Gravitational Waves and Primordial Black Hole Productions from Gluodynamics by Holography Song He12Li Li345Zhibin Li6and Shao-Jiang Wang3

2025-05-06 0 0 1.44MB 17 页 10玖币

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Gravitational Waves and Primordial Black Hole Productions from Gluodynamics by

Holography

Song He1,2,∗Li Li3,4,5,†Zhibin Li6,‡and Shao-Jiang Wang3§

1Center for Theoretical Physics and College of Physics, Jilin University, Changchun 130012, China

2Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muhlenberg 1, 14476 Golm, Germany

3CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China

4School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,

University of Chinese Academy of Sciences, Hangzhou 310024, China

5Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China and

6School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China

Understanding the nature of quantum chromodynamics (QCD) matter is important but

challenging due to the presence of non-perturbative dynamics under extreme conditions. We

construct a holographic model describing the gluon sector of QCD at ﬁnite temperatures in the

non-perturbative regime. The equation of state as a function of temperature is in good accordance

with the lattice QCD data. Moreover, the Polyakov loop and the gluon condensation, which are

proper order parameters to capture the deconﬁnement phase transition, also agree quantitatively

well with the lattice QCD data. We obtain a strong ﬁrst-order conﬁnement/deconﬁnement phase

transition at Tc= 276.5 MeV that is consistent with the lattice QCD prediction. Based on

our model for a pure gluon hidden sector, we compute the stochastic gravitational waves and

primordial black hole (PBH) productions from this conﬁnement/deconﬁnement phase transition in

the early Universe. The resulting stochastic gravitational-wave backgrounds are found to be within

detectability in the International Pulsar Timing Array and Square Kilometre Array in the near

future when the associated productions of PBHs saturate the current observational bounds on the

PBH abundances from the LIGO-Virgo-Collaboration O3 data.

Keywords: AdS/QCD, conﬁnement/deconﬁnement phase transition, gravitational wave, primor-

dial black hole

PACS numbers: 13.40.-f, 25.75.-q, 11.10.Wx

I. INTRODUCTION

The early Universe before the big bang nucleosynthesis is opaque to electromagnetic waves. Thanks to the recent

gravitational-wave detections, future observations of stochastic gravitational wave backgrounds (SGWBs) would reveal

the new physics [1–4] from the early Universe, including various ﬁrst-order phase transitions (FOPTs) beyond the

standard model of particle physics (see [5] and references therein for a model summary). It was recently found that the

FOPT not only associates with SGWBs but also produces primordial black holes (PBHs) in general [6, 7] (see also [8]

for an explicit example from the electroweak phase transition), regardless of the speciﬁc particle physics model for

realizing the FOPTs (see also [9–13] for other speciﬁc mechanisms of PBH productions during some particular kinds

of FOPTs). In particular, for the FOPT around the QCD scale, the associated SGWBs can be probed by the Pulsar

Timing Array (PTA) and Square Kilometre Array (SKA) observations, and the associated PBH abundance could

be constrained by the LIGO-Virgo-Collaboration (LVC) network. While the QCD phase transition in the standard

model at small baryon chemical potentials is cross-over, the pure gluon case features a conﬁnement FOPT. This is a

minimal scenario among many extensions of the standard model and is ideal as a benchmark model. Therefore, we

will study pure gluons in this work for a realization of the FOPT around the QCD scale with associated productions of

SGWBs and PBHs. Note here that the large density perturbations required to form PBHs from FOPTs are generated

during FOPTs in the radiation era. This is totally diﬀerent from other popular PBH production mechanisms with the

large density perturbations induced from the large curvature perturbations originated from the inﬂationary period,

∗Electronic address: hesong@jlu.edu.cn

†Electronic address: liliphy@itp.ac.cn

‡Electronic address: lizhibin@zzu.edu.cn

§Electronic address: schwang@itp.ac.cn

arXiv:2210.14094v2 [hep-ph] 9 Jan 2024

for example, in the non-minimal curvaton models [14, 15].

On the other hand, investigating the pure gluon system is important to understand the nature of hot and dense

QCD matter formed in the early Universe and the laboratory. In particular, the gluon dynamics is dominant during

10−5seconds into the expansion of the early Universe [16–19] and an extremely rapid thermalization [20–22] in

nucleus-nucleus collisions. On the theoretical side, the thermodynamics of the pure-gauge sector can be relevant

to capture the essential qualitative features of the deconﬁnement, which is characterized by center symmetry and

shows all the infrared diﬃculties of QCD. Due to the strong coupling, non-perturbative approaches are necessary

for quantitative studies of its dynamics. In addition to the lattice QCD that relies on massive computing power, an

alternative non-perturbative approach is to employ the gauge/gravity correspondence [23–25] that provides a powerful

way to study strongly coupled non-Abelian gauge theories (see also [26–30] for earlier studies on the pure gluon system

from holography).

In this work, we provide a bottom-up holographic QCD model for the pure gluon QCD system. The equation of

state (EoS) quantitatively matches the pure gluon system in lattice QCD [31, 32]. The conﬁnement phase transition

in gauge theory is characterized by the Polyakov loop operator ⟨P⟩ which is ﬁnite in the deconﬁned phase and

becomes vanishing in the conﬁned phase for pure gluon [33, 34]. The temperature dependence of ⟨P⟩ from our model

matches the lattice simulation [35] perfectly, and the predicted critical temperature Tc= 276.5 MeV agrees with the

expectation in the literature [31, 36]. Moreover, another important quantity characterizing the deconﬁnement phase

transition in a pure gluon system is the gluon condensation, which can be computed to be quantitatively consistent

with the trace anomaly [31]. The strong FOPT in the early Universe is also a potentially important source for the

production of SGWBs and PBHs. Our present model provides a reliable scenario for generating gravitational waves

from a FOPT of a pure SU(3) Yang-Mills sector. The resulting gravitational wave signals could be detected in the

upcoming International PTA (IPTA) and SKA observations for the associated PBH abundance saturating the current

observational bounds from the LVC constraints.

II. MODEL

We now build up a holographic model for the SU(3) pure gluon system with the action of the following form.

S=1

2κ2

NZd5x√−gR − 1

2∇µϕ∇µϕ−V(ϕ)(1)

with the minimal cost of degrees of freedom to capture the essential dynamics. The gravitational theory includes only

two ﬁelds: the spacetime metric gµν , and a real scalar ϕwith its proﬁle breaking conformal invariance that can be

understood roughly as the running coupling of QCD. In addition to κ2

Nthat is the eﬀective Newton constant, the

potential V(ϕ) will be ﬁxed by matching to the lattice QCD data.

The black hole with non-trivial scalar hair reads

ds2=−f(r)e−η(r)dt2+dr2

f(r)+r2dx2

3, ϕ =ϕ(r),(2)

with dx2

3= dx2+ dy2+ dz2and rthe holographic radial coordinate. Denoting rhas the location of the event horizon

where f(rh) = 0, the temperature reads T=f′(rh)e−η(rh)/2/4π. Other thermodynamic quantities can be obtained

straightforwardly using the standard holographic dictionary, see the Supplemental Material [37] for more details. The

next goal is to ﬁnd a potential Vthat can reproduce the EoS of Nc= 3 pure gluon QCD. It comes as a nice surprise

that the simple potential

V(ϕ) = 6γ2−3

2ϕ2−12 cosh(γϕ) (3)

with γ= 0.735 can reproduce the thermodynamics of lattice data for the pure gluon QCD [31, 32, 35] as shown in

Fig. 1. Remarkably, although the error bars of the up-to-date lattice simulation [32] are tiny, our theoretical results

for EoS in the left panel are almost within these error bars. It is obvious from the free energy density Fthat a strong

FOPT takes place at the temperature Tc= 276.5 MeV. We also compare the speed of sound csin the right panel of

Fig. 1. Since csis not provided in [32], we use the early data from lattice QCD [31] and ﬁnd good agreement.

To understand the nature of the FOPT, we compute the expectation value of the Polyakov loop operator ⟨P⟩ [38–

41], which is a good order parameter to the deconﬁnement phase transition for pure gluon system [42]. Surprisingly,

⟨P⟩ by our holographic model quantitatively agrees with the lattice data [35] above Tcand it quickly drops to zero

below Tc, see the right panel of Fig. 1. It suggests that the FOPT from our model is a conﬁnement/deconﬁnement

phase transition. Remarkably, the temperature dependence of the gluon condensation δDβ(g)

2gG2ETcapturing the

1.0 1.5 2.0 2.5

T/Tc

●s/T3

● ϵ /T4

●I/T4

●P/T4

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.0

0.1

0.2

0.3

0.4

T/Tc

270 272 274 276 278 280

-1

-2

-3

-4

-5

×107

T[MeV ]

F[MeV4]

Tc=276.5 MeV

2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

T/Tc

〈〉

FIG. 1: The comparison between the lattice data (with error bar) of the pure gluon thermodynamics and our holographic

calculations (solid curves) on various thermodynamic quantities. Discontinuous changes at the ﬁrst-order phase transition are

represented by dashed lines. Left panel: The temperature dependence of the energy density ϵ, the entropy density s, the

pressure P, and the trace anomaly I= (ϵ−3P) [32]. Right panel: The squared speed of sound c2

s≡dP/dϵ [31] and the

Polyakov loop ⟨P⟩ [35] in function of temperature. Insert: The free energy density Fwith respect to the temperature from

our model. There is a ﬁrst-order conﬁnement/deconﬁnement phase transition at Tc= 276.5 MeV.

δ 〈 β(g)

2 g G2〉T/T4

I/T4lattice data [31]

I/T4lattice data [32]

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

T/Tc

FIG. 2: The temperature dependence of the gluon condensation δDβ(g)

2gG2ETof our pure gluon model, where β(g) is the

β-function with gthe QCD gauge coupling. The data with error bar denotes the trace anomaly I= (ϵ−3P) from lattice

QCD [31, 32].

deconﬁnement phase transition is computed in our holographic model and is found to coincide with the trace anomaly

ϵ−3pfrom EoS [43, 44], see Fig. 2. Therefore, at Tc, we can then read oﬀ some essential quantities that are important

to compute the SGWB and PBH productions associated with our FOPT. The SGWBs generated in cosmological

FOPTs were considered in other holographic models, see e.g. [45–49].

Independent of details of any speciﬁc particle physics model, the PBH production is a universal consequence of the

FOPT [6]. Due to the stochastic nature of bubble nucleations during FOPTs, the progress of populating true-vacuum

bubbles in the false-vacuum background is an asynochronized process. There is always a non-vanishing probability to

ﬁnd some Hubble-sized regions to stay in the false vacuum for a slightly longer period of time than average. Since the

radiation energy density should be rapidly diluted relative to the vacuum energy density in an expanding Universe,

these Hubble-size regions would eventually accumulate enough overdensities in total energy density to ﬁnally reach the

threshold of PBH productions. What is remarkable for this general mechanism of PBH productions during FOPTs is

that the probability to ﬁnd such Hubble-sized regions with postponed vacuum-decay progress can be made of particular

observational interest for both detections from gravitational waves and PBHs, which will be brieﬂy described shortly

below and detailed in the Supplemental Material [37].

III. GRAVITATIONAL WAVE PRODUCTIONS

From the behavior of the free energy density in the insert of the right panel of Fig. 1, it clearly indicates the

occurrence of a ﬁrst-order conﬁnement/deconﬁnement phase transition around the critical temperature Tc= 276.5

MeV, which could be a potentially important source for gravitational waves in the early Universe. The cosmological

FOPT proceeds with stochastic nucleations of true vacuum bubbles in the false vacuum environment followed by the

rapid expansion until percolations via bubble collisions. The bubble wall collision and plasma ﬂuid motion including

sound waves and magnetohydrodynamic (MHD) turbulences would generate the corresponding SGWBs with broken

power-law shapes in their energy density spectra.

Given the expansion history a(t) and vacuum decay rate of form Γ(t)≡A(t)e−B(t)per unit time and unit volume,

the fraction of spatial regions that are still staying at the false vacuum at time tcan be estimated by [50, 51]

F(t;ti) = exp −4π

3Zt

dt′Γ(t′)a(t′)3r(t, t′)3,(4)

where tiis the earliest possible time for the nucleation of the ﬁrst bubble ever, and r(t, t′) = Rt

t′d˜

t/a(˜

t) is the comoving

radius of a bubble at time tnucleated from an earlier time t′. It is obvious that all regions are in the false vacuum

before time ti, namely F(t<ti;ti) = 1. With the help of F(t;ti), the percolation time t∗for the gravitational

wave spectra from the FOPT is then conventionally deﬁned by F(t∗;ti) = 0.7 [52], around which the decay rate

can be expanded linearly in time for its exponent as Γ(t) = A(t∗)e−B(t∗)+β(t−t∗)≡Γ0eβt [53][54]. In this study,

we will simply approximate the percolation temperature by the critical temperature Tc= 276.5 MeV from previous

holographic computations. See [55] for a potential estimation on the eﬀective potential and subsequent nucleation

rate as well as the associated percolation temperature, which will be reserved for more detailed work in future.

The energy density spectra for the prementioned threes sources of SGWBs from a cosmological FOPT are given as

follows. The uncollided part of bubble wall envelopes [56–60] admits an analytic form [60–62] for the gravitational

wave spectrum as

h2Ωenv = 1.67×10−5100

gdof 1

3H∗

β2κϕα

1 + α20.48v3

1+5.3v2

w+ 5v4

Senv(f),(5)

where the shape factor is given by

Senv(f) = 1

clf

fenv −3+ (1 −cl−ch)f

fenv −1+chf

fenv (6)

with cl= 0.064 and ch= 0.48, and the peak frequency is given by

fenv = 1.65 ×10−5Hz gdof

100 1

6T∗

100 GeV 0.35(β/H∗)

1+0.069vw+ 0.69v4

.(7)

Here the eﬃciency factor κϕcharacterizes the amount of released vacuum energy into the kinetic energy of bubble

walls. The dominant contribution to the ﬂuid motions comes from the sound waves [63–65], whose gravitational wave

spectrum is given by

h2Ωsw = 2.65×10−6100

gdof 1

3H∗

βκswα

1 + α277/2vw(f/fsw)3

(4 + 3(f/fsw)2)7/2Υ,(8)

with the peak frequency

fsw = 1.9×10−5Hz gdof

100 1

6T∗

100 GeV  1

vw β

H∗.(9)

Here the eﬃciency factor κsw characterizes the amount of released vacuum energy into the kinetic energy of ﬂuid

motions, and the suppression factor Υ ≡1−(1 + 2τswH∗)−1/2[66] accounts for the ﬁnite lifetime of sound waves from

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GravitationalWavesandPrimordialBlackHoleProductionsfromGluodynamicsbyHolographySongHe1,2,∗LiLi3,4,5,†ZhibinLi6,‡andShao-JiangWang3§1CenterforTheoreticalPhysicsandCollegeofPhysics,JilinUniversity,Changchun130012,China2MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute),AmMuhlenberg1,14...

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Gravitational Waves and Primordial Black Hole Productions from Gluodynamics by Holography Song He12Li Li345Zhibin Li6and Shao-Jiang Wang3

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