# Constraining the Gravitational-Wave Spectrum from Cosmological First-Order
Phase Transitions Using Data from LIGO-Virgo First Three Observing Runs

Yang Jiang<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
No. 19A Yuquan Road, Beijing 100049, China  Qing-Guo Huang  Corresponding
author<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China  School of Fundamental Physics and
Mathematical Sciences Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China

###### Abstract

We search for a first-order phase transition (PT) gravitational wave (GW)
signal from Advanced LIGO and Advanced Virgo’s first three observing runs. Due
to the large theoretical uncertainties, four GW energy spectral shapes from
bubble and sound wave collisions widely adopted in literature are
investigated, separately. Our results indicate that there is no evidence for
the existence of such GW signals, and therefore we give the upper limits on
the amplitude of GW energy spectrum $\Omega_{\text{pt}}(f_{*})$ in the peak
frequency range of $f_{*}\in[5,500]$ Hz for these four theoretical models,
separately. We find that $\Omega_{\text{pt}}(f_{*}\simeq 40\
\text{Hz})<1.3\times 10^{-8}$ at $95\%$ credible level, and roughly
$H_{*}/\beta\lesssim 0.1$ and $\alpha\lesssim 1$ at $68\%$ credible level in
the peak frequency range of $20\lesssim f_{*}\lesssim 100$ Hz corresponding to
the most sensitive frequency band of Advanced LIGO and Advanced Virgo’s first
three observing runs, where $H_{*}$ is the Hubble parameter when PT happens,
$\beta$ is the bubble nucleation rate and $\alpha$ is the ratio of vacuum and
relativistic energy density.

Introduction. The evolution of the Universe can be carried out in a smooth or
abrupt way, and these sudden changes are what we called phase transitions
(PTs) Linde (1979); Kibble (1980); Mazumdar and White (2019). The early
evolution of the Universe might contain a variety of PTs since many extensions
of Standard Model, such as grand unification model Croon  _et al._ (2019),
suppersymmetric model Huber  _et al._ (2016) and so on, predict the occurrence
of PTs. See some other relevant models in Megías  _et al._ (2018); Fromme  _et
al._ (2006); Huang  _et al._ (2016); Hebecker  _et al._ (2016); Jinno and
Takimoto (2017a); Brdar  _et al._ (2019); Cai  _et al._ (2022); Hall  _et al._
(2019, 2020). At QCD scale, PT could happen due to the confinement between
quarks and gluons Schwarz and Stuke (2009); Boeckel and Schaffner-Bielich
(2010). At a higher energy scale, the destruction of electroweak gauge
symmetry may lead to electroweak PT Kajantie  _et al._ (1996). Generally, each
breaking of new symmetry introduced by theoretical models might lead to the
occurrence of PT. Therefore, the observation of PT becomes a way to explore
new physics.

Among all kinds of PTs, the first-order PT is of great significance for
gravitational wave (GW) astronomy, because they are usually strong enough to
be interesting sources of GW radiation Witten (1984); Hogan (1986) in the
Universe. When the temperature of the Universe drops to a certain value,
bubbles containing true vacuum will nucleate in the meta-stable vacuum.
Subsequently, these bubbles expand, merge, fill the Universe in the end and
GWs are produced during this process Kosowsky  _et al._ (1992a, b);
Kamionkowski  _et al._ (1994). Since the process of nucleation and collision
occurs in a random manner, the radiated GWs form a stochastic gravitational
wave background (SGWB). Because the gravitational interaction is very weak,
SGWB decouples from the primordial plasma rapidly. Detecting these GW signals
has become an important way to explore the early Universe which is difficult
to be observed by any other methods due to the longer decoupling time Maggiore
(2000).

At the time when GWs are produced, its frequency depends on the duration of
PT, and then GWs would be redshifted by the expansion of the Universe.
Therefore, the frequency band of SGWB today is also related to the temperature
$T_{*}$ at which PT occurs. Pulsar Timing Arrays (PTAs) Brazier  _et al._
(2019); Desvignes  _et al._ (2016); Kerr  _et al._ (2020); Perera  _et al._
(2019) are used to search for the SGWB at frequencies of several nHz,
corresponding $T_{*}$ at the order of MeV. Even though a stochastic process
has been detected by PTA data sets Arzoumanian  _et al._ (2020); Goncharov
_et al._ (2021); Chen  _et al._ (2021a); Antoniadis  _et al._ (2022), a SGWB
detection consistent with general relativity cannot be claimed because there
is no statistically significant evidence of quadrupolar spatial correlations
Chen  _et al._ (2021b); Wu  _et al._ (2022); Arzoumanian  _et al._ (2021a),
and the constraints on the first-order PT parameters from PTA data sets are
presented in Arzoumanian  _et al._ (2021b); Xue  _et al._ (2021). On the other
hand, the terrestrial GW observatories like Advanced LIGO Aasi  _et al._
(2015) and Advanced Virgo Acernese  _et al._ (2015) are designed to be
sensitive to the frequency of $10\sim 10^{3}$ Hz roughly corresponding to the
PT temperature in the range of $10^{5}\sim 10^{10}$ GeV. So far,
LIGO/Virgo/KAGRA Collaboration has accumulated three generations (O1-O3) of
GWs observing data and there is no evidence for the SGWB signal Abbott  _et
al._ (2021a, b). The related constraints on the GW energy spectrum generated
by the first order PT have been investigated in Romero  _et al._ (2021) based
on the broken power law and two phenomenological PT models.

In this letter, we present a comprehensive analysis of the first-order PT
models utilizing data from Advanced LIGO and Advanced Virgo’s first three
observing runs by extending and improving the analysis in Romero  _et al._
(2021) in several aspects. First of all, we take into account four different
GW energy spectral shapes from bubble and sound wave collisions widely adopted
in literature due to the large theoretical uncertainties, and provide the
constraints on the amplitude of the GW energy spectrum in the peak frequency
range of $f_{*}\in[5,500]$ Hz for these four theoretical models, separately.
Secondly, although compact binary coalescences (CBCs) background is supposed
to contribute SGWB as well, it is not strong enough to produce a detectable
correlation according to the current sensitivities of both Advanced LIGO and
Advanced Virgo Abbott  _et al._ (2021a). Therefore, different from Romero  _et
al._ (2021), we do not include the contribution of CBCs in our analysis.
Thirdly, the parameters related to the amplitude of the GW spectrum are chosen
to be uniform distributions in our analysis for resulting in more conservative
upper limits than the log-uniform priors adopted in Romero  _et al._ (2021).
Fourthly, we do not keep the bubble nucleation rate $\beta$ and the PT
temperature $T_{*}$ fixed for deriving the upper limits on the GW spectrum
because they should be free from the viewpoint of data analysis.

SGWB from the first-order PT. Isotropic SGWB can be described by the
fractional energy density spectrum in the frequency domain:

$\Omega(f)=\frac{1}{\rho_{\text{c}}}\frac{\mathrm{d}\rho_{\text{gw}}}{\mathrm{d}\ln
f},$ (1)

where $\rho_{\text{c}}=3H_{0}^{2}c^{2}/(8\pi G)$ denotes the critical density
of the current universe. It has been known that there are three main sources
of GW produced by a first-order PT: bubble collisions, collisions of sound
waves and magnetohydrodynamic turbulence Hindmarsh  _et al._ (2021); Weir
(2018). In this letter, the contribution from turbulence is not considered due
to the lack of understanding about its energy spectrum Caprini  _et al._
(2016); Roper Pol  _et al._ (2020); Kahniashvili  _et al._ (2008, 2010).
Besides, magnetohydrodynamic turbulence is always subdominant compared with
sound waves. The amplitude of the GW energy spectrum from a first-order PT is
characterized by the bubble nucleation rate $\beta$ and the ratio of vacuum
and relativistic energy density $\alpha$. Quantitatively, the following energy
density spectrum can be used to fit the SGWB from both bubble collisions and
sound waves Caprini  _et al._ (2016); Binetruy  _et al._ (2012); Jinno and
Takimoto (2017b); Hindmarsh  _et al._ (2017); Arzoumanian  _et al._ (2021b):

$h^{2}\Omega_{\text{pt}}(f)=\mathcal{R}g_{*}^{-\frac{1}{3}}\Delta(v_{w})\left(\frac{\kappa\alpha}{1+\alpha}\right)^{p}\left(\frac{H_{*}}{\beta}\right)^{q}\mathcal{S}\left(\frac{f}{f_{*}}\right),$
(2)

where $h$ is the dimensionless Hubble constant, the factor $\mathcal{R}\simeq
7.69\times 10^{-5}$, $\Delta(v_{w})$ is a function of bubble walls velocity
$v_{w}$, $g_{*}$ is the number of relativistic degrees of freedom which is
fixed to be $100$ in this letter, $H_{*}$ is the Hubble parameter when phase
transition happens and $\kappa$ counts for the fraction of vacuum energy
converted. $\mathcal{S}(x)$ represents the shape of the spectrum and it comes
to its maximum value at $x=1$. Here, $f_{*}$ is the peak frequency of SGWB at
present:

$f_{*}\simeq 1.13\times
10^{-7}\left(\frac{\tilde{f_{*}}}{\beta}\right)\left(\frac{\beta}{H_{*}}\right)\left(\frac{T_{*}}{\text{GeV}}\right)\left(\frac{g_{*}}{10}\right)^{1/6}\text{Hz},$
(3)

where $\tilde{f_{*}}$ is the peak frequency when PT happens. For sound waves,
a suppression factor $\Upsilon$ should be multiplied in Eq. (2) counting for
the finite lifetime Ellis  _et al._ (2020); Guo  _et al._ (2021):

$\Upsilon=1-(1+2\tau_{\text{sw}}H_{*})^{-1/2}.$ (4)

The time scale $\tau_{\text{sw}}$ is usually taken to be the timescale for the
onset of turbulence Weir (2018): $\tau_{\text{sw}}\simeq R_{*}/\bar{U}_{f}$,
where $R_{*}=(8\pi)^{1/3}\beta^{-1}\mathrm{Max}(v_{w},c_{s})$ Hindmarsh and
Hijazi (2019); Guo  _et al._ (2021) and $\bar{U}_{f}^{2}\simeq
3\kappa_{\text{sw}}\alpha/[4(1+\alpha)]$ Weir (2018). More details about the
parameters in Eq. (2) and Eq. (3) are listed in Table 1.

| Bubbles Collisions | Sound Waves
---|---|---
$\Delta(v_{w})$ | $\frac{0.48v_{w}^{3}}{1+5.3v_{w}^{2}+5v_{w}^{4}}$ | $0.513v_{w}$
$\kappa$ | $1$ | $f(\alpha,v_{w})$
$p$ | $2$ | $2$
$q$ | $2$ | $1$
$\mathcal{S}(x)$ | $\frac{(a+b)^{c}}{\left(bx^{-a/c}+ax^{b/c}\right)^{c}}$ | $x^{3}\left(\frac{7}{4+3x^{2}}\right)^{7/2}$
$\tilde{f_{*}}/\beta$ | $\frac{0.35}{1+0.07v_{w}+0.69v_{w}^{4}}$ | $\frac{0.536}{v_{w}}$
Table 1: Parameters for the GW energy spectrum in Eq. (2) and Eq. (3)

In fact, there are still large theoretical uncertainties for predicting the PT
GW energy spectrum. Analytically, GW energy spectrum can be calculated
assuming that energy is concentrated on the infinitesimal bubble wall and it
vanishes once the bubbles collide with others. This method is called envelope
approximation Jinno and Takimoto (2017b); Kosowsky  _et al._ (1992b).
Numerically, 3D lattice simulations Cutting  _et al._ (2021, 2018) can be used
to break through these assumptions. However, large solved volume to accomodate
multiple bubbles and very dense lattices to fit thin bubble walls usually lead
to substantial costs. Hence, semi-analytic Lewicki and Vaskonen (2021) methods
are served as alternative ways to do the calculation. Table 2 illustrates the
details of different GW energy spectra and the typical shapes of these spectra
are shown in Fig. 1. In the high frequency band, the numerical simulation
results show faster attenuation than envelope approximation, while the
opposite is true at low frequency band.

| Envelope | Semi-analytic | Numerical
---|---|---|---
$a$ | $3$ | $1\sim 2.3$ | $1.6\sim 0.7$
$b$ | $1$ | $2.2\sim 2.4$ | $1.4\sim 2.3$
$c$ | $1.5$ | $2\sim 4.2$ | $1$
$\tilde{f_{*}}/\beta$ | $\frac{0.35}{1+0.07v_{w}+0.69v_{w}^{4}}$ | $0.1$ | $0.2$
Table 2: Energy spectra of bubble collisions for envelope approximation, semi-
analytic approach and lattice simulations. In this letter, we take
$(a,b,c)=(1,2.2,2)$ and $(a,b,c)=(0.7,2.3,1)$ for semi-analytic and numerical
methods respectively. Figure 1: SGWB energy spectra for bubble collision and
sound wave. The peak frequency $f_{*}$ is chosen to be $25$ Hz.

Method of data analysis. The measurement of SGWB depends on the correlations
Romano and Cornish (2017); Allen and Romano (1999) between multiple detectors
and Bayesian approach is used to calculate the possibility of models Mandic
_et al._ (2012). Correlated magnetic noise budget shows that such signal is
negligible Abbott  _et al._ (2019, 2021a) since the intensity of this
correlation is much lower than the sensitivity of the current detectors.
Therefore, a Gaussian distributed fluctuation is assumed and likelihood is
given by

$p(\hat{C}_{IJ}|\bm{\theta};\lambda)\propto\exp\left[-\frac{1}{2}\sum_{IJ}\sum_{f}\frac{\left(\hat{C}_{IJ}(f)-\lambda\Omega(f;\bm{\theta})\right)^{2}}{\sigma^{2}_{IJ}(f)}\right].$
(5)

The energy spectrum is characterized by parameters $\bm{\theta}$ to be
estimated. $\hat{C}_{IJ}$ denotes the spectrum of correlation and
$\sigma_{IJ}$ is related to the noise intensity of detector pair $IJ$. Summing
the index $IJ$ means multiplying the likelihoods from all detector pairs since
the correlations between different baselines can be neglected Allen and Romano
(1999). Here $\lambda$ denotes the calibration uncertainties Sun  _et al._
(2020) of the detector baselines and should be marginalized by the method
given in Whelan  _et al._ (2014).

The Bayesian analyses for the circumstances of bubble collision dominant and
sound wave dominant cases will be done separately because it is ambiguous to
determine which one plays a leading role. Python software bilby Ashton  _et
al._ (2019) is used for parameter estimation and generate 2D posterior
distribution corner-plots. For the bubble collision dominant case, all three
models are considered, and $\kappa$ and $v_{w}$ are set to unity based on the
assumption that the bubble wall interacts weakly with the plasma and runaway
regime has been reached. For the sound wave dominant case, $\kappa$ is related
to $\alpha$ and $v_{w}$ Espinosa  _et al._ (2010). In our statistical
analysis, $\bm{\theta}=(\alpha,\,H_{*}/\beta,\,f_{*})$ are free parameters to
be determined in the bubble collision dominant cases, and
$\bm{\theta}=(\alpha,\,H_{*}/\beta,\,f_{*},\,v_{w})$ are free parameters in
the sound wave dominant case. The parameters related to the amplitude of the
GW energy spectrum are chosen to be uniform distributions in our analysis
because it results in more conservative upper bounds than the log-uniform
prior. Besides, we note that Eq. (2) may not be applicable for $\alpha\gtrsim
10$ and $H_{*}/\beta\gtrsim 1$ Ellis  _et al._ (2019a, b). Details about the
priors are listed in Table 3.

Parameter | Prior
---|---
$\alpha$ | $\text{Uniform}(10^{-3},\,10)$
$H_{*}/\beta$ | $\text{Uniform}(10^{-3},\,1)$
$v_{w}$ | $\text{Uniform}(10^{-2},\,1)$
$f_{*}$ | $\text{LogUniform}(5,\,500)$
Table 3: Priors distributions of the parameters used for Bayesian analysis,
and the distribution of $v_{w}$ is set to be $\delta(1)$ for the bubble
collision dominant cases.

Results.

Figure 2: Posterior distributions of parameters for bubble collision and sound
wave dominant cases. Here $68\%$ and $95\%$ exclusion contours are shown.
Horizontal solid green lines denote the priors used in analysis.

The posterior distributions of parameters are illustrated in Fig. 2 for all
four theoretical models. Roughly speaking, we find $H_{*}/\beta\gtrsim 0.1$
and $\alpha\gtrsim 1$ are excluded at $68\%$ confidence level in the peak
frequency range of $f_{*}\in[20,100]$ Hz for all four theoretical models
considered in this letter. This frequency range rougly corresponds to the most
sensitive frequency band of Advanced LIGO and Advanced Virgo since $99\%$ of
the sensitivity measuring the SGWB comes from the frequency band of $20\sim
76.6\,\text{Hz}$ during O3 observation run Abbott  _et al._ (2021a). The Bayes
factors between SGWB caused by PT and pure noise
$\log\mathcal{B}_{\text{niose}}^{\text{PT}}$ are $-0.64$, $-0.74$, $-0.70$ and
$-0.63$ for the bubble collision dominant cases fitted by envelope
approximation, semi-analytic, numerical methods, and the sound wave dominant
case, respectively. This result indicates that there is no evidence to claim
such SGWB signals in the data.

In order to effectively demonstrate the constraints on the GW energy spectrum
in the frequency band, we provide the upper limits on the amplitude of the GW
energy spectrum $\Omega_{\text{pt}}(f_{*})$ for the peak frequency $f_{*}$ in
the range of $f_{*}\in[5,500]$ Hz. Our results are illustrated in Fig. 3, and
we find that the constraints are different for these four different
theoretical models. However, the most stringent constraint appears at around
$f_{*}\simeq 40\;\text{Hz}$ with the upper limits of
$\Omega_{\text{pt}}(f_{*}\simeq 40\;\text{Hz})<1.3\times 10^{-8}$ at $95\%$
credible level for all four scenarios.

Figure 3: Upper limit of $\Omega(f_{*})$ at $95\%$ credible level for certain
values of peak frequency $f_{*}$.

Discussion and Conclusions. We search for the GW signals generated by first-
order PT in various theoretical models widely adopted in literature using data
from Advanced LIGO and Advanced Virgo’s first three observing runs. To
demonstrate the influence of theoretical uncertainties, three models of bubble
collision and one of sound wave contribution are investigated separately. The
result of Bayesian analysis does not show preference between different models
and implies that it is impossible to tell which model is more preferred. In
all, we find that, roughly speaking, $H_{*}/\beta\lesssim 0.1$ and
$\alpha\lesssim 1$ at $68\%$ credible level in the peak frequency range of
$20\lesssim f_{*}\lesssim 100$ Hz corresponding to the most sensitive
frequency band of Advanced LIGO and Advanced Virgo’s first three observing
runs.

Generally, a log-uniform prior is preferred for variables spanning several
orders of magnitude. However, the lower bound of a log-uniform distribution
cannot be taken to be zero, and how to choose such a specific value is
unclear. Because there is no correlation found in the raw data, the result of
posterior depends on the choice of priors. The initial sampling points are
mostly concentrated in small values of parameters for log-uniform priors and
might lead to an under-estimation of the upper bound of the GW energy
spectrum. In this sense, we adopt uniform prior distributions to avoid
ambiguities in the choice of priors. Since there is no correlated signal
examined from the observing periods, we put the upper limits on the amplitude
of the GW energy spectrum in the peak frequency range of $f_{*}\in[5,500]$ Hz.
Because the GW spectra of the four theoretical models considered in this
letter are roughly the same around the peaks, the constraints on the
amplitudes of the GW spectra are roughly the same for them if the peak
frequency of the GW energy spectra stays within the most sensitive frequency
band of Advanced LIGO and Advanced Virgo, such as $f_{*}\in[20,80]$ Hz. But,
once the peak frequency stays outside the the most sensitive frequency band,
the shape of GW spectra plays an important role on the data analysis and the
constraints should be different for different theoretical models. The results
in Fig. 3 are consistent with what we expect.

Acknowledgments.  We would like to thank Li Li, Jiang-Hao Yu and Yue Zhao for
their useful conversations. We acknowledge the use of HPC Cluster of ITP-CAS
and HPC Cluster of Tianhe II in National Supercomputing Center in Guangzhou.
This work is supported by the National Key Research and Development Program of
China Grant No.2020YFC2201502, grants from NSFC (grant No. 11975019, 11991052,
12047503), Key Research Program of Frontier Sciences, CAS, Grant NO. ZDBS-
LY-7009, CAS Project for Young Scientists in Basic Research YSBR-006, the Key
Research Program of the Chinese Academy of Sciences (Grant NO. XDPB15).

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