# Detectability of Primordial Black Hole Binaries at High Redshift

Qianhang Ding<EMAIL_ADDRESS>Department of Physics, The Hong Kong
University of Science and Technology, Hong Kong, P.R.China  Jockey Club
Institute for Advanced Study, The Hong Kong University of Science and
Technology, Hong Kong, P.R. China

###### Abstract

We show that the gravitational wave signals from primordial black hole (PBH)
binaries at high redshift can be detected. The detectability of PBH binaries
is enhanced by redshift bias and more PBH binaries at high redshift. The
initial clustering of PBHs is also included and enhances the effectively
detectable mass ranges of PBHs at high redshift. Future observations on the
gravitational wave at high redshift by space-based detectors such as LISA and
SKA can constrain the fraction of PBHs in dark matter and PBHs initial
distribution.

##  I Introduction

Primordial black holes (PBHs) were born from primordial perturbations in
highly overdense regions by gravitational collapse Hawking:1971ei ;
Dovich:1966no ; Carr:1974nx . Among modern mechanisms of PBHs formation, the
inflationary origin Yokoyama:1995ex ; Yokoyama:1998pt ; Kawasaki:2016pql ;
Inomata:2016rbd ; Cai:2018tuh  tightly connects the PBHs properties such as
mass distribution Young:2014ana , abundance distribution Belotsky:2014kca ;
Belotsky:2018wph ; Bringmann:2018mxj ; Ding:2019tjk , the fraction of PBHs in
dark matter Carr:2016drx  with the early universe evolution. Thus, the
observational constraint on PBHs provides important clues for inflationary
model building.

PBHs play an important role in understanding the early universe. Directly
observing signals from PBHs is the key step. In order to achieve that,
distinguishing the PBHs and astrophysical black holes is essential. The method
to distinguish black holes comes from the black holes intrinsic properties
include mass, spin, charge and spacetime properties such as the spatial
distribution and redshift.

One way to distinguish black holes comes from the mass of black holes. The
mass of astrophysical black holes is heavier than a particular mass (around
$3$ solar masses Rhoades:1974fn ; Kalogera:1996ci ). Some specific mechanisms
can produce around $1$ solar mass astrophysical black holes, see
Kouvaris:2018wnh ; Dasgupta:2020mqg . However, PBHs have a different mass
range. The mass of PBHs can be roughly estimated as follows Carr:2016drx ,

$\displaystyle M\sim\frac{c^{3}t}{G}\sim 5.03\times
10^{4}\left(\frac{t}{1\mathrm{s}}\right)M_{\odot}~{},$ (1)

where $t$ denotes the time when the corresponding perturbation returns to the
horizon. Eq. (1) shows that PBHs born before $\mathcal{O}(10^{-4})$ second
since the hot big bang have mass less than $1$ solar mass, which is less than
the minimal mass of astrophysical black holes. Considering Hawking radiation
Hawking:1974rv , PBHs with an initial mass of less than $10^{15}\mathrm{g}$
have already evaporated by now, as a result, PBHs with an initial mass of more
than $10^{15}\mathrm{g}$ and less than $1$ solar mass still exist and produce
various distinct signals in the present universe. In detecting sub-solar-mass
PBHs, microlensing effect Paczynski:1985jf ; Alcock:1993eu ; Aubourg:1993wb ;
Alcock:2000ph , femtolensing effect Barnacka:2012bm , disruption of white
dwarfs Graham:2015apa , disruption of neutron stars Capela:2013yf ;
Kouvaris:2013kra ; Pani:2014rca  and so on put constraints on the fraction of
sub-solar-mass PBHs in dark matter. Some interactions between black holes and
stars can help detect sub-solar-mass PBHs such as superradiance instability
around black holes Baumann:2019ztm ; Ding:2020bnl . However, these
observational effects such as microlensing are extremely weak, only a few
events give a weak constraint on the fraction of PBHs in the dark matter.

The other way to distinguish black holes depends on the redshift of signals
from black holes. The astrophysical black holes forms after the death of
Population III stars which are the first generation stars. Their observational
signals exist at $z<20$ Yoshida:2003ab . In contrast, the redshift of PBHs
goes through the whole history of the universe after the big bang. From this
point, the signals from black holes with $z>20$ should come from the PBHs. In
detecting high redshift signals, gravitational wave is a powerful detection
channel which carries information to travel through cosmological distance due
to weak interaction between gravity and matter. Since the first gravitational
wave event from the merger of binary black holes Abbott:2016blz , the binary
black holes system is the important gravitational wave source, so the
detection of gravitational waves from PBH binaries is significant, which can
bring us high redshift signals.

Recently, stellar mass PBHs have received renewed interest Bird:2016dcv ;
Clesse:2016vqa ; Sasaki:2016jop  since the gravitational wave events detected
by LIGO Abbott:2007kv  and Virgo TheVirgo:2014hva . Intermediate-mass black
holes show clues in LIGO and Virgo detection Abbott:2020tfl . A supermassive
black hole detection is being discussed Arzoumanian:2020vkk . We propose to
search for high redshift gravitational wave signals from PBH binaries, which
covers a large mass range of PBHs at high redshift. Due to the frequency of
gravitational wave from high redshift source is highly redshifted, the usage
of high frequency ground-based gravitational wave detector is limited.
Targeting high redshift signals needs the low frequency space-based
gravitational wave detector such as LISA LISA , DECIGO Kawamura:2006up , SKA
Dewdney et al., (2009). The detectability of PBHs shows the possibility in
gravitational wave astronomy. There are some existing studies Nakamura:2016hna
; Mandic:2016lcn  about gravitational wave from high redshift PBH binaries. In
addition to their studies, we present the potential detectable mass range of
PBH binaries at high redshift and include the effect of initial abundance
distribution of PBHs on detectable mass range.

This paper is organized as follows. In Sect. II, we show that sensitivity of
gravitational wave detectors can detect the high redshift gravitational wave
from binary black holes. In Sect. III, we estimate the event rate of
detectable signals from PBH binaries at high redshift. In Sect. IV, we combine
the gravitational wave detector sensitivity and PBH binaries event rate to
obtain the effectively detectable mass range for PBH binaries at high
redshift. We also discuss potential improvements in detectability by the
initial clustering of PBHs.

##  II GW sensitivity at high redshift

The sensitivity of gravitational wave detectors is determined by the amplitude
of gravitational wave and noise strain of the detector. The amplitude of
gravitational wave from binary system can be expressed as

$\displaystyle
h=\frac{4}{d_{L}(z)}\bigg{(}\frac{G\mathcal{M}_{c}(1+z)}{c^{2}}\bigg{)}^{5/3}\bigg{(}\frac{\pi
f}{c}\bigg{)}^{2/3}~{}.$ (2)

Here, $d_{L}(z)$ is the luminosity distance of gravitational wave source in
flat FRW universe, which can be calculated from
$d_{L}(z)=(1+z)\int_{0}^{z}c/H(z^{\prime})dz^{\prime}$, $G$ is the Newton’s
constant, $c$ is the speed of light, $\mathcal{M}_{c}$ is the chirp mass of
the binary system, which is defined as
$\mathcal{M}_{c}\equiv(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$, and $f$ is the
frequency of the gravitational wave.

The amplitude of the gravitational wave can be boosted by redshifted chirp
mass $(1+z)\mathcal{M}_{c}$, which introduces an effect called redshift bias
Rosado:2015voo . Redshift bias of rest frame chirp mass can improve the
detectability of binary system at high redshift. It works as follows. The
amplitude and frequency of gravitational waves increase with time. The
gravitational wave with higher frequency stays at a later stage in the
inspiral phase, which has a larger amplitude than those with low frequency.
Then, the frequency of high redshift gravitational waves is redshifted to the
observable frequency at present which follows
$f_{\mathrm{rest}}=(1+z)f_{\mathrm{obs}}$. The amplitude of high redshift
gravitational wave also decreases because of large luminosity distance
$d_{L}(z)$. Consequently, at observable frequency band, amplitude of high
redshift gravitational wave depends on two factors: later stage in the
inspiral phase and $d_{L}(z)$, where amplitude firstly increases and then
decreases, which implies amplitude of gravitational wave doesn’t always
decrease monotonically with redshift, it has a minimum value at
$z_{\mathrm{min}}$. In $\Lambda\mathrm{CDM}$ model, $z_{\mathrm{min}}=2.63$
Rosado:2015voo . Redshift bias can effectively improve the detectability of
gravitational wave at high redshift. This effect is shown in Fig. 1.

Figure 1:  Gravitational waves with observed frequency $10^{-4}\mathrm{Hz}$ at
redshift $z=0.03$, $10$, $20$, also see in Rosado:2015voo . The signals at
high redshift stay later stage in binary evolution whose frequency is
$f_{\mathrm{rest}}=(1+z)f_{\mathrm{obs}}$, which increase the amplitude for
observation.

In order to detect a gravitational wave event with a high confidence level,
the signal-to-noise ratio (SNR) should exceed a threshold during observation.
We follow Rosado:2015voo , where the optimal SNR is defined as

$\displaystyle\mathrm{S/N}=\sqrt{4\int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\frac{|\tilde{h}(f)|^{2}}{S_{n}(f)}df}~{}.$
(3)

Here, $\tilde{h}(f)$ is the Fourier transform of $h(t)$ based on the
stationary phase approximation Droz:1999qx , which is

$\displaystyle\tilde{h}(f)=\sqrt{\frac{5}{24}}\frac{(G\mathcal{M}_{c}(1+z))^{5/6}}{\pi^{2/3}c^{3/2}d_{L}(z)}f^{-7/6}~{}.$
(4)

The $S_{n}(f)$ is the noise strain of the detector. The $f_{\mathrm{min}}$ is
the initial observed frequency and $f_{\mathrm{max}}$ is the final frequency
during the observation time, which can be calculated from the evolution of
observed gravitational wave frequency

$\displaystyle\frac{df}{dt}=\frac{96\pi^{8/3}(G\mathcal{M}_{c}(1+z))^{5/3}}{5c^{5}}f(t)^{11/3}~{}.$
(5)

The optimal signal-to-noise ratio should exceed a conservative threshold of
$8$, which makes sure the detection probability $>95\%$ and a false alarm
probability $<0.1\%$ .

Due to the redshift bias effect, $\mathrm{S/N}$ doesn’t always decrease
monotonically with redshift. It also has a minimal value at
$z_{\mathrm{min}}$. When $z<z_{\mathrm{min}}$, optimal SNR decreases with $z$.
Otherwise, optimal SNR increases with $z$ for the case $z>z_{\mathrm{min}}$.
In order to obtain $z_{\mathrm{min}}$, we need calculate
$\partial(\mathrm{S/N})^{2}/\partial z=0$, which can be expressed as
following,

$\displaystyle\frac{\partial\mathrm{S/N}^{2}}{\partial
z}=4\int_{t_{\mathrm{min}}}^{t_{\mathrm{max}}}\frac{1}{S_{n}(f)}\frac{\partial}{\partial
z}(|\tilde{h}(f)|^{2}\frac{df}{dt})dt=0~{}.$ (6)

Applying Eq. (4) and (5) into Eq. (6), we can get equation for
$z_{\mathrm{min}}$,

$\displaystyle(1+z_{\mathrm{min}})\frac{\partial\ln d_{L}(z)}{\partial
z}\bigg{|}_{z_{\mathrm{min}}}=\frac{5}{3}~{}.$ (7)

In $\Lambda\mathrm{CDM}$ model with $\Omega_{\mathrm{m}}=0.315$ and
$\Omega_{\Lambda}=0.685$ Aghanim:2018eyx , $z_{\mathrm{min}}=2.63$. The Fig. 2
of Rosado:2015voo  shows the same result.

Combine Eqs. (3)(4)(5) with the condition $\mathrm{S/N}>8$, we can get
detectable mass ranges at different redshifts in Fig. 2.

Figure 2:  The detectable mass range at different redshifts for gravitational
wave detectors aLIGO, LISA and SKA .The SNR of each binary ($\mathcal{M}_{c}$,
$z$) is larger than the threshold of $8$. The typical observation time
$1\mathrm{month}$ and $10\mathrm{year}$ for LISA, $1\mathrm{year}$ and
$100\mathrm{year}$ for SKA have been shown as dashed lines, where
$\mathrm{S/N}=8$. The shadow regions above dashed lines show the detectable
mass ranges within different observation time.

In order to observe the mass range as large as possible, the frequency range
in Eq. (3) between $f_{\mathrm{min}}$ and $f_{\mathrm{max}}$ should be large
so that $\mathrm{S/N}>8$. We set the $f_{\mathrm{min}}$ as the minimal
detectable frequency of the detector. The $f_{\mathrm{max}}$ is set to the
minimal quantity between the maximal detectable frequency of the detector and
the maximal redshifted inspiral phase frequency
$f_{\mathrm{insp}}^{\mathrm{max}}/(1+z)\mathrm{Hz}$. Here, the maximal
inspiral phase frequency in rest frame can be estimated as
$f_{\mathrm{insp}}^{\mathrm{max}}=(a\eta^{2}+b\eta+c)/\pi GM$ Chernoff:1993th
; Ajith:2009bn ; Zhu:2011bd . The symmetric mass ratio $\eta$ is defined as
$\eta\equiv m_{1}m_{2}/M^{2}$, $M\equiv m_{1}+m_{2}$ and the coefficients
$a=0.29740$, $b=0.04481$, $c=0.09556$ (See Table 1 of Ajith:2007kx ). The
$S_{n}(f)$ for each detector can be obtained from Fig. A2 in Moore:2014lga .

In Fig. 2, there is a redshift range corresponding with a PBH mass. For low
mass PBH binaries, amplitude of gravitational wave is small, which needs a
higher $f_{\mathrm{max}}$ to increase SNR, then the maximal detectable
redshift should be small so that $f_{\mathrm{max}}$ isn’t compressed by
$f_{\mathrm{insp}}^{\mathrm{max}}/(1+z)\mathrm{Hz}$. The boundary of shadow
region corresponds with $\mathrm{S/N}=8$. The PBH binary at higher redshift
can not be detected due to the condition $f_{\mathrm{min}}\geq
f_{\mathrm{max}}$ in Eq. (3), which results from the maximal inspiral
frequency of binary system is redshifted that is less than the minimal
frequency of the detector. For a practical observation, the observation time
$T_{\mathrm{obs}}$ is limited, so we choose different observation time to
determine the frequency range as following,

$\displaystyle
f_{\mathrm{max}}-f_{\mathrm{min}}=\int_{t_{\mathrm{ini}}}^{t_{\mathrm{ini}}+T_{\mathrm{obs}}}\frac{df}{dt}dt~{}.$
(8)

The $f_{\mathrm{min}}$ is chosen from the frequency band of detector, then
$f_{\mathrm{max}}$ can be obtained from Eq. (8). The typical observation time
has been shown as dashed line in Fig. 2. The shadow regions above dashed lines
show detectable mass ranges of detectors within the observation time. We can
find dashed lines are not monotonic, which is caused by the redshift bias
effect. According to Eq. (6) and (7), optimal SNR function is monotonically
decreasing for $z<2.63$ and increasing for $z>2.63$ in our
$\Lambda\mathrm{CDM}$ universe Aghanim:2018eyx , for detecting the signal with
$\mathrm{S/N}=8$, the chirp mass of PBH binary need increase for $z<2.63$ and
decrease for $z>2.63$.

##  III Event rate of PBH binaries

In order to observe the PBHs at high redshift, not only should gravitational
wave sensitivity be high but also the event rate of PBH binaries system should
be large enough. The estimation of PBH binaries event rate follows
Sasaki:2016jop , the condition for decouple of PBH binaries from FRW
background at matter-radiation equality $z=z_{\mathrm{eq}}$ is

$\displaystyle x<f^{1/3}_{\mathrm{PBH}}\bar{x}~{}.$ (9)

Here, $x$ is the physical distance between two nearby black holes,
$f_{\mathrm{PBH}}$ is the fraction of PBHs in the dark matter. $\bar{x}$ is
the mean separation of PBHs at matter-radiation equality, which is determined
by PBH mass $M_{\mathrm{BH}}$ and energy density
$\rho_{\mathrm{BH}}(z_{\mathrm{eq}})$ as
$\bar{x}=(M_{\mathrm{BH}}/\rho_{\mathrm{BH}}(z_{\mathrm{eq}}))^{1/3}$. After
interacting with the third black hole, the probability distribution function
of binary system with respect to the major axes $a$ and eccentricity $e$
Sasaki:2016jop  is

$\displaystyle
P(a,e)=\frac{3}{4}\frac{f^{3/2}_{\mathrm{PBH}}a^{1/2}}{\bar{x}^{3/2}}\frac{e}{(1-e^{2})^{3/2}}~{}.$
(10)

After long time evolution, the major axes shrinks to a suitable quantity that
the frequency of gravitational wave lies in the frequency band of detector.
Following Peters:1963ux ; Peters:1964zz , we assume that $e$ keep the constant
during inspiral phase, the later major axes can be expressed as

$\displaystyle a^{4}=a_{\mathrm{ini}}^{4}-\frac{256}{5}\frac{G^{3}\mu
M^{2}}{c^{5}(1-e^{2})^{7/2}}t~{},$ (11)

where $a_{\mathrm{ini}}$ is the initial major axes of PBH binary, $t$ is the
evolution time, $M\equiv m_{1}+m_{2}$, $\mu\equiv m_{1}m_{2}/M$. Applying the
Kepler’s third law with Eq. (11), Eq. (10) can be written into

$\displaystyle P(t,e)=\frac{48}{5}\frac{f^{3/2}_{\mathrm{PBH}}G^{3}\mu
M^{2}}{c^{5}\bar{x}^{3/2}T^{5/8}}\frac{e}{(1-e^{2})^{5}}~{}.$ (12)

Here, $T$ is defined as

$\displaystyle
T\equiv\left(\frac{GM}{\pi^{2}f_{GW}^{2}}\right)^{4/3}+\frac{256}{5}\frac{G^{3}\mu
M^{2}}{c^{5}(1-e^{2})^{7/2}}t~{}.$ (13)

$f_{GW}$ is the frequency of gravitational waves. Then, integrate the $e$ from
$0$ to the maximum eccentricity to get a probability distribution function of
time $P(t)$, the maximum eccentricity is determined by binary formation
conditions in Sasaki:2016jop .

The event rate of gravitational wave signals from PBH binaries can be
calculated from $P(t)$ and average number density of PBHs in comoving volume
$n_{\mathrm{BH}}$,

$\displaystyle\mathrm{event~{}rate}=n_{\mathrm{BH}}P(t)=\frac{3H_{0}^{2}}{8\pi
G}\frac{\Omega_{\mathrm{BH}}}{M_{\mathrm{BH}}}P(t)~{}.$ (14)

In order to obtain the event rate at different redshifts, the time-redshift
relation along the line of sight is required, which can be calculated from,

$\displaystyle
t(z)=\int_{z}^{\infty}\frac{dz^{\prime}}{H(z^{\prime})(1+z^{\prime})}~{}.$
(15)

Applying Eq. (15) into $P(t)$. Then, we can get the event rate of PBH binary
at different redshifts in Fig. 3.

Figure 3:  The event rate for PBH binaries with $100$ solar masses and the
observed frequency is $10^{-4}\mathrm{Hz}$ at different redshifts. The event
rate is measured in comoving volume with the unit of
$\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$.

It shows the increasing event rate of PBH binaries at higher redshift. This is
caused by $P(a)\propto a^{-1/4}$ Sasaki:2016jop , there are more PBH binaries
having small major axes, when binaries form, which emit high frequency
gravitational wave and merge at high redshift. So more high frequency
gravitational wave events propagate from high redshift to the present that are
redshifted and lied in frequency band of detectors. (also see in Ali-
Haimoud:2017rtz ; Kavanagh:2018ggo ).

More event rates of PBH binaries at high redshift in Fig. 3 shows that it is
possible to detect the signal from PBH binaries at high redshift, even though,
the fraction of PBHs in dark matter $f_{\mathrm{PBH}}$ is strongly constrained
by astronomy observation. Then, we can compare high redshift
$f_{\mathrm{PBH}}-M_{\mathrm{BH}}$ relation with the astronomy observation.
Here, we consider the monochromatic PBH mass function in Eq. (14). Considering
the detectable merger rate of binary black holes is
$2-53\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$ for aLIGO LIGOScientific:2016kwr , a
reasonable and detectable event rate is set to
$1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$. Then we can get the relation among the
fraction of PBHs in dark matter $f_{\mathrm{PBH}}$, PBH mass $M_{\mathrm{BH}}$
and redshift $z$ from Eq. (14), which is shown in Fig. 4.

Figure 4:  The fraction of PBHs in dark matter with respect to PBH mass
$M_{\mathrm{BH}}$ at different redshifts, to reach an event rate of
$1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$. Limits from EROS Tisserand:2006zx ,
Caustic Oguri:2017ock , UFDs Brandt:2016aco , Eridanus II Brandt:2016aco ,
Plank Ali-Haimoud:2016mbv , wide-binary disruption Quinn:2009zg , accretion
limits come from X-ray binaries Inoue:2017csr , millilensing of quasars
Wilkinson:2001vv , generation of large-scale structure through Poisson
fluctuations Afshordi:2003zb  and dynamical friction on halo objects
Carr:1997cn  are also plotted

Three typical redshift curves $z=0,~{}20,~{}2000$ are shown, which corresponds
with a fixed event rate $1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$. We can find
that at the same $M_{\mathrm{BH}}$, the higher redshift, the lower
$f_{\mathrm{PBH}}$ we need to reach an detectable event rate of
$1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$, which means high redshift PBH binaries
can be detected without ruling out by astronomy constraints. It results from
that there are more event rates at high redshift, when we fix
$f_{\mathrm{PBH}}$ as shown in Fig. 3. Then we don’t need a large
$f_{\mathrm{PBH}}$ to produce an detectable event rate of
$1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$ at high redshift.

##  IV Detectable mass range at high redshift

Above all, we have discussed the detectable mass ranges of detectors aLIGO,
LISA, SKA and the event rate of PBH binary at different redshifts. Now, we can
combine the two results to obtain the effectively detectable region for PBHs
at different redshifts as shown in Fig. 5.

Figure 5:  The effectively detectable regions for PBH binaries at different
redshifts. The solid lines give the boundary of parameter regions that
detectors can observe. The black dashed lines give the critical mass with an
event rate of $1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$ that we can observe for
different $f_{\mathrm{PBH}}$. The shadow regions below the black dashed lines
and above the solid lines are the effectively detectable regions for PBH
binaries.

In drawing the effective regions, we assume the effectively detectable range
follows the conditions,

$\displaystyle\mathrm{event~{}rate}>1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}~{},~{}~{}~{}f_{\mathrm{PBH}}<f_{\mathrm{unc}}~{}.$
(16)

Here, we follow Ding:2019tjk ; LIGOScientific:2016kwr , the minimal detectable
event rate is set to $1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$. $f_{\mathrm{unc}}$
is the unconstraint fraction of PBHs in the dark matter where we take
$10^{-3}$, $10^{-4}$, $10^{-5}$ for illustration. The black dashed lines
correspond with an event rate of $1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$. The
shadow regions show the effectively detectable mass ranges at high redshift.
One thing we need to pay attention to is the maximum detectable redshift is
$z=3000$, which comes from the PBH binary model Sasaki:2016jop , the PBH
binaries formed at matter-radiation equality $z_{\mathrm{eq}}=3000$. This
mechanism can hardly produce PBHs in radiation-domination epoch at
$z>z_{\mathrm{eq}}$, which makes it difficult in observing a larger mass range
and detecting PBH binaries on SKA.

The inclusion of initial spatial clustering of PBHs changes the story. The
initial clustering of PBHs forms the high density PBHs regions in radiation-
domination epoch. As we have shown in Ding:2019tjk , the initial clustering of
PBHs can be constructed in multi-stream inflation Li:2009sp ; Li:2009me ;
Wang:2010rs ; Afshordi:2010wn . During multi-stream inflation, inflaton
travels along two inflation trajectories, inflation potential on one
trajectories can produce PBHs, the others can not produce PBHs. In the PBHs
clustering model Ding:2019tjk , the PBHs clustering regions are denoted by
$\beta$, which is the volume fraction of PBHs regions in the observed
universe. The redshift of matter-radiation equality in PBHs rich regions can
be calculated by increasing the PBHs fraction in dark matter in PBHs rich
region $f_{\mathrm{PBH}}\rightarrow f_{\mathrm{PBH}}/\beta$ as following

$\displaystyle
1+z_{\mathrm{PBH},\mathrm{eq}}=(1-f_{\mathrm{PBH}}+\frac{f_{\mathrm{PBH}}}{\beta})(1+z_{\mathrm{eq}})~{}.$
(17)

The high density PBHs region improves the probability of PBH binaries
formation by increasing the PBHs density, then event rate of PBH binaries can
be enhanced due to a larger PBH binaries density, which results in that
massive PBH binaries can meet the Eq. (16), the result is shown in Fig. 6.

Figure 6:  The effectively detectable regions for PBH binaries at different
redshifts with initial clustering. The solid lines give the boundary of
parameter regions that detectors can observe. The black dashed lines give the
critical mass with an event rate of $1\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$ and
$f_{\mathrm{PBH}}=10^{-3}$ that we can observe for different $\beta$. The
shadow regions below the black dashed lines and above the solid lines are the
effectively detectable regions for PBH binaries.

The initial clustering of PBHs expands the detectable mass range for PBH
binaries and improves the detectability of PBH binaries in SKA. However, the
clustering volume fraction $\beta$ cannot be infinitesimal, this could result
in few PBHs surviving at present and most of them merged at very high redshift
which leaves weak signals to detectors.

##  V Conclusion

In this paper, we show that the PBH binaries can be detected by the high
redshift $z>20$ gravitational wave signals. During the propagation of
gravitational wave from high redshift to the present, the frequency of
gravitational wave is decreasing so that lies in the frequency band of
gravitational wave detectors. Then amplitude of redshifted high frequency
gravitational wave can significantly enhance the $\mathrm{S/N}$ at
$z>z_{\mathrm{min}}$, where $z_{\mathrm{min}}=2.63$ in $\Lambda\mathrm{CDM}$
model. This phenomena is detailed studied in Rosado:2015voo . In order to
detect gravitational wave signal from PBH binaries within practical
observation time. The event rate of gravitational wave from PBH binaries is
estimated. We consider the model in Sasaki:2016jop , the PBH binaries events
are enhanced at high redshift which can achieve
$90\mathrm{cGpc}^{-3}\mathrm{yr}^{-1}$ at $z=20$ for PBHs mass $100$ solar
masses with $f_{\mathrm{PBH}}=10^{-3}$ in Fig. 3. The enhancement of
gravitational wave signals and the event rate from PBH binaries at high
redshift improve the detectability of PBH binaries. Then, we study the effect
of initial clustering of PBHs on its detectability as we have studied in
Ding:2019tjk . An enhancement in effectively detectable mass ranges appears by
the initial clustering of PBHs. The effectively observable mass range covers
$20-10^{3}$ solar masses at $z=20$ within ten years observation time on LISA.
This mass range can be enhanced to $20-10^{6}$ solar masses with initial
clustering $\beta=10^{-4}$. The wide mass range shows that it is possible to
detect the PBH binaries on LISA and SKA at high redshift.

Based on the possibility of PBH binaries detection at high redshift, we can
directly observe the signals from the PBHs. This can help us build an early
universe model. The mass function of PBHs can be obtained from the chirp mass
of gravitational wave in the rest frame and gives clues to the inflation
potential Young:2014ana . The abundance distribution of PBHs can be obtained
from the position distribution of gravitational wave events, which gives hint
for inflation model with position properties, such as the initial clustering
of PBHs from multi-stream inflation Ding:2019tjk . The fraction of PBHs in the
dark matter can be constrained by the high redshift gravitational wave event
rate, which can tell us the potential dark matter candidates.

In this work, we only consider the monochromatic PBH mass function. The
extended mass function of PBHs can form extreme mass ratio inspiral, which can
produce different signals. In detecting more PBHs events at high redshift,
stochastic gravitational wave background from PBHs formation need to be
deducted Wang:2016ana ; Cai:2019jah ; Nakama:2020kdc . Due to the high
redshift of the gravitational wave, no electromagnetic counterpart can be
observed along with the gravitational wave signal. We cannot directly observe
the luminosity distance of gravitational wave. However, the interaction
between Population III stars with PBHs could be a potential source for
electromagnetic signals at $z<20$, which helps us understand PBHs evolution at
high redshift.

## Acknowledgements

I thank Yi Wang for detailed suggestions, Lingfeng Li and Xi Tong for useful
comments. I would like to thank Xingwei Tang for her kindly encouragement and
suggestions.

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# Self-supervised learning for gravitational wave signal identification

Hao-Yang<EMAIL_ADDRESS>Yu-Tong
<EMAIL_ADDRESS>1 International Center for Theoretical Physics
Asia-Pacific, Beijing/Hangzhou, China  2 School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  3 School of Physics Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

The computational cost of searching for gravitational wave (GW) signals in low
latency has always been a matter of concern. We present a self-supervised
learning model applicable to the GW detection. Based on simulated massive
black hole binary signals in synthetic Gaussian noise representative of space-
based GW detectors Taiji and LISA sensitivity, and regarding their
corresponding datasets as a GW twins in the contrastive learning method, we
show that the self-supervised learning may be a highly computationally
efficient method for GW signal identification.

##  I Introduction

The field of gravitational wave (GW) detection has seen an explosion of
compact binary coalescence signals over the past several years
LIGOScientific:2016aoc ; LIGOScientific:2016sjg ; LIGOScientific:2017bnn ;
LIGOScientific:2017vwq ; LIGOScientific:2018mvr ; LIGOScientific:2020ibl .

Recently, 93 GW events have been reported LIGOScientific:2021djp . It is
expected that over the coming years, more GW events, including binary black
holes (BBH), binary neutron stars (BNS), as well as other exotic sources will
be observed more frequently. As such, the need for more efficient search
methods will be more important as the detectors improve in sensitivity.

It is well-known that identifying signal is achieved, in part, using a
technique known as template-based matched filtering, which uses a bank
Brown:2012qf ; DalCanton:2017ala ; Harry:2009ea ; DalCanton:2014hxh ;
Ajith:2012mn  of template waveforms Sathyaprakash:1991mt ; Taracchini:2013rva
; Privitera:2013xza ; Blanchet:2013haa  spanning a large astrophysical
parameter space. The corresponding waveform models that cover the inspiral,
merger, and ringdown phases of a compact binary coalescence are based on
combining post-Newtonian theory Arun:2008kb ; Buonanno:2009zt ; Mishra:2016whh
, the effective-one-body method Buonanno:1998gg , and numerical relativity
simulations Pretorius:2005gq . However, the algorithms used by the search
pipelines to make detections are computationally expensive, since the large
parameter space, as well as analysis of the high frequency components of the
waveform, inevitably result in large computational cost.

Recently, the deep learning has been in popularity Goodfellow:2014upx ;
Simonyan:2014cmh ; Yuille:2014lcc ; Fergus:2013mdz ; Szegedy:2014nrf , which
is a subset of machine learning. The advantage of deep learning is that the
computationally intensive stage is pre-computed, so it is able to performing
analyses rapidly. Its successful implementations include image processing
Efros:2016aae ; Karpathy:2014lff , medical diagnosis, in particular GW field,
such as glitch classification Mukund:2016thr ; Zevin:2016qwy ; George:2017fbn
and signal identification George:2016hay ; Wei:2019zlc ; Xia:2020vem ;
Lin:2020aps ; Liao:2021vec ; Ruan:2021fxq ; Bacon:2022lsm ; Zhao:2022qob ;
Schafer:2022dxv ; Murali:2022sba ; Nousi:2022dwh ; Shen:2019vep ; Ren:2022hny
.

However, the deep learning method currently used for searching for the GW
signals is often limited to building supervised learning models, which are
indeed trained on scarcely labelled or simulated data and thus suffer from
biases or small training sizes when applied to new or full datasets. Thus it
is significant to consider the tailored deep learning models in a self-
supervised, semi-supervised or unsupervised manner Hinton:2002knh ;
Maaten:1912mlm .

In this paper, we report the construction of a self-supervised learning that
can reproduce the searching for massive black hole binary (MBHB) GW signals. A
contrastive learning method, the GW twins, which applies redundancy-reduction
to self-supervised learning Deny:2103:mld , is proposed. Based on simulated
MBHB signals in synthetic Gaussian noise representative of space-based GW
detectors Taiji Hu:2017mde  and LISA LISA:2017pwj  sensitivity, we investigate
whether the self-supervised learning can efficiently identify a signal present
in the data, or the data contain only detector noise.

##  II Method

###  II.1 Description of GW twins

In self-supervised learning for GW detection, our GW twins operates on a joint
embedding of augmentation vectors, see Fig.1. In detail, it produces two
augmentation vectors for all data of a batch $h$ sampled from a dataset. In
our case, we consider the batch of $h$ as an injected GW waveform signal. The
augmentation vectors are obtained with a distribution of data augmentations
$\mathcal{T}$ (the light blue area in Fig.1). Two batches of augmentation
vectors $d^{A}$ and $d^{B}$ are fed to a deep network $f_{\theta}$ (covered by
the light yellow area in Fig.1), producing batches of embeddings $h^{A}$ and
$h^{B}$, respectively.

Figure 1: The objective function of GW twins measures the cross-correlation
matrix between the embeddings of two identical networks $h^{A}$ and $h^{B}$
(described as the extraction of GW signal by Net A and Net B) fed with
augmentation versions of a batch of samples $d^{A}$ and $d^{B}$ (covered by
the light blue area). Here, $d^{A}=h+n_{L}$ and $d^{B}=h+n_{T}$, and the
cross-correlation matrix is made close to the identity. This causes the
embedding vectors of distorted versions of a sample (blue and purple signal
waveforms) to be similar, while minimizing the redundancy between the
components of the embedding vectors.

In our GW twins design, the corresponding loss function $\mathcal{L}_{GW}$ is

$\mathcal{L}_{GW}=\underbrace{\sum_{i}\left(1-C^{GW}_{ii}\right)^{2}}_{\text{invariant
part}}+~{}~{}\lambda\underbrace{\sum_{i}\sum_{j\neq
i}\left({C^{GW}_{ij}}\right)^{2},}_{\text{redundancy reduction part}}$ (1)

where

$C^{GW}_{ij}=\frac{\sum_{b}h^{A}_{b,i}h^{B}_{b,j}}{\sqrt{\sum_{b}{(h^{A}_{b,i})}^{2}}\sqrt{\sum_{b}{(h^{B}_{b,j})}^{2}}}$
(2)

is the cross-correlation matrix computed between the outputs of two identical
networks, with the values comprised between -1 (i.e. perfect anti-correlation)
and 1 (i.e. perfect correlation), $\lambda$ is a positive constant, $b$ is
batch samples, and $i,j$ represent the vector dimension of the networks
outputs

Here, the objective function (OF) is understood as an instantiation of the
information bottleneck (IB) objective Tishby:2000:tpb ; Tishby:1503:tzy . It
is easy to see that our OF has similarities with existing OFs for self-
supervised learning, for example, the redundancy reduction part plays a role
similar to the contrastive part in the infoNCE objective Vinyals:1807:olv .

###  II.2 Implementation Details

It is often hard to find a large collection of correctly labelled data. This
scarcity of labelled real data exists for GW events as well. One way to combat
this shortage is to train on simulated data and hope the trained model is
useful for the real dataset as well. The success of such an approach is
dependent on the quality of simulation. However, we will have rich raw data in
GW detection, specially for space-based detectors, such as Taiji and LISA.
Thus the application of self-supervised training can be particularly relevant
for GW signal detection.

The basic idea of our GW twins design for self-supervised training is to have
two dataset which see different versions of GW data and use the OF of GW twins
to learn embeddings. In details, we can feed the data $d^{A}$ from LISA into
Net A and the data $d^{B}$ from Taiji into Net B. In Fig.1, the blue strain
and purple strain in the data augmentation waveform plottings which are
covered by the light blue area correspond to the LISA noise data strain
$n_{L}$ and the Taiji noise data strain $n_{T}$, the orange waveform is the GW
signal $h$. Since the noise in Taiji is different from LISA, we can imagine it
as an augmented sample of the same underlying signal.

It is well-known that most self-supervised works focused on images, and so
usually inputs are images and 2D CNN is used. However, for faster training we
will use a 1D CNN model inspired by LISA and Taiji mock data challengeLISAMDC
; TaijiMDC . Moreover, we observed that adding a GRU layer and reducing the
original pooling size led to better results. We also removed the final fully
connected layers since we require the models to produce embeddings. The
bandpassed waveforms are fed as input to the model and we get embeddings of
size 2048 as output.

In the original paper for calculating $\mathcal{L}_{GW}$ in (1), they used
LARS optimizer You:1708:ygg  but we found that AdamW with an initial learning
rate of $10^{-4}$ also works, though LARS optimizer works better. Currently,
we use LISA mock data challenge and Taiji mock data challenge to generated
noise as an additional augmentation.

The details about the contrastive learning work flow are list as follows:

Data augmentations The input data is first converted to produce the two
augmentation vectors shown in Fig.1. The data augmentation pipeline includes:
random cropping, resizing, horizontal flipping, color jittering, converting to
grayscale, Gaussian blurring, and solarization. The cropping and resizing are
always applied, while the last five are applied randomly, with some
probability. This probability is different for the two augmentation vectors in
the last two conversations (blurring and solarization). In this case we add
the different space-based detectors noise into the input data, this data
augmentation can be viewed as a color jittering. We use the same augmentation
parameters as BYOL.

Architecture The encoder consists of a ResNet-50 network (without the final
classification layer, 2048 output units) followed by a projector network. The
projector network has three linear layers, each with 8192 output units. The
first two layers of the projector are followed by a batch normalisation layer
and rectified linear units. We call the output of encoder the
“representations” and the output of projector the “embeddings”. The
representations are used for downstream tasks and the embeddings are fed to
$\mathcal{L}_{GW}$ of our GW twins.

Optimization We follow the optimization protocol described in BYOL. The LARS
optimizer is used, and we work for 1000 epochs with a batch size of 2048. We
set the learning rate 0.2 for the weights and 0.0048 for the biases and batch
normalization parameters. The learning rate warm-up period of 10 epochs is
adopted, after which we lowered the learning rate by a factor of 1000 using a
cosine decay schedule. The best results of $\lambda$ in $\mathcal{L}_{GW}$ is
$\lambda=5\cdot 10^{-3}$.

##  III Simulated space-based GW datasets

###  III.1 Time-delay interferometry and noise budgets

Time-delay interferometry (TDI) will be employed for both LISA and Taiji to
suppress the laser frequency noise and achieve targeting sensitivity. The
principle of the TDI is to combine multiple time-shifted interferometric links
and obtain an equivalent equal path for two interferometric laser beams. The
GW response of TDI is combined by the response of every single link. For a
LISA-like with six laser links, three optimal TDI channels (A, E, T) could be
constructed from three first-generation Michelson TDI configuration (X, Y, Z),
see Refs. Otto:2012dk ; Otto:2015erp ; Tinto:2018kij ; Wang:2020pkk ;
Wang:2020fwa ; Wang:2021mou ; Prince:2002hp ; Vallisneri:2007xa  for details
about the TDI configuration.

Figure 2: The sensitivity curve of Taiji and LISA. The orange line is the
sensitivity curve of Taiji and the blue line is the sensitivity curve of LISA.

By assuming laser frequency noise is sufficiently suppressed in TDI, the
acceleration noise and optical path noise are considered to be the dominant
noises for GW observation. The budges of acceleration noise for LISA and Taiji
are considered as same LISA:2017pwj ; Yang:2022cgm ,

$S_{\rm acc}=9\ \frac{\rm fm^{2}/s^{4}}{\rm Hz}\left[1+\left(\frac{0.4\ {\rm
mHz}}{f}\right)^{2}\right]\left[1+\left(\frac{f}{8\ {\rm
mHz}}\right)^{4}\right],$ (3)

meanwhile, their optical path noise budgets are slightly different as

$\displaystyle S_{\rm op,LISA}$ $\displaystyle=100\ \frac{\rm pm^{2}}{\rm
Hz}\left[1+\left(\frac{2\ {\rm mHz}}{f}\right)^{4}\right],$ (4) $\displaystyle
S_{\rm op,Taiji}$ $\displaystyle=64\ \frac{\rm pm^{2}}{\rm
Hz}\left[1+\left(\frac{2\ {\rm mHz}}{f}\right)^{4}\right].$ (5)

The power spectrum density (PSD) of noise which is presented in Fig. 2 is
calculated by using the numerical method, see Refs. Wang:2020pkk ;
Wang:2020fwa  for detailed algorithm.

###  III.2 Data Curation

(a) $q=1$, $m_{1}=2\times 10^{6}M_{\odot}$, $m_{2}=2\times 10^{6}M_{\odot}$

(b) $q=1.5$, $m_{1}=3\times 10^{6}M_{\odot}$, $m_{2}=2\times 10^{6}M_{\odot}$

(c) $q=2.5$, $m_{1}=5\times 10^{6}M_{\odot}$, $m_{2}=2\times 10^{6}M_{\odot}$

(d) $q=5$, $m_{1}=10^{7}M_{\odot}$, $m_{2}=2\times 10^{6}M_{\odot}$

Figure 3: Time domain representations of the TDI-X channel with different mass
ratio $\frac{m_{1}}{m_{2}}=q$. The data from the LDC-2 dataset contain one
MBHB signal presented by the orange line, the noise+signal data is presented
by the blue line.

We used the signals extracted by the neural network to create new datasets for
testing our model’s ability to detect GW signals. The test datasets consisted
of 10,000 samples containing signals and 10,000 samples containing only noise.
For the MBHB signals, we set the SNR between 30 to 50, and the false alarm
rate at $1\%$. For simulated MBHB signals, we used SEOBNRv4_opt, which is a
version of the SEOBNRv4 code Bohe:2016gbl  with significant optimizations,
which can bring the signals for a high spin, high mass ratio MBHB system. We
adopted the log-uniform distribution for the parameter $M_{tot}$ from Ref.
Katz:2021uax .To facilitate the visual understanding of the signal amplitude
and SNR trends, we have presented a time-domain illustration of the waveform
in conjunction with the LDC dataset in Fig. 3 and the detailed parameters
range is shown in Table 1.

Table 1: Summary of parameter setups in MBHB signal simulation. Parameter | Lower bound | Upper bound
---|---|---
$M_{tot}$ | $10^{6}M_{\odot}$ | $10^{8}M_{\odot}$
$q$ | $0.1$ | $10$
$s_{1}^{z}$ | $-0.99$ | $0.99$
$s_{2}^{z}$ | $-0.99$ | $0.99$
$\phi_{0}$ | $0$ | $2\pi$
$\iota$ | $0$ | $2\pi$
$\delta$ | $0$ | $2\pi$

Hereafter, we project the signal to space-based detector, and inject the
projected signal to the noise with specific optimal SNR as

$\displaystyle{\rm SNR}=\left(s\mid s\right)^{-1/2},$ (6)

Here, $s$ represents the signal, and $(h\mid s)$ is

$(h\mid
s)=2\int_{f_{min}}^{f_{max}}\frac{\tilde{h}^{*}(f)\tilde{s}(f)+\tilde{h}(f)\tilde{s}^{*}(f)}{S_{n}(f)}\,df,$
(7)

where $f_{min}=3\times 10^{-5}{\rm Hz}$ and $f_{max}=0.05{\rm Hz}$, and
$S_{n}(f)$ is the noise PSD. Here, following the setting of the LDC-2 dataset
and Taiji configuration Ruan:2018tsw ; Ruan:2020smc , we set the SNR to
$20$Gair:2022knq . Then the data was whitened and normalized to $[-1,1]$.
During the whitening procedure, we applied the Tukey window with $\alpha=1/8$.

##  IV Results

Our GW twin design comprises two neural networks, Net A for LISA embedding and
Net B for Taiji embedding. We concatenate the embeddings and use them as input
to the fully connected (FC) layer, where the weights of Net A and Net B are
frozen. We only train the FC layer using a subset of the training dataset. Our
experiments show that training the FC layer for only 7-8 epochs in TDI-1.0
scenario is sufficient, and we found the same for TDI-1.5 and TDI-2.0
scenarios. Therefore, we choose to consider eight epochs for all three
generations.

We evaluate our approach based on five target metrics: AUC score, accuracy
score, precision score, recall score, and F1 score. The AUC score reflects the
area under the receiver operating characteristic (ROC) curve, which plots the
true positive rate (TPR) against the false positive rate (FPR) for all
classification thresholds.

$TPR=\frac{TP}{TP+FN},$ (8)

where TP stands for“True Positive” and FN stands for“False Negative”,

$FPR=\frac{FP}{FP+TN},$ (9)

where FP stands for“False Positive” and TN stands for“True Negative”.

Figure 4: AUC with respect to epoch. The vertical axis represent the value of
AUC for SSL model aim at detecting MBHB and the horizontal axis represent the
sequence label of epoch.

Figure 5: Precision and Recall with respect to epoch under the TDI-1.0(blue),
TDI-1.5(orange) and TDI-2.0(green). The vertical axis of the both subfigure
represent the value of precision score and the value of recall, the horizontal
axis of the both subfigure represent the sequence label of epoch.

Figure 6: F1 score and accuracy with respect to epoch under the TDI-1.0(blue),
TDI-1.5(orange) and TDI-2.0(green). The vertical axis of the both subplotting
represent the value of F1 score and the value of accuracy, the horizontal axis
of the both subplotting represent the sequence label of epoch.

Fig.4 shows the evolution of AUC values with increasing epoch numbers across
TDI-1.0, TDI-1.5, and TDI-2.0 generations. The maximum AUC value ($0.85271$)
occurs in the 7th epoch of the TDI-1.0 generation. Interestingly, the AUC
value for the 8th epoch is slightly smaller than the 7th epoch, which may be
due to overfitting caused by the $1\times 1$ convolution layer between the two
epochs. Our results suggest that our self-supervised learning model’s ability
to extract MBHB GW signals is sufficient after only 7 epochs of training. In
TDI-1.5 and TDI-2.0, the AUC value increases monotonically with an increasing
number of training epochs. Notably, the AUC value for TDI-2.0 is greater than
TDI-1.5, indicating that our SSL model is more accurate under more realistic
TDI assumptions.

(a) JS divergence for $M_{1}$ and $M_{2}$ marginalization posterior between
different TDI generation assumptions with LISA MDC data.

(b) JS divergence for $M_{1}$ and $M_{2}$ marginalization posterior between
different TDI generation assumptions with Taiji MDC data.

Figure 7: JSD for primary mass $M_{1}$ and secondary mass $M_{2}$
marginalization posterior between different TDI generation assumption with
LISA and Taiji MDC data.

(a) JS divergence for $\phi$, $\iota$ and $\delta$ marginalization posterior
between different TDI generation assumptions with LISA MDC data.

(b) JS divergence for $\phi$, $\iota$ and $\delta$ marginalization posterior
between different TDI generation assumptions with Taiji MDC data.

Figure 8: JSD for phase $\phi$, inclination angle $\iota$ and polarization
angle $\delta$ marginalization posterior between different TDI generation
assumption with LISA and Taiji MDC data.

We also show the precision and recall. The Precision attempts to describe the
proportion of actually correct positive identifications,

$Precision=\frac{TP}{TP+FP}.$ (10)

The Recall describes the proportion of identified correctly actual positives,

$Recall=\frac{TP}{TP+FN},$ (11)

Our model achieves high Precision scores in the 7th epoch, suggesting correct
detection of GW signals by our self-supervised learning model. The relatively
high Recall rate indicates a relatively low rate of false negatives; however,
precision and recall are often in tension such that improving one typically
reduces the other. To evaluate the effectiveness of our model, we analyze both
precision and recall, and utilize the F1 score to provide a holistic
assessment in signal retrieval. The F1 score is a frequently-used metric in
machine learning that evaluates the performance of binary classification
models by balancing precision and recall. We define the F1 score using the
equation:

$F1=\frac{2TP}{2TP+FP+FN},$ (12)

In addition to the F1 score, we also calculate Accuracy, a widely used
evaluation metric for classification models. Informally, accuracy measures the
fraction of correct predictions made by our model and is defined as:

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}.$ (13)

In Figure 6, we show the changes in F1 score and accuracy with respect to
epoch number. Both metrics steadily increase, indicating that our self-
supervised learning (SSL) approach supports good generalization for Massive
Binary Black Hole (MBHB) signal searching under any Time Delay Interferometry
(TDI) generation assumption, from a holistic perspective. We also observe that
the F1 score and accuracy levels are similar, suggesting that our SSL model is
capable of handling imbalanced data, which is a significant advantage of SSL.

We investigate the effect of different TDI generations on the SSL method’s
performance in estimating five key parameters ( $M_{1}$, $M_{2}$, $\phi$,
$\iota$, $\delta$), using a modified DINGO-GW code packageGW:DINGO  based on
the neural parameter estimation (NPE) method. Our analysis uses data that we
trained with our SSL method, and we evaluate the influence of different TDI
generations using the Jenson-Shannon (JS) divergence metric. Smaller JS
divergence values indicate that parameter estimations for different TDI
generations are comparable, supporting our inference that the SSL method is
minimally influenced by the TDI generation assumptions. As seen in Figure.7,
the influence of different TDI generation assumptions on the primary and
secondary mass is significant in both the LISA and Taiji MDC datasets.
However, as Figure.8 shows, the effect on $\phi$, $\iota$ and $\delta$ in both
datasets is negligible. This is because the SNR of the MBHB signal is most
strongly influenced by changes in primary and secondary mass, while the
contribution from phase and skymap localization angle is minor. Therefore,
despite the influence of TDI generation assumptions on mass, the SSL method’s
performance in estimating $\phi$, $\iota$ and $\delta$ remains consistent
across different TDI generations.

SSL also offers significant computational efficiency advantages over other
methods. To demonstrate this, we compare SSL to the Long Short-Term Memory
network (LSTM)Hochreiter:1997yld  and LDASoftLDA:LDASoft  methods on the LISA
and Taiji MDC datasets. Using all three methods, we search for specific
injection signals and evaluate the time costs of each. Figure 9 shows that
LDASoft incurs the highest time cost among the three methods in the LISA MDC
data environment, steadily increasing as TDI configuration approaches the true
value. Conversely, SSL and LSTM significantly reduce time costs by over 70% as
compared to LDASoft, with SSL being the most efficient. The annotation costs
and overall computational costs are also lower with SSL. We believe that
further methodological enhancements and the availability of more robust
training organizations will make SSL even more promising for controlling
computational costs.

Figure 9: We compare the time cost for three methods - SSL (blue histogram),
LSTM (yellow histogram), and LDASoft (green histogram) - for different TDI
generations. The left plot displays the results for LISA MDC data, where we
compare the time cost of the three main methods. The right plot shows the
results for TAIJI MDC data, where we compare the time cost of both machine
learning methods.

##  V Conclusion

In this paper, we present a self-supervised learning model for GW detection.
Our model is based on simulated massive black hole binary signals embedded in
synthetic Gaussian noise that matches space-based GW detectors’ Taiji and LISA
sensitivity. Our results demonstrate that the self-supervised learning model
we propose is capable of extracting signals and that it is a highly efficient
method for GW signal identification.

To accomplish this, we utilize a corresponding contrastive learning method in
which we propose the GW twins design operating on a joint embedding of
augmentation vectors (where Net A and Net B represent the LISA and Taiji
embeddings, respectively). This design is conceptually simple, easy to
implement, and benefits from high-dimensional embeddings that do not require
large batches, asymmetric mechanisms such as momentum encoders or non-
differentiable operators, or stop-gradients.

We apply our SSL method to a simple signal search scenario in which we search
for MBHB signals in detecting data. Our results show that the SSL method does
well in recovering the injected MBHB signals from the Gaussian noise with both
the LISA MDC and Taiji data settings. From various metric results, we conclude
that SSL is more effective with a more realistic TDI generation assumption
from an entire angle, and the different TDI generations can influence the
primary mass and secondary mass inference results obviously and slightly for
$\phi$, $\iota$ and $\delta$ when using the SSL method. The time cost of SSL
is advantageous to LDASoft based on globalfit, and it also uses less
computation cost than the LSTM method. It will reduce the computation cost and
time with future optimizations.

Although we applied our method to the MBHB GW events, it could still be
applied to other cases, such as binary white dwarf and extreme mass ratio
inspirals. It might be helpful in searching for the GW echo signals,
particularly the unequal interval echoes Wang:2018mlp ; Wang:2018cum ;
Li:2019kwa . The search for signals in GW data is mainly affected by non-
Gaussian noise. Actually, our method can be applied in realistic non-Gaussian
data. Note that the claim of matched-filtering optimality only applies in the
Gaussian noise case. We will study these interesting issues in later works.

It is worth mentioning that we take Taiji and LISA as GW twins to present our
method; however, self-supervised learning can also be applied in land-based GW
detectors with high-quality dataLIGO:2021ppb ; Virgo:2022fxr ; Virgo:2022ysc .

##  VI Acknowledgments

WYT is supported by National Key Research and Development Program of China
Grant No. 2021YFC2203004. We acknowledge the Tianhe-2 supercomputer for
providing computing resources.

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