# Bubble nucleation and gravitational wave from holography

Yidian Chen<EMAIL_ADDRESS>School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, P.R. China  Danning
Li<EMAIL_ADDRESS>Department of Physics and Siyuan Laboratory, Jinan
University, Guangzhou 510632, P.R. China  Mei Huang<EMAIL_ADDRESS>School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, P.R. China

###### Abstract

We investigate the bounce solution in the holographic QCD and electroweak
models with first-order phase transition. The strength parameter $\alpha$,
inverse duration time $\beta/H$ and bubble wall velocity $v_{w}$ in the
gravitational wave power spectra are calculated by holographic bounce
solution. In contrast to the results of field theory, we find the parameter
$\alpha$ is about $\mathcal{O}(1)$ and $\beta/H$ is about $10^{4}$, which
imply that the phase transition is fast and strong. The critical, nucleation
and percolation temperatures of the phase transition are close to each other
in the holographic model. In addition, the velocity $v_{w}$ is found to be
less than the sound speed of the plasma $c_{s}=1/\sqrt{3}$, which corresponds
to the deflagration scenario. For QCD phase transition, the gravitational wave
power spectrum can reach $10^{-13}-10^{-14}$ around the peak frequency of 0.01
Hz, which can be detected by BBO and Ultimate-DECIGO. For electroweak phase
transition, the gravitational wave power spectrum can reach
$10^{-12}-10^{-16}$ around the peak frequency $1-10$ Hz. Moreover, the
primordial black hole is not favorable for formation due to the large
parameter $\beta/H$ and small velocity $v_{w}$.

##  I introduction

Gravitational waves (GWs) are one of the great predictions of general
relativity [1, 2], which exhibits an effect of the curvature of spacetime. In
1974, Hulse and Taylor discovered the binary system PSR 1913+16 [3], providing
indirect evidence for the existence of GWs, for which they received the 1993
Nobel Prize in Physics. In 2015, astronomy entered the multi-messenger era
with the first observation of a gravitational wave event by LIGO [4, 5].
Nowadays, more and more GW events or possible GW signals are discovered, as in
Refs. [6], which also provide new instruments for understanding cosmology and
astronomy deeply.

The GWs come from the quadrupole moment radiation of the stress-energy tensor,
which can be roughly classified into two categories, that is, cosmological and
astronomical sources (see the reviews Refs. [7, 8, 9]). The stochastic GW
background generated by the first-order phase transition (FOPT) of the early
universe is a significant cosmological source of GWs (see the reviews Ref.
[10]). Different from the transient GWs observed at the present, the
stochastic GW background comes from all directions rather than specific ones.

The detection experiments of GWs mainly include ground-based experiments,
space-based experiments, pulsar timing arrays (PTA), and cosmic microwave
background polarization, (see [11] for review). The ground-based experiments,
such as LIGO [12], Virgo [13], Einstein Telescope (ET) [14], Cosmic Explorer
(CE) [15], etc., mainly observe compact binary systems. The space-based
experiments, such as Laser Interferometer Space Antenna (LISA) [16], Deci-
Hertz Interferometer Gravitational wave Observatory (DECIGO) [17, 18], Big
Bang Observer (BBO) [19], Taiji [20], Tianqin [21], etc., are more sensitive
to the GWs from electroweak phase transition. The PTAs, such as the Parkes PTA
(PPTA) [22], the European PTA (EPTA) [23], the North American Nanohertz
Observatory for Gravitational Waves (NANOGrav) [24], the International PTA
(IPTA) [25] and the Chinese PTA (CPTA), etc., are mainly sensitive to the GWs
from QCD phase transitions.

The dynamic process of the FOPT is described by bubble dynamics. When the
temperature of the system reaches the critical temperature $T_{c}$ for the
phase transition, it does not immediately enter the symmetric broken phase
since the generation of true vacuum bubbles causes additional surface free
energy, which increases the total free energy of the system. As the
temperature decreases to the nucleation temperature $T_{n}$, bubbles are more
likely to be generated due to the increasing probability of thermal
perturbation or quantum tunneling that crosses the free energy barrier. The
temperature drops further to the percolation temperature $T_{p}$, and more
than one-third of the space enters into the true vacuum. During this process,
the bubble expands continuously since the internal pressure is greater than
the false vacuum pressure and surface tension, and it may eventually reach the
final velocity because of the friction of the plasma or accelerate to near the
speed of light. Throughout the process, collisions of bubbles, acoustic modes
of the plasma and turbulences of the magnetohydrodynamics all generate GWs and
eventually contribute to the GW power spectra.

During the evolution of the universe, it may undergo various phase
transitions, such as grand unification phase transition, electroweak phase
transition (EWPT), and QCD phase transition (QCDPT). The FOPT is of interest
since it is related to some physical processes such as baryogenesis, the seeds
of intergalactic magnetic fields and the formation of primordial black holes.
Unfortunately, the electroweak part of the standard model is crossover [26,
27, 28], while the Lattice QCD calculations indicate that the QCD phase
transition of the three flavors is crossover at zero chemical potential and
finite temperature [29, 30]. Of course, many new physical models beyond the
standard model (BSM) predict the FOPT, such as the two-Higgs doublet model
[31, 32, 33], the left-right symmetric model [34], the technicolor model [35,
36, 37], etc. In addition, since QCDPT is flavor-dependent, there is still the
possibility of a first-order QCDPT.

The discovery of the anti-de Sitter/conformal field theory (AdS/CFT)
correspondence [38, 39, 40] has provided a new way to solve strongly coupled
field theory calculations. In the past two decades, holographic QCD has been
widely studied both in top-down [41, 42, 43] and bottom-up models [44, 45, 46,
47, 48, 49, 50]. For beyond the standard model the technicolor model [51, 52,
53, 54, 55, 56] and composite Higgs model [57, 58, 59, 60] have been extended
to the holographic framework.

The intensity of the GW signal is impacted by the strength of the phase
transition, its duration time and the final velocity of the bubble wall. The
calculations in weakly coupled quantum field theory suggest that the strength
parameter $\alpha$ is roughly $\alpha\sim 0.01$ and the inverse of the
duration time $\beta/H$ is roughly $\beta/H\sim 100$ during EWPT. As for the
speed, one would expect that it is close to the speed of light $c$ for
enhancing the GW signal. These quantities are not sufficiently discussed and
understood when the system is strongly coupled. The holographic principle
provides new ways to explore the thermodynamic and kinetic properties of phase
transitions with strong coupling. The bubble nucleation dynamics [61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72] and the GW power spectra [73, 74, 54, 75,
76, 77, 78, 79] are considered in the holographic model. The relation between
the bubble velocity and the pressure difference of the true and false vacuum
is investigated in Refs. [63, 65, 72]. In Refs. [70, 69], the profile of the
fluid velocity is calculated inside and outside the bubble wall.

The paper is organized as follows. The five-dimensional holographic QCD and
electroweak models are introduced in Sec. II. In Sec. III, the bounce solution
and bubble wall velocity are obtained in the holographic model. Using the
bounce solution, the properties of the critical bubble and the thin-wall
approximation are considered in Sec. IV. In Sec. V, the stochastic GW power
spectra generated by the strongly coupled FOPT are calculated. Finally, the
conclusion and discussion are presented in Sec. VI.

##  II 5d setup

In this section, we consider FOPT of QCD or QCD-like electroweak theories,
which corresponds to $SU(N_{f})_{L}\times SU(N_{f})_{R}$ flavor symmetry
breaking to $SU(N_{f})_{V}$ subgroup. Here, only the scalar part of the
flavor-brane is considered under the probe approximation, which has the
following form

$S=-\int d^{5}x\sqrt{-g}e^{-\Phi}{\rm
Tr}[(D^{M}X)^{\dagger}(D_{M}X)+V_{X}(|X|)].$ (1)

Among this, $\Phi$ denotes the dilaton field and the complex scalar field $X$
corresponds to the quark condensation $\langle\bar{q}q\rangle$ or fermionic
condensate of BSMs. The symmetry of the model is spontaneously broken by the
non-zero vacuum expectation value of the scalar field
$X=\frac{\chi(z,x)}{\sqrt{2N_{f}}}I_{N_{f}}$, where $I_{N_{f}}$ is the
$N_{f}\times N_{f}$ identity matrix and $\chi$ depends not only on the fifth
coordinate $z$ but also on the four-dimensional space-time coordinates
$x^{\mu}$.

In this paper, the back-reaction of the dilaton field $\Phi$ and the scalar
field $X$ are not taken into account, so the background geometry remains the
${\rm AdS_{5}}$-Schwarzchild black brane metric

$ds^{2}=\frac{L^{2}}{z^{2}}[-f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+dx_{i}dx^{i}],$
(2)

with the blackening factor $f(z)=1-\frac{z^{4}}{z_{h}^{4}}$ and the horizon
$z_{h}$. For convenience, the AdS radius $L$ is set to 1 in the following. The
Hawking temperature of the system is

$T=\frac{|f^{{}^{\prime}}(z)|}{4\pi}=\frac{1}{\pi z_{h}}.$ (3)

In order to realize the FOPT, the potential of the scalar field is considered
to have the following form[80, 81, 82]

$V(\chi)\equiv{\rm
Tr}[V_{X}(|X|)]=\frac{M_{5}^{2}}{2}\chi^{2}+\upsilon_{3}\chi^{3}+\upsilon_{4}\chi^{4}+\upsilon_{6}\chi^{6},$
(4)

where $M_{5}^{2}$ denotes the square of the five-dimensional mass, and
$\upsilon_{3}$, $\upsilon_{4}$ and $\upsilon_{6}$ are the cubic, quadratic and
sextic terms coupling constants, respectively. According to the AdS/CFT
dictionary, the five-dimensional mass of the scalar field $X$ is
$M_{5}^{2}=(\Delta-p)(\Delta+p-4)=-3$ by taking $p=0$ and $\Delta=3$ for QCD.
For the EW model, a large anomalous dimension $\gamma_{m}\simeq 1$ needs to be
considered[83, 84, 85, 86, 87, 88], so the five-dimensional mass is
$M_{5}^{2}=(\Delta-\gamma_{m})(\Delta-\gamma_{m}-4)=-4$ by taking $\Delta=3$
and $\gamma_{m}=1$. For simplicity, we do not consider all nonzero nonlinear
terms, but instead consider two special cases, i.e., $\upsilon_{6}=0$ (Model
I) or $\upsilon_{3}=0$ (Model II). The correct chiral symmetry breaking in the
chiral limit depends on the form of the dilaton field, which is chosen as in
Refs. [80, 81]

$\Phi=-\mu_{1}z^{2}+(\mu_{1}+\mu_{0})z^{2}\tanh(\mu_{2}z^{2}),$ (5)

where the $\mu_{0}$, $\mu_{1}$, and $\mu_{2}$ parameters determine the
behavior of the dilaton field in the IR and UV.

( a )  ( b )

Figure 1: The condensate $\sigma$ as a function of temperature $T$. Panels (a)
and (b) correspond to the QCD and EW cases, respectively.

In this holographic model, five free parameters
$[\mu_{0},\mu_{1},\mu_{2},\upsilon_{3},\upsilon_{4}]$ or
$[\mu_{0},\mu_{1},\mu_{2},\upsilon_{4},\upsilon_{6}]$ are included. For the
QCD case, we choose
$[\mu_{0},\mu_{1},\mu_{2},\upsilon_{3},\upsilon_{4}]=[(0.43{\rm
GeV})^{2},(0.83{\rm GeV})^{2},(0.176{\rm GeV})^{2},-3,8]$ (Model I) or
$[\mu_{0},\mu_{1},\mu_{2},\upsilon_{4},\upsilon_{6}]=[(0.43{\rm
GeV})^{2},(0.83{\rm GeV})^{2},(0.176{\rm GeV})^{2},-12,100]$ (Model II) with
the critical temperature of the phase transition $T_{c}\simeq 174$ MeV, by
referring to Refs. [80, 81, 82]. For the EW case, $\mu_{0}=\mu_{2}=0$ is
chosen to simplify the model. It is worth noting that if $\Phi=-\mu_{1}z^{2}$
is chosen, there is an additional massless scalar meson in the particle
spectra, as in Ref. [89]. For QCD, this is nonphysical. However, for the EW,
this state can be interpreted as the Higgs boson, which obtains mass by
interacting with other scalar fields. Therefore, for the EW case, the
parameters are chosen as
$[\mu_{0},\mu_{1},\mu_{2},\upsilon_{3},\upsilon_{4}]=[0,(0.28{\rm
TeV})^{2},0,-1.1,3]$ (Model I) or
$[\mu_{0},\mu_{1},\mu_{2},\upsilon_{4},\upsilon_{6}]=[0,(0.28{\rm
TeV})^{2},0,-1.8,4]$ (Model II) with critical temperatures of $T_{c}\simeq
122$ GeV. The Panels. (a) and (b) of Fig. 1 show the QCD and technicolor
condensate as a function of temperature, respectively. In fact, the specific
values of these parameters do not affect the qualitative results in the
following sections. In principle, the same approach can be used for symmetry
breaking of other groups, simply by changing the representation of the group.

##  III bounce solution and bubble velocity

In this section, we will study the kinetics of the FOPT in the holographic
model, which will involve the thermodynamics and dynamics of the bubble,
describing how the phase transition occurs and how long the phase transition
lasts. The phase transition kinetics are characterized by nucleation
temperature, latent heat, transition rate parameters, bubble velocity, etc.,
which will determine the specific magnitude of the GW spectrum.

The kinetic process of the FOPT is described by the nucleation theory, in
which the creation, expansion and fusion of bubbles transform the false vacuum
$\chi_{f}$ into the true vacuum $\chi_{t}$. The bubble nucleation is caused by
quantum tunneling or thermal perturbation. At the critical temperature
$T_{c}$, the true and false vacuums have the same free energy, and the phase
transition is suppressed due to the surface free energy of the bubbles
increasing the total free energy of the system. As the temperature decreases,
the difference between the free energy of the true and false vacuum
compensates for the surface free energy, and the bubble generation becomes
more and more probable until the phase transition is completed.

In the first half of the 20th century, classical and modern nucleation
theories were established[90, 91]. In the 1970s, nucleation theory was
extended to relativistic quantum field theory by Coleman and Callan[92, 93],
while in Refs. [94, 95, 96, 97] and [98] it was further extended to finite
temperature and density, respectively. See Refs. [99, 100, 101] for some
reviews.

###  III.1 Bounce Solution

In this subsection, we will construct bubble solutions in the holographic
model. Considering the spherical symmetry of the bubble, the AdS-Schwarzchild
metric is rewritten in spherical coordinates

$ds^{2}=\frac{1}{z^{2}}[-f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}],\quad(0\leq r\leq R)$ (6)

with azimuth $\varphi$, zenith angle $\theta$ and the edge $R$. Then, the
equations of motion of the scalar field $X$ can be obtained from action (1)

$\partial_{r}^{2}\chi(z,r)+\frac{2\partial_{r}\chi(z,r)}{r}+f(z)\partial_{z}^{2}\chi(z,r)+\left(f^{\prime}(z)-f(z)\Phi^{\prime}(z)-\frac{3f(z)}{z}\right)\partial_{z}\chi(z,r)-\frac{\partial_{\chi}V(\chi)}{z^{2}}=0.$
(7)

Coleman and Callan[92, 93] proposed that the critical bubble is described by
the bounce solution in quantum nucleation theory. The solution requires the
following boundary conditions

$\displaystyle\lim_{\text{$r\to\infty$}}\chi(z,r)$ $\displaystyle=$
$\displaystyle\chi_{f},$ (8)
$\displaystyle\frac{d\chi(z,r)}{dr}\Bigg{|}_{r=0}$ $\displaystyle=$
$\displaystyle 0.$ (9)

For the fifth dimensional direction, the expansion of the scalar field $\chi$
at the conformal boundary has the following form

$\displaystyle\chi|_{z\to 0}$ $\displaystyle=$ $\displaystyle m_{q}\zeta
z+...+\frac{\sigma}{\zeta}z^{3}+...\quad(\text{QCD}),$ (10) $\displaystyle=$
$\displaystyle\sigma z^{2}+...\quad(\text{EW}),$ (11)

with the quark current mass $m_{q}$, the condensate $\sigma$ and constant
$\zeta=\frac{\sqrt{N_{c}}}{2\pi}$. Here, we keep the terms related with the 4D
operators only, and in the following calculation we will take $N_{c}=3$ for
the QCD case. Since the FOPT is considered, it is convenient to set the
current mass $m_{q}$ to 0. At the IR boundary, the natural boundary condition
is selected. Under saddle point approximation, the tunneling rate of the
stochastically generated bubbles is

$\Gamma(T)=A(T)e^{-\frac{S_{b}}{T}},$ (12)

where $S_{b}$ is the Euclidean action evaluated on the bounce solution and the
factor is $A=T^{4}(\frac{S_{b}}{2\pi T})^{3/2}$[95, 96]. According to the
holographic principle, the partition function has the equivalence
$\mathcal{Z}_{\rm QFT}\simeq\mathcal{Z}_{\rm Gra}$, so the action $S_{b}$ can
be calculated by the gravitational part $S_{b}\simeq S_{5}$. The Euclidean
action of the bounce solution has the following form

$\displaystyle S_{b}$ $\displaystyle\simeq S_{5}=$ $\displaystyle
4\pi\int_{0}^{R}dr\int_{0}^{z_{h}}dz\sqrt{-g}e^{-\Phi(z)}(-\frac{\upsilon_{3}}{2}\chi^{3}-\upsilon_{4}\chi^{4}-2\upsilon_{6}\chi^{6}).$
(13)

By solving the equation of motion Eq. (7) with the boundary conditions Eqs.
(8-11), we can obtain the bounce solution in the holographic model. For
simplicity, we focus on the case of the QCD phase transition with
$\upsilon_{3}\neq 0$ (Model I). For other parameter values or holographic EW
models, the qualitative conclusions do not change. The panel (a) of Fig. (2)
shows the scalar field $\chi$ as a function of the fifth dimensional
coordinate $z$ and the radial coordinate $r$ at a temperature of 172 MeV. It
can be seen that the scalar field $\chi$ has a nontrivial structure as a
function of $z$ when the radial $r$ is small, while the profile of $\chi$
varies continuously to the trivial solution as $r$ tends to the edge $R$.

The panel (b) of Fig. 2 represents the condensation as a function of radial
$r$ at different temperatures. It can be seen that the size of the critical
bubble diminishes with decreasing temperature. As the temperature decreases,
the free energy barrier between the true and false vacuums is reduced, and
small-size bubbles are more likely to form. It should be noted that the
condensation values at the center of the bubble do not reach the equilibrium
values when $T\lesssim 170$ MeV. This can be interpreted as the fact that at
that temperature, the bubbles are composed mainly of bubble walls, as can be
seen from panel (b) of Fig. 2. The true vacuum is revealed inside the bubble
when $T\gtrsim 170$ MeV.

( a )  ( b )

Figure 2: The panel (a) represents the bounce solution of the scalar field
$\chi_{b}$ as a function of the fifth dimensional coordinate $z$ and the
radial coordinate $r$. The panel (b) indicates the condensate $\sigma$ as a
function of the radial coordinate $r$ at different temperatures.

###  III.2 Bubble velocity

When a large enough bubble is created in the plasma, it expands, collides with
each other and eventually transforms the false vacuum into a true vacuum.
During this process, some physical phenomena such as baryogenesis, GW
generation, and primordial black hole formation will happen. It can be
expected that the velocity of the bubble wall has an impact on the final
signal. For the GW power spectrum, numerical simulations indicate that faster
bubble wall velocity enhances the signal intensity [102].

The velocity of the bubble wall in the plasma is governed by hydrodynamics and
particle interactions. The way bubbles expand is divided into deflagration,
detonation, hybrid and runaway cases [103], which influence the GW spectra
through bubble collisions and interactions with the hydrodynamics. In order to
understand the details of FOPT more accurately, many methods have been used to
calculate the bubble wall velocity. As in Refs. [104, 105, 106, 107], the
local equilibrium hydrodynamic equations are applied to obtain the velocity.
More generally, considering the effect of out of equilibrium, people need to
solve the distribution function by the Boltzmann equation, which allows to
obtain the velocity containing the backreaction [108, 109, 110, 111, 112, 113,
114, 115]. In addition, the holographic method is also applied to the
calculation of bubble wall velocity. In Refs. [63, 64, 65, 70, 69, 72] ,
applying the gauge/gravity duality, the bubble velocity is calculated for
strongly coupled hydrodynamics.

In this letter, the probe approximation is considered, i.e., we only account
for the obstruction from the plasma fluid without the backaction. To
investigate the real-time evolution, we transform the framework to the ingoing
Eddington-Finkelstein coordinate, then the metric becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{z^{2}}[-f(z)dt^{2}-2dtdz+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}].$ (14)

Under the coordinate transformation, the scalar field $\chi$ is invariant, and
its equation of motion becomes as follows

$\displaystyle\partial_{r}^{2}\chi(t,z,r)+\frac{2\partial_{r}\chi(t,z,r)}{r}+f(z)\partial_{z}^{2}\chi(t,z,r)+\left(f^{\prime}(z)-f(z)\Phi^{\prime}(z)-\frac{3f(z)}{z}\right)\partial_{z}\chi(t,z,r)$
$\displaystyle-\frac{\partial_{\chi}V(\chi)}{z^{2}}+\left(\frac{3}{z}+\Phi^{\prime}(z)\right)\partial_{t}\chi(t,z,r)-2\partial_{t}\partial_{z}\chi(t,z,r)$
$\displaystyle=$ $\displaystyle 0.$ (15)

In order to solve Eq. (15), the equation requires suitable boundary conditions
and initial condition. We choose the following boundary conditions at the
center and edge

$\displaystyle\partial_{t}\partial_{r}\chi|_{r=0}=$
$\displaystyle\partial_{t}\partial_{r}\chi|_{r=R}=$ $\displaystyle 0.$ (16)

This means that the smoothness of the bubbles is ensured at the center and the
edge. Of course, it is also possible to choose $\partial_{t}\chi|_{r=R}=0$ at
the edge, i. e., a false vacuum at the far distance all the time. We find that
there is almost no difference between the two choices when the radial
dimensions are large enough. It can be foreseen that the choice of this paper
is more reasonable when the space contains more than one bubble. At the
conformal boundary, we have

$\displaystyle\partial_{t}\partial_{z}\chi|_{z=0}$ $\displaystyle=$
$\displaystyle 0.$ (17)

This corresponds to the fact that the current mass is fixed and its does not
vary with time.

For the initial condition, we consider adding a perturbation $\delta\chi$ to
the bounce solution of the scalar field $\chi_{b}$. Because the meaning of the
bounce solution is the critical bubble solution, i.e., any bubble larger than
it will expand, and smaller than it will shrink. Since the bounce solution
$\chi_{b}$ does not vary with time, the selection of the perturbation is
important, which will determine whether the bubble expands or shrinks.
Therefore, the perturbation $\delta\chi$ is not completely random, but is
controlled to be positive or negative. In this paper, we choose the
perturbation as follows

$\displaystyle\delta\chi(z,r)=\mathcal{A}\,\chi_{t}(z)\big{\\{}\exp[(r-r_{0})^{2}]+1\big{\\}}^{-1},$
(18)

where $\mathcal{A}$ is a constant that is small enough, $\chi_{t}$ is the true
vacuum solution at that temperature, and $r_{0}$ is an arbitrary number.

The panels (a) and (b) of Fig. 3 show the evolution of the bubble with time at
$T=160$ MeV when positive and negative perturbations are added, respectively.
As seen in panel (a), when positive perturbation is added, the bubble goes
through two steps. First the condensation value at the center of the bubble
keeps growing until the true vacuum. After that, the structure of the bubble
wall stabilizes and gradually expands outward. At times less than about
$t\simeq 15{\rm GeV^{-1}}$, the bubble is still in the first step, and when
the time is greater than 15 ${\rm GeV^{-1}}$ the bubble enters the second
step. If the temperature is $170{\rm MeV}\lesssim T\lesssim 174{\rm MeV}$,
then the first step of bubble expansion does not occur. Of course, the exact
timing of the distinction between the two steps is related to the amplitude of
the perturbation, which is roughly equal in magnitude when the perturbation is
small enough. The panel (b) displays the bubble shrinking with negative
perturbation. As can be seen in the panel, the condensation value at the
center of the bubble gradually decreases until the system completely returns
to the false vacuum at $t\simeq 70{\rm GeV^{-1}}$. The time of the bubble
shrinking process is roughly comparable to the time of the first step of
expansion, for bubbles that are neither too large nor too small.

( a )  ( b )

Figure 3:  The panels (a) and (b) show the profiles of the bubbles at
different times with $T=160$ MeV when positive and negative perturbations are
added, respectively.

To obtain the bubble wall velocity, we used the function
$\sigma(r)=\frac{\sigma_{0}(t)}{2}[-\tanh(\frac{r(t)-r_{w}(t)}{l_{w}(t)})+1]$
to fit the profile of the condensation. Here, $\sigma_{0}$ is the condensation
value at the center, $r_{w}$ denotes the radius of the bubble, and $l_{w}$
represents the thickness of the bubble wall. Fig. 4 shows the numerical and
fitting results. It can be seen that the function fits the numerical results
quite well except at the edges of the bubble walls. Therefore, we can define
the velocity of the bubble wall by $v(t)=dr_{w}(t)/dt$. The panel (a) of Fig.
5 exhibits the bubble wall velocity as a function of time at a temperature of
160 MeV. It can be seen that the velocity gradually increases with time, while
the acceleration gradually decreases. The velocity reaches its final velocity
about $0.31\,c$ when the time reaches about $t\simeq 200{\rm GeV^{-1}}$. As
for the relaxation time for the bubble to reach its final velocity, it depends
on the strength of the perturbation. Numerical calculations show that the
relaxation times with different perturbations are roughly equivalent. Note
that the velocity is not well defined at time $t\lesssim 20{\rm GeV^{-1}}$.
This is because the bubble is still in the first step during this time and the
wall configuration is changing and therefore not shown in the panel.

Figure 4:  The condensate as a function of radial r, where the solid line is
the numerical solution and the dashed line represents the fitting results of
the hyperbolic tangent function.

The final velocity of the bubble wall $v_{w}$ is defined as
$v_{w}=\lim_{t\to\infty}v(t)$. The panel (b) of Fig. 5 shows the final
velocity of the bubble wall as a function of the pressure difference $\Delta
P$ between the inside and outside or the temperature $T$. According to the
holographic principle, the pressure can be obtained from the free energy
$F=-P$, as in Refs. [80, 81]. At low temperatures, the final velocity is
approximately linear with respect to the pressure difference, while at
temperatures close to the critical temperature, their relation exhibits
nonlinear behavior. In Ref. [72], the relation between velocity and pressure
difference has a nonlinear behavior. Although, the method used in Ref. [72] is
different from the one used in this paper, the behavior exhibited is very
similar. It can be seen from the panel that even if the temperature drops to
154 MeV, the final velocity still does not exceed sound speed
$c_{s}=1/\sqrt{3}$ of the plasma, which corresponds to the deflagration case.
This is in agreement with the results of Refs. [63, 64, 69, 72]. It is
reasonable to speculate that in the bottom-up holographic model, the final
velocity is always limited to the deflagrations region because the system
contains a holographic dissipation mechanism [116]. While in Ref. [65], the Dp
brane system can achieve bubble wall velocities close to the speed of light.
Whether detonation and hybrid cases can be implemented in the bottom-up
holographic model is still an open question.

( a )  ( b )

Figure 5:  The panel (a) shows the bubble wall velocity as a function of time.
The panel (b) shows the final velocity of the bubble wall as a function of the
pressure difference $\Delta P$ between the inside and outside or the
temperature $T$.

##  IV bubbles and thin wall approximation

In the previous section, we have obtained the bounce solution. By analyzing
the properties of the holographic bounce solution, the characteristics of the
strong FOPT can be understood qualitatively, which helps to understand its GW
power spectrum. We know that there are three important characteristic
temperatures during the dynamical FOPT that exhibit the details of the phase
transition and are finally reflected in the GW. The first one is the critical
temperature $T_{c}$ of the phase transition, i.e., the temperature at which
the free energy has a degenerate minimum. The rest are the nucleation
temperature $T_{n}$ and percolation temperature $T_{p}$, which represent the
temperature of beginning nucleation and ending of phase transition,
respectively. The temperature $T_{n}$ and $T_{p}$ are mainly determined by the
profile of the Euclidean action $\frac{S_{b}}{T}$ of bounce solution with
temperature. In addition, the bubble wall thickness and bubble radius can be
obtained from the bounce solution. In this section, we analyze the holographic
behavior of the bounce solution with temperature. It should be noted that we
only show the QCDPT with $\upsilon_{3}\neq 0$ (Model I). For $\upsilon_{6}\neq
0$ (Model II) or EWPT, the numerical results change slightly and the
conclusions in the following still apply.

From Eq. (13) we can calculate the Euclidean action $\frac{S_{b}}{T}$. Fig. 6
shows the on-shell action $\frac{S_{b}}{T}$ as a function of $\frac{T}{T_{c}}$
calculated by the bounce solution. It can be seen from the figure that the
value of the action decreases rapidly as the temperature decreases, and at
about $\frac{T}{T_{c}}\sim 0.99$, i.e., at a temperature of about 172 MeV,
$\frac{S_{b}}{T}$ drops to 30. The nucleation temperature $T_{n}$ can be given
by relation $\frac{S_{b}(T_{n})}{T_{n}}\sim 180$ for QCDPT
($\frac{S_{b}(T_{n})}{T_{n}}\sim 140$ for EWPT) (a more rigorous definition is
shown in the next section). Therefore, the nucleation temperature is very
close to the critical temperature $T_{c}\simeq T_{n}$. Furthermore, at low
temperatures, the action does not have a local minimum. Also, since the
nucleation probability $\Gamma$ is proportional to the action, it can be seen
that bubbles are created rapidly and abundantly as the temperature decreases.
Consequently, it is reasonable to suppose that the percolation temperature is
close to the nucleation temperature $T_{n}\simeq T_{p}$. As shown above, the
three characteristic temperatures have relation $T_{c}\simeq T_{n}\simeq
T_{p}$, which corresponds to the supercooling case [117].

Figure 6:  The Euclidean action $\frac{S_{b}}{T}$ as a function of
temperature.

As in the previous section, we use hyperbolic tangent function interpolation
condensation as a function of the radial $r$, shown following

$\displaystyle\sigma(T,r)$ $\displaystyle=$
$\displaystyle\frac{\sigma_{0}(T)}{2}[-\tanh(\frac{r-R_{w}(T)}{L_{w}(T)})+1],$
(19)

where $\sigma_{0}$ is the condensation value at the center, $R_{w}$ denotes
the radius of the bubble, and $L_{w}$ represents the thickness of the bubble
wall. Unlike the previous section, these quantities are independent of time
since the critical bubbles do not evolve with time. Fig. 7 displays the
condensation $\sigma_{0}$, radius $R_{w}$, and thickness $L_{w}$ as functions
of temperature. As seen in panel (a), the condensation at the center increases
with increasing temperature at $T\lesssim 0.975T_{c}\sim 170$MeV and has the
opposite behavior at $T\gtrsim 0.975T_{c}$. Two factors with opposite effects
influence this behavior. As can be seen in Fig. 1, the condensation value
decreases continuously with increasing temperature; in contrast, as in Fig. 2,
the critical bubble is larger at higher temperatures, which favors the
generation of a true vacuum at the center. For the bubble radius $R_{w}$,
panel (b) shows that the critical radius varies slightly at temperature
$T\lesssim 0.98T_{c}\sim 171$MeV. However, at temperature $T\gtrsim
0.98T_{c}$, the critical radius of the bubble increases rapidly, which is not
favorable for bubble generation. In contrast to the behavior of the bubble
radius, the bubble wall thickness essentially does not vary with temperature.
And the thickness changes slightly at low temperature $T\lesssim 0.94T_{c}\sim
164$MeV. This is because the hyperbolic function fitting is not a good choice
at this time.

( a )
( b )  ( c )

Figure 7: The panels (a), (b) and (c) show the condensation value at the
center $\sigma_{0}$, the bubble radius $R_{w}$ and the bubble wall thickness
$L_{w}$ as functions of the temperature, respectively.

###  IV.1 Thin-wall approximation

From the previous discussion, it is clear that in this holographic model, the
FOPT is supercooled. As in Ref. [117], the thin-wall approximation can be
applied in this case. Under this approximation, many physical quantities, such
as latent heat, duration time of phase transition and surface tension of
bubble, have simpler forms and can be obtained more easily. In this section,
we estimate the surface tension and the parameters $\alpha$ and $\beta/H$
using the Euclidean action obtained previously. Among them, the parameters
$\alpha$ and $\beta/H$ are very important for the GW power spectrum.

From Ref. [117], the parameter $\alpha$ is

$\displaystyle\alpha$ $\displaystyle\simeq$
$\displaystyle\frac{15}{2\pi^{2}}\frac{L_{c}}{g_{*}(T_{c})T_{c}^{4}},$ (20)

where $g_{*}$ is the relativistic degree of freedom of the system and $L_{c}$
is the latent heat of phase transition. For QCDPT (EWPT), the degrees of
freedom can be approximated as $g_{*}\simeq 10$ ($g_{*}\simeq 100$). Under the
thin-wall approximation, the latent heat has the following form

$\displaystyle L_{c}$ $\displaystyle=$ $\displaystyle-T\frac{\partial\Delta
V_{\text{eff}}(\langle\phi\rangle_{T},T)}{\partial
T}\bigg{|}_{T=T_{c}}=T\frac{\partial\Delta
F(\langle\phi\rangle_{T},T)}{\partial T}\bigg{|}_{T=T_{c}},$ (21)

with difference of the free energy $\Delta F$. The calculation of the free
energy $F$ can be found in Refs. [80, 81]. Within the approximation, the
three-dimensional Euclidean action $S_{b}$ can be divided into two parts, as
follows

$\displaystyle S_{b}(T)$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{3}R_{w}(T)\varepsilon(T)+4\pi
R_{w}(T)^{2}\sigma_{w}(T),$ (22)

with the $\varepsilon(T)=\Delta V_{\text{eff}}(\langle\phi\rangle_{T},T)$ and
surface tension $\sigma_{w}$ of the bubble wall. From the above equation, it
can be seen that the energy of the critical bubble consists of the vacuum
energy inside the bubble and the surface energy of the bubble wall. With the
bubble radius previously calculated, the surface tension can be obtained as
shown in Fig. 8. For the low-temperature region in the figure, the result is
not reliable because the thin-wall approximation no longer applies.

Figure 8:  The surface tension $\sigma_{w}$ of the bubble wall as a function
of temperature.

Also from Ref. [117], the inverse of duration time $\beta/H$ of the phase
transition is given as

$\displaystyle\frac{\beta}{H}$ $\displaystyle=$
$\displaystyle\left(\frac{3}{4\pi}\frac{T_{c}L_{c}^{2}}{\sigma_{w}(T_{c})^{3}}\right)^{1/2}\left(\frac{S_{b}(T_{p})}{T_{p}}\right)^{3/2},$
(23)

with the percolation temperature $T_{p}$. With the above definitions and the
relation $T_{c}\simeq T_{n}\simeq T_{p}$, the final results of parameters
$\alpha$ and $\beta/H$ are shown in Tab. 1. Here, the velocity $v_{w}$ is
calculated at the percolation temperature $T_{p}$. From the table, we can find
that for QCDPT, the parameter $\alpha$ is greater than 1, about 4-5, which is
strongly FOPT case, and the parameter $\beta/H$ is large, which means that the
phase transition ends rapidly. For EWPT, the strength parameter $\alpha$ is
less than 1, while the inverse of the duration time remains large, which means
that the phase transition is sufficiently fast, that is, the so-called weakly
supercooled FOPT.

Thin-wall approx | QCDPT | EWPT
---|---|---
$\alpha$ | 4-6 | 0.4-0.6
$\beta/H$ | 30000-60000 | 6000-20000
$v_{w}$ | 0.04 | 0.1
Table 1:  The parameters $\alpha$, $\beta/H$ and the bubble wall velocity
$v_{\omega}$ estimated by the thin-wall approximation during QCDPT and EWPT.

##  V stochastic gravitational waves

When the system reaches the nucleation temperature, bubbles will be created,
and they will collide and merge together in the plasma. Then the false vacuum
will be transformed into the true vacuum. In this process, part of the
system’s energy is eventually converted into GW radiation. Within the linear
approximation, the total GW power spectra can be written as

$\displaystyle h^{2}\Omega_{{\rm GW}}$ $\displaystyle\simeq$ $\displaystyle
h^{2}\Omega_{{\rm coll}}+h^{2}\Omega_{{\rm sw}}+h^{2}\Omega_{{\rm turb}},$
(24)

where $\Omega_{{\rm coll}}$ comes from bubble collisions [118, 119, 120, 121,
122, 123, 124, 103, 125, 126], $\Omega_{{\rm sw}}$ from acoustic waves in the
plasma after the collision [127, 128, 129, 130, 131], and $\Omega_{{\rm
turb}}$ from magnetohydrodynamic turbulence in the plasma [132, 133, 134, 135,
136, 137, 138, 139].

In the previous section we have found that for the holographic model, the
final velocity of the bubble wall is less than the speed of sound, i.e., the
non-runaway case. According to Refs. [121, 103, 102, 140, 141, 142], the GWs
generated by collisions in the non-runaway case can be neglected with respect
to acoustic waves and magnetohydrodynamic turbulence. Therefore, the total
power spectrum is approximated to

$\displaystyle h^{2}\Omega_{{\rm GW}}$ $\displaystyle\simeq$ $\displaystyle
h^{2}\Omega_{{\rm sw}}+h^{2}\Omega_{{\rm turb}}.$ (25)

From numerical simulations [130], the power spectrum from sound waves has the
form of

$\displaystyle h^{2}\Omega_{{\rm sw}}(f)$ $\displaystyle=$ $\displaystyle
2.65\times
10^{-6}\left(\frac{H}{\beta}\right)\left(\frac{\kappa_{v}\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}v_{w}S_{{\rm
sw}}(f),$ (26) $\displaystyle S_{{\rm sw}}(f)$ $\displaystyle=$
$\displaystyle(f/f_{{\rm sw}})^{3}\left(\frac{7}{4+3(f/f_{{\rm
sw}})^{2}}\right)^{7/2},$ (27) $\displaystyle f_{{\rm sw}}$ $\displaystyle=$
$\displaystyle 1.9\times 10^{-2}{\rm
mHz}\frac{1}{v_{w}}\left(\frac{\beta}{H}\right)\left(\frac{T_{p}}{100{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}},$ (28)

where the factor $\kappa_{v}$ represents the ratio of vacuum energy
transformed into bulk motion. The specific form of the factor $\kappa_{v}$
depends on the bubble wall velocity and has the following form in the limits
of large and small velocities

$\displaystyle\kappa_{v}$ $\displaystyle\simeq$
$\displaystyle\begin{cases}\alpha\left(0.73+0.083\sqrt{\alpha}+\alpha\right)^{-1},&v_{w}\sim
1,\\\
v_{w}^{6/5}6.9\alpha\left(1.36-0.037\sqrt{\alpha}+\alpha\right)^{-1},&v_{w}\lesssim
0.1.\end{cases}$ (29)

From the Tab. 1, it is obtained that the velocity is about $v_{w}\sim 0.1$,
since the phase transition of the holographic model is supercooled. Therefore,
in this paper, the expression for the factor $\kappa_{v}$ is

$\displaystyle\kappa_{v}$ $\displaystyle=$ $\displaystyle
v_{w}^{6/5}6.9\alpha\left(1.36-0.037\sqrt{\alpha}+\alpha\right)^{-1}.$ (30)

For the power spectrum from the Kolmogorov-type turbulence, numerical
simulations show that it can be given as [132, 138, 143]

$\displaystyle h^{2}\Omega_{{\rm turb}}(f)$ $\displaystyle=$ $\displaystyle
3.35\times 10^{-4}\left(\frac{H}{\beta}\right)\left(\frac{\kappa_{{\rm
turb}}\alpha}{1+\alpha}\right)^{\frac{3}{2}}\left(\frac{100}{g_{*}}\right)^{1/3}v_{w}S_{{\rm
turb}}(f),$ (31) $\displaystyle S_{{\rm turb}}(f)$ $\displaystyle=$
$\displaystyle\frac{(f/f_{{\rm turb}})^{3}}{\left[1+(f/f_{{\rm
turb}})\right]^{\frac{11}{3}}\left(1+8\pi f/h_{*}\right)},$ (32)
$\displaystyle f_{{\rm turb}}$ $\displaystyle=$ $\displaystyle 2.7\times
10^{-2}{\rm
mHz}\frac{1}{v_{w}}\left(\frac{\beta}{H}\right)\left(\frac{T_{p}}{100{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}},$ (33)

where the Hubble rate $h_{*}$ is

$\displaystyle h_{*}$ $\displaystyle=$ $\displaystyle 16.5\times 10^{-3}{\rm
mHz}\left(\frac{T_{p}}{100{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}.$ (34)

In this paper, referring to the numerical results of the Refs. [130, 102], the
factor $\kappa_{{\rm turb}}$ is chosen to be

$\displaystyle\kappa_{{\rm turb}}$ $\displaystyle=$ $\displaystyle
0.05\kappa_{v}.$ (35)

It should be noted that, due to the complexity of turbulence, more research is
still needed on the exact form of the factor $\kappa_{{\rm turb}}$ and its
relation to $\kappa_{v}$. In this holographic model, the choice can be
reasonable considering that the phase transition may end rapidly and therefore
the effect of turbulence is expected to be suppressed.

In the GW power spectrum, the parameters $\alpha$ and $\beta/H$ are important,
which represent the vacuum energy release and duration time of the phase
transition, respectively. The parameter $\alpha$ is defined as

$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\rho_{\text{rad}}}\left(\Delta
V_{\text{eff}}-\frac{T}{4}\frac{\partial\Delta V_{\text{eff}}}{\partial
T}\right)\bigg{|}_{T=T_{p}}=-\frac{1}{\rho_{\text{rad}}}\left(\Delta
F-\frac{T}{4}\frac{\partial\Delta F}{\partial T}\right)\bigg{|}_{T=T_{p}},$
(36)

with radiation energy density

$\displaystyle\rho_{\text{rad}}$ $\displaystyle=$ $\displaystyle
g_{*}\frac{\pi^{2}}{30}T^{4}.$ (37)

The inverse of the duration time is defined as

$\displaystyle\frac{\beta}{H}$ $\displaystyle\equiv$ $\displaystyle
T\frac{d}{dT}\left(\frac{S_{b}}{T}\right)\bigg{|}_{T=T_{p}}.$ (38)

The definitions of the other two characteristic temperatures of the phase
transition, i. e., nucleation temperature $T_{n}$ and percolation temperature
$T_{p}$, are shown below. The nucleation temperature is defined as one bubble
per unit Hubble volume and is written as

$\displaystyle N(T_{n})$ $\displaystyle=$
$\displaystyle\int_{T_{n}}^{T_{c}}\frac{\mathrm{d}T}{T}\frac{\Gamma(T)}{H(T)^{4}}=1,$
(39)

where the nucleation rate $\Gamma(T)$ is given in Eq. (12) and the Hubble
parameter $H(T)$ is

$\displaystyle H(T)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\rho_{\text{rad}}+\rho_{\text{vac}}}{3M_{\text{pl}}^{2}}},$
(40)

with reduced Planck mass $M_{\text{pl}}=2.435\cdot 10^{18}$ GeV. Referring to
Refs. [144, 145, 146], the probability of a false vacuum is defined as

$\displaystyle P(T)$ $\displaystyle=$ $\displaystyle e^{-I(T)},$ (41)

with

$\displaystyle I(T)$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}\int_{T}^{T_{c}}\mathrm{d}T^{\prime}\frac{\Gamma(T^{\prime})}{H(T^{\prime})T^{\prime
4}}\left(\int_{T}^{T^{\prime}}\mathrm{d}T^{\prime\prime}\frac{v_{w}(T^{\prime\prime})}{H(T^{\prime\prime})}\right)^{3}.$
(42)

The percolation temperature is defined as $I(T_{p})\simeq 0.34$, which is the
temperature when the probability of false vacuum is about $P(T_{p})\simeq
0.7$.

The Tab. 2 shows the quantities related to the phase transition in different
holographic models. Here, the velocity $v_{w}$ is selected at the percolation
temperature $T_{p}$. Compare to Table I, we can see that the rigorous results
are very close to that of thin-wall approximation. This further verifies the
reliability of the approximation in the holographic model. It should be noted
that the strength parameter $\alpha$ of QCDPT is greater than 1, which would
be a strongly supercooled FOPT. For EWPT, FOPT is weakly supercooled. It is
worth noting that although QCDPT is strongly supercooled, its properties are
very similar to those of the EWPT and there are no minimum values of the
weights $S_{b}/T$ as mentioned in the Ref. [117]. Furthermore, for parameters
$\alpha$ and $\beta/H$, we find an inverse relation between them, which is
similar to that of Ref. [117]. For Models I and II, we found that Model II
$(\upsilon_{6}\neq 0)$ has lower percolation temperature and larger bubble
wall velocity, which favors a bigger GW signal.

From Tab. 2, we also find that the results of the holographic model differ
significantly from those of the traditional calculations in quantum field
theory. In quantum field theory, the parameter $\alpha$ is generally not
larger than 0.01 and the latent heat released in the phase transition is not
large. However, the strength of the phase transition is greater than 0.1 for
both color brane [54] and flavor brane in the holographic model. For the
parameter $\beta/H$, field theory calculations show that it is generally not
larger than 1000. But for the holographic model, the parameter $\beta/H$ is
larger than 5000 during QCDPT or EWPT. This means that the duration time of
the strongly coupled phase transition is shorter compared to the weakly
coupled case. As for the bubble wall velocity, one would expect it to be close
to the speed of light $c$. Unfortunately, the holographic results suggest that
the bubble wall velocity is smaller than the sound speed of the plasma.

| QCDPT | EWPT
---|---|---
Model | I $(\upsilon_{3}\neq 0)$ | II $(\upsilon_{6}\neq 0)$ | I $(\upsilon_{3}\neq 0)$ | II $(\upsilon_{6}\neq 0)$
$g_{*}$ | 10 | 100
$\alpha$ | 4.881 | 6.142 | 0.238 | 0.763
$\beta/H$ | 41151 | 23276 | 17198 | 7238
$v_{w}$ | 0.027 | 0.041 | 0.063 | 0.125
$T_{c}$[GeV] | 0.1741 | 122.1
$T_{n}$[GeV] | 0.1733 | 0.1712 | 120.7 | 118.1
$T_{p}$[GeV] | 0.1732 | 0.1703 | 120.4 | 117.6
Table 2:  The critical temperature $T_{c}$, nucleation temperature $T_{n}$,
percolation temperature $T_{p}$ of phase transition, parameters $\alpha$,
$\beta/H$, and bubble wall velocity $v_{w}$ in different holographic models.

The GW power spectra of QCDPT and EWPT for different holographic models are
exhibited in Fig. 9. Due to the large value of $\beta/H$, the peak frequency
of GW is larger compared to the results of quantum field theory, causing a
right shift in the sensitive frequency interval of GW. For the holographic
model of QCDPT, the peak frequency is about 0.01 Hz. In addition, since the
bubble wall velocity $v_{w}$ is much less than the speed of light $c$, the GW
signal $h^{2}\Omega$ can only reach about $10^{-13}-10^{-14}$, which can be
detected by $\mu$Ares, BBO and Ultimate-DECIGO for Model II, and by BBO and
Ultimate-DECIGO for Model I. For EWPT, the peak frequency is around $1-10$ Hz,
when the GW spectrum $h^{2}\Omega$ reaches about $10^{-12}-10^{-16}$. For
Model I, the GW can be detected by BBO and Ultimate-DECIGO, but for Model II,
it is not detectable by future experiments.

( a )  ( b )

Figure 9:  The GW power spectra in different holographic models, where the
blue (a) and green (b) lines represent QCDPT and EWPT, respectively.

##  VI conclusion and discussion

In this paper, holographic bounce solutions, bubble wall velocities and GW
power spectra are studied and discussed in holographic QCD and EW models. For
holographic models with first-order phase transitions, holographic bounce
solutions can be obtained by setting appropriate boundary conditions. By
adding positive and negative perturbations to the bounce solution, the bubble
expands or shrinks and the final velocity of the bubble wall can be given. It
turns out that the final velocity is still smaller than the speed of sound in
the plasma, i. e. the deflagration case, even if the phase transition
temperature is very low. Moreover, we find that the critical temperature
$T_{c}$, nucleation temperature $T_{n}$ and percolation temperature $T_{p}$ of
the phase transition are close to each other, which makes the holographic
bubble fit well with the results of thin-wall approximation. Furthermore, the
strength parameter $\alpha$ calculated by the holographic model is about
$\alpha\sim 5$ for QCDPT ($\alpha\sim 0.5$ for EWPT) and the inverse of the
duration time $\beta/H$ is about $\beta/H\sim 10000$, and they are quite
different from the weakly coupled field theory results. For QCDPT, the GW
power spectrum can reach $10^{-13}-10^{-14}$ around the peak frequency 0.01
Hz, which can be detected by $\mu$Ares, BBO and Ultimate-DECIGO for
$\upsilon_{6}\neq 0$ (Model II), and by BBO and Ultimate-DECIGO for
$\upsilon_{3}\neq 0$ (Model I). For EWPT, the GW power spectrum can reach
$10^{-12}-10^{-16}$ around the peak frequency $1-10$ Hz, which can be detected
by BBO and Ultimate-DECIGO for $\upsilon_{6}\neq 0$ (Model II), but not by
future experiments for $\upsilon_{3}\neq 0$ (Model I).

Referring to the quantum bounce solutions proposed by Coleman and Callan[92,
93] with Neumann boundary condition at the center and Dirichlet boundary
condition at the edge, the holographic bounce solution can be yielded. We
found that the radius of the critical bubble and the value of condensation at
the center reduced with decreasing temperature. If a small positive or
negative perturbation is added to the holographic bounce solution, the bubble
starts to expand or shrink. Through a long enough evolution, the velocity of
the bubble wall reaches a constant value. It turns out that in this
holographic model, the bubble velocity is deflagration, i.e., it is less than
the speed of sound of the system. Although our calculations are under the
probe approximation, that is, without a push to the fluid, the conclusions are
similar to the Refs. [63, 65, 64, 70, 69, 72]. How to get detonation and
hybrid cases in the bottom-up model remains an open question.

The radius of the critical bubble and the thickness of the bubble wall can be
obtained by the Euclidean action. We found that the bubble radius drops
rapidly with decreasing temperature while the wall thickness varies little
with temperature. In addition, we find that the critical temperature,
nucleation temperature and percolation temperature are close to each other
during the holographic phase transition. Therefore the thin-wall approximation
is considered to estimate the strength and duration time of the phase
transition. It turns out that the more rigorous calculations agree well with
the approximation results, verifying the reliability of the approximation.

With the holographic bounce solution and the bubble final velocity, we
calculated the GW power spectra of QCDPT and EWPT. We find that the strength
parameter $\alpha$ and the inverse of the duration time $\beta/H$ calculated
by the holographic model differ significantly from the weakly coupled field
theory results, which impact the GW signal. The parameter $\alpha$ can reach
about 5 for QCDPT, which is strong phase transition, and about 0.5 for EWPT.
The parameter $\beta/H$ is about 30,000 for QCDPT and about 10,000 for EWPT,
so the holographic phase transition is sufficiently fast. Due to the large
parameter $\beta/H$, the peak frequency shifts rightward compared to the field
theory results. In addition, the GW spectrum is suppressed due to the small
bubble expansion speed. Eventually, for QCDPT, it can be detected by
$\mu$Ares, BBO and Ultimate-DECIGO for Model II and by BBO and Ultimate-DECIGO
for Model I. For EWPT, Model II is detectable, while Model I cannot be
detected by future experiments. Whether it is QCDPT or EWPT, the FOPT caused
by the sextic term has a larger $\alpha$ and smaller $\beta/H$ and higher GW
energy than that caused by the cubic term.

For primordial black holes, the formula of mechanism [147] cannot be applied
because the speed of the bubble at the phase transition is much smaller than
the speed of light $c$. Therefore, it cannot be determined whether the black
holes have sufficient probability to be produced. However, since the small
bubble wall velocity may lead to a small energy density perturbation during
the collision, we can speculate that the primordial black hole is difficult to
produce in this holographic model. Also, as in Ref. [148], the large parameter
$\beta/H$ is not favorable for the formation of primordial black holes.

###### Acknowledgements.

We thank Anping Huang, Mingqiu Li, Jingdong Shao, Dianwei Wang, and Qi-Shu Yan
for helpful discussions. This work is supported by the China Postdoctoral
Science Foundation under Grant No. 2021M703169, the Fundamental Research Funds
for the Central Universities E2E46303X2, the National Natural Science
Foundation of China (NSFC) Grant Nos:12235016, 12221005, 11725523, 11735007,
12275108, and the Strategic Priority Research Program of Chinese Academy of
Sciences under Grant Nos XDB34030000 and XDPB15, the start-up funding from
University of Chinese Academy of Sciences(UCAS), the Fundamental Research
Funds for the Central Universities, and the Guangdong Pearl River Talents Plan
under Grant No. 2017GC010480.

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