# Non-minimally Coupled
Vector Boson Dark Matter

Basabendu Barman  Nicolás Bernal
Ashmita Das  and Rishav Roshan

###### Abstract

We consider a simple abelian vector dark matter (DM) model, where only the DM
$(\widetilde{X}_{\mu})$ couples non-minimally to the scalar curvature
$(\widetilde{R})$ of the background spacetime via an operator of the form
$\sim\widetilde{X}_{\mu}\,\widetilde{X}^{\mu}\,\widetilde{R}$. By considering
the standard freeze-out scenario, we show, it is possible to probe such a non-
minimally coupled DM in direct detection experiments for a coupling strength
$\xi\sim\mathcal{O}\left(10^{30}\right)$ and DM mass $m_{X}\lesssim 55$ TeV,
satisfying Planck observed relic abundance and perturbative unitarity. We also
discuss DM production via freeze-in, governed by the non-minimal coupling,
that requires $\xi\lesssim 10^{5}$ to produce the observed DM abundance over a
large range of DM mass depending on the choice of the reheating temperature.
We further show, even in the absence of the non-minimal coupling, it is
possible to produce the whole observed DM abundance via 2-to-2 scattering of
the bath particles mediated by massless gravitons.

PI/UAN-2021-697FT

##  1 Introduction

The existence of dark matter (DM) has been extensively proven from several
astrophysical [1, 2, 3, 4] and cosmological [5, 6] evidences (for a review,
see, e.g. Refs. [7, 8, 9]). All these evidences unequivocally point towards
the gravitational interactions of the DM. As far as its fundamental particle
nature goes, it is already established from observations that DM has to be
electrically neutral and stable at least at the scale of lifetime of the
Universe. The measurement of the anisotropy in the cosmic microwave background
(CMB) radiation provides the most precise measurement of the DM relic density,
usually expressed as $\Omega_{\text{DM}}h^{2}\simeq 0.12$ [6], which is an
important constraint to abide by. Since the Standard Model (SM) of particle
physics fails to offer a viable candidate, one has to look beyond the realms
of the SM to explain the particle DM. The weakly interacting massive particle
(WIMP) [7, 10] by far is the most popular DM candidate where one assumes the
DM particles to be in thermal equilibrium in the early Universe due to strong
enough coupling with the visible sector that gives rise to an interaction
strength of the order of the weak scale. The DM abundance then freezes out
once the interaction rate falls out of equilibrium as the Universe expands and
cools down. The weak scale interaction strength with the visible sector
provides a window for WIMP-like DM candidates to be probed in collider,
indirect or scattering experiments (see, e.g., Ref. [11]), however no
significant excess over the background has been found so far to guarantee a
potential discovery in either of these experiments.

Contrary to the vanilla WIMP-paradigm, it is also possible that the DM
particle couples to the visible sector very weakly, so that chemical
equilibrium is never achieved. The DM is then produced by decay or
annihilation processes from the visible sector, until the production ceases
due to the cooling of the primordial thermal bath below the relevant mass
scale connecting the DM particle to the visible sector. Due to the super weak
coupling strength, the DM particles produced via the freeze-in mechanism are
referred to as feebly interacting massive particles (FIMP) [12, 13, 14, 15,
16, 17]. The feeble interaction strength between the DM and the SM sector in
freeze-in scenario implies that these classes of models are inherently very
difficult to search for in direct detection, indirect detection, or collider
experiments. It has further been pointed out, depending on whether the DM
interaction with the visible sector is renormalizable or non-renormalizable,
freeze-in can be either infrared (IR) where the DM abundance becomes important
at a low temperature [12, 16, 18, 17, 19, 20, 21, 22] or ultra-violate (UV)
where the DM genesis takes place at the highest temperature achieved by the
thermal bath [16, 23, 24, 25, 26, 27, 28, 29, 30, 31].111This temperature can
be the reheating temperature in the case of a sudden inflaton decay, but can
also be much larger if the decay of the inflaton is non-instantaneous [32,
33].

Since all the confirmed evidences for DM simply suggest that DM should have
gravitational interaction, the production of gravitational DM and its
detection prospects have been widely studied in the literature in the context
of scalar, fermion and vector boson DM [34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49]. This production mechanism of DM refers to the
particle creation due to the time varying scale factor of the Universe [34,
50, 51, 52]. The production of “supermassive” DM during the transition between
an inflationary and a matter-dominated (or radiation-dominated) Universe due
to the (non-adiabatic) expansion of the background spacetime has been
discussed in Refs. [34, 52]. On a different note, Refs. [53, 39, 54, 42, 55,
46] have studied the production of gravitational DM where only gravity
minimally couples the DM to the visible sector, such as, via the annihilation
of the SM bath particles and/or inflatons mediated via $s$-channel graviton
exchange. Beyond the minimally coupled scenario, it is also possible that the
DM is non-minimally coupled to gravity, characterised by the non-minimal
coupling $\xi$, where $\xi=0$ mimics the minimally coupled scenario. In this
context, it is worth to be mentioned that $\xi=1/6$ is known as conformal
coupling for a massless scalar field. In case where the conformal symmetry is
broken, $\xi$ can well be considered to be a free parameter222In case the SM
Higgs is non-minimally coupled to gravity, $|\xi_{h}|\lesssim 2.6\times
10^{15}$ [56, 57]. that determines the DM-SM interaction strength, albeit
suppressed by the Planck mass. While the phenomenology of non-minimally
coupled DM has been studied both in the context of WIMP [37, 58] and FIMP
[59], considering DM with or without any intrinsic spin, the study of non-
minimally coupled vector DM is rather hard to find.

In this work we have considered a simple scenario, where a massive vector
boson DM $(X_{\mu})$ that originates from an abelian gauge extension of the
SM, has a non-minimal coupling to the gravity. Here we consider only the DM to
be non-minimally coupled to gravity which makes our model construction very
economical in terms of the number of free parameters. Considering dimension-4
operators of the form
$\sim\widetilde{X}_{\mu}\widetilde{X}^{\mu}\widetilde{R}$, we show the vector
DM can either undergo pair-annihilation to the SM final states to produce the
Planck observed relic abundance via freeze-out or can be produced from the
scattering of the bath particles giving rise to out-of-equilibrium production
via freeze-in. It is worth mentioning that a simple construction like
$\sim\widetilde{X}_{\mu}\widetilde{X}^{\mu}\widetilde{R}$ is popular in the
context of inflation [60, 61, 62, 63, 64], whereas its prominence is rarely
explored in the context of DM Physics. Considering the vector DM to be a
standard WIMP, for the freeze-out scenario, one can have a viable parameter
space safe from the stringent (spin-independent) direct search bounds for
$\xi\sim\mathcal{O}(10^{30})$. On the other hand, the freeze-in production can
occur both in the presence or absence of the non-minimal coupling. In contrast
to the freeze-out case, freeze-in production of DM via non-minimal coupling
requires $\xi$ to be smaller by several orders of magnitude, that ensures the
DM production rate remains below the Hubble expansion parameter. For typical
choices of the reheating temperature and the non-minimal coupling $\xi$, it is
possible to produce DM via freeze-in over a wide mass range starting from a
few keV to several orders of TeV, satisfying various theoretical and
observational bounds. Finally, we show, gravitational UV freeze-in,
corresponding to the minimally coupled scenario, can lead to DM overabundance
for a large reheating temperature irrespective of the presence of the non-
minimal coupling. We thus consider DM genesis in the early Universe through:
$i)$ freeze-out via non-minimal coupling, $ii)$ freeze-in via non-minimal
coupling, and $iii)$ minimally coupled gravitational UV freeze-in, and in each
case we illustrate the viable parameter space.

This paper is organised as follows. The model construct is discussed in Sec. 2
elaborating the underlying action in Jordan and in the Einstein frame. The
WIMP phenomenology is discussed in Sec. 3, where the viable parameter space
for the DM satisfying the bounds from Planck observed relic density and spin-
independent direct detection is shown. We then move on to the discussion
regarding freeze-in production of vector DM in Sec. 4, under which freeze-in
via non-minimal coupling is addressed in subsection 4.1 and in subsection 4.2
the gravity mediated (minimal) DM production is addressed. Finally, we
conclude in Sec. 5. Appendices are provided for the paper to be self-
sufficient.

##  2 The framework

###  2.1 Non-minimally coupled vector DM

We consider the vector DM $X_{\mu}$ to be a massive gauge boson under some
abelian $U\left(1\right)_{X}$ symmetry and construct the following action in
the Jordan frame, where the DM is explicitly coupled to the scalar curvature
of the background spacetime ($\widetilde{R}$)

$\mathcal{\widetilde{S}}=\int
d^{4}x\,\sqrt{-\widetilde{g}}\,\bigg{[}\frac{1}{2}\bigg{(}M_{\text{Pl}}^{2}-\xi\,\widetilde{X}_{\mu}\widetilde{X}^{\mu}\bigg{)}\,\widetilde{R}+\,\mathcal{\widetilde{L}}_{\text{DM}}+\,\mathcal{\widetilde{L}}_{\text{SM}}\bigg{]},$
(2.1)

with

$\mathcal{\widetilde{L}}_{\text{DM}}=-\frac{1}{4}\widetilde{X}_{\mu\nu}\widetilde{X}^{\mu\nu}+\frac{1}{2}m_{X}^{2}\widetilde{X}_{\mu}\widetilde{X}^{\mu}.$
(2.2)

Note that $\widetilde{R}$, $\mathcal{\widetilde{L}}_{\text{DM}}$ and
$\mathcal{\widetilde{L}}_{\text{SM}}$ (defined in Appendix A) belong to the
Jordan frame and defined with respect to the metric in the Jordan frame i.e.
$\widetilde{g}_{\mu\nu}$. $M_{\text{Pl}}$ is the reduced Planck scale $\approx
2.4\times 10^{18}$ GeV. Here we assume a Stueckleberg mass term333In abelian
gauge theories, the Stueckelberg mechanism can be taken as the limit of the
Higgs mechanism where the mass of the real scalar is sent to infinity and only
the pseudoscalar is present [65, 66, 67, 68]. for the vector DM which
prohibits any connection of the DM with the visible sector apart from gravity.
This leads to the simplest scenario for a non-minimally coupled abelian vector
DM. We also assume the presence of an unbroken $\mathbb{Z}_{2}$ symmetry under
which the DM is odd while all the SM fields are even, thus ensuring the
stability of the DM by forbidding the kinetic mixing term. In the absence of
the $\mathbb{Z}_{2}$ symmetry, the dark gauge boson can still account for all
of the DM relic abundance if the kinetic mixing is of the order
$\epsilon\lesssim\mathcal{O}\left(10^{-8}\right)$ for DM masses below twice
the electron mass [69, 70], else the cosmological stability condition requires
even smaller values $(\lesssim 10^{-15})$ of the kinetic mixing parameter [71,
72]. Now, to obtain the form of the action in the Einstein frame, we perform a
conformal transformation $g_{\mu\nu}=\omega^{2}\,\widetilde{g}_{\mu\nu}$ to
the action $\mathcal{\widetilde{S}}$, where
$\omega^{2}=1-\frac{\xi}{M_{\text{Pl}}^{2}}\widetilde{X}_{\mu}\widetilde{X}^{\mu}$
and $g_{\mu\nu}$ stands for the spacetime metric in the Einstein frame. This
leads us to the action in the Einstein frame where the gravitational part of
the action turns into the well-known Einstein-Hilbert action. Thus,
considering the metric signature $(+,-)$, and using the above conformal
transformation, we obtain (see Appendix A for details),

$\displaystyle\mathcal{S}=\int d^{4}x\,\sqrt{-g}$
$\displaystyle\Biggl{[}\frac{M_{\text{Pl}}^{2}\,R}{2}+\frac{3\,\omega^{4}}{4\,M_{\text{Pl}}^{2}}\,\nabla_{\alpha}(\xi\,X_{\mu}X^{\mu})\,\nabla^{\alpha}(\xi\,X_{\mu}X^{\mu})-\frac{1}{4}\,X_{\mu\nu}\,X^{\mu\nu}$
(2.3)
$\displaystyle+\frac{1}{2\omega^{2}}m_{X}^{2}\,X_{\mu}\,X^{\mu}+\,\frac{1}{\omega^{4}}\,(\mathcal{L}_{Y}-V(H))+\,\frac{1}{\omega^{2}}(D_{\mu}H)^{\dagger}(D^{\mu}H)$
$\displaystyle+\,\frac{i}{\omega^{3}}\,\bar{f}\,\gamma^{a}\,\partial_{a}\,f-\,\frac{1}{4}\,g^{\mu\nu}\,g^{\lambda\rho}F_{\mu\nu}^{(a)}\,F_{\nu\rho}^{(a)}+\,\frac{3\,i}{\omega^{4}}\,\bar{f}\,(\not{\partial}\,\omega)\,f\Biggr{]},$

where $\mathcal{S}$ represents the action in the Einstein frame. Note that,
all the parameters in the above action such as $R$, $\mathcal{L}_{\text{SM}}$
and as well as the DM sector, now belong to the Einstein frame and defined
with respect to the metric $g_{\mu\nu}$. Now, expanding $\omega$ in the small
field limit $\xi\left(X_{\mu}\,X^{\mu}\right)/M_{\text{Pl}}^{2}\ll 1$ [73] we
obtain

$\omega=\bigg{(}1+\frac{\xi}{M_{\text{Pl}}^{2}}X_{\mu}X^{\mu}\bigg{)}^{-1/2}\simeq
1-\frac{\xi}{2M_{\text{Pl}}^{2}}X_{\mu}X^{\mu}+\,\mathcal{O}\left(M_{\text{Pl}}^{-4}\right).$
(2.4)

Using Eq. (2.4) we find

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Biggl{[}\underbrace{\frac{M_{\text{Pl}}^{2}\,R}{2}}_{{\rm
Pure\,gravity}}+\,\underbrace{(D_{\mu}H)^{\dagger}(D^{\mu}H)+\,(\mathcal{L}_{Y}-V(H))+\,i\,\bar{f}\,\gamma^{\alpha}\,\partial_{\alpha}\,f-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}}_{{\rm
SM}}$ (2.5)
$\displaystyle\underbrace{-\,\frac{1}{4}\,X_{\mu\nu}\,X^{\mu\nu}+\frac{1}{2}m_{X}^{2}\,X_{\mu}\,X^{\mu}}_{{\rm
Free\,DM\,sector}}+\,\bigg{\\{}\frac{\xi\,m_{X}^{2}}{2\,M_{\text{Pl}}^{2}}\,(X_{\alpha}X^{\alpha})^{2}+\,\frac{2\xi}{M_{\text{Pl}}^{2}}\,X_{\alpha}X^{\alpha}\,(\mathcal{L}_{Y}-V(H))$
$\displaystyle+\,\frac{\xi}{M_{\text{Pl}}^{2}}X_{\alpha}X^{\alpha}\,(D_{\mu}H)^{\dagger}(D^{\mu}H)+\,\frac{3i\,\xi}{2\,M_{\text{Pl}}^{2}}\,X_{\alpha}X^{\alpha}\,\bar{f}\,\gamma^{\mu}\,\partial_{\mu}\,f+\,3\,i\,\bar{f}\,(\not{\partial}\,\omega)\,f$
$\displaystyle+\,\frac{6i\,\xi}{M_{\text{Pl}}^{2}}\,X_{\alpha}X^{\alpha}\,\bar{f}\,(\not{\partial}\,\omega)\,f+\,\frac{3\,\omega^{4}\,\xi^{2}}{4\,M_{\text{Pl}}^{2}}\,\nabla_{\alpha}(X_{\mu}X^{\mu})\,\nabla^{\alpha}(X_{\mu}X^{\mu})\bigg{\\}}\biggr{]}.$

From the previous expression, it can be clearly seen that the terms within the
curly bracket are associated with $\xi$, and exhibit all possible DM-SM
interactions in the theory. This also shows that it is not possible for the DM
to decay gravitationally, contrary to Refs. [38, 40, 48]. Note that the mass
term for the DM is also modified by the conformal transformation parameter.
However, such a mass correction is negligible compared to the Stueckleberg
mass because of Planck suppression.

###  2.2 Dark matter production mechanisms

It is clear from Eq. (2.5) that the interaction of the DM with the visible
sector is determined by the non-minimal coupling strength $\xi$. Thus,
depending on the strength, the DM in the present model can be produced via:
$i)$ freeze-out, where the DM $X$ acts as a thermal WIMP that gives rise to
the observed relic abundance for a $\xi\sim\mathcal{O}\left(10^{30}\right)$
via 2-to-2 annihilation with the SM particles in the final states. Such large
coupling is required to ensure that the DM remains in equilibrium with the
thermal bath in the early Universe, and $ii)$ freeze-in, where the DM is
produced from 2-to-2 scattering of the bath particles. Contrary to the freeze-
out scenario, freeze-in requires $\xi\lesssim\mathcal{O}\left(10^{5}\right)$
to ensure the DM production takes place out of equilibrium at large
temperatures. In the former case the DM can be as heavy as $m_{X}\lesssim 55$
TeV satisfying unitarity bound, while in the later case the vector DM becomes
a typical FIMP and can be as heavy as
$m_{X}\sim\mathcal{O}\left(10^{10}\right)$ GeV depending on the choice of the
reheating temperature. Apart from these options, which depend on the non-
minimal coupling strength, the DM can also be produced via the irreducible
gravitational UV freeze-in, as gravitons can still mediate between the DM and
the SM. Such a process is always present since this corresponds to the
minimally coupled scenario with $\xi=0$. Tab. 1 provides a summary of the
different DM genesis mechanisms addressed in this paper.

DM production | $\xi$
---|---
Freeze-out | $\sim\mathcal{O}\left(10^{30}\right)$
(non-minimal) freeze-in | $\lesssim\mathcal{O}\left(10^{5}\right)$
Gravitational UV freeze-in (minimal) | 0
Table 1: Typical non-minimal couplings and their corresponding DM production
mechanisms.

The Boltzmann equation (BEQ) governing the DM number density $n_{X}$ evolution
can be expressed as [10]

$\dot{n}_{X}+3\,H\,n_{X}=\gamma$ (2.6)

where $\gamma$ is the reaction rate density that depends on the underlying DM
production mechanism, while the Hubble parameter reads
$H(T)=\frac{\pi}{3}\sqrt{\frac{g_{\star}}{10}}\frac{T^{2}}{M_{\text{Pl}}}$, in
a SM radiation dominated Universe, and $g_{\star}(T)$ corresponds to the
number of relativistic degrees of freedom contributing to the SM radiation
[74]. It is convenient to express Eq. (2.6) in terms of the dimensionless
quantity $x=m_{X}/T$

$x\,H\,s\,\frac{dY_{X}}{dx}=\gamma(x)\,,$ (2.7)

where we define the DM yield $Y_{X}\equiv n_{X}/s$, as the ratio of DM number
density $n_{X}$ and the comoving entropy density in the visible sector is
defined via $s(T)=\frac{2\pi^{2}}{45}g_{\star s}T^{3}$, with $g_{\star s}(T)$
being the number of relativistic degrees of freedom contributing to the SM
entropy [74]. To match the observed DM abundance
$\Omega_{\text{DM}}h^{2}\simeq 0.12$ at the present epoch $T=T_{0}$, the DM
yield has to be fixed so that
$m_{X}Y_{X}\left(T_{0}\right)=\Omega_{\text{DM}}h^{2}\frac{1}{s_{0}}\frac{\rho_{c}}{h^{2}}\simeq
4.3\times 10^{-10}$ GeV, where $\rho_{c}\simeq 1.1\times 10^{-5}h^{2}$ GeV/cm3
is the critical energy density and $s_{0}\simeq 2.9\times 10^{3}$ cm-3 is the
entropy density at present [6].

##  3 Freeze-out of vector dark matter

The phenomenology of abelian vector boson DM has been widely studied in the
literature both in the context of freeze-out and freeze-in production (see,
e.g., Refs. [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
19, 91, 92, 93, 28]). In all these cases the DM is minimally coupled to
gravity and communicates with the visible sector via the Higgs portal. Here,
however, we consider that both sectors communicate only through gravity by
considering a Stueckelberg mass term for the vector DM. In such a scenario,we
end up with only two free parameters

$\\{m_{X},\,\xi\\}$ (3.1)

that control the resulting parameter space for the DM. This makes the present
model simple yet testable.

Figure 1: Left: Pair annihilation of abelian vector DM into SM final states
leading to DM relic abundance, where SM stands for all SM particles. Right:
Scattering of DM against SM quarks $q$ leading to spin-independent direct
search.

The freeze-out parameter space for the DM is primarily constrained by the
requirement of obtaining the observed relic abundance. One should note here,
due to the absence of Higgs portal, the DM can annihilate to the SM final
states only via contact interaction involving a pair of DM and a pair of SM
particles as shown in Fig. 1. This contact interaction is induced solely by
the gravity, and hence proportional to the strength of the non-minimal
coupling $\xi$. The DM abundance is obtained by numerically solving the BEQ in
Eq. (2.6), where

$\gamma=-\langle\sigma v\rangle\Bigl{(}n_{X}^{2}-n_{X\text{eq}}^{2}\Bigr{)}$
(3.2)

for a standard WIMP scenario [94], where $n_{X\text{eq}}$ is the equilibrium
DM number density given by
$n_{X\text{eq}}(T)\simeq\frac{T}{2\pi^{2}}g_{X}\,m_{X}^{2}\,K_{2}\left(\frac{m_{X}}{T}\right)$,
for non-relativistic DM. In the present case, the pair annihilation cross-
section for the DM to the SM final states (left panel of Fig. 1) is $s$-wave
dominated:

$\left(\sigma
v\right)_{XX\to\text{SM}\text{SM}}\simeq\begin{cases}\frac{4N_{c}m_{X}^{2}\xi^{2}}{M_{\text{Pl}}^{4}\,\pi}\sqrt{1-x^{2}}\,\Bigl{(}4-x^{2}-3x^{4}\Bigr{)}+\,\mathcal{O}[v^{2}]&x\equiv
m_{f}/m_{X}\\\\[8.0pt] \frac{\delta_{V}\,\xi^{2}}{144\,\pi
m_{X}^{2}}\Bigl{(}\frac{g_{2}v_{d}}{c_{w}M_{\text{Pl}}}\Bigr{)}^{4}\sqrt{1-x^{2}}\,\Bigl{(}1+2x^{2}+3x^{4}\Bigr{)}+\,\mathcal{O}[v^{2}]&x\equiv
m_{V}/m_{X}\\\\[8.0pt]
\frac{\xi^{2}m_{X}^{2}}{9\,\pi\,M_{\text{Pl}}^{4}}\sqrt{1-x^{2}}\,\Bigl{(}1+2x^{4}\Bigr{)}+\,\mathcal{O}[v^{2}]&x\equiv
m_{h}/m_{X}\end{cases}$ (3.3)

where $N_{c}=1\,(3)$ for leptonic (quark) final states, $g_{2}$ is the SM
$SU(2)_{L}$ gauge coupling with $s_{w}$ as the sine of the weak mixing angle
and $\delta_{V}=1\,(2)$ for $Z$ $(W^{\pm})$ boson final states. Here
$v_{h}\simeq 246~{}\rm GeV$ is the Higgs vacuum expectation value and $v$ is
the relative velocity between two incoming DM particles. The final DM
abundance can be obtained by solving the BEQ numerically. However, for DM
annihilations dominated by $s$-wave processes, the relic abundance can be
approximated as [10]

$\Omega_{X}h^{2}\simeq 1.07\times
10^{9}\frac{x_{f}~{}\text{GeV}^{-1}}{\left(g_{\star
s}/\sqrt{g_{\star}}\right)\,\langle\sigma v\rangle\,M_{\text{Pl}}}$ (3.4)

where $x_{f}=m_{X}/T_{f}$ is the freeze-out temperature that can be determined
by the condition $H\left(x_{f}\right)=\langle\sigma
v\rangle\left(x_{f}\right)$.

Figure 2: Left: The relic density allowed region is shown in the bi-
dimensional $\xi-m_{X}$ plane. The grey shaded region to the right of the
black dashed line is ruled out by the unitarity bound on WIMP mass. Right: The
relic density satisfied points are shown in the direct search plane where
different coloured regions correspond to different ranges of $\xi$. The spin-
independent direct search limits from Panda-4T and projected XENONnT
experiments are shown via black solid, black dashed and black dot-dashed
curves respectively. The dashed orange curve indicates the expected discovery
limit corresponding to the so called “$\nu$-floor” from CE$\nu$NS of solar and
atmospheric neutrinos for a Ge target.

To obtain the freeze-out parameter space, we implemented this model in LanHEP
[95] and computed the relic abundance numerically in MicrOmegas [96]. The
values of $\xi$ required to match the whole observed DM abundance are shown in
the left panel of Fig. 2. From Eq. (3.3) it is clear that the annihilation
cross-section in all cases is proportional to $\xi^{2}/M_{\text{Pl}}^{4}$.
Thus, with an increase in the strength of the non-minimal coupling, one should
expect DM under abundance, while reducing the non-minimal coupling should lead
to over abundance.

The DM gives rise to a spin-independent direct search cross-section as shown
in the right panel of Fig. 1. Even in the absence of any Higgs portal, the DM
can still recoil against the SM quarks (via contact interaction), giving rise
to direct detection signals that depend only on the non-minimal coupling
$\xi$. On top of relic abundance, the spin-independent direct search exclusion
limit also puts very stringent bound on the DM parameter space, particularly
constraining large values for the non-minimal coupling $\xi$. The effective
DM-quark coupling can be parameterized as

$f_{q}\simeq\frac{\xi\,m_{q}}{M_{\text{Pl}}^{2}}\,,$ (3.5)

which gives rise to a DM-nucleus scattering cross-section for the vector boson
DM [97, 98]

$\sigma_{\text{SI}}^{XN}=\frac{\mu_{XN}^{2}}{\pi\,m_{X}^{2}}\left[Zf_{p}+\left(A-Z\right)f_{n}\right]^{2}$
(3.6)

where $f_{p,n}$ are the effective DM-nucleon coupling and
$\mu_{XN}=m_{X}\,m_{N}/\left(m_{X}+m_{N}\right)$ is the DM-nucleus reduced
mass, with $Z$ being the number of protons and $A-Z$ the number of neutrons.
We rely on the hadronic matrix elements and DM form factors included in
MicrOmegas to compute the direct search cross-section. The right panel of Fig.
2 shows the parameter space in the $\xi-m_{X}$ plane where Planck observed
relic abundance is satisfied, together with constraints from spin-independent
direct detection. Here we see, for a comparatively lower DM mass one needs a
larger $\xi$ to satisfy the observed relic abundance. This can again be
understood from Eq. (3.3), where we find the total annihilation cross-section
to SM final states goes roughly as $\left(\sigma
v\right)_{\text{total}}\sim\mathcal{A}\,\frac{\xi^{2}\,m_{X}^{2}}{M_{\text{Pl}}^{4}}\left(1+\mathcal{C}\,\frac{M_{\text{Pl}}^{4}}{m_{X}^{4}}\right)$
in the limit $x\to 0$, where
$\mathcal{A}=\left(\frac{16N_{c}}{\pi}+\frac{1}{9\pi}\right)$ and
$\mathcal{C}=\mathcal{B}/\mathcal{A}$ with
$\mathcal{B}=\frac{\delta_{v}}{144\pi}\left(\frac{g_{2}\,v_{d}}{c_{w}\,M_{\text{Pl}}}\right)^{4}$.
The relic abundance thus becomes $\Omega_{X}h^{2}\sim\frac{1}{\left(\sigma
v\right)_{\text{total}}}\sim
m_{X}^{2}\,M_{\text{Pl}}^{4}/\left(\mathcal{A}\,\xi^{2}\left[m_{X}^{4}+\mathcal{B}\,M_{\text{Pl}}^{4}\right]\right)$,
which behaves as $\Omega_{X}h^{2}\sim 1/\left(\xi^{2}\,m_{X}^{2}\right)$ for
$m_{X}\gtrsim 20~{}\rm GeV$. Therefore, a larger DM mass requires a smaller
$\xi$ and vice versa, in order to obtain the observed relic abundance. This,
in turn, influences the direct search allowed parameter space for the DM. This
is seen from the right panel of Fig. 2, where higher values of
$10^{30}\lesssim\xi\lesssim 10^{31}$ (red points) are discarded from the
present limit from PandaX-4T experiment [99] (black dashed curve) and mostly
from future projection of XENONnT [100] up to DM mass of $m_{X}\sim 500~{}\rm
GeV$. For DM mass $m_{X}\gtrsim 500$ GeV the direct search bounds are relaxed
(green points) since a smaller $\xi$ is needed to satisfy the desired
abundance for larger DM mass as argued above, which in turn produces a smaller
$\sigma_{\text{SI}}^{XN}$ aiding the direct search. A large part of the viable
parameter space, however, lies beyond the so called “$\nu$-floor” [101], below
which the number of neutrino events due to coherent elastic neutrino-nucleus
scattering (CE$\nu$NS) is expected to be much larger than the number of DM
events, which prevents to identify DM signals with certainty.

Another constraint on the DM mass and non-minimal coupling strength can be
derived from the requirement of perturbative unitarity. Here we restrict
ourselves to the tree-level unitarity bound [102]

$\left|\text{Re}\left(a_{J}\right)\right|<\frac{1}{2}$ (3.7)

where $a_{J}$ is the is the partial-wave amplitude for the total angular
momentum $J$, and is related to the tree-level scattering amplitude
$\mathcal{M}$ via

$a_{J}\left(s\right)=\frac{1}{32\pi}\int_{-1}^{+1}\,d\left(\cos\theta\right)\,P_{J}\left(\cos\theta\right)\,\mathcal{M}$
(3.8)

where $P_{J}\left(\cos\theta\right)$ is the Legendre polynomial of degree $J$.
Here we provide an approximate analytical bound on the DM parameter space for
both the freeze-out and freeze-in scenario. We note, the scattering amplitude
in either case has a dependence of the from

$\mathcal{M}\simeq\xi\,\frac{s}{M_{\text{Pl}}^{2}}\,.$ (3.9)

This leads to

$\sqrt{s}<\sqrt{\frac{8\pi}{\xi}}\,M_{\text{Pl}}$ (3.10)

from the requirement of partial-wave unitarity of the $S$-matrix, which in
turn constraints the annihilation cross section in the early Universe. A part
of the parameter space for the WIMP-like DM is thus excluded, as shown by the
grey shaded region in the left panel of Fig. 2. Before moving on, we would
like to mention that there are limits on the DM annihilation cross-section
from the non-observation of gamma-ray signals in dwarf satellite galaxies from
the MAGIC Cherenkov telescopes and the Fermi Large Area Telescope (LAT) [103].
However, these bounds typically constraint the low DM mass region $m_{X}<50$
GeV in our case , where the direct search bounds are much more severe, hence
we do not show them here.

##  4 Freeze-in production of vector dark matter

Since the effective coupling of the DM with the visible sector in the non-
minimally coupled scenario is suppressed by the Planck mass
($\sim\xi/M_{\text{Pl}}^{2}$), hence it is rather natural to assume the DM
produced out of equilibrium from the SM bath depending on the choice of the
non-minimal coupling $\xi$. In that case, the DM is non-thermally produced in
the early Universe via freeze-in [16, 17]. In this section, we show freeze-in
is a viable set-up for the non-minimally coupled vector DM,444This is in
contrast to Ref. [59] where the authors have considered a conformally induced
Higgs portal. and the observed relic abundance can be produced for a much
lower $\xi$ compared to the freeze-out case, depending on the DM mass and
reheating temperature.

###  4.1 Freeze-in via non-minimal coupling

In the present scenario, the freeze-in production of the DM occurs through the
2-to-2 scattering of the SM particles in the thermal bath via contact
interaction as in Fig. 3. We solve the full BEQ in Eq. (2.7) numerically with
the 2-to-2 annihilation cross-sections collected in Appendix C. The DM yield,
however, can be analytically computed by approximating the annihilation cross-
section to be of the form

$\sigma\left(s\right)\simeq\frac{\xi^{2}}{M_{\text{Pl}}^{4}}\,s$ (4.1)

considering all states to be massless. This gives rise to the DM reaction
density

$\gamma\simeq\frac{24\,g_{a}\,g_{b}}{\pi^{4}}\,\frac{\xi^{2}\,T^{8}}{M_{\text{Pl}}^{4}}$
(4.2)

where $g_{a,b}$ are the degrees of freedom of the incoming SM particles. The
DM yield reads

$Y_{X}\left(T\right)=\frac{45\sqrt{15}}{2\sqrt{2\pi}}\,\frac{\xi^{2}}{\sqrt{g_{\star\rho}}\,g_{\star
s}}\,\frac{\left(T_{\text{RH}}^{3}-T^{3}\right)}{\pi^{7}M_{\text{Pl}}^{3}}$
(4.3)

where assuming an instantaneous decay for the inflaton, $T_{\text{RH}}$
corresponds to the temperature at which the inflaton decays, and therefore the
maximum temperature reached by the thermal bath. We have also neglected the
small deviation due to temperature evolution of the numbers of relativistic
degrees of freedom. Note that the majority of the DM is produced near the
highest temperatures $T\simeq T_{\text{RH}}$ reached by the Universe, which is
the characteristic of UV freeze-in [16, 23, 25, 27, 28].

Figure 3: Freeze-in production of the abelian vector DM, where SM stands for
all the SM particles.

Before closing this section, we would like to comment about the instantaneous
decay approximation of the inflaton. While reheating is commonly approximated
as an instantaneous event, the decay of the inflaton into SM radiation is a
continuous process [104]. Away from this approximation for reheating, the bath
temperature may rise to a value $T_{\text{max}}$ which exceeds $T_{\text{RH}}$
[33]. It is plausible that the DM relic density may be established during this
reheating period, in which case its abundance will significantly differ from
freeze-in calculations assuming radiation domination. In particular, it has
been observed that if the DM is produced during the transition from matter to
radiation domination via an interaction rate that scales like
$\gamma(T)\propto T^{n}$, for $n>12$ the DM abundance is enhanced by a boost
factor proportional to $(T_{\text{max}}/T_{\text{RH}})^{n-12}$ [105], whereas
for $n\leq 12$ the difference between the standard UV freeze-in calculation
differ only by an $\mathcal{O}(1)$ factor from calculations taking into
account non-instantaneous reheating. More recently, it has been highlighted
that the critical mass dimension of the operator at which the instantaneous
decay approximation breaks down depend on the equation of state $\omega$, or
equivalently, to the shape of the inflationary potential at the reheating
epoch [25, 106, 30]. Therefore, the exponent of the boost factor becomes
$(T_{\text{max}}/T_{\text{RH}})^{n-n_{c}}$ with $n_{c}\equiv
6+2\,\left(\frac{3-\omega}{1+\omega}\right)$, showing a strong dependence on
the equation of state [25]. Subsequent papers have explored the impact of this
boost factor in specific models [24, 107, 108, 109, 110, 111, 112, 113, 114,
115, 116, 117, 118]. Finally, another way for enhancing the DM abundance
occurs in cosmologies where inflation is followed by an epoch dominated by a
fluid stiffer than radiation. In such scenarios, even a small radiation
abundance, produced for instance by instantaneous preheating effects, will
eventually dominate the total energy density of the Universe without the need
for a complete inflaton decay. In particular, a strong enhancement if DM
production happens via interaction rates with temperature dependence higher
that $n_{c}=6$ [29].

In the present case, as the interaction rate density $\gamma(T)\propto T^{8}$
(cf. Eq. (4.2)), a sizeable boost factor is not expected, at least in the
standard case where during reheating the inflaton energy density scales like
non-relativistic matter. However, as the precise determination of such boost
factors depends on the details of the inflationary model (in particular on the
energy density carried by the inflaton and its equation of state parameter
previous to its decay), it is beyond the scope of this study.

###  4.2 Gravitational UV freeze-in

In this section we consider the gravitational DM production in the minimal
case where $\xi=0$. In particular, we have already realised in Sec. 4.1 that
as opposed to the freeze-out scenario, freeze-in supports $\xi\ll 1$ depending
on the choice of the DM mass and reheating temperature. However, even if we
set $\xi$ to be exactly zero, still gravity can propagate between the DM and
the SM once we allow a small fluctuation in the background spacetime.
Therefore, in the small $\xi$ limit it is possible that the gravity mediated
interactions may dominate over those due to non-minimal coupling.
Specifically, we consider the weak gravity limit of the Eq. (2.5) and take
$\xi=0$. Subsequently, we expand the free part of the Lagrangian around the
flat Minkowski background which can be realised via
$g_{\mu\nu}=\,\eta_{\mu\nu}+\kappa\,h_{\mu\nu}$, where $h_{\mu\nu}$ is taken
to be a small fluctuation over the flat Minkowski spacetime and
$\kappa=1/M_{\text{Pl}}$. Due to the smallness of the fluctuation, we allow
this perturbative expansion up to the first order in $h_{\mu\nu}$. This also
leads us to $g^{\mu\nu}=\eta^{\mu\nu}-\kappa\,h^{\mu\nu}$ and
$\sqrt{-g}\approx 1+\frac{\kappa\,h}{2}$, where $h=\eta_{\mu\nu}\,h^{\mu\nu}$.
Thus we write

$\mathcal{L}_{\text{gm}}(X,{\rm
SM})=\,\mathcal{L}_{\text{gm}}^{(0)}+\,\kappa\,\mathcal{L}_{\text{gm}}^{(1)}+\mathcal{O}(\kappa^{2})+...$
(4.4)

where $\mathcal{L}_{\text{gm}}(X,{\rm SM})$ stands for the Lagrangian of SM
matter fields and DM field defined with respect to $g_{\mu\nu}$ while $\xi$ is
taken to be zero. Thereafter one can write the perturbed Lagrangian
corresponding to all the SM and DM fields up to the leading order of
$h_{\mu\nu}$ as555For a detailed derivation see Refs. [119, 120]

$\displaystyle\mathcal{L}_{gH}^{(1)}=\,\frac{h}{2}\,(D_{\mu}H)^{\dagger}\,(D^{\mu}H)-\,h^{\mu\nu}\,(D_{\mu}H)^{\dagger}\,(D^{\mu}H)-\,\frac{\kappa\,h}{2}\,V_{H}\,,$
(4.5)
$\displaystyle\mathcal{L}_{gf}^{(1)}=\,\frac{h}{2}\,(i\,\bar{f}\gamma^{\mu}\,\partial_{\mu}f)-\,\frac{i}{2}\,h_{\alpha\beta}\,\bar{f}\,\gamma^{\alpha}\,\partial^{\beta}f\,,$
(4.6)
$\displaystyle\mathcal{L}_{gX}^{(1)}=\,\frac{1}{2}\,h^{\nu}_{\alpha}\,X_{\mu\nu}\,X^{\mu\alpha}-\,\frac{h}{8}\,X_{\mu\nu}X^{\mu\nu}+\,\bigg{(}\frac{h\,m_{X}^{2}}{4}\,X_{\alpha}X^{\alpha}-\,\frac{m_{X}^{2}}{2}\,h^{\mu\alpha}\,X_{\mu}X_{\alpha}\bigg{)}.$
(4.7)

Note that all fields and operators in the above set of equations are now
contracted with respect to the Minkowski metric $\eta_{\mu\nu}$. At this stage
we note:

  * •

The above equations signify that even in the absence of the non-minimal
coupling the DM can still be produced from the visible sector via the
$s$-channel exchange of massless gravitons as in Fig. 4.

Figure 4: Gravitational UV freeze-in production of vector DM via massless
graviton mediation.

  * •

For $\xi\neq 0$, one can similarly employ the metric fluctuation to the terms
within the curly bracket of Eq. (2.5), which gives rise to coupling strength
$\sim\mathcal{O}\left(\xi/M_{\text{Pl}}^{3}\right)$ with gravitational
fluctuations ($h_{\mu\nu},\,h$). This coupling strength is more suppressed
than the effective non-minimal coupling $\left(\xi/M_{\text{Pl}}^{2}\right)$
that corresponds to the background flat metric ($\eta_{\mu\nu}$). Moreover,
this is also suppressed than that due to the metric fluctuations $\left(\sim
1/M_{\text{Pl}}\right)$, emerging from the free part of the Lagrangian in Eq.
(2.5). This leads us to neglect such coupling consistently in the subsequent
analysis.

Therefore, in the presence of both the minimal and non-minimal coupling (for
$\xi\neq 0$), the resulting Lagrangian can be written as

$\mathcal{L}\sim\mathcal{L}_{\rm minimal}+\,\mathcal{L}_{\text{non-
minimal}}\,,$ (4.8)

with the squared amplitude approximated to be

$\left|\mathcal{M}\right|^{2}\simeq\left|\mathcal{M}_{\text{non-
minimal}}\right|^{2}+\left|\mathcal{M}_{\text{minimal}}\right|^{2}+\mathcal{O}\left(\xi/M_{\text{Pl}}^{4}\right).$
(4.9)

We thus separately consider the contributions due to non-minimal coupling and
those due to graviton mediation in the present set-up.

Figure 5: The black straight contours correspond to the central value of
Planck observed DM relic abundance for different choices of $T_{\text{RH}}$
(shown by different patterns) in the left panel and for different choices of
$\xi$ in the right panel. The shaded regions are disallowed from
perturbativity (yellow), DM thermalization: $\xi\gtrsim 10^{5}$ (purple), CMB:
$T_{\text{RH}}>10^{16}$ GeV (magenta), DM production in instantaneous
approximation $T_{\text{RH}}<m_{X}$ (blue), Lyman-$\alpha$ limit:
$m_{X}\lesssim 3.5$ keV (orange) and DM overproduction due to $s$-channel
graviton mediated process (green).

The interaction rate density for such 2-to-2 graviton mediated process reads
[53, 54, 42, 107, 46, 121]

$\gamma(T)=\alpha\,\frac{T^{8}}{M_{\text{Pl}}^{4}}\,,$ (4.10)

with $\alpha\simeq 2.3\times 10^{-3}$. For $T\ll T_{\text{RH}}$, one can
analytically obtain the DM yield at the end of reheating as

$Y_{X}(T_{0})\simeq\frac{45\,\alpha}{2\pi^{3}\,g_{\star
s}}\sqrt{\frac{10}{g_{\star}}}\left(\frac{T_{\text{RH}}}{M_{\text{Pl}}}\right)^{3},$
(4.11)

in the case $m_{X}\ll T_{\text{RH}}$.666Two comments are in order. Firstly, we
note that the DM abundance could also be set entirely in the hidden sector by
the dark freeze-out of 4-to-2 interactions, where four DM particles annihilate
into two of them [122, 123, 124, 125, 46]. However, such a possibility is sub-
dominant due to a strong suppression by higher orders of $M_{\text{Pl}}$.
Secondly, the gravitational production can be enhanced in scenarios with extra
dimensions, see e.g., Refs. [126, 127, 128, 129, 130, 131, 132, 133, 134, 135,
136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146]. In Fig. 5 we
illustrate the parameter space matching the whole observed DM relic density
for the freeze-in scenario. In both the panels, the black straight lines
indicate contours satisfying the observed relic density. From the left panel
we see, for a smaller choice of $T_{\text{RH}}$, a larger $\xi$ is required to
satisfy the observed DM abundance for a fixed DM mass. This is understandable,
since the DM abundance varies as $\Omega_{X}h^{2}\propto
m_{X}\,\xi^{2}\,T_{\text{RH}}^{3}$, hence a larger $T_{\text{RH}}$ calls for a
smaller $\xi$ at a fixed DM mass to produce the right abundance. Thus, the
contours are in increasing order of $T_{\text{RH}}$ from right to left. This
can also be verified from the right panel plot where the black straight line
contours corresponding to right DM abundance are in increasing order of $\xi$
from left to right. Notice, in the freeze-in framework it is possible to have
DM mass from keV up to $\sim\mathcal{O}(10^{10})$ GeV satisfying the observed
abundance, unlike the freeze-out case where the DM mass can be maximum $\sim
55$ TeV. The contribution from the gravitational UV freeze-in is shown in
green, where the coloured region depicts DM overproduction when the DM is
minimally coupled to the SM. We see, from the left panel, for $m_{X}\sim
10^{10}$ GeV and $\xi\sim 10^{-2}$, DM production due to gravitational UV
freeze-in dominates over that due to the non-minimal coupling. The same can
also be found in the right panel, where DM overproduction from gravitational
UV freeze-in can become as important as that due to non-minimal coupling for
large $T_{\text{RH}}$ and smaller $\xi$.

The relic density allowed parameter space for the DM can further be
constrained by several bounds as shown by the coloured regions. Here we
summarise them. First of all, it is important to note that one can not take
$\xi$ arbitrarily large since in that case the reaction rate of DM production
may exceed the Hubble rate at a very high temperature, making the DM thermal.
We find the condition $\Gamma_{\text{int}}=\langle\sigma
v\rangle_{i}n^{i}_{\text{eq}}<H$ (where $i\in\text{SM}$) can be satisfied when
$\xi<10^{5}$, otherwise the DM thermalizes as early as $T\sim 10^{14}$ GeV.
Thus, we confine ourselves to $\xi<10^{5}$ and values above this limit are
discarded, as shown by the purple region in the left panel. A major part of
the viable parameter space is disallowed from the instantaneous inflaton decay
approximation, which does not allow to have a DM mass larger than the
reheating temperature $T_{\text{RH}}$. This is shown by the blue region. The
upper limit on the inflationary scale is constrained from CMB measurements
[147]: $H_{I}^{\text{CMB}}\leq 2.5\times 10^{-5}M_{\text{Pl}}$, which in turn
allows $T_{\text{RH}}\leq 10^{16}$ GeV. DM mass below 3.5 keV is forbidden
from the measurements of the free-streaming of warm DM from Lyman-$\alpha$
flux-power spectra [148, 149, 150]. This is denoted by the thin orange region.
Perturbative unitarity limit (Eq. (3.10)) forbids very small $\xi\lesssim
10^{-5}$ (shown in yellow), which can otherwise spoil the tree-level unitarity
as discussed earlier. Finally, $T_{\text{RH}}\lesssim 4$ MeV can potentially
disturb the measurement of light element yield from big bang nucleosynthesis
(BBN) and hence such a small reheating temperature can not be allowed.

##  5 Conclusions

The gravitational interaction of DM with the SM fields is inevitable, and
supported by all astrophysical and cosmological evidences for the existence of
DM. This leads us to consider a scenario where a vector boson DM couples to
the SM fields through an unique dimension-4 operator, connecting a pair of DM
fields (i.e., $\widetilde{X}_{\mu}\widetilde{X}^{\mu}$) with the scalar
curvature of the background spacetime ($\widetilde{R}$) via a non-minimal
coupling. Unlike most of the cases studied in the literature, here we consider
only the DM fields are non-minimally coupled to gravity. This gives rise to
the non-minimal coupling of DM to the visible sector in the Einstein frame
leading to DM pair annihilation (production) to (from) the SM particles. In
all such cases, the effective DM-SM interaction strength is parametrized by
the coupling $\sim\xi/M_{\text{Pl}}^{2}$, that can further be constrained by
different theoretical and experimental bounds as discussed. We take up the
simplest form of non-minimally coupled abelian vector DM, where the DM is
considered to be $\mathbb{Z}_{2}$-odd (while all the SM fields are
$\mathbb{Z}_{2}$-even) and owns a Stuecklberg mass term. This helps in
reducing the number of free parameters for the theory to only two: the non-
minimal coupling $\xi$ and the DM mass $m_{X}$, which in turn portrays the
minimalistic feature of this model. By considering the DM to be a weakly
interacting massive particle (WIMP), we find, to produce the observed relic
abundance via freeze-out, $\xi$ turns out to be
$\sim\mathcal{O}\left(10^{30}\right)$ for DM mass $m_{X}\lesssim 55$ TeV such
that perturbative unitarity is not violated. We also show, for such choice of
the non-minimal coupling, it is possible to evade strong spin-independent
direct detection bounds arising from PandaX-4T experiment (or even future
projection of XENONnT), however a large part of the viable parameter space
also gets submerged into the so-called neutrino floor. Thus, the model
provides testability for such a gravitationally coupled simple WIMP scenario
in DM scattering experiments.

In the present set-up, it is also interesting to address the freeze-in
production of the vector DM, where only 2-to-2 annihilation of the bath
particles can give rise to the required DM abundance. The freeze-in production
can occur both via the non-minimal coupling to gravity, as well as via
$s$-channel graviton exchange, where the latter is present even in the limit
$\xi\to 0$. We notice that the freeze-in yield replicates the typical UV
nature, where the DM number density reaches maximum at the highest temperature
of the thermal bath (which is the reheating temperature, assuming
instantaneous inflaton decay), and immediately saturates. Contrary to the
freeze-out scenario, a much smaller $\xi\lesssim 10^{5}$ is require to obtain
the Planck observed relic density for DM mass $m_{X}\sim 10^{12}$ GeV,
depending on the choice of the reheating temperature and satisfying several
bounds arising from perturbative unitarity, warm DM limit, scale of inflation
and BBN.

## Acknowledgement

The authors would like to thank Óscar Catà, Kunio Kaneta, Tanmoy Paul, Jing
Ren, Javier Rubio and Soumitra Sengupta for many useful correspondences and
fruitful comments. BB would like to thank Alexander Pukhov for helping with
the freeze-in calculations in MicrOmegas. BB and NB received funding from the
Patrimonio Autónomo - Fondo Nacional de Financiamiento para la Ciencia, la
Tecnología y la Innovación Francisco José de Caldas (MinCiencias - Colombia)
grant 80740-465-2020. NB also received funding from the Spanish FEDER/MCIU-AEI
under grant FPA2017-84543-P. This project has received funding /support from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No 860881-HIDDeN.

##  Appendix A Transformation from Jordan frame to Einstein frame

The action in Jordan frame can be written as follows:

$S=\,\int
d^{4}x\,\sqrt{-\widetilde{g}}\Bigg{[}\frac{1}{2}\bigg{(}M_{\text{Pl}}^{2}-\xi\,\widetilde{X}_{\mu}\widetilde{X}^{\mu}\bigg{)}\,\widetilde{R}+\,\widetilde{\mathcal{L}}_{\text{DM}}+\,\widetilde{\mathcal{L}}_{\text{SM}}\Bigg{]}$
(A.1)

where $\widetilde{g}_{\mu\nu}$ stands for the metric in the Jordan frame while
$\widetilde{\mathcal{L}}_{\text{SM}}$ and
$\widetilde{\mathcal{L}}_{\text{DM}}$ represent the Lagrangian for the SM and
the dark sector respectively, and can be written in explicit form as follows,

$\displaystyle\mathcal{\widetilde{L}}_{\text{DM}}=-\frac{1}{4}\widetilde{X}_{\mu\nu}\widetilde{X}^{\mu\nu}+\frac{1}{2}m_{X}^{2}\widetilde{X}_{\mu}\widetilde{X}^{\mu}.$
(A.2)
$\displaystyle\mathcal{\widetilde{L}}_{\text{SM}}=\widetilde{g}\,^{\mu\nu}\,(\widetilde{D}_{\mu}\widetilde{H})^{\dagger}\,(\widetilde{D}_{\nu}\widetilde{H})-\frac{1}{4}\widetilde{F}_{\mu\nu}\widetilde{F}^{\mu\nu}+\frac{i}{2}\,\,\widetilde{\overline{f}}\,\overleftrightarrow{\widetilde{\not{\nabla}}}\,\widetilde{f}+\,\widetilde{\mathcal{L}}_{Y}.$
(A.3)

As stated earlier, we follow the metric convention
$\eta_{\mu\nu}=\,\\{+,-,-,-\\}$ and consider the conformal transformation as

$g_{\mu\nu}=\,\omega^{2}\,\widetilde{g}_{\mu\nu}$ (A.4)

where $g_{\mu\nu}$ stand for the spacetime metric of Einstein frame and
$\omega$ is known to be the conformal factor. We identify

$1-\frac{\xi\,\widetilde{X}_{\mu}\widetilde{X}^{\mu}}{M_{\text{Pl}}^{2}}\equiv\omega^{2}.$
(A.5)

Note that all the un-tilde quantities belong to the Einstein frame. We mention
that the spacetime coordinates are not altered due to the conformal
transformation. Therefore the ordinary derivative
$\widetilde{\partial}=\partial$, whereas the covariant derivative
$\widetilde{\nabla}_{\mu}$ is defined with respect to the
$\widetilde{g}_{\mu\nu}$. It can also be perceived that when covariant
derivative operates on the scalar it reduces to ordinary derivative and thus
we write:
$\widetilde{\nabla}_{\mu}\phi=\,\widetilde{\partial}_{\mu}\phi=\partial_{\mu}\phi$.
Furthermore note that $\widetilde{H},\,\widetilde{f},\,\widetilde{X}_{\mu}$
and $\widetilde{F}_{\mu\nu}$ all remain unaffected by the conformal
transformation i.e.,
$\widetilde{H}=H,\,\widetilde{f}=f,\,\widetilde{X}_{\mu}=\,X_{\mu}$ and
$\widetilde{F}_{\mu\nu}=\,F_{\mu\nu}$. On the other hand,
$\widetilde{F}^{\mu\nu}=\,\widetilde{g}^{\alpha\mu}\,\widetilde{g}^{\beta\nu}\,\widetilde{F}_{\alpha\beta}=\omega^{4}\,F^{\mu\nu}$.
In the fermionic sector
$\widetilde{\not{\nabla}}=\,\gamma^{\mu}\,\widetilde{\nabla}_{\mu}$, where
$\gamma^{\mu}$ are the gamma matrices in the Jordan frame and can be connected
to the Einstein frame by using the vierbein. Now we analyse term by term of
the action in Eq. (A.1). $\widetilde{R}$ is the Ricci scalar in the Jordan
frame and can be related to the Ricci scalar in the Einstein frame as below
[151]

$\widetilde{R}=\omega^{2}\left[R-6\,g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}({\rm
ln}\,\omega)+6\,g^{\alpha\beta}\nabla_{\alpha}({\rm
ln}\,\omega)\,\nabla_{\beta}({\rm ln}\,\omega)\right].$ (A.6)

Using the above relation we replace $\widetilde{R}$ in the Eq. (A.1). We also
transform all the parameters associated with the term $\widetilde{R}$ to
Einstein frame by using the appropriate transformation relations. Thus the
first two terms in Eq. (A.1) turn out to be

$\int
d^{4}x\,\sqrt{-g}\left[\frac{M_{\text{Pl}}^{2}\,R}{2}+\frac{3\omega^{4}}{4\,M_{\text{Pl}}^{2}}\,\nabla_{\alpha}(\xi\,X_{\mu}X^{\mu})\,\nabla^{\alpha}(\xi\,X_{\mu}X^{\mu})\right].$
(A.7)

Following the discussion above Eq. (A.6), one obtains

$\widetilde{X}_{\mu\nu}\widetilde{X}^{\mu\nu}=\,\tilde{g}^{\alpha\mu}\tilde{g}^{\beta\nu}\,X_{\mu\nu}\,X_{\alpha\beta}=\,\omega^{4}\,g^{\alpha\mu}g^{\beta\nu}\,X_{\mu\nu}X_{\alpha\beta}=\,\omega^{4}\,X_{\mu\nu}X^{\mu\nu}$
(A.8)
$\widetilde{X}_{\mu}\widetilde{X}^{\mu}=\,\widetilde{g}^{\mu\nu}\,X_{\mu}X_{\nu}=\,\omega^{2}\,g^{\mu\nu}\,X_{\mu}X_{\nu}$
(A.9)

and

$\widetilde{g}\,^{\mu\nu}\,(\widetilde{D}_{\mu}\widetilde{H})^{\dagger}\,(\widetilde{D}_{\nu}\widetilde{H})\,=\,\omega^{2}\,g^{\mu\nu}\,(D_{\mu}H)^{\dagger}\,(D_{\nu}H).$
(A.10)

For fermions some comments are in order:

  * •

In case of fermions, the metrics of the two frames are connected by the
vierbein.

$\widetilde{g}_{\mu\nu}=\,e_{\mu}^{a}\,e_{\nu}^{b}\,\eta_{ab}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm
det}(e_{\nu}^{q})=\,\sqrt{-\widetilde{g}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,e_{\mu}^{a}\,e_{b}^{\mu}=\delta_{b}^{a}.$
(A.11)

Furthermore, the vierbein satisfy:
$e_{\mu}^{a}=\,\omega^{-1}\,\delta_{\mu}^{a}$ and
$e^{\mu}_{a}=\,\omega\,\delta^{\mu}_{a}$.

  * •

In the case of the Einstein frame ($g_{\mu\nu}$) we take the background to be
flat and consider $g_{\mu\nu}\to\,\eta_{\mu\nu}$. Consequently the conformal
transformation becomes: $\tilde{g}_{\mu\nu}=\,\omega^{-2}\,\,\eta_{\mu\nu}$
[37].

  * •

Vierbein depicts the connection between two frames and its two indices such as
$(\mu$, $\nu$, $\alpha...)$ represent the indices for Jordan frame and
$(a,\,b,\,c,...)$ stand for Einstein (flat) frame. For example, one can see
below (Eq. (A.12)) that how $\gamma^{\mu}$ and $\gamma^{a}$ are connected by
the vierbein.

  * •

All indices of Jordan and flat frame are contracted with the corresponding
metric such as $\tilde{g}_{\mu\nu}$ and $\eta_{\mu\nu}$, respectively.

Using this relation and following the discussion above Eq. (A.1), we get
partially transformed fermionic action as follows:

$\displaystyle\widetilde{S}_{f}$ $\displaystyle=$ $\displaystyle\,\int
d^{4}x\,\,{\rm
det}\,(e_{\nu}^{q})\,\bigg{[}\frac{i}{2}\,\bar{f}\gamma^{\mu}\,\overleftrightarrow{\widetilde{\nabla}}_{\mu}\,f\bigg{]}$
(A.12) $\displaystyle=$ $\displaystyle\,\int
d^{4}x\,\,\omega^{-4}\sqrt{-g}\,\bigg{[}\frac{i}{2}\,\bar{f}\gamma^{a}\,e_{a}^{\mu}\,\overleftrightarrow{\widetilde{\nabla}}_{\mu}\,f\bigg{]}$
$\displaystyle=$ $\displaystyle\,\int
d^{4}x\,\,\omega^{-4}\sqrt{-g}\,\bigg{[}\frac{i}{2}\,\bar{f}\gamma^{a}\,\omega\,\delta_{a}^{\mu}\,\overleftrightarrow{\nabla_{\mu}}\,f\bigg{]}.$
(A.13)

In case of fermion the covariant derivative is defined as:
$\nabla_{\mu}\,f=\,\partial_{\mu}\,f+\frac{i}{2}\,\omega_{\mu}\,^{mn}\,\sigma_{mn}$,
where $\omega_{\mu}\,^{mn}$ are the antisymmetric coefficients of the spinor
connection and
$\sigma_{mn}=\,\frac{i}{2}\,(\gamma_{m}\gamma_{n}-\gamma_{n}\gamma_{m})$. Thus
putting this in the Eq. (A.13) we get,

$S_{f}=\,\int
d^{4}x\,\,\frac{i\,\sqrt{-g}}{2\,\omega^{3}}\,\bar{f}\,\gamma^{a}\,\overleftrightarrow{\partial_{a}}\,f\,-\,\frac{1}{4}\,\int\,d^{4}x\frac{\sqrt{-g}}{\omega^{3}}\,\bar{f}\,\gamma^{a}\,\delta^{\mu}_{a}\,\omega_{\mu}\,^{mn}\,\sigma_{mn}\,f\,.$
(A.14)

At this stage, let us elaborately analyse the term $\omega_{\mu}\,^{mn}$ as
below:

$\displaystyle\omega_{\mu}\,^{mn}$ $\displaystyle=$
$\displaystyle\,e_{\nu}^{m}\,\Gamma^{\nu}_{\sigma\mu}\,e^{\sigma
n}-\,(\partial_{\mu}\,e^{m}_{\nu})\,e^{\nu n}$ (A.15) $\displaystyle=$
$\displaystyle\,\frac{\omega^{2}}{2}\,\delta_{\nu}^{m}\,\eta^{ab}\,\delta_{a}^{\nu}\,\delta_{b}^{\beta}\,\bigg{[}\partial_{\mu}(\omega^{-2}\eta_{\sigma\beta})+\,\partial_{\sigma}(\omega^{-2}\eta_{\mu\beta})-\,\partial_{\beta}(\omega^{-2}\eta_{\sigma\mu})\bigg{]}\,\eta^{\sigma\alpha}\,\delta_{\alpha}^{n}$
$\displaystyle+$
$\displaystyle\,\frac{\partial_{\mu}\,\omega}{\omega}\,\delta_{\nu}^{m}\,\eta^{\nu\alpha}\,\delta_{\alpha}^{n}.$

Further simplifying the above, we obtain

$\omega_{\mu}\,^{mn}=\,-\,\frac{1}{\omega}\,(\delta_{\mu}^{m}\,\partial^{n}\omega\,-\,\delta_{\mu}^{n}\,\partial^{m}\,\omega).$
(A.16)

Let us put Eq. (A.16) in the second term of Eq. (A.14) and use the following
properties of gamma matrices:

$\left\\{\gamma^{m},\,\gamma^{n}\right\\}=\,2\,\eta^{mn}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma_{m}\,\gamma^{m}=\,4\,I$
(A.17)

Therefore, the second term of Eq. (A.14) becomes:

$\displaystyle-\frac{1}{4}\,\int\,d^{4}x\,\frac{\sqrt{-g}}{\omega^{3}}\,\bar{f}\,\gamma^{a}\,\delta^{\mu}_{a}\,\omega_{\mu}\,^{mn}\,\sigma_{mn}\,f$
$\displaystyle\qquad=\frac{i}{8}\,\int\,d^{4}x\,\sqrt{-g}\,\bar{f}\,\gamma^{p}\,\delta_{p}^{\mu}\,(\delta_{\mu}^{m}\,\partial^{n}\,\omega-\,\delta_{\mu}^{n}\,\partial^{m}\,\omega)\,(\gamma_{m}\gamma_{n}-\gamma_{n}\gamma_{m})\,f$
$\displaystyle\qquad=\,\,\frac{3\,i}{2}\,\int\,d^{4}x\,\frac{\sqrt{-g}}{\omega^{4}}\,\bar{f}\,(\not{\partial}\,\omega)\,f=\,\frac{3\,i}{2}\,\int\,d^{4}x\,\frac{\sqrt{-g}}{\omega^{4}}\,\bar{f}\,(\partial^{m}\,\omega)\,\gamma_{m}\,f\,.$
(A.18)

Therefore Eq. (A.14) becomes:

$S_{f}=\,\int\,d^{4}x\,\sqrt{-g}\,\bigg{[}\frac{i}{\omega^{3}}\,\bar{f}\,\gamma^{a}\,\partial_{a}\,f+\,\frac{3\,i}{\omega^{4}}\,\bar{f}\,(\not{\partial}\,\omega)\,f\bigg{]}.$
(A.19)

Note: $\omega_{\mu}\,^{mn}$ contains $\partial$ operators which also possess
over left right arrow as similar to the $\partial$ operator in the first term
of Eq. (A.14). Now Eq. (A.19), has been written by removing the over left
right arrow from the $\partial$ operators and consequently a $1/2$ factor is
removed from both the terms. Additionally, the SM gauge field sector will
exactly behave as the dark sector under conformal transformation such as shown
in Eq. (A.8). Therefore combining Eqs. (A.7), (A.8), (A.8) (A.9), (A.10) and
(A.19) we get the final form of action in the Einstein frame for the whole
setup as follows:

$\displaystyle\mathcal{S}=\int d^{4}x\sqrt{-g}$
$\displaystyle\Biggl{[}\frac{M_{\text{Pl}}^{2}\,R}{2}+\,\frac{3\,\omega^{4}}{4\,M_{\text{Pl}}^{2}}\,\nabla_{\alpha}(\xi\,X_{\mu}X^{\mu})\,\nabla^{\alpha}(\xi\,X_{\mu}X^{\mu})-\frac{1}{4}\,X_{\mu\nu}\,X^{\mu\nu}+\frac{1}{2\omega^{2}}m_{X}^{2}\,X_{\mu}\,X^{\mu}$
$\displaystyle+\,\frac{1}{\omega^{4}}\,(\mathcal{L}_{Y}-V(H))+\,\frac{1}{\omega^{2}}(D_{\mu}H)^{\dagger}(D^{\mu}H)+\,\frac{i}{\omega^{3}}\,\bar{f}\,\gamma^{\mu}\,\partial_{\mu}\,f$
$\displaystyle-\,\frac{1}{4}\,g^{\mu\nu}\,g^{\lambda\rho}F_{\mu\nu}^{(a)}\,F_{\nu\rho}^{(a)}+\,\frac{3\,i}{\omega^{4}}\,\bar{f}\,(\not{\partial}\,\omega)\,f\Biggr{]},$
(A.20)

where $V\left(H\right)$ is the renormalizable Higgs potential for the SM. Here
$\mathcal{L}_{Y}$ contains the Yukawa interaction terms.

##  Appendix B Interactions with metric fluctuation

Consider the following Lagrangian:

$\mathcal{L}_{\text{gm}}(X,{\rm
SM})=\,\mathcal{L}_{\text{gm}}^{(0)}+\,\kappa\,\mathcal{L}_{\text{gm}}^{(1)}+\mathcal{O}(\kappa^{2})+...$
(B.1)

where $\mathcal{L}_{\text{gm}}(X,{\rm SM})$ stands for the Lagrangian of all
the matter fields, including the SM and the DM, defined with respect to
$g_{\mu\nu}$. This can be written as

$\displaystyle\mathcal{L}_{gH}=\,\sqrt{-g}\,\big{[}\,g^{\mu\nu}(D_{\mu}H)^{\dagger}\,(D_{\nu}H)-\,V_{H}\big{]}$
(B.2)
$\displaystyle\mathcal{L}_{gA}=\,\sqrt{-g}\,\bigg{[}-\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}\bigg{]}$
(B.3)
$\displaystyle\mathcal{L}_{gf}=\,\sqrt{-g}\,\big{[}\,i\,\bar{f}\gamma^{\alpha}\partial_{\alpha}\,f\big{]}$
(B.4)
$\displaystyle\mathcal{L}_{gX}=\,\sqrt{-g}\,\bigg{[}-\frac{1}{4}\,X_{\mu\nu}X^{\mu\nu}+\,\frac{1}{2}\,m_{X}^{2}X_{\mu}\,X^{\mu}\bigg{]}.$
(B.5)

$\mathcal{L}_{\text{gm}}^{(0)}$ depicts the Lagrangian of the SM and DM fields
in the Minkowski ($\eta_{\mu\nu}$) spacetime. We refer the readers to [119]
for detail derivation of the interaction terms. For illustration purposes,
here we analyse $\mathcal{L}_{gH}$ under the metric fluctuation:
$g_{\mu\nu}=\,\eta_{\mu\nu}+\kappa\,h_{\mu\nu}$, where due to the smallness of
the fluctuation $(h_{\mu\nu})$, we allow the perturbative expansion of the
Lagrangian up to the first order in $h_{\mu\nu}$. This also leads us to
$g^{\mu\nu}=\,\eta^{\mu\nu}-\,\kappa\,h^{\mu\nu}$ and
$\sqrt{-g}\approx\,1+\frac{\kappa\,h}{2}$, where
$h=\,\eta_{\mu\nu}\,h^{\mu\nu}$. Using the metric fluctuation and allowing up
to the leading order in $h_{\mu\nu}$, $\mathcal{L}_{gH}$ can be written as

$\displaystyle\mathcal{L}_{gH}$
$\displaystyle=\bigg{(}1+\frac{\kappa\,h}{2}\bigg{)}\,\bigg{[}(\eta^{\mu\nu}-\,\kappa\,h^{\mu\nu})\,(D_{\mu}H)^{\dagger}\,(D_{\nu}H)-\,V_{H}\bigg{]}$
(B.6)
$\displaystyle=\big{[}\eta^{\mu\nu}\,(D_{\mu}H)^{\dagger}\,(D_{\nu}H)-V_{H}\big{]}+\frac{\kappa\,h}{2}(D_{\mu}H)^{\dagger}\,(D^{\mu}H)$
$\displaystyle\quad-\kappa\,h^{\mu\nu}\,(D_{\mu}H)^{\dagger}\,(D_{\nu}H)-\frac{\kappa\,h}{2}\,V_{H}\,$

from which one can find the relevant interaction terms.

##  Appendix C Annihilation cross-section for freeze-in

We have used CalcHEP [152] to calculate the cross-sections for 2-to-2
processes with a pair of DM in the final state as a function of the CM energy
where $f,V$ and $H$ stand respectively for the SM fermions, SM gauge bosons
(massive) and the SM Higgs.

$\displaystyle\sigma\left(s\right)_{ff\to XX}=$
$\displaystyle\frac{\xi^{2}}{480\,M_{\text{Pl}}^{4}\,m_{X}^{4}\,\pi\,s^{6}}\Biggl{[}1152m_{f}^{8}\left(s-4m_{X}^{2}\right)^{4}$
$\displaystyle-64m_{f}^{6}s\left(s-4m_{X}^{2}\right)^{2}\left(288m_{X}^{4}-94m_{X}^{2}s+13s^{2}\right)$
$\displaystyle+32m_{f}^{4}s^{2}\left(3456m_{X}^{8}-2416m_{X}^{6}s+771m_{X}^{4}s^{2}-86m_{X}^{2}s^{3}+6s^{4}\right)$
$\displaystyle-2m_{f}^{2}s^{3}\left(9216m_{X}^{8}-5696m_{X}^{6}s+1596m_{X}^{4}s^{2}-236m_{X}^{2}s^{3}+s^{4}\right)$
$\displaystyle-3s^{4}\left(-384m_{X}^{8}+224m_{X}^{6}s+96m_{X}^{4}s^{2}-6m_{X}^{2}s^{3}+s^{4}\right)\Biggr{]}$
(C.1) $\displaystyle\sigma\left(s\right)_{VV\to XX}=$
$\displaystyle\frac{\xi^{2}}{4320\,M_{\text{Pl}}^{4}\,m_{X}^{4}\,\pi\,s^{6}}\Biggl{[}192m_{V}^{8}\left(s-4m_{X}^{2}\right)^{4}$
$\displaystyle-32m_{V}^{6}s\left(s-4m_{X}^{2}\right)^{2}\left(96m_{X}^{4}-68m_{X}^{2}s+s^{2}\right)$
$\displaystyle+8m_{V}^{4}s^{2}\left(2304m_{X}^{8}-2624m_{X}^{6}s+1304m_{X}^{4}s^{2}-184m_{X}^{2}s^{3}+19s^{4}\right)$
$\displaystyle-4m_{V}^{2}s^{3}\left(768m_{X}^{8}-608m_{X}^{6}s+368m_{X}^{4}s^{2}-148m_{X}^{2}s^{3}+13s^{4}\right)$
$\displaystyle+s^{4}\left(192m_{X}^{8}-32m_{X}^{6}s+152m_{X}^{4}s^{2}-52m_{X}^{2}s^{3}+7s^{4}\right)\Biggr{]}$
(C.2) $\displaystyle\sigma\left(s\right)_{HH\to XX}=$
$\displaystyle\frac{\xi^{2}}{480\,M_{\text{Pl}}^{4}\,m_{X}^{4}\,\pi\,s^{6}}\Biggl{[}192m_{h}^{8}\left(s-4m_{X}^{2}\right)^{4}$
$\displaystyle+128m_{h}^{6}s\left(s-4m_{X}^{2}\right)^{2}\left(-24m_{X}^{4}+12m_{X}^{2}s+s^{2}\right)$
$\displaystyle+32m_{h}^{4}s^{2}\left(576m_{X}^{8}-496m_{X}^{6}s+176m_{X}^{4}s^{2}-31m_{X}^{2}s^{3}+6s^{4}\right)$
$\displaystyle+8m_{h}^{2}s^{3}\left(-384m_{X}^{8}+224m_{X}^{6}s-4m_{X}^{4}s^{2}-31m_{X}^{2}s^{3}+6s^{4}\right)$
$\displaystyle+s^{4}\left(192m_{X}^{8}-32m_{X}^{6}s+152m_{X}^{4}s^{2}-52m_{X}^{2}s^{3}+7s^{4}\right)\Biggr{]}.$
(C.3)

In the massless approximation (Eq. (4.1)), we obtain

$\sigma\left(s\right)\simeq 0.53\,\xi^{2}\,s/M_{\text{Pl}}^{4}.$ (C.4)

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