# Gravitational waves from a curvature-induced phase transition of a Higgs-
portal dark matter sector

Andreas Mantziris  0000-0001-5899-670X Orfeu Bertolami  0000-0002-7672-0560

###### Abstract

The study of interactions between dark matter and the Higgs field opens an
exciting connection between cosmology and particle physics, since such
scenarios can impact the features of dark matter as well as interfering with
the spontaneous breaking of the electroweak symmetry. Furthermore, such Higgs-
portal models of dark matter should be suitably harmonised with the various
epochs of the universe and the phenomenological constraints imposed by
collider experiments. At the same time, the prospect of a stochastic
gravitational wave background offers a promising new window into the
primordial universe, which can complement the insights gained from
accelerators. In this study, we examined whether gravitational waves can be
generated from a curvature-induced phase transition of a non-minimally coupled
dark scalar field with a portal coupling to the Higgs field. The main
requirement is that the phase transition is of first order, which can be
achieved through the introduction of a cubic term on the scalar potential and
the sign change of the curvature scalar. This mechanism was investigated in
the context of a dynamical spacetime during the transition from inflation to
kination, while also considering the possibility for inducing electroweak
symmetry breaking in this manner for a sufficiently low reheating temperature
when the Higgs-portal coupling is extremely weak. We considered a large range
of inflationary scales and both cases of positive and negative values for the
non-minimal coupling, while taking into account the bound imposed by Big Bang
Nucleosythesis. The resulting gravitational wave amplitudes are boosted by
kination and thus constrain the parameter space of the couplings
significantly. Even though the spectra lie at high frequencies for the
standard high inflationary scales, there are combinations of parameter space
where they could be probed with future experiments.

##  1 Introduction

The study of interactions between Dark Matter (DM) and the Higgs field opens
interesting prospects for possible investigations and phenomenology of DM
through collider experiments and cosmological considerations. The first models
[1, 2, 3, 4, 5] along these lines considered an extension of the Standard
Model (SM) with an additional singlet scalar field and a portal coupling to
the Higgs. Several related models have been studied in the literature [6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], with these
scenarios being known as “Higgs-portal” DM. In fact, a putative coupling to
dark energy has also been suggested in Refs. [24, 25]. Naturally, these set-
ups can have an impact on the features of DM itself [26, 27, 28], as well as
interfering with spontaneous breaking of the electroweak (EW) symmetry [29].
In any case, it is crucial that a model of Higgs-portal DM can be suitably
harmonised with the various epochs of our cosmological history and complying
with observational and experimental constraints that it inherits.

At the same time, primordial Gravitational Waves (GW) are a ubiquitous feature
of the early universe, with various production mechanisms during inflation
[30, 31, 32, 33], (p)reheating [34, 35, 36, 37, 38, 39, 40], and the
radiation-domination epoch [41, 42, 43, 44, 45, 46, 47], which cover a large
section of the frequency band, since their scales range from Planckian values
down to the EW scale [48, 49, 50]. Since GWs are ripples in spacetime that
travel unhindered after emission, any detected GW signature of primordial
origin would provide new direct insights into the early universe and the
responsible mechanism for its generation. In this sense, cosmological phase
transitions have been of increasing interest due to their ability to generate
a stochastic GW background that can potentially be detectable with upcoming
surveys [51]. In particular, theories Beyond the Standard Model (BSM) offer a
promising playground for realising strong First-Order Phase Transitions (FOPT)
with prominent GW spectra, where the nature of the ensued phase transition
depends on a conjugation of conditions for the background and the parameters
of the model under study. While the majority of these considerations revolve
around thermal PTs of BSM models at relatively low energy scales of the early
universe, with the temperature decrease of the universe after radiation-
domination inducing the transition, non-thermal PTs [52, 53, 54, 55, 56] can
offer a promising window in the earlier moments of the primordial universe
that can complement the insights from different mechanisms and observables.

The purpose of this study was to combine the models developed in Refs. [29]
and [57] to examine whether GWs can be generated from a curvature-induced
phase transition of a non-minimally coupled scalar DM model with a portal to
the Higgs field. This was explored in a dynamical spacetime during the latest
phase of inflation before reheating, while considering the possibility of a
curvature-induced breaking of the EW symmetry. The cosmological setting
considered was the transition from the inflationary to the kination [58] or
deflation [59] era, where the kinetic energy dominates over the potential of
the inflaton field. Although an amount of kination is unavoidable in most
inflationary models, here we adopt the framework of quintessential-like
inflation that is characterised by an extended period of kination, since the
inflaton rolls unopposed down the approximately flat tail of its potential
after the end of inflation [60, 61, 62]. During long kination periods, the
total energy density is diluted with the scale factor as $\rho\sim a^{-6}$,
resulting in the dominance of any other form of matter over the background
[63]. This behaviour enables an efficient heating of the universe into the
standard Hot Big Bang (HBB) epoch through various mechanisms, although not
along the usual prescriptions used in the typical oscillatory models and
without any direct inflaton-SM couplings,to avoid the destabilisation of the
potential’s late-time plateau [63]. The reason for focussing on prolonged
kination scenarios, which is also motivated in string cosmology [64], is that
it amplifies any corresponding stochastic GW background produced at these
scales, making it a promising context for studying cosmological phase
transitions [57].

The outline of the paper is as follows. In Sec. 2.1, we present an overview of
the cosmological setting for our model, following the evolution of the non-
minimally coupled spectator Higgs-portal scalar field from inflation to
kination. The dynamical features that induce the phase transition and EW
symmetry breaking for both cases of negative and positive non-minimal coupling
are discussed in Sec. 2.2, and Sec. 2.3 includes the necessary consistency
analysis for the parameter space of the couplings. The theoretical framework
for computing GW signals from bubble collisions due to a FOPT in vacuum is
presented in Sec. 3, with the resulting spectra of the parameter space
specified in Sect. 3.3. In Section 4, we present our conclusions for the GW
signatures obtained, together with the phenomenological constraints imposed
that link our model to DM and the breaking of the EW symmetry. In Appendix A,
we offer an overview of the self-interacting massless Higgs-portal DM model of
Ref. [29], as a reference point for comparison. Finally, Appendix B addresses
the necessary transformation of the scalar potential for $\xi_{\phi}<0$, so
that we can apply the standard formalism of cosmological phase transitions of
Sec. 3, when the true vacuum of the PT lies at the origin.

##  2 Cosmological evolution of a spectator scalar field in the early Universe

###  2.1 Non-minimally coupled scalar in a quintessential-like inflationary
context

Following the motivation of Ref. [29], we investigate the cosmological
implications of a dark sector with a portal to the Standard Model via the
Higgs field. In particular, we are interested in the breaking of the
electroweak symmetry via a non-thermal mechanism but also exploring whether
there can be any gravitational wave signatures associated with such models.
Therefore, we extend the SM with the addition of a massless self-interacting
scalar field, $\phi$, which as discussed in Ref. [29], plays the role of dark
matter once it acquires a mass after EW symmetry breaking (EWSB) and
oscillates around the minimum of its potential (see Sec. 2.2), and a decoupled
inflaton field $\varphi$ that drives the accelerated expansion of the
Universe. This theory consists of two scalar fields, the SM Higgs, $h$, and
the dark scalar, which are coupled to each other via the Higgs portal
coupling, $g$, and non-minimally coupled to gravity. The latter terms are
radiatively generated and required for the renormalizability of the stress-
energy tensor in curved spacetime [65], and we cannot disregard the non-
minimal couplings $\xi_{h},\,\xi_{\phi}$ by setting them to zero, as they will
become finite at some other renormalization scale due to their running [66].
It is intuitive to add a renormalizable cubic term for the BSM scalar, which
can easily shape the potential into a double-well that could lead to strong
FOPTs in many contexts (see e.g. Ref. [67]). The full action is given by

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}-\xi_{\phi}\phi^{2}-\xi_{h}h^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{g^{2}}{4}h^{2}\phi^{2}+\frac{\sigma}{3}\phi^{3}-\frac{\lambda_{\phi}}{4}\phi^{4}-\mathcal{L}_{h}-\mathcal{L}_{\varphi}\right]\,,$
(2.1)

where $M_{P}=2.435\times 10^{18}$ GeV is the reduced Planck mass, $\sigma$ is
the dimensionful cubic coupling, $\lambda_{\phi}$ is the dark scalar self-
interaction, $\mathcal{L}_{h}$ is the Higgs Lagrangian, and
$\mathcal{L}_{\varphi}$ is the inflaton Lagrangian density. All couplings are
positive, except for $\xi_{\phi}$ that can have either sign, but note that the
cubic term with the opposite sign from the others places the additional vacuum
of the double-well potential at positive field values.

The interplay of the scalar potential described above with the time-dependent
scalar $R$ from the non-minimally coupled term can lead to the production of
GW signals from a curvature-induced phase transition, as in Ref. [57]. In
order to keep our setup as simple as possible, we limit our model to
renormalizable terms only with no higher-order factors. However, note that
even though there is no $\mathcal{Z}_{2}$ symmetry to prohibit linear terms of
$\phi$ in the BSM potential, we avoid including them, since they can be
eliminated by a suitable redefinition [68]. Any dark scalar-inflaton terms
would be irrelevant for the purposes of this study since they would not affect
the difference between the vacua for the phase transition, only the location
of the potential’s minima, according to the considerations discussed in Sec.
2.2. Higgs-inflaton terms can also be disregarded in this context, given that
during inflation $h=0$ and by the time of EW symmetry breaking, the inflaton
has a negligibly small value. We highlight that we do not specify an
inflationary model in $\mathcal{L}_{\varphi}$, to be as model-independent as
possible, and we just utilise a generic parametrisation of quintessential-like
inflation, as detailed below. Thus, we assume that the decoupled inflationary
sector behaves in the usual manner that provides the necessary exponential
expansion of spacetime in the early universe, without including any inflaton-
curvature terms in Eq. (2.1) for the sake of brevity.

The equation of motion of the inflationary sector,
$\mathcal{L}_{I}=\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-U_{I}$,
in Friedmann– Lemaître–Robertson–Walker (FLRW) spacetime is given by

$\displaystyle\ddot{\varphi}+3H\dot{\varphi}+\frac{dU_{I}}{d\varphi}=0\,,$
(2.2)

where $H=\frac{\dot{a(t)}}{a}$ is the Hubble rate, $a(t)$ is the scale-factor
of the expanding spacetime, and $U_{I}$ is the inflaton’s potential. The
corresponding energy density and pressure of the perfect cosmological fluid
are given by

$\displaystyle\rho_{\varphi}$
$\displaystyle=\frac{1}{2}\dot{\varphi}^{2}+U_{I}(\varphi)\,,$ (2.3)
$\displaystyle p_{\varphi}$
$\displaystyle=\frac{1}{2}\dot{\varphi}^{2}-U_{I}(\varphi)\,,$ (2.4)

respectively, and the dot denotes a derivative with respect to time. These
quantities can be parameterised by the Equation of State (EoS) parameter
$w(t)\equiv\frac{p_{\varphi}}{\rho_{\varphi}}$ and by solving the Friedmann
and Raychaudhuri equations

$\displaystyle H^{2}$ $\displaystyle=\frac{\rho_{\varphi}}{3M_{p}^{2}}\,,$
(2.5) $\displaystyle\dot{H}+H^{2}$
$\displaystyle=-\frac{1}{6M_{p}^{2}}\left(\rho_{\varphi}+3p_{\varphi}\right)\,,$
(2.6)

we can express the first slow-roll parameter in terms of the EoS parameter

$\displaystyle\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}=\frac{3}{2}\left[1+w(t)\right]\,.$
(2.7)

This differential equation can be solved once the dynamics of the EoS are
specified.

As inflation approaches its end, the kinetic term dominates the potential in
the energy density of the inflaton, in contrast to the slow-roll regime. This
results in an epoch with different dynamics, nested between inflation and the
usual radiation-domination period of the HBB era [63]. When focusing on the
transition from inflation to kination, a convenient parametrisation for the
barotropic parameter that exhibits a smooth and continuous function linking
the de Sitter value $w_{\rm inf}=-1$ with $w_{\rm kin}=1$ during kination
[57], without being restricted to a particular $U_{I}$, would be

$\displaystyle w(t)=\tanh{\left[\beta_{w}(t-t_{0})\right]}\,,$ (2.8)

where the transition is centred at $t_{0}$ as shown in Fig. 1, the free
parameter $\beta_{w}>0$ controls its speed and can be adjusted accordingly to
match specific inflationary models111In general, we cannot reverse-engineer to
obtain the inflationary model from the EoS parameter without some additional
information about the inflaton’s behaviour, as
$U_{I}(\varphi,t)=\frac{\dot{\varphi}^{2}}{2}\frac{w(t)-1}{w(t)+1}$.. In this
treatment, $w$ tends asymptotically to $w_{\rm kin}=1$ mimicking
quintessential-like inflationary models, where the inflaton does not oscillate
around a minimum, but instead rolls unobstructed along an additional plateau
around the scale of the cosmological constant [63]. In this case, the Hubble
rate can be calculated analytically from Eq. (2.7) as

$\displaystyle\frac{H(t)}{H_{\textrm{inf}}}=\frac{2}{3}\left[H_{\rm
inf}(t-t_{0})+\left(\frac{H_{\rm
inf}}{\beta_{w}}\right)\ln{\left[2\cosh{\left(\frac{\beta_{w}}{H_{\rm
inf}}H_{\rm inf}(t-t_{0})\right)}\right]}+\frac{2}{3}\right]^{-1}\,,$ (2.9)

where $H_{\rm inf}$ is the scale of inflation, and we set $t_{0}=0$, meaning
that roughly $t<0$ corresponds to inflation and $t>0$ to kination [57].
Finally, it is important to consider the cosmological evolution of spacetime
curvature in this setting, since the Higgs and the dark scalar are non-
minimally coupled to it. A generic expression for the curvature scalar in
terms of the EoS parameter is given by

$\displaystyle R(t)=3\left[1-3w(t)\right]H^{2}(t)\,,$ (2.10)

where its capability for symmetry breaking mechanisms is manifestly evident by
noticing that it switches signs due to evolution of the barotropic parameter
$w$, as depicted in Fig. 1. This feature of the Ricci scalar makes it a core
ingredient of the phase transition triggering mechanism utilised in this
study, which was proposed originally in Ref. [57].

Figure 1: Evolution of the EoS parameter (2.8), the Ricci scalar (2.10), and
the Hubble rate (2.9) from inflation to kination in Hubble time, with
$\beta_{w}=10H_{\rm inf}$.

###  2.2 Curvature-induced phase transitions and electroweak symmetry breaking

In this work, we are interested in the cosmological implications of the dark
scalar field after inflation, and specifically on its dynamic potential from
Eq. (2.1), which is written as

$\displaystyle
V(\phi,h,R)=\frac{1}{2}\left(\xi_{\phi}R+\frac{g^{2}}{2}h^{2}\right)\phi^{2}-\frac{\sigma}{3}\phi^{3}+\frac{\lambda_{\phi}}{4}\phi^{4}\,,$
(2.11)

where we have included the Higgs-portal term that it shares with the Higgs
potential and the explicit dependence on the curvature scalar. The form of the
BSM potential in Eq. (2.11) is chosen as such so that the curvature-induced PT
mechanism of Ref. [57] is used to explore the GW signatures of the oscillating
DM model of Ref. [29], with the minimal addition of a cubic term to the
potential so that it possesses a double-well shape that can readily provide a
FOPT. To simplify the description of the vacuum transition, we use the semi-
analytic expressions from Ref. [68], where any fourth-order polynomial
potential can be written in the reduced dimensionless form as

$\displaystyle\widetilde{V}(\tilde{\phi},\delta)=\frac{\delta}{2}\tilde{\phi}^{2}-\tilde{\phi}^{3}+\frac{1}{4}\tilde{\phi}^{4}\,.$
(2.12)

With this prescription, the dark scalar potential (2.11) resembles the form
above when the field redefinition and its corresponding quadratic coupling are
given by

$\displaystyle\tilde{\phi}$
$\displaystyle=\frac{3\lambda_{\phi}}{\sigma}\phi\,,$ (2.13)
$\displaystyle\delta(h,R)$
$\displaystyle=\frac{9\lambda_{\phi}}{\sigma^{2}}\left(\xi_{\phi}R+\frac{g^{2}}{2}h^{2}\right)\,,$
(2.14)

with the overall rescaling of the potential being
$V=\frac{(\sigma/3)^{4}}{\lambda_{\phi}^{3}}\tilde{V}$. The cubic term gives a
double-well shape to the potential, as long as $4\delta/9<1$, with three
extrema at

$\displaystyle\phi_{1}$
$\displaystyle=0\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\tilde{\phi}_{1}=0\,,$
(2.15) $\displaystyle\phi_{2}(h,R)$
$\displaystyle=\frac{\sigma}{2\lambda_{\phi}}\left(1+\sqrt{1-\frac{4\lambda_{\phi}\left(\xi_{\phi}R+\frac{g^{2}}{2}h^{2}\right)}{\sigma^{2}}}\right)\,,\quad\tilde{\phi}_{2}(\delta)=\frac{3}{2}\left(1+\sqrt{1-\frac{4\delta}{9}}\right)\,,$
(2.16) $\displaystyle\phi_{3}(h,R)$
$\displaystyle=\frac{\sigma}{2\lambda_{\phi}}\left(1-\sqrt{1-\frac{4\lambda_{\phi}\left(\xi_{\phi}R+\frac{g^{2}}{2}h^{2}\right)}{\sigma^{2}}}\right)\,,\quad\tilde{\phi}_{3}(\delta)=\frac{3}{2}\left(1-\sqrt{1-\frac{4\delta}{9}}\right)\,.$
(2.17)

The evolution of the potential is encoded in the time dependence of the Ricci
scalar $R(t)$, and thus the parameter $\delta$ effectively controls the phase
transition. In particular, the range $0\leq\delta\leq 2$ corresponds to
configurations between a vanishing barrier and two degenerate minima. However,
note that the value of $\delta$, as the redefinition of the couplings in Eq.
(2.14), is not strictly restricted to this range. Putting together Eqs. (2.10)
and (2.14), after disregarding the Higgs portal term222When the Higgs field
acquires a non-vanishing vacuum expectation value (vev), Eq. (2.18) admits an
additional term $9\lambda_{\phi}g^{2}v^{2}/2\sigma^{2}$. This is negligible
for the relevant parameter space and in any case irrelevant for the present
calculation, since the FOPTs here take place before EW symmetry breaking.,
allows expressing $\delta$ in a “normalised” form,

$\displaystyle\delta=C\left[\frac{1-3w(t)}{2}\right]\left(\frac{H(t)}{H_{\rm
inf}}\right)^{2}\;,\;\;C=54\lambda_{\phi}\xi_{\phi}\left(\frac{H_{\rm
inf}}{\sigma}\right)^{2}\,,$ (2.18)

where the cosmological evolution is manifestly evident and $C$ is the only
free parameter for a given inflationary scale. The potential possess an
unbroken state (only one global minimum) when $\delta>9/4$ and the couplings
therefore obey $C\left[1-3w(t)\right]>\frac{9}{2}\left(\frac{H_{\rm
inf}}{H(t)}\right)^{2}$, i.e.

$\displaystyle C$
$\displaystyle>\frac{9}{8}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{during\;inflation\,,}$
(2.19) $\displaystyle C$ $\displaystyle<-\frac{9}{4}\left(\frac{H_{\rm
inf}}{H_{\rm
kin}}\right)^{2}\approx-\frac{9}{2}\;\;\;\mathrm{during\;kination\,.}$ (2.20)

Having established a useful framework to express the BSM potential in a
convenient parametrisation, we can study its evolution as inflation comes to
an end and kination commences. We can identify the minima and maxima of the
potential according to the values of its second derivative with respect to the
field at the extrema (2.15)-(2.17),

$\displaystyle\tilde{V}^{\prime\prime}(\tilde{\phi}_{1})$
$\displaystyle\propto\delta\,,$ (2.21)
$\displaystyle\tilde{V}^{\prime\prime}(\tilde{\phi}_{2})$
$\displaystyle>0\;\mathrm{for}\;\delta<9/4\,,$ (2.22)
$\displaystyle\tilde{V}^{\prime\prime}(\tilde{\phi}_{3})$ $\displaystyle\geq
0\;\mathrm{for}\;\delta\leq 0\,,$ (2.23)
$\displaystyle\tilde{V}^{\prime\prime}(\tilde{\phi}_{3})$
$\displaystyle<0\;\mathrm{for}\;0<\delta<9/4\,.$ (2.24)

For $\xi_{\phi}<0$, while $R>0$ during inflation, $\delta$ is negative and
thus $\tilde{\phi}_{3}$ corresponds to the false vacuum (local minimum),
$\tilde{\phi}_{2}$ to the true vacuum (global minimum), and $\tilde{\phi}_{1}$
to the top of the potential barrier (local maximum), as illustrated in Fig. 2.
As $R$ decreases, the increasing $\delta$ reaches zero, where there is an
inflection point at $\tilde{\phi}_{3}=\tilde{\phi}_{1}=0$ and a global minimum
at $\tilde{\phi}_{2}=3$. During kination, $R<0$ and $\delta>0$, so that
$\tilde{\phi}_{3}$ tracks the top of the barrier and $\tilde{\phi}_{1}$ is the
new false vacuum, while $\tilde{\phi}_{2}$ continues to lie at the true
vacuum, until the minima become degenerate
$\tilde{V}(\tilde{\phi}_{1})=\tilde{V}(\tilde{\phi}_{2})=0$ at $\delta=2$, as
in the upper right panel of Fig. 2 for $C=-10$. When reaching the threshold
$\delta=9/4$, the potential has an inflection point at
$\tilde{\phi}_{2}=\frac{3}{2}$ and for $\delta>9/4$ the potential has just a
minimum at $\tilde{\phi}_{1}=0$, as shown in the lower left panel of Fig. 2
for the $C=-10$ case. Since the curvature scalar will oscillate or tend
asymptotically to zero for later times, depending on the inflationary model,
the potential will resemble the shape for $\delta=0$, which is shown in the
lower right plot of Fig. 2.

(a)

(b)

(c)

(d)

Figure 2: Evolution of the reduced dimensionless dark scalar potential for
$\xi_{\phi}<0$, from inflation (upper left) to kination (lower right), where
the vertical dotted and dash-dotted lines follow the extrema
$\tilde{\phi}_{3}$ from being the corresponding false vacua to barriers,
respectively.

According to the above description, the behaviour of the dark scalar is as
follows. During inflation, it has a large vacuum expectation value at
$\phi_{\rm inf}=\phi_{2}$, which contributes to the effective Higgs mass via
the Higgs portal,

$\displaystyle m_{h}^{\rm eff}=\sqrt{\xi_{h}R+\frac{g^{2}}{2}\phi_{\rm
inf}^{2}}=\sqrt{\xi_{h}R+\frac{g^{2}\sigma^{2}}{8\lambda_{\phi}^{2}}\left(1+\sqrt{1-\frac{4\lambda_{\phi}\xi_{\phi}R}{\sigma^{2}}}\right)^{2}}\,,$
(2.25)

and can enhance the stability of the Higgs vacuum at the origin, in addition
to the non-minimal coupling of the Higgs, by suppressing its de Sitter
fluctuations,

$\displaystyle\frac{\langle h^{2}\rangle}{\left(H_{\rm
inf}/2\pi\right)^{2}}\simeq\frac{H_{\rm inf}}{m_{h}^{\rm eff}}=\frac{H_{\rm
inf}/\sigma}{\sqrt{12\xi_{h}\left(\frac{H_{\rm
inf}}{\sigma}\right)^{2}+\frac{1}{2}\left(\frac{g}{\lambda_{\phi}}\right)^{2}\left(\frac{1}{2}+\frac{1}{2}\sqrt{1-48\lambda_{\phi}\xi_{\phi}\left(\frac{H_{\rm
inf}}{\sigma}\right)^{2}}\right)^{2}}}\,,$ (2.26)

if $g>\lambda_{\phi}$ within the relevant parameter space of the couplings
$\sigma\sim H_{\rm inf}\,,\;\lambda_{\phi}<10^{-3}\,,\;\xi_{h}\gtrsim
0.1$333In fact, the Higgs curvature coupling can be much larger,
$\xi_{h}\sim\mathcal{O}\left(1-10^{2}\right)$, if we account for the survival
of the EW vacuum after inflation [69, 70, 71, 72], and thus it can solely
suppress the fluctuations decisively. [29, 57, 73]. During kination, for a
subset of the parameter space of the BSM couplings, it is possible for the
dark scalar to tunnel through the potential barrier from $\phi_{2}$ to
$\phi_{1}=0$, as shown in the lower left panel of Fig. 2 for $C=-5.4$. If this
process happens quickly enough with respect to the evolution of the
background, it can result in a FOPT with an accompanying production of
$\phi_{1}$-bubbles expanding in a $\phi_{2}$-universe. The collisions of these
bubbles in the cold vacuum of the pre-HBB universe produce GW spectra, which
are characterised by the transition strength and duration, as described in
Sec. 3.

In parallel to the BSM field, we have to take into account the dynamics of the
spectator Higgs field, since it is coupled both to the dark scalar via the
Higgs-portal and to the curvature scalar. Following the evolution of the
effective Higgs potential in this cosmological setting is a complicated and
non-trivial matter, due to the renormalization scale dependence of the SM
parameters, pole-matching, loop and curvature corrections, renormalization
group improvement techniques, and higher-order operators [74, 75, 73, 76, 66,
77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]. This is beyond the scope of
this study, and therefore we derive our conclusions considering the simplest
form of the Higgs potential at tree-level,
$U_{H}=\frac{-m^{2}_{h}+\xi_{h}R}{2}h^{2}+\frac{\lambda_{h}}{4}h^{4}$. This
serves as an adequate approximation, given that the phase transition of the
BSM field proceeds much faster than the evolution of the Higgs potential and
the background’s. Hence, for the central values of the SM parameters
$\lambda_{h}<0$ [89] and the true vacuum of the Higgs field during inflation
lies at

$\displaystyle
h=\sqrt{v^{2}+\frac{\xi_{h}R}{|\lambda_{h}|}+\frac{g^{2}\phi^{2}}{2|\lambda_{h}|}}\,,$
(2.27)

where $v=\frac{m_{h}}{\sqrt{\lambda_{h}}}\simeq 246.22$ GeV is the Higgs’ vev
at the EW scale. After inflation, the BSM phase transition takes place and the
dark scalar is found at $\phi=\phi_{1}=0$. We assume that the reheating
temperature is sufficiently low, $T_{\rm reh}\leq 80$ GeV, so that the EW
symmetry breaking cannot be induced thermally, as proposed in Ref. [29].
However in our case, EWSB is induced by the decrease of the curvature scalar
after crossing the threshold

$\displaystyle R_{\rm th}=\frac{|\lambda_{h}|}{\xi_{h}}v^{2}\,,$ (2.28)

unlike Ref. [29], where it is induced post-inflation by the dark scalar
rolling down its potential, as described in Appendix A. Shortly after the PT,
the dark scalar potential develops a high-field vev
$\phi_{2}\approx\frac{\sigma}{2\lambda_{\phi}}$, without a potential barrier
to separate the two vacua. Therefore, since there are no stabilising or
symmetry restoration mechanisms, it is inevitable that the dark scalar will
roll towards the true vacuum in a manner resembling a second-order phase
transition. In this case, we do not expect strong GW signatures to be
generated, but the EW symmetry could be restored, unless the portal coupling
is sufficiently small444For phenomenological flexibility, we quote the weakest
bound on the portal coupling, corresponding to the largest Higgs self-coupling
$\lambda_{h}(\mu_{\rm EW})\approx 0.13$ [66], but our calculation is
technically performed at $\mu_{\rm inf}\gg\mu_{\rm EW}$.,

$\displaystyle
g<\sqrt{\frac{2|\lambda_{h}|\lambda_{\phi}}{\sigma}}v<125\sqrt{\frac{\lambda_{\phi}}{\sigma}}\,,$
(2.29)

so that it suppresses the large vev of the BSM scalar, $\phi_{\rm
kin}=\phi_{2}\propto\frac{\sigma}{\lambda_{\phi}}$.

The situation is different for positive non-minimal coupling $\xi_{\phi}>0$.
During inflation, the dark scalar lies in its true vacuum $\tilde{\phi}_{1}$
at the origin, with the potential being symmetric or having a false vacuum
state at higher field values, depending on the couplings [57]. As inflation
ends and kination proceeds, a deeper vacuum state $\tilde{\phi}_{2}$ develops
with a decreasing potential barrier between the two located at
$\tilde{\phi}_{3}$, as shown in Fig. 3. When the barrier becomes thin enough,
the field can tunnel through resulting in a FOPT and the nucleation of
$\phi_{2}$-bubbles. If the PT proceeds fast enough, the bubbles percolate and
GWs are produced. Since the dark scalar has acquired a large vev, the EW
symmetry cannot be broken unless the portal is weaker that the bound of Eq.
(2.29). Therefore, adopting the same assumption for a low reheating
temperature $T_{\rm reh}\leq 80$ GeV [29], EWSB is again induced by the
evolving curvature crossing the threshold (2.28).

(a)

(b)

(c)

(d)

Figure 3: Evolution of the BSM potential for $\xi_{\phi}>0$, in the reduced
dimensionless form, from inflation (upper left) to kination (lower right). The
vertical dash-dotted and dotted lines follow the extrema $\tilde{\phi}_{3}$
from being the corresponding barriers to false vacua, respectively.

###  2.3 Consistency analysis and phenomenological considerations

Finally, we close this section with a discussion on the constraints that the
BSM scalar has to obey to comply with the assumptions of our cosmological
scenario and particle interactions. Firstly, the dark scalar must behave as a
spectator field at early times without affecting the inflationary dynamics,
$|V(\phi_{\rm inf},0,R=12H_{\rm inf}^{2})|<3M_{P}^{2}H^{2}_{\rm inf}$. For
$\xi_{\phi}<0$, $\phi_{\rm inf}=\phi_{2}$ and thus

$\displaystyle\frac{\sigma^{2}}{\lambda^{3}_{\phi}}\left(1-\frac{4C}{3}+\sqrt{1-\frac{8C}{9}}\right)\left(1+\sqrt{1-\frac{8C}{9}}\right)^{2}<288M_{P}^{2}\left(\frac{H_{\rm
inf}}{\sigma}\right)^{2}\,.$ (2.30)

Three bounds can be inferred that enable the identification of the relevant
parameter space of the couplings, namely

$\displaystyle\lambda^{3}_{\phi}>\frac{1}{36}\left(\frac{\sigma}{M_{P}}\right)^{2}\left(\frac{\sigma}{H_{\rm
inf}}\right)^{2}\quad$ $\displaystyle\mathrm{for}\quad-8C/9\ll 1\,,$ (2.31)
$\displaystyle\lambda^{3}_{\phi}>\frac{1}{26}\left(\frac{\sigma}{M_{P}}\right)^{2}\left(\frac{\sigma}{H_{\rm
inf}}\right)^{2}\quad$ $\displaystyle\mathrm{for}\quad-8C/9\approx 1\,,$
(2.32) $\displaystyle{\lambda_{\phi}}>12\left(\frac{H_{\rm
inf}}{M_{P}}\right)^{2}\xi_{\phi}^{2}\quad$
$\displaystyle\mathrm{for}\quad-8C/9\gg 1\,.$ (2.33)

In the opposite case of $\xi_{\phi}>0$, $\phi_{\rm inf}=\phi_{1}=0$ and so the
dark scalar is by default spectating during inflation, which implies that we
cannot infer any constraints on its couplings.

After EWSB, we assume that for the Higgs branching ratio into invisible
particles, the latter correspond to the dark scalar field. Hence, an upper
bound can be placed on the portal coupling and the corresponding effective
mass [29],

$\displaystyle g<0.13\,,\qquad m_{\phi}\lesssim 22.6\textrm{ GeV}\,,$ (2.34)

where the mass is generated solely by the Higgs portal

$\displaystyle m_{\phi}=\frac{g}{\sqrt{2}}v\,.$ (2.35)

In addition, radiative corrections from the portal to the BSM self-coupling
should be suppressed to avoid fine tuning [29],

$\displaystyle\Delta\lambda_{\phi}\sim\frac{g^{4}}{16\pi^{2}}<\lambda_{\phi}\,.$
(2.36)

By requiring the portal coupling to satisfy the bound (2.29), Eq. (2.36)
provides an intuitive upper constraint on the BSM couplings that allow for
curvature-induced EWSB,

$\displaystyle\lambda_{\phi}<\left(\frac{4\pi^{2}}{\lambda^{2}_{h}v^{4}}\right)\sigma^{2}\approx\left(\frac{H_{\rm
inf}}{10^{3}}\right)^{2}\left(\frac{\sigma}{H_{\rm inf}}\right)^{2}\,,$ (2.37)

which is most constrictive for the largest Higgs quartic coupling [66],
$\lambda_{h}(\mu_{\rm EW})\approx 0.13$ .

Finally, the BSM scalar plays the role of dark matter once it acquires a mass
after EWSB, oscillating around its potential minimum, as discussed in Ref.
[29]. Thus, the abundance of dark matter $\Omega_{\phi,0}\simeq 0.26$ dictates
the value of the portal coupling with respect to the cubic and quartic BSM
couplings,

$\displaystyle
g=\frac{\sqrt{12\Omega_{\phi,0}}H_{0}M_{P}}{v}\left(\frac{g_{*\mathrm{reh}}}{g_{*0}}\right)^{\frac{1}{2}}\left(\frac{T_{\mathrm{reh}}}{T_{0}}\right)^{\frac{3}{2}}\phi_{\rm
reh}^{-1}\simeq 3\times
10^{-17}\left(\frac{T_{\mathrm{reh}}}{80\,\mathrm{GeV}}\right)^{\frac{3}{2}}\frac{\lambda_{\phi}}{\sigma}\,,$
(2.38)

where the Hubble constant is $H_{0}\simeq 1.5\times 10^{-42}$ GeV, the
effective degrees of freedom at reheating was $g_{*\mathrm{reh}}=106.75$ and
at present is $g_{*0}=3.36$, $\phi_{\rm reh}$ is the amplitude of the BSM
scalar at reheating, and the CMB temperature today is $T_{0}=23.48\times
10^{-5}$ GeV [90, 50]. In our model, the dark scalar settles into the true
vacuum $\phi_{\rm reh}=\phi_{2}(R=0,h=v)\approx\frac{\sigma}{\lambda_{\phi}}$
at late times, regardless of the sign of its non-minimal coupling
$\xi_{\phi}$. Therefore, the quartic term always dominates the other terms in
the potential, and the dark scalar oscillates around $\phi_{2}$.

##  3 Primordial gravitational waves from bubble collisions during kination

###  3.1 Framework of cosmological phase transitions in vacuum

As we saw in the previous section, there is a possibility for the BSM scalar
to undergo a FOPT in the early universe because of the evolution of the
curvature scalar. As a result, bubbles of true vacuum would nucleate at the
points where the field has tunnelled through the potential barrier. These
bubbles would grow and quickly reach a relativistic expansion rate due to the
energy difference between the vacua inside and outside of the bubble wall [91,
92]. In the cold post-inflationary scenario considered here, there is no
thermal plasma between the bubbles, and evidently there are no temperature and
friction effects, as reheating takes place at much lower scales.

In the context of cosmological phase transitions, the probability of
tunnelling from false (fv) to true (tv) vacuum is given by the bubble
nucleation rate (per spacetime volume)

$\displaystyle\Gamma=F\left(\frac{S^{E}_{4}}{2\pi}\right)^{2}\exp(-S^{E}_{4})\,,$
(3.1)

where $S_{4}^{E}$ is the Euclidean action for the corresponding
$O(4)$-symmetric bounce solution of a double-well potential in vacuum [93, 94,
95]. In the same fashion as in Refs. [57, 45], we make use of the semi-
analytic approximation for the Euclidean action of the critical bubble,

$\displaystyle
S_{4}^{E}(\delta)=\frac{4\pi^{2}}{3\lambda_{\phi}}(2-\delta)^{-3}\left[\alpha_{1}\delta+\alpha_{2}\delta^{2}+\alpha_{3}\delta^{3}\right]\,,$
(3.2)

where $\alpha_{1}=14.1304$, $\alpha_{2}=-11.1304$, $\alpha_{3}=2.1576$ are
numerically fitted parameters [95], which are updated to the ones derived in
Ref. [68]. This formalism applies to cases where the fv is fixed at the origin
and the tv has a negative potential energy with a positive vev. In the thin-
wall approximation, the prefactor $F$ for cold transitions is given by

$\displaystyle F_{\rm thin}=R_{0}^{-4}\approx\frac{\pi^{2}\Delta
V}{2S_{4}^{E}}\,,$ (3.3)

where $\Delta V=V(\phi_{\rm fv})-V(\phi_{\rm tv})$ is the energy difference
between the two vacua, and the bubble-wall radius $R_{0}$ is estimated after
neglecting the energy stored in the bubble wall [96]. This approach is
generally used when the height of the potential barrier is much larger than
the energy difference between the two vacua, as for example in Fig. 2, but in
scenarios of vanishing potential barriers as in Fig. 3, this approximation is
no longer valid. In such cases, the standard formalism [97, 98, 96, 91, 41]
for the estimation of the prefactor in Eq. (3.1) is purposely avoided, because
the critical radius is not well defined. Hence, the prefactor is typically
estimated with a dimensional approximation involving the field’s vevs

$\displaystyle F_{\rm thick}=\left(\Delta\phi_{v}\right)^{4}=\left(\phi_{\rm
fv}-\phi_{\rm tv}\right)^{4}\,.$ (3.4)

However, a recent study [95] has managed to provide a one-loop result that can
adequately interpolate between thin and thick walls, with the corresponding
prefactor being

$\displaystyle F_{\rm 1-loop}=(m^{2})^{2}e^{-\frac{1}{2}\Sigma_{4}}\,,$ (3.5)

where $m^{2}$ corresponds to the canonically normalised quadratic “mass” term
in the potential of the scalar field undergoing the phase transition, and the
one-loop contribution to the Euclidean action is numerically approximated by
the regularised sum over multipoles

$\displaystyle\Sigma_{4}(\delta)=\frac{53.9926-47.6801\delta+11.0134\delta^{2}+0.3358\delta^{3}+0.4197\delta^{4}-0.2938\delta^{5}}{(2-\delta)^{3}}\,.$
(3.6)

Hence for our study, we will use Eq. (3.5) to derive the resulting GW spectra,
since it can be readily applied to both scenarios of positive and negative
non-minimal coupling and the numerical computations are more stable. To apply
this formalism to the case of $\xi_{\phi}<0$, where the tunnelling takes place
from $\phi_{2}>0$ to $\phi_{1}=0$, we have to perform a redefinition of the
potential, $V\rightarrow W$, that fixes the fv at the origin, as shown
explicitly in Appendix B. The corresponding quadratic “mass” terms are given
according to (B.1) and (2.11) by

$\displaystyle m^{2}_{\xi_{\phi}<0}(R,h=0)$
$\displaystyle=\frac{\sigma^{2}}{4\lambda_{\phi}}\left(1+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)+|\xi_{\phi}|R\,,$
(3.7) $\displaystyle m^{2}_{\xi_{\phi}>0}(R,h=0)$
$\displaystyle=\xi_{\phi}R\,,$ (3.8)

and we have to use the appropriate $\delta$ in each case, i.e. Eq. (B.4) or
Eq. (2.14), respectively. We included the standard prescriptions of Eqns.
(3.3) and (3.4), even though we did not use them for the calculation of the GW
spectra in Sec. 3.3, for the sake of completeness and to provide a reference
point for the comparison of these different approaches.

###  3.2 Bubble nucleation and phase transition parameters

Whether bubble production is efficient enough to result in percolation, within
a particular cosmological setting, is determined by the nucleation condition

$\displaystyle\int_{t_{\rm col}}^{t_{\rm
nuc}}dt\frac{\Gamma(t)}{H(t)^{3}}=1\,,$ (3.9)

where $t_{\rm nuc}$ corresponds to the time of bubble nucleation and $t_{\rm
col}$ to that of collision [41]. In our investigation, any potential FOPTs
would be induced during the transition from inflation to kination, where the
Hubble parameter changes only slightly, as shown in Fig. 1. Therefore, we
consider it being approximately constant for the duration of any phase
transitions taking place during this period [57]. In addition, we assume that
the PT proceeds and completes much faster than the evolution of the
background. This means that the time elapsed from bubble production until
collision is sufficiently small compared to cosmic evolution, so that $t_{\rm
nuc}\approx t_{\rm col}$555For all practical purposes in the remainder of this
work, we will use these two subscripts interchangeably, even though we are
explicitly denoting when each quantity is calculated. [41, 42, 99]. These
assumptions allow us to express the nucleation condition (3.9) in a simpler
form as

$\displaystyle\Gamma(t_{\rm nuc})=H_{\rm nuc}^{4}\,,$ (3.10)

which translates to the formation of one true-vacuum bubble per Hubble volume.
By inserting the decay rate (3.1) in Eq. (3.10), we can estimate the time of
bubble nucleation $t_{\rm nuc}$. However, note that for $\xi_{\phi}<0$, the
decay rate is very sensitive to the choice of the BSM couplings, because of
the required evolution of the $\phi_{2}$-vacuum from a true to a short-lived
false vacuum. Thus, some amount of fine-tuning is required to extract $t_{\rm
nuc}$ effectively.

After the bubbles are formed, they grow rapidly in the post-inflationary
universe and fill the universe, with the BSM scalar field being in its true
vacuum within a cosmological horizon. During percolation, bubble collisions
generate a stochastic GW background, if the PT is strong enough and completes
sufficiently quickly [48, 51]. This is effectively quantified by two PT
parameters. The first is the inverse duration of the transition $\beta_{\rm
col}$, quantifying the inverse of the time interval between bubble formation
and percolation, which can be thought of as the size of the bubbles at the
time of collision, after assuming exponential bubble expansion [48, 57].
Depending on the evolution of the Euclidean action, this parameter can be
evaluated as

$\displaystyle\beta_{\rm
col}^{\xi_{\phi}<0}=\evaluated{\sqrt{\frac{d^{2}S_{4}^{E}}{dt^{2}}}}_{t=t_{\rm
col}}\,,$ (3.11) $\displaystyle\beta_{\rm
col}^{\xi_{\phi}>0}=-\evaluated{\derivative{t}S_{4}^{E}}_{t=t_{\rm col}}\,,$
(3.12)

where in the first case $S_{4}^{E}$ reaches its minimum before the transition
completes, while in the latter the Euclidean action decreases slowly in time
[52], according to the dynamics of the phase transitions shown in Figs. 2 and
3. Note that in each case, we have to use the appropriate expression for
$\delta$, i.e. Eq. (B.4) for $\xi_{\phi}<0$ and Eq. (2.14) for $\xi_{\phi}>0$.
Since we have assumed that the phase transition is sufficiently quick, so that
the background can be considered non-dynamic during the process, the hierarchy
between the respective velocities has to be maintained $\beta_{\rm
col}>\beta_{w}=10H_{\rm inf}$, which places an implicit constraint on the
parameter space of the couplings.

The second parameter is the strength of the PT, which corresponds to the ratio
of latent heat released from tunnelling to the lower ground state over the
background energy at collision time [51, 45, 57], and is given by

$\displaystyle\alpha\equiv\frac{\rho_{\rm PT}}{\rho_{\rm bac}}\,.$ (3.13)

This quantity can be evaluated analytically, as the energy difference between
the two vacua [99] is given by

$\displaystyle|\Delta
V|=|V(\phi_{2})-V(\phi_{1})|=\frac{\sigma^{4}}{96\lambda_{\phi}^{3}}\left(1+\sqrt{1-\frac{4\delta}{9}}\right)^{2}\left(1-\frac{2\delta}{3}+\sqrt{1-\frac{4\delta}{9}}\right)\,,$
(3.14)

and holds for both cases of positive and negative $\xi_{\phi}$ (invariant
under the redefinition of Appendix B), and given that the inflaton field
dominates the energy density and has not decayed during kination, the
background energy density is $\rho_{\rm bac}=\rho_{\rm
kin}=3M_{P}^{2}H^{2}_{\rm inf}$. Thus, the transition strength reads

$\displaystyle\alpha=\frac{|\Delta V|}{3M_{P}^{2}H_{\rm
inf}^{2}}=\frac{1}{72}\left(\frac{H_{\rm
inf}}{M_{P}}\right)^{2}\lambda_{\phi}^{-3}\left(\frac{\sigma}{H_{\rm
inf}}\right)^{4}\left[1+\sqrt{1-\frac{4\delta}{9}}-\frac{2\delta}{3}-\frac{4\delta}{9}\sqrt{1-\frac{4\delta}{9}}+\frac{2\delta^{2}}{27}\right]\,,$
(3.15)

and is evaluated at the time of nucleation. We have purposely written the
above expression in increasing orders of $\delta$ to demonstrate its influence
on $\alpha$, as we roughly have $\delta_{\xi_{\phi}<0}(t_{\rm nuc})\approx 1$
and $\delta_{\xi_{\phi}>0}(t_{\rm nuc})\rightarrow 0$. We restrict ourselves
to sufficiently strong FOPTs that can generate amplified GW signals, without
overtly large values of $\alpha$ where supercooling and additional
considerations must be taken into account [100], i.e.
$10^{-4}\lesssim\alpha\lesssim 10^{-1}$.

###  3.3 Gravitational wave spectra

Any GWs generated in the primordial universe travel unhindered throughout the
fabric of spacetime and even tough they were very energetic events at
emission, their spectra are getting red-shifted as the universe expands.
However, the alternative expansion history of the universe after inflation in
scenarios with prolonged kination affects the spectral shape of the GW signal.
In particular, the tilt of the spectrum is modified as
$\Omega_{\mathrm{GW}}(f)\propto f^{4}$ for the modes entering the horizon
during kination [101], and reverts back to the usual cubic dependence after
reheating [102, 103, 104]. This asymmetric signature in the GW spectra, as
shown also in Ref. [99], acts as a “smoking-gun” feature of an extended period
of kination. Therefore, the GW amplitude from a PT induced during kination
redshifts until the present day [48, 57] is given by

$\displaystyle\Omega_{\rm GW,0}(f)h^{2}=1.67\times 10^{-5}\left(\frac{H_{\rm
col}}{H_{\rm reh}}\right)^{2\frac{3w_{\rm int}-1}{3w_{\rm
int}+3}}\left(\frac{H_{\rm col}}{\beta_{\rm
col}}\right)^{2}\left(\frac{\alpha}{\alpha+1}\right)^{2}S(f)\,,$ (3.16)

where $w_{\rm int}\approx 1$ corresponds to the integrated value of the EoS
parameter over redshift from the time of bubble collision $t_{\rm col}$ to
reheating $t_{\rm reh}$, which approximates $w_{\rm kin}=1$ for quintessential
inflation-like scenarios. The spectral shape $S(f)$ is defined using a
numerically derived broken power law [44, 43]

$\displaystyle S(f)=25.10\,\left[2.41\,\left(\frac{f}{f_{\rm
peak,0}}\right)^{-0.56}+2.34\,\left(\frac{f}{f_{\rm
peak,0}}\right)^{0.57}\right]^{-4.2}\,,$ (3.17)

where $f_{\rm col}=0.13\beta_{\rm col}$ is the peak frequency of the spectrum
at the moment of bubble percolation [48, 57], which at present has redshifted
to

$\displaystyle f_{\rm peak,0}=1.65\times 10^{-5}\left(\frac{H_{\rm
col}}{H_{\rm reh}}\right)^{\frac{3w_{\rm int}-1}{3w_{\rm
int}+3}}\left(\frac{f_{\rm col}}{H_{\rm col}}\right)\left(\frac{T_{\rm
reh}}{100\mbox{ GeV}}\right)\,.$ (3.18)

We adopt the simplified assumption of instantaneous thermalisation after
kination, where the reheating temperature can be related to the corresponding
Hubble rate as

$\displaystyle H_{\rm
reh}=\frac{\pi}{3}\sqrt{\frac{g_{*\mathrm{reh}}}{10}}\frac{T_{\rm
reh}^{2}}{M_{P}}\,.$ (3.19)

This approach purposely avoids dealing with the complicated dynamics of the
reheating process, because the primary focus of our work is on the GW spectra,
which are only affected directly by the order of magnitude of the reheating
scale. Although many different mechanisms with very interesting and rich
phenomenology could be explored here, their overall impact on the GW
signatures would not be significant. In this work, the reheating temperature
is assumed to be $T_{\rm reh}\leq 80$ GeV [29] so that the EW symmetry is
broken by the dynamic curvature, and temperature effects cannot restore it
afterwards. Hence, the scale of reheating is estimated at $H_{\rm reh}\approx
9\times 10^{-15}$ GeV. Finally, we highlight that settings with prolonged
kination can typically constrain the extent of reheating, but for our purposes
here, we avoid complicating our computations by incorporating this effect.

The resulting GW spectra from the phase transitions described in Sec. 2 for
negative non-minimal coupling are shown in Fig. 4 and for $\xi_{\phi}>0$ in
Fig. 5, considering the most relevant region of the parameter space of the BSM
couplings, according to the constraints presented in Sec. 2.3. We have
included two plots in each figure, with the upper one corresponding to signals
originating from typical high inflationary scales $H_{\rm inf}=10^{12}$ GeV,
and the lower one to settings where the scale of inflation approaches the EW
scale $H_{\rm inf}=10^{-8}$ GeV. Since $H_{\rm col}\sim H_{\rm inf}$, this
shows the range of GW signatures between the typically high frequency spectra
associated with the high energy scales of the early universe, and lower
inflationary scales that could be considered, as they are not a priori ruled
out. However, note that the lowest scales that can be accommodated by
realistic quintessential-like inflationary models are around $10^{8}$ GeV.
Therefore, the bottom plots in Figs. 4 and 5 act as illustrative extreme
examples that allow us to visualise approximately the interpolation of GW
spectra from higher scales. Also, it should be pointed out that the energy
density released by GWs cannot exceed the value of $\Omega_{\rm
BBN}h^{2}=1.12\times 10^{-6}$, because it would affect the Big Bang
Nucleosyntheis (BBN). The BBN bound is represented in the plots by the thin
horizontal grey line. We have included the detector sensitivities of LISA
[105, 106], AEDGE [107], the Einstein Telescope (ET) [108, 109] and LIGO [110,
111, 112, 113] to depict the observational prospects of such signatures with
current and future experiments.

In each computation of a GW spectrum, the combination of the BSM couplings was
chosen accordingly so that it provides the strongest possible GW signal,
without crossing the BBN bound or involving supercooling, while also ensuring
that bubble percolation is completed sufficiently quickly so that the
background is approximately static. The low reheating scale, which allows for
a curvature-induced EW symmetry breaking, enhances the GW spectra via the
corresponding denominators in Eq. (3.16) and limits decisively the viable
parameter space of the dark scalar’s couplings. It is worth highlighting that
to compute the GW spectra from PTs where $\xi_{\phi}<0$, which can be reliable
without performing detailed numerical bubble simulations, we limited ourselves
to the area of the parameter space where $C\approx-5.4$, meaning that
$\lambda_{\phi}^{-1}\approx 10|\xi_{\phi}|\left(\frac{H_{\rm
inf}}{\sigma}\right)^{2}$ according to Eq. (2.18). With this choice the
potential maintains always an appropriate double-well shape, that allows the
phase transition to proceed only via tunnelling. For lower values of $C$, the
potential’s symmetry would be restored, as shown in Fig. 4 for $C=-10$. For
higher values, the vacuum at $\phi_{2}$ would not evolve from a tv to a fv
that could tunnel towards $\phi_{1}=0$, and thus a FOPT would not occur.

(a)

(b)

Figure 4: Gravitational wave spectra from phase transitions for $\xi_{\phi}<0$
according to Fig. 2. The scale of reheating is set at $H_{\rm reh}\approx
9\times 10^{-15}$ GeV with high scale inflation $H_{\rm inf}=10^{12}$ GeV and
$\sigma=H_{\rm inf}$ (top) or low scale inflation $H_{\rm inf}=10^{-8}$ GeV
(bottom). The BSM couplings are tuned accordingly in each case, so that they
boost the GW signal while maintaining $C\approx-5.4$ (defined in Eq. (2.18)),
and complying with the constraint (2.30).

(a)

(b)

Figure 5: Gravitational wave spectra from phase transitions for $\xi_{\phi}>0$
according to Fig. 3. The reheating scale is fixed at $H_{\rm reh}\approx
9\times 10^{-15}$ GeV and the scale of inflation at $H_{\rm inf}=10^{12}$ GeV
(top) and $H_{\rm inf}=10^{-8}$ GeV (bottom). The dark scalar’s couplings have
been chosen so that sufficiently strong FOPTs without supercooling are
produced, with boosted GW signals that do not violate the BBN bound.

##  4 Conclusions

In this work, we examined whether gravitational waves can be generated from a
curvature-induced first-order phase transition of a non-minimally coupled
scalar field, that plays the role of dark matter, with a portal to the Higgs
field. This was investigated in the context of a dynamical spacetime during
the transition from inflation to kination, while also considering the
possibility for inducing electroweak symmetry breaking in this manner. We
explored a wide range of inflationary scales and both cases of positive and
negative values for the non-minimal coupling, while taking into account the
phenomenological constraints imposed on the BSM model’s couplings from
observations and experiments. The resulting GW amplitudes are boosted by
extended periods of kination, akin to the quintessential-like inflationary
models, and carry a characteristic tilt on their lower-frequency tail that
serves as evidence of the alternative expansion history after inflation.

More specifically, we analysed the conditions for the realisation of a strong
FOPT in vacuum, which led to the generation of a stochastic GW background from
the collisions of the dark scalar’s true-vacuum bubbles. Similarly as in Ref.
[57], the phase transition complete approximately instantaneously with respect
to the cosmological background, its strength depends on the total difference
of the potential energy between the two vacua, but its inverse duration is
sensitive to the evolution of the corresponding Euclidean action in each case
of $\xi_{\phi}$. We applied the recent result of Ref. [95], using the 1-loop
result for the decay rate in vacuum, which can interpolate between regimes of
thin and thick bubble walls and is thus valid for the PTs of both signs of the
non-minimal coupling. The peak frequency and amplitude of the generated GWs
depend on the ratio of the Hubble scales between bubble percolation and
reheating. In particular, the extremely low reheating scale of $H_{\rm
reh}\sim 10^{-14}$ GeV, corresponding to the assumption of instantaneous
thermalisation with temperature $T_{\rm reh}\leq 80$ GeV, enhanced the GW
signal so much that it strongly confined the parameter space of the BSM
couplings that could comply with the BBN bound. For the specified choices of
couplings depicted in Figs. 4 and 5, it is possible to obtain GW spectra that
could fall within the reach of future detectors. Although realistic high
inflationary scales lead to peak frequencies of GW spectra that are higher
than those probed by the upcoming experiments, the origin of such signatures
is indisputably linked with the early universe, since there is convolution
with other sources at this frequency range. In addition, we can readily
distinguish the almost symmetric GW spectra obtained here from the standard
paradigm of thermal phase transitions, where the interaction of the bubbles
with the surrounding plasma (sound waves, turbulence, etc.) leads to enhanced
high-frequency tails [51, 114].

The BSM theory considered in this study expands the self-interacting model of
Ref. [29], by introducing a cubic term to the dark scalar potential. This
allowed for the realisation of a strong PT due to the evolution of the
curvature scalar, which was coupled to the scalar field, according to the
mechanism developed in Ref. [57]. Although we found agreement between the case
of $\xi_{\phi}>0$ and the behaviour observed in Ref. [57], the additional case
of $\xi_{\phi}<0$ had qualitatively different dynamics. This complicated the
analysis especially when considering the subdominant role that the DM scalar
was expected to have during inflation, which placed a firm constraint on the
couplings according to Eq. (2.30). The presence of the Higgs-portal term was
irrelevant for most of the cosmological evolution explored here, since the EW
symmetry breaking, and therefore the Higgs’ acquisition of its non-zero vev,
was postponed to much later times with a very low reheating temperature
$T_{\rm reh}\leq 80$ GeV. This approach followed the reasoning of Ref. [29],
so that the spontaneous breaking of the EW symmetry is not induced by the
decreasing temperature of the primordial plasma. However, in contrast to Ref.
[29], EWSB was not induced by the rolling of the dark scalar down its
potential towards the origin, but due to the decrease of spacetime curvature
in the post-inflationary universe. This is viable only when the Higgs-portal
coupling is extremely weak, as shown by Eq. (2.29), since the large vev of the
dark scalar could affect symmetry breaking. Coincidentally, the same coupling
has to be extremely weak to allow the BSM scalar to act as DM after EWSB
according to Eq.(2.38),

$\displaystyle 10^{-32}\lesssim g\lesssim 10^{-21}\,,\qquad 10^{-30}\textrm{
GeV}\,\lesssim m_{\phi}\lesssim 10^{-19}\textrm{ GeV}\,,$ (4.1)

for the relevant parameter space across the inflationary scales considered

$\displaystyle-10^{3}\lesssim\xi_{\phi}\lesssim 10^{2}\,,\qquad\sigma\sim
H_{\rm inf}\,,\qquad\lambda_{\phi}\lesssim 10^{-4}\,,$ (4.2)

which trivially satisfies the bounds (2.34) and (2.36). This implies that when
a non-minimally coupled Higgs-portal scalar field with a renormalizable
potential plays the role of very light dark matter in the late universe, it
allows for the breaking of the EW to be induced by the evolution of spacetime
curvature, provided that reheating takes place at sufficiently low
temperatures.

In any case, we believe that it is quite interesting that it is possible to
identify a new observational signature of a Higgs-portal dark scalar field
through its non-minimal coupling to curvature. Even though the prospects for
the detection of the associated gravitational waves are somewhat unfavourable,
it is exciting that they might be, at least theoretically, putative messengers
of the properties of an interesting class of theories involving inflation and
a scalar field DM candidate that can itself undergo a first order phase
transition and have an effect on the spontaneous breaking of the EW symmetry.
This naturally leads to a promising and multi-faceted outlook for future
investigations. The framework developed here could be applied to different BSM
extensions with various field content and couplings. Within the same context,
different inflationary models can be explored, where understanding the
transition from inflation/kination to the radiation-dominated epoch through
reheating being particularly crucial. At the same time, we expect that it
would be fruitful to extend these calculations to theories including non-
minimal couplings of matter to curvature [115, 116, 117], given the pronounced
role that spacetime curvature can play in scenarios like the ones explored
here. These endeavours could offer valuable tools and insights that can
enhance our understanding of the early universe, complimentary to accelerator
experiments and observational surveys.

## Acknowledgements

We would like to thank Marek Lewicki, Mar Bastero-Gil, António P. Morais, and
Marco Finetti for the useful discussions and suggestions, and Maciej Kierkla
for his contribution in the graphic representation of the GW spectra. This
work was supported by FCT - Fundação para a Ciência e Tecnologia, I.P. through
the project CERN/FIS-PAR/0027/2021, with DOI identifier 10.54499/CERN/FIS-
PAR/0027/2021.

##  Appendix A Self-interacting scalar field with a Higgs portal

If we set $\sigma=0$ in the dark scalar potential (2.11), we recover the self-
interacting model from Ref. [29], with its evolution from inflation to
kination shown in Fig. 6. For $\xi_{\phi}<0$, there are two degenerate vacua
at
$\phi=\pm\sqrt{\frac{|\xi_{\phi}|R}{\lambda_{\phi}}-\frac{g^{2}h^{2}}{2\lambda_{\phi}}}$
that become shallower and move closer to the origin during the transition from
inflation to kination, eventually resulting in a single global minimum at
$\phi=0$. This behaviour is reversed for $\xi_{\phi}>0$ albeit not explored in
Ref. [29]. During inflation, the large vev of the spectator scalar field,
$\phi_{\rm inf}=\sqrt{\frac{12|\xi_{\phi}|H^{2}_{\rm inf}}{\lambda_{\phi}}}$,
generates a large contribution to the effective mass of the Higgs field
through the portal, $m_{h}^{\rm eff}\propto\frac{g}{\sqrt{2}}\phi_{\rm
inf}=\sqrt{\frac{6|\xi_{\phi}|}{\lambda_{\phi}}}gH_{\rm inf}$. This keeps the
Higgs field at the origin, by suppressing its de Sitter fluctuations $\langle
h^{2}\rangle\propto H_{\rm inf}/m_{h}^{\rm eff}$ for the appropriate couplings
$g/\sqrt{\lambda_{\phi}}\sim 10^{2}$ and $\xi_{\phi}<-0.1$ [29]. After
symmetry restoration of the potential during kination, the scalar rolls down
towards the minimum and induces EW symmetry breaking when crossing the
critical value $\phi_{\rm crit}=\sqrt{2\lambda_{h}}(v/g)$ according to Eq.
(2.27). Consequently, the Higgs field relaxes around its vev as the
oscillations of the BSM scalar around $\phi=0$ slow down and spacetime
curvature tends to approximate flatness. However, note that this is true only
if the reheating temperature is $T_{\rm reh}\leq 80$ GeV, to avoid thermal
restoration of the EW symmetry.

Figure 6: Evolution of the dark scalar potential from inflation (solid) to
kination (dashed), given by Eq. (2.11) for $\xi_{\phi}=-1\,,\,\,\sigma=0$ and
$\lambda_{\phi}=10^{-5}$.

##  Appendix B Potential redefinition for a phase transition from a positive
true vacuum

The evolution of the BSM potential for $\xi_{\phi}<0$ in Fig. 2 implies that
the phase transition takes place from a positive true vacuum to the false
vacuum located at the origin. However, the prescription for calculating the
decay rate and the PT parameters in Sec. 3 is formulated for trajectories
starting from a fv that is firmly fixed at the origin towards either a
positive [68] or a negative [95] tv. Therefore, in order to accurately
calculate the dynamics of the PT and the corresponding GW spectra, we have to
transform the potential (2.11) accordingly as

$W(\phi,h=0,R)=V(\phi+\phi_{2},h,R)-V(\phi_{2},h,R)=\\\
=\left[\frac{\sigma^{2}}{4\lambda_{\phi}}\left(1+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)+|\xi_{\phi}|R\right]\phi^{2}-\frac{|\sigma|}{2}\left(\frac{1}{3}+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)\phi^{3}+\frac{\lambda_{\phi}}{4}\phi^{4}\,.$
(B.1)

Note that in order to match with the framework of this study, where the tv
lies at positive values, the cubic coupling must have the opposite sign
$\sigma<0$. However, as long as we are consistent, this does not affect the
dynamics and GWs of the PT, it is only a matter of formalism and convenience
for the calculation of the key quantities of the PT. Following the same steps
as in Sec. 2, the potential (B.1) is written in the reduced dimensionless form
as

$\displaystyle\tilde{W}(\chi,\delta)=\frac{\tilde{\delta}}{2}\chi^{2}-\chi^{3}+\frac{1}{4}\chi^{4}\,,$
(B.2)

where the field redefinition and its corresponding quadratic coupling are
given by

$\displaystyle\chi(R)=\frac{2\lambda_{\phi}}{|\sigma|}\left(\frac{1}{3}+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)^{-1}\phi\,,$
(B.3)
$\displaystyle\tilde{\delta}(h=0,R)=\left(\frac{8\lambda_{\phi}}{\sigma^{2}}\right)\frac{\frac{\sigma^{2}}{4\lambda_{\phi}}\left(1+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)+|\xi_{\phi}|R}{\left(\frac{1}{3}+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)^{2}}=\frac{2\left(1-\frac{4\delta}{9}+\sqrt{1-\frac{4\delta}{9}}\right)}{\left(\frac{1}{3}+\sqrt{1-\frac{4\delta}{9}}\right)^{2}}\,,$
(B.4)

where $\delta<0$ during inflation according to Eq. (2.14), but
$\tilde{\delta}>0$. The overall rescaling of the potential is
$W=\left[\frac{|\sigma|}{2}\left(\frac{1}{3}+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)\right]^{4}\lambda_{\phi}^{-3}\tilde{W}$,
and its corresponding extrema lie at

$\displaystyle\phi_{1}^{W}$
$\displaystyle=0\,,\qquad\qquad\qquad\quad\qquad\qquad\qquad\quad\,\chi_{1}=0\,,$
(B.5) $\displaystyle\phi_{2}^{W}(h,R)$
$\displaystyle=\frac{|\sigma|}{2\lambda_{\phi}}\left(1+\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\right)\,,\quad\chi_{2}(\tilde{\delta})=\frac{3}{2}\left(1+\sqrt{1-\frac{4\tilde{\delta}}{9}}\right)\,,$
(B.6) $\displaystyle\phi_{3}^{W}(h,R)$
$\displaystyle=\frac{|\sigma|}{\lambda_{\phi}}\sqrt{1+\frac{4\lambda_{\phi}|\xi_{\phi}|R}{\sigma^{2}}}\,,\quad\qquad\qquad\chi_{3}(\tilde{\delta})=\frac{3}{2}\left(1-\sqrt{1-\frac{4\tilde{\delta}}{9}}\right)\,,$
(B.7)

as long as $4\delta/9>-1$ or equivalently $4\tilde{\delta}/9<-1$ . Finally, we
highlight that the potential difference between the fv and tv of the
potential,

$\displaystyle\Delta W$
$\displaystyle=W\left(\phi_{2}^{W}\right)-W\left(\phi_{1}^{W}\right)=\frac{\sigma^{4}}{96\lambda^{3}}\left(1-\frac{6\lambda\xi_{\phi}R}{\sigma^{2}}+\sqrt{1-\frac{4\lambda\xi_{\phi}R}{\sigma^{2}}}\right)\left(1+\sqrt{1-\frac{4\lambda\xi_{\phi}R}{\sigma^{2}}}\right)^{2}=$
$\displaystyle=\frac{\sigma^{4}}{96\lambda^{3}}\left(1-\frac{2\delta}{3}+\sqrt{1-\frac{4\delta}{9}}\right)\left(1+\sqrt{1-\frac{4\delta}{9}}\right)^{2}\,,$
(B.8)

is equivalent to (3.14). This is expected since the transformation we
performed does not affect the relative depths of the potential vacua, but it
is just a convenient redefinition of the potential that encapsulates the FOPT
of Fig. 2. Therefore, it is trivial that Eq. (3.15) holds in this case, since
the transition strength is the same with either prescription.

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