# Observational prospects of double neutrons star mergers and their multi-
messenger afterglows: LIGO discovery power, event rates and diversity

Maryam Aghaei Abchouyeh  Department of Physics and Astronomy, Sejong
University, 98 Gunja-Dong, Gwangjin-gu, Seoul 143-747, South Korea  Maurice
H.P.M. van Putten  Corresponding author<EMAIL_ADDRESS>Department of
Physics and Astronomy, Sejong University, 98 Gunja-Dong, Gwangjin-gu, Seoul
143-747, South Korea  Lorenzo Amati  INAF-IASF Bologna, via P. Gobetti, 101,
I-40129 Bologna, Italy

###### Abstract

The double neutron star (DNS) merger event GW170817 signifies the first
multimessenger (MM) event with electromagnetic-gravitational (EM-GW)
observations. LIGO-Virgo-KAGRA observational runs O4-5 promise to detect
similar events and as yet unknown GW signals, which require confirmation in
two or more detectors with comparable performance. To this end, we quantify
duty cycles of comparable science quality of data in coincident
H1L1-observations, further to seek consistent event rates of astrophysical
transients in upcoming EM-GW surveys. Quite generally, discovery power scales
with exposure time, sensitivity, and critically depends on the percentage of
time when detectors operate at high quality. We quantify coincident duty
cycles over a time-frequency domain $W\times B$, defined by segments of
duration $W=8$s, motivated by a long-duration descending GW-chirp during
GRB170817A, and the minimum detector noise over about $B=100-250\,$Hz. This
detector yield factor satisfies $1\%-25\%$ in S5-6 and O1-O3ab, significantly
different from duty cycles of H1 and L1 individually with commensurable impact
on consistency in event rates in EM-GW surveys. Significant gain in discovery
power for signals whose frequency varies slowly in time may be derived from
improving detector yield factors by deploying time-symmetric data analysis
methods. For O4-5, these can yield improvements by factors up to
$\mathcal{O}(10^{5})$ relative to existing data and methods. Furthermore, the
diversity of MM afterglows to DNS mergers may be greatest for systems similar
to GW170817 but possibly less so for systems of substantially different mass
such as GW190425. We summarize our findings with an outlook on EM-GW surveys
during O4-5 and perspectives for next-generation GRB missions like THESEUS.

##  1 Introduction

Astronomers have since long waited for independent observation and
identi!cation of astrophysical transients using multimessengers (MMs). These
messengers include high- energy transients in the universe emitting, notably,
cosmic rays, high-energy neutrinos, electromagnetic waves (EMs) from radio to
-ray, and gravitational waves (GWs) (Bartos & Kowalski, 2017; Mészáros et al.,
2019)). GWs have negligible interaction with matter in the universe and
therefore reach us essentially uninhibited making them stand out in this
observa- tional approach. In MM-astronomy (MMA), real-time alerts from
different transient observatories are key to discovery (Dorner et al., 2021)
and there have been numerous efforts to achieve this goal during the past
decades. The concept of MMA, although anticipated for centuries, is a
relatively recent development. The first reported MM event is the observation
of neutrinos associated with Supernova 1987A a few hours before the
observation of visible light of the supernova shock breakout (Arnett et al.,
1989).

The next breakthrough was the gravitational-wave event GW170817 detected by
LIGO-Virgo Abbott et al. (2017a) with an MM afterglow in a short -ray burst
(SGRB) GRB170817A, identified by the Fermi-Gamma-Ray Burst Monitor (GBM) and
INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL; (Connaughton et
al., 2017; Goldstein et al., 2017a; Kasliwal et al., 2017; Savchenko et al.,
2017; Pozanenko et al., 2018)). Importantly, GW170817 is accompanied by a
kilonova AT2017gfo, the latter evidencing cosmic production of elements
heavier than iron in the r-process (Abbott et al., 2017b; Troja et al., 2017;
Hallinan et al., 2017; Abbott et al., 2017c; Alexander et al., 2017; Margutti
et al., 2017; Kasliwal et al., 2022). While this event is expected to be also
luminous in megaelectronvolt-neutrinos, such is not detectable from its
extragalactic distance of about 40 Mpc. The nondetection of high-energy
neutrinos may be due to seeing GRB170817A off- axis (Albert et al., 2017).
GW170817 was the first gravitational- wave event with an EM counterpart,
marking a second birth of MM-astronomy. In the same year, Ice-Cube detected an
extremely high-energy neutrino event with -ray counterpart reported by Fermi-
LAT and the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescope system
from a blazar at the same location as the neutrino source (IceCube
Collaboration, 2018).

Furthering these developments comes from a network of observatories to explore
cosmic transients at high precision including LIGO-Virgo, soon to be including
KAGRA (Akutsu et al., 2021), ENGRAVE, GRANDMA, GROWTH, and GOTO (Ackley et
al., 2020; Antier et al., 2020; Coughlin et al., 2019; Gompertz et al., 2020)
111http://www.engrave-eso.org, https://grandma.ijclab.in2p3.fr. These
observatories were collaborating during the third observational run (O3) of
LIGO-Virgo aiming to detect the counterparts of GW triggered events (Antier,
2020; Levan & Jonker, 2020). Nevertheless, no EM afterglows were detected
during this run because candidate events were relatively distant with rather
wide localization area (Bartos & Kowalski, 2017; Levan & Jonker, 2020; Antier,
2020). Needless to say, at present, the most exciting MM-events include the
double neutron star (DNS) mergers, neutron star-black hole (NS-BH) mergers,
and potentially core-collapse supernova (CC-SNe). While a few observational
candidates exist for the first two, the last one remains enigmatic for their
unknown central engine: a rotating neutron star or black hole (e.g. Soderberg
et al., 2010) with vastly distinct prospects on their potential GW emission in
both energy and frequency (cf. van Putten et al., 2019b; van Putten & Della
Valle, 2023). These key focal points motivate the deployment of a coherent and
reliable network of observatories in the development of EM-GW surveys.

Quite generally, also to set notation, MMA transient events present radiation
energies $\mathcal{E}_{i}$ across multiple energy channels in transition to a
remnant with mass-energy $E_{R}\leq E_{P}$ produced by a progenitor with total
mass-energy $E_{P}$. The $\mathcal{E}_{i}$ satisfies total energy
conservation, i.e.:

$\mathcal{E}_{i}\leq\sum_{k}\mathcal{E}_{k}\leq E_{P}-E_{R}\leq E_{P},$ (1)

where $k$ runs over the different radiation channels in GW, EM, neutrinos,
mass-ejecta and magnetic winds. Among these, GW may be dominant while
neutrino-emission and EM may take second and, respectively, third place (van
Putten & Levinson, 2003).

Already, (1) is illustrated by the very first two MM transients SN1987A and
GW170817 (Fig.6), both with leading energy emission outside the EM channel in
neutrinos (Hirata et al., 1987; Kouveliotou et al., 1993) and, respectively,
GW (Abbott et al., 2017d). Specializing to compact binary coalecences (CBCs)
of DNS and BH-NS, (1) translates to $\mathcal{E}_{i}\leq E_{CBC}-E_{R}\leq
E_{CBC}$ with radiation output distributed over a GRB, kilonova, GWs and
neutrinos (unseen for GW170817). In particular,

$\mathcal{E}_{GW}\leq E_{P}$ (2)

is a secure prior on potential GW emission following final coalescence. We
expect this to be elucidated further in joint EM-GW surveys over the planned
O4-5 observational runs.

An important consideration in MM-observations is simultaneous observing time
in a network of detectors operating at comparable sensitivity and stability,
enabling meaningful cross-correlations. The concept of simultaneous observing
time gains more significance considering the fact that no observatory is
necessarily all-sky and operating continuously at nominal sensitivity at any
given moment. Among the various energy channels, GW-observations stand out to
which the Universe is transparent, particularly distinct from EM-observations.
Presently, the LVK-network of GW-detectors consists of the LIGO detectors H1
and L1 in the US, the Virgo detector in Italy, the KAGRA detector in Japan
(Akutsu et al., 2021) and GEO600 in Germany (Abbott et al., 2008a). Shortly
after the end of the last LVK observational run, KAGRA had a joint
observational run with GEO600 (Abbott et al., 2022). The duty cycle of these
LVK detectors and efficiency of searches close to nominal detector
sensitivities are significant factors in the realization of their full
discovery power - the probability of detecting a signal from an astrophysical
transient in the Local Universe - especially so in a MM context of new GW-
signals hitherto unseen.

With a focus on potential observations of a new GW-signal during O4-5, we here
revisit and quantify the performance of the two LIGO detectors for data-
quality in joint detector operations as part of an effort to connect a LIGO
observations to existing EM-surveys. A first-time observation of a new signal,
possibly at the threshold of detection of each individual detector, requires
confirmation by two (or more) detectors. Quite generally, such requires two
detectors operating at comparable sensitivities. For this reason, the present
study considers H1 and L1, leaving Virgo to be included a future study.

We begin this study with a revisit of duty cycles of individual and joint
detector operations derived from the most sensitive frequency range for LIGO
across the bandwidth

$\displaystyle B=100-250\,\mbox{Hz}$ (3)

for each run S5-6 and O1-O3ab. $B$ is relevant in defining the minimum in
detector noise amplitude, while being largely free of lines. For an equal mass
DNS merger, it captures about one-fourth of the total energy output up to the
characteristic frequency of about 650 Hz before final coalescence (The LIGO
Scientific Collaboration et al., 2017). Hence, this frequency range is
representative for the horizon distance of DNS mergers.

By universality of astrophysical black holes across different mass scales, it
is expected to be relevant to the prospective central engines of GRBs and
energetic CC-SNe of type Ib/c alike, the latter broadly representing the
progenitors of long GRBs (cf. Fryer et al., 2007; Langer et al., 2010).
Scaling of the observed descending GW-chirp in frequency $f_{GW}$ by mass $M$
of the Kerr black hole powering GRB170817A (van Putten & Della Valle, 2023)
gives

$\displaystyle 200\,\mbox{Hz}\left(\frac{2.5M_{\odot}}{M}\right)\lesssim
f_{GW}\lesssim 700\,\mbox{Hz}\left(\frac{2.5M_{\odot}}{M}\right).$ (4)

The scaling relation (4) defines an outlook for similarly powered CC-SNe, upon
estimating the mass $M_{He}$ of the helium core prior to core-collapse and the
mass-fraction $\xi$ thereof retained in collapse to a rotating black hole.
From a He-core mass-function $dN/dM_{He}\propto e^{-M/M_{s}}$ over
$M_{c}<M_{He}<50M_{\odot}$ with cut-off $M_{c}\simeq 6M_{\odot}$ (Kochanek,
2015), we infer $\bar{M}_{He}\simeq 2M_{c}$. Thus, $\bar{M}=2\xi M_{c}$ with
$\xi\simeq 0.4$ for CC-SNe in close stellar binaries produces relatively
rapidly rotating black holes (van Putten, 2004). As a result, we expect
descending GW-chirps also from a fraction of CC-SNe extending over

$\displaystyle 100\,\mbox{Hz}\lesssim f_{GW}\lesssim 360\,\mbox{Hz}.$ (5)

This frequency range (5), similar to (3), appears opportune given the above-
mentioned minimum of LIGO detector strain noise amplitude.

Accurate detector yield factors are essential in seeking consistency in event
rates in EM-GW surveys. Results are particularly relevant for afterglows to
DNS mergers. Considerable diversity in their MM afterglow emission is
expected, both intrinsic and due to observational selection effects. Intrinsic
diversity is expected as a function of total mass of a DNS system, a primary
parameter defining the lifetime of the initial hyper-massive neutron star
(HMNS) (Lucca & Sagunski, 2020). A GRB association such as GRB170817A to
GW170817 is inevitably subject to small observed-to-true event ratios by their
beaming factors. This problem continues to receive broad attention for both
short and long GRBs (e.g. Salafia et al., 2015; Mandhai et al., 2018; Farah et
al., 2020; Dichiara et al., 2020; Fryer et al., 2022). We include these
results in our outlook on O4-5.

In light of the above, we include an overview of LIGO observations S5-6 and
O1-O3ab and their sensitivity, detector duty cycles and horizon distances
(Appendix A). A primary motivation of the present work is probing for the
central engines of extreme transient events such as cosmological GRBs
associated with mergers, normal long GRBs and their parent population of
energetic CC-SNe. Specifically, the present work focuses on searches for

  1. a)

Unknown GW-signals from un-modeled sources requiring confirmation across two
(or more) detectors with comparable performance in sensitivity;

  2. b)

Consistent event rates in EM-GW surveys on astrophysical transient events in
the Local Universe.

Figure 1: Detector duty cycles producing science quality data can be inferred
from detector stain-noise spectra, $\sqrt{S(f)}$ versus $f_{GW}$ (left
bottom). Science quality data is defined by stable and nominal spectra over
segments of predefined duration $W$, evaluated by Welch method over
subintervals of $w$, covering a choice of bandwidth $B$ (3). Here we
schematically show the definition of coincident duty cycle ($\mathcal{U}$)
commonly used by LVK (a) and detector yield factor $y$ used in the present
work (b). Solid lines and filled rectangles are representative for the mean of
$\sqrt{S(f)}$ for a pair of detectors D1 and D2, and, respectively, the
associated standard deviation ($\sigma$) therein. It generally depends on the
choice of time-frequency domain $W\times B$. (a) requires stable performance
for D1 and D2 individually with no regards to comparable performance. In (b),
we require comparable performance highlighted by overlap shown within the
bound (8). Evidently, inequality (9) holds.

In preparing for O4-5, we approach (a-b) in various steps, starting with
detector performance relevant to transient events with the potential to
produce unknown GW-signals lasting several seconds as follows:

  * •

Measuring comparable detector performances by statistics of high-quality data
in coincident H1L1-operation over a time-frequency window $W\times B$ and the
duty cycle thereof (§2), where $W$ refers to a choice of duration of data-
segments representative for long-duration GW-transients;

  * •

Enhancing discovery power when probing long-duration transient events over
several seconds (§3) using time-symmetric methods of analysis at sensitivities
on par with CBC, satisfying (2).

  * •

Diversity in MM-afterglows to DNS mergers as a function of total mass (§4),
supported by progenitor mass-components from existing radio surveys,
illustrated by GW170817 and, possibly, GW190425. Additionally, we discuss the
GRB211227A-kilonova assocation and DNS mass components.

  * •

Outlook on O4-5 focused on diversity in MM-afterglow emissions to mergers
(§5). Discussed is further the planned GRB-mission THESEUS (Amati et al.,
2018, 2021).

##  2 LIGO Detector Performance

We revisit LIGO runs to estimate duty cycles of high-quality data in
coincident detector operation at nominal behavior, essential to searches of
unknown GW-signals in probing central engines of GRBs and CC-SNe in the Local
Universe (e.g. Coward et al. (2002); van Putten et al. (2019b)).

We target GW transients with duration similar to the signals from the central
engine of GRB170817A (van Putten & Della Valle, 2023), i.e., long duration
transients extending over several seconds or more. Therefore, the primary
focus of our analysis is on the segments of

$W=8\ \mbox{seconds}.$ (6)

In particular, our focus is entirely separate from searches for BBH megers,
whose durations are about one order of magnitude shorter.

While high-quality data from one detector may suffice to probe mergers by
their by now familiar ascending GW-chirps, this is not the case for new
sources. To assess sensitivity for the latter, we perform a broad study over
S5, S6, O1, O2, O3ab using above-mentioned frequency range (3) as a proxy.
According to Appendix A (Table A.1), results of this revisit are expected to
be illustrative for upcoming observational runs O4-5.

To this end, we extract statistically representative samples of data-frames of
4096 seconds (down-sampled to 4096 Hz) by randomly selecting 112 central GPS
times in each run. Thus, a total of 672 randomly selected frames are used for
analysis by the following steps:

Step (a): Table 1, column 2 reveals the existence of gaps gaps in each run.
Listed are the number of available data following the randomly selected GPS
times.

Step (b): Available data are down-selected to paired H1L1-data. In this
process, we note there are some incomplete frames, not covering the whole
duration of 4096 s and there are instances when only one of the detectors is
in observing mode. Notably, there is a probability of having too many gaps,
inhibiting extraction of spectra (and spectrograms). These results are
presented in column 3 of Table 1, roughly equivalent to column 4 of Table A.1,
i.e., the coincidence duty cycle of H1 and L1 ($\mathcal{U}$) during each run.

Step (c): Since a full data does not guarantee quality, we next check for the
joint H1L1-data segments with at least one of the detectors collecting data
without significant gaps (Column 4 of Table 1). As our target signals are GWs
transients with the duration in (6), to do this analysis we split data into
segments of 8 seconds for checking the quality of data. We determine data-
quality for each segment according to detector sensitivity over $B$ by the
Welch method applied to Fourier windows of size

$w=W/2=4\ \mbox{s}.$ (7)

For S6 with $w=64$s over the whole of 4096 s data, our statistical results are
in quantitative agreement with van Putten (2017), in which $all$ 12728 frames
were analyzed. This evidences that our random selection of 112 GPS times is
sufficient.

Step (d): As a prerequisite for coincidence high-quality segments of
H1L1-data, both H1 and L1 must be free of significant gaps. The number of
segments that satisfy this condition is quite small. The least and largest are
found in S6 and O3a (S5), respectively (Table 1, column 5). Notably absent is
a discernible trend that might point to significant improvements of detectors
in later runs.

| Joint | joint H1-L1 4096 s: | Segment | Joint | Joint
---|---|---|---|---|---
| H1-L1 data | Not all zero | Densities | H1-L1 segments | H1-L1 segments at
|  |  |  |  | high quality ($y$)
S5 | $80\%$ | $78\%$ | $77\%$ | $63\%$ | $13\%$
S6 |  $40\%$ | $37\%$ | $34\%$ | $20\%$ | $8\%$
O1 | $45\%$ | $42\%$ | $41\%$ | $34\%$ | $1\%$
O2 | $49\%$ | $44\%$ | $42\%$ | $36\%$ | $19\%$
O3a | $75\%$ | $70\%$ | $69\%$ | $54\%$ | $14\%$
O3b | $58\%$ | $57\%$ | $55\%$ | $44\%$ | $25\%$
Table 1: Summary of data-quality in LIGO runs . Results derived from
statistics of 112 frames randomly selected by GPS times, evaluated for
availability and quality of data. A segment of (8 s) is considered in searches
of long duration transients. Joint H1L1-operation at comparable sensitivities
is crucial in discovery of unknown GW signals from un-modeled sources,
requiring coincident high-quality data. Column 2 lists gaps in the data.
Deviation from $100\%$ indicates the fraction of GPS times in a run with no
available data. Even when frames exist, they may not cover the whole 4096 s.
Column 3 indicates the fraction of joint full frame out of 112 (independent
from target of search i.e., $w$-independent), roughly equivalent to Column 4
of Table A.1. A full frame does not guarantee quality data or the absence of
significant gaps. Gap statistics is $w$-dependent, i.e., it generally depends
on search target. To this end, we consider segments of duration $\Delta
t=W=2w=8$ seconds for further analysis (6). Column 4 lists the number of joint
H1L1-data segments in which at least one of the detectors is free of
significant gaps. The fraction of segments with no gaps listed in Column 5
significantly decreases compared to Column 4 with no evident trend with runs.
Finally, column 6 lists joint H1L1-data segments both in observing mode at
high-quality, defined by $\sigma/\mu<1$ simultaneously for both detectors
across the frequency range (3). This measure can be further restricted or
relaxed depending on search scope. The results illustrate that the fraction of
observational time with high-quality data simultaneously for both detectors of
interest to un-modeled searches over $W=8\,$s is roughly $10-20\%$ with no
significant improvement across the different runs.

Step (e): Finally, we consider both detectors in observing mode collecting
high-quality data, defined when H1 and L1 show comparable nominal performance
permitting cross-correlation of candidate observational events. Specifically,
we consider an overlap of detectors strain-noise amplitude within $0.5\sigma$,
$1\sigma$ or $2\sigma$ band in above-mentioned most sensitive frequency
interval and, correspondingly,

$\dfrac{\sigma}{\mu}<\dfrac{1}{2},1\mbox{\leavevmode\nobreak\ or}\ 2,$ (8)

where $\mu$ is the mean value of the spectrum (one-sided PSD) based on Welch
method and $\sigma$ is the standard deviation of the spectrum at each
frequency calculated over two half-segments of FFT window size $w=4$s conform
(7). Fig. 1 schematically shows the difference between the definition of high-
quality data required in search of unknown GW signals, and coincidence duty
cycle (used by LVK) which is defined based on individual stable performance in
L1 and H1.

Figure 2: Sample spectra of O2 (left) and O3 (right) for the segments of
$W=4096$ s and $W=8$ s, respectively. These samples represent high-quality
data with small standard deviation for both H1 and L1. Their nominal behavior
shows overlap within the most sensitive frequency interval of LIGO over about
$B$ (3). The right panel corresponds to our target in searches of un-modeled
signals. At lower frequencies seismic noise is dominant, while at higher
frequencies, optical shot-noise is dominant. The difference in the line widths
is due to different sizes of data segments and smoothing by a moving average
over frequency.

Fig. 2 shows an example of coincidence high-quality data-segment (6) in joint
H1L1-operation with $\sigma/\mu<1$ over the most sensitive frequency interval
(3). Applying these restrictions on the remaining pair of segments, LIGO is
found to be observing in high-quality regime generally up to $25\%$ (Table 1,
column 6). Hereafter, we refer to these numbers as the detector yield factor
($y$) of each run, satisfying (Fig. 2)

$0\leq y\leq\mathcal{U}\leq 100\%,$ (9)

where $\mathcal{U}$ refers to the coincidence duty cycle (Table A.1).

Repeating the above with $\sigma/\mu<2$ or $0.5$, results vary: the latter is
at the edge with S5, S6 and O1, leaving almost no high-quality data. The
former slightly increases detector yields by a few percent compared to
$\sigma/\mu<1$.

It is remarkable that there is no discernible trend across the different runs,
with $y$ of O3a to be similar to that of S5. Fig. 3 presents a summary of this
section covering S5 to O3b. Our results illustrate that (9) is $W$-dependent
i.e. different search targets will have different detector yield factors.

Comparing the $y$-values of each run, presented in the last column of Table.
1, with coincidence duty cycles ($\mathcal{U}$) presented in column 4 of Table
A.1 and the summary of LVK performance during O2 and O3 in (Davis et al.,
2021), it is observed that $y$ is generally less than $\mathcal{U}$ by a
factor of about $3-5$. This significant discrepancy is due to two reasons:

  * •

Detector yield factors and coincidence duty cycles presented here depend on
target of search. We focus on unknown GW transient in the time-frequency
domain $W\times B$ with $W$ in (6) and $B$ in (3) representative for the GW
signals with the duration similar to that of the descending chirp in GW170817
in the most sensitive frequency band of LIGO. Changing the target of search to
signals with shorter duration (e.g., BBH mergers), or longer duration (e.g.,
signals from pulsars), will change the associated detector yield factor. For
instance, seeking for long duration signals with $W=4096$, $y=28\%$ for O2
while $y=9\%$ for O3.

  * •

In Davis et al. (2021) coincidence duty cycles are defined by stable
performance of individual detectors in their observing mode. In our approach
we look for simultaneous comparable nominal performance according to (8). Fig.
1 is illustrative for the distinction between these two approaches.

To provide science quality data, LVK applies vetoes to exclude the data that
are seriously affected by noise or glitch (Davis et al., 2021). The criteria
for high quality data automatically filters out the moments that the detectors
are performing outside the bounds of (8).

Figure 3: Histogram of LIGO performance from S5 to O3b derived from 112
randomly selected frames by GPS time. The darkest shade is for high-quality
data segments with $\sigma/\mu<1$ and steady phase at the most sensitive
frequency range 3 in both H1 and L1. The fraction of segments providing high-
quality data is representative for the detector yield factor. O3b features the
largest yield factor in search of unknown long duration transients.

###  2.1 Detector yield factors $y$

The coincidence high-quality observing time of LIGO runs presented here is
essential to the conversion of observed-to-true astrophysical event rates. The
reported coincidence duty cycle of LIGO detectors spans from $25\%$ to $68\%$
in S5 to O3b and from $40\%$ to $62\%$ in O1 to O3b. The resulting yield
factors span from $1\%$ for O1 to $25\%$ for O3b, with no discernible trend
across different runs as shown in Fig. 3.

Fig. 3 summarizes yield factors of the detectors during all LIGO runs, derived
from the mean $\mu$ of the spectrum of each segment (6) for the frequency
range (3) and the ratios of the standard deviations to the mean for each
segment.

Fig. 4 shows that in general, the data in S5 and S6 are widely scattered,
expected as these are scientific runs and before aLIGO. For the four later
runs, surprisingly, the data for O1, O3a and O3b are scattered, while the data
for O2 are placed on a block around the mean value as depicted in Fig. 4.
During O2 detector performance was substantially more stable than during other
runs, although sensitivity was less than that of O3 by a factor of about 1.5.
This perhaps explains the paradox that the most exciting event, GW170817, was
detected during O2 rather than O3.

The detector yield factor $y$ is important for true-to-observed event rates of
astrophysical transients in EM surveys. It evidently affects observing volume

$\left<VT\right>=\left<(\mbox{sensitivity})^{3}\times
T\times\mbox{coincidence\ duty\ cycle}\right>,$ (10)

where $T$ is the total duration of the run. Equation (10) is to be modified by
a $y$ to include the true coincidence duty cycle,

$\left<VT^{\prime}\right>=\left<VT\right>\times y,$ (11)

providing a more precise measure for observational performance of a detector.
Based on (11), for the same total observing time, if the sensitivity and
detector yield factor go as a “seesaw” - sensitivity improves but duty cycle
decreases- the total observing volume can remain the same. Therefore, detector
performance is not necessarily enhanced even as sensitivity improves.

Fig. 5 shows true observing volume for the LIGO runs in searches of unknown
long duration transients across segments (6) based on Table 3, here normalized
to O2. Importantly, astrophysical event rate calculations are sensitive to the
same $y$ relevant, for instance, to heavy element production rates that may
further be compared with abundances provided by EM-surveys, population
statistics of SGRBs and their properties (Kasliwal et al., 2022; Gillanders et
al., 2021; Dado et al., 2022). Many other astronomical transients provide
existing event rates from EM observations (Table 2).

Astrophysical Transient Event |  | Event rate
---
$\mbox{yr}^{-1}\mbox{Gpc}^{-3}$
| Event rate in Local Universe
---
$(100\mbox{Mpc})^{-3}$
Binary Neutron Star (DNS) merger |  $\sim 10-1700$ (1,2,3) | $\sim 0.1-2\ (\mbox{yr})^{-1}$
Binary Black Holes (BBH) merger |  $\sim 17.9-44$ (1,2,3) | $\sim 1\ (\mbox{40yr})^{-1}$
Double White Dwarf (DWD) merger |  $\sim 10^{5}$ (4,5) | $\sim 10^{2}\ (\mbox{1yr})^{-1}$
SNe Ibc |  $\sim 10^{4}$ (6,7,9,10) | $\sim 10\ (\mbox{1yr})^{-1}$
Short Gamma-Ray Bursts (SGRB) |  $\sim 1000$ ${}^{(11)\ \ddagger}$ | $\sim 1\ (\mbox{yr})^{-1}$
Long Gamma-Ray Burst (LGRB) | $\sim 1.5\times\mbox{SGRB}^{\ \ddagger}$ | $\sim 2\ (\mbox{yr})^{-1}$
Core-Collapse SNe |  $\sim 10^{5}$ (4,12) | $\sim 10^{2}\ (\mbox{yr})^{-1}$
Low Luminosity GRB |  $\sim 100-1800$ (12) | $\sim 0.1-2\ (\mbox{yr})^{-1}$
High Luminosity GRB |  $\sim 100-550$ (12) | $\sim 1-5(\mbox{10yr})^{-1}$
Low red-shift Long duration GRB |  $\sim 380$ (6,12) | $\sim 1\ (\mbox{3yr})^{-1}$
Super Luminous SNe |  $\sim 35-150$ (6,7,8) | $\sim 1-5\ (\mbox{30yr})^{-1}$
Fast Radio Bursts (FRBs) |  $\sim 10^{3}-10^{5}$ (13,14,15,16) | $\sim 1-10^{2}\ (\mbox{yr})^{-1}$
X-Ray Flashes |  $\sim 2$ (17,18) | $\sim 1\ (\mbox{500yr})^{-1}$
Table 2: Estimated event rates of transient events based on EM surveys.
‡ GRB rates provided are corrected for beaming (Nakar, 2020). Precise
measurements of true-to-observed GRB rates due to beaming are not available
despite multiple dedicated missions (e. g. BATSE, BeppoSax, Swift, Fermi).
There appears to be some discrepancy in observed GRB event rates inferred from
the BATSE 4B catalog and Swift. BATSE suggests LGRBs to be somewhat more
numerous compared to SGRBs (Paciesas et al., 1999; Kouveliotou et al., 1993).
This seems reversed in Swift (Wanderman & Piran, 2010, 2015), although with
relatively large uncertainties.
(1)Abbott et al. (2021);(2)The LIGO Scientific Collaboration et al. (2021a) ;(3)The LIGO Scientific Collaboration et al. (2021b) ;(4)Heo et al. (2016);(5)Maoz et al. (2018) ;(6)Gal-Yam (2019) ;(7)Li et al. (2011); (8)Frohmaier et al. (2021);(9)Dahlen et al. (2012);(10)Cappellaro et al. (2015) ;(11)Nakar (2020) ;(12)Guetta & Della Valle (2007) ;(13)Fialkov et al. (2018) ;(14)CHIME/FRB Collaboration et al. (2021);(15)Ravi et al. (2019);(16)Bhandari et al. (2018) ;(17)Yamazaki et al. (2002) ;(18)Yamazaki et al. (2003). Figure 4: Detector performance over S5-O3ab in terms of variations($\sigma$) in sensitivity versus sensitivity($\mu$). Shown is a scatter plot of the spectrum($\mu$) vs. for each 4096s data and for the most sensitive frequency range in (3). Percentages in the legend refer to the fraction of non-zero mean values. There is a significant change from iLIGO to aLIGO (lower panel). Interestingly, O2 shows a highly compact region of small standard deviations and stable performance around the mean. O3a and O3b are similar to each other and despite the improvement in the sensitivity compared to O2, they are more scattered from the mean by having larger $\bar{\sigma}/\bar{\mu}$.  Figure 5: Observation volume improvement in LIGO runs from S5 to O3ab, using the inverse of detector noise spectrum representative for the last row of Table 3. It shows aLIGO improvement over iLIGO (lower panel), and discovery potential of O3ab similar to O2 (in search of long-duration transients) by true observational volume - essential for detecting unknown GW signals and event rate estimation (upper panel). Run |  | S5 | S6 | O1 | O2 | O3a | O3b
---|---|---|---|---|---|---|---
Sensitivity ($1/\sqrt{Hz}$) | H1 | $1.02\times 10^{-21}$ | $7.27\times 10^{-22}$ | $1.037\times 10^{-23}$ | $7.3\times 10^{-24}$ | $5.23\times 10^{-24}$ | $4.86\times 10^{-24}$
L1 | $3.3\times 10^{-22}$ | $5.9\times 10^{-22}$ | $1.52\times 10^{-23}$ | $7.2\times 10^{-24}$ | $5.07\times 10^{-24}$ | $4.88\times 10^{-24}$
Detector Yield ($y_{D}$)(%) |  | 13 | 8 | 1 | 18.7 | 13.6 | 25.3
$VT^{\prime}=VT\times y$ | H1 | $2.6\times 10^{-7}$ | $2.09\times 10^{-10}$ | $5\times 10^{-7}$ | $1.2\times 10^{-4}$ | $1.03\times 10^{-4}$ | $1.9\times 10^{-4}$
L1 | $7.9\times 10^{-6}$ | $3.9\times 10^{-10}$ | $1.6\times 10^{-7}$ | $1.3\times 10^{-4}$ | $1.14\times 10^{-4}$ | $1.9\times 10^{-5}$
($\mbox{Gpc}^{3}\mbox{yr}$)
$VT^{\prime}$/$(VT^{\prime})_{O2}$ | H1 | $2.2\times 10^{-3}$ | $1.7\times 10^{-6}$ | $4.2\times 10^{-3}$ | 1 | $0.86$ | $1.61$
L1 | $0.620$ | $3.07\times 10^{-6}$ | $1.2\times 10^{-3}$ | 1 | $0.90$ | $1.50$
Table 3: Summary of LIGO detector performance in terms of observing volume
$\left<VT^{\prime}\right>$. LIGO detectors observational volume improvements
in the most sensitive frequency range ($100-250\mbox{Hz}$) during the 6 runs
S5-O3b. Observational volume is defined as (11), where $V$ is the spatial
volume defined by the sensitivity of the detector in each run, $T$ is the
observing time and $T^{\prime}$ is the $y$-corrected observational time
considering the dimensionless detector yield factor $y$ in each run. The last
row provides the $<VT>\times\mbox{yield}$ values normalized to O2 as it was
the host for the only multi-messenger LIGO event GW170817. To be more clear,
S5 and S6 was scientific runs during iLIGO, but the other four are
observational runs in aLIGO. The last two rows show that O2 and O3ab are
almost on par with each other as they have similar observing volume.

As per Table 2, EM-surveys offer comprehensive yet somewhat imprecise
statistics of our transients of interest. This is apparent in considerable
uncertainties in event rates that appear to be statistic and/or systematic.
Future joint EM-GW surveys may come to the rescue. This will pose additional
challenges (if not opportunities) for consistent event rates. Inevitably, this
calls for accurate and high detector yield factors $y$.

GW170817 serves as a case at hand. Detected in O2 at a distance of
$40\mbox{Mpc}$, it represents a DNS coalescence followed by an SGRB, X-ray,
UV, optical and radio-emissions. This was the only event of the type within
about 269 days of O2. During O2 the horizon distance of LIGO was about
$100\mbox{Mpc}$ with a coincidence duty cycle of $45\%$. Assuming GW170817 is
representative of typical DNS merger, these numbers give a rough estimate of
$3\mbox{yr}^{-1}(\mbox{100Mpc})^{-3}$ for DNS coalescences. Including accurate
detector yield factors $y$, the event rate increases by a factor of about
three. If all SGRBs to be counterparts of DNS mergers, this affects the
estimated event rates for SGRBs and kilonovae associated with DNS mergers. A
comparison with EM-surveys shows the resulting event rates to have some
tension, though inconclusively due to the $n=1$ event count with GW170817 in
O2 and the somewhat uncertain estimates from EM-surveys. Further comments are
deferred to §5.

##  3 LIGO Improvement Factors

Our results in the previous section point toward appreciable room to enhance
discovery power, some of which may be realizable in advance of much
anticipated next generation detectors (Abbott et al., 2017e, a; Torres-Rivas
et al., 2019). Alternatively, next generation detectors, such as the Cosmic
Explorer and Einstein Telescope, are anticipated to have a horizon that covers
high redshift cosmological distances (at least an order of magnitude larger
than aLIGO), essential for detecting numerous events (Dwyer et al., 2015;
Regimbau et al., 2017). Consequently, they promise to provide precise
measurements of transients event rates.

Focusing on O4-5, we highlight some factors to enhance the discovery potential
of our current generation of detectors.

###  3.1 Unrealized Sensitivity

As one of the most important factors in detection performance and discovery
potential, sensitivity has been always been a primary objective in each new
run. From S5 to O3b, detector sensitivity has seen improvements by over two
orders of magnitude. Further improvements are expected by a factor of about
2-2.5 in O5 about 2027 (Abbott et al., 2020). This is expected to result in an
additional increase of discovery power by a factor of about 10 for
observational times similar to existing runs.

###  3.2 Detector yield factor in EM-GW surveys

Our focus is on true duty cycle ($y$), defined by a detector yield factors key
to discovery power of new and unknown signals. Currently, $y$-values are
optimistically about $10-20\%$ \- considerably below that of conventional EM-
observatories (e.g. NASA, 2008). It also challenges searches for MM-signals by
missed events when continuous coverage is poor. In particular, this would
apply to transient events of interest, notably SGRBs associated with DNS
mergers and GW-emission from central engines of CC-SNe. According to EM-
observations, the event rate of SGRBs is $\mbox{1000 yr}^{-1}\mbox{Gpc}^{-3}$
(Nakar, 2020). If all SGRBs derive from DNS mergers are similar to GW170817
and $y\simeq 20\%$ (similar to O2-O3), it might take some a half-century to
see another GW170817-like event at similar proximity - or more if $y\simeq
10\%$ (Table 1). Even though $y\simeq 100\%$ may not be realistic, yields
similar or better than O2/O3ab seem desirable. High detector yield factors
seems pertinent to discovery of the unknown in joint EM-GW surveys.

###  3.3 Time-symmetric data analysis

Data analysis methods are essential to realizing discovery power, even for
resolving known events for their detailed physical and astronomical aspects.
Quite generally, astrophysical transient sources tend to produce ascending GW-
chirps, as in CBC, or descending GW-chirps, in spin-down of a compact central
engine powering a GRB or CC-SN. In searching for the unknown, an optimal
analysis method should be equally sensitive to these two classes of signals:
the method of analysis should be time-symmetric.

In contrast, LIGO currently employs highly optimized methods to detect CBC
(Allen et al., 2012), notably GW170817 with ${\cal E}_{GW}\simeq
2.5\%M_{\astrosun}c^{2}$ by essentially ideal matched filtering against known
wave-forms. LIGO has also multiple pipelines for searching for unknown signals
for instance cWB (Drago et al., 2021), Hidden Markov Model (Suvorova et al.,
2016) and Bayeswave (Cornish & Littenberg, 2015), among others. Yet, a LIGO
threshold of about ${\cal E}_{th,GW}\geq 650\%M_{\astrosun}c^{2}$ is reported
for unknown GW-signals from a putative long-lived compact remnant (Sun &
Melatos, 2019); see further Fig. 1 of (Abbott et al., 2017a). Over two times
the total mass-energy of the DNS GW170817, this threshold violates (2).

Fig. 6 shows search domains for descending signals defined by (2), indicating
“Allowed” and “Excluded” zones. LIGO explorations to-date for un-known GW-
signals have been performed mostly in the Excluded zone, leaving most of the
Allowed region “Uncharted”. In fact, for long-lived remnants, the entire
Allowed region remains Uncharted by the above mentioned LIGO search algorithms
(Abbott et al., 2019).

When ${\cal E}_{th,GW}\gtrsim 650\%M_{\astrosun}c^{2}$, (2) is not satisfied.
Furthermore, it is ostensibly unbalanced, limiting sensitivity to an un-
modeled descending signal to about $0.3\%$ of a preceding merger measured by
energy output ${\cal E}_{GW}$. By amplitude, the corresponding discrepancy is
a factor of 17 between ascending merger chirps and any potential descending
signal from spin-down of a compact remnant.

A time-symmetric method of analysis principally uses some form of matched
filtering over a bank of time-symmetric templates. Notable examples are time-
sliced Fourier-based methods and butterfly matched filtering (van Putten et
al., 2019b, a).

To illustrate time-symmetry, we apply conventional time-sliced Fast Fourier
Transform (FFT) (Fourier, 1822; Gauss, 1866 [1805]; Cooley & Tukey, 1965) to
GW170817. GW170817 is presently the only event to date of interest by the
associated GRB170817A with contemporaneous emission in a descending GW-chirp,
recently identified with spin-down of a Kerr black hole (van Putten & Della
Valle, 2023). Fig. B.1, clearly confirms the merger chirp from $\sim
100\mbox{Hz}$ to $\sim 300$, representing ${\cal E}_{GW}\simeq
2.25\%M_{\astrosun}c^{2}$ output in gravitational radiation
($2.5\%M_{\odot}c^{2}$ in a run-up to 650 Hz (The LIGO Scientific
Collaboration et al., 2017)). The result is evidently invariant by doing the
same calculation following a time reversal. Importantly, the signal is
detectable in both detectors. The glitch in L1 during the merger is perhaps
illustrative for the need of having (at least) two detectors working at the
same time producing high-quality data.

Interestingly, Fig. 7 shows a faint yet discernible trace of the above-
mentioned (descending) GW-signature of GRB170817A in the time-sliced FFT
spectrogram. Notably, we have consistency in representing a total output of
${\cal E}_{GW}\simeq 3.5\%M_{\astrosun}c^{2}$ for the descending chirp with a
more detailed analysis of the descending chirp of black hole spin-down,
recently confirmed at the 5.5$\sigma$ confidence level (Fig. 8 and van Putten
& Della Valle (2023). Here, the latter is also seen to be similar noting
${\cal E}_{GW}\simeq 2.25M_{\odot}c^{2}$ of the merger chirp at slightly lower
frequencies, where LIGO sensitivity is relatively high.

The results of these two independent approaches reveal a descending GW-chirp
during GRB170817A at $\mathcal{E}_{GW}\simeq 3.5\%M_{\odot}c^{2}$: the
Uncharted region is not empty. Therefore, (2) needs to be satisfied by a
detection threshold of order $1\%M_{\odot}c^{2}$, far below the threshold
$E_{GW}=650\%M_{\odot}c^{2}$ of aforementioned power-excess methods (Sun &
Melatos, 2019).

###  3.4 Improving discovery power

Following §3.1, §3.2 and §3.3 we estimate potential room for an improvement in
discovery power to the unknown in the current generation of LIGO detectors,
calculated by the four-dimensional volume $\left<VT\right>$ over O4-5.
Compared to O3, we estimate:

$\dfrac{(VT^{\prime})_{f}}{(VT^{\prime})^{LIGO}_{today}}=\mathcal{A}\times\mathcal{B}\sim\mathcal{O}\left(10^{5}\right),\vspace{0.2cm}$
(12)

where $\mathcal{A}$ and $\mathcal{B}$ are representative for potential
improvements in LIGO hardware and, respectively, software,

$\displaystyle\begin{array}[]{lll}&&\mathcal{A}\simeq\left(\dfrac{h_{O5}}{h_{On}}\right)^{3}\times\left(\dfrac{100\%}{Y_{On}}\right)\sim\left\\{\begin{aligned}
\mathcal{O}\left(10^{4}\right)\hskip 5.69046ptn=1\\\
\mathcal{O}\left(10^{2}\right)\hskip 5.69046ptn=2\\\
\mathcal{O}\left(10^{2}\right)\hskip 5.69046ptn=3\end{aligned}\right.\\\ \\\
&&\mathcal{B}\simeq\left(\dfrac{\mbox{LIGO sensitivity to CBC}}{\mbox{LIGO
sensitivity to descending
signals}}\right)^{3/2}\sim\mathcal{O}\left(10^{3}\right).\end{array}$ (16)

Here, ${\cal A}$ is derived from potential improvements in the detector yield
factor; ${\cal B}$ is derived from the above-mentioned ${\cal E}_{GW}\simeq
2.5\%M_{\astrosun}c^{2}$ in GW170817 and a reported threshold of ${\cal
E}_{GW}={\rm few}\times M_{\astrosun}c^{2}$ (Abbott et al., 2017a; Sun &
Melatos, 2019). Bringing down the latter to a few percent on par with CBC by
time-symmetric data-analysis shows ${\cal B}\simeq 4200$. Estimating
$\mathcal{B}$ during O3 gives the same result as the same analysis methods
were employed compared to O2 (Fig. 2 and Table I in Abbott et al. (2021)).

Up to an order of magnitude, $\mathcal{A}$ and $\mathcal{B}$ each point to
considerable expansion of discovery power in the coming years. Even if
hardware and software improve at a fraction of their limits, considerable
improvements in discovery power are expected. For instance, $\mathcal{B}$
combined with joint H1 and L1 duty cycles for quality data at about $60\%$
presents an outlook on discovery power (12) improving by about 393000 in the
planned O5 run, compared to O3. On this basis, present observations appear to
be just the tip of the iceberg.

Figure 6: The search domains for a GW170817 descending signal according to
Abbott et al. (2017a). (Left.) Radiation energies emitted following a merger
must satisfy energy conservation (2). Specialized to GW170817, $E_{P}$ is
given by the total mass-energy $E_{DNS}\simeq 2.7M_{\astrosun}c^{2}$ is the
total mass-energy of the progenitor and (2) defines a required energy
threshold of detection $\mathcal{E}_{th}\leqq E_{DNS}$: $\mathcal{E}>E_{DNS}$
represents an “Excluded zone” and $\mathcal{E}<E_{DNS}$ a zone of “Allowed”
radiation energies. Possibly, the merger output $\mathcal{E}_{GW}\simeq
2.5\%M_{\astrosun}c^{2}$ sets a scale for potential descending chirps in GW-
emission, further suggesting a reduced search domain of interest $\leq
10\%M_{\astrosun}c^{2}$. Notably, EM-radiation is only $\mathcal{E}_{EM}\simeq
0.5\%M_{\astrosun}c^{2}$ (Mooley et al., 2017, 2018). (Right.) The Excluded
(grey) and Allowed (green) zones are shown in the frequency-noise amplitude
spectrum of LIGO during O2, based on Fig. 1 in Abbott et al. (2017a). High-
ligthed is the “Uncharted zone” covering most of the Allowed zone, not
explored by any of the LIGO searches. Notably, LIGO searches for long-lived
remnants are entirely in the Excluded zone in light of a search threshold
$\mathcal{E}_{th,GW}\geq 6.5M_{\astrosun}c^{2}$ (Sun & Melatos, 2019; Abbott
et al., 2019), explaining the non-detection in these earlier post-merger
searches. Figure 7: The upper and middle panels show spectrograms of 32 s of
data around GW170817 for H1 and L1 by GWXplore. Discernible though faint is a
descending chirp following the merger. This benchmark, together with Fig. B.1,
is illustrative for a time-symmetric exploration of the Uncharted region in
Fig. 6. The lower panel shows the time delay time of 1.7 s between the
GW170817 trigger and the time-of-onset of GRB170817A (Fermi-GBM panel after
van Putten & Della Valle (2023), and the moment that the descending chirp
appears visible. Although the exact starting point of the descending signal is
not well resolved in the present spectrogram, it appears contemporaneous with
GRB170817A attributed to MM-emission from a black hole disk or torus system
for the lifetime of black hole-spin (van Putten & Della Valle, 2023). Notably,
the lifetime of the primary HMNS is of about 0.92 s, when it experiences
delayed gravitational collapse to a Kerr black hole - in the 1.7 s gap between
GW170817 and GRB170817A (see also Murguia-Berthier et al., 2021). Figure 8:
The descending GW-chirp during GRB170817A observed at high contrast in a
merged (H1,L1)-spectrogram, produced by butterfly matched filtering. This
descending signal starts about 0.92 s post-merger, covering the frequency
range $700\sim 200$Hz. It is confirmed to be associated with the spin-down of
the central engine of GRB170817A with a confidence level of $5.5\sigma$
(Reprinted from van Putten & Della Valle 2023).

##  4 Diversity in DNS and MM-afterglows

Neutron stars are the remnant of the majority of massive stars producing CC-
SNe. In general, the lifetime of a (proto-)neutron star depends on its mass
(e.g. Lucca & Sagunski, 2020). Hyper-massive neutron stars (HMNS) that may be
produced in the immediate aftermath of a DNS merger such as GW170817 tend to
have a relatively short lifetime (Lippuner et al., 2017) before experiencing
gravitational collapse to a black hole, while supermassive neutron stars
(SMNS) may live forever.

DNS systems are interesting as progenitors of MM-astronomical transients.
GW170817 produced the SGRB (GRB170817A) and kilonova (AT2017gfo). Normal long
GRBs, on the other hand, are believed to originate from CC-SNe (Troja et al.,
2017; Eichler et al., 1989; Rezzolla et al., 2011; Nakar et al., 2006; Levan
et al., 2016; Berger, 2014). A DNS merger sequence can be complex, depending
on total mass of the system, producing an SMNS or HMNS, the latter with prompt
or delayed collapse to a black hole. As the total mass of the system given by
the two neutron star progenitors is key factor to the final outcome, we here
first recall the mass distribution of neutron stars provided by existing
radio-surveys.

Radio surveys reveal a bimodal distribution for both neutron stars in DNS
systems, and total mass of the binary system (Özel et al., 2012; Schwab et
al., 2010). Neutron stars masses range from about $1M_{\astrosun}$ to slightly
more than $2M_{\astrosun}$. The available EM-data derives from neutron stars
associated with another neutron star or white dwarf, also from optical
binaries (Özel et al., 2012; Tauris et al., 2017; Rocha et al., 2019; Linares,
2020).

As DNS mergers are of particular interest, we will continue with only these
included in radio-selected surveys. Neutron stars in DNS systems usually
consist of a recycled fast spinning neutron star with a slowly spinning
companion neutron star. These two populations were reported to have distinct
mass distributions, the first with a bimodal mass-distribution, the second
with a uniform mass-distribution (Farrow et al., 2019).

DNS systems are often classified in two categories, according to merger times
shorter or longer than the Hubble time (Andrews & Mandel, 2019). This can be
inferred from the relation between the total mass of a DNS system and its
orbital period by orbital separation. Fig. 9 shows the statistics of the known
DNS systems, showing no further dependency between the orbital separation and
mean mass or, equivalently, lifetime of the progenitor DNS system.
Intrinsically, these two categories are on the same footing.

Considering all of the neutron stars in DNS systems with precisely measured
components and total mass (Tauris et al., 2017; Sengar et al., 2022; Stovall
et al., 2018; Martinez et al., 2017; Lynch et al., 2018; Cameron et al.,
2018), the mass-distribution of DNS components and merger remnants is plotted
in Fig. 10. Excluded is a $10\%$ energy loss of the system for the remnant
mass-distribution, representative for gravitational radiation and mass loss in
the process of coalescence. Both distributions are representative for a normal
mass-distribution function (Linares, 2020), consistent with the first peaks in
Fig. 10.

While DNS systems are represented by the first peak, the second peak
corresponds to neutron stars in a binary with a white dwarf or X-ray binaries
, not included in this study. Fig. 10 suggests that a DNS merger of two
neutron stars at the left tail of the first distribution can form an SMNS
belonging to the same mass-distribution with a long lifetime and possibly so
without any major afterglow.

Figure 9: A scatter plot of total mass vs. orbital period (top panel) and
orbital separation (lower panel) of known DNS systems Sengar et al. (2022);
Tauris et al. (2017). Results show no distinction between DNS systems merging
within or longer than a Hubble time. Figure 10: Mass-distribution of NS in DNS
systems from radio-surveys and GWs and their merger remnant. The pink curve is
for mass-components while the blue curve shows merger remnant mass ($10\%$ of
their total mass is to account for GW-emission and mass-loss). The second peak
in pink curve represents the mass distribution of NS in binaries with a WD or
in X-ray binaries, which are not shown here. Considering both populations, and
$2.4\mbox{M}_{\astrosun}$ and $2.7\mbox{M}_{\astrosun}$ as a putative lower
and upper threshold for HMNS masses, $47\%$ of DNS mergers are expected to
produce an SMNS while $42\%$ are expected to produce an HMNS with a relatively
short lifetime as in GW170817. A remaining $11\%$ of the DNS mergers are
expected to experience prompt gravitational collapse to a black hole.

Marked in different colors in Fig. 10, we provide a first GW-survey of DNS
systems consisting of GW170817 ($M_{1}=1.46M_{\astrosun},\
M_{2}=1.27M_{\astrosun},\ M_{T}=2.73M_{\astrosun}$) and GW190425
($M_{1}=2.1M_{\astrosun},\ M_{2}=1.3M_{\astrosun},\ M_{T}=3.4M_{\astrosun}$)
(Abbott et al., 2019a; The LIGO Scientific Collaboration et al., 2021b). Shown
is a significant difference in the distribution pattern. Apart from a slight
change in the first peak, GW190425 seems to support a bimodal distribution of
DNS systems and remnant mass, while GW170817 follows the original radio-survey
based mass-distribution.

###  4.1 GW170817 and GW190425

While GW170817 and GW190425 are both derived from DNS mergers, they are
nevertheless distinct.

GW170817 has an associated SGRB GRB170817A observed 1.7 s after the merger
(Goldstein et al., 2017b). Within 100 days it featured emission in X-rays,
optical and radio seen by INTEGRAL, Fermi-GBM, the Hubble Space Telescope
(HST), the Chandra X-ray Observatory and the Karl G. Jansky Very Large Array
(Abbott et al., 2017b; Troja et al., 2017; Hallinan et al., 2017; Abbott et
al., 2017c; Goldstein et al., 2017b; Margutti et al., 2018). These MM-
observations provide strong evidence for the GW170817-GRB170817A association.
The kilonova AT2017gfo reveals a further production of heavy elements
(Gillanders et al., 2021). The central engine of GRB170817A is identified with
a Kerr black hole through a contemporaneous descending chirp in gravitational
radiation, attributed to delayed gravitational collapse of the initial HMNS
(van Putten et al., 2019a; van Putten & Della Valle, 2023).

GW190425 presented a second DNS merger candidate. Searches for GW170817-like
MM-afterglow emission produced no confirmed results. This may, in part, be due
to the large (imprecise) localization area, a significantly larger distance
compared to GW170817 and a relatively heavy remnant with possibly prompt
gravitational collapse to a black hole (Levan & Jonker, 2020; Antier, 2020;
Coughlin et al., 2019). GW190425 is detected by L1 when H1 was not in
observation mode (The LIGO Scientific Collaboration et al., 2020). While
GW190425 is subject to considerable observational uncertainties, it
nevertheless appears to be distinct from GW170817 by no detection of MM-
afterglow emission and relatively high total mass.

As mentioned, the second peak in the first bimodal distribution (pink) in Fig.
10 shows NS in WD-NS binaries. GW190425 appears to represent the merger of a
low mass companion $M_{2}=1.3M_{\astrosun}$ from the first with a massive
member of this second peak. The sum is possibly sufficiently massive to result
in prompt gravitational collapse of the initial HMNS, leaving negligible
window for an accompanying kilonova, while a GRB would unlikely be observable
due to beaming.

From Fig. 10, we can infer the lifetime of the initial remnant of DNS
coalescence, arbitrarily long for an SMNS and finite for an HMNS before
gravitational-collapse to a black hole. Fig. 10 suggests SMNS below
$2.4M_{\astrosun}$ are relatively long-lived. Heavier yet below
$2.7-3M_{\astrosun}$, HMNS are formed with finite but short lifetimes. The
lifetime of more massive HMNS is expected to be essentially zero, marking
prompt gravitational-collapse to a BH in the immediate aftermath of the
merger. These possible regimes are highlighted with dashed lines in Fig. 10.

Figure 11: Merger sequence of a GW170817-like event. The merger signal (GW+)
terminates in the formation of the initial HMNS powering the kilonova
AT2017gfo. In delayed gravitational collapse, it produces a Kerr black hole
powering GRB170817A, signaled by a descending gravitational wave signal (GW-)
(van Putten & Della Valle, 2023).  Figure 12: Schematic of the lifetime of an
HMNS as a function of total mass of the progenitor DNS merger. Here, $M_{1}$
refers to the cutoff at the left tail of the blue curve in Fig. 10, while
$M_{2}$ refers to the cutoff at the right tail of the same curve.

The expected dependence of the lifetime of the initial remnant HMNS on total
mass of the binary system (e.g. Lucca & Sagunski, 2020) is consistent with
GW170817 and GW190425. Specifically, GW170817 has a total mass is
$M_{T}=2.73M_{\astrosun}$ ($M_{T}=2.45M_{\astrosun}$ reduced by $10\%$ due to
losses in gravitational-radiation and mass ejecta), leaving an initial HMNS
remnant with relatively short but non-zero lifetime. Based on the kilonova
AT2017gfa, the estimated lifetime of the HMNS is $t_{w}\left(0.98\pm
0.3\right)$ s (Gill et al., 2019), consistent with the time of onset
$t_{s,GW}$ of a descending chirp in GW-emission from a black hole-disk or
torus system following gravitational-collapse van Putten & Della Valle (2023),

$t_{s,GW}-t_{m}\simeq(0.92\pm 0.08)\ {\rm s},$ (17)

where $t_{m}$ is the merger time. (17) improves on a previous estimate of the
same van Putten & Valle (2019); van Putten et al. (2019a) reviewed in Murguia-
Berthier et al. (2021). In contrast, GW190425 has a total mass of
$M_{T}=3.4M_{\astrosun}$ in the second peak (blue curve, Fig. 10) and above
the threshold of $3M_{\astrosun}$, even after the reduction of $10\%$. Prompt
gravitational collapse of its initial remnant naturally explains the non-
detection of a kilonova, while the non-detection of a GRB may be attributed to
beaming. A black hole torus system may have formed with weak kilonova by the
relatively low MeV-temperatures of the torus compared to the temperature of
the HMNS in GW170817 (further below) with accompanying SGRB or SGRBEE
depending on black hole-spin (van Putten & Ostriker, 2001) \- GW190425 is too
far out to probe for a contemporaneous descending chirp in gravitational
radiation.

###  4.2 Kilonovae from HMNS or BH-torus systems?

Conceivably, kilonovae may derive from MeV-neutrino driven winds from a newly
formed massive NS (cf. Gill et al., 2019) or, following its collapse, from a
disk or torus around a newly formed black hole. The difference, if any, is
related to duration and temperature. Torus-driven kilonovae driven by
neutrino-winds at $T_{D}\simeq 2$ MeV are powered by outflows of about
$\dot{m}\simeq 0.05\%M_{\odot}$s-1 (see Fig 10 of van Putten & Levinson
(2003)). Torus-driven kilonovae may appear moderate compared to those driven
by neutrino-winds at temperatures $\gg 10$ MeV of an HMNS newly formed (cf.
MeV-neutrino light curve of SN1987A) in the immediate aftermath of a merger.
The significance to the r-process in kilonovae may be comparable if such
torus-wind is sufficiently long-lived.

GRB211211A might be illustrative, upon identifying its duration $T_{90}\sim
50$ s with the lifetime of a black hole-torus system - post-merger to a DNS or
BH-NS merger. A kilonova derived from a mass-ejecta

$\displaystyle m_{ej}=\dot{m}T_{90}\simeq 2.5\%M_{\odot}$ (18)

will be on par with AT2017gfo, derived from the HMNS produced in GW170817A
over its relatively short lifetime of $\lesssim 1$ s (Margutti et al., 2017).

Torus-driven kilonovae may be challenging to detect at the distance of
GW190425, however. This scenario appears consistent with numerical simulations
for GW190425 with no definite EM counterpart (T. Dietrich and W. Tichy and S.
Ossokine and A. Buonanno and the CoRe-collaboration, 2020).

The above is based solely on the mass-distribution of DNS remnants in Fig. 10,
$46\%$, allowing the formation of an SMNS from the left tail of the
distribution (pink curve). $42\%$ are GW170817-like, producing an HMNS with
finite lifetime with kilonova output and, subsequent to delayed collapse to a
black hole surrounded by a finite amount of merger debris, an accompanying
descending GW-signal representing spin-down of a Kerr black hole; and $11\%$
promptly producing a black hole at final coalescence - relatively quiet and
dark. Any possible post-merger signal will have no time delay in prompt
formation of a BH-torus system.

Fig. 11 schematically shows the merger sequence of GW170817-like events which
include a diverse MM-afterglow events (van Putten & Della Valle, 2023).

###  4.3 The GRB211211A-kilonova

EM-GW afterglows may be derived from mergers of an NS with a companion NS, BH
or white dwarf (WD). While kilonovae such as AT2017gfo to GW170817 represent
the ejection of a few $\%M_{\odot}c^{2}$ in neutron-rich matter, considerably
less is involved in GRBs derived from ultra-relativistic baryon-poor jets.

GRB211211A is a case in point given the associated kilonova. A WD-NS merger
progenitor (Yang et al., 2022) can, in principle, be identified by the GW-
frequency of its precursor merger chirp. By the relatively large radius of the
WD, this frequency is bounded by some four orders of magnitude below that of
DNS mergers, i.e., about 0.01 Hz. Indirect evidence might be obtained from
non-detection of a merger chirp, provided the source is within the horizon
sensitivity distance of canonical DNS and NS-BH mergers. An associated GRB
cannot be ruled out, potentially accompanied by a descending GW-chirp during
spin-down of an NS or BH remnant. The degeneracy between the two may be broken
by GW-calorimetry, by the ample energy reservoir in angular momentum of a Kerr
BH compared to the same in an NS (van Putten & Della Valle, 2023).

Perhaps GRB211211A is derived from an NS-BH merger (Rastinejad et al., 2022).
In general, however, NS-BH mergers are diverse by expected broad distributions
in mass and spin of the BH (van Putten, 1999). For a BH sufficiently massive
and/or slowly spinning ($\gtrsim 28M_{\odot}$ for extremal Kerr black hole
down to $3.7\,M_{\odot}$ for a Schwarzschild black hole), it may swallow the
NS whole “in one shot” sans tidal break-up, possibly producing a FRB without
the usual post-merger drama (Table 4). Otherwise, a BH-torus system may form
producing a short or long GRB depending on BH-spin (van Putten & Ostriker,
2001; van Putten, 2009).

An important open question is a possible diversity in kilonovae powered by (a)
relatively high temperature MeV-neutrino winds from an SMNS or HMNS in the
immediate aftermath of a DNS merger such as GW170817 or (b) relatively low-
temperature MeV-neutrino winds from a BH-torus system (van Putten & Levinson
(2003), §6) by its much lower temperature of a few MeV. The simplest scenario
for GRB211211A appears to be another DNS merger, whose long duration
represents the lifetime of spin of its BH-central engine - like GRB170817A.

Table 4 provides a summary of the above providing a prospect of MM-afterglows
to CBC involving an NS.

###  4.4 DNS binary masses

Considering the mass-distribution of NS based on EM-surveys alongside
GW170817, GW190425, GRB060614 and GRB211227, this phenomenology suggests that
the mass-components and consequently the mass of the system may be decisive in
post-merger evolution. Fig. 10 is illustrative. The total mass of GW170817 was
$2.7M_{\astrosun}$ placing it in HMNS category after $10\%$ reduction due to
gravitational radiation. This HMNS had a lifetime of about $0.92$ s (e.g.
Murguia-Berthier et al., 2021; van Putten et al., 2019a). On the other hand,
the total mass of GW190425 was $3.4M_{\astrosun}$ pointing to immediate
collapse to BH remnant, i.e., the lifetime of the HMNS formed in the immediate
aftermath of the merger was (essentially) zero. We conjecture that as the
total mass gets larger, the lifetime of the immediate merger product is
shorter and, consequently, the probability of a powerful kilonova is
commensurably smaller (Fig. 12), though a kilonova may still emerge from a
black hole-torus system.

###  4.5 Classification of progenitor masses

Our conjecture suggests that not all DNS mergers successfully produce an SGRB
and/or kilonova. To clarify, we categorize the NS mass-distribution to six
different regions: $A^{\pm}_{0}$ and $B^{\pm}_{0}$ considering that the mass
distribution of NS in DNS systems is within the first peak of the pink part of
Fig. 10. $A$ and $B$ refer to the first and, respectively, second peak.
Superscripts $\pm$ represent heaver and lighter masses relative to the central
value (“zero”) in each peak. Accordingly,

$\displaystyle\begin{array}[]{lll}{\rm GW170817}:&&2A_{0}\rightarrow B^{+}\\\
\\\ {\rm GW190425}:&&A_{0}+B_{0}\rightarrow B^{+}.\end{array}$ (22)

By mass, $B^{+}$ in the second is at the tail of NS mass distribution, almost
at the edge of being a BH. While $2A^{-}$ results in an SMNS that lives
forever and can be the progenitor of NS-driven kilonova, $2B^{+}$ is expected
to produce a BH instantly and possibly a faint torus-driven kilonova.

Any other combination ends up somewhere in between with opportunities for a
full story of MM-afterglows to GW170817-like events or partially so like
GRB211227. The latter may include a torus-driven kilonova, relatively less
energetic than an HMNS-driven kilonova and hence may not be detectable
(similar to GW190425). Mid-range events may feature a kilonova similar to
AT2017gfo associated with GW170817 but with noticeably weaker/more powerful
high-energy EM radiation depending on the total mass of the system.

A similar argument can be applied to CC-SNe event suggesting that the event
rate of CC-SNe featuring descending GW-chirps powered by Kerr black holes may
be more numerous than their relatively small branching ratio to normal long
GRBs.

Meanwhile a general classification based on Fig. 10, including the GW surveys,
shows that, considering the lower mass threshold of SMNSs to be
$1.7M_{\astrosun}$, $46.8\%$ of DNS mergers end up with an SMNS, while $42\%$
produce an HMNS with a finite lifetime giving the opportunity for part of this
class to have observable EM counterparts. The remaining $11.2\%$ of DNS
mergers have a BH as a prompt remnant, meaning that they would have a silent
merger without any (observable) afterglow.

##  5 Discussion and Conclusions

The planned LVK O4-5 observations offer unprecedented opportunities to probe
new astrophysical processes in extreme transient events in the Local Universe.
Detecting new GW-signals requires confirmation cross-correlating output across
two (or more) GW-detectors with comparable sensitivities. Currently, the LIGO
H1 and L1 detectors show nominally the same sensitivity. Sensitivity of Virgo
is slightly less, while sensitivity of KAGRA in track to become comparable in
the near future. GW170817-GRB170817A demonstrates the power of joint detection
in EM-GW, opening a radically new window of MM-observations. We discussed the
opportunity of joint GW-EM observations to further our understanding of MM-
afterglows to mergers involving a neutron star and CC-SNe. Those leading to
the formation of a rapidly spinning Kerr black hole may produce long-duration
descending chirps in gravitational radiation.

GW170817-GRB170817A stands out as a template for what may be expected in MM-
afterglows to mergers and, by mass-scaling (4-5), for an outlook on luminous
GW-emission from a small fraction of energetic CC-SNe in the Local Universe.
GW170817 was triggered during O2 with the second highest detector yield factor
at a relatively close distance of $40\mbox{Mpc}$ with mid-range mass
components (Fig. 10). These three factors - detector yield factor, close
proximity and mid-range mass components - have significant impact on the
probability and localization of detecting such event.

A second DNS candidate GW190425 revealed no confirmed EM counterparts. This
event, however, was relatively far out (at a distance of about
$150\mbox{Mpc}$) and the localization area was significantly wider than the
field of view (FOV) of EM observatory. Its total mass is considerably larger
than that of GW170817, suggesting a prompt gravitational collapse of the
initial HMNS to a black hole-disk or torus system. Non-detection of a kilonova
similar to AT2017gfo is therefore not unexpected, also due to relatively
inexact localization, nor the non-detection of a GRB due to beaming.

Non-detection of kilonovae can be challenging by distance. For instance, we
recall GRB060614 at $z=0.125$, attributed to a DNS merger with an associated
kilonova, and GRB211227, observed at $z=0.228$, possibly likewise a DNS merger
based on EM-observations (Lü et al., 2022) (This event was detected after the
completion of O3, however). Although GRB211227 and GRB060614 are similar, the
first showed no detectable kilonova. This seems natural, however, as the
kilonova associated with GRB060614 was barely observed (one data point) while
the redshift of GRB211227 is essentially twice that of GRB060614 which may
explain any associated kilonova to be practically undetectable.

We summarize the results of this study as follows:

  * •

Discovery power of GW-observations depends critically on true observation
time, especially in searches for unknown GW-signals from un-modeled sources,
requiring confirmation across two (or more) detectors operating at nominal
performance. We quantify this over a choice of time-frequency domain $W\times
B$ in terms of detector yield factors of joint H1L1-observations. These yields
are also essential to converting observed to true astrophysical event rates,
which is crucial when seeking consistency with existing event rates from EM
surveys;

  * •

Discovery power for unknown GW-signals depends critically on using time-
symmetric methods of analysis with equal sensitivity to ascending and
descending chirps, anticipated from transient emission in mergers and spin-
down of compact objects and/or in the central engine of GRBs and CC-SNe;

  * •

DNS mergers may be intrinsically diverse in their MM-afterglow emission. Of
particular interest is dependency of mass components (§4.4) and, consequently,
kilonova characteristics;

  * •

Kerr black holes may be universal in powering the most extreme GW-transients
such as GRBs and energetic CC-SNe. For O4, the horizon distance to a
descending GW-chirp from the latter may be in excess of 100Mpc, based on a
relatively low frequency (5) commensurate with $B$ and a relatively large
energy output ${\cal E}_{GW}$, implied by a black hole mass appreciably larger
than that powering GW170817A.

###  5.1 Detector yield factors and event rates

The true duty cycle of LIGO has an important impact on the MMA related
processes and results. The concept of true duty cycle translates to a detector
yield factor when both detectors are collecting high-quality data with small
standard deviation.

The importance of the detector yield factor is more clear if we consider the
relatively high yield factors of EM-observatories (e.g. NASA, 2008) and the
fact that it is deterministic in estimating the astronomical event rates,
i.e.: accurate yield factors are required for consistent event rate in EM-GW
surveys. Event rates are crucial in observational astronomy because they are
to set the waiting time to capture an event.

Since LVK is not operating continuously, precise measurement about their
detector yield factors is crucial. A revisit of LIGO runs shows results far
different from mere coincidence duty cycles. The coincidence duty cycle of
LIGO ranged from $25\%$ to $68\%$ during S5 to O3b, increasing in transition
during aLIGO (Table A.1). Detector yield factors of coincident
H1L1-observations (important for MMA and in search of long duration unknown
transients) is roughly $1\%$ up to $25\%$ (Table 1, Fig. 3).

Fig. 4 is illustrative for the H1L1-performance during the runs at their most
sensitive frequencies, showing the exceptional status of O2. We herein define
high quality data as the segments with $\sigma/\mu<1$ and steady phase at the
most sensitive frequency of LIGO ($100-250\mbox{Hz}$), in both H1 and L1 and
overlapping within $\sigma>0$. Operating both detectors in high quality state
is essential when searching for unknown GW-signals, requiring confirmation
across two independent detectors.

From a MM point of view, the number of GW-events associated with EM-surveys
(Table 2) follows from row 3 of Table 3. These GW-EM survey numbers as they
are today provide only a rough estimates of waiting time for a next event. For
instance, based on the event rates of SGRB (Table 2), GW170817 associated with
an SGRB had a probability of about $20\%$ to be detected during O2 (Table 3).
This has a corresponding estimated event rate of one in 5yr, where time refers
to high data quality observing time, at a sensitivity similar to O2 and O3.
GW190425 with no EM counterpart makes this result more pertinent based on DNS
events seen in GWs.

In the previous section, we estimate potential LVK discovery power for unknown
GW-signals by extrapolating performance of current LIGO runs. According to
(12), improvement in duty-cycle and data-analysis combined may improve our
chances of detecting a transient event of unknown origin by a factor up to
655,000, compared to O3, courtesy of cubic scaling with (realized)
sensitivity. Even a fraction thereof, e.g., at realistic yield factors below
$100\%$, a detector sensitivity improvement of a factor of $2.5$ by O5 and by
employing time-symmetric data-analysis methods (equal sensitivity to ascending
and descending chirps), points to improvements of about ${\cal
O}\left(10^{5}\right)$. For the latter, Fig. B.1 is illustrative, which
depicts identical results for ascending and descending chirps in a spectrogram
of GW170817 by the time-sliced FFT-based MatLab routine spectrogram.

To estimate SGRB event rates using GW-observations, we can use
GW170817-GRB170817A. By (10), during O2 the observational satisfies

$\left<VT\right>=\left<(100\mbox{Mpc})^{3}(269\ \mbox{days})\times
0.45\right>\simeq(100\mbox{Mpc})^{3}(4\ \mbox{months}).$ (23)

Given the unique event GRB170817A during O2, (23) implies an SGRB event rate
of $\sim 3\ \mbox{yr}^{-1}(100\mbox{Mpc})^{-3}$. Taking into account $y\simeq
0.19$, the SGRB event rate increases about a factor of two to $7\
\mbox{yr}^{-1}(100\mbox{Mpc})^{-3}$. However, none of these are consistent
with $1\ \mbox{yr}^{-1}(\mbox{100Mpc})^{-3}$ from EM-surveys. Instead, it
suggests that some DNS systems may merge with no GRB and/or hard to detect
sub-luminous EM counterparts.

Further including O3, 601 operating days, one GRB170817A and GW190824 having
an EM counterpart suggests an even more modest estimation of the true SGRB
event rate. Considering the total coincidence duty cycle of O2 and O3 to be
their geometric mean $52\%$, we obtain $1\ \mbox{yr}^{-1}(100\
\mbox{Mpc})^{-3}$, apparently consistent with EM-surveys. Given the detector
yield factor of about $20\%$, the true event rate is $2-3\
\mbox{yr}^{-1}(100\mbox{Mpc})^{-3}$ \- now perhaps above EM event rates. The
detector yield factor lowers our expectation for another GW170817-like event
in O4, provided the O4 detector yield factor is similar to O3. Conversely, if
O4 sensitivity is about $160\mbox{Mpc}$, another GW170817-like event may be
detected, provided the detector yield factor is better than $24\%$.

Nevertheless, it appears GW170817 and GW190425 are to be seen jointly for our
current survey of DNS events with GW190425 representative for having no EM-
counterparts. This supports the idea that not all DNS mergers produce EM
afterglows.

###  5.2 Relativistic time delay in GRB170817A

For DNS mergers, the life-time of HMNS remnants, dependent on total mass of
the system, is expected to determine diversity of MM-emission that may include
a descending chirp in the case of delayed gravitational collapse to a Kerr
black hole. This delay time, if finite, sets the start time of the central
engine of a putative GRB in (17). By relativistic beaming, an additional delay
time $\Delta t$ appears to the observer,

$\Delta t=t_{GRB}-t_{s,GW}\approx(0.5\ {\rm s})\
r_{10}\left(\dfrac{\Gamma}{70}\right)^{-2},$ (24)

where $r=r_{10}10\mbox{AU}$, $t_{GRB}$ is the starting time of GRB (associated
with GW event), produced by an ultra-relativistic baryon-poor jet with Lorentz
factor $\Gamma$. According to (24), more energetic GRBs are detectable in a
shorter time interval after their production in the central engine.

In (24), for GRB170817A observed is $t_{GRB}-t_{m}=1.7\,$s. According to (17)
the life-time of the initial HMNS is $\sim 0.92\,{\rm s}$ marking the birth
time of a Kerr black hole central engine. This defines a relativistic time-
delay of $\Delta t\sim 0.8\,{\rm s}$ before detecting GRB170817A. Thus,
$\Delta t$ is consistent with the recent HST estimate $\Gamma\simeq 70$
(Mooley et al., 2022), providing additional consistency between GW and EM-
timing highlighted in Table 2 of van Putten & Della Valle (2023). With
$\Gamma$ and $t_{s,GW}$ at hand, for the first time, we can determine

$r_{10}=\left(\dfrac{\Delta
t}{0.5\,s}\right)\left(\dfrac{\Gamma}{70}\right)^{2}\approx 1.6,$ (25)

providing a new observational constraint on external shock models for the
prompt emission of GRB170817A (cf. van Putten & Levinson (2012)). (25) is
discussed in more detail including off-axis seeing of GRB170817A in van Putten
(2023).

| Merger type | Companion | EM | GW
---|---|---|---|---
| FRB | GRB | Kilonova | QNM | descending | $\mathcal{E}_{GW}$
| Breakup | NS | -1 | 1 | 1 | -1 | 1 | NS limited
NS+ | BH | KBH limited
| One-shot | BH | 1 | -1 | -1 | 1 | -1 | 1
| Sweet | WD | -1 | 0 | 0 | -1 | 0 | NS limited
| KBH limited
Table 4: High-energy MM-afterglows to CBC involving an NS. Breakup mergers are
those forming a BH-torus system. When initially rapidly spinning, this may
produce a descending chirp during BH spin-down powering an associated GRB of
similar duration and, possibly, a kilonova. One-shot mergers represent CBC in
which the BH swallows the NS whole in one-shot sans breakup, when the BH is
sufficiently massive and/or spins slowly. The result may be a FRB without the
usual post-merger drama such as GW170817. Sweet mergers refer to NS-WD
mergers. While expected to be more frequent than DNS mergers, their
coalescence is hard to detect by GW-frequencies some four orders of magnitude
below that of NS-NS or NS-BH mergers. Here, “-1”, “0” and “1” indices
represent “impossible”, “possible but without observational evidence”,
“possible with observational evidence”, respectively. QNM refers to Quasi-
Normal Mode ringing of the event horizon of the black hole.

###  5.3 Advances in GW over EM$\nu$ observations

GW170817 is a breakthrough MM-event by the associated GRB170817A and kilonova
AT2017gfo. The key open question is the evolution of the initial HMNS remnant
and the ensuing trigger of the birth of the central engine of GRB170817A. This
poses inevitably the challenge to break the degeneracy between this NS and the
BH that may form by delayed gravitational collapse for which the EM
observations are inadequate (Lazzati, 2020; Piro et al., 2019; Metzger et al.,
2018; Pooley et al., 2018):

  * •

According to (24-25), the prompt GRB emission of GRB170817 originates at a
distance of about 16 AU away from the source, inhibiting a direct view on its
central engine.

  * •

The total energy radiated in the EM channel is $\mathcal{E}_{EM}\simeq
0.5\%M_{\odot}c^{2}$ (Mooley et al., 2022). This is about five times smaller
than the energy $\mathcal{E}_{GW}\simeq 2.5\%M_{\odot}c^{2}$ radiated in
GW170817: EM radiation is vastly subdominant.

  * •

While the kilonova AT2017gfo is representative of neutron-rich mass ejecta
largely from the HMNS, it perhaps includes contributions from the BH-turos
system formed following a potential delayed gravitational collapse of the HMNS
to a Kerr BH given the estimated mass of $\sim 2.4M_{\odot}$. If so, the
inferred life-time of the HMNS (Gill et al., 2019) reduces to an upper bound.

An additional neutrino detection - which is absent for GW170817 - is not
likely to change these conclusions. After all, SN1987A revealed a dominant
neutrino channel, yet its remnant is still unknown (Alp et al., 2018).

Note that this is not so for GWs. A revisit of data around GW170817, detects a
descending chirp (Figs. 7 and 8) at $5.5\sigma$ C.L. signaling the delayed
gravitational collapse of HMNS to a Kerr BH. This interpretation derives from
the energy $\mathcal{E}_{GW}\simeq 3.5\%M_{\odot}c^{2}$ making GW emission the
dominant radiation channel, over the frequency range (4). By the start-time of
(17), it represents the trigger leading to GRB170817A (van Putten & Della
Valle, 2023; van Putten & Levinson, 2003). As elucidated in Fig. 6, this
signal would not be detectable by earlier LIGO searches (Abbott et al., 2017a)
in light of the sensitivity threshold of $650\%M_{\odot}c^{2}$ exceeding the
total mass-energy of the system of about $270\%M_{\odot}c^{2}$ (Sun & Melatos,
2019).

###  5.4 THESEUS: a next-generation GRB mission

The MM-afterglow to GW170817 - a descending branch in GW-emission, GRB170817
and kilonova AT2017gfo - marks a complex transition of the DNS progenitor
involving an HMNS with delayed gravitational collapse to its BH remnant. As
indicated in Fig. 12, GW170817 may be part of a one-parameter family of DNS
mergers in which the massive NS formed in the immediate aftermath of the
merger shows different lifetimes ranging from infinity to zero, marking the
formation of an SMNS and, respectively, prompt collapse to a BH.

Furthermore, GW170817 EM afterglow is a subdominant radiation channel with
$\mathcal{E}_{EM}=0.5\%M_{\astrosun}c^{2}\ll 2.5\%M_{\astrosun}c^{2}$ (Fig. 6,
left panel). For SN1987A the dominant radiation channel is MeV-neutrinos with
somewhat similar energy output $\mathcal{E}_{\nu}\simeq 4\%M_{\astrosun}c^{2}$
(Hirata et al., 1987). Thus, GW170817 and SN1987A demonstrate dominant
emission channels beyond EM-radiation. For total calorimetry, evidently, a
comprehensive EM-GW survey seems pertinent.

A statistically significant sample of MM-observations may resolve the
anticipated diversity in DNS merger afterglow phenomenology, including NS-BH
events whose post-merger evolution lacks an (intermediate) massive NS.
Realizing an EM-GW survey of DNS and NS-BH mergers promises to unambiguously
identify post-merger evolution, derived from total calorimetry across all
radiation channels and detailed event timing in both EM-GW radiation channels.
In fact, consistency in EM-GW timing will be crucial to confirm identification
of the source and central engine of an associated kilonova and (short) GRB
(the latter may further be with or without extended emission, known to satisfy
the Amati-relation of LGRBs (e.g. Norris & Bonnell, 2006; Amati et al., 2002;
Amati, 2006; van Putten et al., 2014)). Notably this applies to determining
the total contribution of merger-induced kilonovae to the cosmic abundances of
the heavy elements produced in the r-process.

Significant prospects to realize such surveys are expected from next-
generation GRB space mission concepts, currently being developed and proposed
by large international collaborations, notably THESEUS (Transient High-Energy
Sky and Early Universe Surveyor) (Amati et al., 2018, 2021) and the Gamov
Explorer (White et al., 2021). These projects aim at overcoming some of the
current limitations in GRB detection, localization, characterization and
multi-wavelength follow-up by extending GRB monitoring down to soft X-rays and
performing autonomous near infrared (NIR) follow-up, arcsec localization and
redshift measurement.

In particular, an unprecedented GRB and transient surveyor like THESEUS would
be capable of providing a substantial advancement in the aforementioned
observational dimensions across all classes of GRBs (Stratta et al., 2018;
Ciolfi et al., 2021) with particular emphasis on events at high-$z$, on- and
off-axis (such as GRB170817A), short GRBs with associated GW-emission, under-
luminous GRBs associated with CC-SNe, which may nevertheless feature
detectable GW-emission from their central engines, and the like.

For short GRBs, the THESEUS X-Gamma ray Imaging Spectrometer (XGIS, 2 keV-20
MeV) and Soft X-ray Imager (SXI, 0.3-5 keV) may detect and localize $\sim
40\,$ events at $<\,15\,$arcmin (90%) and $<\,7\,$arcmin (50%) events over a
3.5 yr time of observation, accounting for all payload observational
constraints. At higher energies ($>\,150$ keV), short GRBs are expected to be
detected at a rate of $\sim 28$/yr outside the FOV of XGIS. While at
relatively course sky localization ($\sim 500\,$deg2), this nevertheless may
combine with GW-observations by temporal coincidences.

THESEUS is ideally equipped to provide accurate redshifts from detections at
low energy from optical/IR afterglow synchrotron emission in the propagation
of the short GRB jet, propagating into the interstellar medium (ISM). A fair
percentage of prompt short GRB detections ($\sim 3$/yr) are expected to be
accompanied by Infrared Telescope (IRT) afterglow detection by THESEUS
InfraRed Telescope (IRT covering 0.7-1.8$\mu$m with a 0.7 m class IR-telescope
over an FoV 15${}^{\prime}\times 15^{\prime}$ with spectroscopic
capabilities). Accurate arcminute sky localization by THESEUS-XSI with, where
available, IRT, will allow facilities such as Athena (Nandra et al., 2013),
E-ELT (Gilmozzi & Spyromilio, 2007) and the SquareKilometreArray (SKA) (e.g.
Janssen et al., 2015) to deeply monitor and further characterize the source,
significantly improving chances to identify the host galaxy with redshift.
Conservative estimated point to a yield of $>26$, assuming follow-up by other
facilities within hours.

Notable is that mere duration of GRBs cannot be representative for all the
properties of these relativistic transients (Amati et al., 2021). A fraction
of technically long GRBs may be produced by DNS mergers and be classified as
SGRB with Extended Emission (SGRBEE). GRB211211A may just be such an SGRBEE at
a distance of 346Mpc with a duration of about 50 seconds, with an associated
kilonova similar to the one during GW170817 (Troja et al., 2022; Mei et al.,
2022; Rastinejad et al., 2022; Yang et al., 2022). Such might also be derived
from some of the NS-BH or WD-NS mergers (Table 4, §4.4). This outlook, taking
us beyond SGRBs alone, suggests a possible increase in rates of joint GW-GRB
detection in joint LVK-THESEUS operation.

For DNS mergers, these redshift determinations supplementing accurate
measurement of chirp mass and associated GW-luminosity provide a potentially
powerful means for a determination of the Hubble parameter in late-time
cosmology (e.g. Abbott et al., 2017f; Guidorzi et al., 2017), independent of
the Local Distance Ladder. Such may be crucial in confirming the present
$H_{0}$-tension between the latter and what is inferred from Planck
$\Lambda$CDM analysis of the CMB (Riess et al., 2022). Above-mentioned EM-GW
surveys promise detailed parameterization of MM-afterglows to DNS events,
which might lead the way to normalize descending branches in GW-emission,
possibly improving the potential of DNS mergers as standard sirens of GW-
emission. A decisive determination of the Hubble parameter in late-time
cosmology may provide us with a probe of (in)stability of de Sitter space in
low-energy quantum cosmology (e.g. van Putten, 2021).

## Acknowledgment

We are thankful to the anonymous reviewer for providing insightful comments on
this manuscript. The authors are also grateful to Cristiano Guidorzi for his
constructive comments on this manuscript, specifically on event rates, and to
Peter Hoeflich for useful information about r-process and heavy elements
abundances. This research is supported by NRF of Korea Nos. 2018044640. M. van
Putten thanks LIGO-Caltech for hospitality over a sabbatical visit, where some
of this work was performed. L. Amati acknowledges support from the Italian
Ministry of Research through grant PRIN MIUR 2020 – 2020KB33TP METE. This
research has made use of S5-O3ab data obtained from the Gravitational Wave
Open Science Center (gw-openscience.org), a service of LIGO Laboratory, the
LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. LIGO
Laboratory and Advanced LIGO are funded by the United States National Science
Foundation (NSF) as well as the Science and Technology Facilities Council
(STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of
Niedersachsen/Germany for support of the construction of Advanced LIGO and
construction and operation of the GEO600 detector. Additional support for
Advanced LIGO was provided by the Australian Research Council. Virgo is
funded, through the European Gravitational Observatory (EGO), by the French
Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions
by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, Spain. The construction and operation of KAGRA are
funded by Ministry of Education, Culture, Sports, Science and Technology
(MEXT), and Japan Society for the Promotion of Science (JSPS), National
Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea,
Academia Sinica (AS) and the Ministry of Science and Technology (MoST) in
Taiwan.

##  Appendix A Observations by LIGO

With the first direct detection of gravitational waves GW150914, LIGO opened a
radically new window of observation to the Universe. The LIGO project has
evolved through various generations of engineering, science and observational
runs since 1995. A primary objective has been to detect gravitational waves
from astrophysical transients events sources with rigorous confirmation across
the two independent detectors at Hanford (H1) and Livingston (L1). LIGO is now
jointly operated with Virgo (EU), KAGRA (Japan), GEO600 (Germany)(Somiya,
2012; Aso et al., 2013; Abbott et al., 2020; Schutz, 2011; Abbott et al.,
2008a), expected to include IndIGO in the near future (Fairhurst, 2014; Riles,
2013).

LIGO has performed six scientific runs (Abbott, B. P. et al., 2009; Abbott et
al., 2008b, 2009a, 2009b; Aasi et al., 2014; Abbott et al., 2008c). During S5
(4/11/2005-1/10/2007), H1 and L1 had duty cycles of $78\%$ and, respectively,
$67\%$ with a coincidence duty cycle of $60-68\%$ (Abbott et al., 2009b;
Abadie et al., 2010). Duty cycle refers to the fraction of total run time with
science quality data (Aasi et al., 2015). For a fiducial $1\%$ output in
gravitational radiation in a CBC relative to the total mass-energy of the
system, the horizon distance of S5 for DNS was estimated to be $\sim
30-35\mbox{Mpc}$, while for binary black holes (BBH) of
$M_{1}=M_{2}=5M_{\astrosun}$ it was about $\sim 100\mbox{Mpc}$ (The LIGO
Scientific Collaboration et al., 2010; Abadie et al., 2010). S6
(7/7/2009-20/10/2010) - the last run before Advanced LIGO - shows improvements
in sensitivity at a reduced duty cycle (Fig. 3, Table A.1). Duty cycles of H1
and L1 during S6 are $50\%$ and, respectively, $47\%$, giving a simultaneous
observing time of 117 days ($25\%$). The S6 horizon distance is estimated to
be $\sim 35-45\mbox{Mpc}$ for equal mass binary neutron stars (Abadie et al.,
2012), and $\sim 300\mbox{Mpc}$ for BBH of $M_{1}=M_{2}=20M_{\astrosun}$ (Aasi
et al., 2013). Nevertheless, no event candidates have been identified during
S5 and S6.

During O1 of Advanced LIGO (aLIGO), GW150914 presented the first-ever direct
detection of gravitational waves. The detectors were three to five times more
sensitive in the frequency range of $100-300\mbox{Hz}$ compared to initial
LIGO (iLIGO) prior (Abbott et al., 2017d, g). O1 extends over 130
observational days Abbott et al. (2016a) (12/09/2015-19/01/2016). Detailed
data-analysis shows GW150914 to be a BBH merger with masses
$M_{1}=35^{+5}_{-3}M_{\astrosun}$ and $M_{2}=30^{+3}_{-4}M_{\astrosun}$ at a
luminosity distance of $440^{+160}_{-180}\mbox{Mpc}$ (Abbott et al., 2016b).
The remnant of this merger is a black hole of mass
$M=62^{+4}_{-3}M_{\astrosun}$. It indicates a total energy release
$\mathcal{E}_{GW}\cong 3.0\pm 0.5M_{\astrosun}c^{2}$ in gravitational waves
(Castelvecchi & Witze, 2016; Abbott et al., 2016c), i.e., a few percent of the
total mass-energy of the system, confirmed by similar events in subsequent
observational runs (Aasi et al., 2014). Following a normalization to the unit
of luminosity

$\displaystyle L_{0}=\frac{c^{5}}{G}\simeq 200,000\
M_{\astrosun}c^{2}/\mbox{s}$ (A1)

of general relativity, where $c$ is the velocity of light and $G$ is Newton’s
constant, GW150914 features a peak GW-luminosity of about (van Putten et al.,
2019b)

$\displaystyle\hat{L}_{GW}\simeq 0.1\%L_{0}$ (A2)

characteristic for BBH events.

Run O1 added a few more BBH coalescence events to the transient catalog of
LIGO. The O1 duty cycle of H1 was $62-70\%$ (LIGO Scientific Collaboration et
al., 2016; Buikema et al., 2020). The observing time of L1 covers $55\%$ of
the total time (LIGO Scientific Collaboration et al., 2016), giving a
coincidence duty cycle of $40-48\%$ (Abbott et al., 2016d; Buikema et al.,
2020). The O1 horizon distance was $\sim 1.3\mbox{Gpc}$ and
$\leavevmode\nobreak\ 70-80\mbox{Mpc}$ for BBH similar to GW150914 and,
respectively, DNS mergers (Martynov et al., 2016). This clearly evidences
aLIGO improvement over iLIGO. The spectral density of LIGO detectors during O1
was $\sim 10^{-23}\,\sqrt{1/\mbox{Hz}}$ (Abbott et al., 2016a). An improvement
by a factor of 3 was expected for the next observational runs (Abbott et al.,
2017d).

By improved sensitivity and spectral density, O2 showed a horizon distance of
$70-100\mbox{Mpc}$ for DNS and $\sim 8\times 10^{-24}\,\sqrt{1/\mbox{Hz}}$
(Davis et al., 2021; Martynov et al., 2016). O2 extended over 269 days
(30/11/2016-25/08/2017) (Abbott et al., 2017e) with an H1 duty cycle of
$60-70\%$ (Buikema et al., 2020; Davis et al., 2021; Abbott et al., 2019a,
2021) and an L1 duty cycle of $60-65\%$ (Davis et al., 2021; Buikema et al.,
2020; Abbott et al., 2019a, 2021), leaving a joint H1L1-duty cycle of
$45-46\%$ (Abbott et al., 2019a; Buikema et al., 2020).

During O2, LIGO successfully detected GW170817: the first DNS coalescence
event with a short gamma ray burst GRB170817A, detected by INTEGRAL and Fermi-
GBM 1.7 seconds after the coalescence. LIGO and Fermi-GBM detections at the
same location in the sky and time provides evidence for a secure association
of GRB170817A with GW170817 at a significance of $4.8\sigma$. This marks
GW170817-GRB170817A as a genuine multi-messenger event further by the
associated kilonova AT2017gfo (Abbott et al., 2017b, e, h; Goldstein et al.,
2017b). GW170817 was detected at the luminosity distance of
$40^{+8}_{-14}\mbox{Mpc}$. This is quite fortuitous considering the DNS merger
rate of $320^{+490}_{-240}\mbox{Gpc}^{-3}\mbox{yr}^{-1}$ obtained from EM
surveys (Abbott et al., 2017e, b). It should be mentioned, however, that
estimating event rates in the Local Universe by extrapolation of rates on
cosmological scales tends to be notoriously uncertain.

Although there has been some discussion about the binary GW170817 (Rueda et
al., 2018; Coughlin & Dietrich, 2019; Tsai et al., 2021), a DNS is the main
candidate (van Putten et al., 2019a; van Putten & Della Valle, 2023; Coughlin
& Dietrich, 2019; Abbott et al., 2019b). The alternative of an NS-BH is
effectively ruled out by the delayed collapse of an HMNS in the initial
aftermath of this merger (van Putten & Della Valle (2023); Fig. 11). GW170817
is a most exceptional event also for a first-ever demonstration of measurement
of the Hubble constant $H_{0}$ by a DNS merger (Abbott et al., 2017a; The LIGO
Scientific Collaboration et al., 2017; Fishbach et al., 2019), independent of
Planck-$\Lambda$CDM and the Local Distance Ladder. O2 increased the total
number of detected CBCs by LVK to 11 events (Abbott et al., 2019a). Two
additional events were reported later by Zackay et al. (2021).

The most recent LIGO run O3 - O3a (01/04/2019-01/10/2019, 184 days) and O3b
(01/11/2019-27/03/2020, 148 days) (Buikema et al., 2020) \- extended over 11
months. O3 represents some notable improvements in duty cycles and sensitivity
(Davis et al., 2021; Abbott et al., 2019). During O3a, the duty cycles of H1
and L1 are $71\%$ (Buikema et al., 2020; Abbott et al., 2021) and,
respectively, $76\%$ (Buikema et al., 2020; Abbott et al., 2021). For O3b, H1
and L1 have duty cycles of $78\%-79\%$ (Buikema et al., 2020; The LIGO
Scientific Collaboration et al., 2021c). Detector sensitivity improved by a
factor of two over O2, i.e. to $\sim 4\times 10^{-24}$ (Davis et al., 2021).
Accordingly, the BBH horizon distance for $M_{1}=M_{2}=30M_{\astrosun}$
increased to $1.15\mbox{Gpc}$ for H1 and $1.425\mbox{Gpc}$ for L1 (Davis et
al., 2021). For DNS, the O3ab horizon distances are $108-115\mbox{Mpc}$ for H1
and $133-135\mbox{Mpc}$ for L1 (Buikema et al., 2020; Abbott et al., 2021; The
LIGO Scientific Collaboration et al., 2021c). Table A.1 lists some statistics
of duty cycles, coincidence duty cycles ($\mathcal{U}$), sensitivity and
horizon distances. These continuous improvements have produced an extensive
LIGO-Virgo catalog of CBC events (The LIGO Scientific Collaboration et al.,
2021c, b; Abbott et al., 2019a, 2021). This catalog was updated later to
include 7 additional BBH (Nitz et al., 2021).

| Total | Individual | Coincidence | Sensitivity | Horizon | Horizon
---|---|---|---|---|---|---
Run | number of | duty cycles |  duty cycles ($\mathcal{U}$) | (spectral density) | distance BBH | distance DNS
| days |  [$\%$] |  [$\%$] |  [$\sqrt{1/Hz}$] | [Mpc] | [Mpc]
S5 | $697^{(1)}$ | H1: 78 | $60^{(1)}$ | $\sim 3\times 10^{{-22}^{(1)}}$ |  3 ($10M_{\astrosun}$)(1) |  30-35 (2,3,4)
|  |  L1: 67 (1) |  68 (4) |  |  90 (4) |
|  |  |  |  |  100 ($5M_{\astrosun}$) (3) |
|  |  |  |  |  120 ($50M_{\astrosun}$) (1) |
S6 |  471 (5) | H1: 50 |  25 (6) | $\sim 2\times 10^{{-23}^{(6)}}$ |  300 ($20M_{\astrosun}$) (7) |  40 (8)
|  |  L1: 47 (9) |  |  |  90 ($5M_{\astrosun}$) (5) |  35-45 (10)
O1 |  130 (11) |  H1: 62-70 (12,13,14) |  40-48 (12,14,15) |  $\sim 8\times 10^{-24}$ (11,16) |  1300 ($30M_{\astrosun}$) (16) |  70-80 (16)
|  |  L1: 55-57 (13,14) |  |  |  |
O2 |  269 (17) | H1: 60-70 |  45 (14,18) |  $\sim 7\times 10^{-24}$ (19) | Farthest event |  70-100 (18)
|  | L1: 60-65 |  |  |  : 2840 (18) |
|  | (11,12,14,19) |  |  |  |
O3a |  184 (12) | H1: 71 |  62 (12) |  $\sim 5\times 10^{-24}$ (19) |  H1: 1150 ($30M_{\astrosun}$) | H1: 108,111
|  |  L1: 75-76 (12,20) |  |  |  L1: 1425 ($30M_{\astrosun}$) | L1: 134,135
|  |  |  |  | (19) | (12,20)
O3b |  148 (12) | H1: 78-79 |  62 (12) |  $\sim 5\times 10^{-24}$ (19) |  H1: 1150 ($30M_{\astrosun}$) | H1: 115
|  |  L1: 78-79 (12,21) |  |  |  L1: 1425 ($30M_{\astrosun}$) |  L1: 133 (21)
|  |  |  |  | (19) |

Table A.1: Summary of some properties of LIGO runs. Listed are statistical
results of S5, S6, O1, O2, O3ab. The H1-L1 detectors have undergone major
improvements during 16 years of operation, most importantly migrating to aLIGO
from iLIGO in sensitivity. The second column shows the total number of days
covered by each run. For instance S5 extended over 697 days
(04.11.2005-01.10.2007). Individual duty cycles of H1 and L1 are presented in
the third column, indicating the fraction of time of operation and in taking
science data (Abbott et al., 2009b). Times when operations overlap defines
coincidence duty cycles, included in the fourth column. A slight improvement
in duty cycles is noticeable during aLIGO, starting from O1. The fifth column
presents LIGO sensitivity during each run based on the spectral density of the
strain ($\sqrt{S_{n}(f)}$) and the improvement therein. Sensitivity
improvements can be expressed by horizon distance out to which can explore a
standard event (BBH or DNS merger) or, equivalently, by detectors sensitivity
defined by strain-noise amplitude (columns 6-7). Values are calculated for
standard events mentioned in parentheses.

(1)Abbott et al. (2009b);(2)The LIGO Scientific Collaboration et al.
(2010);(3)Abbott, B. P. et al. (2009);(4)Abadie et al. (2010);(5)Abadie et al.
(2012);(6)Aasi et al. (2015);(7)Aasi et al. (2013);(8)Abadie et al.
(2012);(9)Aasi et al. (2014);(10)The LIGO Scientific Collaboration & The Virgo
Collaboration (2012);(11)Abbott et al. (2016a);12Buikema et al. (2020);13LIGO
Scientific Collaboration et al. (2016);(14)Abbott et al. (2020),(15)Abbott et
al. (2016d);(16)Martynov et al. (2016);(17)Abbott et al. (2017e);(18)Abbott et
al. (2019a);(19)Davis et al. (2021);(20)Abbott et al. (2021);(21)The LIGO
Scientific Collaboration et al. (2021c)

##  Appendix B Time-symmetric analysis method

An important component in studying transients data is employing a method of
analysis with equal sensitivity to ascending and descending signals: a time-
symmetric method. This is essential in un-modeled searches of GW signals from
unknown sources. A prototype of time-symmetric method is time-sliced FFT
showing identical results for the strength of a merger signal at each
frequency in comparison with its time reversal, illustrated in Fig. B.1.
Optimal performance of time-sliced FFT depends on the density of the filter in
each frequency. But FFT is most sensitive to narrow band frequencies.
Specifically the time-frequency uncertainty equation $\Delta t\Delta f\simeq
1$ puts restrictions on the capability of this method in detecting signals
with frequencies varying in time over an interval of phase-coherence $\Delta
t$. Butterfly matched filtering is a time-symmetric method of analysis, going
one step further, that covers the time-varying frequencies with a filter dense
in both $f$ and $\dot{f}$ (van Putten et al., 2014; van Putten, 2023).

Figure B.1: (Left panel.) The spectrogram of 32 s of data around GW170817
produced by GWXplore, a user-friendly laptop App employing conventional time-
sliced FFT. Here, the FFT window size is set to be $0.25\,{\rm s}$, whitening
bandwidth $B=4\mbox{Hz}$ and low-pass and high-pass frequencies are
$10\,\mbox{Hz}$ and $1700\,\mbox{Hz}$, respectively. (Right panel.) FFT-based
analysis of time-reversed data produces the same, demonstrating time-symmetry
in the analysis relevant to targeting ascending and descending chirps from CBC
including the potential descending emission from the spin-down of merger
remnants and central engines of CC-SNe. We have made partial use of the color
maps in Thyng et al. (2016).

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