# Cosmology in theories with spontaneous scalarization of neutron stars

Ratchaphat Nakarachinda1, Sirachak Panpanich2, Shinji Tsujikawa2, and Pitayuth
Wongjun1 1The Institute for Fundamental Study, Naresuan University,
Phitsanulok, 65000, Thailand
2Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo
169-8555, Japan

###### Abstract

In a model of spontaneous scalarization of neutron stars proposed by Damour
and Esposite-Farese, a general relativistic branch becomes unstable to trigger
tachyonic growth of a scalar field $\phi$ toward a scalarized branch. Applying
this scenario to cosmology, there is fatal tachyonic instability of $\phi$
during inflation and matter dominance being incompatible with solar-system
constraints on today’s field value $\phi_{0}$. In the presence of a four-point
coupling $g^{2}\phi^{2}\chi^{2}/2$ between $\phi$ and an inflaton field
$\chi$, it was argued by Anson et al. that a positive mass squared heavier
than the square of a Hubble expansion rate leads to the exponential
suppression of $\phi$ during inflation and that $\phi_{0}$ can remain small
even with the growth of $\phi$ after the radiation-dominated epoch. For
several inflaton potentials approximated as $V(\chi)=m^{2}\chi^{2}/2$ about
the potential minimum, we study the dynamics of $\phi$ during reheating as
well as other cosmological epochs in detail. For certain ranges of the
coupling $g$, the homogeneous field $\phi$ can be amplified by parametric
resonance during a coherent oscillation of the inflaton. Incorporating the
backreaction of created particles under a Hartree approximation, the maximum
values of $\phi$ reached during preheating are significantly smaller than
those obtained without the backreaction. We also find that the minimum values
of $g$ consistent with solar system bounds on $\phi$ at the end of reheating
are of order $10^{-5}$ and hence there is a wide range of acceptable values of
$g$. Thus, the scenario proposed by Anson et al. naturally leads to the viable
cosmological evolution of $\phi$ consistent with local gravity constraints,
without modifying the property of scalarized neutron stars.

††preprint: WUCG-22-10

##  I Introduction

The physics in strong gravity regimes can be now probed by the observations of
gravitational waves (GWs) emitted from a binary system containing black holes
or neutron stars (NSs) Abbott  _et al._ (2016, 2017, 2020). With the future
high precision data of GWs, it will be possible to probe the accuracy of
General Relativity (GR) and the possible deviation from it Berti  _et al._
(2015); Barack  _et al._ (2019, 2019); Berti  _et al._ (2018). In theories
beyond GR, there are usually additional degrees of freedom that can be coupled
to gravity Copeland  _et al._ (2006); De Felice and Tsujikawa (2010); Clifton
_et al._ (2012); Joyce  _et al._ (2015); Heisenberg (2019); Kase and Tsujikawa
(2019a); Baker  _et al._ (2021). A simplest example is known as scalar-tensor
theories, in which a scalar field $\phi$ has nonminimal or derivative
couplings to gravity Brans and Dicke (1961); Fujii (1982); Fujii and Maeda
(2007); Horndeski (1974); Deffayet  _et al._ (2011); Kobayashi  _et al._
(2011); Charmousis  _et al._ (2012).

In the presence of a nonminimal coupling $F(\phi)R$, where $F$ is a function
of $\phi$ and $R$ is the Ricci scalar, it is known that NSs can have scalar
hairs through an indirect coupling between the scalar field and matter
mediated by gravity Damour and Esposito-Farese (1993, 1996); Harada (1998);
Novak (1998); Sotani and Kokkotas (2004); Cooney  _et al._ (2010); Arapoglu
_et al._ (2011); Orellana  _et al._ (2013); Aparicio Resco  _et al._ (2016);
Kase and Tsujikawa (2019b). For the exponential coupling
$F(\phi)=e^{-2Q\phi/M_{\rm Pl}}$, where $Q$ is a constant and $M_{\rm Pl}$ is
the reduced Planck mass, the fifth force propagates around weak gravitational
objects like the Sun. From solar-system experiments, there is a tight bound
$|Q|<2\times 10^{-3}$ on the coupling strength Will (2014); Tsujikawa  _et
al._ (2008). Then, the deviation from GR in the vicinity of NSs is also
suppressed, so probing the modification of gravity from the GW observations is
challenging for the nonminimal coupling $F(\phi)=e^{-2Q\phi/M_{\rm Pl}}$.

If we consider nonminimal couplings containing even power-law functions of
$\phi$, it is possible to have a nontrivial branch $\phi^{\prime}(r)\neq 0$
besides a GR branch $\phi^{\prime}(r)=0$ on a static and spherically symmetric
background with the radial distance $r$. A typical example is the coupling
$F(\phi)=e^{-\beta\phi^{2}/(2M_{\rm Pl}^{2})}$ proposed by Damour and
Esposite-Farese (DEF), where $\beta$ is a constant Damour and Esposito-Farese
(1993, 1996). For strong gravitational objects like NSs, the GR branch can be
unstable to trigger tachyonic instability of the scalar field to reach a
nontrivial branch of $\phi$. This phenomenon, which is called spontaneous
scalarization, occurs for negative coupling constants in the range
$\beta\leq-4.35$ Harada (1998); Novak (1998); Silva  _et al._ (2015); Barausse
_et al._ (2013). Since the gravitational interaction for such scalarized NSs
exhibits the appreciable deviation from that in GR, it is possible to probe
the modification of gravity from binary pulsar measurements Freire  _et al._
(2012); Shao  _et al._ (2017); Anderson  _et al._ (2019) as well as the
observations of GWs emitted from a binary system containing at least one NS
Niu  _et al._ (2021); Higashino and Tsujikawa (2022).

When spontaneous scalarization of NSs occurs, the asymptotic value of the
scalar field $\phi_{0}$ needs to be in the range $|\phi_{0}|\lesssim
10^{-3}M_{\rm Pl}|\beta|^{-1}$ for the consistency with local gravity
constraints Damour and Esposito-Farese (1993). This asymptotic value should be
determined by the cosmological evolution of $\phi$ from the past to today. In
the original DEF model, however, the background cosmological scalar field
largely deviates from 0 for the coupling range $\beta$ allowing for
spontaneous scalarization Damour and Nordtvedt (1993); Anderson  _et al._
(2016). This is attributed to the fact that the negative coupling $\beta$
leads to tachyonic growth of $|\phi|$ during the cosmological evolution.
Without severely fine-tuned initial conditions, the amplitude of today’s field
value exceeds the upper limit constrained by solar system tests of gravity. We
note that the similar type of instabilities is also present for spontaneously
scalarized BHs arising from a scalar Gauss-Bonnet coupling Anson  _et al._
(2019a); Franchini and Sotiriou (2020); Antoniou  _et al._ (2021).

On the other hand, Anson et al. Anson  _et al._ (2019b) proposed a mechanism
of reconciling spontaneous scalarization with cosmology by incorporating a
coupling $g^{2}\phi^{2}\chi^{2}/2$ between the scalar field $\phi$ and an
inflaton field $\chi$. Since the field $\phi$ can have a positive effective
mass squared $g^{2}\chi^{2}$ larger than the squared Hubble expansion rate
during inflation, the amplitude of the homogeneous field $\phi$ exponentially
decreases. Even though $|\phi|$ increases after the onset of the radiation era
to today, it is expected that today’s field value is still in the range
$|\phi_{0}|\lesssim 10^{-3}M_{\rm Pl}|\beta|^{-1}$. However, the analysis of
Ref. Anson  _et al._ (2019b) does not accommodate the evolution of $\phi$
during the post inflationary reheating period. Indeed, for certain ranges of
the coupling $g$, the four-point coupling $g^{2}\phi^{2}\chi^{2}/2$ leads to
parametric resonance of the homogeneous field $\phi$ and its perturbations
during a preheating stage after inflation Kofman  _et al._ (1994); Shtanov
_et al._ (1995); Kaiser (1996); Khlebnikov and Tkachev (1997a, b); Prokopec
and Roos (1997); Kofman  _et al._ (1997); Bassett  _et al._ (2006). Even for
small couplings $g$ without the period of preheating, it can happen that the
amplitude of $\phi$ grows during reheating by the dominance of the negative
nonminimal coupling over the positive mass term $g^{2}\chi^{2}$. Hence it is
important to clarify the coupling range of $g$ in which the model can be at
work. We note that there are other mechanisms for reconciling spontaneous
scalarization with cosmology Anderson  _et al._ (2016); Silva and Minamitsuji
(2019); Minamitsuji and Tsujikawa (2022); Higashino and Tsujikawa (2022), but
it is typically nontrivial to realize the acceptable cosmological evolution of
$\phi$.

In the DEF model with the coupling $g^{2}\phi^{2}\chi^{2}/2$, we will study
the cosmological evolution of the scalar field $\phi$ responsible for
spontaneous scalarization. We pay particular attention to the dynamics during
reheating in which the further growth of $|\phi|$ can be expected. Since the
inflaton potentials are approximated as $V(\chi)\simeq m^{2}\chi^{2}/2$ around
$\chi=0$, the reheating period corresponds to an effective matter era driven
by the oscillation of a massive inflaton field.

For coupling ranges of $g$ in which the preheating stage is present, we need
to take into account the backreaction of created $\phi$ particles that leads
to the violation of coherent oscillations of $\chi$. Without the backreaction,
the maximum amplitude $\phi_{\rm max}$ of the field $\phi$ reached during
preheating can exceed a value constrained by solar-system tests of gravity at
the end of reheating ($|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$). However,
we will show that implementing the backreaction under the Hartree
approximation Khlebnikov and Tkachev (1997b); Kofman  _et al._ (1997) leads to
$\phi_{\rm max}$ significantly smaller than $10^{-11}M_{\rm Pl}$. For two
inflaton potentials considered in this paper, $\phi_{\rm max}$ is less than
the order of $10^{-38}M_{\rm Pl}$. After the system reaches an equilibrium
state with the violation of coherent oscillations of the inflaton, the further
significant amplification of $\phi$ is not expected by the end of reheating
because the negative nonminimal coupling is suppressed compared to
$g^{2}\chi^{2}$ in the background equation of $\phi$. Thus, even with the
preheating epoch, the presence of the coupling $g^{2}\phi^{2}\chi^{2}/2$
allows the cosmological evolution of $\phi$ consistent with local gravity
constraints on $\phi_{0}$.

For small coupling ranges of $g$ in which preheating does not occur, we do not
need to implement the backreaction of created particles. In such cases, we
will solve the background equations of motion by the end of reheating with a
Born decay constant $\Gamma$ taken into account. Depending on the form of
inflaton potentials, the nonminimal coupling can overwhelm the term
$g^{2}\chi^{2}$ during reheating. Since this leads to the growth of $|\phi|$
by the end of reheating, it can happen that $|\phi_{\rm R}|$ exceeds the upper
bound $10^{-11}M_{\rm Pl}$. This is especially the case for a low-scale
reheating scenario with the reheating temperature of order MeV. For two
inflaton potentials, we will put lower bounds on the coupling $g$ consistent
with solar-system constraints. In both cases, the minimum values of $g$ are of
order $10^{-5}$, so the mechanism proposed by Anson et al. Anson  _et al._
(2019b) is at work for wide ranges of the coupling $g$ (including the case
where preheating occurs).

This paper is organized as follows. In Sec. II, we briefly review the DEF
model and derive the background equations of motion on the spatially-flat
Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime in the presence of the
coupling $g^{2}\phi^{2}\chi^{2}/2$. In Sec. III, we discuss the cosmological
evolution of $\phi$ from the radiation era to today and interpret a solar-
system bound on $\phi_{0}$ as the constraint on $\phi$ at the onset of
radiation dominance. In Sec. IV, we study the dynamics of the scalar field
during inflation for several inflaton potentials and find minimum values of
$g$ above which the amplitude of $\phi$ exponentially decreases. In Sec. V, we
analyze the evolution of the homogeneous field $\phi$ and its perturbations
during the reheating stage and derive minimum values of the coupling constant
$g$ consistent with solar-system constraints. Sec. VI is devoted to
conclusions.

##  II Theories with spontaneous scalarization

We consider theories given by the action

$\displaystyle{\cal S}$ $\displaystyle=$ $\displaystyle\int{\rm
d}^{4}x\sqrt{-g_{J}}\left[\frac{M_{\rm
Pl}^{2}}{2}F(\phi)R+\omega(\phi)X+L_{\rm inf}\right]$ (1) $\displaystyle+{\cal
S}_{m}(g_{\mu\nu},\Psi_{m})\,,$

where $g_{J}$ is a determinant of metric tensor $g_{\mu\nu}$ in the Jordan
frame, and $X=-(1/2)g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ is a scalar
kinetic term with $\nabla_{\mu}$ being a covariant derivative operator. The
$\phi$-dependent function in front of $X$ is chosen to be Kase  _et al._
(2020a, b); Higashino and Tsujikawa (2022)

$\omega(\phi)=\left(1-\frac{3M_{\rm
Pl}^{2}F_{,\phi}^{2}}{2F^{2}}\right)F(\phi)\,,$ (2)

where $F_{,\phi}={\rm d}F/{\rm d}\phi$. Brans-Dicke (BD) theories Brans and
Dicke (1961) correspond to the particular nonminimal coupling
$F(\phi)=e^{-2Q\phi/M_{\rm Pl}}$, where a coupling constant $Q$ is related to
the BD parameter $\omega_{\rm BD}$ as $3+2\omega_{\rm BD}=1/(2Q^{2})$
Tsujikawa  _et al._ (2008); Khoury and Weltman (2004). In theories of
spontaneous scalarization, $F(\phi)$ contains even power-law functions of
$\phi$.

We take into account the contribution of an inflaton field $\chi$ as the
Lagrangian

$L_{\rm
inf}=-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\chi\nabla_{\nu}\chi-V(\chi)-\frac{1}{2}g^{2}\phi^{2}\chi^{2}\,,$
(3)

where $V$ is the potential of $\chi$. The last term in Eq. (3) characterizes
an interaction between $\phi$ and $\chi$ with a coupling constant $g$. During
inflation, this can generate a large effective positive mass squared of $\phi$
relative to the square of the Hubble expansion rate. Then, it can compensate a
negative mass squared induced by the nonminimal coupling $F(\phi)R$
responsible for spontaneous scalarization. This allows a possibility for
avoiding the tachyonic growth of $\phi$ during inflation Anson  _et al._
(2019b).

In the presence of the four-point coupling $g^{2}\phi^{2}\chi^{2}/2$, it is
known that a phenomenon called preheating Kofman  _et al._ (1994, 1997) can
occur after inflation during the coherent oscillation of inflaton. In this
stage, the scalar field $\phi$ and its perturbations $\delta\phi$ can be
amplified by parametric resonance. Since the dynamics of the field $\phi$
during preheating were not addressed in Ref. Anson  _et al._ (2019b), we will
study whether the presence of this stage is harmless or not for the
compatibility with local gravity constraints on today’s field value
$\phi_{0}$. For this purpose, we also scrutinize the scalar-field dynamics in
other cosmological epochs from the onset of inflation to today.

In Eq. (1), the action ${\cal S}_{m}$ incorporates the contributions of matter
fields $\Psi_{m}$ such as radiation, nonrelativistic matter, and dark energy.
We assume that matter fields are minimally coupled to gravity.

Under the conformal transformation $\hat{g}_{\mu\nu}=F(\phi)g_{\mu\nu}$, the
action (1) is transformed to De Felice and Tsujikawa (2010); Kase  _et al._
(2020a)

$\displaystyle\hat{{\cal S}}$ $\displaystyle=$ $\displaystyle\int{\rm
d}^{4}x\sqrt{-\hat{g}}\left[\frac{M_{\rm
Pl}^{2}}{2}\hat{R}-\frac{1}{2}\hat{g}^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+\hat{L}_{\rm
inf}\right]$ (4) $\displaystyle+{\cal
S}_{m}(F^{-1}(\phi)\hat{g}_{\mu\nu},\Psi_{m})\,,$

where a hat represents quantities in the Einstein frame. In the transformed
frame, the scalar field $\phi$ is coupled to matter fields through the metric
tensor $\hat{g}_{\mu\nu}$.

We deal with the Jordan frame as a physical one and perform all analyses in
this frame by exploiting the action (1). We consider the nonminimal coupling
chosen by DEF Damour and Esposito-Farese (1993, 1996)

$F(\phi)=e^{-\beta\phi^{2}/(2M_{\rm Pl}^{2})}\,,$ (5)

where $\beta$ is a constant. In this case, we have

$\omega(\phi)=\left(1-\frac{3\beta^{2}\phi^{2}}{2M_{\rm
Pl}^{2}}\right)e^{-\beta\phi^{2}/(2M_{\rm Pl}^{2})}\,.$ (6)

Spontaneous scalarization of NSs can occur for the coupling $\beta\leq-4.35$
Harada (1998); Novak (1998); Silva  _et al._ (2015); Barausse  _et al._
(2013). In such cases, the GR branch $\phi=0$ can be unstable to trigger
tachyonic instability toward the other nontrivial branch $\phi\neq 0$. From
binary pulsar measurements of the energy loss through dipolar radiation, the
coupling $\beta$ was constrained to be $\beta\geq-4.5$ Freire  _et al._
(2012); Shao  _et al._ (2017). Thus, the coupling constant $\beta$ is
restricted in a limited range.

We consider a spatially-flat FLRW background given by the line element

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\delta_{ij}{\rm d}x^{i}{\rm d}x^{j}\,,$
(7)

where $a(t)$ is a time-dependent scale factor. Incorporating the Born decay
term $\Gamma\dot{\chi}$ to complete the reheating process ($\Gamma$ is a decay
constant and a dot represents a derivative with respect to $t$) into the
inflaton equation of motion, it follows that

$\ddot{\chi}+\left(3H+\Gamma\right)\dot{\chi}+V_{,\chi}+g^{2}\phi^{2}\chi=0\,,$
(8)

where $H=\dot{a}/a$ is the Hubble expansion rate. We will consider the case
$H\gg\Gamma$ during inflation, so the decay term $\Gamma\dot{\chi}$ is
important only at the late stage of reheating. Due to the energy transfer from
the inflaton to radiation, the radiation energy density $\rho_{r}$ satisfies
the differential equation

$\dot{\rho}_{r}+4H\rho_{r}=\Gamma\dot{\chi}^{2}\,.$ (9)

The energy density $\rho_{m}$ of nonrelativistic matter (cold dark matter and
baryons) obeys the continuity equation

$\dot{\rho}_{m}+3H\rho_{m}=0\,,$ (10)

with a vanishing pressure. As a source for dark energy, we take the
cosmological constant $\Lambda$ into account.

The (00) and (11) components of gravitational field equations following from
the action (1) are given, respectively, by

$\displaystyle 3M_{\rm
Pl}^{2}H(FH+F_{,\phi}\dot{\phi})=\frac{1}{2}\dot{\chi}^{2}+V+\frac{1}{2}\omega\dot{\phi}^{2}+\frac{1}{2}g^{2}\phi^{2}\chi^{2}$
$\displaystyle+\rho_{r}+\rho_{m}+\Lambda\,,$ (11) $\displaystyle-M_{\rm
Pl}^{2}F(2\dot{H}+3H^{2})=M_{\rm
Pl}^{2}\left(F_{,\phi}\ddot{\phi}+F_{,\phi\phi}\dot{\phi}^{2}+2F_{,\phi}H\dot{\phi}\right)$
$\displaystyle+\frac{1}{2}\dot{\chi}^{2}-V+\frac{1}{2}\omega\dot{\phi}^{2}-\frac{1}{2}g^{2}\phi^{2}\chi^{2}+\frac{1}{3}\rho_{r}-\Lambda\,.$
(12)

The scalar field $\phi$ obeys the differential equation

$\ddot{\phi}+3H\dot{\phi}+m_{\rm eff}^{2}\phi=0\,,$ (13)

where

$m_{\rm eff}^{2}\equiv\frac{1}{\omega}\left[g^{2}\chi^{2}+3\beta
F\left(2H^{2}+\dot{H}\right)-\frac{\beta\omega\dot{\phi}^{2}}{2M_{\rm
Pl}^{2}}-\frac{3\beta^{2}F\dot{\phi}^{2}}{2M_{\rm Pl}^{2}}\right].$ (14)

To discuss the dynamics of inflation and reheating, we consider the inflaton
potentials of $\alpha$-attractors Kallosh  _et al._ (2013) given by

$V(\chi)=\frac{3}{4}\alpha m^{2}M_{\rm
Pl}^{2}\left[1-\exp\left(-\sqrt{\frac{2}{3\alpha}}\frac{\chi}{M_{\rm
Pl}}\right)\right]^{2}\,,$ (15)

where $\alpha$ is a positive dimensionless constant, and $m$ is a constant
having a dimension of mass. In the limit $\alpha\to\infty$, the potential (15)
reduces to that in chaotic inflation, i.e., $V(\chi)=m^{2}\chi^{2}/2$ Linde
(1983). Starobinsky’s model with the Lagrangian $L=R+R^{2}/(6m^{2})$
Starobinsky (1980) gives rise to the potential (15) with $\alpha=1$ after a
conformal transformation to the Einstein frame De Felice and Tsujikawa (2010).
For the numerical simulation performed in Secs. IV and V, we will consider the
two potentials: (i) $V(\chi)=m^{2}\chi^{2}/2$, and (ii) the potential (15)
with $\alpha=1$.

##  III Cosmological dynamics from radiation domination to today

In this section, we investigate a bound of the field value $\phi$ at the onset
of radiation dominance constrained from solar system tests of gravity. In
theories given by the action (1), the post-Newtonian parameter $\gamma_{\rm
PPN}$ is Damour and Esposito-Farese (1992, 1993)

$\gamma_{\rm PPN}-1=-\frac{2\alpha_{\rm PPN}^{2}(\varphi_{0})}{1+\alpha_{\rm
PPN}^{2}(\varphi_{0})}=-\frac{2\beta^{2}\varphi_{0}^{2}}{1+\beta^{2}\varphi_{0}^{2}}\,,$
(16)

where $\varphi=\phi/(\sqrt{2}M_{\rm Pl})$ is a dimensionless field with
today’s value $\varphi_{0}$, and $\alpha_{\rm
PPN}=-F_{,\varphi}/(2F)=\beta\varphi$ with $F=e^{-\beta\varphi^{2}}$. The
Shapiro time delay measurements have given the bound $\gamma_{\rm
PPN}-1=(2.1\pm 2.3)\times 10^{-5}$ Bertotti  _et al._ (2003). Since
$\gamma_{\rm PPN}-1$ is negative in the current theory, we adopt the limit
$|\gamma_{\rm PPN}-1|\leq 0.2\times 10^{-5}$. This corresponds to the bound
$|\beta\varphi_{0}|\leq 1\times 10^{-3}$, so today’s field value $\phi_{0}$ is
constrained to be

$|\phi_{0}|\leq 1.4\times 10^{-3}M_{\rm Pl}|\beta|^{-1}\,.$ (17)

For $\beta=-4.4$, we have $|\phi_{0}|\leq 3.2\times 10^{-4}M_{\rm Pl}$. In the
following, we will study how this constraint translates to the upper limit of
$|\phi|$ at the onset of radiation era.

After the reheating period ends, we can neglect the contribution of the
inflaton field $\chi$ in Eq. (14). Moreover, provided that the energy density
of $\phi$ is negligible relative to that of the background and that $|\phi|\ll
M_{\rm Pl}$, the effective mass squared (14) is approximated as $m_{\rm
eff}^{2}\simeq 3\beta(2H^{2}+\dot{H})$. Then, the scalar-field equation (13)
approximately reduces to

$\phi^{\prime\prime}+\frac{3}{2}\left(1-w_{\rm
eff}\right)\phi^{\prime}+\frac{3}{2}\left(1-3w_{\rm eff}\right)\beta\phi\simeq
0\,,$ (18)

where a prime represents the derivative with respect to $N=\ln a$, and $w_{\rm
eff}=-1-2H^{\prime}/(3H)$ is the effective equation of state. If $w_{\rm
eff}={\rm constant}$, there is the following growing-mode solution

$\phi\propto a^{\lambda}\,,$ (19)

where

$\lambda=\frac{3}{4}\left(1-w_{\rm
eff}\right)\left[\sqrt{1-\frac{8\beta(1-3w_{\rm eff})}{3(1-w_{\rm
eff})^{2}}}-1\right].$ (20)

During the radiation era ($w_{\rm eff}=1/3$), we have $\lambda=0$ and hence
$\phi={\rm constant}$. This property is attributed to the fact that $R$
vanishes on the exact radiation-dominated background. In the matter-dominated
era ($w_{\rm eff}=0$), the negative mass squared of $\phi$ leads to the
following tachyonic growth of the field

$\phi\propto a^{(3/4)\left(\sqrt{1-8\beta/3}-1\right)}\,.$ (21)

During the epoch of cosmological constant domination, the growth of $\phi$ is
even stronger: $\phi\propto a^{(3/2)\left(\sqrt{1-8\beta/3}-1\right)}$.
However, the dominance of dark energy over nonrelativistic matter occurs only
at low redshifts $z\lesssim 0.3$, so we can approximately use Eq. (21) for the
evolution of $\phi$ from radiation-matter equality to today. Then, the field
value at radiation-matter equality can be estimated as

$\phi_{\rm eq}=\phi_{0}(z_{\rm
eq}+1)^{-(3/4)\left(\sqrt{1-8\beta/3}-1\right)}\,,$ (22)

where $z_{\rm eq}$ is determined by $z_{\rm eq}=\Omega_{m0}/\Omega_{r0}-1$,
with $\Omega_{m0}$ and $\Omega_{r0}$ being today’s density parameters of
nonrelativistic matter and radiation respectively.

In Fig. 1, we plot the evolution of $\phi/M_{\rm pl}$ and
$\Omega_{r}=\rho_{r}/(3FH^{2}M_{\rm Pl}^{2})$,
$\Omega_{m}=\rho_{m}/(3FH^{2}M_{\rm Pl}^{2})$, and $\Omega_{\rm
DE}=\Lambda/(3FH^{2}M_{\rm Pl}^{2})$ for $\beta=-4.4$ between the radiation
era and today. We choose the initial conditions around redshift $z=10^{9}$ to
realize today’s value $\phi_{0}=3.2\times 10^{-4}M_{\rm Pl}$, which
corresponds to the upper limit consistent with local gravity constraints. In
this case the redshift at radiation-matter equality is $z_{\rm eq}\simeq
3470$, so the analytic estimation (22) gives $\phi_{\rm eq}=4.8\times
10^{-11}M_{\rm Pl}$. This is fairly close to the numerical value $\phi_{\rm
eq}=6.0\times 10^{-11}M_{\rm Pl}$, even though we ignored the epoch of late-
time cosmic acceleration for the analytic estimation of $\phi_{0}$.

Figure 1:  Evolution of $\phi/M_{\rm pl}$ and $\Omega_{r}$, $\Omega_{m}$, and
$\Omega_{\rm DE}$ versus $z+1~{}(=1/a)$ for $\beta=-4.4$. The initial
conditions are chosen to realize today’s values $\phi_{0}/M_{\rm pl}=3.2\times
10^{-4}$, $\Omega_{r0}=9.2\times 10^{-5}$, and $\Omega_{{\rm DE}0}=0.68$.

In Fig. 1, we observe that $\phi$ slowly grows even at the late stage of
radiation era. This reflects the fact that $R=6(2H^{2}+\dot{H})$ does not
completely vanish due to the presence of nonrelativistic matter. Numerically,
we find that $\phi_{\rm eq}$ is larger than the initial value $\phi_{\rm R}$
in the radiation era by one order of magnitude. This is consistent with the
analytic estimation given in Ref. Anson  _et al._ (2019b). Multiplying the
analytic value (22) by a factor 0.2, it is possible to reproduce the numerical
value of $\phi_{\rm R}$ approximately. Then, we have the following analytic
formula

$|\phi_{\rm R}|\simeq
0.2|\phi_{0}|\left(\frac{\Omega_{m0}}{\Omega_{r0}}\right)^{-(3/4)\left(\sqrt{1-8\beta/3}-1\right)}\,.$
(23)

For the model parameters used in the numerical simulation of Fig. 1, i.e.,
$\phi_{0}=3.2\times 10^{-4}M_{\rm Pl}$, $\Omega_{m0}=0.32$, and
$\Omega_{r0}=9.2\times 10^{-5}$, the analytic estimation (23) gives
$|\phi_{\rm R}|=9.6\times 10^{-12}M_{\rm Pl}$. This is close to the numerical
value $|\phi_{\rm R}|=9.2\times 10^{-12}M_{\rm Pl}$.

Applying the formula (23) to Eq. (17), we obtain the following upper bound

$|\phi_{\rm R}|\leq 2.8\times 10^{-4}M_{\rm
Pl}|\beta|^{-1}\left(\frac{\Omega_{m0}}{\Omega_{r0}}\right)^{-(3/4)\left(\sqrt{1-8\beta/3}-1\right)}.$
(24)

For $\beta=-4.5$ and $\beta=-4.35$, the criterion (24) gives $|\phi_{\rm
R}|\leq 7.5\times 10^{-12}M_{\rm Pl}$ and $|\phi_{\rm R}|\leq 1.1\times
10^{-11}M_{\rm Pl}$, respectively, where we used the same values of
$\Omega_{m0}$ and $\Omega_{r0}$ mentioned above. These bounds are close to the
numerically derived upper limits $|\phi_{\rm R}|\leq 7.1\times 10^{-12}M_{\rm
Pl}$ and $|\phi_{\rm R}|\leq 1.1\times 10^{-11}M_{\rm Pl}$, respectively.
Thus, in the coupling range $-4.5\leq\beta\leq-4.35$, the initial value of
$|\phi_{\rm R}|$ at the onset of radiation era needs to be smaller than the
order $10^{-11}M_{\rm Pl}$ for the consistency with local gravity constraints.

##  IV Inflationary epoch

In this section, we study the evolution of $\phi$ during inflation in the
presence of the coupling $(1/2)g^{2}\phi^{2}\chi^{2}$ besides the nonminimal
coupling (5) with $\beta<0$. We consider two inflaton potentials: (i)
$V(\chi)=m^{2}\chi^{2}/2$ and (ii) $\alpha$-attractor potential (15) with
$\alpha=1$. The potential (i), which corresponds to the limit
$\alpha\to\infty$ of Eq. (15), leads to the scalar spectral index $n_{s}\simeq
1-2/N$ and the tensor-to-scalar ratio $r\simeq 8/N$, where $N$ is the number
of e-foldings backward from the end of inflation to the epoch at which the
perturbations relevant to observed Cosmic Microwave Background (CMB)
temperature anisotropies crossed the Hubble radius Bassett  _et al._ (2006).
For $N=60$ we have $n_{s}\simeq 0.967$ and $r=0.133$, so the model is in
tension with the Planck2018 bound of the tensor-to-scalar ratio $r<0.066$ (95
% CL) Akrami  _et al._ (2020).

Nevertheless, the potential (i) of chaotic inflation with the four-point
interaction $g^{2}\phi^{2}\chi^{2}/2$ is a baseline model widely studied in
the context of preheating after inflation Kofman  _et al._ (1994); Shtanov
_et al._ (1995); Kaiser (1996); Khlebnikov and Tkachev (1997a, b); Prokopec
and Roos (1997); Kofman  _et al._ (1997); Bassett  _et al._ (2006). We will
accommodate this case in our analysis for the purpose of understanding the
difference from the $\alpha$-attractor with $\alpha=1$. In the model (ii) with
$\alpha\lesssim{\cal O}(1)$, we have $n_{s}\simeq 1-2/N$ and $r\simeq
12\alpha/N^{2}$ Kallosh  _et al._ (2013), and hence $r=3.3\times 10^{-3}$ for
$\alpha=1$ and $N=60$. If $\alpha\lesssim 40$, the $\alpha$-attractor model is
compatible with the Planck CMB bound $r<0.066$ and $n_{s}=0.9661\pm 0.0040$
(68 % CL) Heisenberg  _et al._ (2019).

In the current theory the field $\phi$ is present besides the inflaton $\chi$,
so the existence of the former field can modify the prediction of inflationary
observables like $n_{s}$ and $r$. To avoid this, we consider the case in which
the contribution of $\phi$ to the background equations of motion is suppressed
around the e-folding $N=60$ backward from the end of inflation. This amounts
to the conditions $\beta^{2}\phi^{2}/(2M_{\rm Pl}^{2})\ll 1$,
$\dot{\phi}^{2}\ll H^{2}M_{\rm Pl}^{2}$, and $g^{2}\phi^{2}\chi^{2}\ll V$.
Considering the coupling $\beta$ in the range $-4.5\leq\beta\leq-4.35$ with
$|\dot{\phi}|$ at most of order $|H\phi|$, the first two conditions can be
satisfied for $|\phi|\lesssim 0.1M_{\rm Pl}$. Then, the field value $\phi_{\rm
inf}$ about the $60$ e-folding backward from the end of inflation is in the
range

$|\phi_{\rm inf}|\lesssim 0.1M_{\rm Pl}\,,\quad{\rm and}\quad|\phi_{\rm
inf}|\lesssim 0.1\frac{\sqrt{V}}{g|\chi|}\,.$ (25)

For the potential $V(\chi)=m^{2}\chi^{2}/2$, the latter condition translates
to $|\phi_{\rm inf}|\lesssim 0.1m/g$.

Under the conditions (25), the effective mass squared (14) during inflation
($|\dot{H}|\ll H^{2}$ and $3H^{2}M_{\rm Pl}^{2}\simeq V$) approximately
reduces to

$m_{\rm eff}^{2}\simeq g^{2}\chi^{2}+6\beta H^{2}\simeq
g^{2}\chi^{2}+\frac{2\beta V}{M_{\rm Pl}^{2}}\,.$ (26)

When $g=0$, we have $m_{\rm eff}^{2}<0$ and hence there is the tachyonic
growth of $\phi$ Anson  _et al._ (2019b). On the exact de-Sitter background,
the growing-mode solution to Eq. (13) for $g=0$ is given by

$\phi\propto\exp\left[\frac{3}{2}\left(\sqrt{1-\frac{8}{3}\beta}-1\right)Ht\right]\,.$
(27)

During the time interval $t=10H^{-1}$, for instance, the field $\phi$ is
amplified by a factor $5\times 10^{16}$ for $\beta=-4.4$. This enhancement of
$\phi$ destroys the inflationary period driven by the potential energy of
$\chi$. Even if the initial field value $\phi_{\rm inf}$ is fine-tuned to be
extremely close to 0, there exists a field perturbation $\delta\phi$ whose
amplitude does not vanish due to the uncertainty principle. After the Hubble
radius crossing, the perturbation $\delta\phi$ is amplified in a manner
analogous to the homogeneous field $\phi$ discussed above Anson  _et al._
(2019b). Hence the existence of the nonminimal coupling (5) with $\beta=-{\cal
O}(1)$ violates the successful inflationary prediction of primordial density
perturbations sourced by the perturbations of $\chi$.

The nonvanishing coupling $g$ gives rise to a positive contribution to the
mass squared (26). If $g$ is in the range

$g>\frac{\sqrt{2|\beta|V}}{M_{\rm Pl}|\chi|}\,,$ (28)

we have $m_{\rm eff}^{2}>0$ and hence the exponential growth of $|\phi|$ can
be avoided.

Let us first consider the effective mass in the range $0<m_{\rm eff}^{2}\leq
9H^{2}/4$. Neglecting the variation of $m_{\rm eff}^{2}$ during inflation, the
growing-mode solution to Eq. (13) is given by

$\phi\propto\exp\left[-\frac{3}{2}\left(1-\sqrt{1-\frac{4m_{\rm
eff}^{2}}{9H^{2}}}\right)Ht\right]\,.$ (29)

When $m_{\rm eff}^{2}=0$ we have $\phi={\rm constant}$, while, for increasing
$m_{\rm eff}^{2}$, the decreasing rate of $\phi$ gets larger.

For the effective mass satisfying $m_{\rm eff}^{2}>9H^{2}/4$, the coupling $g$
is in the range

$g>\frac{\sqrt{(3-8\beta)V}}{2M_{\rm Pl}|\chi|}\,,$ (30)

whose lower limit is larger than that in Eq. (28). In this mass range, the
field $\phi$ evolves as

$\phi\propto\exp\left(-\frac{3}{2}Ht\right)\cos\left(\sqrt{1-\frac{9H^{2}}{4m_{\rm
eff}^{2}}}\,m_{\rm eff}t\right)\,.$ (31)

Then, the amplitude of $\phi$ decreases as $|\phi|\propto e^{-3Ht/2}\propto
a^{-3/2}$ with the oscillation induced by the effective mass $m_{\rm eff}$
Jedamzik and Sigl (2000); Ivanov (2000); Liddle  _et al._ (2000); Tsujikawa
and Bassett (2002), where we used the approximate solution $a\propto e^{Ht}$
during inflation. On using the number of e-foldings $N_{\rm inf}$ relevant to
the observation of CMB temperature anisotropies, the amplitude of $\phi$ at
the end of inflation (denoted as $|\phi_{I}|$) can be estimated as

$|\phi_{I}|=|\phi_{\rm inf}|e^{-3N_{\rm inf}/2}\,.$ (32)

For $N_{\rm inf}=60$ and $\phi_{\rm inf}=0.1M_{\rm Pl}$, the estimation (32)
gives $|\phi_{I}|=8\times 10^{-41}M_{\rm Pl}$, so there is the strong
suppression of $|\phi|$ during inflation. Notice that, for $m_{\rm
eff}^{2}>9H^{2}/4$, the decreasing rate of $|\phi|$ does not depend on $g$.

###  IV.1 Quadratic potential

For the inflaton potential $V(\chi)=m^{2}\chi^{2}/2$, the inequalities (28)
and (30) translate to $g>\sqrt{|\beta|}m/M_{\rm Pl}$ and
$g>\sqrt{3/8-\beta}\,m/M_{\rm Pl}$, respectively. We consider the mass
$m=6\times 10^{-6}M_{\rm Pl}$ constrained from the Planck normalization of CMB
temperature anisotropies Akrami  _et al._ (2020). For $\beta=-4.4$, the
conditions (28) and (30) correspond to $g>1.26\times 10^{-5}$ and
$g>1.31\times 10^{-5}$, respectively.

Figure 2:  Evolution of $|\phi|/M_{\rm Pl}$ versus $\ln a$ during inflation
for the potential $V(\chi)=m^{2}\chi^{2}/2$ with $\beta=-4.4$ and $m=6.0\times
10^{-6}M_{\rm Pl}$. The initial field values of scalar fields are chosen to be
$\chi=15.38M_{\rm Pl}$, with $\phi$ corresponding to the maximum of Eq. (25).
In this case, the number of e-foldings acquired inflation is $N=60$. Each line
corresponds to (a) $g=1.27\times 10^{-5}$, (b) $g=1.29\times 10^{-5}$, (c)
$g=1.31\times 10^{-5}$, and (d) $g=1.0\times 10^{-3}$.

In Fig. 2, we plot the evolution of $|\phi|/M_{\rm Pl}$ for four different
coupling constants $g$. We choose the initial field value $\phi_{\rm inf}$ as
a maximum satisfying the two conditions given in Eq. (25). In case (a) the
coupling $g=1.27\times 10^{-5}$ is slightly larger than the value
$g=1.26\times 10^{-5}$ corresponding to $m_{\rm eff}^{2}=0$, so the field
$\phi$ mildly decreases according to Eq. (29). As $g$ increases in the range
$1.26\times 10^{-5}<g<1.31\times 10^{-5}$, the decreasing rate of $\phi$ tends
to be larger as we observe in cases (b) and (c) of Fig. 2. For $g>1.31\times
10^{-5}$, the amplitude of $\phi$ decreases in proportion to $a^{-3/2}$ with
oscillations. Thus, provided the condition (30), i.e.,

$g>\sqrt{\frac{3}{8}-\beta}\,\frac{m}{M_{\rm Pl}},$ (33)

is satisfied, the field value at the end of inflation is suppressed to be in
the range $|\phi_{I}|\lesssim 10^{-40}M_{\rm Pl}$. For such couplings, unless
$|\phi|$ is amplified by a factor more than $10^{29}$ during reheating, the
bound $|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$ can be satisfied at the
onset of radiation era. For the coupling $g$ in the range
$\sqrt{2|\beta|V}/(M_{\rm Pl}|\chi|)<g<\sqrt{(3-8\beta)V}/(2M_{\rm
Pl}|\chi|)$, the suppression of $\phi$ during inflation is not necessarily
significant, so the growth of $\phi$ during reheating matters to satisfy the
bound $|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$.

###  IV.2 $\alpha$-attractor with $\alpha=1$

Let us consider the $\alpha$-attractor potential (15) with $\alpha=1$. In this
case, the right hand-sides of Eqs. (28) and (30) depend on the value of
$\chi$. Under the slow-roll approximation, the number of e-foldings backward
from the end of inflation (inflaton value $\chi_{f}$) can be computed by
$N=M_{\rm Pl}^{-2}\int_{\chi_{f}}^{\chi}V/V_{,\chi}\,{\rm d}\chi$. In the
present case, we have $N=(3/4)[1/y-1/y_{f}+\ln(y/y_{f})]$, where
$y=e^{-\sqrt{2/3}\chi/M_{\rm Pl}}$ and $y_{f}=e^{-\sqrt{2/3}\chi_{f}/M_{\rm
Pl}}$. The field value at the end of inflation is determined by the condition
$\epsilon_{V}=(M_{\rm Pl}^{2}/2)(V_{,\chi}/V)^{2}=1$, and hence
$\chi_{f}=0.940M_{\rm Pl}$ and $y_{f}=0.464$. For $N=60$, we have
$y=1.165\times 10^{-2}$ and $\chi=5.453M_{\rm Pl}$, in which case the Planck
normalization of primordial curvature perturbations gives the constraint
$m\simeq 1.1\times 10^{-5}M_{\rm Pl}$.

Since $\chi\gtrsim{\cal O}(M_{\rm Pl})$ during the inflationary period, the
potential (15) is nearly constant in this regime and hence the term
$g^{2}\chi^{2}$ in Eq. (26) decreases faster than $2|\beta|V/M_{\rm Pl}^{2}$.
Unlike the quadratic potential, there are some ranges of the coupling $g$ for
which $g^{2}\chi^{2}$ is initially larger than $2|\beta|V/M_{\rm Pl}^{2}$, but
the latter dominates over the former during inflation. On using the value
$\chi=5.453M_{\rm Pl}$ at $N=60$ with $\beta=-4.4$ and $m=1.1\times
10^{-5}M_{\rm Pl}$, the condition (28) for the realization of positive $m_{\rm
eff}^{2}$ translates to $g>5.12\times 10^{-6}$.

Figure 3:  Evolution of $|\phi|/M_{\rm Pl}$ versus $\ln a$ during inflation
for the $\alpha$-attractor potential (15) with $\alpha=1$, $\beta=-4.4$, and
$m=1.1\times 10^{-5}M_{\rm Pl}$. The initial field values of scalar fields are
chosen to be $\chi=5.418M_{\rm Pl}$, with $\phi$ corresponding to the maximum
of Eq. (25). Each case corresponds to (a) $g=5.5\times 10^{-6}$, (b)
$g=6.5\times 10^{-6}$, (c) $g=1.68\times 10^{-5}$, and (d) $g=1.0\times
10^{-3}$.

In case (a) of Fig. 3, we plot the evolution of $|\phi|/M_{\rm Pl}$ during
inflation for $g=5.5\times 10^{-6}$. In this case the field initially
decreases because $m_{\rm eff}^{2}>0$, but it starts to grow at some point
because $g^{2}\chi^{2}$ drops below $2|\beta|V/M_{\rm Pl}^{2}$. This
enhancement of $\phi$ largely modifies the dynamics of inflation in such a way
that the total number of e-foldings does not reach even 40. As we observe in
case (b) of Fig. 3, the growth of $\phi$ also occurs for the coupling
$g=6.5\times 10^{-6}$. In this case, the field value $\phi$ at the end of
inflation is of order $10^{-11}M_{\rm Pl}$, so the dynamics of inflation
driven by the field $\chi$ is hardly modified. In Sec. V, we will study
whether or not the existence of a subsequent reheating stage leads to
additional growth of $\phi$ exceeding the bound $|\phi_{\rm R}|<10^{-11}M_{\rm
Pl}$.

If we demand that the condition (28), i.e., $m_{\rm eff}^{2}>0$, holds by the
end of inflation (at which $\chi_{f}=0.940M_{\rm Pl}$), then the coupling $g$
is constrained to be

$g>0.698\sqrt{|\beta|}\frac{m}{M_{\rm Pl}}\,.$ (34)

Substituting the values $\beta=-4.4$ and $m=1.1\times 10^{-5}M_{\rm Pl}$ into
Eq. (34), we have $g>1.61\times 10^{-5}$. Similarly, the condition (30)
translates to $g>1.68\times 10^{-5}$. The evolution of $|\phi|/M_{\rm pl}$ for
$g=1.68\times 10^{-5}$ is plotted as case (c) in Fig. 3. We observe that the
field exhibits exponential decrease $|\phi|\propto a^{-3/2}$ by the end of
inflation. Numerically, we find that, even for the marginal coupling
$g=1.61\times 10^{-5}$, $\phi$ decreases in a similar manner to case (c).
Hence the condition (34) is practically sufficient to ensure the exponential
suppression of $\phi$ during the whole stage of inflation. As we see in case
(d), the decreasing rate of $\phi$ for $g=1.0\times 10^{-3}$ is practically
the same as in case (c). The only difference between cases (c) and (d) is the
choice of initial conditions, where we have selected the maximum value of
$|\phi_{\rm inf}|$ satisfying the conditions (25). For larger $g$, $|\phi_{\rm
inf}|$ tends to be smaller.

##  V Reheating epoch

After the end of inflation, the Universe enters a reheating stage in which the
inflaton field $\chi$ oscillates around the potential minimum. The field value
$\chi_{I}$ at the onset of reheating is determined by the condition
$\epsilon_{V}=(M_{\rm Pl}^{2}/2)(V_{,\chi}/V)^{2}=1$, so that
$\chi_{I}=1.414M_{\rm Pl}$ for $V(\chi)=m^{2}\chi^{2}/2$ and
$\chi_{I}=0.940M_{\rm Pl}$ for the $\alpha$-attractor potential (15) with
$\alpha=1$. Around $\chi=0$, the potential (15) approximately reduces to the
quadratic one: $V(\chi)\simeq m^{2}\chi^{2}/2$.

In the presence of the coupling $g^{2}\phi^{2}\chi^{2}/2$, the background
field $\phi$ and its perturbations can be resonantly amplified by broad
parametric resonance during reheating for the coupling $g$ in the range
$g^{2}\chi^{2}\gg m^{2}$ Kofman  _et al._ (1994, 1997). The resonant growth of
the variance of $\phi$ affects the evolution of the homogeneous mode of $\chi$
through the backreaction of created particles Khlebnikov and Tkachev (1997b);
Kofman  _et al._ (1997); Tsujikawa  _et al._ (1999). To incorporate this
effect, we expand the inhomogeneous field $\delta\phi(t,{\bm{x}})$ in terms of
the Fourier series, as

$\delta\phi(t,{\bm{x}})=\frac{1}{(2\pi)^{3}}\int{\rm
d}^{3}k\,\delta\phi_{k}(t)e^{i{\bm{k}}\cdot{\bm{x}}}\,,$ (35)

where ${\bm{k}}$ is a comoving wavenumber with $k=|{\bm{k}}|$. Under a Hartree
approximation, the zero-momentum mode of $\chi$ is the only nonvanishing
component of inflaton Khlebnikov and Tkachev (1997b). Using this approximation
with $V(\chi)\simeq m^{2}\chi^{2}/2$ in the reheating stage, Eq. (8) is
modified to

$\ddot{\chi}+\left(3H+\Gamma\right)\dot{\chi}+\left[m^{2}+g^{2}\left(\phi^{2}+\langle\delta\phi^{2}\rangle\right)\right]\chi=0\,,$
(36)

where $\phi$ is the homogeneous value of the field, and
$\langle\delta\phi^{2}\rangle$ is the vacuum expectation value given by

$\langle\delta\phi^{2}\rangle=\int\frac{{\rm d}k}{k}{\cal
P}_{\delta\phi_{k}}\,,$ (37)

with the power spectrum

${\cal P}_{\delta\phi_{k}}=\frac{k^{3}}{2\pi^{2}}|\delta\phi_{k}|^{2}\,.$ (38)

The Fourier modes $\delta\phi_{k}(t)$ obey the differential equation

$\ddot{\delta\phi}_{k}+\left(3H+\frac{\omega_{,\phi}\dot{\phi}}{\omega}\right)\dot{\delta\phi}_{k}+\left(\frac{k^{2}}{a^{2}}+M_{\rm
eff}^{2}\right)\delta\phi_{k}=0\,,$ (39)

where

$\displaystyle M_{\rm eff}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\omega}\biggl{[}g^{2}\chi^{2}-3F_{,\phi\phi}M_{\rm
Pl}^{2}(2H^{2}+\dot{H})$ (40)
$\displaystyle~{}~{}~{}~{}+(\ddot{\phi}+3H\dot{\phi})\omega_{,\phi}+\frac{\dot{\phi}^{2}}{2}\omega_{,\phi\phi}\biggr{]}.$

To the right hand-sides of Eqs. (11) and (12), we take into account the
perturbed density $\langle\rho_{\delta\phi}\rangle$ and the pressure $\langle
P_{\delta\phi}\rangle$ of produced $\phi$ particles, respectively. Assuming
that the conditions $m\gg H$ and $\langle\delta\phi^{2}\rangle\ll M_{\rm
Pl}^{2}$ hold during reheating, we have

$\displaystyle\langle\rho_{\delta\phi}\rangle$ $\displaystyle=$
$\displaystyle\frac{\omega(\phi)}{2}\left[\langle\dot{\delta\phi}^{2}\rangle+\frac{1}{a^{2}}\langle\partial\delta\phi\rangle^{2}\right]+\frac{1}{2}g^{2}\chi^{2}\langle\delta\phi^{2}\rangle,$
(41) $\displaystyle\langle P_{\delta\phi}\rangle$ $\displaystyle=$
$\displaystyle\left[\frac{\omega(\phi)}{2}-\beta
F(\phi)\right]\left[\langle\dot{\delta\phi}^{2}\rangle+\frac{1}{a^{2}}\langle\partial\delta\phi\rangle^{2}\right]$
(42) $\displaystyle-\frac{1}{2}g^{2}\chi^{2}\langle\delta\phi^{2}\rangle\,,$

where

$\displaystyle\langle\dot{\delta\phi}^{2}\rangle$ $\displaystyle=$
$\displaystyle\int{\rm d}k\frac{k^{2}}{2\pi^{2}}|\dot{\delta\phi}_{k}|^{2}\,,$
(43) $\displaystyle\langle\partial\delta\phi\rangle^{2}$ $\displaystyle=$
$\displaystyle\int{\rm
d}k\frac{k^{4}}{2\pi^{2}}\left|\delta\phi_{k}\right|^{2}\,.$ (44)

Defining the rescaled perturbed field

$\delta\varphi_{k}=a^{3/2}\omega^{1/2}\delta\phi_{k}\,,$ (45)

Eq. (39) can be expressed in the form

$\ddot{\delta\varphi}_{k}+\Omega_{k}^{2}\,\delta\varphi_{k}=0\,,$ (46)

where

$\displaystyle\Omega_{k}^{2}$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{a^{2}}-\frac{9}{4}H^{2}-\frac{3}{2}\dot{H}+\frac{1}{\omega}\biggl{[}g^{2}\chi^{2}-3F_{,\phi\phi}M_{\rm
Pl}^{2}(2H^{2}+\dot{H})$ (47)
$\displaystyle+\frac{1}{2}(\ddot{\phi}+3H\dot{\phi})\omega_{,\phi}+\frac{\dot{\phi}^{2}\omega_{,\phi}^{2}}{4\omega}\biggr{]}\,.$

The typical wavenumber relevant to the parametric excitation of $\phi$
particles is $k/a_{I}\sim m$, where $a_{I}$ is the scale factor at the onset
of reheating. The perturbations with $k/a_{I}\sim m$ are deep inside the
Hubble radius during inflation ($k/a\gg H$), so the dominant contribution to
$\Omega_{k}^{2}$ is the term $k^{2}/a^{2}$. For such modes, we choose a
positive-frequency solution in the Bunch-Davies vacuum state as

$\delta\varphi_{k}=\frac{1}{\sqrt{2\Omega_{k}}}e^{-i\int\Omega_{k}{\rm
d}t}\,,$ (48)

which corresponds to an initial condition of $\delta\varphi_{k}$ at the onset
of reheating. For the modes $k/a_{I}\sim m$, the frequency $\Omega_{k}$ can be
approximated as $\Omega_{k}\simeq k/a$ during inflation except for the last
short period in which $k^{2}/a^{2}$ drops below $M_{\rm eff}^{2}$. Then, the
amplitude of $\delta\phi_{k}$ around the beginning of reheating is estimated
as

$|\delta\phi_{k}(t_{I})|\simeq\frac{1}{a_{I}\sqrt{\omega}\sqrt{2k}}\,.$ (49)

The square root of the power spectrum (38) for the mode $k/a_{I}\simeq m$ is
given by

$\sqrt{{\cal
P}_{\delta\phi_{k}}(t_{I})}\simeq\frac{1}{\sqrt{\omega}}\frac{k}{2\pi
a_{I}}\simeq\frac{m}{2\pi}\,,$ (50)

where we used the approximation $\omega\simeq 1$ in the second equality. The
perturbation $\delta\phi_{k}$ excited by parametric resonance has a typical
initial amplitude $m/(2\pi)$. For the quadratic potential $V=m^{2}\chi^{2}/2$
and the $\alpha$-attractor potential (15) with $\alpha=1$, we have
$\sqrt{{\cal P}_{\delta\phi_{k}}(t_{I})}\approx 10^{-6}M_{\rm Pl}$.

Let us also estimate the power spectrum of larger-scale modes of
$\delta\phi_{k}$ that exit the Hubble radius during inflation. We are
interested in the range of coupling $g$ where parametric resonance occurs
during preheating, in which case $M_{\rm eff}^{2}>9H^{2}/4$ during inflation.
After the Hubble radius crossing during inflation ($k<aH$), $\Omega_{k}^{2}$
is of order $M_{\rm eff}^{2}\simeq g^{2}\chi^{2}/\omega$. Around the onset of
reheating, the amplitude of $|\delta\phi_{k}|$ can be estimated as

$|\delta\phi_{k}|(t_{I})\simeq\frac{1}{a_{I}^{3/2}\sqrt{\omega}\sqrt{2M_{\rm
eff}}}\,.$ (51)

This corresponds to the power spectrum

${\cal P}_{\delta\phi_{k}}(t_{I})\simeq\frac{1}{4\pi^{2}\omega M_{\rm
eff}}\left(\frac{k}{a_{I}}\right)^{3}\simeq\frac{1}{4\pi^{2}}\frac{H_{\rm
inf}^{3}}{M_{\rm eff}}e^{-3N_{\rm inf}}\,,$ (52)

where $H_{\rm inf}$ is the value of $H$ at the Hubble radius crossing. In the
second equality of Eq. (52), we have substituted $k=a_{\rm inf}H_{\rm inf}$
and used the number of e-foldings $N_{\rm inf}=\ln(a_{I}/a_{\rm inf})$. For
the perturbations relevant to the observed CMB temperature anisotropies, we
have $N_{\rm inf}=55\sim 60$. Taking the value $N_{\rm inf}=60$, the amplitude
of perturbations $\delta\phi_{k}$ at the beginning of reheating is of order

$\sqrt{{\cal P}_{\delta\phi_{k}}(t_{I})}\simeq 10^{-40}\frac{H_{\rm
inf}}{\sqrt{r_{M}}}\,,$ (53)

where $r_{M}\equiv M_{\rm eff}/H_{\rm inf}$. For the inflationary scale
$H_{\rm inf}\simeq 10^{-4}M_{\rm Pl}$, we have $\sqrt{{\cal
P}_{\delta\phi_{k}}(t_{I})}\simeq 10^{-44}M_{\rm Pl}/\sqrt{r_{M}}$. This
suppression of the large-scale modes of $\delta\phi_{k}$ is analogous to what
happens for the homogeneous field $\phi$, see Eq. (32). The positive mass
squared $M_{\rm eff}^{2}$ greater than the order $H_{\rm inf}^{2}$ leads to
the exponential decrease of $\delta\phi_{k}$ for the perturbations that exit
the Hubble radius long before the end of inflation. From the above discussion,
the large-scale perturbations $\delta\phi_{k}$ with wavenumbers $k$ in the
range $k/a_{I}\ll m$ hardly contribute to the vacuum expectation value
$\langle\delta\phi^{2}\rangle$.

Provided the background field $\phi$ is suppressed during inflation, its
contributions to Eqs. (8) and (11)-(12) can be ignored in the early stage of
reheating. The initial reheating period in which $\chi$ oscillates coherently
can be identified by a temporal matter era where the scale factor evolves as
$a\propto t^{2/3}$ with $H=2/(3t)$. For the approximate potential $V\simeq
m^{2}\chi^{2}/2$ around $\chi=0$, the background Eqs. (8) and (11) reduce to
$\ddot{\chi}+3H\dot{\chi}+m^{2}\chi\simeq 0$ and $4M_{\rm
Pl}^{2}/(3t^{2})\simeq\dot{\chi}^{2}/2+m^{2}\chi^{2}/2$, respectively. The
solution consistent with the virial relation
$\langle\dot{\chi}^{2}/2\rangle=\langle m^{2}\chi^{2}/2\rangle$ averaged over
oscillations is given by

$\chi=\chi_{I}\frac{t_{I}}{t}\sin(mt)\,,\quad{\rm
with}\quad\chi_{I}\equiv\sqrt{\frac{8}{3}}\frac{M_{\rm Pl}}{mt_{I}}\,,$ (54)

where $\chi_{I}$ is the amplitude at the onset of reheating ($t=t_{I}$). The
inflaton field oscillates with the amplitude decreasing in proportion to
$1/t$.

Ignoring the contributions of time derivatives of $\phi$ and terms
$-9H^{2}/4-3\dot{H}/2$ in Eq. (47) during the coherent oscillation of $\chi$,
the perturbation $\delta\varphi_{k}$ obeys the Mathieu equation

$\frac{{\rm d}^{2}\delta\varphi_{k}}{{\rm
d}z^{2}}+\left[A_{k}-2q\cos(2z)\right]\delta\varphi_{k}=0\,,$ (55)

where

$\displaystyle A_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{k}{ma}\right)^{2}+2q+\frac{2\beta}{3z^{2}}\,,$ (56)
$\displaystyle q$ $\displaystyle=$ $\displaystyle
q_{I}\left(\frac{t_{I}}{t}\right)^{2}\,,\qquad
q_{I}=\frac{g^{2}\chi_{I}^{2}}{4m^{2}}\,,$ (57) $\displaystyle z$
$\displaystyle=$ $\displaystyle mt\,.$ (58)

As we see in Eq. (13), the homogeneous mode $\varphi=a^{3/2}\phi$ satisfies
the same form of equation as (55) with the limit $k/(ma)\to 0$ in Eq. (56). If
the parameter $q_{I}$ is in the range $q_{I}\gg 1$, it is known that there is
an epoch of the preheating period in which $\delta\phi_{k}$ and $\phi$ are
amplified by broad parametric resonance Kofman  _et al._ (1994, 1997). Since
$\chi_{I}$ is of order $M_{\rm Pl}$ for the inflaton potentials discussed in
Sec. IV, preheating can occur in the coupling range

$g\gg\frac{m}{M_{\rm Pl}}\,.$ (59)

For the exact quadratic potential $V=m^{2}\chi^{2}/2$, the mass $m$ is
constrained to be $m=6.0\times 10^{-6}M_{\rm Pl}$ from the Planck
normalization, so the condition (59) translates to $g\gg 6.0\times 10^{-6}$.
Since the parameter $q$ decreases as $q\propto 1/t^{2}$ due to cosmic
expansion, the growth of the variance $\langle\delta\phi^{2}\rangle$ actually
occurs for $g\gtrsim 10^{-4}$ Kofman  _et al._ (1997); Khlebnikov and Tkachev
(1997b, a). In the $\alpha$-attractor model with $\alpha=1$, we have
$m=1.1\times 10^{-5}M_{\rm Pl}$ from the Planck normalization, so the bound
(59) corresponds to $g\gg 1.1\times 10^{-5}$. In this case, the field value
$\chi_{I}$ at the onset of reheating is smaller than that for the exact
quadratic potential, so the larger coupling $g$ is required for the
realization of broad parametric resonance.

For the coupling $g$ in the range (59), the homogeneous field $\phi$ is
exponentially suppressed ($|\phi|\propto a^{-3/2}$) during inflation. Since we
are considering the coupling constant around $\beta\sim-4$, the term
$2\beta/(3z^{2})$ in Eq. (56) is less than the order 1 for $z\gtrsim 1$.
Provided that $g\gg m/M_{\rm Pl}$, the condition $2q\gg 2|\beta|/(3z^{2})$ is
satisfied during preheating, so the nonminimal coupling $\beta$ hardly affects
the dynamics of $\delta\phi_{k}$. If $g\sim m/M_{\rm Pl}$, the parameter $2q$
is of the same order as $2|\beta|/(3z^{2})$ and hence the coupling $\beta$
cannot be ignored. In such cases, however, the absence of parametric resonance
does not lead to the growth of the field variance
$\langle\delta\phi^{2}\rangle$. For the range of $g$ where the preheating does
not occur, we do not take the backreaction into account.

For wavenumbers in the range $k/a_{I}\gg m$, the term $(k/ma_{I})^{2}$ is much
larger than 1. Although $k/(ma)$ decreases during reheating, the parametric
excitation of $\delta\phi_{k}$ is not significant for $k/a_{I}$ very much
larger than $m$. We recall that the perturbations $\delta\phi_{k}$ with
$k/a_{I}\ll m$ as well as the homogeneous mode $\phi$ are subject to the
exponential suppression during inflation. Then, the main contribution to
$\langle\delta\phi^{2}\rangle$ at the beginning of preheating comes from the
modes distributed around $k/a_{I}\sim m$. We compute the variances (37) and
(43)-(44) for the wavenumbers up to $k/a_{I}\lesssim 10^{2}m$ by using the
initial condition (48) at the onset of reheating.

###  V.1 Quadratic potential

Let us first study the reheating dynamics for the inflaton potential
$V(\chi)=m^{2}\chi^{2}/2$. As we showed in Sec. IV.1, the amplitude of $\phi$
decreases as $|\phi|\propto a^{-3/2}$ during inflation for $g>1.31\times
10^{-5}$. Parametric resonance occurs for $g\gtrsim 10^{-4}$, in which regime
the field value $\phi_{I}$ at the onset of reheating is determined by Eq.
(32), where $\phi_{\rm inf}$ is limited as Eq. (25). When $g\gtrsim 10^{-4}$,
the second condition of (25) gives the upper bounds of $\phi_{\rm inf}$ and
$\phi_{I}$. For larger $g$, the maximum values of $\phi_{\rm inf}$ and
$\phi_{I}$ tend to be smaller. In the numerical simulation given below, we use
the maximum allowed values of $\phi_{I}$ constrained by Eq. (25) as the
initial condition of reheating.

Figure 4:  Evolutions of $|\phi|,\chi^{2},\langle\delta\phi^{2}\rangle$ during
the early stage of reheating for the quadratic potential
$V(\chi)=m^{2}\chi^{2}/2$ with $g=2.0\times 10^{-3}$, $\beta=-4.4$,
$m=6.0\times 10^{-6}M_{\rm Pl}$, and $\Gamma=1.0\times 10^{-13}M_{\rm Pl}$
($x=|\phi|,\chi^{2},\langle\delta\phi^{2}\rangle$ in the figure). Here $\phi$
is normalized by $M_{\rm Pl}$, while $\chi^{2}$ and
$\langle\delta\phi^{2}\rangle$ are normalized by $M_{\rm Pl}^{2}$. The initial
field value $\phi_{I}$ at the onset of reheating is determined by the
evolution of $\phi$ during inflation discussed in Sec. IV. The red and green
lines correspond to the cases in which the backreaction of created particles
is included and neglected, respectively.

In Fig. 4, we show the evolutions of $|\phi|$, $\chi^{2}$, and
$\langle\delta\phi^{2}\rangle$ during the early stage of reheating for the
coupling $g=2.0\times 10^{-3}$ with $\beta=-4.4$, $m=6.0\times 10^{-6}M_{\rm
Pl}$, and $\Gamma=1.0\times 10^{-13}M_{\rm Pl}$, in which case
$q_{I}=5.6\times 10^{4}$. In this case, the variance
$\langle\delta\phi^{2}\rangle$ starts to increase by parametric resonance and
eventually catches up with the background inflaton density $\chi^{2}$. After
$\langle\delta\phi^{2}\rangle$ grows to the order $m^{2}/g^{2}$, the
backreaction of created particles starts to violate the coherent oscillation
of $\chi$. In Fig. 4, this property can be seen after $\chi^{2}$ drops below
$\langle\delta\phi^{2}\rangle$. The growth of $\langle\delta\phi^{2}\rangle$
terminates by the violation of coherent oscillations of $\chi$. In the regime
where $q_{I}$ is sufficiently larger than 1, our numerical analysis shows that
the maximum values of $\langle\delta\phi^{2}\rangle$ reached during preheating
have the approximate dependence $\langle\delta\phi^{2}\rangle_{\rm max}\propto
1/\sqrt{q_{I}}$. This property agrees with what was found for a minimally
coupled theory ($\beta=0$) Khlebnikov and Tkachev (1997b). This means that,
for increasing $g$ in the range $q_{I}\gg 1$,
$\langle\delta\phi^{2}\rangle_{\rm max}$ tends to be suppressed.

In Fig. 4, we observe that the homogeneous field $\phi$ stops growing after
$\langle\delta\phi^{2}\rangle$ catches up with $\chi^{2}$. Hence the
backreaction induced by the growth of $\langle\delta\phi^{2}\rangle$
terminates the parametric excitation of $\phi$ as well. For $g=2.0\times
10^{-3}$, the maximum value of $|\phi|$ reached during preheating is of order
$\phi_{\rm max}=10^{-40}M_{\rm Pl}$. If we do not take the backreaction of
created particles into account, $|\phi|$ continues to grow by the time at
which the resonance parameter $q$ drops below the order 1. Without the
backreaction effect, the maximum field value $\phi_{\rm max}$ for $g=2.0\times
10^{-3}$ is of order $10^{-21}M_{\rm Pl}$, which is enormously larger than the
value $10^{-40}M_{\rm Pl}$ obtained by implementing the backreaction (see Fig.
4).

In Fig. 5, we plot the evolution of $|\phi|$ for four different values of $g$.
Provided that $g\gtrsim 10^{-4}$, $|\phi|$ is initially amplified by
parametric resonance. For $g=1.0\times 10^{-4}$, which corresponds to the
parameter $q_{I}=1.4\times 10^{2}$, there is an initial short period in which
$|\phi|$ is enhanced, but the growth is limited due to the early entry to the
region $q\lesssim 1$. For $g\gtrsim 10^{-2}$, the growth of $|\phi|$ is
terminated by the backreaction of created particles. When $g\simeq 10^{-2}$,
we find that $\phi_{\rm max}$ is of order $10^{-38}M_{\rm Pl}$. In this case,
the maximum value of $|\phi|$ obtained without the backreaction is of order
$10^{-3}M_{\rm Pl}$, which is $10^{35}$ times as large as the value
$10^{-38}M_{\rm Pl}$. This shows the importance of properly implementing the
backreaction effect to estimate the maximum value of $\phi_{\rm max}$ reached
during preheating.

Figure 5:  Evolution of $|\phi|/M_{\rm Pl}$ versus $mt$ during the early stage
of reheating for the quadratic potential $V(\chi)=m^{2}\chi^{2}/2$ with
$\beta=-4.4$, $m=6.0\times 10^{-6}M_{\rm Pl}$, and $\Gamma=1.0\times
10^{-13}M_{\rm Pl}$. Each line corresponds to (a) $g=1.0\times 10^{-4}$
(black), (b) $g=3.0\times 10^{-3}$ (red), (c) $g=1.5\times 10^{-2}$ (blue),
and (d) $g=5.0\times 10^{-2}$ (green).

As $g$ increases in the coupling range $g\gtrsim 10^{-2}$, $\phi_{\rm max}$
tends to be suppressed in comparison to the value around $g=10^{-2}$. Indeed,
we can confirm this property in Fig. 5 for the coupling $g=5.0\times 10^{-2}$.
Then, for $g\gtrsim 10^{-4}$, the maximum values of $|\phi|$ obtained under
the Hartree approximation are in the range

$\phi_{\rm max}\lesssim 10^{-38}M_{\rm Pl}\,.$ (60)

This is significantly smaller than the upper limit $|\phi_{\rm R}|\simeq
10^{-11}M_{\rm Pl}$ constrained from the post-Newtonian bound at the end of
reheating.

We note that our approximation scheme does not incorporate nonlinear effects
like rescattering. However, it is known that the rescattering of $\phi$
particles off the inflaton condensate tends to limit the growth of
$\langle\delta\phi^{2}\rangle$ further Khlebnikov and Tkachev (1997a). This
means that the growth of the homogeneous field $\phi$ should be also limited,
so it is expected that the maximum values of $\phi$ do not exceed those
derived under the Hartree approximation. After the rescattering of produced
particles, the variance $\langle\delta\phi^{2}\rangle$ reaches an equilibrium
state. At this stage, the significant amplification of the homogeneous field
from $|\phi|\lesssim 10^{-38}M_{\rm Pl}$ to the value exceeding the order
$10^{-11}M_{\rm Pl}$ by the end of reheating is unexpected.

In particular, the effective mass squared (14) is approximately given by
$m_{\rm eff}^{2}\simeq g^{2}\chi^{2}+3\beta H^{2}/2$ in the temporal matter
era during reheating. Until the onset of the radiation era at which the
inflaton field $\chi$ completely decays to radiation, the negative coupling
term $3\beta H^{2}/2$ does not completely dominate over the positive
contribution $g^{2}\chi^{2}$ to $m_{\rm eff}^{2}$. Hence the strong tachyonic
instability of $\phi$ induced by the negative nonminimal coupling constant
$\beta$ is not expected at the late stage of reheating. Eventually, the Born
decay term $\Gamma\dot{\chi}$ in Eq. (8) starts to work to convert the
inflaton density to the radiation density.

Thus, for $g\gtrsim 10^{-4}$, the growth of $|\phi|$ saturated by the
backreaction during preheating allows the field value $\phi_{\rm R}$ at the
end of reheating consistent with the post-Newtonian constraint.

For $g\lesssim 10^{-4}$, the preheating stage is absent and hence we do not
need to incorporate the backreaction effect. For $g\lesssim 10^{-5}$, the
coupling $g$ does not overwhelm the negative nonminimal coupling $\beta$ in
the effective mass of $\phi$. Then, $\phi$ is not subject to the strong
suppression during inflation. In Fig. 6, we plot the evolution of
$|\phi|/M_{\rm Pl}$ during inflation and reheating for three different values
of $g$ with the decay constant $\Gamma=1.0\times 10^{-9}M_{\rm Pl}$. We
integrate the background equations of motion by the end of reheating (time
$t_{\rm R}$) at which $\rho_{r}$ catches up with the inflaton density
$\rho_{\chi}=\dot{\chi}^{2}/2+V(\chi)$. In case (a) the field value at
$t=t_{\rm R}$ is of order $10^{-10}M_{\rm Pl}$, so it exceeds the solar-system
bound $|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$. In cases (b) and (c), the
field values at $t=t_{\rm R}$ are of orders $10^{-15}M_{\rm Pl}$ and
$10^{-20}M_{\rm Pl}$, respectively, which are well within the bound
$|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$. This means that, for
$\Gamma=1.0\times 10^{-9}M_{\rm Pl}$, the coupling in the range $g\geq
1.28\times 10^{-5}$ can be consistent with the solar-system constraint.

Figure 6:  Evolution of $|\phi|/M_{\rm Pl}$ versus $\ln(mt)$ during inflation
and reheating for the quadratic potential $V(\chi)=m^{2}\chi^{2}/2$ with
$\beta=-4.4$, $m=6.0\times 10^{-6}M_{\rm Pl}$, and $\Gamma=1.0\times
10^{-9}M_{\rm Pl}$. Each line corresponds to (a) $g=1.27\times 10^{-5}$ (red),
(b) $g=1.28\times 10^{-5}$ (blue), and (c) $g=1.29\times 10^{-5}$ (green). Two
black dashed lines show the evolution of the inflaton density $\rho_{\chi}$
and radiation density $\rho_{r}$. We integrate the background equations by the
time at which $\rho_{r}$ catches up with $\rho_{\chi}$.

In Fig. 6, we observe that, after the initial rapid decrease of $|\phi|$
during inflation, $|\phi|$ exhibits mild decrease in the reheating period.
This reflects the fact that $g^{2}\chi^{2}$ is larger than the term
$3|\beta|H^{2}/2$ during reheating. The field value $\phi_{\rm R}$ at the end
of reheating depends on the decay constant $\Gamma$. For decreasing $\Gamma$,
$|\phi_{\rm R}|$ tends to be smaller because of the longer period of
reheating. We also note that the field value $\phi_{I}$ at the beginning of
reheating depends on the duration of inflation. For the minimum number of
e-foldings $N=60$, the criterion consistent with the bound $|\phi_{\rm
R}|\lesssim 10^{-11}M_{\rm Pl}$ is given by

$g\geq 1.3\times 10^{-5}\,,$ (61)

irrespective of the values of $\Gamma$ smaller than $H$ at the end of
inflation. Thus, the mechanism proposed by Anson et al. Anson  _et al._
(2019a) works even for small couplings of order $10^{-5}$.

###  V.2 $\alpha$-attractor with $\alpha=1$

We also study the dynamics of $\phi$ in the $\alpha$-attractor model with
$\alpha=1$. In the $\alpha$-attractor model, the field value
$\chi_{I}=0.940M_{\rm Pl}$ at the onset of reheating is smaller than that for
the exact quadratic potential. Hence we require larger couplings $g$ to
enhance both $\phi$ and $\langle\delta\phi^{2}\rangle$ by parametric resonance
in comparison to the case of quadratic potential. In the numerical simulation
of Fig. 7, we observe that $|\phi|$ does not grow for $g=1.0\times 10^{-4}$,
but parametric resonance occurs for $g=1.0\times 10^{-3}$.

Numerically, we find that the maximum value of $|\phi|$ reached for
$g=5.0\times 10^{-3}$ is of order $\phi_{\rm max}=10^{-38}M_{\rm Pl}$ under
the Hartree approximation. As $g$ increases in the range $g\gtrsim 5\times
10^{-3}$, $\phi_{\rm max}$ tends to be decreased. This property can be seen in
Fig. 7 for the couplings $g=1.0\times 10^{-2}$ and $g=1.0\times 10^{-1}$.
Thus, for any couplings with $g\gtrsim 10^{-4}$, the maximum values of
$|\phi|$ reached during preheating are in the range

$\phi_{\rm max}\lesssim 10^{-38}M_{\rm Pl}\,,$ (62)

which are again much smaller than the post-Newtonian upper limit $\phi_{\rm
R}\simeq 10^{-11}M_{\rm Pl}$ at the end of reheating.

Figure 7:  Evolution of $|\phi|/M_{\rm Pl}$ versus $mt$ during the early stage
of reheating for the $\alpha$-attractor potential (15) with $\alpha=1$,
$\beta=-4.4$, $m=1.1\times 10^{-5}M_{\rm Pl}$, and $\Gamma=1.0\times
10^{-13}M_{\rm Pl}$. Each line corresponds to (a) $g=1.0\times 10^{-4}$
(black), (b) $g=1.0\times 10^{-3}$ (red), (c) $g=5.0\times 10^{-3}$ (blue),
(d) $g=1.0\times 10^{-2}$ (green), and (e) $g=1.0\times 10^{-1}$ (brown).

If we ignore the backreaction of created $\phi$ particles, $\phi_{\rm max}$
can be significantly larger than the upper limit (62). When $g=1.0\times
10^{-1}$, for example, the numerical value of $\phi_{\rm max}$ derived by
neglecting the backreaction effect is of order $10^{-2}M_{\rm Pl}$, which is
very much larger than $10^{-11}M_{\rm Pl}$. Thus, inclusion of the
backreaction is crucially important for the proper estimation of $\phi_{\rm
max}$. The upper limit $\phi_{\rm max}={\cal O}(10^{-38}M_{\rm Pl})$ derived
in the presence of the preheating stage is similar to that obtained for the
quadratic inflaton potential.

Figure 8:  Evolution of $|\phi|/M_{\rm Pl}$ versus $\ln(mt)$ during inflation
and reheating for the $\alpha$-attractor potential (15) with $\alpha=1$,
$\beta=-4.4$, $m=1.1\times 10^{-5}M_{\rm Pl}$, and $\Gamma=1.0\times
10^{-9}M_{\rm Pl}$. Each line corresponds to (a) $g=6.7\times 10^{-6}$ (red),
(b) $g=8.0\times 10^{-6}$ (blue), and (c) $g=1.6\times 10^{-5}$ (green). Two
black dashed lines show the evolution of the inflaton density $\rho_{\chi}$
and radiation density $\rho_{r}$. The background equations are integrated by
the time at which $\rho_{r}$ catches up with $\rho_{\chi}$.

For the coupling $g$ smaller than the order $10^{-5}$, the field $\phi$ is not
subject to strong suppression during inflation. Since parametric resonance
does not occur for such small couplings, we do not need to implement the
backreaction of created $\phi$ particles. In Fig. 8, we show the evolution of
$|\phi|$ during inflation and reheating for three different values of $g$ with
the decay constant $\Gamma=1.0\times 10^{-9}M_{\rm Pl}$. In cases (a) and (b)
the field values at the end of reheating (time $t=t_{\rm R}$) are of order
$10^{-11}M_{\rm Pl}$ and $10^{-25}M_{\rm Pl}$, respectively, so case (b) is
consistent with the solar system limit $|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm
Pl}$.

We need to caution that, in both cases (a) and (b), $|\phi|$ grows during
reheating due to the dominance of the negative nonminimal coupling $3\beta
H^{2}/2$ relative to $g^{2}\chi^{2}$ in $m_{\rm eff}^{2}$ (whose dominance
starts to occur during inflation). If we consider a very low energy-scale
reheating where the reheating temperature is of order MeV Hannestad (2004);
Hasegawa  _et al._ (2019), the decay constant is of order $\Gamma\simeq
1~{}{\rm sec}^{-1}\simeq 10^{-43}M_{\rm Pl}$. In this case, the time at the
end of reheating is estimated as $t_{\rm R}\simeq 1/\Gamma\simeq 10^{43}M_{\rm
Pl}^{-1}$ and hence $\ln(mt_{\rm R})\simeq 88$. In case (b), for example, the
order of $|\phi|$ increases by one order of magnitude during the time interval
$\ln(m\Delta t)=2$. Then, for the MeV scale reheating, the field value
$\phi_{\rm R}$ exceeds the limit even in case (b).

To avoid the increase of $|\phi|$ during reheating, we require that $m_{\rm
eff}^{2}$ is positive by the end of inflation. As we discussed in Sec. IV.2,
this translates to the condition $g>1.61\times 10^{-5}$ for the
$\alpha$-attractor with $\alpha=1$. In case (c) of Fig. 8, we plot the
evolution of $|\phi|$ for $g=1.6\times 10^{-5}$. This corresponds to the
marginal case in which the monotonic growth of $|\phi|$ during reheating can
be avoided. Then, provided that

$g\geq 1.6\times 10^{-5}\,,$ (63)

the field value at the end of reheating does not exceed the solar-system limit
irrespective of the decay constant $\Gamma$. For the coupling range $g\gtrsim
10^{-3}$ in which the preheating epoch is present, the maximum field value is
limited as Eq. (62). In this coupling regime, $g^{2}\chi^{2}$ dominates over
$3|\beta|H^{2}/2$ after inflation and hence the growth of $|\phi|$ from the
end of preheating to the onset of radiation era is not expected to occur.

##  VI Conclusions

In this paper, we studied the cosmological evolution of a scalar field $\phi$
in the presence of a nonminimal coupling $e^{-\beta\phi^{2}/(2M_{\rm
pl}^{2})}R$ and a four-point coupling $g^{2}\phi^{2}\chi^{2}/2$ between $\phi$
and the inflaton field $\chi$. This nonminimal coupling gives rise to the
phenomenon of spontaneous scalarization of NSs for $\beta\leq-4.35$. If we
apply the original DEF scenario to cosmology, the scalar field $\phi$ is
subject to tachyonic instability during the periods of cosmic acceleration and
matter dominance. In Ref. Anson  _et al._ (2019b), it was argued that the
coupling $g^{2}\phi^{2}\chi^{2}/2$ allows a possibility for curing this
problem by realizing a positive mass squared during inflation.

Since the dynamics of the field $\phi$ during the post inflationary reheating
period was not addressed in the literature, we have studied the cosmological
evolution from the onset of inflation to today including the reheating stage.
To satisfy solar-system constraints, the field value at the beginning of
radiation era is constrained as Eq. (24). For the coupling $\beta$ relevant to
the occurrence of spontaneous scalarization, this bound corresponds to
$|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$.

Provided that the effective mass squared (14) is larger than the order of
$H^{2}$ during inflation, the field $\phi$ is subject to the exponential
suppression ($|\phi|\propto a^{-3/2}$) by the end of inflation. For the
quadratic and $\alpha$-attractor potentials with $\alpha=1$, we showed that
this suppression of $\phi$ occurs for the coupling $g$ in the ranges (33) and
(34), respectively, whose minimum values are both of order $10^{-5}$. With
these two potentials, the parametric excitation of $\phi$ and its
perturbations during preheating can occur for $g\gtrsim 10^{-4}$ and $g\gtrsim
10^{-3}$, respectively. If we do not take the backreaction of created $\phi$
particles into account, the maximum values of $|\phi|$ reached during
preheating can exceed the order of $10^{-11}M_{\rm Pl}$. Incorporating the
backreaction under the Hartree approximation, however, we found that
$\phi_{\rm max}$ is smaller than the order of $10^{-38}M_{\rm Pl}$. After the
termination of parametric resonance, the further growth of $|\phi|$ is
unexpected by the end of reheating because the $g^{2}\chi^{2}$ term dominates
over the negative nonminimal coupling in the equation of motion of $\phi$.

In the regime of small couplings $g$ without the preheating stage, we also
numerically solved the background equations of motion by the end of reheating
with the Born decay term taken into account. For the quadratic inflaton
potential, the amplitude of $\phi$ decreases during reheating in the range of
couplings $g$ that gives a positive mass squared during inflation. In this
case, provided $g\geq 1.3\times 10^{-5}$, the model is consistent with the
solar-system bound $|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$. For the
$\alpha$-attractor with $\alpha=1$, $m_{\rm eff}^{2}$ can change its sign
during inflation by the presence of a negative nonminimal coupling even if
$m_{\rm eff}^{2}>0$ at the onset of inflation. In such cases, the growth of
$|\phi|$ also occurs in the reheating period, see Fig. 8. If we consider a
low-scale reheating scenario with the reheating temperature of order MeV, the
monotonic growth of $|\phi|$ during a long period of the reheating era can
conflict with the limit $|\phi_{\rm R}|\lesssim 10^{-11}M_{\rm Pl}$. Provided
$g\geq 1.6\times 10^{-5}$, we found that the $\alpha$-attractor with
$\alpha=1$ can be consistent with the solar-system bound irrespective of the
decay constant $\Gamma$.

We thus showed that, for natural couplings in the range $g\gtrsim 10^{-5}$,
the scenario proposed by Anson et al. leads to the viable cosmological
evolution of $\phi$ consistent with today’s local gravity constraints. Since
the inflaton field decays to radiation by the onset of radiation era, it does
not affect the process of spontaneous scalarization of NSs which can occur in
later cosmological epochs. Interestingly, the field $\phi$ responsible for
spontaneous scalarization can also exhibit the phenomenon of parametric
resonance in the early Universe. It is then possible to probe this scenario
not only from the gravitational waveform emitted from compact binaries but
also from the gravitational wave background.

While we focused on the cosmology in the DEF model with the coupling
$g^{2}\phi^{2}\chi^{2}/2$, it may be of interest to extend the analysis to the
cases in which higher-order $\phi$-dependent terms like $\phi^{4}$ in
$F(\phi)$ Anderson  _et al._ (2016) or k-essence terms like $X^{2}$ Higashino
and Tsujikawa (2022) are also present. This may widen parameter spaces of the
coupling constant $\beta$ consistent with binary pulsar constraints. These
issues are left for future works.

## ACKNOWLEDGMENTS

We thank Masato Minamitsuji for useful discussions. This research project is
supported by National Research Council of Thailand (NRCT): NRCT5-RGJ63009-110.
SP is supported by a Waseda University Grant for Special Research Project (No.
2022C- 632). ST is supported by the Grant-in-Aid for Scientific Research Fund
of the JSPS Nos. 19K03854 and 22K03642.

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