TU-1197

KEK-QUP-2023-0013

#  Quantum decay of scalar and vector boson stars and oscillons into gravitons

Kazunori Nakayama1,2, Fuminobu Takahashi1, Masaki Yamada1,3
1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan
2International Center for Quantum-field Measurement Systems for Studies of the
Universe and Particles (QUP, WPI), KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan
3FRIS, Tohoku University, Sendai, Miyagi 980-8578, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We point out that a soliton such as an oscillon or boson star inevitably
decays into gravitons through gravitational interactions. These decay
processes exist even if there are no apparent self-interactions of the
constituent field, scalar or vector, since they are induced by gravitational
interactions. Hence, our results provide a strict upper limit on the lifetime
of oscillons and boson stars including the dilute axion star. We also
calculate the spectrum of the graviton background from decay of solitons.

##  1 Introduction

Bosons are ubiquitous in physics beyond the Standard Model. For example, the
Peccei-Quinn (PQ) mechanism predicts a pseudo-Nambu-Goldstone boson, the QCD
axion Peccei:1977hh ; Peccei:1977ur . There may also exist a large number of
axion-like particles in the low-energy effective field theory of the string
theory. We generally call them axions for both cases. Even if their
interaction rates with other particles are suppressed by a large decay
constant, they can be produced via coherent oscillations in the early Universe
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah  and are a good candidate for
dark matter (DM). On another hand, massive vector fields in the dark sector
can also be considered as a candidate for DM (see Ref. Antypas:2022asj  and
references therein). However, the production mechanism of vector DM is non-
trivial, and includes the misalignment mechanism Nelson:2011sf ; Arias:2012az
(see, however, Ref. Nakayama:2019rhg ; Nakayama:2020rka  for its difficulty
and Ref. Kitajima:2023fun  for a viable scenario), emission from cosmic
strings Long:2019lwl ; Kitajima:2022lre , phase transition Nakayama:2021avl ,
scalar-field dependent gauge kinetic coupling Salehian:2020asa ;
Firouzjahi:2020whk , preheating and decay of axion-like particles
Agrawal:2018vin ; Dror:2018pdh ; Co:2018lka ; Bastero-Gil:2018uel ; Co:2021rhi
; Kitajima:2023pby , and gravitational production during and after inflation
Graham:2015rva ; Ema:2019yrd ; Ahmed:2020fhc ; Nakayama:2020ikz ; Kolb:2020fwh
; Arvanitaki:2021qlj ; Sato:2022jya . The cosmological implications of those
bosons are therefore important to understand the consistent scenario of
particle physics throughout the history of the Universe.

In cosmology, light bosons are usually produced with extremely large occupancy
numbers, which imply the formation of Bose-Einstein condensate Guth:2014hsa .
In fact, gravitationally bound clumps, called boson stars or oscillatons
Kaup:1968zz ; Ruffini:1969qy ; Colpi:1986ye ; Seidel:1991zh ; Tkachev:1991ka ;
Kolb:1993zz ; Kolb:1993hw , are formed via the growth of perturbations during
the matter dominated era Schive:2014hza ; Levkov:2018kau ; Widdicombe:2018oeo
; Eggemeier:2019jsu ; Chen:2020cef . In the context of axion cosmology,
gravitationally bounded clumps of axion are sometimes called dilute axion
stars or axiton Kolb:1993hw ; Braaten:2015eeu ; Braaten:2016dlp ;
Visinelli:2017ooc ; Schiappacasse:2017ham . In this paper, we simply call them
as boson stars. The formation of such objects is supported by detailed
numerical simulations Veltmaat:2018dfz ; Eggemeier:2019jsu ; Chen:2020cef ,
where it was found that localized objects of scalar fields are formed at the
center of DM halo, for the QCD axion and ultralight (string) axion. The
configuration and properties of the boson star can be calculated by the
approach used in Refs. Chavanis:2011zi ; Chavanis:2011zm  (see also Ref.
Eby:2018ufi ). The boson star is not completely stable but has a long lifetime
in vacuum on cosmological time scales Mukaida:2016hwd ; Braaten:2016dlp ;
Visinelli:2017ooc ; Eby:2018ufi . The evaporation and growth of boson stars in
the DM halo was investigated in Ref. Chan:2022bkz . Those localized objects
may affect the image of gravitational lensing Fujikura:2021omw ; Ellis:2022grh
. Those objects are therefore relevant in cosmology, especially when they are
dominant components of DM in the Universe.

On the other hand, if the attractive self interaction of the scalar field is
stronger than the gravitational interaction, the localized object is called an
oscillon or I-ball (or dense axion star when the scalar is axion)
Bogolyubsky:1976nx ; Segur:1987mg ; Gleiser:1993pt ; Copeland:1995fq ;
Gleiser:1999tj ; Honda:2001xg ; Kasuya:2002zs ; Fodor:2006zs ; Fodor:2008du ;
Gleiser:2008ty ; Gleiser:2009ys ; Amin:2010jq ; Amin:2011hj ; Salmi:2012ta ;
Saffin:2014yka ; Mukaida:2014oza ; Kawasaki:2015vga . A perturbative expansion
in terms of the spatial gradient or binding energy is often used to determine
the configuration of oscillons Fodor:2008es , whereas some systematic
approaches to calculate the configuration in the non-relativistic effective
field theory are also proposed in Refs. Braaten:2016kzc ; Mukaida:2016hwd ;
Namjoo:2017nia ; Braaten:2018lmj ; Levkov:2022egq . Those objects are realized
by the QCD axion after the QCD phase transition in the post-inflationary PQ
symmetry breaking scenario Vaquero:2018tib  and in a large misalignment
scenario Arvanitaki:2019rax . The inflaton may also form oscillons after
inflation, depending on models (see, e.g., Ref. Copeland:2002ku ; Amin:2011hj
). However, the lifetime of oscillons tends to be much shorter than the
cosmological time scale. It was noted that the amplitude of the primordial
gravitational waves may be enhanced if the oscillons dominate the Universe
before they decay Inomata:2019ivs ; Lozanov:2022yoy . Therefore, determining
the lifetime of those localized objects has important implications for
cosmological observations.

Recently, the formation of similar extended objects from vector bosons, which
we call the vector boson star or the vector oscillon depending on the
attractive force that stabilizes the configuration, has been discussed
Adshead:2021kvl ; Jain:2021pnk ; Zhang:2021xxa ; Amin:2023imi . Detailed
numerical simulations Gorghetto:2022sue ; Amin:2022pzv ; Jain:2023ojg
supports the formation of vector boson stars. They may also have lots of
phenomenological impacts, and it is important to understand the stability of
these objects.

The classical/quantum stability of the boson star or the oscillon is
extensively studied in the literature. In Refs. Mukaida:2016hwd ; Eby:2018ufi
, a diagrammatic technique to calculate the classical decay rate of oscillons
and axion stars was developed. The resulting decay rate is consistent with
that of numerical simulations of oscillons Salmi:2012ta ; Mukaida:2016hwd .
See also Refs. Eby:2015hyx ; Eby:2017azn ; Ibe:2019vyo ; Ibe:2019lzv ;
Zhang:2020bec ; Zhang:2020ntm  for the analysis of the oscillon decay, which
agrees with Refs. Mukaida:2016hwd ; Eby:2018ufi  and a similar formula was
derived there from different perspectives. In Refs. Grandclement:2011wz ;
Fodor:2009kg ; Page:2003rd , they considered the classical decay rate of
oscillatons. In particular, the dilute axion star is expected to have a very
long lifetime. In Ref. Zhang:2020ntm ; Eby:2020ply , they discussed the effect
of gravity on the classical decay rate of axion stars.

On the other hand, the quantum decay of oscillons and axion stars was
considered in Refs. Hertzberg:2010yz ; Kawasaki:2013awa ; Braaten:2016dlp ;
Hertzberg:2018zte  with and without parametric resonance. Those works
particularly considered the decay into generic scalar fields or photons. The
quantum decay process corresponds to the imaginary part of the loop diagrams
in the above diagrammatic technique.

In this paper, we consider the quantum decay of soitons, which collectively
represent scalar/vector boson stars and oscillons, into gravitons via
gravitational interactions. Although a soliton is spherically symmetric, it
can produce gravitons (or gravitational waves) quantum mechanically. This is
similar to the graviton emission from the spherically symmetric black holes
via Hawking radiation, which is a semi-classical process and is classically
forbidden. The quantum decay into graviton is inevitable even if the boson has
only the quadratic potential. A free real scalar field is the simplest model
of quantum field theory but is non-trivial due to the gravitational
interactions, where it is localized by its self gravitational potential and
decays into gravitons.

Gravitational particle production due to an oscillating scalar field in a
cosmological setup has been discussed in Ref. Ema:2015dka , which is also
interpreted as the gravitational annihilation of the scalar field into any
field. Later this and related subjects have been studied in many papers in the
context of dark matter production and reheating Ema:2015dka ; Ema:2016hlw ;
Tang:2017hvq ; Ema:2018ucl ; Chung:2018ayg ; Ema:2019yrd ; Mambrini:2021zpp ;
Basso:2021whd ; Clery:2021bwz ; Haque:2021mab , graviton production
Ema:2015dka ; Schiappacasse:2016nei ; Ema:2016hlw ; Ema:2020ggo ,
gravitational annihilation of Standard Model particles in thermal bath
Garny:2015sjg ; Tang:2016vch ; Tang:2017hvq ; Garny:2017kha ; Gross:2020zam
.111See also Refs. Ford:1986sy ; Chung:2001cb ; Chung:2011ck ; Graham:2015rva
; Hashiba:2018tbu ; Ema:2018ucl ; Ema:2019yrd ; Ahmed:2020fhc ; Kolb:2020fwh
for gravitational production during inflation or transition from the
inflationary epoch. With this knowledge, it is not surprising that the
oscillating scalar field inside the soliton leads to the production of any
particle through the gravitational process like $\phi\phi\to XX$, where $\phi$
is a scalar or vector that constitutes the soliton and $X$ denotes any light
field since they are inevitably coupled to $\phi$ through gravity. In
practice, production of fermions and massless vector bosons are suppressed and
dominant process is that into the graviton pair or the Higgs boson (or
longitudinal gauge boson) pair if $\phi$ is heavier than them. In particular,
the graviton production process is always present no matter how $\phi$ is
light. Thus this process provides a strict upper limit on the lifetime of
solitons. We stress that this process is inevitable even if there is no
explicit scalar self-interaction term in the action. We estimate the graviton
production rate by deriving the graviton equation of motion under the
scalar/vector soliton background and finding that it is the same as the
Mathieu equation. The source of the oscillating term is identified with the
oscillating gravitational potential in the presence of soliton.

To the best of our knowledge, the only previous work that explicitly pointed
out the quantum graviton production from the scalar oscillon or boson star is
Ref. Page:2003rd . There the graviton production rate is calculated by
interpreting it as the annihilation process $\phi\phi\to hh$, where $h$
denotes the graviton. In this paper we estimate the graviton production rate
by using the time-dependent gravitational potential with the classical soliton
configuration. Our approach is useful for more general objects such as Q-balls
Coleman:1985ki , vector boson stars or vector oscillons, since in these multi-
field solitons it is sometimes nontrivial whether such a perturbative
annihilation picture is applicable or not Page:2003rd . In particular, we will
see an example in which even the graviton production is absent and our
formulation makes it clear.

This paper is organized as follows. In Sec. 2 we briefly review properties of
scalar solitons, i.e. oscillons/boson stars for a scalar field, and estimate
the time-dependence of the gravitational potential. In Sec. 3 we discuss the
case of vector solitons. In Sec. 4 we estimate the gravitational quantum decay
rate of a scalar/vector soliton, by using the information on the gravitational
potential derived in preceding sections. In Sec. 5 we point out several
phenomenological impacts. Sec. 6 is devoted to conclusions and discussion.

##  2 Scalar solitons

We mainly focus on a spherically symmetric localized configuration of a free
real scalar or vector field with gravity, which is sometimes called a boson
star. In the context of axion, this corresponds to a dilute axion star. Our
qualitative result for quantum decay rate can be applied to almost any
localized scalar-field configurations, including oscillons and axion stars.
For concreteness, we mainly consider boson stars in this paper and briefly
comment on the case of oscillons. We consider a scalar field in this section
and a vector field in Sec. 3.

###  2.1 Setup

We consider a system with the action of

$\displaystyle S=\int\sqrt{-g}\,d^{4}x\left[-\frac{1}{16\pi
G}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]\,,$
(1)

where $R$ is the Ricci scalar, $g$ is the determinant of the metric, $G$
($=(8\pi M_{\rm Pl}^{2})^{-1}$) is the gravitational constant, and $M_{\rm
Pl}$ is the reduced Planck mass. The scalar potential is given by

$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}\phi^{2}-\frac{1}{24}\lambda\phi^{4}+\dots\,,$
(2)

where $m$ is the scalar mass and $\lambda$ is a quartic coupling. We consider
the case with a sufficiently small $\lambda$. For the case of axion with a
sine-Gordon potential, one can identify $\lambda=m^{2}/f_{a}^{2}$ with $f_{a}$
being the axion decay constant, neglecting higher-dimensional terms.

The gravitational interaction is a long-range force, whereas the self-
interaction is a point interaction. The former one can be dominant to
determine the localized configuration, if the density at the center of the
boson star is sufficiently low. The extent to which this approximation is
valid can be understood by estimating the gradient energy, the gravitational
potential energy, and the potential energy of the self-interaction,

$\displaystyle\delta_{x}\sim\frac{\left(\nabla\phi\right)^{2}}{m^{2}\phi^{2}}\sim\frac{1}{(mR)^{2}}\,,$
(3) $\displaystyle\delta_{g}\sim\frac{1}{M_{\rm Pl}^{2}}\int
d^{3}x\frac{m^{2}\phi^{2}}{r}\sim\frac{m^{2}\phi^{2}}{M_{\rm Pl}^{2}}R^{2}\,,$
(4) $\displaystyle\delta_{\phi}\sim\frac{\lambda\phi^{4}}{m^{2}\phi^{2}}\,,$
(5)

where we have normalized the energy densities by $m^{2}\phi^{2}$, $R$
represents the typical size of the boson star and $\phi$ represents the boson
field value at the center of boson star.222 This is an abuse of notation, but
this $R$ should not be confused with the Ricci scalar; the Ricci scalar only
appears in the Hilbert-Einstein action in this paper.  The self-interaction is
negligible if $\delta_{g}\gg\delta_{\phi}$. In this case, the size of boson
star is determined by $\delta_{x}\sim\delta_{g}$ and we obtain $R\sim(M_{\rm
Pl}/\phi)^{1/2}/m$. Using this relation, the consistency condition for this
approximation, $\delta_{g}\gg\delta_{\phi}$, is written as

$\displaystyle\lambda\ll\frac{m^{4}R^{2}}{M_{\rm Pl}^{2}}\sim\frac{m^{2}}{\phi
M_{\rm Pl}}\,.$ (6)

For the case of axion, this condition implies

$\displaystyle f_{a}\gg\sqrt{\phi M_{\rm Pl}}\,.$ (7)

We assume that those conditions are met when we discuss boson stars.

Since the gravitational interaction is important, we write the metric around
the boson star such as

$\mathrm{d}s^{2}=e^{-2\Psi(t,r)}\,\mathrm{d}t^{2}-e^{2\Phi(t,r)}\left[\mathrm{d}r^{2}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\varphi^{2}\right)\right]\,.$
(8)

Here, we adopt the isotropic coordinates, which should not be confused with
the spherical coordinates Friedberg:1986tp .333 Note that the spherical
coordinates has been used in Refs. Eby:2018ufi ; Zhang:2020ntm . The reason
why we are using the isotropic coordinates is that it is directly compared
with the Cartesian coordinates, which will be used in the analysis of particle
production in Sec. 4.  The energy-momentum tensor is written as

$\displaystyle T_{\mu\nu}$
$\displaystyle=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\phi\partial_{\sigma}\phi+g_{\mu\nu}V(\phi)\,,$
(9)

where $V(\phi)\simeq\frac{1}{2}m^{2}\phi^{2}$.

###  2.2 Configuration of scalar boson star

The Einstein equations read

$\displaystyle\displaystyle{G^{0}}_{0}=-e^{-2\Phi}\left[2\frac{\partial^{2}\Phi}{\partial
r^{2}}+\frac{1}{r}\left(4+r\frac{\partial\Phi}{\partial
r}\right)\frac{\partial\Phi}{\partial
r}\right]+3e^{2\Psi}\dot{\Phi}^{2}=\frac{\rho}{M_{\rm Pl}^{2}},$ (10)
$\displaystyle\displaystyle{G^{0}}_{r}=-2e^{2\Psi}\left(\frac{\partial\Psi}{\partial
r}\dot{\Phi}+\frac{\partial\dot{\Phi}}{\partial
r}\right)=\frac{e^{2\Psi}}{M_{\rm Pl}^{2}}\frac{\partial\phi}{\partial
t}\frac{\partial\phi}{\partial r},$ (11)
$\displaystyle\displaystyle{G^{r}}_{r}=e^{-2\Phi}\left[\frac{2}{r}\frac{\partial\Psi}{\partial
r}-\frac{2}{r}\frac{\partial\Phi}{\partial r}+2\frac{\partial\Psi}{\partial
r}\frac{\partial\Phi}{\partial r}-\left(\frac{\partial\Phi}{\partial
r}\right)^{2}\right]+e^{2\Psi}\left(3\dot{\Phi}^{2}+2\dot{\Phi}\dot{\Psi}+2\ddot{\Phi}\right)=-\frac{p_{r}}{M_{\rm
Pl}^{2}}\,.$ (12)

The energy density and pressure are defined as

$\displaystyle\displaystyle\rho\equiv{T^{0}}_{0}=\frac{e^{2\Psi}}{2}\dot{\phi}^{2}+\frac{e^{-2\Phi}}{2}\left(\frac{\partial\phi}{\partial
r}\right)^{2}+\frac{1}{2}m^{2}\phi^{2}\,,$ (13) $\displaystyle\displaystyle
p_{r}\equiv-{T^{r}}_{r}=\frac{e^{2\Psi}}{2}\dot{\phi}^{2}+\frac{e^{-2\Phi}}{2}\left(\frac{\partial\phi}{\partial
r}\right)^{2}-\frac{1}{2}m^{2}\phi^{2}\,,$ (14)

where we have used $V(\phi)\simeq\frac{1}{2}m^{2}\phi^{2}.$

We are interested in the weak gravity limit, where we can treat the gravity as
perturbations. We thus write $e^{-2\Psi}\simeq 1-2\Psi(t,r)$ and
$e^{2\Phi}\simeq 1+2\Phi(t,r)$ and treat $\Psi$ and $\Phi$ as small
parameters. From the Einstein equation, they obey

$\displaystyle\Psi(t,r)=G\int_{r}^{\infty}\frac{\mathrm{d}r^{\prime}}{r^{\prime
2}}\left(M(r^{\prime})+4\pi r^{\prime
3}p_{r}(t,r^{\prime})\right)+\int_{r}^{\infty}\mathrm{d}r^{\prime}\,r^{\prime}\ddot{\Phi}\,,$
(15)
$\displaystyle\Phi(t,r)=G\int_{r}^{\infty}\frac{\mathrm{d}r^{\prime}}{r^{\prime
2}}M(r^{\prime})\,,$ (16)

where

$\displaystyle M(r)\equiv\int_{0}^{r}dr^{\prime}\,4\pi r^{\prime
2}\rho(r^{\prime})\,,$ (17)

is the energy enclosed within the radius $r$. As we shall see later, the time
dependence of $M(r)$ is negligible for our purpose. Also, while it slowly
changes with time via the decay of the scalar field as we will see in Sec. 5,
its time scale is much longer. Thus we can neglect $\dot{\Phi}$ to derive the
boson star configuration.

The equation of motion for a free scalar field $\phi$ is given by

$\displaystyle
e^{2\Psi}\left[\ddot{\phi}+(\dot{\Psi}+3\dot{\Phi})\dot{\phi}\right]-e^{-2\Phi}\left[\frac{\partial^{2}\phi}{\partial
r^{2}}+\frac{2}{r}\frac{\partial\phi}{\partial
r}-\left(\frac{\partial\Psi}{\partial r}-\frac{\partial\Phi}{\partial
r}\right)\frac{\partial\phi}{\partial r}\right]+\frac{\partial
V}{\partial\phi}=0\,.$ (18)

The scalar field can have a localized configuration thanks to the
gravitational attractive interaction. This implies that the spatial gradient
energy of the scalar field is proportional to the gravitational potential. We
thus neglect the cross term between the spatial gradient and the gravitational
potential in the weak gravity limit. Moreover, as we will see shortly, the
time-dependent parts of $\Psi$ and $\Phi$ are next-to-leading terms and we can
neglect their time derivatives in the equation of motion. We then obtain the
simplified equation of motion,

$\displaystyle(1+2\Psi_{0})\ddot{\phi}-\left(\frac{\partial^{2}\phi}{\partial
r^{2}}+\frac{2}{r}\frac{\partial\phi}{\partial r}\right)+m^{2}\phi=0\,,$ (19)

where we denote the leading-order (time-independent) part of $\Psi$ as
$\Psi_{0}$.

Now, let us take an ansatz $\phi(t,r)=\phi_{r}(r)\cos(\omega t)$ to solve the
equation of motion. The equation of motion for $\phi_{r}(r)$ is then given by

$\displaystyle\frac{\partial^{2}\phi_{r}}{\partial
r^{2}}+\frac{2}{r}\frac{\partial\phi_{r}}{\partial
r}-\left[m^{2}-\omega^{2}(1+2\Psi_{0}(r))\right]\phi_{r}=0\,,$ (20)

where $\Psi_{0}(r)$ is determined from

$\displaystyle\frac{1}{4\pi G}\left(\frac{\partial^{2}\Psi_{0}}{\partial
r^{2}}+\frac{2}{r}\frac{\partial\Psi_{0}}{\partial
r}\right)=-\frac{1}{2}m^{2}\phi_{r}^{2}(r)\,.$ (21)

To obtain the configuration $\phi_{r}(r)$ through numerical calculations, a
convenient approach is to rescale the variables into dimensionless units. The
set of equations (20) and (21) can be rewritten as

$\displaystyle\frac{\partial^{2}\varphi}{\partial
z^{2}}+\frac{2}{z}\frac{\partial\varphi}{\partial
z}+\left[1-g(z)\right]\varphi=0,$ (22)
$\displaystyle\frac{\partial^{2}g}{\partial z^{2}}+\frac{2}{z}\frac{\partial
g}{\partial z}=\varphi^{2}(z),$ (23)

where we define

$\displaystyle
R\equiv\frac{1}{\sqrt{\omega^{2}-m^{2}+2\omega^{2}\Psi_{0}(0)}}\,,$ (24)
$\displaystyle r=Rz\,,$ (25)
$\displaystyle\phi_{r}(r)=\frac{\varphi(z)}{\sqrt{4\pi G}\,m\omega R^{2}}\,,$
(26) $\displaystyle\Psi_{0}(r)=\Psi_{0}(0)-\frac{g(z)}{2\omega^{2}R^{2}}\,.$
(27)

Here we note that $\Psi_{0}=0$ at an asymptotic infinity, which implies
$g_{\infty}\equiv g(\infty)=2\omega^{2}R^{2}\Psi_{0}(0)$. We numerically solve
the above equations from $z=z_{0}\ll 1$ to $z=z_{1}\gg 1$ with the boundary
condition of $\varphi(z_{0})=\varphi_{*}$,
$\varphi^{\prime}(z_{0})=-\varphi_{*}z_{0}/2$, $g(z_{0})=0$,
$g^{\prime}(z_{0})=\varphi_{*}^{2}z_{0}/2$, and $\varphi(\infty)=0$. The
localized configuration is realized only by a certain value of $\varphi_{*}$,
which can be determined by the shooting algorithm. The result is given by

$\displaystyle\varphi_{*}\simeq 1.089,$ (28) $\displaystyle g(z\gg 1)\simeq
2.066-\frac{3.619}{z}\,.$ (29)

In particular, we obtain $g_{\infty}\simeq 2.066$, which implies

$\displaystyle
R\simeq\frac{\sqrt{g_{\infty}-1}}{\epsilon\,m}\simeq\frac{1.0}{\epsilon\,m}\,,$
(30) $\displaystyle\phi_{r}(0)\simeq\frac{0.29\,\epsilon^{2}}{\sqrt{G}}\,,$
(31)

where we define $\epsilon^{2}\equiv(m^{2}-\omega^{2})/m^{2}$ ($\ll 1$).

We define $R_{99}$ by the radius at which 99% of total energy of the axion
star is enclosed. The total energy and $R_{99}$ are given by

$\displaystyle E\equiv M(\infty)\simeq 1.8\,\frac{\epsilon}{Gm}\,,$ (32)
$\displaystyle R_{99}\simeq 5.2\,R\simeq\frac{5.4}{\epsilon\,m}\,.$ (33)

We note that the amplitude $\phi_{r}(0)$ has to be much larger than $m$, since
otherwise the number density is too small for the classical picture to be
valid. This requires $\epsilon\gg\sqrt{m/M_{\rm Pl}}$.

###  2.3 Oscillating behavior of the gravitational potential

Here we derive the the behavior of gravitational potential that helps to
understand the quantum decay process. The potential $\Phi$ is given in Eq.
(16) as a solution to Eq. (10). To find a time dependence of $\Phi$, it is
convenient to look at Eq. (11). By substituting an ansatz for the
configuration $\phi(t,r)=\phi_{r}(r)\cos(\omega t)$, it is easy to find that
the solution is given in the form of

$\displaystyle\Phi(t,r)\simeq\Phi_{0}(r)-\frac{\phi_{r}(r)^{2}}{16M_{\rm
Pl}^{2}}\cos(2\omega t)\,,$ (34)

where $\Phi_{0}(r)$ is time-independent part. Parametrically
$|\Phi_{0}|\sim(R\omega\phi_{r}/M_{\rm Pl})^{2}$ and it is $(R\omega)^{2}$
times larger than the oscillating term for a typical oscillon radius $R$.444
The expression (34) is independent of whether the object is stabilized by the
self interaction (oscillon) or gravitational force (boson star). Thus we can
use the oscillating gravitational potential of $\sim(\phi_{r}(r)^{2}/16M_{\rm
Pl}^{2})\cos(2\omega t)$ in both cases to estimate the graviton production
rate. See Sec. 4.  As a consistency check, the same expression can be directly
derived from (16). To see this, let us note that the energy density $\rho$
satisfies the equation

$\displaystyle\dot{\rho}=e^{-2\Phi}\left[\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\dot{\phi}\frac{\partial\phi}{\partial
r}\right)+\frac{\partial\phi}{\partial
r}\left(\dot{\phi}\left(\frac{\partial\Phi}{\partial
r}-\frac{\partial\Psi}{\partial
r}\right)-\dot{\Phi}\frac{\partial\phi}{\partial
r}\right)\right]-e^{2\Psi}3\dot{\Phi}\dot{\phi}^{2}\,.$ (35)

Neglecting terms involving $\Psi$ or $\Phi$, we find

$\displaystyle\rho(t,r)\simeq\rho_{0}(r)+\frac{1}{8r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial\phi_{r}^{2}}{\partial r}\right)\cos(2\omega t)\,.$
(36)

By substituting this into Eq. (16) we obtain the same expression as (34). From
Eq. (12), we also obtain the time-dependence of $\Psi$ as

$\displaystyle\Psi(t,r)\simeq\Psi_{0}(r)+\Psi_{1}(r)\cos(2\omega t)\,,$ (37)

where $|\Psi_{0}|\sim(R\omega\phi_{r}/M_{\rm Pl})^{2}$ and
$|\Psi_{1}|\sim(\phi_{r}/M_{\rm Pl})^{2}$. The time dependence of the
gravitational potential is essential to understand the quantum decay of the
boson star, i.e., the production of gravitationally coupled particles such as
gravitons as we will see in Sec. 4.

###  2.4 The case of complex scalar: Q-balls

Here we give comments on the case of a complex scalar $\phi$ with global U(1)
symmetry $\phi\to\phi e^{i\theta}$ with an arbitrary constant $\theta$. The
action is

$\displaystyle S=\int\sqrt{-g}\,d^{4}x\left[-\frac{1}{16\pi
G}R+g^{\mu\nu}\partial_{\mu}\phi^{*}\partial_{\nu}\phi-V(|\phi|)\right]\,.$
(38)

In this case, it is known that there appear soliton-like objects, called
Q-balls Coleman:1985ki ; Cohen:1986ct , if the potential is shallower than the
quadratic potential. Note that, in contrast to oscillons/I-balls, the
potential does not have to be dominated by a quadratic term Kasuya:2002zs .
Q-balls are known to play essential roles in the dynamics of supersymmetric
flat directions Kusenko:1997ad ; Kusenko:1997zq ; Kusenko:1997si  in the
context of Affleck-Dine baryogenesis Affleck:1984fy ; Dine:1995kz .

The Q-ball solution is written as $\phi(t,r)=\phi_{r}(r)e^{i\omega t}$ with a
real function $\phi_{r}(r)$. A crucial difference from the case of real scalar
is that it allows a solution in which the gravitational potential $\Phi$ does
not oscillate with time. It is easily seen from the complex scalar version of
(11):

$\displaystyle\displaystyle{G^{0}}_{r}=-2e^{2\Psi}\left(\frac{\partial\Psi}{\partial
r}\dot{\Phi}+\frac{\partial\dot{\Phi}}{\partial
r}\right)=\frac{e^{2\Psi}}{2M_{\rm
Pl}^{2}}\left(\frac{\partial\phi^{*}}{\partial t}\frac{\partial\phi}{\partial
r}+\frac{\partial\phi}{\partial t}\frac{\partial\phi^{*}}{\partial
r}\right)\,.$ (39)

For the Q-ball solution, the right hand side is zero and hence the
gravitational potential is constant with time. Therefore, no gravitational
particle production happens in this limit and such a configuration is stable
even against to the graviton production (see Sec. 4). It can be understood as
a consequence of the U(1) symmetry and the conserved global charge associated
with it. If the orbit is initially elliptical in the complex plane, it will
approach to the circular orbit by quantum decay process, including the
graviton emission even if there are no explicit U(1) breaking terms.

One may add Planck-suppressed higher-dimensional terms that explicitly break
U(1) symmetry. They are motivated by a low-energy effective field theory for
quantum gravity, where there should be no global symmetry in nature. Those
U(1) symmetry breaking terms distort the dynamics of the complex scalar field
from the circular orbit in the complex plane. Then the gravitational potential
$\Phi$ has an oscillating term that can result in gravitational emission. As
the amplitude of scalar field decreases, the effect of higher-dimensional U(1)
breaking terms becomes weaker. Therefore the graviton emission rate becomes
smaller as the Q-ball emits more gravitons.

##  3 Vector solitons

In this section we consider vector solitons and discuss the properties of the
gravitational potentials, particularly focusing on their time dependence.
While the most discussion is parallel to the case of scalar solitons discussed
in the previous section, a crucial difference is that there are polarization
degrees of freedom in the case of vector solitons. We follow the analysis in
Ref. Adshead:2021kvl  (see also Ref. Jain:2021pnk ) and extend it to calculate
the time-dependence of gravitational potential.

###  3.1 Setup

The action we consider is given by

$\displaystyle S=\int d^{4}x\sqrt{-g}\left[-\frac{R}{16\pi
G}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}m^{2}A^{\mu}A_{\mu}\right]\,,$
(40)

where $m$ is the Stuckelberg mass for the dark photon $A^{\mu}$ and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is its field
strength. The energy-momentum tensor is then given by

$\displaystyle
T^{\mu}{}_{\nu}=m^{2}A^{\mu}A_{\nu}-F^{\mu\alpha}F_{\nu\alpha}+\delta^{\mu}_{\nu}\left[-\frac{m^{2}}{2}A_{\alpha}A^{\alpha}+\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}\right]\,.$
(41)

The Maxwell equations read

$\displaystyle\nabla_{\mu}F^{\mu\nu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}F^{\mu\nu}\right)=-m^{2}A^{\nu}\,,$
(42)

where we note that

$\displaystyle
F^{\mu\nu}=g^{\mu\alpha}\left(\partial_{\alpha}A^{\nu}-A_{\beta}\partial_{\alpha}g^{\nu\beta}\right)-g^{\nu\alpha}\left(\partial_{\alpha}A^{\mu}-A_{\beta}\partial_{\alpha}g^{\mu\beta}\right)\,.$
(43)

This particularly implies the Lorentz condition

$\displaystyle\nabla_{\nu}A^{\nu}=0\,.$ (44)

Since we consider a massive gauge field, there are three degrees of freedom
for the gauge field and $A^{0}$ is a non-dynamical field.

We are interested in the weak-gravity limit, where the metric is treated as a
perturbation from the Minkowski metric. In the Newtonian gauge, we write

$\displaystyle
ds^{2}=(1-2\Psi)d\tau^{2}+2V_{i}dx^{i}d\tau-\left[(1+2\Phi)\delta_{ij}-h_{ij}\right]dx^{i}dx^{j}\,,$
(45)

where $\partial_{i}V_{i}=0$ and we also choose a gauge to eliminate the
transverse vector degree of freedom. The transverse-traceless metric
perturbation $h_{ij}$ satisfies $\partial_{i}h_{ij}=\partial_{i}h_{ji}=0$, and
$h_{ii}=0$. This completely fixes the gauge: $\Psi$, $\Phi$ and $V_{i}$ are
given in terms of $A_{\mu}$. We first neglect $h_{ij}$ to determine the
profile of vector star and later take it into account to discuss the quantum
decay. In particular, we have

$\displaystyle\sqrt{-g}=1+(3\Phi-\Psi)\,,$ (46)

for $h_{ij}=0$.

###  3.2 Configuration of vector boson star

The Lorentz constraint Eq. (44) is written as

$\displaystyle\partial_{0}A^{0}+\partial_{i}A^{i}+A^{0}\partial_{0}\left(3\Phi-\Psi\right)+A^{j}\partial_{j}(3\Phi-\Psi)=0\,.$
(47)

We are interested in the mode that oscillates with a frequency of
$\omega\simeq m$. This implies $A^{0}=\mathcal{O}(A^{i}/(mR))$ at the leading
order, where $R$ is the typical size of the object. For $mR\gg 1$, the Maxwell
equations and the Lorentz condition are greatly simplified such as

$\displaystyle\partial_{0}^{2}A^{j}-\Delta
A^{j}+\partial_{0}\left(5\Phi+\Psi\right)\partial_{0}A^{j}+\left[(1-2\Psi)m^{2}+2\partial_{0}^{2}\Phi\right]A^{j}\simeq
0\,,$ (48) $\displaystyle\partial_{0}A^{0}+\partial_{i}A^{i}\simeq 0\,,$ (49)

where $\Delta=\partial_{i}\partial_{j}\delta^{ij}$. Here we included the
leading-order terms that has time derivatives of gravitational potentials,
which are negligible compared with the other terms, though. The Einstein
equation leads to

$\displaystyle-\Delta\Phi=\frac{1}{2M_{\rm Pl}^{2}}T^{0}{}_{0}\,,$ (50)
$\displaystyle-\frac{2}{3}\Delta^{2}(\Psi-\Phi)=\frac{1}{M_{\rm
Pl}^{2}}\left[\delta^{il}\delta^{jk}\partial_{i}\partial_{j}-\frac{\delta^{kl}\Delta}{3}\right]T^{k}{}_{l}\,,$
(51) $\displaystyle\Delta V_{i}=\frac{2}{M_{\rm
Pl}^{2}}T^{0}{}_{i}^{\rm(T)}\,,$ (52)
$\displaystyle-\partial_{i}\dot{\Phi}=\frac{1}{2M_{\rm
Pl}^{2}}T^{0}{}_{i}^{\rm(L)}\,,$ (53)

where the superscript (T) and (L) indicate the transverse and longitudinal
component, respectively. Here, the $(0,0)$ and $(0,i)$ component of the
energy-momentum tensor are given by

$\displaystyle
T^{0}{}_{0}\simeq\frac{m^{2}}{2}\left(A^{0}\right)^{2}+\frac{m^{2}}{2}\sum_{i}\left(A^{i}\right)^{2}(1+2\Phi)+\frac{1}{2}\sum_{i}(\partial_{0}A^{i})^{2}(1+2\Phi+2\Psi)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{i}\partial_{0}A^{i}\partial_{i}A^{0}+\frac{1}{2}\sum_{i,j}\left(\partial_{i}A^{j}\partial_{i}A^{j}-\partial_{i}A^{j}\partial_{j}A^{i}\right),$
(54) $\displaystyle
T^{0}{}_{i}\simeq-m^{2}A^{0}A^{i}-\sum_{j}\partial^{0}A^{j}(\partial_{j}A^{i}-\partial_{i}A^{j})\,,$
(55)

where we include terms up to $\mathcal{O}(k^{2}(A^{i})^{2})$ for $T^{0}{}_{0}$
and leading-order terms for $T^{0}{}_{i}$.

Now, let us suppose

$\displaystyle A^{i}=n^{i}f(r)\cos(\omega t+\theta^{i})\,,$ (56)

where $n_{i}$ is a unit vector representing the polarization and $\theta^{i}$
is the phase. Then, at the leading order, the equation of motion for $f$ is
independent of the polarization and is given by

$\displaystyle\frac{\partial^{2}f}{\partial r^{2}}+\frac{2}{r}\frac{\partial
f}{\partial r}-\left[m^{2}-\omega^{2}(1+2\Psi_{0}(r))\right]f=0\,,$ (57)

where $\Psi_{0}(r)$ is the leading-order term of $\Psi$ and is given by

$\displaystyle\frac{1}{4\pi G}\left(\frac{\partial^{2}\Psi_{0}}{\partial
r^{2}}+\frac{2}{r}\frac{\partial\Psi_{0}}{\partial
r}\right)=-\frac{1}{2}m^{2}f^{2}(r)\,.$ (58)

The latter equation comes from Eq. (50) by noting $\Psi_{0}\simeq\Phi_{0}$
from Eq. (51) at the leading order. These are identical to the equations for
the scalar fields (see Eqs. (20) and (21)). So the mass, size, and amplitude
at the center of the vector boson star are the same with those for scalar
boson star.

###  3.3 Oscillating behavior of the gravitational potential

Similarly to the case of scalar boson star studied in the previous section, we
are interested in the behavior of the gravitational potential $\Psi$ and
$\Phi$ and its time dependence since it is essential to evaluate the quantum
decay rate. Compared with the scalar case, it is nontrivial since there are
four components $A_{\mu}$ and several configurations of the vector
polarization are known Adshead:2021kvl ; Jain:2021pnk ; Zhang:2021xxa .555
Recall that the gravitational potential is inevitably oscillating for a real
scalar case, but it can be time-independent for a complex scalar as shown in
Sec. 2.4.  Our short conclusion is that in the massive vector case, at least
for known configurations, the gravitational potential oscillates with a
similar amplitude. In deriving it, $A^{0}$ plays an important role since it is
related to $A^{i}$ through the Lorentz constraint (49).

Using the ansatz Eq. (56), we obtain the constraint from the Lorentz condition
Eq. (49) such as

$\displaystyle A^{0}=-\frac{1}{\omega}\sum_{i}n^{i}\partial_{i}f(r)\sin(\omega
t+\theta^{i})\,.$ (59)

Substituting these into $T^{0}{}_{0}$, we obtain

$\displaystyle T^{0}{}_{0}$
$\displaystyle\simeq\frac{m^{2}}{2\omega^{2}}\left(\sum_{i}n^{i}\partial_{i}f\sin(\omega
t+\theta^{i})\right)^{2}+\frac{m^{2}}{2}\sum_{i}\left(n^{i}f(r)\cos(\omega
t+\theta^{i})\right)^{2}(1+2\Phi)$
$\displaystyle+\frac{\omega^{2}}{2}\sum_{i}(n^{i}f(r)\sin(\omega
t+\theta^{i}))^{2}(1+2\Phi+2\Psi)$ $\displaystyle+\sum_{i}n^{i}f(r)\sin(\omega
t+\theta^{i})\left(\sum_{j}n^{j}\partial_{i}\partial_{j}f\sin(\omega
t+\theta^{j})\right)$
$\displaystyle+\frac{1}{2}\sum_{i,j}\left(n^{j}n^{j}\partial_{i}f\partial_{i}f\cos^{2}(\omega
t+\theta^{j})-n^{i}n^{j}\partial_{i}f\partial_{j}f\cos(\omega
t+\theta^{i})\cos(\omega t+\theta^{j})\right)\,.$

In the following, we consider two extremely polarized configurations to see
the time-dependence of gravitational potential.

####  3.3.1 Linear polarization

Let us consider the case with a linear polarization: $n^{1}=1$, $n^{2}=0$,
$n^{3}=0$, $\theta^{i}=0$. We then obtain

$\displaystyle T^{0}{}_{0}$
$\displaystyle\simeq\frac{1}{2}\hat{x}^{2}(f^{\prime})^{2}\sin^{2}(\omega
t)+\frac{m^{2}}{2}(1+2\Phi)f^{2}+\frac{(\omega^{2}-m^{2})+2\omega^{2}\Psi}{2}f^{2}\sin^{2}(\omega
t)$ (61)
$\displaystyle+f\left(f^{\prime}/r+\hat{x}^{2}(f^{\prime\prime}-f^{\prime}/r)\right)\sin^{2}(\omega
t)+\frac{1}{2}f^{\prime 2}(1-\hat{x}^{2})\cos^{2}(\omega t)\,,$
$\displaystyle=\frac{m^{2}}{2}(1+2\Phi)f^{2}+\left(-\frac{ff^{\prime\prime}}{2}+\hat{x}^{2}\left(ff^{\prime\prime}+\frac{f^{\prime}f^{\prime}}{2}-\frac{ff^{\prime}}{r}\right)\right)\sin^{2}(\omega
t)$ $\displaystyle+\frac{1}{2}f^{\prime 2}(1-\hat{x}^{2})\cos^{2}(\omega
t)\,,$

where $\hat{x}\equiv x/r$, the prime denotes the derivative with respect to
$r$ and we used Eq. (57) in the last line. The large parenthesis in front of
$\sin^{2}(\omega t)$ is given by $\sim f^{2}/R^{2}$ with $R$ being the typical
size of the vector boson star. Noting that

$\displaystyle\Delta f^{2}(r)=2\left(ff^{\prime\prime}+f^{\prime
2}+\frac{2ff^{\prime}}{r}\right)\sim\frac{f^{2}}{R^{2}}\,,$ (62)

and Eq. (50), we approximate the gravitational potential as

$\displaystyle\Phi(t,r)\sim\Phi_{0}(r)+\frac{f^{2}}{16M_{\rm
Pl}^{2}}\cos(2\omega t)\,,$ (63)

which should be regarded as an order-of-magnitude estimate. Here, we
particularly omit the deviation from the spherical symmetry.

####  3.3.2 Circular polarization

Let us consider the case with a maximally circular polarization:
$n^{1}=1/\sqrt{2}$, $n^{2}=1/\sqrt{2}$, $n^{3}=0$, $\theta^{1}=0$,
$\theta^{2}=\mp\pi/2$, $\theta^{3}=0$. We first note that
$\sum_{i}n^{i}\partial_{i}f\sin(\omega
t+\theta^{i})=f^{\prime}(\hat{x}\sin(\omega t)\mp\hat{y}\cos(\omega
t))/\sqrt{2}=f^{\prime}\sin(\omega t\mp\phi_{x})\sin\phi_{z}/\sqrt{2}$, where
$x=r\cos\phi_{x}\sin\phi_{z}$ and $y=r\sin\phi_{x}\sin\phi_{z}$. We then
obtain

$\displaystyle T^{0}{}_{0}$
$\displaystyle\simeq\frac{m^{2}}{2}(1+2\Phi)f^{2}+\frac{(\omega^{2}-m^{2})+2\omega^{2}\Psi}{4}f^{2}+\frac{1}{4}f^{\prime}f^{\prime}$
(64)
$\displaystyle+\frac{1}{4}\sin^{2}\phi_{z}f^{\prime}f^{\prime}\left(\sin^{2}(\omega
t\mp\phi_{x})-\cos^{2}(\omega t\mp\phi_{x})\right)$
$\displaystyle+\frac{1}{2}f\left[\partial_{x}^{2}f\sin^{2}(\omega t)\mp
2\partial_{x}\partial_{y}f\cos(\omega t)\sin(\omega
t)+\partial_{y}^{2}f\cos^{2}(\omega t)\right].$

Here, the third line can be rewritten as

$\displaystyle\frac{1}{2}\left[\frac{ff^{\prime}}{r}+\left(ff^{\prime\prime}-\frac{ff^{\prime}}{r}\right)\sin^{2}(\omega
t\mp\phi_{x})\sin^{2}\phi_{z}\right].$ (65)

Using Eq. (57), we finally obtain

$\displaystyle T^{0}{}_{0}$
$\displaystyle\simeq\frac{m^{2}}{2}(1+2\Phi)f^{2}+\frac{1}{4}\left(f^{\prime}f^{\prime}-ff^{\prime\prime}\right)-\frac{1}{4}f^{\prime}f^{\prime}\cos^{2}(\omega
t\mp\phi_{x})\sin^{2}\phi_{z}$ (66)
$\displaystyle+\frac{1}{2}\left(ff^{\prime\prime}+\frac{1}{2}f^{\prime}f^{\prime}-\frac{ff^{\prime}}{r}\right)\sin^{2}(\omega
t\mp\phi_{x})\sin^{2}\phi_{z}\,.$

Thus we have the oscillating term.666 Recently Ref. Amin:2023imi  studied the
real photon emission due to the higher dimensional interactions between the
photon and dark photon. It has been found that some of the operators do not
contribute to the photon emission for circular polarized vector solitons,
since $|A_{i}|^{2}={\rm const.}$, neglecting the small $A_{0}$ contribution.
For the graviton emission case, we do not have a freedom to choose such
effective operators and the inclusion of the $A_{0}$ contribution is
important.  Again, from Eq. (50), we approximate the gravitational potential
as

$\displaystyle\Phi(t,r)\sim\Phi_{0}(r)+\frac{f^{2}}{16M_{\rm
Pl}^{2}}\cos(2\omega t\pm 2\phi_{x})\,.$ (67)

This should be regarded as an order-of-magnitude estimate because we
particularly omit the deviation from the spherical symmetry. This is the same
orders-of-magnitude result as the linear polarized case.

##  4 Quantum decay into gravitons

Now let us estimate the quantum production rate of the graviton in the boson
star. As far as we consider the graviton wavelength much shorter than the
typical size of the boson star $R$, we can approximate the metric as if it is
spatially flat.777 See Ref. Kawasaki:2013awa  for more precise discussion
about this approximation for the case of scalar particle production.  Then the
the equation of motion of the gravitational wave $h_{\lambda}$, with
$\lambda=+,\times$ being the polarization, is given by

$\displaystyle\ddot{h}_{\lambda}+(3\dot{\Phi}+\dot{\Psi})\dot{h}_{\lambda}+e^{-2(\Psi+\Phi)}k^{2}h_{\lambda}=0\,,$
(68)

where $k$ denotes the wavenumber. One may think that both the oscillating
$\Psi$ and $\Phi$ may induce the particle production, but we need to take care
of the $e^{-2(\Psi+\Phi)}$ factor in front of the $k^{2}$ term. Actually it
turns out that oscillation of $\Psi$ does not lead to particle production. In
order to see this, let us define a new time variable $d\tau\equiv
e^{-(\Psi+\Phi)}dt$. Then the equation of motion (68) is rewritten as

$\displaystyle\partial^{2}_{\tau}h_{\lambda}+2(\partial_{\tau}\Phi)(\partial_{\tau}h_{\lambda})+k^{2}h_{\lambda}=0\,.$
(69)

With this time variable, $\Psi$ disappeared and it is time dependence of
$\Phi$ that leads to particle production. This is further simplified by
defining $\widetilde{h}_{\lambda}\equiv e^{\Phi}h_{\lambda}$ as888 A new time
variable is the analogue of the conformal time and $b$ is the analogue of the
cosmic scale factor in the Freedman-Robertson-Walker universe.

$\displaystyle\partial^{2}_{\tau}{\widetilde{h}}_{\lambda}+\left(k^{2}-\frac{\partial^{2}_{\tau}b}{b}\right)\widetilde{h}_{\lambda}=0\,,$
(70)

where $b\equiv e^{\Phi}$. The time-dependent effective mass term is given by

$\displaystyle\frac{\partial^{2}_{\tau}b}{b}\simeq\left(\frac{\omega\phi_{r}}{2M_{\rm
Pl}}\right)^{2}\cos(2\omega t)~{}~{}~{}{\rm for~{}scalar}\,,$ (71)
$\displaystyle\frac{\partial^{2}_{\tau}b}{b}\simeq\left(\frac{\omega
f}{2M_{\rm Pl}}\right)^{2}\cos(2\omega t)~{}~{}~{}{\rm for~{}vector}\,,$ (72)

from Eq. (34) for the scalar case. Since the scalar and vector cases give
similar results, below we collectively denote by $\phi$ both the scalar and
vector boson for notational simplicity.

Equation (70) is the same as the Mathieu equation and hence the modes with
$k\simeq\omega$ are enhanced. Once the equation of motion is identified with
the Mathieu equation, we can use the same technique to solve this equation
(e.g. the Bogoliubov coefficient technique) and derive the production rate in
the narrow width limit, as found in literature Dolgov:1989us ; Traschen:1990sw
; Shtanov:1994ce ; Kofman:1997yn . In the case of purely gravitational
production, more details are found in Refs. Ema:2015dka ; Ema:2018ucl ;
Chung:2018ayg . Although we do not repeat the calculation details here, the
point is that it is regarded as a perturbative annihilation of the scalar or
vector into the graviton pair999 One should be careful about this
interpretation. It is justified after deriving the time dependence of $\Phi$,
which exhibits $\cos(2\omega t)$ behavior. For the case of Q-ball, as shown in
Sec. 2.4, the gravitational potential does not oscillate and this
interpretation is not justified.  $\phi\phi\to hh$, since $(\phi_{r}/2M_{\rm
Pl})^{2},(f/2M_{\rm Pl})^{2}\ll 1$. The graviton production rate from a boson
star is given by

$\displaystyle\Gamma_{\rm grav}=\Gamma(\phi\phi\to
hh)=\frac{\mathcal{C}}{256\pi}\frac{\phi_{R}^{2}m^{3}}{M_{\rm Pl}^{4}}\,,$
(73)

where $\phi_{R}$ denotes the typical amplitude of $\phi$ in the scalar boson
star or $A_{i}$ in the vector boson star and we approximated $\omega\simeq m$,
and $\mathcal{C}$ is an $\mathcal{O}(1)$ coefficient that represents the
effect of radial shape of the boson star.

Note that this production mechanism is purely quantum. The production through
(70) requires initial seed of the graviton field $\widetilde{h}_{\lambda}$,
which is provided by the zero-point fluctuation in the vacuum. Thus it is not
understood as a gravitational wave production from classical sources. Although
a spherically symmetric boson star cannot be a classical source of
gravitational waves, it can enhance the graviton quantum fluctuations existing
there.

Here we comment on the possibility of parametric resonance. The equation of
motion (70) is exactly the form of the Mathieu equation, and hence one may
wonder whether the parametric resonant amplification of the graviton can
happen or not. To see this, let us compare the growth rate of the graviton
amplitude and the escape rate of the produced graviton from the soliton. If
the former is larger than the latter, the parametric resonance can happen.
Otherwise gravitons do not accumulate in the phase space with narrow momentum
rage around $k\sim\omega$ and no Bose enhancement would be expected
Hertzberg:2010yz ; Kawasaki:2013awa . The growth rate is characterized by the
parameter $\mu$, which is defined as $h_{\lambda}\propto e^{\mu t}$. In our
case it is given by $\mu\sim\omega\phi_{R}^{2}/M_{\rm Pl}^{2}$ Kofman:1997yn .
On the other hand, the escape rate is simply given by $\sim
1/R\sim\omega\phi_{R}/M_{\rm Pl}$ for the self-gravitating boson star, which
is larger than the growth rate $\mu$. For the oscillon with self-interactions,
the escape rate is even larger. Therefore, the escape rate is always larger
than the growth rate for $\phi_{R}\ll M_{\rm Pl}$. Thus we conclude that there
is no parametric resonance effect and the graviton production rate is linear
in time with the perturbative rate (73).

All the Standard Model particles are also gravitationally coupled to $\phi$
and $A^{\mu}$ and hence their pair production can happen through the
$s$-channel graviton exchange. Among them, massless fermion and transverse
vector boson production are suppressed due to their conformal nature. The
production rate of fermions is $(m_{f}/m)^{2}$ times smaller than the graviton
production rate, where $m_{f}$ is the fermion mass. On the other hand, scalar
production is not suppressed unless it is conformally coupled to the Ricci
scalar. The equation of motion of a free scalar $\chi$, which is minimally
coupled to gravity, is given by

$\displaystyle
e^{2\Psi}\left[\ddot{\chi}+(3\dot{\Phi}+\dot{\Psi})\dot{\chi}\right]-e^{-2\Phi}\vec{\nabla}^{2}\chi+m_{\chi}^{2}\chi=0\,,$
(74)

where $m_{\chi}$ is the mass of $\chi$. Similarly to the graviton case, it is
simplified by introducing $\widetilde{\chi}\equiv e^{\Phi}\chi$ as

$\displaystyle\partial^{2}_{\tau}\widetilde{\chi}+\left(k^{2}-\frac{\partial^{2}_{\tau}b}{b}+b^{2}m_{\chi}^{2}\right)\widetilde{\chi}=0\,,$
(75)

where we have moved to the Fourier space. Therefore, if $k\simeq m\gg
m_{\chi}$, the scalar production rate is the same as the graviton production
rate up to a factor $2$ coming from the graviton polarization degrees of
freedom. In the Standard Model, the production rate of the Higgs boson is the
same order of the graviton production rate if $m$ is much larger than the
Higgs mass. In the electroweak symmetry breaking phase, the production of
longitudinal massive vector bosons is also the same order. If $0.1$ GeV
$\lesssim m\lesssim 1$ GeV, the boson stars may also efficiently produce pions
or other mesons.101010 The gravitational gluon production for $m\gtrsim
1\,{\rm GeV}$ may not be much suppressed compared with the graviton production
due to the relatively strong breaking of scale invariance in the strong
dynamics.  If there exist other scalar fields lighter than $\phi$ or
$A^{\mu}$, such as axions, they should also be produced with the same
production rate.

##  5 Phenomenological implications of gravitational decay

In this section, we discuss time evolution of the boson star with a
gravitational decay rate of (73) and then calculate the spectrum of graviton
background. The following discussion applies to the case of both scalar and
vector. For notational simplicity we simply express them by $\phi$, but it can
be as either scalar or vector.

###  5.1 Decay of boson star

If there are no direct interactions between $\phi$ (or $A^{\mu}$) and other
lighter fields, the gravitational processes are the only processes that cause
the decay of boson star. Classical decay processes such as $3\phi\to\phi$ are
possible mediated by the gravitational interaction Page:2003rd , but it is
likely to be exponentially suppressed for $\omega R\gg 1$. In this limit, the
graviton production is an inevitable decay process and it gives a strict lower
bound on the lifetime of oscillons/boson stars.

Via the graviton production, a boson star mass $M$ decreases according to

$\displaystyle\dot{M}(t)=-\Gamma_{\rm grav}(t)M(t)\,.$ (76)

Note that the decay rate (73) depends on the amplitude of the scalar/vector in
the boson star $\phi_{R}$ and hence the decay rate decreases as the boson star
mass decreases. By using the relation $M\sim m^{2}\phi_{R}^{2}R^{3}$ and the
equilibrium condition $GM\sim(Rm^{2})^{-1}$, we find $\Gamma_{\rm grav}\propto
M^{4}$. Then we can solve (76) as

$\displaystyle M^{4}(t)=\frac{M_{i}^{4}}{1+\Gamma_{\rm
grav}^{0}t},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma_{\rm
grav}^{0}\equiv\frac{\mathcal{C}}{1024\pi}\frac{\phi_{R,i}^{2}m^{3}}{M_{\rm
Pl}^{4}}\,,$ (77)

where we assumed the initial boson star mass $M_{i}$ in the limit $t\to 0$ and
$\phi_{R,i}$ denotes the initial scalar/vector amplitude at the boson star
formation. Correspondingly, we obtain

$\displaystyle\phi_{R}^{2}(t)=\frac{\phi_{R,i}^{2}}{1+\Gamma_{\rm
grav}^{0}t}\,.$ (78)

Therefore, the boson star mass decreases rather slowly as $M\propto t^{-1/4}$
for $t\gtrsim(\Gamma_{\rm grav}^{0})^{-1}$. We call $\tau_{\rm
grav}\equiv(\Gamma_{\rm grav}^{0})^{-1}$ as the boson star lifetime, although
it does not decay exponentially. Numerically it is estimated as

$\displaystyle\tau_{\rm grav}=(\Gamma_{\rm grav}^{0})^{-1}\simeq 1\times
10^{16}\,{\rm sec}\,\left(\frac{M_{\rm
Pl}}{\phi_{R,i}}\right)^{2}\left(\frac{1\,{\rm GeV}}{m}\right)^{3}\,.$ (79)

It easily exceeds the present age of the universe for, e.g., light axion-like
particles $m\lesssim 1\,{\rm eV}$. If $m>\mathcal{O}(100)$ GeV, the boson
stars may efficiently produce the Higgs boson and longitudinal weak bosons
after the Big-Bang Nucleosynthesis epoch with a rate similar to the graviton
production. This gives a constraint on the abundance of boson star.111111 If
$0.1$ GeV $\lesssim m\lesssim 1$ GeV, the boson stars may also efficiently
produce pions or other mesons after the recombination epoch, which would also
give a tight constraint.  We will come back to this issue in detail elsewhere.

Once the amplitude $\phi_{R}(t)$ becomes smaller than $m$, the classical
picture of the boson star may no longer be valid. This happens at
$t\sim\left[\Gamma_{\rm grav}^{0}(m/\phi_{R,i})^{2}\right]^{-1}$. Then one has
to treat the bosons as individual particles with negligible Bose enhancement
effect.

###  5.2 Decay of oscillon and Q-ball

Here we comment on the case with oscillon, where the self-interactions support
the localized configuration. In most cases, oscillons classically decay into
individual particles via self-interactions. In a certain case, however, the
classical decay processes mediated by the self-interactions can be
exponentially suppressed for $\omega R\gg 1$ Mukaida:2016hwd .121212 For the
Gaussian profile, the suppression factor should read $\sim\exp(-(\omega
R)^{2})$ Eby:2018ufi ; Mukaida:2016hwd . One should note that the decay rate
is sensitive to the exact radial profile, since a small change in the radial
profile may result in completely different functional form for its Fourier
component, which is essential for the estimation of the classical decay rate.
In this paper we do not go into details of this subject and just assume that
it is exponentially suppressed for $\omega R\gg 1$.  In this limit, the
graviton production is an inevitable decay process and it gives strict lower
bound on the lifetime of oscillon.

The graviton production rate is given by the same formula (73). In order to
discuss the evolution of the oscillon, we need to know the relation between
$\phi_{R}$ and $R$. It is nontrivial and highly model-dependent. Nevertheless,
we can make an orders-of-magnitude estimate for the typical time scale at
which the gravitational process becomes important. Independently of the
relation between $\phi_{R}$ and $R$, we can repeat a similar procedure to the
case of boson star, and find a typical time scale $\tau_{\rm grav}$, the same
as (79), although the time dependence of the boson star mass $M(t)$ at
$t>\tau_{\rm grav}$ may differ from the boson star case. Thus Eq. (79) gives a
reasonable estimate for the oscillon lifetime for $\omega R\gg 1$ when the
classical decay processes are exponentially suppressed.131313 For the so-
called exact I-ball/oscillon Kawasaki:2015vga , there are no classical decay
processes. Although fluctuations with particular modes may grow in analogy
with the parametric resonance, still the final state is considered to be
stable Ibe:2019lzv . In such a case, the graviton production is the only
energy loss channel.

As we discussed in Sec. 2.4, the quantum decay of Q-ball into graviton is
forbidden by the conserved global charge for the case with the exactly
spherical orbit in the complex plane. If one adds higher-dimensional
U(1)-breaking terms, the orbit in the complex plane becomes elliptical and the
quantum decay process to the graviton becomes open. The amplitude of scalar
field decreases as it decays, which makes decay slower. The resulting time-
dependence for the graviton emission rate is different from the case with
boson stars and provides an unique signals in gravitational wave
spectrum.141414 The presence of U(1) breaking term itself tends to make
Q-balls unstable. For the gravity mediation type Q-ball, for example, it has
been shown that Q-balls decay within a simulation time if the magnitude of the
U(1) breaking term is sizable Kawasaki:2005xc . However, still it is difficult
to estimate the instability time scale for general (or much smaller) value of
the U(1) breaking term. See also Refs. Hiramatsu:2010dx ; Hasegawa:2019bbo
for related works. Moreover, dissipation via scatterings with thermal plasma
makes the orbit in the complex plane being elliptical Co:2019wyp ; Co:2019jts
. This depends on the size of Q-balls and temperature of the ambient plasma.
Below we make an assumption that the elliptical orbit becomes circular only
through the graviton emission. This gives a strict lower limit on the lifetime
of the “elliptical” Q-ball, while it gives an upper bound on the graviton
abundance.

Below we estimate the effect of graviton production on the Q-ball. Let us
suppose that the orbit of the complex scalar field $\phi$ is given by
$\phi(t)=\varphi_{R}\cos(\omega t)+i\varphi_{I}\sin(\omega t)$ inside the
Q-ball with $\varphi_{R}$ and $\varphi_{I}$ being the amplitude of the real
and imaginary component.151515 The following estimate formally applies to the
case of I-ball/oscillon by taking $\varphi_{R}\gg\varphi_{I}\to 0$.  Without
loss of generality, we take $\varphi_{R}>\varphi_{I}$ initially. From Eq.
(39), the gravitational potential $\Phi(t,r)$ is given by

$\displaystyle\Phi(t,r)\simeq\Phi_{0}(r)-\frac{\varphi_{R}^{2}(t,r)-\varphi_{I}^{2}(t,r)}{16M_{\rm
Pl}^{2}}\cos(2\omega t).$ (80)

Repeating the same discussion for the case of real scalar, we obtain the
graviton production rate due to the oscillating gravitational potential as

$\displaystyle\Gamma_{\rm
grav}=\frac{\mathcal{C}}{256\pi}\frac{(\varphi_{R}^{2}-\varphi_{I}^{2})\,\omega^{3}}{M_{\rm
Pl}^{4}}.$ (81)

It is expected that $\varphi_{R}^{2}-\varphi_{I}^{2}$ decreases through the
graviton production and this process stops when $\varphi_{R}=\varphi_{I}$. On
the other hand, the Q-ball charge $Q$ should be conserved as far as there is
no explicit U(1) breaking term. It is roughly estimated as
$Q\sim\omega\varphi_{R}\varphi_{I}\times
R^{3}\sim\varphi_{R}\varphi_{I}/(\omega^{2}K^{3})$ where we assumed that the
Q-ball radius is given by $R\sim 1/(\omega K)$ with some numerical constant
$K$. Thus the final value will be
$\varphi_{R}=\varphi_{I}\sim\omega\sqrt{K^{3}Q}\equiv\bar{\varphi}$. Keeping
this in mind, one can solve (76) in the case of Q-ball.

For $\varphi_{R}\gg\varphi_{I}$, the situation is similar to the case of real
scalar. We obtain

$\displaystyle\varphi_{R}^{2}(t)\simeq\frac{\varphi_{R,i}^{2}}{1+\Gamma^{0}_{\rm
grav}t},~{}~{}~{}~{}~{}~{}M(t)\simeq\frac{M_{i}}{1+\Gamma^{0}_{\rm
grav}t},~{}~{}~{}~{}~{}~{}\Gamma^{0}_{\rm
grav}=\frac{\mathcal{C}}{256\pi}\frac{\varphi_{R,i}^{2}\,\omega^{3}}{M_{\rm
Pl}^{4}}.$ (82)

Note that we have assumed that the Q-ball mass is given by
$M\simeq\omega^{2}(\varphi_{R}^{2}+\varphi_{I}^{2})\times
R^{3}\sim(\varphi_{R}^{2}+\varphi_{I}^{2})/(\omega K^{3})$. The amplitude
$\varphi_{R}$ decreases according to this result. Eventually $\varphi_{R}$
becomes close to $\varphi_{I}$ at $t=\bar{t}$ and the above approximation
breaks down. After this epoch, by defining
$\varphi_{R}=\bar{\varphi}+\delta\varphi$ and
$\varphi_{I}=\bar{\varphi}-\delta\varphi$ and noting that
$M\sim(\bar{\varphi}^{2}+\delta\varphi^{2})/(\omega K^{3})$, we obtain

$\displaystyle\delta\varphi(t)=\delta\varphi(\bar{t})-2\overline{\Gamma}_{\rm
grav}\bar{\varphi}(t-\bar{t}),~{}~{}~{}~{}~{}~{}M(t)\simeq
M(\bar{t}),~{}~{}~{}~{}~{}~{}\overline{\Gamma}_{\rm
grav}=\frac{\mathcal{C}}{256\pi}\frac{\bar{\varphi}^{2}\,\omega^{3}}{M_{\rm
Pl}^{4}}\sim\bar{t}^{-1}.$ (83)

Note that $\overline{\Gamma}_{\rm grav}$ is defined by $\Gamma_{\rm
grav}\simeq\overline{\Gamma}_{\rm grav}4\delta\varphi/\bar{\varphi}$, so the
graviton production vanishes for $\delta\varphi\to 0$. Since
$\delta\varphi(\bar{t})\sim\bar{\varphi}$, we can easily see that
$\delta\varphi(t)$ becomes zero after the time $t\sim\overline{\Gamma}_{\rm
grav}^{-1}$ and the graviton production stops thereafter. The above results
are approximately combined in a convenient way as

$\displaystyle M(t)\simeq M_{i}\frac{1+\overline{\Gamma}_{\rm
grav}t}{1+\Gamma_{\rm grav}^{0}t},~{}~{}~{}~{}~{}~{}\Gamma_{\rm
grav}(t)\simeq\Gamma_{\rm grav}^{0}\frac{1-\overline{\Gamma}_{\rm
grav}t}{1+\Gamma_{\rm grav}^{0}t},$ (84)

for $t<\overline{\Gamma}_{\rm grav}^{-1}$.

###  5.3 Graviton background

One of the immediate consequences of the gravitational soliton decay is the
formation of cosmic graviton background with a characteristic frequency
spectrum.161616Gravitational waves from the density perturbation during the
formation of solitons such as oscillons Zhou:2013tsa ; Antusch:2016con ;
Liu:2017hua ; Lozanov:2017hjm ; Amin:2018xfe ; Kitajima:2018zco ; Liu:2018rrt
; Lozanov:2019ylm ; Hiramatsu:2020obh ; Kou:2021bij ; Garcia:2023eol  and
Q-balls Kusenko:2008zm ; Kusenko:2009cv ; Chiba:2009zu  have been studied. Our
graviton production mechanism is completely different from them. Even if they
do not completely decay within the age of the universe due to the relatively
slow decrease of the soliton mass $M$, the most fraction of their energy goes
into the graviton radiation within the time scale of $\Gamma_{\rm grav}$. The
present energy density of the graviton per logarithmic energy is calculated as

$\displaystyle\frac{d\rho_{\rm GW}}{d\ln E}$ $\displaystyle=E^{2}\int
dz\frac{N(z)n_{\rm sol}(z)a^{3}(z)\Gamma_{\rm
grav}(z)}{H(z)}\frac{dN_{h}}{dE^{\prime}}$ (85)
$\displaystyle=\frac{2E^{4}}{m^{4}}\frac{\Gamma_{\rm grav}(z_{E})\rho_{\rm
sol}(z_{E})}{H(z_{E})}\,,$ (86)

where $z$ denotes the redshift, $N(z)=M(z)/m$ is the number of $\phi$ in the
soliton, $n_{\rm sol}(z)$ is the soliton number density, $a(z)=(1+z)^{-1}$ is
the cosmic scale factor, $H(z)$ is the Hubble constant, $E^{\prime}=(1+z)E$
and we approximate the produced graviton spectrum as a line:
$dN_{h}/dE^{\prime}=2\delta(E^{\prime}-m)$. In the second line we have
performed the integral and $z_{E}=m/E-1$ and defined the soliton energy
density $\rho_{\rm sol}(z)=M(z)n_{\rm sol}(z)$.

Fig. 1 shows the graviton spectrum in the present universe from the decay of
boson stars, assuming that the boson stars formed in the early universe lose
their energy only through the graviton production.171717 It may be true for
boson stars without self-interactions or for the exact I-balls/oscillons
Kawasaki:2015vga .  The spectrum is represented in terms of $\Omega_{\rm
GW}\equiv\frac{d\rho_{\rm GW}}{d\ln E}/\rho_{\rm crit}$ with $\rho_{\rm crit}$
being the present critical density of the universe, normalized by the soliton
energy density parameter $\Omega_{\rm sol}$. The soliton energy density
parameter is defined by $\Omega_{\rm sol}\equiv\rho_{\rm sol}/\rho_{\rm crit}$
with neglecting the effect of decay. In the left panel, the five lines show
the case of $m=100,10,1,10^{-1},10^{-2}\,{\rm GeV}$ with $\phi_{R,i}=M_{\rm
Pl}$. Looking at the line of $m=100\,{\rm GeV}$, for example, the low
frequency part corresponds to the gravitons produced at an early epoch when
$t\Gamma_{\rm grav}^{0}<1$, which exhibits the scaling $\Omega_{\rm GW}\propto
E^{3}$. The middle part corresponds to those produced when $t\Gamma_{\rm
grav}^{0}>1$. Since the decay is not exponential but power-law as shown in
Sec. 5.1, we obtain a spectrum with a power-law exponent different from the
low-frequency part. It behaves as $\Omega_{\rm GW}\propto E^{5/8}$ in the
matter-dominated universe and $\Omega_{\rm GW}\propto E^{1/2}$ in the
radiation-dominated universe at the time of production. Finally there is a
cutoff at $E=m$ and a softening of the spectrum around the cutoff frequency is
caused by the effect of cosmological constant. These are unique features of
the gravitational wave spectrum from the solitons, which is distinguished
from, for example, a homogeneous scalar field decaying into two gravitons
Ema:2021fdz . The right panel shows the $\phi_{R,i}$ dependence on the
spectrum for $m=100\,{\rm GeV}$. As is clear from Eq. (79), the peak amplitude
is maximized around $\phi_{R,i}=10^{-4}M_{\rm Pl}$ since $\tau_{\rm grav}$
becomes close to the present age of the universe. For smaller $\phi_{R,i}$,
$\tau_{\rm grav}$ becomes even longer and the graviton production efficiency
is suppressed.

Fig. 2 shows the graviton spectrum in the present universe from the Q-ball
decay. Note that similar results also apply to the case of oscillon, if the
oscillon lifetime is long enough, as in the case of exact I-ball/oscillon. In
the left panel we have taken $m=10^{6},10^{8},10^{10}\,{\rm GeV}$ with
$\varphi_{R,i}=M_{\rm Pl}$ and $\bar{\varphi}=0.1M_{\rm Pl}$, i.e.,
$r\equiv\varphi_{R,i}/\bar{\varphi}=10$. In the right panel we have taken
$m=10^{6}\,{\rm GeV}$ and $r=1.5,10,50$ with $\varphi_{R,i}=M_{\rm Pl}$ fixed.
In this case, the graviton production stops in the early universe when the
orbit of the complex scalar becomes circular and the cutoff frequency roughly
corresponds to the gravitons produced at this epoch. Looking at the line of
$r=50$ in the right panel, for example, the low frequency part corresponds to
those produced at $t<(\Gamma_{\rm grav}^{0})^{-1}$ that exhibits the scaling
$\Omega_{\rm GW}\propto E^{3}$. A small modulation is caused by the change of
relativistic degrees of freedom around the graviton emission epoch. The higher
frequency part corresponds to those produced at $(\Gamma_{\rm
grav}^{0})^{-1}<t<(\overline{\Gamma}_{\rm grav})^{-1}$ and the spectrum
behaves as $\Omega_{\rm GW}\propto E^{-1}$. This is the characteristic
signature of the gravitational wave spectrum from the Q-ball decay. The
transition energy between these two regimes is given by $E\sim 10^{-6}\,{\rm
GeV}\,(10^{10}\,{\rm GeV}/m)^{1/2}(M_{\rm Pl}/\varphi_{R,i})$. Finally there
is a cutoff around $E\sim 10^{-6}\,{\rm GeV}\,(10^{10}\,{\rm
GeV}/m)^{1/2}(M_{\rm Pl}/\bar{\varphi})$, corresponding to
$t=(\overline{\Gamma}_{\rm grav})^{-1}$, after which the orbit of the complex
scalar becomes completely circular and the graviton production stops. For
reference, this final epoch corresponds to the temperature $T\sim 10^{4}\,{\rm
GeV}\,(m/10^{10}\,{\rm GeV})^{3/2}(\bar{\varphi}/M_{\rm Pl})$ if the universe
is radiation-dominated.

In both cases, the predicted typical frequency of the stochastic gravitational
wave is very high. Recently there are many proposals to detect such high
frequency gravitational waves Ejlli:2019bqj ; Ringwald:2020ist ;
Aggarwal:2020olq ; Berlin:2021txa ; Domcke:2022rgu ; Tobar:2022pie ;
Dolgov:2012be ; Domcke:2020yzq ; Ramazanov:2023nxz ; Liu:2023mll ; Ito:2023fcr
, although it is still challenging to detect it.

---
Figure 1: (Left) The present graviton spectrum from the boson star decay for
$\phi_{R,i}=M_{\rm Pl}$ with $m=100,10,1,10^{-1},10^{-2}\,{\rm GeV}$. (Right)
The spectrum for $m=100\,{\rm GeV}$ with
$\phi_{R,i}=(1,10^{-2},10^{-4},10^{-6})\times M_{\rm Pl}$. We are assuming
that the boson star is long-lived and the dominant part comes from the boson
star decaying at the present universe.
---
Figure 2: The present stochastic gravitational wave spectrum from the Q-ball
decay. (Left) We have taken $m=10^{6},10^{8},10^{10}\,{\rm GeV}$ with
$\varphi_{R,i}=M_{\rm Pl}$ and $\bar{\varphi}=0.1M_{\rm Pl}$, i.e.,
$r\equiv\varphi_{R,i}/\bar{\varphi}=10$. (Right) We have taken $m=10^{6}\,{\rm
GeV}$ and $r=1.5,10,50$ with $\varphi_{R,i}=M_{\rm Pl}$ fixed. The graviton
production stops in the early universe when the orbit of the complex scalar
becomes circular and the cutoff frequency roughly corresponds to the gravitons
produced at this epoch.

##  6 Discussion and conclusions

It is known that the cosmological dynamics of scalar or vector fields can lead
to the formation of soliton-like objects such as an oscillon, Q-ball or boson
stars. Oscillons/Q-balls are stabilized by self-interactions and boson stars
are stabilized by gravitational interactions. They may have significant
impacts on the early universe history, although still complete understandings
of their dynamics have not been reached yet. One of the important properties
of the soliton is its lifetime. There have been many studies for estimating
the soliton lifetime, most of which focused on the emission of relativistic
particles due to self-interactions.

In this paper we point out that there is a universal decay process of the
solitons due to the gravitational interactions, i.e., quantum decay of a
soliton into the graviton pair. This decay channel exists even for a free
(scalar or vector) field with a minimal coupling to gravity and provides a
strict upper limit on their lifetime. Although the solitons are spherically
symmetric, its gravitational potential has an oscillating component in time,
which leads to the production of gravitational waves (gravitons) via quantum
decay process. The quantum decay rate into gravitons has a power-law
dependence on the size of solitons, which should be compared with the
exponentially-suppressed classical decay rate. Therefore, it can be the
dominant channel for boson stars consisting of fuzzy dark matter or vector
dark matter without a kinetic mixing. We have discussed the evolution of
solitons via the quantum decay and calculated the spectrum of stochastic
gravitational waves. Although the amplitude of gravitational waves is too
small to be detected in the near future, the spectrum has an unique shape with
a broken power law at a low frequency and a sharp cutoff at a high frequency.

## Acknowledgments

This work was supported by MEXT Leading Initiative for Excellent Young
Researchers (M.Y.), and by JSPS KAKENHI Grant No. 20H01894 (F.T.), 20H05851
(F.T. and M.Y.), and 23K13092 (M.Y.), and JSPS Core-to-Core Program (grant
number: JPJSCCA20200002) (F.T.) and World Premier International Research
Center Initiative (WPI), MEXT, Japan. This article is based upon work from
COST Action COSMIC WISPers CA21106, supported by COST (European Cooperation in
Science and Technology).

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