††institutetext: Department of Physics and Astronomy, Rice University,
Houston, TX 77005, USA

# Gravitational effects on oscillon lifetimes

Hong-Yi Zhang<EMAIL_ADDRESS>
###### Abstract

Many scalar field theories with attractive self-interactions support
exceptionally long-lived, spatially localized and time-periodic field
configurations called oscillons. A detailed study of their longevity is
important for understanding their applications in cosmology. In this paper, we
study gravitational effects on the decay rate and lifetime of dense oscillons,
where self-interactions are more or at least equally important compared with
gravitational interactions. As examples, we consider the $\alpha$-attractor
T-model of inflation and the axion monodromy model, where the potentials
become flatter than quadratic at large field values beyond some characteristic
field distance $F$ from the minimum. For oscillons with field amplitudes of
$\mathcal{O}(F)$ and for $F\ll 0.1M_{\mathrm{pl}}$, we find that their
evolution is almost identical to cases where gravity is ignored. For $F\sim
0.1M_{\mathrm{pl}}$, however, including gravitational interactions reduces the
lifetime slightly.

##  1 Introduction

Oscillons are exceptionally long-lived, spatially-localized and time-periodic
field configurations that exist in real-valued scalar field theories with
attractive self-interactions osti_4051808 ; Bogolyubsky:1976yu ;
Copeland:1995fq ; Amin:2010jq ; Amin:2013ika . The natural emergence of
oscillons from general initial conditions Seidel:1993zk ; Farhi:2007wj ;
Amin:2010xe ; Levkov:2018kau  makes them relevant to various cosmological
contexts, e.g. reheating after inflation Amin:2010dc ; Amin:2011hj ;
Lozanov:2014zfa ; Lozanov:2016hid ; Lozanov:2017hjm ; Kou:2019bbc , scalar
dark matter Olle:2019kbo ; Amin:2019ums ; Arvanitaki:2019rax ;
Kawasaki:2020jnw , phase transitions in the early universe Dymnikova:2000dy ;
Gleiser:2010qt ; Bond:2015zfa , gravitational wave production Zhou:2013tsa ;
Helfer:2018vtq ; Liu:2017hua ; Amin:2018xfe ; Lozanov:2019ylm ;
Dietrich:2018jov , electromagnetic bursts Dietrich:2018jov ; Hook:2018iia ;
Clough:2018exo ; Levkov:2020txo ; Prabhu:2020yif ; Amin:2020vja , the 21cm
forest Kawasaki:2020tbo , and even formation of black holes Helfer:2016ljl ;
Muia:2019coe ; Nazari:2020fmk ; Widdicombe:2019woy  and primordial black holes
Khlopov:1985jw ; Cotner:2018vug ; Cotner:2019ykd . For a quantitative
understanding of their relevance to cosmology, a detailed study of oscillon
decay rates and lifetimes is important.

In the absence of gravity, the decay rate of small-amplitude oscillons
Segur:1987mg ; Fodor:2008du ; Hertzberg:2010yz  and large-amplitude ones in
some polynomial models Mukaida:2016hwd ; Ibe:2019vyo  has been characterized
in detail. In a recent paper Zhang:2020bec , we calculated the decay rate of
oscillons without restricting them to small amplitudes or to polynomial
potentials. By investigating the relation between decay rates and fundamental
frequencies of oscillons in a given theory, see the right panel of figure 1,
we found that two major features were responsible for their particularly long
lifetimes: (1) Some exceptionally stable intermediate configurations exist,
e.g. a dip structure of the blue curve around $\omega_{\mathrm{dip}}\approx
0.82m$. For such configurations, the leading decay channel into scalar
radiation vanishes. (2) The decay rate is dramatically suppressed just before
their final collapse at the critical frequency $\omega_{\mathrm{crit}}\approx
0.98m$, such as the orange curve.

Figure 1: We will consider oscillons in two flattened potentials in this paper
(left panel), and semi-analytical calculation of their decay rate is presented
for Minkowski spacetime (right panel). The dashed lines indicate the location
of $\omega_{\mathrm{crit}}$, at which oscillons finally decay away due to
instabilities to small fluctuations.

The gravitational effects on oscillon decay rates are expected to depend on
the relative magnitude between gravitational and self- interactions. In 3+1
dimensions, the lifetime of small-amplitude _dilute_ oscillons (whose self-
interactions are negligible) was shown to exceed the present age of the
universe Grandclement:2011wz ; Eby:2015hyx ; Eby:2020ply .111The stability of
dilute oscillons is ensured by gravitational attraction, for example,
oscillatons Seidel:1991zh ; UrenaLopez:2002gx ; Alcubierre:2003sx  and dilute
axion stars Visinelli:2017ooc ; Eby:2019ntd . Their size can be cosmological
scales, e.g. $\sim\mathrm{kpc}$ for fuzzy dark matter Hui:2016ltb . What was
not clear to us is whether the existence of gravity stablizes large-amplitude
_dense_ ones, whose self-interactions are more or at least equally important.
In this paper, we will study two well-motivated examples to explore this
impact, the $\alpha$-attractor T-model of inflation Kallosh:2013hoa ;
Lozanov:2017hjm  and the axion monodromy model Silverstein:2008sg ;
Amin:2011hj ; McAllister:2014mpa

$\displaystyle V(\phi)=\frac{m^{2}F^{2}}{2}\tanh^{2}\frac{\phi}{F}~{},\quad
V(\phi)=m^{2}F^{2}\left(\sqrt{1+\frac{\phi^{2}}{F^{2}}}-1\right)~{},$ (1)

where $F$ is the amplitude scale that indicates a significant deviation from a
quadratic minimum, see the left panel of figure 1.

Throughout the paper we will assume classical field theory and spherical
symmetry. The first assumption is a standard procedure to deal with compact
objects like dense oscillons due to their particularly large occupation number
Guth:2014hsa . We also stick to the second one because it simplifies the
problem quite a bit and non-spherical components usually decay rapidly with
little disturbance on oscillon evolution Seidel:1993zk ; Hindmarsh:2006ur .
Even if the early inhomogeneities forming oscilllons possess angular momentum,
a non-axisymmetric instability seems to develop and ejects all the angular
momentum from the scalar star Sanchis-Gual:2019ljs .

The rest of this paper is organized as follows. In section 2, we briefly
review the method developed in Zhang:2020bec  to calculate oscillon decay
rates and lifetimes in Minkowski spacetime. In section 3, we introduce a
Lagrangian mechanism and define the mass and energy of oscillons in curved
spacetime. In section 4, we generalize the method to calculate oscillon decay
rates and lifetimes in the non-relativistic limit and weak-field limit of
gravity. Then we discuss the gravitational effects on oscillon lifetimes in
section 5 and make conclusions in section 6. In appendix A and B, we provide
analytical expressions for cosine series of the scalar potential and describe
numerical algorithms for full GR simulations respectively. We will adopt the
natural units $c=\hbar=1$, and frequently use the reduced Planck mass defined
by $M_{\mathrm{pl}}\equiv(8\pi G)^{-1/2}\sim 10^{18}\mathrm{GeV}$.

##  2 Oscillons in Minkowski spacetime

We study oscillon dynamics in Minkowski spacetime in this section. In section
2.1, we briefly review a model-independent calculation of oscillon decay rates
and lifetimes following our earlier work Zhang:2020bec . Then in section 2.2,
we derive a virial theorem and introduce some small parameters that will allow
us to simplify equations when gravitational effects are included.

###  2.1 Profiles, decay rates and lifetimes

Let us begin with a real-valued scalar field with the Lagrangian given by

$\displaystyle\mathcal{L}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^{2}\phi^{2}-V_{\mathrm{nl}}(\phi)~{},$
(2)

where $g_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$ is the Minkowski metric and
$V_{\mathrm{nl}}(\phi)$ is the nonlinear part of the potential $V(\phi)$. The
equation of motion is the Klein-Gordon equation

$\displaystyle\left[-\partial_{t}^{2}+\nabla^{2}-m^{2}\right]\phi-
V_{\mathrm{nl}}^{\prime}(\phi)=0~{},$ (3)

where $\nabla^{2}\equiv\partial_{r}^{2}+(2/r)\partial_{r}$. For simplicity we
will only consider symmetric potentials, but the method developed in this
paper should also be applicable to asymmetric ones. As suggested in
Seidel:1991zh , we approximate oscillons by a cosine series

$\displaystyle\phi(t,r)=\phi_{\mathrm{osc}}(t,r)+\xi(t,r)=\phi_{1}(r)\cos(\omega
t)+\sum_{j=3}^{\infty}\xi_{j}(r)\cos(j\omega t)~{},$ (4)

where $j$ is odd, $\phi_{\mathrm{osc}}$ is a single-frequency profile and
$\xi(t,r)$ includes all the radiating modes.222The $\xi_{j}(r)\cos(j\omega t)$
mode is a radiating mode if $j\omega>m$. Notice that this expansion is
actually a balance between ingoing and outgoing waves, and we must manually
ignore the ingoing contributions in the end. Typically
$|\xi|\ll|\phi_{\mathrm{osc}}|$ inside oscillons.333The assumption of a
single-frequency profile becomes invalid when $\omega\ll m$, however we never
consider the $\omega\ll m$ limit because: (1) The particles that make up the
oscillons in this regime are relativistic as you will see in section 2.2. In
this case, a large number of particles can easily pop in and out of the
condensate and we are unlikely to have a stable long-lived condensate. (2) The
radiating modes, and hence decay rates, are typically too large for the
oscillon to maintain a stable configuration and (3) The size of the oscillon
approaches the Schwarzschild radius and the nonlinearity of gravity becomes
important, which is beyond the scope of this paper. As a result, the potential
and its derivatives can also be written in terms of the Fourier cosine series

$\displaystyle U$ $\displaystyle\equiv
V_{\mathrm{nl}}(\phi_{\mathrm{osc}})=\frac{1}{2}U_{0}(r)+\sum_{j=2}^{\infty}U_{j}(r)\cos(j\omega
t)\quad\text{and}\quad
U_{j}=\frac{\omega}{\pi}\int_{-\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}V_{\mathrm{nl}}(\phi_{\mathrm{osc}})\cos(j\omega
t)dt~{},$ (5) $\displaystyle M$ $\displaystyle\equiv
V_{\mathrm{nl}}^{\prime\prime}(\phi_{\mathrm{osc}})=\frac{1}{2}M_{0}(r)+\sum_{j=2}^{\infty}M_{j}(r)\cos(j\omega
t)\quad\text{and}\quad
M_{j}=\frac{\omega}{\pi}\int_{\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}V_{\mathrm{nl}}^{\prime\prime}(\phi_{\mathrm{osc}})\cos(j\omega
t)dt~{}.$ (6)

where $j$ is even, and

$\displaystyle J\equiv
V_{\mathrm{nl}}^{\prime}(\phi_{\mathrm{osc}})=\sum_{j=1}^{\infty}J_{j}(r)\cos(j\omega
t)$ $\displaystyle\quad\text{and}\quad
J_{j}=\frac{\omega}{\pi}\int_{-\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}V_{\mathrm{nl}}^{\prime}(\phi_{\mathrm{osc}})\cos(j\omega
t)dt~{},$ (7)

where $j$ is odd. The $U_{j}$ is a functional of $\phi_{1}$ hence a function
of $r$, namely $U_{j}(\phi_{1})\equiv U_{j}(r)$. We will mix the notation
$U_{j}(\phi_{1})$ and $U_{j}(r)$, and similarly for $J_{j}$ and $M_{j}$. For
polynomial potentials, it is possible to find analytical expressions for
$U_{j},M_{j}$ and $J_{j}$. These are provided in appendix A.

Plugging the single-frequency profile $\phi_{\mathrm{osc}}$ into the Kelin-
Gordon equation and collecting the coefficient of $\cos(\omega t)$, we obtain
the radial profile equation

$\displaystyle(\nabla^{2}+\kappa_{1}^{2})\phi_{1}(r)=J_{1}(r)~{},$ (8)

where $\kappa_{j}^{2}\equiv(j\omega)^{2}-m^{2}$ is the square of the momentum
for a particular mode and we have used
$J_{1}(\phi_{1})=U_{0}^{\prime}(\phi_{1})$. Oscillon profiles can be found by
using the numerical shooting method and by demanding a localized, smooth and
no-node solution for each $\omega$. Without loss of generality, we assume that
the minimum of $V(\phi)$ is located at $\phi=0$ thus the boundary condition is
$\phi_{1}(\infty)=0$. Once the profile is found, the energy of oscillons can
be obtained by time averaging the energy density over a period, and is given
by

$\displaystyle
E_{\mathrm{osc}}=\int_{0}^{\infty}\left[\frac{1}{4}\left(\partial_{r}\phi_{1}\right)^{2}+\frac{1}{4}(\omega^{2}+m^{2})\phi_{1}^{2}+\frac{1}{2}U_{0}\right]4\pi
r^{2}dr~{}.$ (9)

For these single frequency objects, we can define the particle number
Mukaida:2016hwd

$\displaystyle
N_{\mathrm{osc}}=\frac{\omega}{2}\int_{0}^{\infty}\phi_{1}^{2}~{}4\pi
r^{2}dr~{}.$ (10)

The stability condition of oscillons against small perturbations is given by
Friedberg:1976me

$\displaystyle\frac{dN_{\mathrm{osc}}}{d\omega}<0\quad\text{or}\quad\frac{dE_{\mathrm{osc}}}{d\omega}<0~{},$
(11)

and the critical frequency $\omega_{\mathrm{crit}}$ can be obtained by setting
$dE_{\mathrm{osc}}/d\omega=0$.

Radial equations for radiating modes can be obtained by plugging the full
expansion (4) into the Klein-Gordon equation and collecting the coefficient of
$\cos(j\omega t)$, i.e.

$\displaystyle\left[\nabla^{2}+\kappa_{j}^{2}\right]\xi_{j}(r)=S_{j}(r)\equiv
J_{j}+\frac{1}{2}\sum_{k=3}^{\infty}\xi_{k}\left(M_{|k+j|}+M_{|k-j|}\right)~{},$
(12)

where $j$ and $k$ are both odd, $S_{j}$ is the effective source and $M_{j}$ is
the effective mass. For $\kappa_{j}^{2}>0$, the central amplitude of a
radiating mode is

$\displaystyle\xi_{j}(0)=-\int_{0}^{\infty}dr^{\prime}~{}S_{j}(r^{\prime})~{}r^{\prime}\cos(\kappa_{j}r^{\prime})~{}.$
(13)

This is not in closed form, and we can find the solution of $\xi_{j}(r)$ by
using the _iterative method_ developed in Zhang:2020bec .444We will no longer
call this method “shooting” as we did in Zhang:2020bec , because technically
we are not solving a boundary value problem. The energy loss rate of oscillons
is determined by radiation at large radius

$\displaystyle\frac{dE_{\mathrm{osc}}}{dt}=-4\pi
r^{2}\tensor{T}{{}^{1}_{0}}|_{r\rightarrow\infty}=\left.-4\pi
r^{2}\partial_{t}\xi\partial_{r}\xi\right|_{r\rightarrow\infty}~{}$

where
$\tensor{T}{{}^{\mu}_{\nu}}=\partial^{\mu}\phi\partial_{\nu}\phi+\delta^{\mu}_{\nu}\mathcal{L}$
is the energy-momentum tensor and the radiation is given by

$\displaystyle\left.\xi(t,r)\right|_{r\rightarrow\infty}=-\frac{1}{4\pi
r}\sum_{j=3}^{\infty}\tilde{S}_{j}(\kappa_{j})\cos(\kappa_{j}r-j\omega t)~{}.$
(14)

Here $\tilde{S}_{j}(p)$ is the Fourier transform of the effective source

$\displaystyle\tilde{S}_{j}(p)=\int_{0}^{\infty}dr~{}4\pi
r^{2}\operatorname{sinc}(pr)S(r)~{}.$ (15)

The (absolute value of) decay rate is defined

$\displaystyle\Gamma\equiv\left|\langle\dot{E}_{\mathrm{osc}}/E_{\mathrm{osc}}\rangle_{T}\right|=\frac{1}{8\pi
E_{\mathrm{osc}}}\sum_{j=3}^{\infty}\left[\tilde{S}_{j}(\kappa_{j})\right]^{2}j\omega~{}\kappa_{j}\equiv\sum_{j=3}^{\infty}\Gamma_{j}~{},$
(16)

where $\Gamma_{j}$ is the contribution due to $S_{j}$. Typically the radiating
mode safisfies $|\xi_{j}|\gg|\xi_{j+2}|$, which means only finite terms are
needed in the radial equation (12). However, the leading channel of decay
rates $\Gamma_{3}$ might vanish for some $\omega$, causing a dip structure
seen in figure 1.

If we start with an oscillon with $\omega<\omega_{\rm dip}$, the frequency of
the oscillons evolves to larger values by slowly emitting scalar radiation
(and the profile changes correspondingly). Since the leading decay channel
$\Gamma_{3}$ is vanishing in the dip, the oscillon configuration with
$\omega\approx\omega_{\rm dip}$ is expected to have a long lifetime. When
considering the total lifetime of oscillons that start out at
$\omega<\omega_{\rm dip}$, the oscillon will spend most of its lifetime in
such a dip. Generally speaking, we will use $\Gamma_{3}+\Gamma_{5}$ to
estimate the lifetime of oscillons, which is just the area enclosed by the
evolution curve in $dt/dE_{\mathrm{osc}}$ versus $E_{\mathrm{osc}}$ plot,
where $dt/dE_{\mathrm{osc}}=1/(\Gamma E_{\mathrm{osc}})$. For future
reference, if we keep only $\xi_{3}$ and $\xi_{5}$, the effective source
becomes

$\displaystyle S_{3}$
$\displaystyle=J_{3}+\frac{1}{2}\xi_{3}(M_{0}+M_{6})+\frac{1}{2}\xi_{5}(M_{2}+M_{8})~{},$
(17) $\displaystyle S_{5}$
$\displaystyle=J_{5}+\frac{1}{2}\xi_{3}(M_{2}+M_{8})+\frac{1}{2}\xi_{5}(M_{0}+M_{10})~{}.$
(18)

The generalization to including more $\xi_{j}$ terms is straightforward. Note
that the effective source $S_{j}$ receives a contribution from the oscillon
background $J_{j}$ as well as corrections due to radiation $\xi_{j}$.

###  2.2 A virial theorem and small parameters

Assume that oscillons are single-frequency objects like Q-balls and Boson
stars Lee:1991ax , then one way to derive a virial theorem is to use the
variational principle. The Legendre transformation

$\displaystyle F_{\mathrm{osc}}=\omega N_{\mathrm{osc}}-E_{\mathrm{osc}}$ (19)

defines a functional of $\phi_{1}$ and a function of $\omega$ (one may
recognize that $F_{\mathrm{osc}}$ is just the Lagrangian). The variation of
$F_{\mathrm{osc}}$ in terms of $\phi_{1}$ by keeping $\omega$ fixed gives the
profile equation of oscillons (8). A virial theorem can be obtained by
considering the variation $\phi_{1}(r)\rightarrow\phi_{1}(\lambda r)$ for an
oscillons solution. By setting $\left(\partial
F_{\mathrm{osc}}/\partial\lambda\right)_{\omega}=0$ at $\lambda=1$, we find

$\displaystyle E_{\mathrm{S}}/3+E_{\mathrm{V}}=E_{\mathrm{K}}~{},$ (20)

where the surface energy, potential energy and kinetic energy are defined

$\displaystyle
E_{\mathrm{S}}=\int\frac{1}{4}(\partial_{r}\phi_{1})^{2}d^{3}r~{},\quad
E_{\mathrm{V}}=\int\left(\frac{1}{4}m^{2}\phi_{1}^{2}+\frac{1}{2}U_{0}\right)d^{3}r~{},\quad
E_{\mathrm{K}}=\int\frac{1}{4}\omega^{2}\phi_{1}^{2}~{}d^{3}r~{}.$ (21)

For oscillons with $\omega\lesssim m$, we can identify three small parameters
immediately

$\displaystyle\epsilon_{r}\equiv
1-\omega^{2}/m^{2}~{},\quad\epsilon_{V}\sim\frac{U_{0}}{m^{2}\phi_{1}^{2}/2}~{},\quad\epsilon_{\xi}\sim\frac{\xi_{j}}{\phi_{1}}~{}.$
(22)

The parameter $\epsilon_{r}$ is a measure of how relativistic the particles
inside the oscillon are. Equation (8) implies that at large radius
$\phi_{1}(r)\propto r^{-1}\exp\left[-(m^{2}-\omega^{2})^{1/2}r\right]$, hence
a typical spatial derivative brings a factor $\sqrt{\epsilon_{r}}m$, that is,
$\partial_{r}\phi_{1}\sim-\sqrt{\epsilon_{r}}m\phi_{1}$, and

$\displaystyle\nabla^{2}\phi_{1}\ll\omega^{2}\phi_{1}~{}.$ (23)

This means that the particles that make up the oscillon are _non-
relativistic_. And from equations (12) and (20), we see
$\epsilon_{\xi}\sim-\epsilon_{V}$ and $\epsilon_{V}\sim-\epsilon_{r}$.

For oscillons with $\omega\ll m$, we may not regard surface energy as a small
quantity anymore. Take the tanh potential in (1) for example and assume a
Gaussian profile

$\displaystyle\phi_{1}(r)=C~{}e^{-r^{2}/R^{2}}~{},$ (24)

where $C\gg F$. Then each energy component becomes

$\displaystyle E_{\mathrm{S}}\sim C^{2}R~{},\quad E_{\mathrm{K}}\sim
C^{2}R^{3}\omega^{2}\sim\frac{N_{\mathrm{osc}}^{2}}{C^{2}R^{3}}~{},\quad
E_{\mathrm{V}}\sim R_{\mathrm{V}}^{3}m^{2}F^{2}~{},$ (25)

where we have taken advantage of the flatness of the potential at large $\phi$
and $R_{\mathrm{V}}$ is the length scale satisfying
$\phi_{1}(R_{\mathrm{V}})\sim\mathcal{O}(1)$, i.e.

$\displaystyle R_{\mathrm{V}}\sim R\log^{1/2}\left(\frac{C}{F}\right)\sim
R~{}.$ (26)

By setting $\partial E_{\mathrm{osc}}/\partial R=\partial
E_{\mathrm{osc}}/\partial C=0$ and keeping $N_{\mathrm{osc}}$ fixed, we obtain

$\displaystyle R\sim\omega^{-1}~{},\quad C\sim\omega^{-1}~{}.$ (27)

We see that three components of energy now are all comparable, and thus
oscillon particles are relativistic. This phenomenon has been witnessed
numerically in the context of dense axion stars Visinelli:2017ooc ;
Eby:2019ntd .

##  3 Full GR formalism

So far our analysis does not include gravity. In this section, the scalar
field is assumed to be minimally coupled to gravity. We introduce a Lagrangian
mechanism that can convert the Hilbert action into one that contains only the
first derivative of the metric in section 3.1, then we use it to generalize
the definition of energy and mass into curved spacetime in section 3.2.

###  3.1 A Lagrangian mechanism

The simplest choice of metric to describe oscillons is the spherical
coordinates

$\displaystyle
ds^{2}=-e^{2\Phi(t,r)}\mathrm{d}t^{2}+e^{-2\Psi(t,r)}\mathrm{d}r^{2}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta~{}\mathrm{d}\varphi^{2})~{},$
(28)

where $\theta,\varphi$ are the polar and azimuthal angles, $r$ is
$(2\pi)^{-1}$ times the circumference of a two-sphere. The action of our
theory is composed of the action of gravity and matter
$S=S_{\mathrm{G}}+S_{\mathrm{M}}$, specifically

$\displaystyle S_{\mathrm{G}}$ $\displaystyle=\frac{1}{16\pi
G}\left(\int_{\Omega}R\sqrt{-g}~{}d^{4}x+\int_{\partial\Omega}K\sqrt{|h|}~{}d^{3}x\right)~{},$
(29) $\displaystyle S_{\mathrm{M}}$
$\displaystyle=\int\left[-\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-V(\phi)\right]\sqrt{-g}~{}d^{4}x~{},$
(30)

where $\phi_{,\mu}\equiv\partial_{\mu}\phi$ is defined for notation
convenience, $R$ is the Ricci scalar and $g$ is the determinant of
$g_{\mu\nu}$, i.e.

$\displaystyle R=$ $\displaystyle
2r^{-2}-2e^{-2\Phi}\left[\Psi_{,00}-\Psi_{,0}(\Phi+\Psi)_{,0}\right]$ (31)
$\displaystyle-2e^{2\Psi}\left[r^{-2}+2r^{-1}(\Phi+\Psi)_{,1}+\Phi_{,1}(\Phi+\Psi)_{,1}+\Phi_{,11}\right]~{},$
(32) $\displaystyle\sqrt{-g}~{}d^{4}x=$ $\displaystyle
e^{\Phi-\Psi}r^{2}\sin\theta~{}dt~{}dr~{}d\theta~{}d\varphi~{}.$ (33)

Apart from the standard Hilbert action, $K$ is a surface term and $h$ is the
induced metric on the boundary $\partial\Omega$ Gibbons:1976ue  that can be
appropriately chosen Lee:1988av , i.e.

$\displaystyle K$ $\displaystyle=2e^{\Psi}\Phi_{,1}+4r^{-1}(e^{\Psi}-1)~{},$
(34) $\displaystyle\sqrt{|h|}~{}d^{3}x$
$\displaystyle=e^{\Phi}r^{2}\sin\theta~{}dt~{}d\theta~{}d\varphi~{},$ (35)

for the three dimensional surface at $r=r_{0}$ and

$\displaystyle K$ $\displaystyle=2e^{-\Phi}\Psi_{,0}~{},$ (36)
$\displaystyle\sqrt{|h|}~{}d^{3}x$
$\displaystyle=e^{-\Psi}r^{2}\sin\theta~{}dr~{}d\theta~{}d\varphi~{},$ (37)

for that which is bounded by $t=\pm T$. The inclusion of surface terms will
not change the Einstein equations, but can convert the Hilbert action into one
that contains only the first derivative of the metric so that the usual
Lagrangian mechanics can be applied. After setting $r_{0}$ and
$T\rightarrow\infty$, the Lagrangian, i.e. $S=\int L~{}dt$, becomes

$\displaystyle L_{\mathrm{G}}$
$\displaystyle=(2G)^{-1}\int_{0}^{\infty}\left[e^{\Phi-\Psi}+e^{\Phi+\Psi}(1+2r\Phi_{,1})-2e^{\Phi}(1+r\Phi_{,1})\right]dr~{},$
(38) $\displaystyle L_{\mathrm{M}}$
$\displaystyle=\int_{0}^{\infty}(X-Y-V)~{}e^{\Phi-\Psi}4\pi r^{2}dr~{},$ (39)

where we have defined

$\displaystyle X\equiv\frac{1}{2}e^{-2\Phi}\phi_{,0}^{2}~{},\quad
Y\equiv\frac{1}{2}e^{2\Psi}\phi_{,1}^{2}~{}.$ (40)

The gravity and matter are related by the Einstein equation

$\displaystyle\tensor{G}{{}_{\mu}{}_{\nu}}=\tensor{R}{{}_{\mu}{}_{\nu}}-\frac{1}{2}g_{\mu\nu}R=8\pi
G~{}\tensor{T}{{}_{\mu}{}_{\nu}}~{},$ (41)

where $R_{\mu\nu}$ is the Ricci tensor and the energy-momentum tensor $\tensor
T{{}^{\mu}_{\nu}}$ is given by

$\displaystyle T_{\mu\nu}=-2\frac{1}{\sqrt{-g}}\frac{\delta
S_{\mathrm{M}}}{\delta
g^{\mu\nu}}=\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\phi_{,\rho}\phi_{,\sigma}-g_{\mu\nu}V(\phi)~{}.$
(42)

More specifically

$\displaystyle\tensor{G}{{}^{0}_{0}}$
$\displaystyle=r^{-2}\left(e^{2\Psi}-1+2r\Psi_{,1}e^{2\Psi}\right)=-8\pi
G(X+Y+V)~{},$ (43) $\displaystyle\tensor{G}{{}^{1}_{0}}$
$\displaystyle=-2r^{-1}\Psi_{,0}e^{2\Psi}=8\pi
G~{}e^{2\Psi}\phi_{,0}\phi_{,1}~{},$ (44)
$\displaystyle\tensor{G}{{}^{1}_{1}}$
$\displaystyle=r^{-2}\left(e^{2\Psi}-1+2r\Phi_{,1}e^{2\Psi}\right)=8\pi
G(X+Y-V)~{},$ (45) $\displaystyle\tensor{G}{{}^{2}_{2}}$
$\displaystyle=e^{-2\Phi}\left[\Psi_{,00}-\Psi_{,0}(\Phi+\Psi)_{,0}\right]+e^{2\Psi}\left[\Phi_{,11}+(\Phi_{,1}+r^{-1})(\Phi+\Psi)_{,1}\right]=8\pi
G(X-Y-V)~{},$ (46)

where $\tensor{G}{{}^{3}_{3}}=\tensor{G}{{}^{2}_{2}}$ and all other components
vanish. The $\tensor{G}{{}^{0}_{0}}$ and $\tensor{G}{{}^{1}_{1}}$ equations
can be alternatively obtained by varying the Lagrangian with respect to $\Phi$
and $\Psi$. The others can be derived using the contracted Bianchi identity
$\tensor{G}{{}^{\mu}_{\nu;\mu}}=0$, i.e. $\nu=0$ gives (44), $\nu=2$ gives
$\tensor{G}{{}^{2}_{2}}=\tensor{G}{{}^{3}_{3}}$ and $\nu=1$ gives (46). Some
combanitions will be useful, for example, equations (43) and (45) give

$\displaystyle r^{-1}e^{2\Psi}(\Phi-\Psi)_{,1}=8\pi G(X+Y)~{},$ (47)

and equations (46) and (47) give

$\displaystyle
e^{-2\Phi}\left[\Psi_{,00}-\Psi_{,0}(\Phi+\Psi)_{,0}\right]+e^{2\Psi}\left[\nabla^{2}\Phi+\Phi_{,1}(\Phi+\Psi)_{,1}\right]=8\pi
G(2X-V)~{},$ (48)

where $\nabla^{2}\equiv\partial_{r}^{2}+(2/r)\partial_{r}$. The equation of
motion of $\phi$ is

$\displaystyle
e^{-2\Phi}\left[-\phi_{,00}+(\Phi+\Psi)_{,0}\phi_{,0}\right]+e^{2\Psi}\left[\nabla^{2}\phi+(\Phi+\Psi)_{,1}\phi_{,1}\right]-V^{\prime}(\phi)=0~{},$
(49)

which is obtained by varying the Lagrangian with respect to $\phi$.

###  3.2 Mass and energy

Following Lee:1988av , we distinguish between the mass and energy of
oscillons. At large radius the mass density vanishes exponentially, hence
$\Phi$ and $\Psi$ scale as $r^{-1}$. The mass then must satisfy

$\displaystyle
M_{\mathrm{osc}}=-G^{-1}\lim\limits_{r\rightarrow\infty}r\Psi=-G^{-1}\lim\limits_{r\rightarrow\infty}r\Phi~{},$
(50)

to be consistent with the static Schwarzschild solution. There are other ways
to express the same mass. For example, the LHS of 00 component of Einstein
equation can be rewritten into $r^{-2}[r\left(e^{2\Psi}-1\right)]_{,1}$ hence
the mass is also given by

$\displaystyle M_{\mathrm{osc}}=\int_{0}^{\infty}(X+Y+V)~{}4\pi
r^{2}dr=-\int_{0}^{\infty}\tensor{T}{{}^{0}_{0}}~{}4\pi r^{2}dr~{},$ (51)

which is in agreement with the Schwarzschild mass (50).

A more enlightening way to describe the mass is to use the Hamiltonian
formalism. The Lagrangian of matter (39) indicates that the _energy_ of the
oscillon is

$\displaystyle E_{\mathrm{osc}}=\int_{0}^{\infty}(X+Y+V)~{}e^{\Phi-\Psi}4\pi
r^{2}dr=-\int\tensor{T}{{}^{0}_{0}}\sqrt{-g}~{}d^{3}x~{}.$ (52)

There is no kinetic term in the Lagrangian of gravity (38) thus the energy of
gravity is $E_{\mathrm{G}}=-L_{\mathrm{G}}$. Then we define the _mass_ of
oscillons

$\displaystyle M_{\mathrm{osc}}\equiv E_{\mathrm{osc}}+E_{G}~{}.$ (53)

Combining the energy expression and the 00 component of Einstein equations, we
find

$\displaystyle[r(e^{\Phi}-e^{\Phi+\Psi})]_{,1}=G(E_{\mathrm{M}}+E_{\mathrm{G}})_{,1}~{},$
(54)

in agreement with the Schwarzschild mass (50) and the ADM mass Arnowitt:1962hi
. After understanding what we mean by energy, the decay rate of oscillons can
be studied numerically in appendix B and analytically in the next section.

##  4 Oscillons with linearized gravity

The typical central amplitude, radius and mass of dense oscillons in the non-
relativistic limit are

$\displaystyle\phi_{1}(0)\sim F~{},\quad R_{\mathrm{osc}}\sim 5m^{-1}~{},\quad
M_{\mathrm{osc}}\sim 100F^{2}/m~{}.$ (55)

Nonlinear effects of gravitational interactions are not important if the size
of oscillons is much smaller than their Schwarzschild radius

$\displaystyle R_{\mathrm{osc}}\ll
GM_{\mathrm{osc}}\Rightarrow\epsilon_{\phi}\sim\phi_{1}/M_{\mathrm{pl}}\ll
1~{},$ (56)

which is satisfied by a number of cosmological models. In this section,
therefore, we study the decay rate and lifetime of oscillons in the non-
relativistic limit and weak-field limit of gravity, specifically those with
$\epsilon_{r}\lesssim 0.1$ and $\epsilon_{\phi}\lesssim 0.1$.555Examples of
the effective field theory that focuses on such low-energy phenomena includes
Mukaida:2016hwd ; Eby:2018ufi ; Braaten:2018lmj ; Namjoo:2017nia ;
Salehian:2020bon . The basic idea is similar to what we have done in section
2.

###  4.1 Profiles

In the weak-field approximation, the spherical metric (28) reduces to

$\displaystyle
ds^{2}=-(1+2\Phi)\mathrm{d}t^{2}+(1-2\Psi)\mathrm{d}r^{2}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta~{}\mathrm{d}\varphi^{2})~{}.$
(57)

So far we have encountered five sets of small dimensionless parameters, recall
equations (22) and (56), i.e. the spatial derivative parameter $\epsilon_{r}$,
the nonlinear potential parameter $\epsilon_{V}$, the radiation parameter
$\epsilon_{\xi}$, the amplitude parameter $\epsilon_{\phi}$ and the
gravitational potentials $\Phi$ and $\Psi$ (denoted by $\epsilon_{g}$). To be
consistent, we will keep all small quantities to 1st order, and to 2nd order
if spatial derivatives of small parameters are involved. Then the equation of
motion of $\phi$ (49) becomes

$\displaystyle-(1-2\Phi)\phi_{,00}+(\Phi+\Psi)_{,0}\phi_{,0}+\nabla^{2}\phi-V^{\prime}(\phi)=\mathcal{O}(\epsilon^{2}m^{2}\phi)~{}.$
(58)

To be consistent with the field expansion (4), we expand the gravitational
potentials in terms of a Fourier cosine series

$\displaystyle\Phi=\frac{1}{2}\Phi_{0}+\sum_{j=2}^{\infty}\Phi_{j}\cos(j\omega
t)~{},\quad\Psi=\frac{1}{2}\Psi_{0}+\sum_{j=2}^{\infty}\Psi_{j}\cos(j\omega
t)~{},$ (59)

where $j$ is even. We can solve these radial modes by plugging the expansion
into the Einstein equations and collecting the coefficient for each Fourier
mode. Then equations (44) and (45) give

$\displaystyle\Psi_{0}=-r\Phi_{0,1}+\mathcal{O}(\epsilon^{2})~{},\quad\Psi_{2}=\mathcal{O}(\epsilon^{2})~{},\quad\Psi_{j\geq
4}=\mathcal{O}(\epsilon^{3})~{},$ (60)

and equations (47) and (48) give

$\displaystyle\nabla^{2}\Phi_{0}=4\pi
G~{}m^{2}\phi_{1}^{2}+\mathcal{O}(\epsilon^{3}m^{2})~{},\quad\Phi_{2,1}/r=-2\pi
G~{}m^{2}\phi_{1}^{2}+\mathcal{O}(\epsilon^{3}m^{2})~{},\quad\Phi_{j\geq
4}=\mathcal{O}(\epsilon^{2})~{}.$ (61)

Therefore,
$-\Phi_{0}\sim-\Psi_{0}\sim\Phi_{2}\sim\epsilon_{g}\sim\epsilon_{\phi}^{2}/\epsilon_{r}$.666For
the quadratic potential, there is no mass scale $F$ and the central amplitude
of oscillons satisfies $\phi_{1}\sim\mathcal{O}(\epsilon_{r}M_{\mathrm{pl}})$
Lee:1991ax , hence $\epsilon_{\phi}\sim\epsilon_{g}\sim\epsilon_{r}$. Somewhat
surprisingly, we find $\Phi_{2}$ and $\Psi_{2}$ are not the same order of
magnitude, in constrast with the common results of the Newtonian gauge. We
will ignore $\Phi_{j\geq 4}$ and $\Psi_{j\geq 2}$ in future calculations.

In order to find the profile of oscillons, we plug the field expansion (4) and
(59) into equation (58) and collect the coefficient of $\cos(\omega t)$ to get

$\displaystyle[\nabla^{2}+\omega^{2}(1-\Phi_{0})-m^{2}]\phi_{1}=J_{1}+\mathcal{O}(\epsilon^{2}m^{2}\phi_{1})~{}.$
(62)

Here $\omega(1-\Phi_{0})^{1/2}$ can be regarded as an effective frequency and
is larger than $\omega$. This equation can be solved by numerical shooting
method.777The boundary conditions are $\phi_{1}(\infty)\rightarrow 0$,
$\Phi_{0}(\infty)\propto 1/r$ and $\Phi_{2}(\infty)\rightarrow 0$. The time-
averaged formula for the oscillon energy (52) is

$\displaystyle
E_{\mathrm{osc}}=\int_{0}^{\infty}\left[\frac{1}{4}(\partial_{r}\phi_{1})^{2}+\frac{1}{4}(\omega^{2}+m^{2})\left(1-\frac{1}{2}\Psi_{0}+\frac{1}{2}\Phi_{2}\right)\phi_{1}^{2}+\frac{1}{2}U_{0}\right]4\pi
r^{2}dr+\mathcal{O}\left(\frac{\epsilon^{1/2}\phi_{1}^{2}}{m}\right)~{}.$ (63)

As long as the oscillating part of the gravitational potentials is not too
important, namely $\epsilon_{g}\lesssim 0.1$, oscillons share great
similarities with mini-boson stars Friedberg:1986tp , and we assume the
stability condition is still valid Lee:1988av

$\displaystyle\frac{dE_{\mathrm{osc}}}{d\omega}<0~{}.$ (64)

This is confirmed by comparing analytical predictions of
$\omega_{\mathrm{crit}}$ with numerical results in the left panel of figure 4.

###  4.2 Decay rates

Plugging the field expansion (4) and (59) into equation (58) and collecting
the coefficient of $\cos(j\omega t)$, we obtain the radial equation of
radiating modes

$\displaystyle\left[\nabla^{2}+\kappa_{j}^{2}\right]\xi_{j}(r)=S_{j}(r)+\mathcal{O}(\epsilon^{3}m^{2}\phi_{1})~{},$
(65)

where $j\geq 3$ and $j$ is odd, and $S_{j}$ is the effective source. We can
keep finite terms of $\xi_{j}$ as we did in Minkowski spacetime. In
particular, if we keep only $\xi_{3}$ and $\xi_{5}$, the effective source
becomes

$\displaystyle S_{3}$
$\displaystyle=J_{3}+\frac{1}{2}\xi_{3}(M_{0}+M_{6})+\frac{1}{2}\xi_{5}(M_{2}+M_{8})+2\omega^{2}\phi_{1}\Phi_{2}+9\omega^{2}\xi_{3}\Phi_{0}+20\omega^{2}\xi_{5}\Phi_{2}~{},$
(66) $\displaystyle S_{5}$
$\displaystyle=J_{5}+\frac{1}{2}\xi_{3}(M_{2}+M_{8})+\frac{1}{2}\xi_{5}(M_{0}+M_{10})+12\omega^{2}\xi_{3}\Phi_{2}+25\omega^{2}\xi_{5}\Phi_{0}~{},$
(67)

where we have kept higher-order perturbations $\xi_{5}\Phi_{j}$ because of
their large coefficients. Comparing (66) with the corresponding expression in
Minkowski spacetime (17), new corrections are introduced due to the coupling
of gravity to oscillons and their radiation. The radiation equation can be
solved by the iterative method Zhang:2020bec .888The participation of
$\Phi_{0}$ is possible to make the iteration divergent (since both
$\Phi_{0},\xi_{3}\propto 1/r$ at large radius), in which case $\xi_{j}$ can be
easily found by adjusting initial values and matching the central amplitudes
calculated by equations (13) and (65).

The conservation law of the energy-momentum tensor is

$\displaystyle\tensor{T}{{}^{\mu}_{\nu;\mu}}=0\Rightarrow(\tensor{T}{{}^{\mu}_{\nu}}\sqrt{-g})_{,\mu}=0~{},$
(68)

which implies

$\displaystyle\frac{dE_{\mathrm{osc}}}{dt}=\int(\tensor{T}{{}^{i}_{0}}\sqrt{-g})_{,i}~{}d^{3}x=\left.-4\pi
r^{2}\tensor{T}{{}^{i}_{0}}(1+\Phi-\Psi)\right|_{r\rightarrow\infty}=\left.-4\pi
r^{2}\xi_{,0}\xi_{,1}\right|_{r\rightarrow\infty}~{},$ (69)

where we have used Gauss’s divergence theorem to obtain the second equal sign.
The $\xi$ at infinity is calculated in (14), then the decay rate expression is
just the same as the one in Minkowski spacetime (16).

##  5 Examples

In this section, we will explore gravitational effects on oscillon lifetimes
by studying two examples, recall figure 1 and equation (1), and comparing
numerical results from full GR simulations with our analytical predictions.
The numerical algorithm is described in appendix B.

###  5.1 The $\alpha$-attractor T-model of inflation

The longevity of oscillons in this model (whose lifetime $\sim 10^{6}\,m^{-1}$
in Minkowski spacetime) is characterized by a dip structure in decay rates,
see figure 1. In the dip, the leading channel of decay rates $\Gamma_{3}$
vanishes and the scalar radiation is dominated by the subleading channel
$\Gamma_{5}$. Now we argue that the existence of gravity reduces the lifetime
of oscillons.

Figure 2: Evidence that the existence of gravity reduces the lifetime of
oscillons in the $\alpha$-attractor T-model. In the left panel, we show the
dependence of dip locations on values of the characteristic mass scale $F$.
The smaller value of $\omega_{\mathrm{dip}}$ is an indication of a shorter
lifetime of oscillons, since the amplitude of radiating modes (in unit of $F$)
is inversely related to frequencies and thus larger decay rates are expected.
In the right panel, we confirm this expectation by explicitly calculating the
decay rate at $\omega_{\mathrm{dip}}$.

We first study how the location of the dip is affected both analytically and
numerically in figure 2 (left panel). The value of $\omega_{\mathrm{dip}}$ is
significantly reduced when the mass scale $F$ approaches the reduced Planck
mass. A smaller value of $\omega_{\mathrm{dip}}$ is an indication of shorter
lifetimes, since the amplitude of radiating modes (in unit of $F$) is
inversely related to frequencies and thus larger decay rates are expected. The
comparison between analytics and numerics also provides a chance to test our
formalism, which correctly captures the gravitational effect on
$\omega_{\mathrm{dip}}$ as long as the assumptions of weak-field gravity and
non-relativistic limit remain valid.

To exclude the possibility that the subleading channel of decay rates also
vanishes, we explicitly calculate $\Gamma_{5}$ around $\omega_{\mathrm{dip}}$
as shown in figure 2 (right panel). As a determinant factor, the increasing of
$\Gamma_{\mathrm{dip}}\approx\Gamma_{5}$ is a clear evidence that the
existence of gravity reduces the lifetime of oscillons, which has also been
confirmed numerically. For convenience, we present a direct visualization of
these two factors in figure 3.999We thank Andrew Long for a suggestion of
making this plot.

Figure 3: A comparison of decay rates for two different values of $F$ in the
$\alpha$-attractor T-model.

###  5.2 The axion monodromy model

In constrast with the $\alpha$-attractor T-model, the subleading channel of
decay rates $\Gamma_{5}$ is never comparable with the leading channel
$\Gamma_{3}$ in the axion monodromy model. The longevity of oscillons in this
case (whose lifetime $\sim 10^{8}\,m^{-1}$ in Minkowski spacetime) is due to
the dramatic suppresion of $\Gamma_{3}$ just before their final collapse at
$\omega_{\mathrm{crit}}$. Consequently, there are two factors that determine
the lifetime of oscillons, the value of $\omega_{\mathrm{crit}}$ and the decay
rate around $\omega_{\mathrm{crit}}$.

In figure 4 (left panel), we show that the values of $\omega_{\mathrm{crit}}$
are inversely related to $F$. This seems an indication of shorter lifetimes
for stronger gravitational effects because oscillons now spend less time
around $\omega_{\mathrm{crit}}$. A good match between analytics and numerics
implies that the stability condition (64) is still valid as long as gravity is
not too important.

Based on our semi-analytical framework, we calculate oscillon decay rates for
$F=0.0001M_{\mathrm{pl}}$ and $F=0.07M_{\mathrm{pl}}$ in the right panel.
Compared with the very weak-field gravity, it is shown that the decay rate
around $\omega_{\mathrm{crit}}$ is more suppressed for
$F=0.07M_{\mathrm{pl}}$. This can be qualitatively understood by inspecting
equations (17) and (66). Since $J_{3}$ at $\omega_{\mathrm{crit}}$ typically
has a Gaussian-like shape with a negative amplitude, the introduction of the
positive term $2\omega_{2}\phi_{1}\Phi_{2}$ diminishes the magnitude of the
effective source, and thus tends to reduce the decay rate and increase the
total lifetime.

To determine which factor dominates, we integrate out the decay rate and show
the lifetime in an inset of the right panel of figure 4. Our results imply
that the first factor plays a more important role and the oscillon lifetime is
shorter for stronger-field gravity. Nevertheless, the oscillon is still too
long-lived to be simulated with our current numerical algorithms. And we leave
this as a testable prediction for a future numerical experiment.

Figure 4: Two determinant factors of oscillon lifetimes in the axion monodromy
model, values of the critical frequency (left panel) and decay rates around
$\omega_{\mathrm{crit}}$ (right panel). It is shown that gravitational effects
decrease the value of $\omega_{\mathrm{crit}}$, which tends to reduce the
lifetime, while suppressing the decay rate around $\omega_{\mathrm{crit}}$,
which tends to stablize oscillons. In the inset, we show that oscillon
lifetimes are reduced slightly for the stronger-field gravity.

##  6 Conclusions

In this paper we generalize the method developed in our previous work
Zhang:2020bec  to include gravitational interactions, by expanding equations
in terms of five small parameters

$\displaystyle\epsilon_{r}\sim\frac{\partial_{r}^{2}}{m^{2}}~{},\quad\epsilon_{V}\sim\frac{V_{\mathrm{nl}}}{m^{2}\phi_{\mathrm{osc}}^{2}}~{},\quad\epsilon_{\xi}\sim\frac{\xi}{\phi_{\mathrm{osc}}}~{},\quad\epsilon_{\phi}\sim\frac{\phi_{\mathrm{osc}}}{M_{\mathrm{pl}}}~{},\quad\epsilon_{g}\sim\Phi~{},$
(70)

where $\epsilon_{r}\sim-\epsilon_{V}\sim\epsilon_{\xi}$ and
$\epsilon_{g}\sim\epsilon_{\phi}^{2}/\epsilon_{r}$ for dense oscillons. We
have shown in Zhang:2020bec  that keeping just the leading order of
$\epsilon_{r},\epsilon_{V}$ and $\epsilon_{\xi}$ gives accurate predictions of
oscillon decay rates and lifetimes in Minkowski spacetime. Thus in this paper,
we explore the impacts of $\epsilon_{\phi}$ and $\epsilon_{g}$ by keeping all
other parameters fixed. The application of our semi-analytical and
perturbative framework is based on two assumptions, the non-relativistic limit
of oscillons and the weak-field limit of gravity, namely $\epsilon_{r}\lesssim
0.1$ and $\epsilon_{\phi}\lesssim 0.1$.

In order to explore the gravitational effects on oscillon lifetimes, we
discuss two well-motivated and representative models in detail, the
$\alpha$-attractor T-model of inflation and the axion monodromy model with
potentials given by (1). As presented in figure 2, 3 and 4, our results for
$F\sim 0.1M_{\mathrm{pl}}$ show the following:

  * •

In the $\alpha$-attractor T-model, oscillon lifetimes are dominated by an
exceptionally stable intermediate configuration, which is visualized as a dip
structure in figure 1. We show that the existence of gravity decreases the
value of $\omega_{\rm dip}$ and increases the decay rate at $\omega_{\rm
dip}$. As a result, oscillon lifetimes are reduced when gravity becomes more
important.

  * •

In the axion monodromy model, oscillons are the most stable just before their
final collapse at $\omega_{\mathrm{crit}}$. It is shown that stronger
gravitational effects suppress the decay rate around $\omega_{\mathrm{crit}}$,
which tends to stablize oscillons, while diminishing the value of
$\omega_{\mathrm{crit}}$, which tends to decrease the lifetime. By explicitly
integrating out the decay rate, we find that the latter factor dominates and
oscillon lifetimes are reduced slightly.

In both examples we have considered, the evolution of an oscillon is almost
identical to that in Minkowski spacetime if $F\ll 0.1M_{\mathrm{pl}}$.
Therefore, in such cases, one may study the decay rate and dynamics of at
least single oscillons by ignoring gravity.

For stronger-field gravity, i.e. $\epsilon_{\phi}\gtrsim 0.1$ or $F\gtrsim
0.1M_{\mathrm{pl}}$, our equations with leading-order corrections are no
longer accurate. Also note that both $\epsilon_{\phi}$ and $\epsilon_{g}$
increase rapidly when we probe smaller frequences. Thus one should keep
higher-order perturbations of $\epsilon_{\phi}$ and $\epsilon_{g}$ to obtain
more reliable conclusions. But in this regime, exotic phenomena such as black
hole formation Helfer:2016ljl ; Muia:2019coe ; Nazari:2020fmk  might be more
interesting than oscillon lifetimes. In future work, we will further show that
a new phenomenon of migration from dense to dilute oscillons makes the notion
of oscillon lifetimes less meaningful.

###### Acknowledgements.

We would especially like to thank Mustafa Amin for many stimulating
discussions, initial collaboration and insightful advice. We would like to
thank Paul Saffin for a careful reading of the manuscript and suggestions for
its improvement, and Borna Salehian for helpful discussions on scalar
perturbation theory. We also thank Mudit Jain, Andrew Long, Kaloian Lozanov
and Zong-Gang Mou for useful comments. This work is supported by a NASA ATP
theory grant NASA-ATP Grant No. 80NSSC20K0518.

##  Appendix A Fourier cosine coefficients of scalar potentials

In order to build some intuition of how $U_{j},J_{j}$ and $M_{j}$ behave, let
us consider a symmetric polynomial potential of the general form

$\displaystyle
V_{\mathrm{nl}}(\phi)=\sum_{n=4}^{\infty}\frac{g_{n}}{n!}\phi^{n}~{},$ (71)

where $n$ is even. By using the oscillon profile
$\phi_{\mathrm{osc}}=\phi_{1}\cos(\omega t)$ and the identity

$\displaystyle\cos^{n}(\omega
t)=\sum_{k=0}^{n}\frac{1}{2^{n}}\frac{n!}{(n-k)!k!}\cos[(n-2k)\omega t]~{},$
(72)

we find the potentials can be written in terms of Fourier cosine series

$\displaystyle U=$
$\displaystyle\sum_{n=4}^{\infty}\frac{g_{n}}{2^{n}\left(\frac{n}{2}!\right)^{2}}\phi_{1}^{n}+\sum_{n=4}^{\infty}\frac{g_{n}}{2^{n-1}\left(\frac{n+2}{2}\right)!\left(\frac{n-2}{2}\right)!}\phi_{1}^{n}\cos(2\omega
t)$ (73)
$\displaystyle+\sum_{j=4}^{\infty}\sum_{n=j}^{\infty}\frac{g_{n}}{2^{n-1}\left(\frac{n+j}{2}\right)!\left(\frac{n-j}{2}\right)!}\phi_{1}^{n}\cos(j\omega
t)~{},$ (74) $\displaystyle M=$
$\displaystyle\sum_{n=2}^{\infty}\frac{g_{n+2}}{2^{n}\left(\frac{n}{2}!\right)^{2}}\phi_{1}^{n}+\sum_{j=2}^{\infty}\sum_{n=j}^{\infty}\frac{g_{n+2}}{2^{n-1}\left(\frac{n+j}{2}\right)!\left(\frac{n-j}{2}\right)!}\phi_{1}^{n}\cos(j\omega
t)~{},$ (75)

where $j$ and $n$ are even, and

$\displaystyle
J=\sum_{n=3}^{\infty}\frac{g_{n+1}}{2^{n-1}\left(\frac{n+1}{2}\right)!\left(\frac{n-1}{2}\right)!}\phi_{1}^{n}\cos(\omega
t)+\sum_{j=3}^{\infty}\sum_{n=j}^{\infty}\frac{g_{n+1}}{2^{n-1}\left(\frac{n+j}{2}\right)!\left(\frac{n-j}{2}\right)!}\phi_{1}^{n}\cos(j\omega
t)~{},$ (76)

where $j$ and $n$ are odd.

##  Appendix B Numerical algorithms

In this appendix we introduce briefly our numerical algorithm. For convenience
we will set $M_{\mathrm{pl}}\equiv 1$ so that all the mass is in unit of
reduced Planck mass. And to make the equations more accessible to coding,
define

$\displaystyle A\equiv e^{-\Psi}~{},\quad B\equiv e^{\Phi+\Psi}~{},\quad
v\equiv B^{-1}\phi_{,0}~{},\quad u\equiv\phi_{,1}~{}.$ (77)

Then the equation of motion of $\phi$ becomes

$\displaystyle\phi_{,0}=vB~{},\quad u_{,0}=(vB)_{,1}~{},\quad
v_{,0}=r^{-2}(r^{2}Bu)_{,1}-A^{2}BV^{\prime}(\phi)~{},$ (78)

where the metric can be given by 00 and 11 components of Einstein equations,
i.e.

$\displaystyle A_{,1}$
$\displaystyle=\frac{A}{2}\left[\frac{1-A^{2}}{r}+\frac{r}{2}\left(v^{2}+u^{2}+2A^{2}V\right)\right]~{},$
(79) $\displaystyle B_{,1}$
$\displaystyle=B\left(\frac{A^{2}-1}{r}-rA^{2}V\right)~{}.$ (80)

One is also recommended to rescale the fields in unit of $F$ to reduce the
roundoff errors brought by small numbers when $F\ll M_{\mathrm{pl}}$. The
oscillon energy can be calculated by

$\displaystyle
E_{\mathrm{osc}}=\int_{0}^{\infty}\frac{1}{2}B(v^{2}+u^{2}+2A^{2}V)~{}4\pi
r^{2}dr~{}.$ (81)

To maintain the smoothness at the center $r=0$, we must require

$\displaystyle u=v_{,1}=0~{},\quad A=1~{},\quad A_{,1}=B_{,1}=0~{}.$ (82)

To appropriately account for the origin, we set a spatial grid
$r_{n}=(n-1/2)dr$ with a fictitious point $r_{0}=-dr/2$. The inner boundary
conditions then become a parity condition: $\phi,v,A,B$ are even and $u$ is
odd. For the outer boundary conditions we adopt the radiative boundary
conditions Alcubierre:2000xu  by assuming that the dynamical variables $\phi$
and $v$ behave like spherical waves $\phi(t,r)=f(r-t)/r$. In practice, we will
use this in the differential form

$\displaystyle\phi_{,0}+\phi_{,1}+\phi/r=0~{},\quad
v_{,0}+v_{,1}+v/r=0~{},\quad v+u+\phi/r=0~{},$ (83)

for $\phi,v,u$ repectively. Note that one actually does not need boundary
conditions for $\phi$, but we find that integrating $\phi$ all the way to the
boundary point will inevitably generate instabilities in long-time
simulations. Finally, we note that the Schwarzchild metric should be recovered
at large $r$ hence $A$ and $B$ satisfy

$\displaystyle B(r\rightarrow\infty)=1/A^{2}(r\rightarrow\infty)~{}.$ (84)

We adopt 4th-order Runge-Kutta method as the time integrator to evolve
equations (78), while the spatial derivatives are discretized by the standard
4th-order centered difference Zlochower:2005bj . The 6th-order Kreiss-Oliger
dissipation with strength parameter $\epsilon=0.001$
alcubierre2008introduction  is added except boundary points to avoid shock
waves while the accuracy is still remained. The boundary condition (83) is
discretized by finite difference methods, specifically a 2nd-order upwind
method alcubierre2008introduction

$\displaystyle
v^{m+1}_{n}=\frac{1}{1+dt/dr}\left[\left(1-\frac{dt}{dr}-\frac{2dt}{r_{n}}\right)v^{m}_{n}+\left(1+\frac{dt}{dr}\right)v^{m}_{n-1}-\left(1-\frac{dt}{dr}\right)v^{m+1}_{n-1}\right]~{},$
(85)

where $m,n$ are the temporal and spatial indices respectively. Once $\phi,u,v$
have been advanced for one time level, we use the 4th-order Runge-Kutta method
to integrate (79) outwards and (80) inwards to get $A,B$. The initial values
of $B$ are obtained by using the Schwarzchild condition (84).

We typically set the boundary at a finite value $r_{f}$, which should be
significantly larger than the oscillon size, and $dr=2dt$ to satisfy the
Courant-Friedrichs-Lewy condition. All the physical quantities are averaged
over a time window $T_{\mathrm{ave}}=300~{}\mathrm{m^{-1}}$, which is much
larger than one period $2\pi/\omega$ but smaller than oscillon lifetime,
unless otherwise stated. We have checked that slight changes of parameters
(i.e. $dt,dr,r_{f},\epsilon,T_{\mathrm{ave}}$) do not affect the results
significantly.

Figure 5: Convergence test for axion monodromy-type potential with
$F=10^{-3}M_{\mathrm{pl}}$. Three colors refer to three different resolutions:
$dt=0.025m^{-1}$ (blue), $dt=0.05m^{-1}$ (orange) and $dt=0.1m^{-1}$ (green).
The initial conditions correspond to a Gaussian profile
$\phi(r)=1.43F\exp(-m^{2}r^{2}/5.2^{2})$. And we have set
$T_{\mathrm{ave}}=20m^{-1}$ to average the data. The fact that the L2 norm
values increase by a factor of 4 every time we double the time step shows that
the code maintains 2nd-order convergent.

Although we tried to implement 4th-order methods to evolve the system, the
$1/r$ terms at small $r$ and the boundary discretization will destroy the 4th-
order accuracy, hence we expect our algorithm to maintain 2nd-order. The codes
can be tested by an exact equation

$\displaystyle\delta_{10}\equiv A_{,0}-rABuv/2=0~{},$ (86)

which is just the $10$ component of Einstein equations. As an example, we plot
L2 norm of $\delta_{10}$ across the grids in figure 5, where the L2 norm is
defined by $||\phi||_{2}\equiv\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}\phi_{n}^{2}}$
, and $\phi_{n}$ denotes the value of $\phi$ at the spatial point $r_{n}$.

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