aainstitutetext: Harish-Chandra Research Institute, A CI of Homi Bhabha
National Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211019,
Indiabbinstitutetext: Regional Centre for Accelerator-based Particle Physics,
Harish-Chandra Research Institute,
Prayagraj (Allahabad) 211019, India

# Dilution of Dark Matter Relic abundance due to First Order Electroweak Phase
Transition in the singlet scalar extended type-II seesaw model

Subhojit Roy<EMAIL_ADDRESS>
###### Abstract

We investigate the effect of a first-order electroweak phase transition
(FOEWPT), which is one of the prerequisites for electroweak baryogenesis, on
the thermal relic abundance of the dark matter (DM) that freezes out before
the phase transition occurs in the complex singlet scalar extended
$Z_{3}$-invariant type-II seesaw model that can simultaneously provide a DM
candidate, explain the non-vanishing neutrino masses and the baryon asymmetry
of the Universe. Such a phase transition around the electroweak scale leaves
an impact on the relic density due to the release of entropy, particularly for
a TeV-scale DM. We thus concentrate on the region of parameter space of the
said model which favors an FOEWPT in the early Universe and for which the DM
is heavy such that its freeze-out temperature turns out to be larger than the
phase transition temperature. We further study the dependencies of the
dilution factor of the DM relic density on the model parameters, the
nucleation temperature, the strength and the duration of the phase transition.
Such a dilution might retrieve some of the regions of parameter space that
were previously ruled out by the measured value of the DM relic density and/or
the latest constraints from the DM direct-detection experiments. Furthermore,
a direct connection is drawn between the dilution factor and the generation of
stochastic gravitational waves as a result of an FOEWPT.

###### Keywords:

Beyond Standard Model, Electroweak Phase transition, Dark Matter,
Gravitational waves

††preprint: HRI-RECAPP-2022-015

##  1 Introduction

The relic density of dark matter (DM) in the present Universe is precisely
measured from various astrophysical and cosmological experiments, i.e.,
$\Omega_{\rm DM}h^{2}=0.120\pm 0.001$ Planck:2018vyg . The particle
responsible for the observed DM in the Universe is likely the most wanted new
particle after the latest finding of the Standard Model (SM)-like Higgs boson
mass around 125 GeV. Among the various possibilities of the nature of the DM,
the weakly interacting massive particle (WIMP) still remains a rather viable
and the well-motivated DM candidate despite null results from numerous DM
experiments and various searches at the Large Hadron Collider (LHC).

The discovery of the Higgs boson ATLAS:2012yve ; CMS:2012qbp  at the LHC
completes the findings of all the particles appearing in the SM particle
physics which vindicates the scheme of spontaneous electroweak symmetry
breaking (SSB) as incorporated in the SM. Nevertheless, the electroweak
symmetry is restored in the early Universe, i.e., at high temperatures
Kirzhnits:1972ut . It is generally understood that SSB transition occurs in
the SM via a smooth cross-over Kajantie:1996qd . However, in the presence of
additional scalars in a scenario beyond the SM (BSM), a first-order phase
transition (FOPT) could occur. FOPT at the electroweak scale has an advantage
in that it can provide one of the required conditions to explain the baryon
asymmetry of the Universe (BAU). The conventional measure of BAU, $Y_{B}$, is
the ratio of the difference in the number density of baryon and anti-baryon
($n_{B}$ and $n_{\bar{B}}$, respectively) and the entropy density ($s$), i.e.,
$Y_{B}=(n_{B}-n_{\bar{B}})/s$. The most accurate value of $Y_{B}$ ($=8.65\pm
0.09\times 10-11$) comes from the Planck experiment Planck:2015fie  from its
measurement of the baryon acoustic oscillations in the power spectrum of the
cosmic microwave background.

In order for baryogenesis to take place, the following three Sakharov
requirements Sakharov:1967dj  must be met: (i) non-conservation of baryon
number ($\not{B}$), (ii) $C$ and $CP$ violations ($\not{C}$, $\cancel{CP}$)
and (iii) departure from thermal equilibrium. Electroweak baryogenesis (EWBG)
is one of the most attractive mechanisms to explain the BAU. It has been
widely studied in the literature Trodden:1998ym ; Anderson:1991zb ;
Huet:1995sh ; Morrissey:2012db  and it requires an FOPT in the electroweak
Higgs sector. FOPT fulfils the third criterion of Sakharov’s conditions, i.e.,
the system has to go out of thermal equilibrium. In addition, the phase
transition has to be strongly first-order to avoid a wash-out of the created
baryons inside the broken EW phase Quiros:1999jp .

The study of FOPT at around the electroweak scale has recently gained
heightened attention, particularly after the recent observation of
gravitational waves (GW) from LIGO and VIRGO collaboration
LIGOScientific:2016aoc ; LIGOScientific:2017vwq ; LIGOScientific:2018mvr ;
LIGOScientific:2020ibl . This is since FOPT would be responsible for possible
stochastic GW which might act as a background for the various future
space/ground-based GW experiments, viz., LISA LISA:2017pwj , ALIA Gong:2014mca
, Taiji Hu:2017mde , TianQin TianQin:2015yph , aLigo+ Harry:2010zz , Big Bang
Observer (BBO) Corbin:2005ny  and Ultimate(U)-DECIGO Kudoh:2005as . Note that
EWPT in the SM is unable to generate any GW since the transition is of a
cross-over type. Therefore, detection of such type of a stochastic GW would
likely point to BSM physics.

In addition, an FOPT in the early Universe can significantly impact the
thermal history of those particle(s) species that freeze out before the phase
transition. During the phase transition of the Universe from a false to the
true vacuum, the relic density of those species is diluted due to an injection
of entropy and a release of latent heat. Today’s relic abundance of the
thermal species that decouple after the phase transition would not be affected
as a thermal equilibrium attained after the phase transition would erase the
effect of dilution.

The effect of dilution on the DM relic density would be significant when the
temperature of the FOPT happens to be smaller than the DM freeze-out
temperature. Such an effect would therefore be important in the context of a
first-order electroweak phase transition (FOEWPT) for DM candidates with TeV-
scale masses ($m_{\chi}$) which are expected to get frozen out from the
thermal equilibrium at a freeze-out temperature ($T_{\rm FO}$) around the
electroweak scale ($T_{\rm FO}\backsimeq\frac{m_{\chi}}{25}$). The dilution of
thermal relic abundance of DM due to FOPT has been studied in the past by
considering some toy-model-like effective potential such as adding a larger
degree of freedom from the hidden sector Megevand:2003tg ; Megevand:2007sv ;
Wainwright:2009mq ; Chung:2011hv . Recently, it has been studied in more
realistic models such as a singlet extension of the SM (xSM) and two-Higgs-
doublet model (2HDM+S) in reference Xiao:2022oaq .

Apart from the issues pertaining to DM and BAU, another shortcoming of the SM
is that it considers the neutrinos to be massless while various experiments
have established that neutrinos indeed have mass Super-Kamiokande:1998kpq ;
SNO:2001kpb . In this work, we consider $Z_{3}$-symmetric complex singlet
scalar extension type-II seesaw model which has a natural explanation of the
origin of finite neutrino mass, provides a singlet scalar DM candidate and can
generate the observed baryon asymmetry of the Universe via the EWBG. In
contrast to the type-I and type-III seesaw models, type-II seesaw model
permits large neutrino Yukawa couplings via a small triplet vacuum expectation
value concurrently even with a sub-TeV seesaw. In addition, new interactions
between the SM Higgs doublet and the complex triplet could modify the Higgs
potential in favour of an FOPT. The interplay among the production of GW, DM
and physics at the LHC has been studied in this model in reference
Ghosh:2022fzp . Recently, it has been shown that this model can explain the
observed excess of electron-positron flux from AMS-02 AMS:2014xys , DAMPE
DAMPE:2017fbg  and Fermi-LAT Fermi-LAT:2017bpc  collaborations Yang:2021buu .
The present work aims to investigate a possible dilution of the relic
abundance of a heavy singlet-like scalar DM due to an FOPT in the Higgs sector
around the electroweak scale. We calculate the magnitude of the dilution
factor and point out the region of the relevant parameter space where such a
dilution could be significant. Furthermore, a direct connection between the
dilution factor and the generation of stochastic GW as a result of an FOEWPT
is also discussed.

The present work is organized as follows. We briefly review how an FOPT
affects the relic abundance of DM particles that freeze out later than the
transition temperature in section 2. The detailed description of the model is
discussed in section 3. We present our results in section 4. In section 5, we
conclude. The estimation of the production of stochastic GW from sound waves
is discussed in the appendix A.

##  2 Dilution of DM relic density due to FOPT

In this section, we discuss the dilution of thermal DM due to the release of
entropy during an FOPT that occurs at a considerably lower temperature than
$T_{\rm FO}$. The dilution factor depends critically on the temperature at
which FOPT occurs. Thus, we will start off by talking about the study of phase
transition of the effective scalar potential at finite temperature. We will
later go over how to estimate the dilution factor due to the release of
entropy during the FOPT.

When the finite-temperature effective Higgs potential, $V(\phi,T)$, exhibits
(at least) two local minima that are already separated by a barrier, an FOEWPT
can start to occur in the early Universe. In the hot and radiation-dominated
era of the early Universe at high temperatures, the minimum of the scalar
potential is at the origin in the field space ($\phi=0$), i.e., the
electroweak symmetry is unbroken. As the temperature drops due to the
expansion of the Universe, a second minimum away from the origin in the field
value ($\phi=\phi_{\text{broken}}$) may develop at which
$V(\phi_{\text{broken}},T)>V(0,T)$. Note that the value of
$\phi_{\text{broken}}$ varies with temperature. With the drop in the
temperature of the Universe, $V(\phi_{\text{broken}},T)$ starts to decrease
relative to $V(0,T)$ and at some temperature $T=T_{c}$, where $T_{c}$ is known
as the ‘critical temperature of the phase transition, both the two minima
become degenerate. As the temperature continues to drop further below $T_{c}$,
the minimum at $\phi=\phi_{\text{broken}}$ becomes the global minimum of the
effective potential while the minimum at the origin is now considered to be a
false minimum. When there is a barrier between the two minima, the system can
start to tunnel from the false minimum to the true global minimum of the
potential. As a result, at a certain temperature $T<T_{c}$, a phase transition
like this one, known as a first-order type phase transition, can take place in
the early Universe.

The model of bubble nucleation is an effective way to describe this type of
tunnelling process. Because of the large surface tension of the bubble walls
compared with the energy difference between the two minima, most of the
bubbles of the broken minimum remain small and finally collapse. The bubble of
the broken phase nucleates in the cosmological plasma, in which the
electroweak symmetry is unbroken, when the rate of nucleation per unit volume
($\Gamma$) is larger than the Hubble expansion rate. A bubble that is already
formed in the plasma continues to develop, collide, and coalesce with other
bubbles which are also forming until a big bubble is created. In this manner,
the whole space is engulfed and EWSB spreads across the entire field-space. At
finite temperatures ($T$), in the semi-classical approximation the tunnelling
probability from the false vacuum to the true vacuum per unit time per unit
volume is given by Turner:1992tz

$\Gamma(T)\simeq T^{4}\left(\frac{S_{3}\left(T\right)}{2\pi
T}\right)^{3/2}e^{-S_{3}\left(T\right)/T},$ (1)

where the three-dimensional Euclidean action $S_{3}\left(T\right)$ of the
background field ($\phi$), in the spherical polar coordinate, is given by

$S_{3}\left(T\right)=4\pi\int dr\hskip
2.84526ptr^{2}\left[\dfrac{1}{2}\left(\partial_{r}\vec{\phi}\right)^{2}+V\right]\,\,.$
(2)

By extending this Euclidean action, it is possible to determine the
prerequisite of the formation of a bubble of a critical size. In this work, we
consider the publicly accessible package $\mathbf{CosmoTransitions}$
Wainwright:2011kj  for this investigation and to calculate the bounce solution
for the aforementioned Euclidean action. The nucleation temperature can be
found by solving the following equation:

$\int_{T_{n}}^{\infty}\frac{dT}{T}\frac{\Gamma(T)}{H(T)^{4}}\simeq 1.$ (3)

The above relation indicates that the probability of a single bubble
nucleating inside one horizon volume is around 1. It estimates the required
condition $S_{3}(T)/T\approx 140$ to obtain the nucleation temperature
($T_{n}$) of the phase transition Apreda:2001us . $T_{n}$ is the highest
temperature at which $S_{3}/T\lesssim 140$. The absolute values of $T_{c}$,
$T_{n}$ and their difference ($\Delta T_{cn}=T_{c}-T_{n}$) have a significant
impact on the dilution factor of the relic density of DM due to FOPT.

Depending on the value of $\Delta T_{cn}$, the study of dilution of DM relic
density due to the release of entropy caused by an FOPT can be classified into
two cases. First, consider the case where $\Delta T_{cn}$ is small. Here, the
temperature at which the phase transition occurs is close to the critical
temperature of the effective potential which corresponds to negligible
supercooling Megevand:2007sv ; Anderson:1991zb . In this scenario, the total
entropy is conserved as the system is almost in the equilibrium, i.e.,
$sa^{3}$ is conserved where ‘$s$’ denotes the entropy density and ‘$a$’
represents the scale factor of the expansion of the Universe. Therefore the
entropy density of a phase can be found using this conservation relation. In
the following discussion, $+$ ($-$) indicates the high-temperature symmetric
(low-temperature broken) phase and the subscript $i$ ($f$) denotes the
beginning (end) of the FOPT. Due to the release of entropy during an FOPT, the
entropy density changes as

$s_{-}=\bigg{(}\frac{a_{i}}{a_{f}}\bigg{)}^{3}s_{+}\,\,.$ (4)

Thus, in this scenario, the dilution factor due to an FOPT is given by

$d\equiv\bigg{(}\frac{a_{f}}{a_{i}}\bigg{)}^{3}=\bigg{(}\frac{s_{+}(T_{c})}{s_{-}(T_{n})}\bigg{)}\,\,.$
(5)

The second case corresponds to the larger $\Delta T_{cn}$ scenario, i.e., when
$T_{n}<<T_{c}$. Equation 4 does not hold as the symmetric and the broken
phases are no longer in equilibrium at the beginning of the phase transition.
The evolution of the phase transition in this category can be described in
three distinct stages: supercooling, reheating, and phase coexistence. The
high-temperature phase dominates the Universe during the supercooling stage
wherein the entropy remains conserved. Therefore, similar to equation 4 the
following relation can be found

$\bigg{(}\frac{a_{i}}{a_{s}}\bigg{)}^{3}=\bigg{(}\frac{s_{+}(T_{n})}{s_{+}(T_{c})}\bigg{)}\,\,,$
(6)

where $a_{s}$ is the scale factor at the end of the supercooling stage, i.e.,
near $T\simeq T_{n}$. At the end of the supercooling stage, the release of the
latent heat reheats the Universe. The overall energy density of the Universe,
$\rho$, does not change if the reheating proceeds far faster than the Universe
expansion rate. However, in this situation entropy does not continue to be
preserved. The Universe can reach the stage of phase coexistence at $T=T_{c}$
if a significant amount of reheating takes place. Its energy density would
then be given by

$\rho_{-}(T_{c})=f\rho_{-}(T_{c})+(1-f)\rho_{+}(T_{c}),$ (7)

where $f$ is the volume fraction of the plasma of the low-temperature phase at
the beginning of the phase coexistence which can be found from the energy
conservation relation given by

$\rho_{+}(T_{n})=\rho_{-}(T_{c})\,.$ (8)

Using equations 7 and 8, we can find the following relation for $f$, i.e.,

$f=\frac{\rho_{+}(T_{c})-\rho_{+}(T_{n})}{L},$ (9)

where $L~{}(=\rho_{+}(T_{c})-\rho_{-}(T_{c}))$ denotes the transition latent
heat at $T=T_{c}$. Finally, the process of the third stage, i.e., the phase
coexistence stage, occurs very quickly such that the total entropy again
remains conserved which leads to the following relation:

$a_{s}^{3}[(1-f)s_{+}(T_{c})+fs_{-}(T_{c})]=a_{f}^{3}s_{-}(T_{c}).$ (10)

The left- and right-hand sides of the above relation correspond to the total
entropy at the beginning and at the end of this stage, respectively. It can be
expressed as

$\bigg{(}\frac{a_{s}}{a_{f}}\bigg{)}^{3}=\frac{1-\Delta
s/s_{+}(T_{c})}{1-f\Delta s/s_{+}(T_{c})},$ (11)

where $\Delta s=s_{+}(T_{c})-s_{-}(T_{c})$. Therefore, by combining equations
6 and 11, it is possible to find the dilution factor for this scenario and it
is given by

$d\equiv\bigg{(}\frac{a_{f}}{a_{i}}\bigg{)}^{3}=\bigg{(}\frac{1-f\Delta
s/s_{+}(T_{c})}{1-\Delta
s/s_{+}(T_{c})}\bigg{)}\bigg{(}\frac{s_{+}(T_{c})}{s_{+}(T_{n})}\bigg{)}.$
(12)

In this work, we use this relation to estimate the dilution of the DM relic
density caused by an FOPT in section 4.

In addition, FOPT in the early Universe could generate stochastic GW. Such a
GW signal can be estimated from the four portal parameters that are related to
the phase transition in the scalar potential and those are
$T_{n},\alpha,\beta/H_{n}$ and $v_{w}$ where ‘$\alpha$’ is a dimensionless
quantity which is related to the energy budget of the FOPT, $\beta/H_{n}$
estimates the inverse duration of the phase transition and $v_{w}$ is the
bubble wall velocity. Definitions of these parameters are shown in Appendix A.
Such a GW from an FOPT is generally produced via three distinct processes:
bubble wall collisions Kosowsky:1992rz ; Kosowsky:1991ua , long-standing sound
waves in the plasma Hindmarsh:2013xza ; Giblin:2013kea ; Giblin:2014qia ;
Hindmarsh:2015qta  and magnetohydrodynamic (MHD) turbulence Caprini:2006jb ;
Kahniashvili:2008pf ; Kahniashvili:2008pe ; Kahniashvili:2009mf ;
Caprini:2009yp ; Kisslinger:2015hua . Among all these, the sound wave
contribution is the dominant contribution to the peak of the GW intensity.
Various relations from that computation are presented in Appendix A. We use
these relations to study the connection between the dilution of the DM relic
density and the produced stochastic GW resulting from an FOPT.

Before ending this section, note that there are some prerequisites in
estimating the dilution factor. It is related to the assumption that the
released latent heat after the supercooling reheats the Universe which brings
the temperature back to a temperature just close to $T_{c}$. Observe that the
value of the variable $f$, that estimates the plasma volume fraction of the
low-temperature phase at the starting of the phase coexistence stage, can not
be larger than 1 in equations 7 and 8. However, in some situations,
particularly when the dilution factor is large, the value of $f$ can exceed
the value 1 using the relation in equation 9. The underlying assumption made
above has the explanation behind this. When $T_{n}<<T_{c}$, the large amount
of released latent heat cannot bring the system to a temperature close to
$T_{c}$. In this scenario, for the correct estimation of the dilution factor,
$T_{n}$ needs to be replaced by the reheating temperature $T_{r}$ in equation
12. Although $T_{r}$ estimation is beyond the scope of this work as such a
calculation consider specific kinetic processes Megevand:2007sv . If we
neglect the contributions from the terms that includes $f$, i.e., considering
$f=1$, in equation 12 the relation of dilution factor becomes
$d\equiv\bigg{(}\frac{a_{f}}{a_{i}}\bigg{)}^{3}=\bigg{(}\frac{s_{+}(T_{c})}{s_{+}(T_{n})}\bigg{)}=\bigg{(}\frac{T_{c}}{T_{n}}\bigg{)}^{3}=\bigg{(}1+\frac{\Delta
T_{\text{cn}}}{T_{n}}\bigg{)}^{3}$ Xiao:2022oaq . This relation actually
estimates a much-enhanced dilution factor. In addition, in the realistic
scenario, friction will drastically reduce the mobility of bubble walls in the
plasma-wall system. The collisions of the walls of the bubbles of the true
broken phase also release entropy and enhance the dilution factor
Wainwright:2009mq . Therefore, refuting the underline assumption does not
imply a reduction in the dilution factor that we estimate in this work.

##  3 The theoretical scenario

The current model is an extension of the SM with a $SU(2)_{L}$ scalar triplet,
$\Delta$, with hyper-charge, $Y=2$, and a complex scalar singlet, $S$, which
transforms non-trivially under a $Z_{3}$-discrete symmetry Ghosh:2022fzp ;
Yang:2021buu . The discrete $Z_{3}$\- symmetry stabilises the singlet field,
i.e., it does not develop ${\it vev}$ on EWSB and becomes the DM candidate of
the model. In Table 1, we present various quantum numbers of various scalars
of the scenario under the extended gauge group $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times Z_{3}$. The Lagrangian of the model is given by

$\displaystyle{\cal{L}}_{\mathrm{tot}}={\cal{L}}_{\mathrm{Kinetic}}+{\cal{L}}_{\mathrm{Yukawa}}-{\cal{V}}(H,\Delta,S).$
(13)

where

$\displaystyle{\cal{L}}_{\mathrm{Kinetic}}$ $\displaystyle=$
$\displaystyle(D^{\mu}H)^{\dagger}D_{\mu}H+(D^{\mu}\Delta)^{\dagger}D_{\mu}\Delta+(\partial^{\mu}{S})^{\dagger}\partial_{\mu}S\
,$ (14) $\displaystyle{\cal{L}}_{\mathrm{Yukawa}}$ $\displaystyle=$
$\displaystyle{\cal{L}}^{\mathrm{SM}}_{\mathrm{Yukawa}}-(y_{L})_{\alpha\beta}~{}\overline{L_{\alpha}^{c}}~{}i\sigma^{2}\Delta~{}L_{\beta}+h.c.\
,$ (15)

with

$\displaystyle D_{\mu}H$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}-ig_{2}\frac{\sigma^{a}}{2}W_{\mu}^{a}-ig_{1}\frac{Y_{H}}{2}B_{\mu}\Big{)}H,$
$\displaystyle D_{\mu}\Delta$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Delta-
ig_{2}\left[\frac{\sigma^{a}}{2}W_{\mu}^{a},\Delta\right]-{ig_{1}}\frac{Y_{\Delta}}{2}B_{\mu}\Delta~{}.$
(16)

Here, $y_{L}$ is a $3\times 3$ complex symmetric matrix. $g_{1}$ and $g_{2}$
are the $U(1)$ and the $SU(2)$ gauge couplings, respectively. The heavy
triplet scalar helps us to generate neutrino masses through the standard type-
II seesaw mechanism. Neutrino masses are generated once the neutral triplet
scalar gets non-zero ${\it vev}$ after the EWSB. The representations of the
Higgs doublet ($H$), the triplet scalar ($\Delta$) and the singlet scalar
($S$) are shown in table 1. After the EWSB, ${\it vev}$ of the neutral
$CP$-even field of $H$ ($\Delta$) is denoted by $v_{d}$ ($v_{t}$) where
$v=\sqrt{v_{d}^{2}+2v_{t}^{2}}=246$ GeV.

Fields | $\underbrace{SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ $\otimes Z_{3}$
---|---
Complex Scalar DM | $s=\frac{1}{\sqrt{2}}(h_{s}+ia_{s})$ |  1 1 0 $e^{i\frac{2\pi}{3}}$
Scalar Triplet | $\Delta=\left(\begin{matrix}\frac{\Delta^{+}}{\sqrt{2}}&\Delta^{++}\\\ \frac{1}{\sqrt{2}}\big{(}h_{t}+ia_{t}\big{)}&-\frac{\Delta^{+}}{\sqrt{2}}\end{matrix}\right)$ |  1 3 2 1
Higgs doublet | $H=\left(\begin{matrix}G^{+}\\\ \frac{1}{\sqrt{2}}\big{(}h_{d}+ia_{d}\big{)}\end{matrix}\right)$ |  1 2 1 1

Table 1:  Assignments of charges on the scalar fields under the gauge group
$\mathcal{G}\equiv\mathcal{G}_{\rm SM}\otimes Z_{3}$ where $\mathcal{G}_{\rm
SM}\equiv SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$. The hypercharges ($Y$)
are obtained by using the relation: $Q=I_{3}+\frac{Y}{2}$, where $Q$ is the
electromagnetic charge and $I_{3}$ is the third component of isospin.

The scalar potential of the model is given by:

$\displaystyle V(H,\Delta,S)$ $\displaystyle=$
$\displaystyle-\mu_{H}^{2}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}~{}$
(17) $\displaystyle+\mu_{\Delta}^{2}{\rm
Tr}\left[\Delta^{\dagger}\Delta\right]+\lambda_{1}\left(H^{\dagger}H\right){\rm
Tr}\left[\Delta^{\dagger}\Delta\right]+\lambda_{2}\left({\rm
Tr}[\Delta^{\dagger}\Delta]\right)^{2}$ $\displaystyle+\lambda_{3}~{}{\rm
Tr}[\left(\Delta^{\dagger}\Delta\right)^{2}]+\lambda_{4}~{}\left(H^{\dagger}\Delta\Delta^{\dagger}H\right)+\left[\mu\left(H^{T}i\sigma^{2}\Delta^{\dagger}H\right)+h.c.\right]+\mu_{S}^{2}~{}|S|^{2}~{}$
$\displaystyle+\lambda_{S}|S|^{4}+\frac{\mu_{3}}{3!}\Big{(}S^{3}+{S^{*}}^{3}\Big{)}+\lambda_{SH}H^{\dagger}H(S^{*}S)+\lambda_{S\Delta}~{}{\rm
Tr}\left[\Delta^{\dagger}\Delta\right](S^{*}S)~{}~{}.$

In this work, we consider all the Lagrangian parameters to be real. Thus,
there is no spontaneous $CP$ violation in our scenario. The bare masses
($\mu_{H}^{2}$ and $\mu_{\Delta}^{2}$) can be obtained by minimising the
scalar potential for the field values at the vacuum. The mass of the singlet
scalar DM after the EWSB is given by

$\displaystyle
m_{S}^{2}=\mu_{S}^{2}+\frac{1}{2}\lambda_{SH}v_{d}^{2}+\frac{1}{2}\lambda_{S\Delta}v_{t}^{2}~{}.$
(18)

On EWSB, the doublet and the triplet scalars would mix. The neutral scalar
mass-matrix, on diagonalization, gives rise to two $CP-$even scalars $h^{0}$
and $H^{0}$. These are given by,

$\displaystyle
h^{0}=h_{d}~{}\cos\theta_{t}+h_{t}~{}\sin\theta_{t}~{},~{}~{}~{}~{}H^{0}=-h_{d}~{}\sin\theta_{t}+h_{t}~{}\cos\theta_{t}\,,$
(19)

where $\theta_{t}$ represents the mixing angle of the diagonalizing matrix and
is given by

$\displaystyle\tan 2\theta_{t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\mu
v_{d}-(\lambda_{1}+\lambda_{4})v_{d}~{}v_{t}}{M_{\Delta}^{2}-\frac{1}{4}\lambda_{H}v_{d}^{2}+(\lambda_{2}+\lambda_{3})v_{t}^{2}}\,\,,$
(20)

where $M_{\Delta}^{2}=\frac{\mu v_{d}^{2}}{\sqrt{2}v_{t}}$. The corresponding
mass eigenvalues of these physical eigenstates are given by

$\displaystyle m_{h^{0}}^{2}$ $\displaystyle=$
$\displaystyle\left(M_{\Delta}^{2}+2v_{t}^{2}(\lambda_{2}+\lambda_{3})\right)\sin^{2}\theta_{t}+2\lambda_{H}v_{d}^{2}\cos^{2}\theta_{t}-\frac{v_{t}\sin
2\theta_{t}\left(2M_{\Delta}^{2}-v_{d}^{2}(\lambda_{1}+\lambda_{4})\right)}{v_{d}}\,,$
$\displaystyle m_{H^{0}}^{2}$ $\displaystyle=$
$\displaystyle\left(M_{\Delta}^{2}+2v_{t}^{2}(\lambda_{2}+\lambda_{3})\right)\cos^{2}\theta_{t}+2\lambda_{H}v_{d}^{2}\sin^{2}\theta_{t}+\frac{v_{t}\sin
2\theta_{t}\left(2M_{\Delta}^{2}-v_{d}^{2}(\lambda_{1}+\lambda_{4})\right)}{v_{d}}.$

For $m_{h^{0}}<m_{H^{0}}$, the lighter eigenstate $h^{0}$ mimics the SM Higgs
boson with mass $m_{h_{\mathrm{SM}}}=125$ GeV. Similarly, the mixing between
two $CP$-odd states ($a_{d}$, $a_{t}$) would project out the massless Nambu-
Goldstone mode and one massive pseudoscalar, $A^{0}$ with its mass given by

$\displaystyle m_{A^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{M_{\Delta}^{2}\left(4v_{t}^{2}+v_{d}^{2}\right)}{v_{d}^{2}}~{}~{}.$
(22)

A similar orthogonal rotation in the singly-charged scalar sector comprising
of $G^{\pm}$ and $\Delta^{\pm}$ fields would lead to one massless Nambu-
Goldstone state which shows up as the longitudinal components of the massive
$W^{\pm}$-bosons and the charged scalar $H^{\pm}$. The mass of $H^{\pm}$ is
given by

$\displaystyle m_{H^{\pm}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\left(2v_{t}^{2}+v_{d}^{2}\right)\left(4M_{\Delta}^{2}-\lambda_{4}v_{d}^{2}\right)}{4v_{d}^{2}}.$
(23)

The mass of the doubly charged scalar $H^{\pm\pm}(\equiv\Delta^{\pm\pm})$ is
given by

$\displaystyle m_{H^{\pm\pm}}^{2}$ $\displaystyle=$ $\displaystyle
M_{\Delta}^{2}-\lambda_{3}v_{t}^{2}-\frac{\lambda_{4}v_{d}^{2}}{2}~{}~{}.$
(24)

Note that the constraint on $m_{H^{\pm\pm}}$ from the various heavy Higgs
searches at the LHC mostly depends on $v_{t}$, $m_{H^{\pm\pm}}$ and the mass
difference $\Delta m$ $(=m_{H^{\pm\pm}}-m_{H^{\pm}}$). At the limit
$v_{t}^{2}/v_{d}^{2}<<1$, $\Delta m$ can be found from the following relation:

$\displaystyle\big{(}m_{H^{\pm\pm}}^{2}-m_{H^{\pm}}^{2}\big{)}\approx-\frac{\lambda_{4}v_{d}^{2}}{4}~{};~{}~{}\big{(}m_{H^{\pm}}^{2}-m_{A^{0}}^{2}\big{)}\approx-\frac{\lambda_{4}v_{d}^{2}}{4}~{};~{}~{}m_{H}\approx
m_{A}^{0}\approx M_{\Delta}.$ (25)

Thus, sign of $\lambda_{4}$ generates two types of the mass hierarchy among
$H^{\pm\pm}$, $H^{\pm}$, $H^{0}/A^{0}$. For $\lambda_{4}$ $<0$ ($>0$) the mass
ordering is $m_{H^{\pm\pm}}>m_{H^{\pm}}>m_{H^{0},A^{0}}$
($m_{H^{\pm\pm}}<m_{H^{\pm}}<m_{H,A^{0}}$).

Various theoretical constraints such as stability, unitarity, perturbativity
and the latest experimental constraints from the dark sector as well as from
the various LHC searches, the constraints from the flavor and neutrino sectors
and the constraints from the electroweak precision data observables such as
$\rho$, $S$, $T$, $U$ parameters have been discussed in detail in reference
Ghosh:2022fzp .

In the present work, we focus on that kind of parameter space region
exhibiting an FOEWPT in the early Universe. Such a scenario prefers the
triplet sector scalars to be lighter. However, the null results from various
collider searches that have been carried out over the years at the LHC
ATLAS:2012hi ; Chatrchyan:2012ya ; ATLAS:2014kca ; Khachatryan:2014sta ;
CMS:2016cpz ; CMS:2017pet ; Aaboud:2017qph ; CMS:2017fhs ; Aaboud:2018qcu ;
Aad:2021lzu  to hunt those triplet-like scalar excitations, strongly constrain
the parameter space of the type-II seesaw sector of the present model. Such a
bound on $m_{H^{\pm\pm}}$ mostly depend on $\Delta m$, $v_{t}$ and the value
of $m_{H^{\pm\pm}}$. In the scenario when $\Delta m=0$, the ATLAS
collaboration finds the most stringent bound to be $m_{H^{\pm\pm}}\gtrsim 870$
GeV. This was accomplished by making the assumption that
BR($H^{\pm\pm}\rightarrow\mu^{\pm}\mu^{\pm}$) is 100$\%$ Aaboud:2017qph ,
which corresponds to a smaller $v_{t}$ region, using 36.1 fb-1 data at
$\sqrt{s}=13$ TeV. On the other hand, a different ATLAS analysis assuming
100$\%$ branching fraction of $H^{\pm\pm}$ to the $W^{\pm}W^{\pm}$ final
state, that corresponds to the larger $v_{t}$ region, excludes on
$m_{H^{\pm\pm}}\gtrsim 350$ GeV Aad:2021lzu  with 139 fb-1 data. In the case
of $\Delta m<0$ the bound is much stronger. In this scenario, the effective
production cross-section of $H^{\pm\pm}$ at the LHC increases as it can
produce from the cascade decays of $H^{0}/A^{0}$ and $H^{\pm}$. By recasting
several ATLAS and CMS results, it has been shown in a recent study
Ashanujjaman:2021txz  that the existing exclusion limit for $m_{H^{\pm\pm}}$
is 1115 (1076) GeV for $\Delta m=-10$ ($-30$) GeV with $v_{t}\sim
10^{-5}-10^{-6}$ GeV. On the contrary, the bound on $m_{H^{\pm\pm}}$ relaxes a
lot in the case of $\Delta m>0$.

The scenario with $\Delta m>$ 0, the neutral triplet-like scalar $H^{0}$ and
$A^{0}$ mostly decay to invisible neutrinos for $v_{t}<$ $10^{-4}$ GeV
Ashanujjaman:2021txz . Thus the produced $H^{\pm\pm}H^{\mp\mp}$ and
$H^{++}H^{--}$ yields soft-leptons/jets in the final states. Since the
particles are very soft, it is difficult to reconstruct them at the LHC.
Recently, by recasting various ATLAS and CMS analyses in reference
Ashanujjaman:2021txz , it has been shown that LHC would be unable to constrain
the parameter space of $\Delta m=$ 10 GeV (30 GeV), $v_{t}$ around $10^{-3}$
GeV to $10^{-5}$ GeV ($10^{-3}$ GeV to $10^{-6}$ GeV) and
$m_{H^{\pm\pm}}\gtrsim$ 200 GeV at 3 ab-1 integrated luminosity data.

Therefore, the region of parameter space of the triplet sector that we
consider for this work corresponds to, 111In this article, we focus on a
similar region of parameter space of the type-II seesaw model that is
considered in reference Ghosh:2022fzp . Note that, from the EW precision data
$\Delta m$ is upper bounded and it has to be below 40 GeV.

$\displaystyle\\{10\,\text{GeV}<\Delta
m<40\,\text{GeV}\\}\Rightarrow\lambda_{4}<0~{};\,\\{10^{-6}\,\text{GeV}<v_{t}<10^{-3}\
\,\text{GeV}\\};\,\\{m_{H^{\pm\pm}}>200\,\text{GeV}\\}.~{}~{}$ (26)

The precision study of $h_{\rm SM}$ at the LHC restricts the large mixing of
the doublet-like scalar with the triplet-like neutral scalar as the observed
$\sim$ 125 GeV Higgs boson is more SM-like. Therefore, we limit our scan to
tiny $\sin\theta_{t}$ values to allow minimum mixing and maintain the nature
of the Higgs boson SM-like. However, loop-induced decays of the SM-like Higgs
boson ($h_{\rm SM}$), such as $h_{\rm SM}\rightarrow\gamma\gamma$, can still
deviate significantly from the experimentally observed limits of the signal
strength of the same decay channel even at small $\sin\theta_{t}$ value. The
variation of the signal strength of the $h_{\rm SM}\rightarrow\gamma\gamma$
decay mode in the above-discussed region of parameter space in equation 26 is
discussed in reference Ghosh:2022fzp ; Zhou:2022mlz . Note that, at relatively
small triplet-like charged Higgs bosons and larger portal couplings
($\lambda_{1},\lambda_{4}$), the decay width of the SM Higgs into a photon
pair significantly deviates from the observed limits ATLAS:2022tnm ;
CMS:2021kom .

In this article, we analyze the thermal history of the scalar potential and
determine the region of the parameter space that features an FOEWPT in the
early Universe. After such an FOEWPT, the broken phase eventually evolves to
the EW minimum of the potential at $T=0$. We use $\mathbf{CosmoTransitions}$
Wainwright:2011kj  to study the evolution of various minima with temperature
as well as the transition between the various minima of the scalar potential
at finite temperature. The theoretical details for the study of phase
transition in the early Universe, such as the field-dependent mass relations
of various degrees of freedom and the daisy coefficients (to estimate the
thermal mass corrections) are described in detail in reference Ghosh:2022fzp .
Furthermore, we analyse the dilution factor of the thermal relic density
caused by such an FOEWPT. Various DM observables, such as its relic density,
its freeze-out temperature and its spin-independent direct detection cross-
section have been estimated using micrOMEGAs (v4.3) Belanger:2006is .

##  4 Result

With the description of the model and various experimental constraints in
section 3, we move on to present our results. we discuss the feasibility of a
scalar DM with a mass around/above 1 TeV in our scenario, as well as the range
of the freeze-out temperatures for such heavy DM in subsection 4.1.
Furthermore, in subsection 4.2, we estimate the dilution factor of the DM
relic density due to an FOEWPT. We discuss the region of parameter space of
this model where the dilution factor becomes significantly large. As pointed
out in the introduction, we further study the dependencies of the dilution
factor on the model parameters, the nucleation temperature and on the strength
and the duration of the phase transition. Finally, in subsection 4.3, we
discuss the connection between the production of stochastic GW and the
dilution factor as a result of an FOEWPT.

###  4.1 TeV-scale scalar DM and the freeze-out temperature

Figure 1: [Left plot:] Variation of relic density ($\Omega_{s}h^{2}$) as a
function of DM mass ($m_{s}$) in GeV for different ranges $\lambda_{S\Delta}$
: $0.01\leq\lambda_{S\Delta}\leq 0.1$ (green), $0.1<\lambda_{S\Delta}\leq 1.0$
(red), $1.0<\lambda_{S\Delta}\leq 2.0$ (blue) and $0.001<\lambda_{SH}\leq
0.5$. [Right plot:] Variation of freeze-out temperature ($T_{\text{FO}}$) with
$m_{s}$ for same range of $\lambda_{S\Delta}$ and $\lambda_{SH}$. In these two
plots $\mu_{3}=m_{s}$ and the triplet sector parameters are fixed at
$m_{H^{0},A^{0}}=367.7$ GeV , $m_{H^{\pm}}=387.5$ GeV, $m_{H^{\pm\pm}}=406.4$
GeV, $\sin\theta_{t}=-1.03\times 10^{-6}$. In the left plot, on the vertical
axis $\Omega_{s}h^{2}$ is truncated at the value of 50 and on the horizontal
axis $m_{s}$ is considered up to 5 TeV. Black horizontal lines indicates the
Planck observed DM relic density, i.e., $\Omega_{\text{DM}}h^{2}=0.120\pm
0.001$. The vertical balck lines indicate opening of various DM annihilation
channels. In the right plot, the horizontal axis is truncated at the value of
$T_{\text{FO}}=200$ GeV.

The phenomenology of the scalar DM in our situation is discussed in this
subsection. The DM annihilates in three ways to freeze out in the early
Universe and those processes are given by: DM annihilation$\hskip
8.5359pt(a)~{}S~{}S^{*}~{}\to{\rm SM~{}~{}SM}$, (b) $S~{}S^{*}~{}\to{\rm
X~{}~{}Y};~{}~{}~{}\\{\rm X,Y\\}=\\{H^{0},A^{0},H^{\pm},H^{\pm\pm}\\}$ and DM
semi-annihilation (c) $~{}S~{}S~{}\to S~{}U;~{}~{}\\{U\\}=\\{h_{\rm
SM},H^{0}\\}$. The DM Higgs portal interactions
$\lambda_{SH}(S^{*}S)(H^{\dagger}H)$ and $\lambda_{S\Delta}(S^{*}S){\rm
Tr}[\Delta^{\dagger}\Delta]$ control the DM-annihilation processes (a) and
(b), respectively, whereas the DM semi-annihilation processes are initiated
via the cubic interaction term $\mu_{3}(S^{3}+{S^{*}}^{3})$ in the scalar
potential.222Note that at small $v_{t}$-limit the DM semi-annihilation process
$S~{}S\to S~{}H^{0}$ and the $S$-channel neutral triplet-like scalar
$H^{0}$-mediated DM-annihilation process are high suppressed. Therefore, the
DM relic density varies as:

$\displaystyle\Omega_{S}h^{2}\propto\frac{1}{\langle\sigma v\rangle_{SS\to{\rm
SM~{}SM}}+\langle\sigma v\rangle_{SS\to SU}+\langle\sigma v\rangle_{SS\to
X~{}Y}}~{}.$ (27)

In the left plot of figure 1, we illustrate the variation of $\Omega_{S}h^{2}$
as a function of $m_{S}$ while fixing the triplet-sector parameters at
$m_{H^{0},A^{0}}=367.7$ GeV , $m_{H^{\pm}}=387.5$ GeV, $m_{H^{\pm\pm}}=406.4$
GeV and $\sin\theta_{t}=-1.03\times 10^{-6}$ while the DM-Higgs interaction
couplings are varied in the range of $0.001<\lambda_{SH}<0.5$,
$0.01<\lambda_{S\Delta}<2$ and $0<m_{s}<5$ TeV. Such a choice of parameters is
considered to understand the dependencies of the phenomenology of DM on the
model parameters. In the right plot of figure 1, we present the variation of
the freeze-out temperature ($T_{\text{FO}}$) with $m_{s}$. Different colours
show different ranges of $\lambda_{S\Delta}$ in these two plots. Colors in
green, red and blue indicate $0.01\leq\lambda_{S\Delta}\leq 0.1$,
$0.1<\lambda_{S\Delta}\leq 1.0$ and $1.0<\lambda_{S\Delta}\leq 2.0$,
respectively. The black dotted horizontal lines indicate the measured density
of DM relics from the PLANCK experiment, i.e.,
$\Omega_{\text{DM}}h^{2}=0.120\pm 0.001$ Planck:2018vyg . The equation 1
suggests that the relic density might rapidly decrease with growing DM mass as
a result of the opening of new number-changing processes at a certain
threshold DM mass. Such a phenomenon shows up in the left plot.

In the scenario where the DM mass is larger than the triplet-like scalar
masses, the DM annihilation processes mentioned in the category ‘b’ open up
and the four-point contact interaction term, $\lambda_{S\Delta}(S^{*}S){\rm
Tr}[\Delta^{\dagger}\Delta]$, starts to contribute. A larger
$\lambda_{S\Delta}$ enhances the DM annihilation cross-sections and beyond a
certain value, the DM annihilation cross-section of the processes of the
category ‘b’ becomes the dominant one compared with the other annihilation
processes. Note that in the limit of small $v_{t}$ and $\sin\theta_{t}$, the
contribution of the $H^{0}$-mediated $t$-channel process to the DM direct
detection spin-independent (DMDD-SI) cross-section is negligible. Therefore,
the DMDD-SI cross-section mostly depends on $\lambda_{SH}$ (mediated via
$h_{\rm SM}$) and has an insignificant effect on the variation of
$\lambda_{S\Delta}$. Thus, for a heavier DM, the region of parameter space
with larger $\lambda_{S\Delta}$ can meet the experimentally observed DM relic
density criteria without increasing the DMDD-SI cross-section significantly.
As a result, this model can accommodate a heavy ($\sim$TeV-scale) DM complying
with various DM experimental constraints. The variation of $T_{\text{FO}}$
with $m_{S}$ in the right plot of figure 1 indicates that heavy $m_{S}$
corresponds to larger $T_{\text{FO}}$. It follows roughly $T_{\text{FO}}\sim
m_{s}/25$ relation. The ratio $m_{s}/T_{\text{FO}}$ changes with the variation
of various portal couplings. It is worth noting that the DM mass has to be
large ($\gtrsim 1$ TeV) for the situation in which the DM freeze-out
temperature is higher than the temperature at which the FOEWPT occurs in the
scalar potential.

In order to obtain the DM abundance and the freeze-out temperature we first
implement the model in ${\tt FeynRules}$ Alloul:2013bka  and then the outputs
are fed into ${\tt micrOMEGAs}$ Belanger:2006is . Note that in the situation
where an FOPT occurs around the electroweak scale in the scalar potential and
the DM freeze-out temperature is larger than the phase transition temperature,
the DM relic density and freeze-out temperature that are estimated using ${\tt
micrOMEGAs}$ no longer remain valid. Such a scenario may appear for heavier DM
($\gtrsim 1$ TeV). However, such a computation is outside the scope of this
study, we preserve it for future work. The present work aims to quantify the
dilution factor of the DM relic density caused by the FOEWPT due to the
release of entropy, which is discussed in the next subsection.

###  4.2 Estimation of the dilution factor

In this subsection, we estimate the dilution factor of the DM relic density
due to an FOPT along the $h_{d}$-field direction. In the region of parameter
space we scan for this work, the triplet field values, $h_{t}$, at the various
minima of the effective potential at finite temperature always remain tiny.
The FOPT happens only along the $h_{d}$-field direction. Therefore, we only
discuss the dilution factor due to this one-step FOPT. However, an FOPT can
occur in the early Universe along the singlet-field direction, i.e., along the
$h_{s}$-field direction. Such a phase transition demands a relatively low DM
mass Ghosh:2022fzp . On the other hand, the dilution effect caused by the FOPT
on the relic density becomes significant only when the DM mass is relatively
large $\gtrsim$ 1 TeV. An FOPT along the $h_{s}$-field direction is unlikely
to occur for such a heavy DM. Therefore, in the context of this work, the
relevant FOPT is the transition that happens only along the $h_{d}$-field
direction.

Figure 2: Variation of the dilution factor ($d$) with the effective quartic
coupling $\lambda_{1}+\lambda_{4}$. The palette-color indicates
$m_{H^{\pm\pm}}$.

The variation of the dilution factor caused by an FOPT, calculated using
equation 12, with the effective quartic coupling of the potential
($\lambda_{1}+\lambda_{4}$) has been shown in figure 2. The color in the
palette indicates the mass of the doubly-charged Higgs. Note that the dilution
factor increases with the increase of the effective quartic coupling
$\lambda_{1}+\lambda_{4}$. The color variation indicates that a decrease in
$m_{H^{\pm\pm}}$ enhances the dilution factor for a given value of
$\lambda_{1}+\lambda_{4}$. Thus, the parameter space with larger effective
quartic couplings ($\lambda_{1}$, $\lambda_{4}$) and lower $m_{H^{\pm\pm}}$
are preferred for larger dilution factors resulting from an FOPT along the
$h_{d}$-field direction.

One of the most critical parameters of the estimation of dilution is the
temperature at which the FOPT occurs. Here, we discuss the correlations
between the nucleation temperature ($T_{n}$) of the phase transition and the
dilution factor. Figure 3 represents the variation of the dilution factor with
$T_{n}$. It indicates that the dilution factor increases with decreasing
$T_{n}$. The palette color in the left plot indicates the difference between
$T_{c}$ and $T_{n}$, i.e., $\Delta T_{cn}$. The variation in color indicates
that the dilution factor also increases with an increasing $\Delta T_{cn}$. In
the scenario with a small $\Delta T_{cn}$, the impact of dilution becomes
insignificant due to negligible supercooling. As opposed to that, significant
supercooling can occur for the large separations between $T_{c}$ and $T_{n}$,
which enhances the dilution factor. This result is consistent with the
findings in references Megevand:2007sv ; Wainwright:2009mq ; Bian:2018mkl ;
Bian:2018bxr . The dependence of the supercooling with the dilution factor can
be understood from the relation in equation 12. In the right plot of figure 3,
the strength of the phase transition ($\xi_{n}$) is defined as,
$\xi_{n}=\frac{\sqrt{<h_{d}(T_{n})>^{2}_{\text{BP}}-<h_{d}(T_{n})>^{2}_{\text{SP}}}}{T_{n}}$
where ‘BP’ and ‘SP’ represent the broken phase and the symmetric phase,
respectively. $\xi_{n}$ is indicated by the palette color. The color variation
indicates that the dilution factor increases with $\xi_{n}$. It is expected
since a stronger FOPT often results in substantial supercooling, which
enhances the dilution factor.

Figure 3: Variation of dilution factor ($d$) with the nucleation temperature
($T_{n}$) of the FOEWPT. The variation of the difference between the critical
temperature and the nucleation temperature $\Delta T_{cn}$ and the strength of
the phase transition $\xi_{n}$ are indicated via the palettes in the left and
right plots, respectively.

We find that the dilution factor can increase up to around 4 at $T_{n}\sim 40$
GeV. On the other hand, the effect of dilution is almost negligible for
$T_{n}\gtrsim$ 120 GeV. Therefore, the relic density would not be diluted if
the DM freeze-out temperature, $T_{\text{FO}}$, is lower than 40 GeV. On the
contrary, there would always be a considerable dilution of the relic density
for any DM candidate that freezes out at temperatures larger than 120 GeV. As
a result, the dilution effect would only be significant for significantly
heavy DM.

The DMDD cross-section depends on the relic density of the DM in the present
Universe. For the underabundant DM, the DMDD-SI cross-section scales with the
factor $f_{\text{scale}}=\frac{\Omega_{\text{DM}}h^{2}}{0.120}$. The study of
the dilution of relic density is crucial as it can rescue some of the samples
which are otherwise excluded from the DMDD experimental constraints from the
various experiment such as XENON-1T Aprile:2018dbl  and PANDA-4T
PandaX-4T:2021bab . Additionally, such a dilution caused by FOPT can bring
down the DM relic density value below the Planck experiment measured value and
save the point that would otherwise be excluded due to the overabundance.
Therefore, such an estimation of the dilution factor due to FOEWPT is crucial
for the study of DM phenomenology. In addition, note that such an FOEWPT can
generate a stochastic GW. Thus, in the following subsection, we shall discuss
the connection between the dilution factor and the produced GW due to the
FOEWPT.

###  4.3 Connection between the produced GW and the Dilution factor

Future proposed GW experiments might detect stochastic GW generated during an
FOEWPT which also dilutes the relic density of DM. The dominant contribution
of the GW peak intensity among the three mechanisms of the production of GW,
as described in section 2, comes from the sound waves. The details of the
estimation of GW intensity from the sound waves are presented in Appendix A.
The GW peak amplitude can be found by putting $f=f_{\text{SW}}$ in equation
28. The peak frequency ($f_{\text{SW}}$) is given in equation 29.

In figure 4, we present the variation of the peak amplitudes with the peak
frequencies of the produced GWs as a result of an FOEWPT. The dilution factor
of the DM relic density caused by an FOEWPT is presented by the palette-color.
Note that various future proposed space/ground-based GW experiments, such as
LISA LISA:2017pwj , ALIA Gong:2014mca , Taiji Hu:2017mde , TianQin
TianQin:2015yph , aLigo+ Harry:2010zz , Big Bang Observer (BBO) Corbin:2005ny
and Ultimate(U)-DECIGO Kudoh:2005as , etc., have different regions of
sensitivity over the peak intensity and the peak frequency range. Therefore,
the study of the variation of the peak frequency and the peak amplitude of the
produced GW is crucial for detecting the spectrum in future proposed GW
experiments. The color variation in figure 4 indicates that the dilution
factor increases as both GW peak amplitude and peak frequency increase. Such a
connection between GW intensity and dilution factor can be realised on their
dependence on the strength of the phase transition, $\xi_{n}$. Thus, a
stronger FOEWPT forecasts a bigger GW peak amplitude as well as a larger
dilution factor. Therefore, observing the GW peak amplitude and the peak
frequency in some of the future proposed GW experiments can reveal the range
of the DM dilution factor of the DM relic density due to a release of entropy
during the FOEWPT.

Figure 4: Variation of the GW peak amplitude
($\Omega_{\text{GW}}h^{2})_{peak}$ with the peak frequency
($f_{{\text{GW}}_{peak}}$) estimated from the sound wave contribution. The
palette-color shows the variation of the dilution factor ($d$) .

##  5 Conclusions

Despite many successes of the SM, it cannot explain the origin of the observed
baryon asymmetry of the Universe, the non-vanishing masses of neutrinos and
accommodate a DM candidate. In this article, we consider the $Z_{3}$-symmetric
complex singlet scalar extended type-II seesaw model that can simultaneously
provide solutions to these issues within a single framework. An SFOEWPT is one
of the essential requirements for triggering EWBG that can generate the
observed BAU at the electroweak scale. Such an FOEWPT could dilute the relic
density of DM due to an injection of entropy and a release of latent heat.

In the current model, the triplet sector can modify the SM Higgs potential in
favor of an FOEWPT along the $h_{d}$-field direction even when the singlet
scalar fields are heavy. As a result, the singlet scalar DM can get heavier
while still allowing for an FOEWPT in the scalar potential. In such a
scenario, a significant dilution in the DM relic abundance is possible for a
heavier DM that freezes out much before the FOEWPT takes place.

We find the regions of parameter space of the triplet sector of the scenario
that corresponds to an FOEWPT and can generate a sizable amount of dilution.
Such a region prefers relatively light $m_{H^{\pm\pm}}$ and larger effective
quartic couplings $\lambda_{1}+\lambda_{4}$. In the case of supercooling, the
released entropy caused by the FOEWPT could dilute the DM relic density up to
a factor of around 4 in our scenario. Furthermore, we show the variation of
the dilution factor with the temperature at which the phase transition occurs.
We note that the dilution factor increases with decreasing $T_{n}$. The
difference between $T_{c}$ and $T_{n}$ is also an important factor in this
estimation. A larger gap between $T_{c}$ and $T_{n}$ leads to a significant
amount of supercooling during the phase transition which, in turn, enhances
the dilution factor. DM relic density dilution starts only for $T_{n}\lesssim
120$ GeV. Thus, the DM with a freeze-out temperature much larger than this,
which is possible in this model, could experience significant dilution in its
relic abundance. Such a dilution might retrieve some of the regions of
parameter space that were previously ruled out by the measured value of the DM
relic density and/or the latest constraints from the DM direct-detection
experiments. On the contrary, there will be no dilution in the abundance of DM
whose freeze-out temperature is $\lesssim$ 40 GeV and such a scenario
corresponds to a relatively light DM ($\lesssim$ 1 TeV).

As is well known we also find that the dilution factor increases with the
strength of the FOEWPT. Furthermore, in the early Universe, such an FOEWPT
could have generated stochastic GW. Hence a connection between the dilution
factor and the generation of the stochastic GW, as a result of an FOEWPT, is
discussed. The dilution factor increases with GW peak amplitude and peak
frequency. Therefore, observing peak amplitude and peak frequency of GW
produced from an FOEWPT in some of the future proposed space/ground-based GW
experiments can reveal the range of the DM dilution factor of the DM relic
density due to the release of entropy during the same phase transition.

Dilution of the DM relic density is only possible in this model when the DM is
heavy and an FOEWPT occurs in the early Universe. The type-II seesaw sector of
this current model that can alter the SM-like Higgs potential in favor of an
FOEWPT would get tested by studying the di-photon decays of the SM-like Higgs
boson at the HL-LHC. A null-finding of new physics at the HL-LHC would rule
out the possibility of an FOEWPT triggered by the triplet sector and hence the
possibility of dilution of relic density of a TeV-scale DM in such a scenario.

## Acknowledgments

The work of SR is supported by the Department of Atomic Energy (DAE),
Government of India for the Regional Centre for Accelerator-based Particle
Physics (RECAPP) at Harish-Chandra Research Institute (HRI). SR is also
supported by the excellence in research award from Infosys. SR would like to
thank AseshKrishna Datta and Yang Zhang for useful discussions. SR also
acknowledges the use of High-Performance Scientific cluster computing
facilities at HRI and RECAPP.

## Appendix

##  Appendix A Contribution of sound waves to gravitational waves power
spectrum

The dominant contribution to the total GW power spectrum coming from the sound
waves, $\Omega_{\text{sw}}{\rm h^{2}}$, is given by Hindmarsh:2013xza ;
Giblin:2013kea ; Giblin:2014qia ; Hindmarsh:2015qta

$\Omega_{\text{SW}}{\rm h}^{2}=2.65\times
10^{-6}\Upsilon(\tau_{SW})\left(\dfrac{\beta}{H_{\star}}\right)^{-1}v_{w}\left(\dfrac{\kappa_{v}\alpha}{1+\alpha}\right)^{2}\left(\dfrac{g_{*}}{100}\right)^{-\frac{1}{3}}\left(\dfrac{f}{f_{\text{SW}}}\right)^{3}\left[\dfrac{7}{4+3\left(\dfrac{f}{f_{\text{SW}}}\right)^{2}}\right]^{\frac{7}{2}}\,\,,$
(28)

where $f_{\text{SW}}$ is the present day peak frequency and is given by

$f_{\text{SW}}=1.9\times 10^{-5}\hskip
2.84526pt\text{Hz}\left(\dfrac{1}{v_{w}}\right)\left(\dfrac{\beta}{H_{\star}}\right)\left(\dfrac{T_{n}}{100\hskip
2.84526pt\text{GeV}}\right)\left(\dfrac{g_{*}}{100}\right)^{\frac{1}{6}}\,\,.$
(29)

In the above relations, $H_{\star}$ is the Hubble rate just after the end of
GW production and $T_{\star}$ is the temperature at that time. For this work,
we consider $T_{\star}\approx T_{n}$. The dimensionless quantity that is
related to the energy budget of the phase transition, $\alpha$, is defined as
Espinosa:2010hh

$\alpha=\frac{\rho_{\text{vac}}}{\rho^{*}_{\text{rad}}}=\frac{1}{\rho^{*}_{\text{rad}}}\left[T\frac{\partial\Delta
V(T)}{\partial T}-\Delta V(T)\right]\Bigg{|}_{T_{*}}.$ (30)

The parameter $\beta$ that estimates the inverse time duration of the phase
transition is given by

$\beta=-\frac{dS_{3}}{dt}\Bigr{|}_{t_{*}}\simeq\dfrac{\dot{\Gamma}}{\Gamma}=H_{*}T_{*}\frac{d(S_{3}/T)}{dT}\Bigr{|}_{T_{*}}.$
(31)

The parameter $\kappa_{v}$ estimates the fraction of latent heat energy
transferred into the bulk motion of the fluid and is given by Espinosa:2010hh

$\kappa_{v}\simeq\left[\dfrac{\alpha}{0.73+0.083\sqrt{\alpha}+\alpha}\right]\,\,.$
(32)

The suppression factor $\Upsilon(\tau_{SW})$ appears due to the finite
lifetime of sound waves and is given by Guo:2020grp ; Hindmarsh:2020hop

$\Upsilon(\tau_{SW})=1-\frac{1}{\sqrt{1+2\tau_{\text{sw}}H_{\ast}}}\,\,,$ (33)

Where $\tau_{\text{sw}}$ is the lifetime of the sound waves. It can be
estimated from the time scale in which the turbulence develops. It is
approximately given by Pen:2015qta ; Hindmarsh:2017gnf :

$\displaystyle\tau_{\text{sw}}\sim\frac{R_{\ast}}{\bar{U}_{f}}\,\,,$ (34)

where $R_{\ast}$ denotes the mean bubble separation related to the phase
transition duration. It is given by $R_{\ast}=(8\pi)^{1/3}v_{w}/\beta$
Hindmarsh:2019phv ; Guo:2020grp . The quantity $\bar{U}_{f}$ represents the
root-mean-squared (RMS) velocity and it is given by
$\bar{U}_{f}=\sqrt{3\kappa_{v}\alpha/4}$ Hindmarsh:2019phv ; Weir:2017wfa .
More details related to the computation of the GW power spectrum from the FOPT
in the present model have been discussed in reference Ghosh:2022fzp .

## References

  * (1) N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * (2) G. Aad et al. [ATLAS], Phys. Lett. B 716 (2012), 1-29 doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]].
  * (3) S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012), 30-61 doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]].
  * (4) D. A. Kirzhnits and A. D. Linde, Phys. Lett. B 42 (1972), 471-474 doi:10.1016/0370-2693(72)90109-8
  * (5) K. Kajantie, M. Laine, K. Rummukainen and M. E. Shaposhnikov, Nucl. Phys. B 493 (1997), 413-438 doi:10.1016/S0550-3213(97)00164-8 [arXiv:hep-lat/9612006 [hep-lat]].
  * (6) P. A. R. Ade et al. [Planck], Astron. Astrophys. 594 (2016), A13 doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]].
  * (7) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967), 32-35 doi:10.1070/PU1991v034n05ABEH002497
  * (8) M. Trodden, Rev. Mod. Phys. 71 (1999), 1463-1500 doi:10.1103/RevModPhys.71.1463 [arXiv:hep-ph/9803479 [hep-ph]].
  * (9) G. W. Anderson and L. J. Hall, Phys. Rev. D 45 (1992), 2685-2698 doi:10.1103/PhysRevD.45.2685
  * (10) P. Huet and A. E. Nelson, Phys. Rev. D 53 (1996), 4578-4597 doi:10.1103/PhysRevD.53.4578 [arXiv:hep-ph/9506477 [hep-ph]].
  * (11) D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012), 125003 doi:10.1088/1367-2630/14/12/125003 [arXiv:1206.2942 [hep-ph]].
  * (12) M. Quiros, [arXiv:hep-ph/9901312 [hep-ph]].
  * (13) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 116 (2016) no.6, 061102 doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
  * (14) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 119 (2017) no.16, 161101 doi:10.1103/PhysRevLett.119.161101 [arXiv:1710.05832 [gr-qc]].
  * (15) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. X 9 (2019) no.3, 031040 doi:10.1103/PhysRevX.9.031040 [arXiv:1811.12907 [astro-ph.HE]].
  * (16) R. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. X 11 (2021), 021053 doi:10.1103/PhysRevX.11.021053 [arXiv:2010.14527 [gr-qc]].
  * (17) P. Amaro-Seoane et al. [LISA], [arXiv:1702.00786 [astro-ph.IM]].
  * (18) X. Gong, Y. K. Lau, S. Xu, P. Amaro-Seoane, S. Bai, X. Bian, Z. Cao, G. Chen, X. Chen and Y. Ding, et al. J. Phys. Conf. Ser. 610 (2015) no.1, 012011 doi:10.1088/1742-6596/610/1/012011 [arXiv:1410.7296 [gr-qc]].
  * (19) W. R. Hu and Y. L. Wu, Natl. Sci. Rev. 4 (2017) no.5, 685-686 doi:10.1093/nsr/nwx116
  * (20) J. Luo et al. [TianQin], Class. Quant. Grav. 33 (2016) no.3, 035010 doi:10.1088/0264-9381/33/3/035010 [arXiv:1512.02076 [astro-ph.IM]].
  * (21) G. M. Harry [LIGO Scientific], Class. Quant. Grav. 27 (2010), 084006 doi:10.1088/0264-9381/27/8/084006
  * (22) V. Corbin and N. J. Cornish, Class. Quant. Grav. 23 (2006), 2435-2446 doi:10.1088/0264-9381/23/7/014 [arXiv:gr-qc/0512039 [gr-qc]].
  * (23) H. Kudoh, A. Taruya, T. Hiramatsu and Y. Himemoto, Phys. Rev. D 73 (2006), 064006 doi:10.1103/PhysRevD.73.064006 [arXiv:gr-qc/0511145 [gr-qc]].
  * (24) A. Megevand, Phys. Rev. D 69 (2004), 103521 doi:10.1103/PhysRevD.69.103521 [arXiv:hep-ph/0312305 [hep-ph]].
  * (25) A. Megevand and A. D. Sanchez, Phys. Rev. D 77 (2008), 063519 doi:10.1103/PhysRevD.77.063519 [arXiv:0712.1031 [hep-ph]].
  * (26) C. Wainwright and S. Profumo, Phys. Rev. D 80 (2009), 103517 doi:10.1103/PhysRevD.80.103517 [arXiv:0909.1317 [hep-ph]].
  * (27) D. Chung, A. Long and L. T. Wang, Phys. Rev. D 84 (2011), 043523 doi:10.1103/PhysRevD.84.043523 [arXiv:1104.5034 [astro-ph.CO]].
  * (28) Y. Xiao, J. M. Yang and Y. Zhang, [arXiv:2207.14519 [hep-ph]].
  * (29) Y. Fukuda et al. [Super-Kamiokande], Phys. Rev. Lett. 81 (1998), 1562-1567 doi:10.1103/PhysRevLett.81.1562 [arXiv:hep-ex/9807003 [hep-ex]].
  * (30) Q. R. Ahmad et al. [SNO], Phys. Rev. Lett. 87 (2001), 071301 doi:10.1103/PhysRevLett.87.071301 [arXiv:nucl-ex/0106015 [nucl-ex]].
  * (31) P. Ghosh, T. Ghosh and S. Roy, [arXiv:2211.15640 [hep-ph]].
  * (32) M. Aguilar et al. [AMS], Phys. Rev. Lett. 113 (2014), 121102 doi:10.1103/PhysRevLett.113.121102
  * (33) G. Ambrosi et al. [DAMPE], Nature 552 (2017), 63-66 doi:10.1038/nature24475 [arXiv:1711.10981 [astro-ph.HE]].
  * (34) S. Abdollahi et al. [Fermi-LAT], Phys. Rev. D 95 (2017) no.8, 082007 doi:10.1103/PhysRevD.95.082007 [arXiv:1704.07195 [astro-ph.HE]].
  * (35) X. Qi and H. Sun, [arXiv:2104.01045 [hep-ph]].
  * (36) M. S. Turner, E. J. Weinberg and L. M. Widrow, Phys. Rev. D 46 (1992), 2384-2403 doi:10.1103/PhysRevD.46.2384
  * (37) C. L. Wainwright, Comput. Phys. Commun. 183 (2012), 2006-2013 doi:10.1016/j.cpc.2012.04.004 [arXiv:1109.4189 [hep-ph]].
  * (38) R. Apreda, M. Maggiore, A. Nicolis and A. Riotto, Nucl. Phys. B 631 (2002), 342-368 doi:10.1016/S0550-3213(02)00264-X [arXiv:gr-qc/0107033 [gr-qc]].
  * (39) A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. Lett. 69 (1992), 2026-2029 doi:10.1103/PhysRevLett.69.2026
  * (40) A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. D 45 (1992), 4514-4535 doi:10.1103/PhysRevD.45.4514
  * (41) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Phys. Rev. Lett. 112 (2014), 041301 doi:10.1103/PhysRevLett.112.041301 [arXiv:1304.2433 [hep-ph]].
  * (42) J. T. Giblin, Jr. and J. B. Mertens, JHEP 12 (2013), 042 doi:10.1007/JHEP12(2013)042 [arXiv:1310.2948 [hep-th]].
  * (43) J. T. Giblin and J. B. Mertens, Phys. Rev. D 90 (2014) no.2, 023532 doi:10.1103/PhysRevD.90.023532 [arXiv:1405.4005 [astro-ph.CO]].
  * (44) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Phys. Rev. D 92 (2015) no.12, 123009 doi:10.1103/PhysRevD.92.123009 [arXiv:1504.03291 [astro-ph.CO]].
  * (45) C. Caprini and R. Durrer, Phys. Rev. D 74 (2006), 063521 doi:10.1103/PhysRevD.74.063521 [arXiv:astro-ph/0603476 [astro-ph]].
  * (46) T. Kahniashvili, A. Kosowsky, G. Gogoberidze and Y. Maravin, Phys. Rev. D 78 (2008), 043003 doi:10.1103/PhysRevD.78.043003 [arXiv:0806.0293 [astro-ph]].
  * (47) T. Kahniashvili, L. Campanelli, G. Gogoberidze, Y. Maravin and B. Ratra, Phys. Rev. D 78 (2008), 123006 [erratum: Phys. Rev. D 79 (2009), 109901] doi:10.1103/PhysRevD.78.123006 [arXiv:0809.1899 [astro-ph]].
  * (48) T. Kahniashvili, L. Kisslinger and T. Stevens, Phys. Rev. D 81 (2010), 023004 doi:10.1103/PhysRevD.81.023004 [arXiv:0905.0643 [astro-ph.CO]].
  * (49) C. Caprini, R. Durrer and G. Servant, JCAP 12 (2009), 024 doi:10.1088/1475-7516/2009/12/024 [arXiv:0909.0622 [astro-ph.CO]].
  * (50) L. Kisslinger and T. Kahniashvili, Phys. Rev. D 92 (2015) no.4, 043006 doi:10.1103/PhysRevD.92.043006 [arXiv:1505.03680 [astro-ph.CO]].
  * (51) G. Aad et al. [ATLAS], Eur. Phys. J. C 72 (2012), 2244 doi:10.1140/epjc/s10052-012-2244-2 [arXiv:1210.5070 [hep-ex]].
  * (52) S. Chatrchyan et al. [CMS], Eur. Phys. J. C 72 (2012), 2189 doi:10.1140/epjc/s10052-012-2189-5 [arXiv:1207.2666 [hep-ex]].
  * (53) G. Aad et al. [ATLAS], JHEP 03 (2015), 041 doi:10.1007/JHEP03(2015)041 [arXiv:1412.0237 [hep-ex]].
  * (54) V. Khachatryan et al. [CMS], Phys. Rev. Lett. 114 (2015) no.5, 051801 doi:10.1103/PhysRevLett.114.051801 [arXiv:1410.6315 [hep-ex]].
  * (55) [CMS], CMS-PAS-HIG-14-039.
  * (56) [CMS], CMS-PAS-HIG-16-036.
  * (57) M. Aaboud et al. [ATLAS], Eur. Phys. J. C 78 (2018) no.3, 199 doi:10.1140/epjc/s10052-018-5661-z [arXiv:1710.09748 [hep-ex]].
  * (58) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 120 (2018) no.8, 081801 doi:10.1103/PhysRevLett.120.081801 [arXiv:1709.05822 [hep-ex]].
  * (59) M. Aaboud et al. [ATLAS], Eur. Phys. J. C 79 (2019) no.1, 58 doi:10.1140/epjc/s10052-018-6500-y [arXiv:1808.01899 [hep-ex]].
  * (60) G. Aad et al. [ATLAS], JHEP 06 (2021), 146 doi:10.1007/JHEP06(2021)146 [arXiv:2101.11961 [hep-ex]].
  * (61) S. Ashanujjaman and K. Ghosh, JHEP 03 (2022), 195 doi:10.1007/JHEP03(2022)195 [arXiv:2108.10952 [hep-ph]].
  * (62) R. Zhou, L. Bian and Y. Du, JHEP 08 (2022), 205 doi:10.1007/JHEP08(2022)205 [arXiv:2203.01561 [hep-ph]].
  * (63) [ATLAS], [arXiv:2207.00348 [hep-ex]].
  * (64) A. M. Sirunyan et al. [CMS], JHEP 07 (2021), 027 doi:10.1007/JHEP07(2021)027 [arXiv:2103.06956 [hep-ex]].
  * (65) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 176 (2007), 367-382 doi:10.1016/j.cpc.2006.11.008 [arXiv:hep-ph/0607059 [hep-ph]].
  * (66) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185 (2014), 2250-2300 doi:10.1016/j.cpc.2014.04.012 [arXiv:1310.1921 [hep-ph]].
  * (67) L. Bian and Y. L. Tang, JHEP 12 (2018), 006 doi:10.1007/JHEP12(2018)006 [arXiv:1810.03172 [hep-ph]].
  * (68) L. Bian and X. Liu, Phys. Rev. D 99 (2019) no.5, 055003 doi:10.1103/PhysRevD.99.055003 [arXiv:1811.03279 [hep-ph]].
  * (69) E. Aprile et al. [XENON], Phys. Rev. Lett. 121 (2018) no.11, 111302 doi:10.1103/PhysRevLett.121.111302 [arXiv:1805.12562 [astro-ph.CO]].
  * (70) Y. Meng et al. [PandaX-4T], Phys. Rev. Lett. 127 (2021) no.26, 261802 doi:10.1103/PhysRevLett.127.261802 [arXiv:2107.13438 [hep-ex]].
  * (71) J. R. Espinosa, T. Konstandin, J. M. No and G. Servant, JCAP 06 (2010), 028 doi:10.1088/1475-7516/2010/06/028 [arXiv:1004.4187 [hep-ph]].
  * (72) H. K. Guo, K. Sinha, D. Vagie and G. White, JCAP 01 (2021), 001 doi:10.1088/1475-7516/2021/01/001 [arXiv:2007.08537 [hep-ph]].
  * (73) M. B. Hindmarsh, M. Lüben, J. Lumma and M. Pauly, SciPost Phys. Lect. Notes 24 (2021), 1 doi:10.21468/SciPostPhysLectNotes.24 [arXiv:2008.09136 [astro-ph.CO]].
  * (74) U. L. Pen and N. Turok, Phys. Rev. Lett. 117 (2016) no.13, 131301 doi:10.1103/PhysRevLett.117.131301 [arXiv:1510.02985 [astro-ph.CO]].
  * (75) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Phys. Rev. D 96 (2017) no.10, 103520 [erratum: Phys. Rev. D 101 (2020) no.8, 089902] doi:10.1103/PhysRevD.96.103520 [arXiv:1704.05871 [astro-ph.CO]].
  * (76) M. Hindmarsh and M. Hijazi, JCAP 12 (2019), 062 doi:10.1088/1475-7516/2019/12/062 [arXiv:1909.10040 [astro-ph.CO]].
  * (77) D. J. Weir, Phil. Trans. Roy. Soc. Lond. A 376 (2018) no.2114, 20170126 doi:10.1098/rsta.2017.0126 [arXiv:1705.01783 [hep-ph]].