi: 10.1086/155360
  * Cobos et al. (2002) Cobos, F. J., Gonzalez, J. J., Tejada, C., Cepa, J., & Rasilla, J. L. 2002, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4832, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 249–257, doi: 10.1117/12.486467
  * Coulter et al. (2017a) Coulter, D. A., Kilpatrick, C. D., Siebert, M. R., et al. 2017a, GRB Coordinates Network, 21529, 1
  * Coulter et al. (2017b) Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017b, Science, 358, 1556, doi: 10.1126/science.aap9811
  * Coulter et al. (2024) Coulter, D. A., Kilpatrick, C. D., Jones, D. O., et al. 2024, arXiv e-prints, arXiv:2404.15441, doi: 10.48550/arXiv.2404.15441
  * Cowperthwaite et al. (2017) Cowperthwaite, P. S., Berger, E., Villar, V. A., et al. 2017, ApJ, 848, L17, doi: 10.3847/2041-8213/aa8fc7
  * Dalton et al. (2006) Dalton, G. B., Caldwell, M., Ward, A. K., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6269, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. I. S. McLean & M. Iye, 62690X, doi: 10.1117/12.670018
  * Dálya et al. (2018) Dálya, G., Galgóczi, G., Dobos, L., et al. 2018, MNRAS, 479, 2374, doi: 10.1093/mnras/sty1703
  * DePoy et al. (2003) DePoy, D. L., Atwood, B., Belville, S. R., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes, ed. M. Iye & A. F. M. Moorwood, 827–838, doi: 10.1117/12.459907
  * Díaz et al. (2017) Díaz, M. C., Macri, L. M., Garcia Lambas, D., et al. 2017, ApJ, 848, L29, doi: 10.3847/2041-8213/aa9060
  * Djupvik & Andersen (2010) Djupvik, A. A., & Andersen, J. 2010, in Astrophysics and Space Science Proceedings, Vol. 14, Highlights of Spanish Astrophysics V, 211, doi: 10.1007/978-3-642-11250-8_21
  * Drake et al. (2009) Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009, ApJ, 696, 870, doi: 10.1088/0004-637X/696/1/870
  * Drout et al. (2017) Drout, M. R., Piro, A. L., Shappee, B. J., et al. 2017, Science, 358, 1570, doi: 10.1126/science.aaq0049
  * Dudi et al. (2022) Dudi, R., Adhikari, A., Brügmann, B., et al. 2022, Phys. Rev. D, 106, 084039, doi: 10.1103/PhysRevD.106.084039
  * Dyer et al. (2022) Dyer, M. J., Ackley, K., Lyman, J., et al. 2022, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 12182, Ground-based and Airborne Telescopes IX, ed. H. K. Marshall, J. Spyromilio, & T. Usuda, 121821Y, doi: 10.1117/12.2629369
  * Eichler et al. (1989) Eichler, D., Livio, M., Piran, T., & Schramm, D. N. 1989, Nature, 340, 126, doi: 10.1038/340126a0
  * Emerson et al. (2006) Emerson, J., McPherson, A., & Sutherland, W. 2006, The Messenger, 126, 41
  * Evans et al. (2017) Evans, P. A., Cenko, S. B., Kennea, J. A., et al. 2017, Science, 358, 1565, doi: 10.1126/science.aap9580
  * Fabricant et al. (2019) Fabricant, D., Fata, R., Epps, H., et al. 2019, PASP, 131, 075004, doi: 10.1088/1538-3873/ab1d78
  * Farah et al. (2010) Farah, A., Barojas, E., Butler, N. R., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 77357Z, doi: 10.1117/12.857828
  * Filippenko et al. (2001) Filippenko, A. V., Li, W. D., Treffers, R. R., & Modjaz, M. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 246, IAU Colloq. 183: Small Telescope Astronomy on Global Scales, ed. B. Paczynski, W.-P. Chen, & C. Lemme, 121
  * Flaugher et al. (2015) Flaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, AJ, 150, 150, doi: 10.1088/0004-6256/150/5/150
  * Fletcher et al. (2019) Fletcher, C., Fermi-GBM Team, & GBM-LIGO/Virgo Group. 2019, GRB Coordinates Network, 24185, 1
  * Foley et al. (2020) Foley, R. J., Coulter, D. A., Kilpatrick, C. D., et al. 2020, Monthly Notices of the Royal Astronomical Society, 494, 190, doi: 10.1093/mnras/staa725
  * Foley et al. (2020) Foley, R. J., Coulter, D. A., Kilpatrick, C. D., et al. 2020, MNRAS, 494, 190, doi: 10.1093/mnras/staa725
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091
  * Gompertz et al. (2020) Gompertz, B. P., Cutter, R., Steeghs, D., et al. 2020, MNRAS, 497, 726, doi: 10.1093/mnras/staa1845
  * Górski et al. (2005) Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759, doi: 10.1086/427976
  * Greiner et al. (2008) Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032
  * Grossman et al. (2014) Grossman, D., Korobkin, O., Rosswog, S., & Piran, T. 2014, Monthly Notices of the Royal Astronomical Society, 439, 757, doi: 10.1093/mnras/stt2503
  * Guillochon et al. (2018) Guillochon, J., Nicholl, M., Villar, V. A., et al. 2018, ApJS, 236, 6, doi: 10.3847/1538-4365/aab761
  * Guo et al. (2022) Guo, B., Peng, Q., Chen, Y., et al. 2022, Research in Astronomy and Astrophysics, 22, 055007, doi: 10.1088/1674-4527/ac5959
  * Han et al. (2005) Han, W., Mack, P., Lee, C.-U., et al. 2005, PASJ, 57, 821, doi: 10.1093/pasj/57.5.821
  * Han et al. (2021) Han, X., Xiao, Y., Zhang, P., et al. 2021, Publications of the Astronomical Society of the Pacific, 133, 065001, doi: 10.1088/1538-3873/abfb4e
  * Hiramatsu et al. (2019a) Hiramatsu, D., Arcavi, I., Burke, J., et al. 2019a, GRB Coordinates Network, 24194, 1
  * Hiramatsu et al. (2019b) Hiramatsu, D., Pellegrino, C., Burke, J., et al. 2019b, GRB Coordinates Network, 24225, 1
  * Hosseinzadeh et al. (2019a) Hosseinzadeh, G., Cowperthwaite, P. S., Gomez, S., et al. 2019a, ApJ, 880, L4, doi: 10.3847/2041-8213/ab271c
  * Hosseinzadeh et al. (2019b) Hosseinzadeh, G., Berger, E., Blanchard, P. K., et al. 2019b, GRB Coordinates Network, 24182, 1
  * Hotokezaka et al. (2013) Hotokezaka, K., Kiuchi, K., Kyutoku, K., et al. 2013, Phys. Rev. D, 87, 024001, doi: 10.1103/PhysRevD.87.024001
  * Howell et al. (2019) Howell, E., Christensen, N., Blazek, M., et al. 2019, GRB Coordinates Network, 24256, 1
  * Hu et al. (2017) Hu, L., Wu, X., Andreoni, I., et al. 2017, Science Bulletin, 62, 1433, doi: 10.1016/j.scib.2017.10.006
  * Hu et al. (2019a) Hu, Y. D., Li, X. Y., Carrasco, I., et al. 2019a, GRB Coordinates Network, 24270, 1
  * Hu et al. (2019b) Hu, Y. D., Castro-Tirado, A. J., Li, X. Y., et al. 2019b, GRB Coordinates Network, 24324, 1
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Im et al. (2019) Im, M., Paek, G. S. H., Lim, G., et al. 2019, GRB Coordinates Network, 24183, 1
  * Izzo et al. (2019) Izzo, L., Malesani, D. B., & Carini, R. 2019, GRB Coordinates Network, 24319, 1
  * Kann et al. (2019) Kann, D. A., Thoene, C., Stachie, C., et al. 2019, GRB Coordinates Network, 24459, 1
  * Kasen et al. (2013) Kasen, D., Badnell, N. R., & Barnes, J. 2013, The Astrophysical Journal, 774, 25, doi: 10.1088/0004-637X/774/1/25
  * Kasen et al. (2017) Kasen, D., Metzger, B., Barnes, J., Quataert, E., & Ramirez-Ruiz, E. 2017, Nature, 551, 80, doi: 10.1038/nature24453
  * Kashikawa et al. (2002) Kashikawa, N., Aoki, K., Asai, R., et al. 2002, PASJ, 54, 819, doi: 10.1093/pasj/54.6.819
  * Kasliwal et al. (2017) Kasliwal, M. M., Nakar, E., Singer, L. P., et al. 2017, Science, 358, 1559, doi: 10.1126/science.aap9455
  * Kasliwal et al. (2019) Kasliwal, M. M., Cannella, C., Bagdasaryan, A., et al. 2019, PASP, 131, 038003, doi: 10.1088/1538-3873/aafbc2
  * Keller et al. (2007) Keller, S. C., Schmidt, B. P., Bessell, M. S., et al. 2007, PASA, 24, 1, doi: 10.1071/AS07001
  * Kim et al. (2019) Kim, J., Im, M., Lee, C. U., et al. 2019, GRB Coordinates Network, 24216, 1
  * Kim et al. (2016) Kim, S.-L., Lee, C.-U., Park, B.-G., et al. 2016, Journal of Korean Astronomical Society, 49, 37, doi: 10.5303/JKAS.2016.49.1.37
  * Kong (2019) Kong, A. 2019, GRB Coordinates Network, 24303, 1
  * Kulkarni (2005) Kulkarni, S. R. 2005, arXiv e-prints, astro, doi: 10.48550/arXiv.astro-ph/0510256
  * Kyutoku et al. (2020) Kyutoku, K., Fujibayashi, S., Hayashi, K., et al. 2020, ApJ, 890, L4, doi: 10.3847/2041-8213/ab6e70
  * Li et al. (2019a) Li, B., Xu, D., Zhou, X., & Lu, H. 2019a, GRB Coordinates Network, 24285, 1
  * Li & Paczyński (1998) Li, L.-X., & Paczyński, B. 1998, The Astrophysical Journal, 507, L59, doi: 10.1086/311680
  * Li et al. (2019b) Li, W. X., Chen, J. C., Xiaofeng, W. X., et al. 2019b, GRB Coordinates Network, 24267, 1
  * Ligo Scientific Collaboration & VIRGO Collaboration (2019a) Ligo Scientific Collaboration, & VIRGO Collaboration. 2019a, GRB Coordinates Network, 24168, 1
  * Ligo Scientific Collaboration & VIRGO Collaboration (2019b) —. 2019b, GRB Coordinates Network, 24228, 1
  * LIGO Scientific Collaboration et al. (2015) LIGO Scientific Collaboration, T., Aasi, J., Abbott, B. P., et al. 2015, Classical and Quantum Gravity, 32, 074001, doi: 10.1088/0264-9381/32/7/074001
  * Lipunov et al. (2010) Lipunov, V., Kornilov, V., Gorbovskoy, E., et al. 2010, Advances in Astronomy, 2010, 349171, doi: 10.1155/2010/349171
  * Lipunov et al. (2017) Lipunov, V. M., Gorbovskoy, E., Kornilov, V. G., et al. 2017, ApJ, 850, L1, doi: 10.3847/2041-8213/aa92c0
  * Lundquist et al. (2019) Lundquist, M. J., Paterson, K., Fong, W., et al. 2019, ApJ, 881, L26, doi: 10.3847/2041-8213/ab32f2
  * Margutti & Chornock (2021) Margutti, R., & Chornock, R. 2021, ARA&A, 59, 155, doi: 10.1146/annurev-astro-112420-030742
  * Masci et al. (2018) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2018, Publications of the Astronomical Society of the Pacific, 131, 018003, doi: 10.1088/1538-3873/aae8ac
  * Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, The Astrophysical Journal, 702, 791, doi: 10.1088/0004-637X/702/1/791
  * Metzger et al. (2010) Metzger, B. D., Martínez-Pinedo, G., Darbha, S., et al. 2010, Monthly Notices of the Royal Astronomical Society, 406, 2650, doi: 10.1111/j.1365-2966.2010.16864.x
  * Morihana et al. (2019a) Morihana, K., Jian, M., & Nagayama, T. 2019a, GRB Coordinates Network, 24219, 1
  * Morihana et al. (2019b) —. 2019b, GRB Coordinates Network, 24328, 1
  * Nagayama et al. (2003) Nagayama, T., Nagashima, C., Nakajima, Y., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes, ed. M. Iye & A. F. M. Moorwood, 459–464, doi: 10.1117/12.460770
  * Nakar (2020) Nakar, E. 2020, Phys. Rep., 886, 1, doi: 10.1016/j.physrep.2020.08.008
  * Nakar & Piran (2016) Nakar, E., & Piran, T. 2016, The Astrophysical Journal, 834, 28, doi: 10.3847/1538-4357/834/1/28
  * Nicholl et al. (2021) Nicholl, M., Margalit, B., Schmidt, P., et al. 2021, MNRAS, 505, 3016, doi: 10.1093/mnras/stab1523
  * Nicholl et al. (2021) Nicholl, M., Margalit, B., Schmidt, P., et al. 2021, Monthly Notices of the Royal Astronomical Society, 505, 3016, doi: 10.1093/mnras/stab1523
  * Paek et al. (2019) Paek, G. S. H., Im, M., Lim, G., et al. 2019, GRB Coordinates Network, 24188, 1
  * Park et al. (2012) Park, W.-K., Pak, S., Im, M., et al. 2012, PASP, 124, 839, doi: 10.1086/667390
  * Perley & Copperwheat (2019) Perley, D. A., & Copperwheat, C. M. 2019, GRB Coordinates Network, 24202, 1
  * Perley et al. (2019) Perley, D. A., Copperwheat, C. M., & Taggart, K. L. 2019, GRB Coordinates Network, 24314, 1
  * Pian et al. (2017) Pian, E., D’Avanzo, P., Benetti, S., et al. 2017, Nature, 551, 67, doi: 10.1038/nature24298
  * Piro & Kollmeier (2018) Piro, A. L., & Kollmeier, J. A. 2018, ApJ, 855, 103, doi: 10.3847/1538-4357/aaaab3
  * Pozanenko et al. (2020) Pozanenko, A. S., Minaev, P. Y., Grebenev, S. A., & Chelovekov, I. V. 2020, Astronomy Letters, 45, 710, doi: 10.1134/S1063773719110057
  * Pozanenko et al. (2018) Pozanenko, A. S., Barkov, M. V., Minaev, P. Y., et al. 2018, ApJ, 852, L30, doi: 10.3847/2041-8213/aaa2f6
  * Radice et al. (2024) Radice, D., Ricigliano, G., Bhattacharya, M., et al. 2024, Monthly Notices of the Royal Astronomical Society, 528, 5836, doi: 10.1093/mnras/stae400
  * Roming et al. (2005) Roming, P. W. A., Kennedy, T. E., Mason, K. O., et al. 2005, Space Sci. Rev., 120, 95, doi: 10.1007/s11214-005-5095-4
  * Rosswog et al. (1999) Rosswog, S., Liebendörfer, M., Thielemann, F. K., et al. 1999, A&A, 341, 499, doi: 10.48550/arXiv.astro-ph/9811367
  * Sagués Carracedo et al. (2021) Sagués Carracedo, A., Bulla, M., Feindt, U., & Goobar, A. 2021, MNRAS, 504, 1294, doi: 10.1093/mnras/stab872
  * Sakamoto et al. (2019) Sakamoto, T., Barthelmy, S. D., Lien, A. Y., et al. 2019, GRB Coordinates Network, 24184, 1
  * Sasada et al. (2019) Sasada, M., Akitaya, H., Nakaoka, T., et al. 2019, GRB Coordinates Network, 24192, 1
  * Savchenko et al. (2019) Savchenko, V., Ferrigno, C., Martin-Carillo, A., et al. 2019, GRB Coordinates Network, 24178, 1
  * Schady et al. (2019) Schady, P., Chen, T. W., Schweyer, T., Malesani, D. B., & Bolmer, J. 2019, GRB Coordinates Network, 24229, 1
  * Schechter (1976) Schechter, P. 1976, ApJ, 203, 297, doi: 10.1086/154079
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
  * Shappee et al. (2017) Shappee, B. J., Simon, J. D., Drout, M. R., et al. 2017, Science, 358, 1574, doi: 10.1126/science.aaq0186
  * Smartt et al. (2017) Smartt, S. J., Chen, T. W., Jerkstrand, A., et al. 2017, Nature, 551, 75, doi: 10.1038/nature24303
  * Smartt et al. (2024) Smartt, S. J., Nicholl, M., Srivastav, S., et al. 2024, MNRAS, 528, 2299, doi: 10.1093/mnras/stae100
  * Steele et al. (2004) Steele, I. A., Smith, R. J., Rees, P. C., et al. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5489, Ground-based Telescopes, ed. J. Oschmann, Jacobus M., 679–692, doi: 10.1117/12.551456
  * Sun et al. (2019) Sun, T., Chen, J., Hu, L., et al. 2019, GRB Coordinates Network, 24234, 1
  * Swift et al. (2022) Swift, J. J., Andersen, K., Arculli, T., et al. 2022, Publications of the Astronomical Society of the Pacific, 134, 035005, doi: 10.1088/1538-3873/ac5aca
  * Tan et al. (2019a) Tan, H. J., Yu, P. C., Ngeow, C. C., & Ip, W. H. 2019a, GRB Coordinates Network, 24193, 1
  * Tan et al. (2019b) Tan, H. J., Yu, P. C., Patil, A. S., et al. 2019b, GRB Coordinates Network, 24274, 1
  * Tanaka & Hotokezaka (2013) Tanaka, M., & Hotokezaka, K. 2013, The Astrophysical Journal, 775, 113, doi: 10.1088/0004-637X/775/2/113
  * Tanvir et al. (2019) Tanvir, N. R., Gonzalez-Fernandez, C., Levan, A. J., Malesani, D. B., & Evans, P. A. 2019, GRB Coordinates Network, 24334, 1
  * Tanvir et al. (2017) Tanvir, N. R., Levan, A. J., González-Fernández, C., et al. 2017, ApJ, 848, L27, doi: 10.3847/2041-8213/aa90b6
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505, doi: 10.1088/1538-3873/aabadf
  * Troja et al. (2017) Troja, E., Piro, L., van Eerten, H., et al. 2017, Nature, 551, 71, doi: 10.1038/nature24290
  * Troja et al. (2019) Troja, E., Watson, A. M., Becerra, R. L., et al. 2019, GRB Coordinates Network, 24335, 1
  * Utsumi et al. (2017) Utsumi, Y., Tanaka, M., Tominaga, N., et al. 2017, PASJ, 69, 101, doi: 10.1093/pasj/psx118
  * Valenti et al. (2017) Valenti, S., Sand, D. J., Yang, S., et al. 2017, ApJ, 848, L24, doi: 10.3847/2041-8213/aa8edf
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
  * Vinko et al. (2019) Vinko, A., Bodi, L., Kriskovics, K., Sarneczky, K., & Pal, A. 2019, GRB Coordinates Network, 24367, 1
  * Wang et al. (2019) Wang, C.-J., Bai, J.-M., Fan, Y.-F., et al. 2019, Research in Astronomy and Astrophysics, 19, 149, doi: 10.1088/1674-4527/19/10/149
  * Waratkar et al. (2019) Waratkar, G., Kumar, H., Bhalerao, V., Stanzin, J., & Anupama, G. C. 2019, GRB Coordinates Network, 24304, 1
  * Watson et al. (2016) Watson, A. M., Cuevas Cardona, S., Alvarez Nuñez, L. C., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami, 99085O, doi: 10.1117/12.2233000
  * Watson et al. (2019) Watson, A. M., Butler, N., Troja, E., et al. 2019, GRB Coordinates Network, 24239, 1
  * Waxman et al. (2018) Waxman, E., Ofek, E. O., Kushnir, D., & Gal-Yam, A. 2018, MNRAS, 481, 3423, doi: 10.1093/mnras/sty2441
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9, doi: 10.1051/aas:2000332
  * Winkler et al. (2003) Winkler, C., Courvoisier, T. J. L., Di Cocco, G., et al. 2003, A&A, 411, L1, doi: 10.1051/0004-6361:20031288
  * Wyatt et al. (2020) Wyatt, S. D., Tohuvavohu, A., Arcavi, I., et al. 2020, ApJ, 894, 127, doi: 10.3847/1538-4357/ab855e
  * Xin et al. (2019) Xin, L. P., Han, X. H., Wei, J. Y., et al. 2019, GRB Coordinates Network, 24315, 1
  * Xu et al. (2019) Xu, D., Zhu, Z. P., Yu, B. Y., et al. 2019, GRB Coordinates Network, 24190, 1
  * Yoshida et al. (2000) Yoshida, M., Shimizu, Y., Sasaki, T., et al. 2000, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4009, Advanced Telescope and Instrumentation Control Software, ed. H. Lewis, 240–249, doi: 10.1117/12.388394
  * Zheng et al. (2019) Zheng, W., Zhang, K., Vasylyev, S., & Filippenko, A. V. 2019, GRB Coordinates Network, 24179, 1

##  Appendix A Healpy Exclusive vs. Inclusive Methods

We present an illustration of the inclusive vs. the exclusive methods of
healpy pixel queries in Figure 8. The inclusive method can significantly
overestimate the area covered by a footprint with a field of view smaller than
the pixel size. The exclusive method is much more conservative, and hence we
chose it for our analysis.

Figure 8:  An illustration of the exclusive and inclusive healpy pixel query
methods. Each black square represents a pixel centered around a black dot and
the yellow rectangles are hypothetical observational footprints. Grey pixels
are the ones returned according to each method and white pixels are ignored.
The inclusive method (left) returns every pixel that intersects with the
observed footprint, while the exclusive method (right) returns only pixels
with centers included in the observed footprint. For footprints with a field
of view (FOV) much smaller than the pixel area (bottom), the inclusive method
can significantly overestimate the area covered.


# Persistence of the Pattern in the Interior of 5d Moduli Spaces

Tom Rudelius<EMAIL_ADDRESS>Department of Mathematical
Sciences, Durham University, Durham DH1 3LE United Kingdom

###### Abstract

Castellano, Ruiz, and Valenzuela recently observed a remarkable “pattern” in
infinite-distance limits of moduli spaces in quantum gravity, which relates
the field space variation of the mass of the lightest tower of particles to
the field space variation of the species scale. In this work, we show how a
version of this pattern can be proven to hold for BPS particles and strings
throughout the vector multiplet moduli space of a 5d supergravity theory, even
in regions where the particle masses and string tensions are substantially
modified relative to their asymptotic behavior in the infinite-distance
limits. This suggests that a suitably defined version of the pattern may hold
not merely in the asymptotic limits of moduli space, but in the interior as
well.

###### pacs:

##  I Introduction

Recent explorations of moduli spaces in quantum gravity have uncovered
evidence of universal structures within infinite-distance limits. As pointed
out long ago by Ooguri and Vafa [1], such limits seem to feature a tower of
light particles, whose masses decay exponentially with geodesic distance
$||\phi||$ as $m\sim\exp(-\alpha||\phi||)$. Such a tower of light particles
implies that quantum gravity becomes strongly coupled at an energy scale
$\Lambda_{\rm QG}$ (often referred to as the “species scale”) which is
parametrically smaller than the Planck scale, and which itself decays as
$\Lambda_{\rm QG}\sim\exp(-\lambda_{\rm QG}||\phi||)$.

It has long been understood that these exponential decay coefficients
$\alpha$, $\lambda_{\rm QG}$ should be order-one numbers in Planck units,111We
will set $8\pi G=M_{\textrm{Pl};d}^{2-d}=1$ throughout this letter. but
recently these values have been pinned down with greater precision. Reference
[2] presented strong evidence that the coefficient $\alpha$ of the lightest
tower in a given infinite-distance limit must satisfy
$\frac{1}{\sqrt{d-2}}\leq\alpha\leq\sqrt{\frac{d-1}{d-2}}$ in $d$ spacetime
dimensions, and several subsequent works [3, 4, 5] argued that the species
scale should decay with a coefficient
$\frac{1}{\sqrt{(d-1)(d-2)}}\leq\lambda_{\rm QG}\leq\frac{1}{\sqrt{d-2}}$.

Finally, [6, 7] provided strong evidence for not merely an inequality, but
actually an _equality_ , which relates the two decay coefficients mentioned
above. In the case of a 1-dimensional moduli space, this equality takes the
form $\alpha\lambda_{\rm QG}=1/(d-2)$. More generally, the equality takes the
form

$\frac{\vec{\nabla}m}{m}\cdot\frac{\vec{\nabla}\Lambda_{\rm QG}}{\Lambda_{\rm
QG}}=\frac{1}{d-2}\,,$ (1)

where $m$ is the characteristic mass scale of the lightest tower in the
infinite distance limit, the gradient $\vec{\nabla}$ involves the derivative
with respect to every massless modulus of the theory, and the inner product is
taken with respect to the metric $g_{ij}$ on moduli space, which is
parametrized by vacuum expectation values of the massless scalar fields.
Plugging in $m=m_{0}\exp(-\alpha\phi)$ and $\Lambda_{\rm
QG}=\Lambda_{0}\exp(-\lambda_{\rm QG}\phi)$, we see that (1) indeed reduces to
$\alpha\lambda_{\rm QG}=1/(d-2)$ for a 1-dimensional moduli space.

In [6, 7], equation (1)–which the authors referred to as the “pattern”–was
conjectured to apply in any infinite-distance limit in the moduli space.
Within such limits, the pattern may be viewed as a consequence of the Emergent
String Conjecture [8], which leads to tight constraints on the set of
exponentially light towers and the relationships between them [9]. The
Emergent String Conjecture holds that every infinite-distance limit is either
a decompactification limit or an emergent string limit in an appropriate
duality frame, which in turn implies that the lightest tower of particles is
either a tower of Kaluza-Klein modes or a tower of string oscillation modes.
In addition, it suggests that the scale $\Lambda_{\rm QG}$ should be (roughly)
identified with either a string scale or a higher-dimensional Planck scale,
which can be used to justify the pattern (1), as well as many of the other
bounds on light towers and the species scale mentioned above [9].

It is not so clear, however, how this whole story carries over to the interior
of moduli space, which is the region of greatest practical importance for
phenomenology and cosmology. As pointed out in [7], even the notion of the
“lightest tower” in moduli space is not generically well-defined outside the
asymptotic regimes of moduli space. One exception to this, however, comes from
towers of BPS particles, which remain well-defined throughout moduli space.
Similarly, although the definition of the species scale may be subtle, the
tension $T$ of a BPS string is well-defined throughout moduli space, and from
it we may define a string scale via $M_{\rm string}\equiv\sqrt{2\pi T}$.

One setting in which both BPS particles and BPS strings appear is supergravity
in five dimensions, which offers a rich and diverse landscape of quantum
gravity theories arising from M-theory compactifications on Calabi-Yau
threefolds. In this work, we explore this landscape, demonstrating that a
suitably refined version of (1) persists even into the interior of moduli
space, where other universal features of infinite-distance limits cease to be
valid.

In particular, we focus on vector multiplet moduli spaces of 5d supergravity
theories that feature emergent string limits, in which a BPS string becomes
tensionless in the asymptotic limit. In such theories, we find that the
pattern in (1) is satisfied throughout vector multiplet moduli space provided
that (a) $m$ is defined to be the mass scale associated with a tower of BPS
particles which become massless in some infinite-distance limit, and (b) the
quantum gravity scale $\Lambda_{\rm QG}$ is defined to be the string scale
$M_{\rm string}=\sqrt{2\pi T}$ of an emergent BPS string of tension $T$.
Remarkably, we find that (1) remains satisfied even when the lengths of the
individual vectors ${\vec{\nabla}m}/{m}$ and ${\vec{\nabla}\Lambda_{\rm
QG}}/{\Lambda_{\rm QG}}$ vary significantly throughout moduli space, and it
remains satisfied even after passing (via flop transitions of the Calabi-Yau
threefold) into distinct phases of the theory.

Our analysis suggests that the pattern is indeed pointing toward a universal
feature of quantum gravity, and it further suggests a possible refinement of
the pattern. Namely, our results suggest that one should set $m$ in (1) to be
the mass scale associated with the lightest tower of Kaluza-Klein modes or
string oscillation modes (which become massless in an appropriate infinite-
distance limit), and one should take the quantum gravity scale $\Lambda_{\rm
QG}$ to be a string scale or higher-dimensional Planck scale (each of which
also vanish in an infinite-distance limit). This choice for the quantum
gravity scale $\Lambda_{\rm QG}$ agrees with the species scale in asymptotic
limits, but in the interior of moduli space the two may differ in subtle but
important ways, as we discuss in §IV below. Although the full scope of this
refinement is unclear at present, it is encouraging that some features of the
asymptotic limits of moduli space seem to apply in the interior as well.
Clearly, further research is needed.

The remainder of this letter is structured as follows: in §II, we review
relevant aspects of 5d supergravity. In §III, we show how equation (1), when
properly defined, persists throughout vector multiplet moduli spaces of 5d
supergravity theories. In §IV, we end with a discussion of implications and
possible refinements of the pattern.

##  II Basics of 5d Supergravity

In this section, we review relevant aspects of 5d supergravity. For further
details, see [10].

Many features of a 5d supergravity theory are captured by its prepotential, a
cubic function of the coordinates $Y^{I}$:

$\mathcal{F}=\frac{1}{6}C_{IJK}Y^{I}Y^{J}Y^{K}\,,$ (2)

where here and henceforth repeated indices are summed. In an M-theory
compactification to 5d on a Calabi-Yau threefold $X$, indices $I,J,K$ run from
$0$ to $h^{1,1}(X)-1$, the constants $C_{IJK}$ are the triple intersection
numbers of the manifold, and the moduli $Y^{I}$ are volumes of certain two-
cycles. The vector multiplet moduli space is given by the slice
$\mathcal{F}=1$, which means that this moduli space has dimension $n\equiv
h^{1,1}(X)-1$.

At a generic point in moduli space, the gauge group is $U(1)^{h^{1,1}(X)}$,
and the gauge kinetic matrix is given by

$a_{IJ}=\mathcal{F}_{I}\mathcal{F}_{J}-\mathcal{F}_{IJ},$ (3)

with

$\mathcal{F}_{I}=\partial_{I}\mathcal{F}=\frac{1}{2}C_{IJK}Y^{J}Y^{K},~{}~{}\mathcal{F}_{IJ}=\partial_{I}\partial_{J}\mathcal{F}=C_{IJK}Y^{K}.$
(4)

The metric on moduli space is given (up to a factor of $1/2$) by the pullback
of $a_{IJ}$ to the $\mathcal{F}=1$ slice:222Note that our definition of the
moduli space metric $g_{ij}$ differs from that of [10] by a factor of $2$:
$g_{ij}^{\rm here}=\frac{1}{2}\mathfrak{g}_{ij}^{\rm there}$.

$g_{ij}=\frac{1}{2}a_{IJ}\partial_{i}Y^{I}\partial_{j}Y^{J},$ (5)

where $\partial_{i}\equiv\frac{\partial}{\partial\phi^{i}}$, and we have
parametrized $Y^{I}=Y^{I}(\phi)$ in terms of the $n$ moduli $\phi^{i}$,
$i=1,...,n$.

Using the definitions above, it is possible to prove the following useful
relation between the inverse metric $g^{ij}$ and the inverse gauge kinetic
matrix $a^{IJ}$ [10]:

$\displaystyle
a^{IJ}=\frac{1}{2}g^{ij}\partial_{i}Y^{I}\partial_{j}Y^{J}+\frac{1}{3}Y^{I}Y^{J},$
(6)

Similarly, using the simple identities
$Y^{J}\mathcal{F}_{IJ}=2\mathcal{F}_{I}$,
$Y^{I}\mathcal{F}_{I}=3\mathcal{F}=1$, and $a_{IJ}a^{JK}=\delta_{I}^{K}$, it
is easy to see that

$a_{IJ}Y^{J}=\mathcal{F}_{I}~{}~{}\Rightarrow~{}~{}a^{IJ}\mathcal{F}_{I}=Y^{J}\,.$
(7)

A particle is labeled by a vector of electric charges, $q_{I}$. The mass of
such a particle is bounded from below by the BPS bound:

$\displaystyle
m(q_{I})\geq\left(\sqrt{2}\pi\right)^{1/3}|Z|=\left(\sqrt{2}\pi\right)^{1/3}|q_{I}Y^{I}|\,,$
(8)

Particles that saturate the BPS bound are called BPS particles.

Meanwhile, a string may carry magnetic charge under the gauge fields, which is
labeled by a magnetic charge vector $\tilde{q}^{I}$. The tension of such a
charged string is bounded from below by the BPS bound for strings:

$\displaystyle
T(\tilde{q}^{I})\geq\frac{1}{2}\left(\frac{1}{\sqrt{2}\pi}\right)^{1/3}|\tilde{Z}|=\frac{1}{2}\left(\frac{1}{\sqrt{2}\pi}\right)^{1/3}|\tilde{q}^{I}\mathcal{F}_{I}|\,.$
(9)

Strings that saturate this bound are called BPS strings.

##  III The Pattern in 5d Supergravity

Suppose there exists an infinite-distance limit in vector multiplet moduli
space in which a tower of BPS particles of mass $m_{k}=km$ become light, and a
BPS string of tension $T$ becomes tensionless. The results of [2, 11, 9]
indicate that in the asymptotic limit, the tension of the BPS string and the
mass of the BPS particles will satisfy

$\displaystyle
g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}m}{m}=\frac{4}{3}\,,~{}~{}~{}~{}~{}g^{ij}\frac{\partial_{i}T}{T}\frac{\partial_{j}T}{T}=\frac{4}{3}\,,$
(10)

and

$\displaystyle
g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{i}T}{T}=\frac{2}{3}\,.$ (11)

Here, (11) matches the pattern equation (1) after setting $d=5$, $\Lambda_{\rm
QG}=M_{\rm string}\equiv\sqrt{2\pi T}$. In what follows, we will prove that
(11) remains true at _all_ points of the vector multiplet moduli space, even
points located in different phases of the theory. This is remarkable because
the relations in (10) do not, in general, hold beyond the asymptotic regimes
of moduli space.

To prove this, we must use one important fact: the tower of light particles
(of charge $kq_{I}$, for increasing $k$) and the asymptotically tensionless
BPS string (of magnetic charge $\tilde{q}^{I}$) have vanishing Dirac pairing,
i.e., $q_{I}\tilde{q}^{I}=0$. This fact follows from the discussion in §5 of
[10], which showed that all asymptotic limits of the moduli space feature a
tensionless BPS string and a tower of light BPS particles with vanishing Dirac
pairing.333At least, any such limit features a tensionless BPS string of
charge $\tilde{q}^{I}$ and a set of charges $kq_{I},k\in\mathbb{Z}$ with
$q_{I}\tilde{q}^{I}=0$ and $Z=q_{I}Y^{I}\rightarrow 0$. That these charges are
occupied by BPS particles remains to be proven in full generality, but this is
true in all known examples and follows from many well-supported quantum
gravity conjectures [10]. In contrast, boundaries of moduli space involving
light particles and tensionless strings of nontrivial Dirac pairing
($q_{I}\tilde{q}^{I}\neq 0$) lie at finite distance rather than infinite
distance.444Geometrically, such finite-distance boundaries correspond to the
collapse of a divisor to a point, whereas infinite-distance boundaries
correspond to the collapse of the entire Calabi-Yau threefold [12].

With this fact in hand, the relation (11) follows straightforwardly from the
definitions and identities of §II. To begin, we plug the formulas for the mass
(tension) of BPS particles (strings) into the left-hand side of (11):

$\displaystyle
g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}T}{T}=\frac{\partial_{i}(q_{I}Y^{I})}{q_{K}Y^{K}}\frac{\partial_{j}(\tilde{q}^{J}\mathcal{F}_{J})}{\tilde{q}^{L}\mathcal{F}_{L}}\,.$
(12)

From here, we use the definition of $\mathcal{F}_{IJ}$ in (4) to set
$\partial_{j}\mathcal{F}_{J}=\partial_{j}Y^{P}\mathcal{F}_{PJ}$, and we use
the identity of (6) to rewrite this as

$\displaystyle g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}T}{T}$
$\displaystyle=\frac{2q_{I}a^{IP}\mathcal{F}_{PJ}\tilde{q}^{J}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}-\frac{2}{3}\frac{(q_{I}Y^{I})\mathcal{F}_{PJ}Y^{P}\tilde{q}^{J}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}$
$\displaystyle=\frac{2q_{I}a^{IP}\mathcal{F}_{PJ}\tilde{q}^{J}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}-\frac{4}{3}\,,$
(13)

where in the second line we have used the fact that
$\mathcal{F}_{PJ}Y^{P}=2\mathcal{F}_{J}$. Next, by the definition of $a_{IJ}$
in (3), we have

$\displaystyle\delta^{I}_{J}=a^{IP}a_{PJ}=a^{IP}(\mathcal{F}_{P}\mathcal{F}_{J}-\mathcal{F}_{PJ})\,,$
(14)

which implies

$\displaystyle
a^{IP}\mathcal{F}_{PJ}=-\delta^{I}_{J}+a^{IP}\mathcal{F}_{P}\mathcal{F}_{J}=-\delta^{I}_{J}+Y^{I}\mathcal{F}_{J}\,.$
(15)

where in the last equation we have used the identity in (7). Plugging this
into (13), we arrive finally at

$\displaystyle g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}T}{T}$
$\displaystyle=\frac{2}{3}-\frac{2q_{I}\tilde{q}^{I}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}\,.$
(16)

Imposing the vanishing of the Dirac pairing between the BPS particles and the
BPS string, $q_{I}\tilde{q}^{I}=0$, we find the desired result (11).

##  IV Discussion

Within 5d supergravity, we have shown that _any_ BPS particle of charge
$q_{I}$ and _any_ BPS string of charge $\tilde{q}^{I}$ with vanishing Dirac
pairing $q_{I}\tilde{q}^{I}=0$ satisfy the relation (11). This implies the
pattern of (1) after setting $\Lambda_{\rm QG}=M_{\rm string}=\sqrt{2\pi T}$.
It is relatively well-established that this relation is satisfied in the
asymptotic limits of moduli space; our result shows that it extends to the
_entire_ moduli space. This agrees with observations made in [7], which showed
that the pattern is satisfied exactly (i.e., not merely at leading order in an
expansion) in certain infinite-distance limits in vector multiplet moduli
spaces of 4d supergravity theories.

Our result is nontrivial for a couple of reasons. To begin, it is not hard to
find examples in which the closely related formulas of (10) are badly violated
in the interior of moduli space, even though they are indeed satisfied in the
asymptotic limits.555Type I’ string theory in 9d is another well-studied case
where formulas analogous to those of (10) are violated–see [13] for further
details. In some cases, the length of the vector $\partial_{i}m/m$ may even
diverge in certain limits of moduli space, yet even then the relation (11)
remains valid. (The $h^{1,1}=3$ geometry of [14], studied extensively in [10],
is one example where this occurs.) Secondly, the interior of a 5d vector
multiplet moduli space can feature complicated phase transitions
(geometrically realized by flop transitions or Weyl reflections [15]), each of
which involves a modification of the triple intersection numbers $C_{IJK}$ and
hence of the prepotential $\mathcal{F}$. Remarkably, our derivation of (11) is
unaffected by such modifications, and thus the relation persists even across
the distinct phases of vector multiplet moduli space. This suggests that,
whereas many features of infinite-distance limits of moduli spaces in quantum
gravity may be consequences of the weakly coupled lamppost, the pattern of
(1), suitably interpreted, may have a wider range of applicability.

In addition, our work suggests a possible modification of the pattern of (1),
in which the species scale is replaced by a string scale or higher-dimensional
Planck scale. One possibility takes the form

$\displaystyle\frac{\vec{\nabla}m}{m}\cdot\frac{\vec{\nabla}M}{M}$
$\displaystyle=\frac{1}{d-2}\,,$
$\displaystyle\text{with}~{}m=\min(\mathcal{T})~{}$ $\displaystyle\text{for
some}~{}M\in\mathcal{S}\cup\mathcal{P},$ (17)

where $\mathcal{T}$ is the set comprised of the mass scales of all of the
Kaluza-Klein towers or string oscillator towers which become light in the
various asymptotic limits of the theory, $\mathcal{S}$ is the set of all of
the string scales $M_{\rm string}=\sqrt{2\pi T}$ of the fundamental strings
which become light in the emergent string limits of the theory, and
$\mathcal{P}$ is the set of higher-dimensional Planck scales that arise in the
decompactification limits of the theory. In many cases, including in
asymptotic limits of moduli space, $M$ can in fact be chosen to be the
smallest mass scale in the set $\mathcal{S}\cup\mathcal{P}$.

This modified version of the pattern is more or less equivalent to (1) in
infinite-distance limits of moduli space. The advantage of the modification
can be seen, for instance, in Type IIB string theory in 10 dimensions. Here,
the species scale $\Lambda_{\rm sp}$ (defined as the coefficient of the
$R^{4}$ term in the action) varies smoothly over moduli space, reaching a
critical point with $\vec{\nabla}\Lambda_{\rm sp}=0$ at $\tau=i$ [16]. As a
result, (1) breaks down in a neighborhood of this point if the quantum gravity
scale is simply defined to be the species scale, $\Lambda_{\rm
QG}=\Lambda_{\rm sp}$, and $m$ is simply defined to be the mass scale of the
lightest tower. A similar breakdown may be expected to occur near critical
points of the species scale in 5d supergravity as well, which can occur at a
boundary between two phases of the theory. (The symmetric flop geometry of
[10] is one example where this occurs.)

In contrast, the choice of the scale $M$ used in (17) has the advantage that
it, like the mass scale of the lightest tower $m$, can have discontinuous
first derivative across the boundaries between different regions of moduli
space, as different string scales/higher-dimensional Planck scales may satisfy
the bound at different points in moduli space. As a result, (17) is satisfied
at all points in Type IIB moduli space and all points in the vector multiplet
moduli space of the aforementioned symmetric flop example of [10].

The full scope of the relation proposed in (17) remains to be understood.
Indeed, it is not even clear that it holds in 5d supergravity theories when
the lightest tower consists of non-BPS particles (such as the Kaluza-Klein
modes for a decompactification limit to 11d M-theory). Nonetheless, it is
exciting to think that the pattern first observed in [6, 7] may have a broader
range of applicability than previously realized, and it is refreshing to see
that some of the conjectured features of quantum gravity may extend beyond the
infinite-distance lamppost.

Acknowledgments. We thank Muldrow Etheredge, Ben Heidenreich, Damian van de
Heisteeg, Jacob McNamara, Ignacio Ruiz, Cumrun Vafa, Irene Valenzuela, and Max
Wiesner for useful discussions, and we further thank Ben Heidenreich, Ignacio
Ruiz, and Irene Valenzuela for comments on a draft. This work was supported in
part by STFC through grant ST/T000708/1.

## References

  * Ooguri and Vafa [2007] H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl.Phys. B766, 21 (2007), arXiv:hep-th/0605264 [hep-th] .
  * Etheredge  _et al._ [2022] M. Etheredge, B. Heidenreich, S. Kaya, Y. Qiu, and T. Rudelius, Sharpening the Distance Conjecture in Diverse Dimensions, (2022), arXiv:2206.04063 [hep-th] .
  * van de Heisteeg  _et al._ [2023a] D. van de Heisteeg, C. Vafa, and M. Wiesner, Bounds on Species Scale and the Distance Conjecture, (2023a), arXiv:2303.13580 [hep-th] .
  * Calderón-Infante  _et al._ [2023] J. Calderón-Infante, A. Castellano, A. Herráez, and L. E. Ibáñez, Entropy Bounds and the Species Scale Distance Conjecture, (2023), arXiv:2306.16450 [hep-th] .
  * van de Heisteeg  _et al._ [2023b] D. van de Heisteeg, C. Vafa, M. Wiesner, and D. H. Wu, Bounds on Field Range for Slowly Varying Positive Potentials, (2023b), arXiv:2305.07701 [hep-th] .
  * Castellano  _et al._ [2023a] A. Castellano, I. Ruiz, and I. Valenzuela, A Universal Pattern in Quantum Gravity at Infinite Distance, (2023a), arXiv:2311.01501 [hep-th] .
  * Castellano  _et al._ [2023b] A. Castellano, I. Ruiz, and I. Valenzuela, Stringy Evidence for a Universal Pattern at Infinite Distance, (2023b), arXiv:2311.01536 [hep-th] .
  * Lee  _et al._ [2019] S.-J. Lee, W. Lerche, and T. Weigand, Emergent Strings from Infinite Distance Limits, (2019), arXiv:1910.01135 [hep-th] .
  * [9] M. Etheredge, B. Heidenreich, T. Rudelius, I. Ruiz, and I. Valenzuela, To appear, .
  * Alim  _et al._ [2021] M. Alim, B. Heidenreich, and T. Rudelius, The Weak Gravity Conjecture and BPS Particles, (2021), arXiv:2108.08309 [hep-th] .
  * Rudelius [2023] T. Rudelius, Gopakumar-Vafa Invariants and the Emergent String Conjecture, (2023), arXiv:2309.10024 [hep-th] .
  * Witten [1996] E. Witten, Phase transitions in M theory and F theory, Nucl. Phys. B471, 195 (1996), arXiv:hep-th/9603150 [hep-th] .
  * Etheredge  _et al._ [2023] M. Etheredge, B. Heidenreich, J. McNamara, T. Rudelius, I. Ruiz, and I. Valenzuela, Running Decompactification, Sliding Towers, and the Distance Conjecture, (2023), arXiv:2306.16440 [hep-th] .
  * Klemm  _et al._ [1997] A. Klemm, P. Mayr, and C. Vafa, BPS states of exceptional noncritical strings,  _Advanced quantum field theory. Proceedings, European Research Conference in the Memory of Claude Itzykson, La Londe les Maures, France, August 31-September 5, 1996_ , Nucl. Phys. Proc. Suppl. 58, 177 (1997), [,177(1996)], arXiv:hep-th/9607139 [hep-th] .
  * Gendler  _et al._ [2022] N. Gendler, B. Heidenreich, L. McAllister, J. Moritz, and T. Rudelius, Moduli Space Reconstruction and Weak Gravity, (2022), arXiv:2212.10573 [hep-th] .
  * van de Heisteeg  _et al._ [2023c] D. van de Heisteeg, C. Vafa, M. Wiesner, and D. H. Wu, Species Scale in Diverse Dimensions, (2023c), arXiv:2310.07213 [hep-th] .