# NANOGrav hints for first-order confinement-deconfinement phase transition
in different QCD-matter scenarios

Zu-Cheng Chen  0000-0001-7016-9934<EMAIL_ADDRESS>Department of
Physics and Synergetic Innovation Center for Quantum Effects and Applications,
Hunan Normal University, Changsha, Hunan 410081, China  Institute of
Interdisciplinary Studies, Hunan Normal University, Changsha, Hunan 410081,
China  Department of Astronomy, Beijing Normal University, Beijing 100875,
China  Advanced Institute of Natural Sciences, Beijing Normal University,
Zhuhai 519087, China  Shou-Long Li  0000-0002-7192-8717
<EMAIL_ADDRESS>Department of Physics and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China  Institute of Interdisciplinary Studies, Hunan Normal
University, Changsha, Hunan 410081, China  Puxun Wu  0000-0002-9188-7393
<EMAIL_ADDRESS>Department of Physics and Synergetic Innovation Center for
Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China  Institute of Interdisciplinary Studies, Hunan Normal
University, Changsha, Hunan 410081, China  Hongwei Yu  0000-0002-3303-9724
Corresponding author<EMAIL_ADDRESS>Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha, Hunan 410081, China  Institute of Interdisciplinary
Studies, Hunan Normal University, Changsha, Hunan 410081, China

###### Abstract

Recent observations from several pulsar timing array (PTA) collaborations have
unveiled compelling evidence for a stochastic signal in the nanohertz band.
This signal aligns remarkably with a gravitational wave (GW) background,
potentially originating from the first-order color charge confinement phase
transition. Distinct quantum chromodynamics (QCD) matters, such as quarks or
gluons, and diverse phase transition processes thereof can yield disparate GW
energy density spectra. In this paper, employing the Bayesian analysis on the
NANOGrav 15-year data set, we explore the compatibility with the observed PTA
signal of the GW from phase transitions of various QCD matter scenarios in the
framework of the holographic QCD. We find that the PTA signal can be
effectively explained by the GW from the confinement-deconfinement phase
transition of pure quark systems in a hard wall model of the holographic QCD
where the bubble dynamics, one important source of the GWs, is of the Jouguet
detonations. Notably, our analysis decisively rules out the plausibility of
the pure gluon QCD-matter scenario and the non-runaway bubble dynamics model
for the phase transition in explaining the observed PTA signal.

##  I Introduction

The detections of gravitational waves (GWs) from compact binary coalescences
by ground-based detectors Abbott  _et al._ (2019a, 2021a, 2023a) in the
culmination of a decades-long quest confirm directly a prediction of Einstein
based on his classical theory of general relativity. These detections not only
have been utilized to test general relativity in the strong-field regime
Abbott  _et al._ (2019b, 2021b, 2021c), but also have allowed us to explore
the properties of the broader population of GW sources Abbott  _et al._
(2019c, 2021d); Chen  _et al._ (2022a); Abbott  _et al._ (2023b); Chen  _et
al._ (2023a). Despite the great achievement of LIGO-Virgo-KAGRA in detecting
individual merger events, the quest to detect a stochastic GW background
(SGWB), another type of GW source, remains an ongoing pursuit. By inspecting
the spatially correlated fluctuations induced by an SGWB on the arrival times
of radio pulses from an array of pulsars Sazhin (1978); Detweiler (1979);
Foster and Backer (1990), we can use a pulsar timing array (PTA) as a valuable
tool to search for the SGWB in the nanohertz frequency band. Recently, several
PTA collaborations, including the NANOGrav Agazie  _et al._ (2023a, b), the
PPTA Zic  _et al._ (2023); Reardon  _et al._ (2023), the EPTA (in combination
with the InPTA) Antoniadis  _et al._ (2023a, b), and the CPTA Xu  _et al._
(2023), have independently detected a stochastic signal shared by all pulsars.
Furthermore, these PTAs found evidence, with varying levels of significance,
which supports the Hellings-Downs Hellings and Downs (1983) spatial
correlations and suggests the SGWB origin of the signal. While various
potential sources exist within the PTA window Chen  _et al._ (2020); Vagnozzi
(2021); Chen  _et al._ (2021); Sakharov  _et al._ (2021); Benetti  _et al._
(2022); Chen  _et al._ (2022b); Ashoorioon  _et al._ (2022); Wu  _et al._
(2022); Chen  _et al._ (2023b); Falxa  _et al._ (2023); Dandoy  _et al._
(2023); Madge  _et al._ (2023); Bi  _et al._ (2023a); Wu  _et al._ (2023a, b),
the nature of this signal as to whether it arises from astrophysical or
cosmological phenomena remains a subject of extensive investigation Afzal  _et
al._ (2023); Antoniadis  _et al._ (2023c); Athron  _et al._ (2023); Niu and
Rahat (2023); Bi  _et al._ (2023b); Wang  _et al._ (2023a); Liu  _et al._
(2023a); Vagnozzi (2023); Fu  _et al._ (2023); Han  _et al._ (2023); Li  _et
al._ (2023); Franciolini  _et al._ (2023a); Shen  _et al._ (2023); Kitajima
_et al._ (2023); Franciolini  _et al._ (2023b); Addazi  _et al._ (2023); Cai
_et al._ (2023); Inomata  _et al._ (2023); Murai and Yin (2023); Li and Xie
(2023); Anchordoqui  _et al._ (2023); Liu  _et al._ (2023b); Figueroa  _et
al._ (2023); Yi  _et al._ (2023a); Zhao  _et al._ (2023); Wu  _et al._
(2023c); Bian  _et al._ (2023); Geller  _et al._ (2023); Antusch  _et al._
(2023); Ye and Silvestri (2023); Hosseini Mansoori  _et al._ (2023); Jin  _et
al._ (2023); Zhang  _et al._ (2023); Choudhury (2023); Yi  _et al._ (2023b);
Gorji  _et al._ (2023); Das  _et al._ (2023); Ellis  _et al._ (2023); Balaji
_et al._ (2023); Maji and Park (2024); Bhaumik  _et al._ (2023); Basilakos
_et al._ (2023); Huang  _et al._ (2023); Jiang  _et al._ (2023); Wang  _et
al._ (2023b); Di Bari and Rahat (2023); Aghaie  _et al._ (2023); Harigaya  _et
al._ (2023); Kawai and Kim (2023); Salvio (2023); Lozanov  _et al._ (2023);
Choudhury  _et al._ (2023a); Cang  _et al._ (2023); Liu  _et al._ (2023c);
Lazarides  _et al._ (2023); Choudhury  _et al._ (2023b); Fei (2023); Chen  _et
al._ (2024); You  _et al._ (2023); Yi  _et al._ (2023c).

One potential source for this signal is the SGWB generated by a first-order
phase transition in the early Universe, occurring as the temperature decreases
to the scale of quantum chromodynamics (QCD) He  _et al._ (2023); Zu  _et al._
(2023); Abe and Tada (2023); Ghosh  _et al._ (2023); Gouttenoire (2023);
Bigazzi  _et al._ (2021); Li  _et al._ (2021). Here, the first-order
cosmological phase transition refers to the transitions of the Universe from a
metastable vacuum state to a stable vacuum state through quantum tunneling or
thermal fluctuations via the nucleation of bubbles in a sea of metastable
phase. These bubbles expand and eventually collide with each other. The phase
transition processes, especially the collision of the bubble walls and the
subsequent plasma phenomena like sound waves and magnetohydrodynamic (MHD)
turbulence, which are closely related to the dynamics of the bubble models,
can lead to a significant SGWB Kosowsky  _et al._ (1992a, b); Kosowsky and
Turner (1993); Kamionkowski  _et al._ (1994); Caprini  _et al._ (2008);
Hindmarsh  _et al._ (2014); Giblin and Mertens (2013, 2014); Kosowsky  _et
al._ (2002); Caprini and Durrer (2006); Kahniashvili  _et al._ (2008a, b,
2010); Kisslinger and Kahniashvili (2015); Huber and Konstandin (2008);
Caprini  _et al._ (2009, 2016); Schwaller (2015); Hindmarsh  _et al._ (2015);
Ellis  _et al._ (2019, 2020); Morgante  _et al._ (2023); Lewicki and Vaskonen
(2023). The model for the bubble collision, which is one important source of
the GWs generated in the phase transition processes, may encompass both the
Jouguet detonation scenario Steinhardt (1982); Caprini  _et al._ (2016);
Kamionkowski  _et al._ (1994); Nicolis (2004); Espinosa  _et al._ (2010);
Hindmarsh  _et al._ (2015), characterized by strong detonations where the
phase transition proceeds at the speed of sound, and the non-runaway bubble
scenario Espinosa  _et al._ (2010); Caprini  _et al._ (2016); Hindmarsh  _et
al._ (2015); Kamionkowski  _et al._ (1994), involving a more moderate
transition pace without extreme velocities.

Specifically, the Universe may undergo, as the temperature decreases to the
QCD scale, a first-order confinement-deconfinement phase transition,
transitioning from a quark-gluon plasma (QGP) phase to a hadron phase. The
transition temperatures can slightly vary for different QCD-matter plasma He
_et al._ (2023); Zu  _et al._ (2023); Abe and Tada (2023); Ghosh  _et al._
(2023); Gouttenoire (2023); Li  _et al._ (2021), such as pure quark or pure
gluon systems. The diverse dynamics of the bubble evolution and the
temperatures at which phase transitions occur can both imprint variations in
the power spectrum of the SGWB generated by the phase transitions. Despite
these variations, the frequencies of the SGWB consistently fall within the
observable frequency range of PTAs. Therefore, we can use the PTA data to
study the specific scenarios of the cosmic evolution at the QCD scale.

In a previous study Li  _et al._ (2021) utilizing NANOGrav 12.5-year data set,
it was suggested that the PTA signal could be potentially generated by the
first-order confinement-deconfinement transitions of heavy static quarks
plasma with a zero baryon chemical potential or quarks with a finite baryon
chemical potential in both the Jouguet detonation and non-runaway bubble
dynamics models. In this paper, by using Bayesian analysis on the NANOGrav
15-year data set, we delve deeper into exploring the compatibility with the
PTA signal of the GWs resulting from phase transitions in various scenarios of
QCD-matter plasma. We examine three scenarios: (i) heavy static quarks at zero
baryon chemical potential, (ii) quarks with a finite baryon chemical
potential, and (iii) a pure gluon system. We show that the PTA signal can be
effectively explained by the GW from the phase transition of pure quark
systems within the hard wall model of the holographic QCD in which the bubble
dynamics is of Jouguet detonations, irrespective of the chemical potential’s
value. Notably, the PTA signal is incompatible with the GW generated by the
phase transition of a pure gluon plasma and the non-runaway bubble dynamics
model.

Model | QCD matter | Holographic QCD-like model | $\alpha$ | $T_{*}$
---|---|---|---|---
$S_{1}$ | Heavy static quarks with a zero chemical potential | Hard wall | $13.5$ |  $122$ MeV Herzog (2007); Ahmadvand and Bitaghsir Fadafan (2017)
$S_{2}$ | Heavy static quarks with a zero chemical potential | Soft wall | $4.27$ |  $191$ MeV Herzog (2007); Ahmadvand and Bitaghsir Fadafan (2017)
$S_{3}$ | Quarks with a finite chemical potential | Hard wall | $32.2$ |  $112$ MeV Ahmadvand and Bitaghsir Fadafan (2018)
$S_{4}$ | Quarks with a finite chemical potential | Soft wall | $4.56$ |  $192$ MeV Ahmadvand and Bitaghsir Fadafan (2018)
$S_{5}$ | Pure gluons |  Quenched dynamical holographic QCD |  $0.611$ |  $255$ MeV Chen  _et al._ (2018)
Table 1: The ratio of the vacuum energy density $\alpha$ and critical
temperature $T_{*}$ from five holographical QCD-like models.

##  II GWs from first-order QCD phase transitions

The spectrum of an isotropic SGWB is characterized by the dimensionless GW
energy density parameter $\rho_{\mathrm{GW}}$ per logarithm frequency $f$
Allen and Romano (1999)

$\Omega_{\mathrm{GW}}(f)=\frac{1}{\rho_{c}}\frac{d\rho_{\mathrm{GW}}}{d\ln
f},$ (1)

where $\rho_{c}=3H_{0}^{2}/8\pi G$ is the critical energy density of the
Universe in which $H_{0}$ is the Hubble constant. As mentioned earlier, the
total energy density of GWs, $\rho_{\mathrm{GW}}$, mainly arises from three
processes: collisions of vacuum bubble walls, and the subsequent sound waves
and the MHD turbulence in the plasma Kosowsky  _et al._ (1992a, b); Kosowsky
and Turner (1993); Kamionkowski  _et al._ (1994); Caprini  _et al._ (2008);
Hindmarsh  _et al._ (2014); Giblin and Mertens (2013, 2014); Kosowsky  _et
al._ (2002); Caprini and Durrer (2006); Kahniashvili  _et al._ (2008a, b,
2010); Kisslinger and Kahniashvili (2015); Huber and Konstandin (2008);
Caprini  _et al._ (2009, 2016); Hindmarsh  _et al._ (2015). So, we have

$h^{2}\Omega_{\mathrm{GW}}(f)\simeq
h^{2}\Omega_{\textup{en}}(f)+h^{2}\Omega_{\textup{sw}}(f)+h^{2}\Omega_{\textup{tu}}(f)\,,$
(2)

where $h\equiv
H_{0}/(100\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1})=0.674$ Aghanim
_et al._ (2020) is the dimensionless Hubble constant, and Huber and Konstandin
(2008); Caprini  _et al._ (2009, 2016); Hindmarsh  _et al._ (2015)

$\displaystyle h^{2}\Omega_{\textup{en}}$ $\displaystyle\\!=\\!3.6\times
10^{-5}\\!\left(\\!\frac{H_{\ast}}{\beta}\\!\right)^{\\!\\!2}\\!\\!\left(\\!\frac{\kappa_{1}\alpha}{1+\alpha}\\!\right)^{\\!\\!2}\\!\\!\left(\\!\frac{10}{g_{\ast}}\\!\right)^{\\!\\!\frac{1}{3}}\\!\\!\left(\\!\frac{0.11v_{w}^{3}}{0.42+v_{w}^{2}}\\!\right)\\!\\!S_{\textup{en}},$
(3) $\displaystyle h^{2}\Omega_{\textup{sw}}$ $\displaystyle\\!=\\!5.7\times
10^{-6}\\!\left(\frac{H_{\ast}}{\beta}\right)\\!\\!\left(\frac{\kappa_{2}\alpha}{1+\alpha}\right)^{\\!2}\\!\left(\frac{10}{g_{\ast}}\right)^{\\!\\!\frac{1}{3}}v_{w}S_{\textup{sw}},$
(4) $\displaystyle h^{2}\Omega_{\textup{tu}}$ $\displaystyle\\!=\\!7.2\times
10^{-4}\\!\left(\frac{H_{\ast}}{\beta}\right)\\!\left(\frac{\kappa_{3}\alpha}{1+\alpha}\right)^{\\!2}\\!\\!\left(\frac{10}{g_{\ast}}\right)^{\\!\frac{1}{3}}\\!v_{w}S_{\textup{tu}},$
(5)

are the contributions respectively from collisions of the bubble walls and the
sound waves, as well as from the turbulence. Here, $\alpha$ is the ratio of
the vacuum energy density to the radiation energy density, $\beta$ is the
inverse time duration of the phase transition, and $v_{w}$ denotes the
velocity of the bubble wall. Meanwhile, $g_{\ast}$ and $H_{\ast}$ correspond
to the number of active degrees of freedom and the Hubble parameter at the
time when GWs are produced. We set $g_{\ast}=10$ and $\beta/H_{\ast}=10$
Ahmadvand and Bitaghsir Fadafan (2017, 2018); Chen  _et al._ (2018) at the QCD
scale. The spectral shapes of GWs are expressed by numerical fits as Huber and
Konstandin (2008); Caprini  _et al._ (2016); Hindmarsh  _et al._ (2015);
Caprini  _et al._ (2009)

$\displaystyle S_{\textup{en}}$
$\displaystyle=\frac{3.8\left(f/f_{\mathrm{en}}\right)^{2.8}}{1+2.8\left(f/f_{\mathrm{en}}\right)^{3.8}},$
(6) $\displaystyle S_{\textup{sw}}$
$\displaystyle=\left(f/f_{\mathrm{sw}}\right)^{3}\left[\frac{7}{4+3\left(f/f_{\mathrm{sw}}\right)^{2}}\right]^{\frac{7}{2}},$
(7) $\displaystyle S_{\textup{tu}}$
$\displaystyle=\frac{\left(f/f_{\mathrm{tu}}\right)^{3}}{\left[1+\left(f/f_{\mathrm{tu}}\right)\right]^{\frac{11}{3}}\left(1+8\pi
f/h_{\ast}\right)},$ (8)

where
$h_{\ast}=11.2\,\mathrm{nHz}\,[T_{\ast}/(100\,\mathrm{MeV})](g_{\ast}/10)^{1/6}$
is the Hubble rate at $T_{\ast}$, the temperature of the thermal bath when GWs
are generated. We assume that the GWs are generated promptly after the phase
transition, hence $T_{\ast}$ approximates the critical phase transition
temperature. The peak frequencies are given by Kamionkowski  _et al._ (1994);
Caprini  _et al._ (2016, 2009)

$\displaystyle f_{\textup{en}}$
$\displaystyle\\!=\\!11.2\,\textup{nHz}\\!\left(\\!\frac{0.62}{1.8-0.1v_{w}+v_{w}^{2}}\\!\right)\\!\\!\\!\left(\\!\frac{\beta}{H_{\ast}}\\!\right)\\!\\!\\!\left(\\!\frac{T_{\ast}}{100\textup{MeV}}\\!\right)\\!\\!\left(\\!\frac{g_{\ast}}{10}\\!\right)^{\\!\frac{1}{6}},$
(9) $\displaystyle f_{\textup{sw}}$
$\displaystyle\\!=\\!12.9\,\textup{nHz}\\!\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}},$
(10) $\displaystyle f_{\textup{tu}}$
$\displaystyle\\!=\\!18.4\,\textup{nHz}\\!\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}.$
(11)

The coefficients $\kappa_{1},\kappa_{2}$, and $\kappa_{3}$ in Eqs. (3-5) are
model dependent. These coefficients represent fractions of the vacuum energy
converted to the kinetic energy of the bubbles, bulk fluid motion, and the MHD
turbulence, respectively. In this work, we examine two bubble scenarios:
Jouguet detonations and non-runaway bubbles. For Jouguet detonations
Steinhardt (1982); Caprini  _et al._ (2016); Kamionkowski  _et al._ (1994);
Nicolis (2004); Espinosa  _et al._ (2010); Hindmarsh  _et al._ (2015), we have
$\kappa_{1}=(0.715\alpha+0.181\sqrt{\alpha})/(1+0.715\alpha)$,
$\kappa_{2}=\sqrt{\alpha}/(0.135+\sqrt{\alpha+0.98})$,
$\kappa_{3}=0.05\kappa_{2}$, and
$v_{w}=(\sqrt{1/3}+\sqrt{\alpha^{2}+2\alpha/3})/(1+\alpha)$. For non-runaway
bubbles Espinosa  _et al._ (2010); Caprini  _et al._ (2016); Hindmarsh  _et
al._ (2015); Kamionkowski  _et al._ (1994), we have $\kappa_{1}=0$,
$\kappa_{2}=\alpha/(0.73+0.083\sqrt{\alpha}+\alpha)$,
$\kappa_{3}=0.05\kappa_{2}$, and $v_{w}=0.95$.

Figure 1: Left panel: Bayesian posteriors for model parameters $\alpha$ and
$T_{*}$ in the Jouguet detonation bubble scenario, using the NANOGrav 15-year
data set. We show the $1\sigma$, $2\sigma$, and $5\sigma$ contours in the two-
dimensional plot. The five holographical models are also indicated in the
parameter space. Right panel: Posterior predictive distributions for the GW
spectrum derived from the NANOGrav 15-year data set. The grey region
represents the $5\sigma$ confidence interval, while the brown violins depict
free spectrum data from the NANOGrav 15-year data set. Additionally, We show
the GW spectra from the QCD-matter confinement-deconfinement phase transition
in the Jouguet detonation bubble case from five holographical models.

The expression in Eq. (2) relies entirely on $\alpha$ and $T_{\ast}$, which
are related to the QCD matter scenario and the phase transition under
consideration. Normally, one may expect to determine the specific values of
$\alpha$ and $T_{*}$ by the standard lattice QCD simulations. However, these
simulations encounter limitations due to the sign problem, yielding very few
applicable results Takeda  _et al._ (2012); Fromm  _et al._ (2012); Ding
(2017). To circumvent these difficulties, one then adopts the Anti-de Sitter
(AdS)/QCD framework that is established upon the AdS/conformal field theory
(CFT) correspondence principle Maldacena (1998); Gubser  _et al._ (1998);
Witten (1998). This framework offers novel insights into non-perturbative
hadron dynamics through the dual gravitational field Erlich  _et al._ (2005).
Specifically, it interprets the first-order confinement-deconfinement phase
transitions via Hawking-Page (HP) transitions Hawking and Page (1983) in five-
dimensional spacetime within AdS/QCD models Herzog (2007). In this framework,
the high-temperature quark-gluon plasma corresponds to the AdS black hole,
while the low-temperature hadron phase aligns with thermal AdS space. The
generation of GWs from the first-order QCD phase transition has been studied
within this framework Randall and Servant (2007), considering systems with
heavy static quarks under both zero Ahmadvand and Bitaghsir Fadafan (2017) and
finite baryon chemical potentials Ahmadvand and Bitaghsir Fadafan (2018);
Rezapour  _et al._ (2022), as well as in pure gluon systems Chen  _et al._
(2018), and employing various AdS/QCD models. Following Li  _et al._ (2021),
we explore the three types of QCD matter systems across five distinct
holographic models: (i) heavy static quarks with a zero baryon chemical
potential in hard-wall model $S_{1}$ and (ii) soft-wall model $S_{2}$, (iii)
quarks with a finite baryon chemical potential in hard-wall model $S_{3}$ and
(iv) soft-wall model $S_{4}$, and (v) pure gluons in the quenched dynamical
holographic model $S_{5}$. For a specific model, one can obtain the value of
$\alpha$ and $T_{\ast}$ via some holographic techniques. Table 1 summarizes
the corresponding values, and we refer our readers to Herzog (2007); Ahmadvand
and Bitaghsir Fadafan (2018, 2017); Chen  _et al._ (2018) for detailed
calculations.

Figure 2:  Same as Fig. 1 but for the non-runaway bubble case.

##  III Data analyses and results

In this work, we analyze the NANOGrav 15-year data set Agazie  _et al._
(2023b) to estimate model parameters $\alpha$ and $T_{*}$. Specifically, we
utilize the free spectrum amplitudes derived by NANOGrav, considering the full
spatial correlations following the Hellings-Downs pattern. The sensitivity of
a PTA’s observation begins at a frequency of $1/T_{\mathrm{obs}}$, with
$T_{\mathrm{obs}}=16.03\,\mathrm{yr}$ representing the observational time span
of NANOGrav. Following NANOGrav Agazie  _et al._ (2023a), we employ $14$
frequencies for Bayesian inference.

We start with the posterior data for time delay, $d(f)$, released by NANOGrav
Agazie  _et al._ (2023a). The power spectrum $S(f)$ is linked to the time
delay $d(f)$ through

$S(f)=d(f)^{2}\,T_{\mathrm{obs}}.$ (12)

By using the time delay data, we can further compute the GW energy density of
the free spectrum as

$\hat{\Omega}_{\mathrm{GW}}(f)=\frac{2\pi^{2}}{3H_{0}^{2}}f^{2}h_{c}^{2}(f)=\frac{8\pi^{4}}{H_{0}^{2}}T_{\mathrm{obs}}f^{5}d^{2}(f),$
(13)

where $h_{c}(f)$ is the characteristic strain defined by

$h_{c}^{2}(f)=12\pi^{2}f^{3}S(f).$ (14)

For each observed frequency $f_{i}$, we employ the posteriors of
$\hat{\Omega}_{\mathrm{GW}}(f_{i})$ obtained in Eq. (13) to estimate the
corresponding kernel density, $\mathcal{L}_{i}(\hat{\Omega}_{\mathrm{GW}})$.
Consequently, the total log-likelihood is the summation of individual log-
likelihoods given by Moore and Vecchio (2021); Lamb  _et al._ (2023); Liu  _et
al._ (2023b); Wu  _et al._ (2023c); Jin  _et al._ (2023); Liu  _et al._
(2023a)

$\ln\mathcal{L}(\Lambda)=\sum_{i=1}^{66}\ln\mathcal{L}_{i}(\Omega_{\mathrm{GW}}(f_{i},\Lambda)),$
(15)

where $\Lambda\equiv\\{\alpha,T_{*}\\}$ represents the two model parameters.
We utilize the dynesty Speagle (2020) sampler available within the Bilby
Ashton  _et al._ (2019); Romero-Shaw  _et al._ (2020) package to explore the
parameter space. To evaluate the model fits, we calculate the Bayesian
Information Criterion (BIC) Schwarz (1978), defined as

$\mathrm{BIC}=k\ln n-2\ln\hat{\mathcal{L}},$ (16)

where $n=66$ corresponds to the number of data points in the frequency space,
$k$ is the number of parameters in the model, and $\hat{\mathcal{L}}$ is the
maximum likelihood. The BIC facilitates the comparison of two competing models
Kass and Raftery (1995): if the difference in BICs between the two models is
$0-2$, it constitutes ‘weak’ evidence in favor of the model with the smaller
BIC; a difference in BICs between 2 and 6 indicates ‘positive’ evidence; a
difference in BICs between 6 and 10 signifies ‘strong’ evidence; and a
difference in BICs greater than 10 indicates ‘very strong’ evidence in favor
of the model with a smaller BIC.

For the model parameters, we sample them in logarithm space. Specifically, we
employ priors such that $\log_{10}\alpha$ is uniformly distributed in
$[-1,3]$, and $\log_{10}(T_{*}/\mathrm{MeV})$ is uniformly distributed in
$[0.5,2.5]$. For the Jouguet detonation case, the NANOGrav data yields
$\alpha=1.22^{+351}_{-0.86}$ and $T_{*}=42^{+76}_{-31}\,\mathrm{MeV}$. Unless
specified otherwise, we report the median value and $90\%$ equal-tail
uncertainties for each parameter. Fig. 1 shows the posteriors of model
parameters and the corresponding GW energy density in the Jouguet detonation
case. Notably, the NANOGrav 15-year data set is in agreement with the $S_{1}$
and $S_{3}$ models, suggesting that the confinement-deconfinement phase
transition in pure quark systems (specifically, cases involving heavy static
quarks with a zero baryon chemical potential and quarks with a finite baryon
chemical potential) can serve as the cosmological sources of the PTA signal in
the Jouguet detonation scenario. Furthermore, both $S_{1}$ and $S_{3}$ belong
to the hard wall holographic model, thereby excluding the soft wall
holographic model. In contrast, the confinement-deconfinement transition in
the pure gluon system is unlikely the source of the NANOGrav signal. Since the
critical phase transition temperatures for the cases of a finite chemical
potential and a zero chemical potential are very close, the chemical potential
exerts negligible influence on the power spectrum of GWs. Thus, the current
NANOGrav observation indicates that quark confinement dominates the
cosmological QCD transition. For the non-runaway bubble case, the posteriors
are $\alpha=0.63^{+35}_{-0.23}$ and $T_{*}=17^{+42}_{-6.6}\,\mathrm{MeV}$.
Fig. 2 displays the results for the non-runaway bubble case. It is evident
that all five holographic models are ruled out by the NANOGrav 15-year data
set at a $5\sigma$ confidence level. Therefore, the NANOGrav data decisively
excludes the non-runaway bubble case.

In Table 2, we present the differences in BIC among the ten QCD-like models
with $\beta/H_{*}=10$. Remarkably, the $S_{1}$ and $S_{3}$ models within the
Jouguet bubble scenario exhibit superior performance compared to all other
models. Furthermore, there is ‘positive’ evidence supporting the preference
for the $S_{3}$ model over the $S_{1}$ model in this scenario. Extending the
parameter $\beta/H_{*}$ to a larger value, such as $\beta/H_{*}=15$, as shown
in Table 3, the $S_{1}$ and $S_{3}$ models in the Jouguet bubble scenario
maintain their superior performance over other models, with ‘very strong’
evidence favoring the $S_{3}$ model over the $S_{1}$ model. Therefore, a
larger value of $\beta/H_{*}$ does not significantly alter our findings.

| $S_{1}$ | $S_{2}$ | $S_{3}$ | $S_{4}$ | $S_{5}$
---|---|---|---|---|---
Jouguet | 0 | 42.5 | -3.1 | 41.2 | 63.4
non-runaway | 43.0 | 60.2 | 34.0 | 60.1 | 59.2
Table 2: The difference of BIC among the ten QCD-like models with fixing $\beta/H_{*}=10$. We have set the $S_{1}$ from the Jouguet bubble case as the fiducial model. | $S_{1}$ | $S_{2}$ | $S_{3}$ | $S_{4}$ | $S_{5}$
---|---|---|---|---|---
Jouguet | 0 | 17.5 | -12.9 | 17.4 | 17.0
non-runaway | 16.6 | 18.6 | 19.9 | 18.6 | 16.3
Table 3: Same as Table 2, but fixing $\beta/H_{*}=15$.

##  IV Summary and discussion

In this paper, we firstly demonstrate that the stochastic signal detected in
the NANOGrav 15-year data set can be successfully attributed to an SGWB
arising from the first-order cosmological confinement-deconfinement phase
transition of either heavy static quarks with zero baryon chemical potential
or quarks with finite baryon chemical potential, aligning with the results in
our previous work Li  _et al._ (2021). The baryon chemical potential exerts
negligible influence on the power spectra of GWs generated by confinement-
deconfinement phase transitions. This is attributed to the close proximity of
the phase transition temperatures in both the finite baryon chemical potential
and zero chemical potential scenarios. Furthermore, it is worth noting that,
based on the Bayesian method, our analysis has revealed that the phase
transition of a pure gluon plasma cannot be regarded as the potential source
of the NANOGrav signal, and the non-runaway bubble dynamics model is
effectively ruled out by the NANOGrav data at a significance level of
$5\sigma$. Secondly, we obtain a remarkably distinct result compared to the
previous work, i.e., the bubble dynamics in the phase transition should be
characterized by the Jouguet detonation rather than a non-runaway scenario.
Thirdly, our results provide some insights into the applicability of
holographic descriptions of the quark phase transitions, in which the models
of hard and soft walls relate to the introduction of sharp and smooth cutoffs
respectively. According to our analysis, the holographic description using the
hard wall model for the quark phase transition is more likely to be consistent
with PTA data as compared to that using the soft wall model.

In this work, we only examine the first-order confinement-deconfinement phase
transitions. However, it is also worth considering other types of the first-
order phase transition at the QCD scale, such as the chiral symmetry-breaking
Reya (1974). Moreover, relying solely on PTA observations might not be
sufficient to distinguish the various details of different first-order phase
transitions. Combining additional constraints Bai and Korwar (2022) related to
first-order phase transitions will help us gain further insights into the
processes of cosmic phase transitions. We leave these aspects for future
research.

_Acknowledgments_ ZCC is supported by the National Natural Science Foundation
of China (Grant No. 12247176 and No. 12247112) and the China Postdoctoral
Science Foundation Fellowship No. 2022M710429. SLL is supported by the
National Natural Science Foundation of China (Grant No. 12105098, No.
11947216, No. 12005059) and the Natural Science Foundation of Hunan Province
(Grant No. 2022JJ40264). HWY and PXW are supported by the National Key
Research and Development Program of China Grant No. 2020YFC2201502, and by the
National Natural Science Foundation of China under Grants No. 12275080 and No.
12075084.

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