Measuring properties of primordial black hole mergers at cosmological distances effect of higher order modes in gravitational waves

2025-05-02 0 0 1.89MB 15 页 10玖币
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ET-0231A-22, CE-P2200006
Measuring properties of primordial black hole mergers at cosmological distances:
effect of higher order modes in gravitational waves
Ken K. Y. Ng,
1, 2, 3, ∗
Boris Goncharov,
4, 5
Shiqi Chen,
2, 3
Ssohrab Borhanian,
6, 7
Ulyana
Dupletsa,
4, 5
Gabriele Franciolini,
8, 9
Marica Branchesi,
4, 5
Jan Harms,
4, 5
Michele
Maggiore,
10, 11
Antonio Riotto,
10, 11
B. S. Sathyaprakash,
6, 12, 13
and Salvatore Vitale
2, 3
1
William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, Maryland 21218, USA
2
LIGO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
3
Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
4
Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy
5
INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy
6
Institute for Gravitation and the Cosmos, Department of Physics,
Pennsylvania State University, University Park, PA, 16802, USA
7
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, 07743, Jena, Germany
8
Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy
9
INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy
10
D´epartement de Physique Th´eorique, Universit´e de Gen`eve,
24 quai E. Ansermet, CH-1211 Geneva, Switzerland
11
Gravitational Wave Science Center (GWSC), Universit´e de Gen`eve, CH-1211 Geneva, Switzerland
12
Department of Astronomy & Astrophysics, Pennsylvania State University, University Park, PA, 16802, USA
13
School of Physics and Astronomy, Cardiff University, Cardiff, UK, CF24 3AA
(Dated: October 10, 2022)
Primordial black holes (PBHs) may form from the collapse of matter overdensities shortly after
the Big Bang. One may identify their existence by observing gravitational wave (GW) emissions
from merging PBH binaries at high redshifts
z&
30, where astrophysical binary black holes (BBHs)
are unlikely to merge. The next-generation ground-based GW detectors, Cosmic Explorer and
Einstein Telescope, will be able to observe BBHs with total masses of
O
(10
−
100)
M
at such
redshifts. This paper serves as a companion paper of Ref. [
1
], focusing on the effect of higher-order
modes (HoMs) in the waveform modeling, which may be detectable for these high redshift BBHs,
on the estimation of source parameters. We perform Bayesian parameter estimation to obtain the
measurement uncertainties with and without HoM modeling in the waveform for sources with different
total masses, mass ratios, orbital inclinations and redshifts observed by a network of next-generation
GW detectors. We show that including HoMs in the waveform model reduces the uncertainties
of redshifts and masses by up to a factor of two, depending on the exact source parameters. We
then discuss the implications for identifying PBHs with the improved single-event measurements,
and expand the investigation of the model dependence of the relative abundance between the BBH
mergers originating from the first stars and the primordial BBH mergers as shown in Ref. [1].
I. INTRODUCTION
An interesting possibility is that a fraction of the merger
events detected by the LIGO-Virgo-KAGRA (LVK) Col-
laboration may be due to primordial BHs (PBHs) [
2
–
5
]
formed from the collapse of sizable overdensities in the
radiation-dominated early universe [
6
–
9
]. In this scenario,
PBHs are not clustered at formation [
10
–
15
], they are
born spinless [
16
,
17
] and may assemble in binaries via
gravitational decoupling from the Hubble flow before the
matter-radiation equality [
18
,
19
] (see [
20
–
24
] for reviews).
After their formation, PBH binaries may be affected by
a phase of baryonic mass accretion at redshifts smaller
than
z∼
30, which would modify the PBH masses, spins
and merger rate [25,26].
∗kng15@jhu.edu
Analysing the population properties of masses, spins,
and redshifts of binary black holes (BBHs) in the
LVK’s second catalog [
27
], several studies constrained
the potential contribution from PBH binaries to current
data [
26
,
28
–
35
]. However, these analyses require precise
knowledge of the astrophysical BBH “foreground” in or-
der to verify if there is a PBH subpopulation within the
BBHs observed at low redshifts [
33
,
36
]. Such analyses
are limited by the horizon of current GW detectors,
z.
3
at their design sensitivity [
37
], and are subject to signifi-
cant uncertainties on the mechanisms of BBH formation
in different astrophysical environments, such as galactic
fields [
38
–
48
], dense star clusters [
49
–
56
], active galactic
nuclei [
57
–
65
], or from the collapse of Population III (Pop
III) stars [66–69].
Instead, searching for PBHs at high redshifts where
astrophysical BHs have not merged yet may mitigate most
of the issues caused by the astrophysical foreground. The
PBH merger rate increases with redshift [
70
], while the
arXiv:2210.03132v1 [astro-ph.CO] 6 Oct 2022
2
astrophysical contribution decreases at
z.
30 [
69
,
71
,
72
]. The proposed next-generation detectors, such as the
Cosmic Explorer (CE) [
73
–
75
] and the Einstein Telescope
(ET) [
76
,
77
], whose horizons are up to
z∼
100 for stellar-
mass BBHs [
37
,
78
], may provide a unique opportunity
to test and shed light on the primordial origin of BH
mergers at high redshifts. A key question is therefore to
understand the uncertainties related to the measurements
of the source parameters, such as the redshift, masses and
spins.
In Ref. [
1
], we established the possibility of identify-
ing the PBH mergers with masses of 20 and 40
M
at
z≥
40 using single-event redshift measurements. We also
discussed how the prior knowledge of relative abundance
between Pop III and PBH mergers affects the statisti-
cal significance, assuming that there is a critical red-
shift,
zcrit
= 30, above which no astrophysical BBHs
are expected to merge. The results were based on full
Bayesian parameter estimation with a waveform model,
IMRPhenomXPHM
, which includes the effects of spin pre-
cession and higher-order modes (HoMs) [
79
–
81
]. In this
paper, we show the importance of HoMs to the param-
eter estimation of the high redshift BBHs at
z≥
10
in the context of PBH detections. We compare the
Bayesian posteriors of the relevant parameters obtained by
IMRPhenomXPHM
and the similar waveform family without
HoMs,
IMRPhenomPv2
[
82
–
84
] to systematically study the
improvement on measurements due to the HoM modeling
in the waveform.
We first recap the details of our simulations and the
settings of the parameter estimation in Sec. II. Then, we
show whether and how
IMRPhenomXPHM
performs better
when measuring redshift (Sec. III), as well as masses and
spins (Sec. IV), for BBHs with different sets of the source-
frame total mass, mass ratio, orbital inclination, and
redshift. Finally, in Sec. V, we re-examine the estimation
of the probability that a single source originated from
PBHs using redshift measurements under different choices
of
zcrit
, and discuss the possible implications of the mass
and spin measurements for PBH detections.
II. SIMULATION DETAILS
As in Ref. [
1
], we simulate BBHs at five different red-
shifts,
ˆz
= 10
,
20, 30, 40 and 50. The hat symbol denotes
the true value of a parameter here and throughout the pa-
per. To encompass the detectable mass range, we choose
the total masses in the source frame to be
ˆ
Mtot
= 5, 10,
20, 40, and 80
M
, with mass ratios
ˆq
= 1, 2, 3, 4 and 5.
Here, we define q≡m1/m2for m1> m2, where m1and
m2
are the primary and secondary mass, respectively. For
each mass pair, we further choose four orbital inclination
angles,
ˆι
= 0 (face-on),
π/
6,
π/
3, and
π/
2 (edge-on). All
simulated BBHs are non-spinning, as we expect that PBHs
are born with negligible spins [
16
,
17
,
85
] and may be spun-
up by accreting materials at later times [
25
,
26
,
85
,
86
].
However, we do not assume zero spins when performing
parameter estimation of the source parameters and in-
stead allow for generic spin-precession. For each of these
500 sources, the sky location and polarization angle are
chosen to maximize the signal-to-noise ratio (SNR) for
each source. The reference orbital phase and GPS time
are fixed at 0 and 1577491218, respectively. The baseline
detector network is a 40-km CE in the United States, 20-
km CE in Australia, and ET in Europe. We only analyze
simulated sources whose network SNRs are larger than
12. We use Planck 2018 Cosmology when calculating the
luminosity distance dLat a given redshift [87].
We employ a nested sampling algorithm [
88
,
89
] pack-
aged in Bilby [
90
] to obtain posterior probability densi-
ties. As we are only interested in the uncertainty caused
by the loudness of the signal, we use a zero-noise real-
ization [
91
] for the Bayesian inference and mitigate the
offsets potentially caused by Gaussian fluctuations [
92
].
To ensure our results are free from the systematics due
to the difference in the two waveform families, we use
the same waveform family for both simulating the wave-
forms and calculating the likelihood. That is, we use the
IMRPhenomXPHM
(
IMRPhenomPv2
) waveform template to
analyze the
IMRPhenomXPHM
(
IMRPhenomPv2
) simulated
waveforms
1
. The low-frequency cut off in the likelihood
calculations is 5 Hz for all sources.
As in Ref. [
1
], we first sample the parameter space
with uniform priors on the detector-frame total mass,
MD
tot
=
Mtot
(1+
z
), between [0
.
5
,
1
.
5]
ˆ
MD
tot
, and
q
between
[1
,
10]. The prior on redshift is uniform in the comoving
rate density,
∝dVc
dz
1
1+z
, between [
z
(
ˆ
dL/
10)
, z
(5
ˆ
dL
)]. In
Sec. V, we will revisit the physically motivated prior on
redshift. We use uniform priors for other parameters: the
sky position, the polarization angle, the orbital inclination,
the spin orientations, the spin magnitudes, the arrival
time and the phase of the signal at the time of arrival.
Then, we reweigh the posteriors into uniform prior on
the source-frame primary mass,
m1
, and the inverse mass
ratio 1
/q
(which is between [0
.
1
,
1]). Strictly speaking,
the marginalized one-dimensional priors on
m1
and 1
/q
are not exactly uniform after the reweighing because the
boundary of the square domain of (
MD
tot, q
) transforms
into a different shape according to the Jacobian. For ex-
ample, the marginalized prior on the redshift and that on
the inverse mass ratio have additional factors of 1
/
(1 +
z
)
and
q/
(
q
+ 1), respectively, upon the coordinate trans-
formation. However, we find that such boundary effect
has negligible effect on the posteriors. As we will discuss
below, the degeneracy among different parameters and
the scaling in p0(z) is more significant.
1
See Ref. [
93
] for the analysis of waveform systematics for the
next-generation GW detectors.
3
III. REDSHIFT MEASUREMENT IN
PRESENCE OF HIGHER-ORDER MODES
Since each HoM has a different angular emission spec-
trum, including HoMs in the waveform model breaks
the distance-inclination degeneracy characteristic of the
dominant (2
,
2) harmonic mode [
94
,
95
]. The interfer-
ence of additional HoMs can result in amplitude mod-
ulation, similar to what can be induced by spin pre-
cession [
80
,
81
,
96
]. For example, in the top panel of
Fig. 1we show the Fourier amplitude of a BBH with
(
ˆ
Mtot,ˆz, ˆq
) = (80
M,
30
,
1) and
ˆι
= 0
◦,
30
◦,
60
◦
and 90
◦
(blue, orange, green and red, respectively). To reduce the
systematics between waveform families due to differences
in precessing frame mapping, we compare
IMRPhenomXPHM
(solid lines) and
IMRPhenomXP
[
79
] (dotted lines) instead.
The amplitude modulation of the waveforms with HoMs –
which is stronger for inclination angles close to 90
◦
– is
apparent and helps improving the estimation of the dis-
tance and the inclination. By contrast, for the waveforms
without HoMs the main effect of increasing the incli-
nation angle is to reduce the Fourier amplitude, which
qualitatively shows why the two parameters are partially
degenerate when only the (2,2) mode is used
2
. The other
contribution is the phase modulation in the later part
of the waveform due to HoMs. To visualize this effect,
we show the phase difference between
IMRPhenomXP
and
IMRPhenomXPHM
, ∆Φ(
f
), in the bottom panel of Fig. 1.
Whereas the (2,2) mode of the inspiral defines the wave-
form up to
≈
8 Hz, after which the ringdown takes over,
HoMs of the inspiral extend to higher frequencies. The
interference of the HoMs and the (2,2) ringdown piles up
a significant phase modulation, and hence improves the
measurement of inclination.
Moreover, the parameters,
q
,
ι
and
Mtot
, determine the
amplitude of each mode. The uncertainties of distance
(and hence redshift) are thus sensitive to the values of
(
q, ι, Mtot
) with other parameters fixed. In this section,
we will quantify the variation of the redshift uncertainty
due to each intrinsic parameter one at a time. We will
also show which region of redshift gain the most from the
presence of HoMs in the waveform model. In the following
figures, blue (red) violins represent the posteriors obtained
by IMRPhenomXPHM (IMRPhenomPv2).
A. Orbital inclination
We first discuss the role of orbital inclination in the
redshift measurements using HoM waveforms. Waveform
models which only contain (2, 2) mode suffer from the
distance-inclination degeneracy of the mode, especially
for nearly face-on (
ι'
0) systems whose amplitude scales
as
∼
(1
−ι2/
2)
/dL
. On the other hand, each HoM cor-
responds to spherical harmonics with a different angular
2
The degeneracy is worst at small inclination angles, see Sec. III A.
10.0
5.0 6.0 7.0 8.0 9.0
f(Hz)
10−27
10−26
10−25
10−24
10−23
A(f)
ι= 0◦
ι= 30◦
ι= 60◦
ι= 90◦
XPHM
XP
ET
10.0
5.0 6.0 7.0 8.0 9.0
f(Hz)
−6
−4
−2
0
2
4
6
∆Φ(f) (rad)
ι= 0◦
ι= 30◦
ι= 60◦
ι= 90◦
FIG. 1. Comparison between waveforms without HoMs
(
IMRPhenomXP
) and with HoMs (
IMRPhenomXP
) for BBHs with
(
ˆ
Mtot,ˆz, ˆq
) = (80
M,
30
,
1) and
ˆι
= 0
◦,
30
◦,
60
◦
and 90
◦
(blue, orange, green and red, respectively). Top panel: Strain
amplitudes for
IMRPhenomXPHM
(solid lines) and
IMRPhenomXP
(dotted lines) projected on ET’s detector frame. Bottom panel:
Phase difference between
IMRPhenomXPHM
and
IMRPhenomXP
at each frequency. In all systems, the right ascension angle,
declination angle and polarization angle are 110
◦
, 45
◦
and 93
◦
,
respectively.
response as a function of
ι
. If the waveform models are
sensitive to HoMs, measuring the relative amplitudes of
the HoMs provides better constraints on the orbital in-
clination angle, and thus reduces the distance-inclination
degeneracy.
We now quantify the improvement on the redshift mea-
surements due to the presence of HoMs with varying
inclination angles. In Fig. 2, we show the redshift posteri-
ors obtained by the two waveform models for sources with
(
ˆ
Mtot,ˆz, ˆq
) = (40
M,
30
,
1) at
ˆι
= 0
◦,
30
◦,
60
◦
and 90
◦
.
Indeed, the redshift uncertainties are generally smaller in
the cases of
IMRPhenomXPHM
than those of
IMRPhenomPv2
.
The decrease in the uncertainties is about
∼
30%
−
50%.
Notably, the lower bound of redshift uncertainties in-
creases from
z∼
10 in the cases of
IMRPhenomPv2
to
z∼
20 in the cases of
IMRPhenomXPHM
. This improvement
due to HoMs is particularly interesting for determining
the astrophysical or primordial origin of BBHs. If the
redshift measurement of a system is precise enough to rule
out the epoch of the astrophysical BBHs, one may even
use a single measurement to identify the existence of pri-
摘要：

ET-0231A-22,CE-P2200006Measuringpropertiesofprimordialblackholemergersatcosmologicaldistances:e ectofhigherordermodesingravitationalwavesKenK.Y.Ng,1,2,3,BorisGoncharov,4,5ShiqiChen,2,3SsohrabBorhanian,6,7UlyanaDupletsa,4,5GabrieleFranciolini,8,9MaricaBranchesi,4,5JanHarms,4,5MicheleMaggiore,10,11An...

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