# Gravitational SIMPs

Basabendu Barman  and Nicolás Bernal

###### Abstract

We study the impact of thermalization and number-changing processes in the
dark sector on the yield of gravitationally produced dark matter (DM). We take
into account the DM production through the $s$-channel exchange of a massless
graviton both from the scattering of inflatons during the reheating era, and
from the Standard Model bath via the UV freeze-in mechanism. By considering
the DM to be a scalar, a fermion, and a vector boson we show, in a model-
independent way, that DM self-interaction gives rise to a larger viable
parameter space by allowing lower reheating temperature to be compatible with
Planck observed relic abundance. As an example, we also discuss our findings
in the context of the $\mathbb{Z}_{2}$-symmetric scalar singlet DM model.

PI/UAN-2021-688FT

##  1 Introduction

The existence of dark matter (DM) has been proven beyond any doubt from
several astrophysical [1, 2, 3, 4] and cosmological [5, 6] evidences (for a
review, see, e.g. Refs. [7, 8, 9]). Its fundamental nature, however, still
remains elusive. As it is already established from observations, DM as a
fundamental particle has to be electrically neutral and stable at the scale of
lifetime of the Universe. The measurement of the anisotropy in the cosmic
microwave background radiation (CMB) 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 absence of any significant excess in direct search experiments like XENON
[10] have cornered DM as a weakly interacting massive particle (WIMP) (see,
e.g., Refs. [11, 12]) which, by far, is the most popular DM candidate that can
be found in myriad extensions to the SM. Here one assumes that the DM
particles were in thermal equilibrium in the early Universe due to non-
negligible coupling with the visible sector. The DM abundance then freezes out
once the interaction rate falls out of equilibrium as the Universe expands and
cools down. It is also quite possible that DM particles never attain thermal
equilibrium because either they are too massive or they have very feeble
couplings with the SM. In such a case DM is dubbed as a feebly interacting
massive particle (FIMP). Such nonthermal DM yield freezes in to provide the
observed DM abundance. Depending on whether the DM interaction with the
visible sector is renormalizable or non-renormalizable, freeze-in in principle
can be of two kinds: $i)$ infrared (IR) where the DM abundance becomes
important at a low temperature [13, 14, 15, 16, 17, 18, 19, 20], or $ii)$
ultra-violate (UV) where the DM genesis takes place at the highest temperature
achieved by the thermal bath [13, 21, 22, 23, 24, 25, 26, 27]. This
temperature can be the reheating temperature in the case of an instantaneous
and complete inflaton decay, but can also be much larger if the decay of the
inflaton is non-instantaneous [28, 29]. During inflationary reheating the DM
can also be produced from the decay of the inflaton itself, either directly
[30, 31, 32, 33] or radiatively [34]. On top of decay, pair annihilation of
inflatons mediated by gravity can also lead to sufficient DM abundance during
inflation [35, 36, 37].

The irreducible gravitational interaction can lead to DM production which, in
general, can be dubbed as the gravitational production. Several mechanisms for
the gravitational particle production exist. For example, particles are
generated due to time variations in the background metric [38, 39, 40, 41,
42], with a maximal efficiency when the particle mass $m\simeq H_{I}$ [43],
where $H_{I}$ denotes the Hubble scale at the end of inflation. Production of
superheavy (quasi-) stable particles with masses $m\gg H_{I}$ during the
transition from the inflationary phase to either a matter- or a radiation-
dominated phase as a result of the expansion of the background spacetime has
been studied in the context of a scalar field conformally coupled to gravity
[44, 45], and for a wide range of models with minimal and/or conformal
coupling to gravity, where it has been shown such particles can constitute a
considerable fraction of the DM [40]. The production of purely gravitational
DM due to expansion of the Universe has been extensively discussed in Refs.
[46, 35, 47], while DM production through $s$-channel graviton exchange has
been studied in Refs. [43, 46, 48, 49]. The latter may as well be regarded as
gravitational particle creation since no direct or indirect interaction, apart
from gravity, between the DM and the SM field is considered. It has been
pointed out that (small) oscillations of the Hubble parameter or the scale
factor caused by inflaton oscillation induces an effective coupling between
the inflaton and a (massive) particle of arbitrary spin, leading to particle
creation in the inflaton oscillation regime [50, 51]. Such a particle
production mechanism can also be interpreted as the decay or annihilation of
the inflaton through the gravitational interaction. In the present context,
however, the DM particle production takes place via gravitational interaction
which is described in terms of coupling of the energy-momentum tensor
$T^{\mu\nu}$ to the metric perturbation $h^{\mu\nu}$ [52, 53, 54] where the
latter is identified as the quantum field for the spin-2 massless graviton.
This gives rise to interactions of all matter fields with the graviton and
subsequent pair production of DM particles. Note that, the present framework
differs from those where non-minimal couplings are introduced [55, 56], and
also from Refs. [57, 40] which rely on the dynamics of quantum field theory on
curved background spacetimes.

The DM particles, on the other hand, can have strong self-interactions among
themselves leading to the strongly interacting massive particle (SIMP)
paradigm that can have significant impact on the resulting phenomenology [58].
The long-standing puzzles of the collisionless cold DM paradigm can be
addressed by invoking these interactions. Two such examples are the ‘core vs
cusp’ [59, 60, 61] and the ‘too big to fail’ [62] problems which can be
alleviated if at the scale of galaxies there exists a large self-scattering
cross section. DM self-interaction can also accommodate the observed diversity
in galaxy rotation curves with the same circular velocity [63, 64, 65].
Although a large self-scattering cross section is constrained by Bullet
cluster spherical halo shapes [66, 67, 68], it can lead to distinct signatures
in galaxies and galaxy clusters, such as the offset of the dark matter subhalo
from the galaxy centre as hinted in Abell-3827 [69]. However, the dynamics
within the dark sector can be non-trivial in case there is a
$N$-to-$N^{\prime}$ number changing process involving DM particles with
$N>N^{\prime}\geq 2$. Typically such a number-changing process corresponds to
3-to-2 scenario [70, 58, 71, 72, 73] but an unavoidable 4-to-2 process may
dominate depending on the stabilizing symmetry of the dark sector [74, 75, 76,
77]. If the number-changing processes within the dark sector reach
equilibrium, DM forms a thermal bath with a temperature $T^{\prime}$ in
general different from $T$ of the SM. Such number-changing interactions
naturally affect the DM abundance once the DM is produced.

Motivated from these, in this work we investigate the effect of thermalization
due to number-changing processes within the dark sector on the abundance of
gravitationally produced DM, where gravity minimally couples to all matter
particles through the energy-momentum tensor.111The effect of strong self-
interactions has also been studied in the case of conformal [78, 79, 80] and
non-minimally coupled [81] dark sectors. We focus on the DM production $a)$
during the reheating epoch when the inflatons dominate the energy density of
the Universe, and $b)$ through the UV freeze-in mechanism via 2-to-2
annihilation of the SM states, both mediated by the $s$-channel exchange of
massless gravitons.222Two comments are in order: First, one could also
conceive scenarios where gravity and another portal are effective, see e.g.,
Refs. [82, 83, 84, 85, 86]. Second, the gravitational production can be
enhanced in scenarios with extra dimensions, see e.g., Refs. [87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106]. In a
model-independent way, considering the DM to be a scalar, a fermion or a
vector boson, we show that thermalization broadens the viable parameter space
allowing lower reheating temperature compatible with the DM relic abundance.
We also show that, for the fermionic case, DM production from the thermal bath
(mediated by gravity) could dominate for $T_{\text{max}}/T_{\text{rh}}\lesssim
20$ once self-interaction is invoked. In the later part of the analysis, as an
example, we demonstrate in detail the consequences of thermalization on the
gravitationally produced scalar singlet DM (SSDM) [107, 108, 109] , where we
see the DM abundance typically increases on increasing self-interaction
strength before non-relativistic freeze-out occurs.

The paper is organized as follows. In section 2 we review in detail the DM
gravitational production, i.e., via 2-to-2 annihilations of inflatons and the
SM particles mediated by $s$-channel exchange of gravitons. In section 3 the
role of DM self-interaction is presented in a model independent way
emphasizing its impact on the DM relic abundance. In section 4 we briefly
review the singlet scalar DM model as an example of gravitational DM with
sizeable self-interactions. Finally, we conclude in section 5.

##  2 Dark Matter Gravitational Production

Gravity is the only interaction that is guaranteed to mediate between the DM
and the visible sector. In that context, as mentioned in the introduction, DM
can be produced by scattering of inflatons or SM particles, mediated by
$s$-channel exchange of gravitons. In such a scenario, the Boltzmann equation
(BEQ) governing the evolution of the DM number density $n$ is given by

$\frac{dn}{dt}+3H\,n=\gamma\,,$ (2.1)

where $\gamma$ corresponds to the DM production rate density and $H$ to the
Hubble expansion rate given by $H^{2}=(\rho_{R}+\rho_{\phi})/(3M_{P}^{2})$,
with $M_{P}$ as the reduced Planck mass. The evolution of the inflaton and SM
radiation energy densities ($\rho_{\phi}$ and $\rho_{R}$, respectively) can be
tracked via the set of BEQs

$\displaystyle\frac{d\rho_{\phi}}{dt}+3H\,\rho_{\phi}=-\Gamma_{\phi}\,\rho_{\phi}\,,$
(2.2)
$\displaystyle\frac{d\rho_{R}}{dt}+4H\,\rho_{R}=+\Gamma_{\phi}\,\rho_{\phi}\,,$
(2.3)

where $\Gamma_{\phi}$ is the total inflaton decay width. We note that, by
construction, the inflaton does not decay into DM particles as DM is assumed
to only have gravitational interactions with the visible sector.333Here we are
assuming that in the reheating era the inflaton energy density scales like
non-relativistic matter. This can be achieved in quadratic inflation, or in
non-minimally coupled scenarios like in the case of Starobinsky inflation.

The photon temperature can be extracted from the SM energy density
$\rho_{R}=\frac{\pi^{2}}{30}\,g_{\star}\,T^{4}$, where $g_{\star}(T)$
corresponds to the SM relativistic degrees of freedom contributing to the
energy density [110]. We note that the SM particles do not necessarily
thermalize instantaneously, and thus the decay products could be initially
distributed with smaller occupation numbers and harder momenta [111, 30, 112].
However, here we assume that the SM thermalizes rapidly. The reheating
temperature $T_{\text{rh}}$ can then be defined as the temperature at which
the equality $H(T_{\text{rh}})=\Gamma_{\phi}$ holds, and is given by

$T_{\text{rh}}^{2}=\frac{3}{\pi}\sqrt{\frac{10}{g_{\star}}}\,M_{P}\,\Gamma_{\phi}\,.$
(2.4)

It is also a proxy of the onset of the radiation domination era. As the
inflaton does not decay instantaneously, the SM bath temperature may rise to a
value $T_{\text{max}}$ much higher than $T_{\text{rh}}$. The value of
$T_{\text{max}}$ depends both on $\Gamma_{\phi}$ and on the inflationary
scale, and is given in Eq. (A.3). During the heating era, i.e., while
$T_{\text{max}}>T>T_{\text{rh}}$, the inflaton decays into SM radiation. As a
result, the SM energy density does not scale as free radiation but rather like
$\rho_{R}(a)\propto a^{-3/2}$, with $a$ as the scale factor. Figure 1 shows
the SM temperature evolution as a function of the scale factor obtained from
the numerical solution of Eqs. (2.2) and (2.3),444It is useful to make the
variable transformation $\Phi\equiv\rho_{\phi}a^{3}$ and
$R\equiv\rho_{R}a^{4}$ to solve the system of BEQs [29]. for
$T_{\text{max}}=10^{12}$ GeV and $T_{\text{rh}}=10^{9}$ GeV shown by the red
dashed lines. During reheating $T\propto a^{-3/8}$, whereas once the reheating
is completed, $T\propto a^{-1}$ as the inflaton has already decayed and hence
the entropy injection has ceased.555We note that the scaling $T\propto
a^{-3/8}$ characterizes scenarios where a non-relativistic particle that
dominates the total energy density (here the inflaton) decays into SM states.

Figure 1: Evolution of the SM temperature as a function of the scale factor
for $T_{\text{max}}=10^{12}$ GeV and $T_{\text{rh}}=10^{9}$ GeV.

###  2.1 Inflaton Scatterings

The total energy density of the Universe is dominated by inflatons during
reheating. Taking into account that $T\propto a^{-3/8}$ and the inflaton
energy density scales as non-relativistic matter,

$\rho_{\phi}(T)=\frac{\pi^{2}\,g_{\star}}{30}\,\frac{T^{8}}{T_{\text{rh}}^{4}}\,,$
(2.5)

where $\rho_{\phi}(T_{\text{rh}})=\rho_{R}(T_{\text{rh}})$ is assumed.
Therefore, the Hubble parameter in an inflaton-dominated Universe takes the
form

$H(T)=\frac{\pi}{3}\sqrt{\frac{g_{\star}}{10}}\,\frac{T^{4}}{M_{P}\,T_{\text{rh}}^{2}}\,.$
(2.6)

During reheating, the whole observed DM abundance can be generated via 2-to-2
annihilations of inflatons, mediated by the $s$-channel exchange of gravitons.
The interaction rate density for DM production out of non-relativistic
inflatons then reads [36, 37]

$\gamma=\frac{\rho_{\phi}^{2}}{4096\pi\,M_{P}^{4}}\,f\left(\frac{m_{\text{DM}}}{m_{\phi}}\right)=\frac{\pi^{3}\,g_{\star}^{2}}{3686400}\,\frac{T^{16}}{M_{P}^{4}\,T_{\text{rh}}^{8}}\,f\left(\frac{m_{\text{DM}}}{m_{\phi}}\right),$
(2.7)

where $m_{\text{DM}}$ and $m_{\phi}$ are the masses of the DM and the
inflaton, respectively, and we have defined

$f(x)\equiv\begin{cases}\left(x^{2}+2\right)^{2}\sqrt{1-x^{2}}&\mbox{for real
scalars},\\\\[8.0pt] x^{2}\left(1-x^{2}\right)^{3/2}&\mbox{for Dirac
fermions},\\\\[8.0pt]
\frac{1}{8}\sqrt{1-x^{2}}\left(4+4x^{2}+19x^{4}\right)&\mbox{for vector
bosons.}\end{cases}$ (2.8)

For completeness, in Appendix C we have gathered the relevant vertices between
the graviton and particles with different spins (Tab. 1). Additionally, the
details of the computation of the rates are presented in Appendix D.

The evolution of the DM number density given in Eq. (2.1) can be recasted in
terms of the co-moving number density $N\equiv n\,a^{3}$, considering the fact
that during the reheating era the SM entropy is not conserved due to the
inflaton decay

$\frac{dN}{dT}=-\frac{8}{\pi}\sqrt{\frac{10}{g_{\star}}}\,\frac{M_{P}\,T_{\text{rh}}^{10}}{T^{13}}\,a^{3}(T_{\text{rh}})\,\gamma\,.$
(2.9)

This equation integrated in the range $T_{\text{max}}\geq T\geq T_{\text{rh}}$
gives the final DM number density. The DM yield $Y(T)\equiv n(T)/s(T)$ is
defined as a function of the SM entropy density
$s\equiv\frac{2\pi^{2}}{45}g_{\star s}\,T^{3}$ with $g_{\star s}(T)$ being the
number of relativistic degrees of freedom contributing to the SM entropy
[110]. The yield $Y_{0}$ at the end of reheating (i.e., at $T=T_{\text{rh}}$)
can be analytically computed and reads

$Y_{0}=\frac{g_{\star}^{2}}{81920\,g_{\star
s}}\sqrt{\frac{10}{g_{\star}}}\,\left(\frac{T_{\text{rh}}}{M_{P}}\right)^{3}\left[\left(\frac{T_{\text{max}}}{T_{\text{rh}}}\right)^{4}-1\right]f\left(\frac{m_{\text{DM}}}{m_{\phi}}\right).$
(2.10)

Figure 2 shows, with black solid lines, the evolution of the comoving DM yield
$N$ as a function of the scale factor $a$ (left panel) and the SM temperature
$T$ (right panel) for DM produced via inflaton scatterings during reheating.
Here we assume $T_{\text{max}}=10^{12}$ GeV, $T_{\text{rh}}=10^{9}$ GeV shown
by the red dashed vertical lines, and a scalar DM particle with mass
$m_{\text{DM}}\ll m_{\phi}$. The bulk of the DM is produced promptly at the
beginning of reheating, when $T\simeq T_{\text{max}}$.

Figure 2: Evolution of the comoving DM yield $N$ as a function of the SM
temperature $T$ (black solid line), for $T_{\text{max}}=10^{12}$ GeV and
$T_{\text{rh}}=10^{9}$ GeV. The blue lines correspond to the equilibrium
yields, for different DM masses. The vertical red dotted lines correspond to
$T=T_{\text{max}}$ and $T=T_{\text{rh}}$.

To match the observed DM abundance $\Omega_{\text{DM}}h^{2}\simeq 0.12$, the
DM yield has to be fixed so that
$m_{\text{DM}}Y_{0}=\Omega_{\text{DM}}h^{2}\frac{1}{s_{0}}\frac{\rho_{c}}{h^{2}}\simeq
4.3\times 10^{-10}$ 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]. Figure 3 shows the regions of the parameter
space that generate the observed DM abundance for scalar (left panels),
fermionic (central panels) and vector (right panels) DM. We consider
$m_{\phi}=3\times 10^{13}$ GeV, and $T_{\text{max}}/T_{\text{rh}}=10^{1}$
(upper panels) or $T_{\text{max}}/T_{\text{rh}}=10^{2}$ (lower panels). Blue
lines correspond to the DM production via inflaton annihilation as in Eq.
(2.10). The strong cuts at $m_{\text{DM}}=m_{\phi}$ correspond to the
kinematical production threshold. Furthermore, regions above the lines
generate a DM overdensity, overclosing the Universe. Large values of the ratio
$r\equiv T_{\text{max}}/T_{\text{rh}}$ increase the DM production as
$Y_{0}\sim T_{\text{rh}}^{3}\,r^{4}$ following Eq. (2.10) for a fixed DM mass
and reheating temperature. As a result, for a given $r$, a lower
$T_{\text{rh}}$ is required to obtain the correct DM abundance. Large values
of $T_{\text{max}}/T_{\text{rh}}$ are also in conflict with the scale of
inflation $H_{I}^{\text{CMB}}\leq 2.5\times 10^{-5}~{}M_{P}$ [113] coming from
the CMB observations by the Planck satellite. The upper green bands (labeled
CMB) correspond to the reheating temperatures that are in tension with the CMB
observations, as derived in Eq. (A.5) of Appendix A. It is interesting to note
that, if $T_{\text{max}}/T_{\text{rh}}\gtrsim 3\times 10^{5}$, the whole
parameter space where DM can be produced via the scattering of inflatons is in
tension with the upper bound of the inflationary scale. Finally, we note that
DM lighter than few keV is in tension with the Lyman-$\alpha$ bound [114, 115,
116] if produced via SM scatterings. Alternatively, if it is produced out of
inflaton scatterings, this constraint can be eased as discussed in Appendix B.

Figure 3: Parameter space that generates the observed DM abundance for scalar
(left panels), fermionic (central panels) and vector (right panels) DM, for
$m_{\phi}=3\times 10^{13}$ GeV, and $T_{\text{max}}/T_{\text{rh}}=10^{1}$
(upper panels) or $T_{\text{max}}/T_{\text{rh}}=10^{2}$ (lower panels). DM is
produced by inflaton (blue areas) or SM (red areas) scatterings mediated by
the $s$-channel exchange of gravitons. Blue and red lines correspond to the
case without thermalization, whereas the width of the bands to the effect of
thermalization and number-changing processes. Above (below) the bands DM is
overabundant (underabundant)

###  2.2 SM Scatterings

Apart from the inflaton scattering, the DM is also gravitationally produced
from the UV freeze-in mechanism via 2-to-2 annihilation of the SM particles
mediated by $s$-channel exchange of gravitons. The interaction rate density
for such a process reads [43, 49, 48, 117]

$\gamma(T)=\alpha\,\frac{T^{8}}{M_{P}^{4}}\,,$ (2.11)

with $\alpha\simeq 1.9\times 10^{-4}$ (real scalar DM), $\alpha\simeq
1.1\times 10^{-3}$ (Dirac DM) or $\alpha\simeq 2.3\times 10^{-3}$ (vector DM).
In the sudden decay approximation for the inflaton, Eq. (2.1) can be recasted
as

$\frac{dY}{dT}=-\frac{\gamma(T)}{H(T)\,T\,s(T)}\,,$ (2.12)

where $H(T)=\frac{\pi}{3}\sqrt{\frac{g_{\star}}{10}}\frac{T^{2}}{M_{P}}$ in a
radiation-dominated Universe. For $T\ll T_{\text{rh}}$, one can analytically
obtain

$Y_{0}=\frac{45\,\alpha}{2\pi^{3}\,g_{\star
s}}\sqrt{\frac{10}{g_{\star}}}\left(\frac{T_{\text{rh}}}{M_{P}}\right)^{3},$
(2.13)

in the case $m_{\text{DM}}\ll T_{\text{rh}}$. Instead, if the produced
particle is heavier than the reheating temperature (but still below
$T_{\text{max}}$), it cannot be generated after but during reheating. In that
case, the yield can be computed by integrating Eq. (2.9) in the range
$T_{\text{max}}\geq T\geq m_{\text{DM}}$:

$Y_{0}=\frac{45\,\alpha}{2\pi^{3}\,g_{\star
s}}\sqrt{\frac{10}{g_{\star}}}\frac{T_{\text{rh}}^{7}}{M_{P}^{3}\,m_{\text{DM}}^{4}}\,.$
(2.14)

Away from the instantaneous decay approximation, the DM yield is only boosted
by a small factor of order $\mathcal{O}(1)$ for an inflaton behaving as non-
relativistic matter [31, 23].666We note that big boost factors can appear when
considering nonthermal effects [112], or expansion eras dominated by a fluid
component stiffer than radiation [23, 118].

Figure 3 also shows, this time with red lines, the parameter space reproducing
the observed DM abundance via the UV freeze-in, i.e., by annihilation of SM
particles mediated by gravitons. Two slopes are clearly visible: DM lighter
than $m_{\text{DM}}\lesssim 10^{13}$ GeV is mainly produced at $T\simeq
T_{\text{rh}}$ and its yield is given by Eq. (2.13), whereas in the opposite
case ($m_{\text{DM}}\gtrsim 10^{13}$ GeV) the bulk of DM is generated at
$T\simeq m_{\text{DM}}$ and the corresponding yield is given by Eq. (2.14).
Again, regions above the lines generate a DM overdensity, overclosing the
Universe. We emphasize, even if this channel is largely independent on
$T_{\text{max}}$, the bound on the inflationary scale makes it viable only if
$T_{\text{max}}/T_{\text{rh}}\lesssim 30$. Finally, we note that this
production mechanism is typically subdominant with respect to that due to the
scattering of inflatons. However, it can dominate if $m_{\text{DM}}>m_{\phi}$
and $T_{\text{max}}/T_{\text{rh}}\lesssim 20$, where the production by
inflaton annihilation is kinematically forbidden. Before moving on, we would
like to clarify that the gravitational production of DM at the end of
inflation due to time varying background metric has negligible contribution in
the present framework compared to the production due to inflaton annihilation
via $s$-channel graviton mediation. As it has been pointed out in Refs. [43,
48], the gravitational production due to metric fluctuations is most relevant
for $m_{\text{DM}}\simeq H_{I}$, and depending on the efficiency of reheating
$\gamma_{\text{rh}}=\sqrt{\Gamma/H_{I}}$ can become important only within a
relatively narrow window: $m_{\text{DM}}\sim 10^{-10}-10^{-7}~{}M_{P}$.
Although it could dominant over the UV freeze-in (which has also been shown in
Ref. [35]), it is always sub-dominant with respect to the production via
inflaton scattering.

##  3 Self-interacting Dark Matter

Even in the present scenario, where gravity is the only portal mediating
between the visible and the dark sectors, DM can feature sizeable (i.e., non-
gravitational) self-interactions. In that case, thermalization within the dark
sector can occur inducing a potentially strong impact on the DM abundance
[119, 72, 74, 75, 120, 121, 122, 123, 124, 125, 126].

Figure 4: Evolution of the ratio of the DM to SM temperatures $T^{\prime}/T$
for the case without chemical equilibrium for different DM masses, taking
$T_{\text{max}}=10^{12}$ GeV, $T_{\text{rh}}=10^{9}$ GeV and $m_{\phi}=3\times
10^{13}$ GeV.

If the dark sector features sizeable self-interactions, DM could reach kinetic
equilibrium with itself. As a consequence, the DM distribution is
characterized by a temperature $T^{\prime}$ which is in general different from
the SM temperature $T$. The evolution of the ratio of the DM to SM
temperatures $T^{\prime}/T$ is shown in Fig. 4 for DM produced from the
scattering of inflatons. Here we assume $T_{\text{rh}}=10^{9}$ GeV,
$T_{\text{max}}=10^{12}$ GeV, and inflatons with mass $m_{\phi}=3\times
10^{13}$ GeV. During reheating, i.e., while $T_{\text{max}}>T>T_{\text{rh}}$,
relativistic DM cools down much faster than radiation, $T^{\prime}\propto
a^{-1}$ whereas $T\propto a^{-3/8}$, and therefore $T^{\prime}/T\propto
T^{5/3}$. We note that as inflatons annihilate at rest, initial DM temperature
is of the order of inflaton mass. As a result, at $T\simeq T_{\text{max}}$ one
has $T^{\prime}/T\sim m_{\phi}/T_{\text{max}}\sim 30$. After reheating, the
relativistic DM and free SM radiation have the same scaling ($T\propto
T^{\prime}\propto a^{-1}$) due to which $T^{\prime}/T$ remains constant.
Eventually, the DM becomes non-relativistic. As it is out-of-chemical
equilibrium, its temperature scales as $T^{\prime}\propto a^{-2}$, and
therefore $T^{\prime}/T\propto T^{13/3}$ or $T^{\prime}/T\propto T$ for $T\gg
T_{\text{rh}}$ or $T\ll T_{\text{rh}}$ respectively. This implies that the DM
cools down much faster than the SM. In Fig. 4 we see this nature where the
$10^{2}$ GeV DM is always relativistic for $T\gtrsim 10^{6}$ GeV. Finally, the
right panel of Fig. 2 shows the equilibrium comoving DM yield (in blue) for
different DM masses. For relativistic DM (i.e., $T^{\prime}\gg
m_{\text{DM}}$), the yield at equilibrium is constant, however it rapidly
decreases while becoming nonrelativistic due to Boltzmann suppression. One may
note here, the DM number density is much smaller at production than that at
equilibrium.

In addition to kinetic equilibrium, self-interaction can also bring DM into
chemical equilibrium with itself. The maximal effect of thermalization and
number-changing interactions can be computed following a model-independent
approach [124]. The DM energy density at production, i.e., when $T\simeq
T_{\text{rh}}$, can be estimated to be

$\rho(T_{\text{rh}})\simeq Y_{0}\,s(T_{\text{rh}})\,\langle E\rangle\,,$ (3.1)

where $\langle E\rangle$ corresponds to the mean DM energy at production. If
DM is produced by annihilation of inflatons, $\langle E\rangle\simeq
m_{\phi}\,(T_{\text{rh}}/T_{\text{max}})^{8/3}$ as the inflatons annihilate at
rest.777At $T\simeq T_{\text{max}}$, $\langle E\rangle\simeq m_{\phi}$.
However, $\langle E\rangle\simeq
m_{\phi}\,(a_{\text{max}}/a_{\text{rh}})\simeq
m_{\phi}\,(T_{\text{rh}}/T_{\text{max}})^{8/3}$ at $T=T_{\text{rh}}$. However,
if UV freeze-in is responsible for the DM genesis, $\langle E\rangle\simeq
T_{\text{rh}}$ or $\langle E\rangle\simeq m_{\text{DM}}$ for $m_{\text{DM}}\ll
T_{\text{rh}}$ or $m_{\text{DM}}\gg T_{\text{rh}}$, respectively. In the case
where DM self-interactions guarantee kinetic equilibrium in the dark sector,
DM is characterized by a temperature $T^{\prime}$ which is in general
different from the SM one. Assuming the DM instantaneously thermalizes, its
temperature ${T^{\prime}_{\text{rh}}}$ at $T=T_{\text{rh}}$ is given by

${T^{\prime}_{\text{rh}}}\equiv
T^{\prime}(T_{\text{rh}})\simeq\begin{cases}\left[\frac{30}{\pi^{2}\,\mathcal{C}_{\rho}\,g}\,\rho(T_{\text{rh}})\right]^{1/4}&\text{for
}m_{\text{DM}}\ll{T^{\prime}_{\text{rh}}}\,,\\\\[8.0pt]
\frac{2}{3}\,m_{\text{DM}}\,W_{0}^{-1}\left[\frac{g^{2/3}m_{\text{DM}}^{8/3}}{3\pi\,\rho(T_{\text{rh}})^{2/3}}\right]&\text{for
}m_{\text{DM}}\gg{T^{\prime}_{\text{rh}}}\,,\end{cases}$ (3.2)

where $\mathcal{C}_{\rho}=1$ or $\mathcal{C}_{\rho}=7/8$ for bosonic or
fermionic DM respectively, $g$ corresponds to the internal DM degrees of
freedom and $W_{0}$ is the $0^{\text{th}}$ branch of the $W$-Lambert function.
Additionally, if DM also reaches chemical equilibrium via number-changing
interaction, its equilibrium number density turns out to be

$n({T^{\prime}_{\text{rh}}})=\begin{cases}\frac{\zeta(3)}{\pi^{2}}\,\mathcal{C}_{n}\,g\,{T^{\prime}_{\text{rh}}}^{3}&\text{for
}m_{\text{DM}}\ll{T^{\prime}_{\text{rh}}}\,,\\\\[8.0pt]
\frac{\rho(T_{\text{rh}})}{m_{\text{DM}}}&\text{for
}m_{\text{DM}}\gg{T^{\prime}_{\text{rh}}}\,,\end{cases}$ (3.3)

where $\mathcal{C}_{n}=1$ or $3/4$ respectively for bosonic or fermionic DM.
Therefore, the maximal yield after thermalization is given by

$Y_{0}^{\text{max}}=\frac{n({T^{\prime}_{\text{rh}}})}{s(T_{\text{rh}})}\simeq\begin{cases}\frac{15\sqrt{2}\,\mathcal{C}_{n}\,\zeta(3)}{\pi^{4}}\left(\frac{3\,g}{g_{\star
s}}\right)^{1/4}\left[\frac{1}{\mathcal{C}_{\rho}}\,\frac{\langle
E\rangle}{T_{\text{rh}}}\,Y_{0}\right]^{3/4}&\text{for
}m_{\text{DM}}\ll{T^{\prime}_{\text{rh}}}\,,\\\\[8.0pt] Y_{0}\,\frac{\langle
E\rangle}{m_{\text{DM}}}&\text{for
}m_{\text{DM}}\gg{T^{\prime}_{\text{rh}}}\,.\end{cases}$ (3.4)

It is important to emphasize that the maximal yield presented in the previous
expression may not always be reached. On one hand, DM self-interactions may
not be strong enough to guarantee chemical equilibrium, especially for heavy
DM, where the unitarity sets upper bounds to the DM self-couplings. On the
other hand, if those couplings are strong enough to make DM number-changing
interactions freeze-out when DM is non-relativistic, a cannibalization phase
occurs, depleting the DM number density [70, 58, 127, 128].

The blue- and red-colored bands in Fig. 3 show the parameter space that
generates the entire observed DM abundance once thermalization and number-
changing processes are taken into account. The blue bands correspond to the
production via scattering of inflatons while UV freeze-in via scattering of
the SM particles correspond to the red bands. The bands feature two slopes
corresponding to the cases whether the DM is relativistic or not after
thermalization. Following Eq. (3.4), at high reheating temperatures light DM
is produced relativistically, whereas heavy DM at small $T_{\text{rh}}$ is
non-relativistic. We observe that, the inclusion of DM self-interaction
broadens the allowed parameter space. It is also intriguing to note that, for
fermionic DM with $T_{\text{max}}/T_{\text{rh}}\lesssim 20$, DM production via
scattering of the SM particles can dominate for masses in the range $10^{6}$
GeV$\lesssim m_{\text{DM}}\lesssim 10^{10}$ GeV.

In the standard SIMP scenario DM cannibalization leads to a relative rise of
the DM temperature with respect to the SM, and therefore poses a potential
problem due to the bound on warm DM. This could be avoided in different ways,
for example: $i)$ by demanding kinetic equilibrium between the dark and the
visible sectors at the moment of the freeze-out of the number-changing
interactions [58], $ii)$ by assuming that the dark sector contains additional
states that are relativistic when the number-changing interactions freeze-out
[71, 129], or $iii)$ by assuming that DM is originally produced much colder
than the SM [72, 74]. Here, however, the bound on warm DM is naturally
satisfied since during reheating DM cools down much faster than the SM as the
inflaton keeps injecting entropy to the SM thermal bath, cf. Fig. 4.

##  4 Singlet Scalar Dark Matter

As an example of gravitational DM with sizeable self-interactions, in this
section the scalar singlet DM (SSDM) model [107, 108, 109] is studied. In
addition to the SM particle spectrum, the SSDM model contains only a real
scalar $s$, singlet under the SM gauge group. The scalar $s$ is assumed to be
odd under a $\mathbb{Z}_{2}$ symmetry that guarantees its stability. Thus, the
most general renormalizable scalar potential is given by

$V=\mu_{\mathcal{H}}^{2}\,|\mathcal{H}|^{2}+\lambda_{\mathcal{H}}\,|\mathcal{H}|^{4}+\mu_{s}^{2}\,s^{2}+\lambda_{s}\,s^{4}+\lambda_{\mathcal{H}s}\,|\mathcal{H}|^{2}\,s^{2},$
(4.1)

where $\mathcal{H}$ is the SM Higgs doublet. The phenomenology of this model
is completely determined by three parameters

$m_{\text{DM}}\,,\,\lambda_{\mathcal{H}s}\,,\,\lambda_{s}\,,$ (4.2)

corresponding to the DM mass, the Higgs portal and the DM quartic coupling,
respectively. One should note here, the role of the DM self-coupling
$\lambda_{s}$ does not influence the DM abundance if it is produced via the
standard WIMP or FIMP mechanisms.888It has been very recently pointed out that
in this model DM could be also be produced via inverse phase transitions
[130]. However, the self-interaction strength plays a deterministic role in
the case where the DM genesis occurs via the SIMP paradigm [74, 131, 77, 132].
Due to simplicity and predictability, the phenomenology of SSDM as a WIMP has
been widely studied, where it is assumed that the scalar $s$ has a sizeable
mixing with the SM Higgs. This scenario has been found to be highly
constrained from collider searches [133, 134, 135, 136, 137, 138, 139], DM
direct detection [140, 141, 142, 143, 144] and indirect detection [145, 146,
147, 148, 149, 150, 151, 152]. In contrast, scenarios with a very suppressed
Higgs portal are much less constrained, and could also lead to a vast
phenomenology, such as the DM production via freeze-in [153, 154, 155, 16] or
via Hawking evaporation of primordial black holes [85]. Recent studies have
also focused the effect of a non-standard cosmology on this model [77, 156,
157, 158].

In the singlet scalar model, DM can reach kinetic equilibrium by efficient
2-to-2 elastic scatterings via its quartic interaction (by construction the
exchange of a Higgs boson is assumed to be subdominant). The reaction rate for
2-to-2 DM scattering is given by

$\Gamma_{2\to
2}(T^{\prime})=\frac{\lambda_{s}^{2}}{64\pi^{3}}\frac{{T^{\prime}}^{2}}{m_{\text{DM}}}\frac{K_{1}(2\,m_{\text{DM}}/T^{\prime})}{K_{2}(m_{\text{DM}}/T^{\prime})}\,,$
(4.3)

as derived in Appendix E.1. Apart from kinetic equilibrium, chemical
equilibrium is also required to have reproductive DM interactions. In the
singlet scalar DM model, this comes mainly from 4-to-2 scatterings, the 3-to-2
processes being forbidden by the $\mathbb{Z}_{2}$ symmetry. The interaction
rate depends on the quartic coupling $\lambda_{s}$, and is estimated to be

$\Gamma_{2\to 4}(T^{\prime})\simeq\begin{cases}2.4\times
10^{-5}\,\lambda_{s}^{4}\,T^{\prime}&\text{ for }m_{\text{DM}}\ll
T^{\prime},\\\\[8.0pt]
\frac{27}{128\,\pi^{11/2}}\sqrt{\frac{3}{2}}\,\lambda_{s}^{4}\,T^{\prime}\left(\frac{T^{\prime}}{m_{\text{DM}}}\right)^{7/2}e^{-3\frac{m_{\text{DM}}}{T^{\prime}}}&\text{
for }m_{\text{DM}}\gg T^{\prime},\end{cases}$ (4.4)

as reported in Appendix E.2. It is worth mentioning here that the self-
interaction strength $\lambda_{s}$ is bounded from above from the requirement
of perturbative unitarity [159, 160, 161] that necessarily demands
$\lambda_{s}<4\pi$.

In order to account for the evolution of DM number density with self-
interaction we include both the gravitational production and number-changing
processes corresponding to 2-to-4 and 4-to-2 reactions to the BEQ in Eq. (2.1)

$\frac{dn}{dt}+3H\,n=\gamma+\gamma_{2\to
4}\left(\frac{n}{n_{\text{eq}}}\right)^{2}\,\left[1-\left(\frac{n}{n_{\text{eq}}}\right)^{2}\right],$
(4.5)

where $\gamma_{2\to 4}\equiv n_{\text{eq}}\,\Gamma_{2\to 4}\equiv
n_{\text{eq}}^{2}\,\langle\sigma v\rangle_{2\to 4}$ is the 2-to-4 rate
density, and $n_{\text{eq}}(T^{\prime})$ is the equilibrium DM number density.

Figure 5: Evolution of the comoving DM yield $N$ (left panel) and the ratio of
temperatures $T^{\prime}/T$ (right panel) as a function of the SM temperature
$T$ for the SSDM model for $T_{\text{rh}}=6\times 10^{12}$ GeV and
$T_{\text{max}}/T_{\text{rh}}=10$, assuming $m_{\phi}=3\times 10^{13}$ GeV. We
have also taken $m_{\text{DM}}=8\times 10^{4}$ GeV, and $\lambda_{s}=10$. The
vertical red dotted lines correspond to $T=T_{\text{max}}$ and
$T=T_{\text{rh}}$. The solid thick black lines correspond to the case with
chemical equilibrium and yields the observed DM abundance, whereas the dashed
black lines correspond to the case without chemical equilibrium (with
$\lambda_{s}\to 0$) that gives rise to an underabundant DM density. The black
dot-dashed line in the left panel plot denotes $N_{\text{eq}}(T^{\prime})$
corresponding to the equilibrium DM number density.

Figure 5 shows the evolution of the comoving DM yield $N$ (left panel) and the
ratio of temperatures $T^{\prime}/T$ (right panel) as a function of the SM
temperature $T$ for $T_{\text{rh}}=6\times 10^{12}$ GeV and
$T_{\text{max}}/T_{\text{rh}}=10$, assuming $m_{\phi}=3\times 10^{13}$ GeV. We
have also considered $m_{\text{DM}}=8\times 10^{4}$ GeV and $\lambda_{s}=10$.
The vertical red dotted lines correspond to $T=T_{\text{max}}$ and
$T=T_{\text{rh}}$. The solid thick black curves denote the scenario with
chemical equilibrium that gives rise to the observed DM abundance, while the
black dashed lines indicate the case without chemical equilibrium (similar to
Fig. 2 and 4) that generate a DM underabundance. The evolution in the case
where chemical equilibrium is reached contrasts with the one without in two
main aspects: firstly, there is a fast increase of the DM number density due
to the out-of-equilibrium 2-to-4 processes at the cost of reducing its
temperature. This period ends at $T\simeq 2\times 10^{9}$ GeV when chemical
equilibrium is reached as seen from the left panel of Fig. 5 where the
equilibrium number density of the DM is shown by the black dot-dashed curve.
Secondly, once the DM becomes non-relativistic, 4-to-2 cannibalizations become
effective. This causes the DM number density to reduce and the DM temperature
falls less rapidly since $T^{\prime}\propto 1/\log a$ [70]. As a result, the
DM is heated up with respect to the SM and the ratio $T^{\prime}/T$ rises at
$T\simeq 3\times 10^{8}$ GeV. The number-changing interactions eventually
freeze out at $T\simeq 7\times 10^{7}$ GeV, breaking the chemical equilibrium.

Finally, we note that DM self-interactions are bounded. In fact, Bullet
cluster measurements point towards DM self-interactions below 1.25 cm2/g at
$68\%$ CL [66, 67, 68]. Moreover, recent observational data on cluster
collisions have led to a more stringent bound on $\sigma_{2\to
2}/m_{\text{DM}}<0.47$ cm2/g at 95% CL [162], where for the SSDM model

$\frac{\sigma_{2\to
2}}{m_{\text{DM}}}\simeq\frac{9}{8\,\pi}\frac{\lambda_{s}^{2}}{m_{\text{DM}}^{3}}\,.$
(4.6)

However, this bound constrains sub-GeV DM and not reachable in this context of
gravitational production.

##  5 Conclusions

Dark matter (DM) produced from inflaton scattering mediated by massless
gravitons during the era of inflation can account for the entire observed DM
abundance ranging from DM masses of a few TeV to as large as a few PeV
depending on the choice of the reheating temperature, and the maximum
temperature achieved during reheating. At the same time, DM can also be
produced from 2-to-2 annihilation of the Standard Model (SM) bath particles
via graviton exchange leading to the standard UV freeze-in mechanism. DM
production via such gravity mediated processes is unavoidable and can yield
the whole observed relic density. Now, even if gravity remains as the only
portal between the dark and the visible sector, the DM particles still can
have sizeable interactions among themselves which can lead to kinetic
equilibrium within the dark sector. In that case, the dark sector can evolve
with a distribution that is characterized by a temperature generally different
from the SM bath. On top of kinetic equilibrium, number-changing interactions
can result in chemical equilibrium of the DM with itself.

In the present work, by considering the thermalization due to number-changing
interactions within the dark sector, we have shown that the gravitationally
produced DM can give rise to the observed relic abundance over a broader
region of the parameter space which is otherwise absent. The allowed parameter
space remains safe from the Lyman-$\alpha$ bound on warm DM mass but conflicts
with CMB measurements on the upper limit on the inflationary scale for large
reheating temperatures. The analysis is model independent and applies to DM
particles with 0, 1/2, or 1 intrinsic spin. Although the gravitational DM
production from the SM bath is generally subdominant, in the presence of self-
interactions it could dominate for fermionic DM in the mass window of $10^{6}$
to $10^{10}$ GeV for $T_{\text{max}}/T_{\text{rh}}\lesssim 20$.

We finally take up the $\mathbb{Z}_{2}$ symmetric scalar singlet DM model as
an example of a minimal extension of the SM where self-interaction amongst the
DM particles can arise naturally. In this case, the DM reaches kinetic
equilibrium via 2-to-2 scattering process which is controlled by the quartic
interaction strength. The chemical equilibrium, on the other hand, is achieved
via the number-changing 2-to-4 and 4-to-2 processes. Considering the
gravitational production of the scalar DM, to obtain the yield, we numerically
solve a set of coupled Boltzmann equations by introducing DM self-
interactions. We find, for large enough quartic coupling satisfying
perturbative unitarity, the scalar DM can produce the required abundance by
remaining well within chemical equilibrium.

## Acknowledgments

The authors would like to acknowledge discussions with Xiaoyong Chu and
Purusottam Ghosh. NB received funding from Universidad Antonio Nariño grants
2019101, and 2019248, the Spanish FEDER/MCIU-AEI under grant FPA2017-84543-P,
and 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. 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 Bound on the Reheating Temperature

The evolution of the inflaton and SM radiation energy densities can be tracked
via the set of Boltzmann equations given in Eqs. (2.2) and (2.3). During
reheating, the inflaton dominates the total energy density (i.e.,
$H^{2}\simeq\rho_{\phi}/(3M_{P}^{2})$) and evolves as

$\rho_{\phi}(a)=\rho_{\phi}(a_{I})\left[\frac{a_{I}}{a}\right]^{3}=3M_{P}^{2}\,H_{I}^{2}\left[\frac{a_{I}}{a}\right]^{3},$
(A.1)

with $a$ the scale factor and $H_{I}\equiv H(a_{I})$, where $a_{I}$
corresponds to the scale factor at the beginning of the reheating era. The
evolution of the radiation energy density can be extracted from Eqs. (2.3) and
(A.1), and reads

$\rho_{R}(a)=\frac{6}{5}\,M_{P}^{2}\,\Gamma_{\phi}\,H_{I}\,\left(\frac{a_{I}}{a}\right)^{4}\,\left[\left(\frac{a}{a_{I}}\right)^{5/2}-1\right].$
(A.2)

It follows that the thermal bath reaches a maximum temperature
$T=T_{\text{max}}$ when only a small fraction of the inflaton has decayed
[163, 29] at $a=a_{\text{max}}\equiv(8/3)^{2/5}\,a_{I}$ that corresponds to

$T_{\text{max}}^{4}=\frac{60}{\pi^{2}\,g_{\star}}\left(\frac{3}{8}\right)^{8/5}M_{P}^{2}\,\Gamma_{\phi}\,H_{I}\,.$
(A.3)

Finally, the Hubble expansion rate at the beginning of the heating era takes
the form [23]

$H_{I}=\frac{\pi}{2}\sqrt{\frac{g_{\star}}{10}}\left(\frac{8}{3}\right)^{8/5}\frac{T_{\text{max}}^{4}}{M_{P}\,T_{\text{rh}}^{2}}\,,$
(A.4)

which, taking into account the upper limit on the inflationary scale
$H_{I}^{\text{CMB}}\leq 2.5\times 10^{-5}~{}M_{P}$ [113], allows to extract an
upped bound on $T_{\text{rh}}$:

$T_{\text{rh}}\leq\sqrt{\frac{2}{\pi}}\left(\frac{10}{g_{\star}}\right)^{\frac{1}{4}}\left(\frac{3}{8}\right)^{\frac{4}{5}}\left(\frac{T_{\text{rh}}}{T_{\text{max}}}\right)^{2}\sqrt{M_{P}\,H_{I}^{\text{CMB}}}\lesssim
2.5\times
10^{9}~{}\text{GeV}\left[\frac{10^{3}}{T_{\text{max}}/T_{\text{rh}}}\right]^{2}.$
(A.5)

##  Appendix B Lyman-$\alpha$ Constraint

Due to their large initial momentum, DM particles could have a large free-
streaming length leading to a suppression on the structure formation at small
scales. In the present scenario where DM has no interactions with the SM or
with itself, the DM momentum simply redshifts, and its value $p_{0}$ at
present is [164]

$p_{0}=\frac{a_{\text{in}}}{a_{0}}\,p_{\text{in}}=\frac{a_{\text{in}}}{a_{\text{eq}}}\frac{\Omega_{R}}{\Omega_{m}}\,p_{\text{in}}=\begin{cases}\left[\frac{g_{\star
s}(T_{\text{eq}})}{g_{\star
s}(T_{\text{in}})}\right]^{1/3}\frac{T_{\text{eq}}}{T_{\text{in}}}\frac{\Omega_{R}}{\Omega_{m}}\,p_{\text{in}}\quad&\text{for
}T_{\text{in}}\leq T_{\text{rh}}\,,\\\\[8.0pt] \left[\frac{g_{\star
s}(T_{\text{eq}})}{g_{\star
s}(T_{\text{rh}})}\right]^{1/3}\left(\frac{T_{\text{rh}}}{T_{\text{in}}}\right)^{8/3}\frac{T_{\text{eq}}}{T_{\text{rh}}}\frac{\Omega_{R}}{\Omega_{m}}\,p_{\text{in}}\quad&\text{for
}T_{\text{in}}\geq T_{\text{rh}}\,,\end{cases}$ (B.1)

where $p_{\text{in}}$ is the mean initial momentum at production
($a=a_{\text{in}}$), $T_{\text{eq}}$ and $a_{\text{eq}}$ correspond to the
temperature and the scale factor at the matter-radiation equality,
respectively. For DM produced by the scattering of SM particles
$T_{\text{in}}\simeq p_{\text{in}}\simeq T_{\text{rh}}$, and

$p_{0}\simeq\left[\frac{g_{\star s}(T_{\text{eq}})}{g_{\star
s}(T_{\text{rh}})}\right]^{1/3}\frac{\Omega_{R}}{\Omega_{m}}\,T_{\text{eq}}\,.$
(B.2)

In this case, the usual lower bound from warm DM applies, corresponding to
$m_{\text{DM}}\simeq 3.5$ keV [114]. However, if DM is produced by the
scattering of inflatons then $T_{\text{in}}\simeq T_{\text{max}}$ and
$p_{\text{in}}\simeq m_{\phi}$, therefore the typical momentum for DM at
present epoch can be estimated as

$p_{0}\simeq\left[\frac{g_{\star s}(T_{\text{eq}})}{g_{\star
s}(T_{\text{rh}})}\right]^{1/3}\left(\frac{T_{\text{rh}}}{T_{\text{max}}}\right)^{8/3}\frac{T_{\text{eq}}}{T_{\text{rh}}}\frac{\Omega_{R}}{\Omega_{m}}\,m_{\phi}\,.$
(B.3)

A lower bound on the DM mass can be obtained from the upper bound on a typical
velocity of warm DM at the present time. Taking $v_{\text{DM}}\lesssim
1.8\times 10^{-8}$ [165] for $m_{\text{DM}}\simeq 3.5$ keV [114], one gets

$m_{\text{DM}}\gtrsim
0.8~{}\text{eV}\left(\frac{10^{3}}{T_{\text{max}}/T_{\text{rh}}}\right)^{8/3}\frac{10^{9}~{}\text{GeV}}{T_{\text{rh}}}\,\frac{3\times
10^{13}~{}\text{GeV}}{m_{\phi}}\,,$ (B.4)

where $T_{\text{eq}}\simeq 0.8$ eV, $\Omega_{R}\simeq 5.4\times 10^{-5}$ and
$\Omega_{m}\simeq 0.315$ [6, 166] were used. It is interesting to note that in
this case DM in the eV ballpark can be explored, easing the tension with
structure formation. In fact, DM naturally tends to be much colder than the SM
due to the fact that during reheating SM radiation is not free and scales as
$T\propto a^{-3/8}$ whereas DM momentum simply redshifts.

##  Appendix C Relevant Vertices

In Tab. 1 we have collected the relevant vertex factors [53] for graviton-DM
interaction for the case of spin-0 $S$, spin-1/2 $\psi$, and spin-1 $X_{\mu}$
DM.

Interaction Vertex | Vertex factors
---|---
|
$-\frac{i}{2M_{P}}\left[p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}-\eta_{\mu\nu}\left(p_{1}\cdot
p_{2}-m_{S}^{2}\right)\right]$
|
$-\frac{i}{8M_{P}}\left[\left(p_{1}+p_{2}\right)_{\mu}\gamma_{\nu}+\left(p_{1}+p_{2}\right)_{\nu}\gamma_{\mu}-2\eta_{\mu\nu}\left(\not{p_{1}}+\not{p_{2}}-2m_{\psi}\right)\right]$
| $\begin{aligned}
-\frac{i}{2M_{P}}\Bigl{[}\eta_{\lambda\kappa}\eta_{\mu\nu}\left(p_{1}\cdot
p_{2}-m_{X}^{2}\right)-\eta_{\lambda\kappa}p_{1\nu}p_{2\mu}+\eta_{\kappa\mu}p_{1\nu}p_{2\lambda}-\eta_{\mu\nu}p_{1\kappa}p_{2\lambda}&\\\
+\eta_{\lambda\nu}p_{1\kappa}p_{2\mu}-\eta_{\kappa\mu}\eta_{\lambda\nu}\left(p_{1}\cdot
p_{2}-m_{X}^{2}\right)-\eta_{\mu\nu}p_{1\lambda}p_{2\kappa}+\eta_{\lambda\mu}p_{1\nu}p_{2\kappa}\\\
-\eta_{\kappa\nu}\eta_{\lambda\mu}\left(p_{1}\cdot
p_{2}-m_{X}^{2}\right)\Bigr{]}\end{aligned}$
Table 1: Relevant graviton-DM vertices for scalar $(S)$, fermion $(\psi)$ and
vector boson $(X)$ DM, from top to bottom. For the inflaton-graviton
interaction one has to replace $m_{S}$ with $m_{\phi}$.

##  Appendix D Dark Matter Production Rates from Inflaton Annihilations

For 2-to-2 process we can define the following kinematics in the center-of-
mass (CM) frame for a $1,2\to 3,4$ scattering considering all states to be
massive:

$\begin{split}&E_{\text{1CM}}=\frac{s+m_{1}^{2}-m_{2}^{2}}{2\sqrt{s}},~{}E_{\text{2CM}}=\frac{s+m_{2}^{2}-m_{1}^{2}}{2\sqrt{s}},\\\
&E_{\text{3CM}}=\frac{s+m_{3}^{2}-m_{4}^{2}}{2\sqrt{s}},E_{\text{4CM}}=\frac{s+m_{4}^{2}-m_{3}^{2}}{2\sqrt{s}},\end{split}$
(D.1)

where $\sqrt{s}$ is the CM energy for the process. Now, the relevant amplitude
for $\phi,\phi\to\text{DM},\text{DM}$ is given by

$\mathcal{M}^{\phi
j}\propto\mathcal{M}_{\phi}^{\mu\nu}\Pi^{\mu\nu\rho\sigma}\mathcal{M}_{\rho\sigma}^{j}$
(D.2)

where $j$ denotes the spin of the DM involved in the production process. The
graviton propagator is defined via

$\Pi^{\mu\nu\rho\sigma}=\frac{1}{2k^{2}}\Bigl{(}\eta^{\rho\nu}\eta^{\sigma\mu}+\eta^{\rho\mu}\eta^{\sigma\nu}-\eta^{\rho\sigma}\eta^{\mu\nu}\Bigr{)},$
(D.3)

with $k=p_{1}+p_{2}=\sqrt{s}$ as the sum of the four-momenta of the initial
state (inflatons). We can write the partial amplitudes following Tab. 1 as

$\begin{split}&\mathcal{M}_{\phi(S)}^{\mu\nu}=\frac{1}{2}\left(p_{1(3)\mu}p_{2(4)\nu}+p_{1(3)\nu}p_{2(4)\mu}-\eta_{\mu\nu}p_{1(3)}\cdot
p_{2(4)}-\eta_{\mu\nu}m_{\phi(S)}^{2}\right)\\\
&\mathcal{M}_{\psi}^{\mu\nu}=\frac{1}{4}\,\overline{v}\left(p_{4}\right)\Bigl{[}\gamma_{\mu}\left(p_{3}-p_{4}\right)_{\nu}+\gamma_{\nu}\left(p_{3}-p_{4}\right)_{\mu}\Bigr{]}u\left(p_{3}\right)\\\
&\mathcal{M}_{X}^{\mu\nu}=\frac{1}{2}\Biggl{(}\left(p_{3}\cdot
p_{4}-m_{X}^{2}\right)\eta^{\mu\nu}\eta^{\rho\sigma}-\left(p_{3}\cdot
p_{4}-m_{X}^{2}\right)\eta^{\mu\sigma}\eta^{\nu\rho}-\left(p_{3}\cdot
p_{4}-m_{X}^{2}\right)\eta^{\mu\rho}\eta^{\nu\sigma}\\\
&-p_{3}^{\sigma}p_{4}^{\rho}\eta^{\mu\nu}+p_{3}^{\sigma}p_{4}^{\nu}\eta^{\mu\rho}+p_{3}^{\nu}p_{4}^{\rho}\eta^{\mu\sigma}+p_{3}^{\sigma}p_{4}^{\mu}\eta^{\nu\rho}+p_{3}^{\mu}p_{4}^{\rho}\eta^{\nu\sigma}-p_{3}^{\nu}p_{4}^{\mu}\eta^{\rho\sigma}-p_{3}^{\mu}p_{4}^{\nu}\eta^{\rho\sigma}\Biggr{)}\epsilon\left(p_{3}\right)_{\rho}^{\star}\epsilon\left(p_{4}\right)_{\sigma}^{\star},\end{split}$
(D.4)

where $p_{3,4}$ is the four momenta of the outgoing states (DM). The 2-to-2
cross-section in the CM frame can be expressed as

$\begin{split}\Bigl{(}\sigma
v_{\text{rel}}\Bigr{)}_{\phi\phi\to\text{DM}\text{DM}}=\frac{1}{64\pi^{2}m_{\phi}^{2}}\frac{\left|\vec{p_{f}}\right|}{\sqrt{s}}\int\overline{\mathcal{M}^{2}}_{\phi\phi\to\text{DM}\text{DM}}d\Omega\end{split}$
(D.5)

with

$\left|\vec{p_{f}}\right|=\frac{\sqrt{s}}{2}\sqrt{1-\frac{4m_{\text{DM}}^{2}}{s}}$
(D.6)

as the three-momentum of the final state. Substituting Eq. (D.6) in Eq. (D.5)
we then find

$\Bigl{(}\sigma
v_{\text{rel}}\Bigr{)}_{\phi\phi\to\text{DM}\text{DM}}=\frac{\overline{\mathcal{M}^{2}}_{\phi\phi\to\text{DM}\text{DM}}}{32\pi
m_{\phi}^{2}}\sqrt{1-\frac{m_{\text{DM}}^{2}}{m_{\phi}^{2}}}\,.$ (D.7)

###  D.1 Scalar Dark Matter

For the scalar DM case $S$, using Eq. (D.1) we can write the spin-averaged
squared amplitude as

$\overline{\mathcal{M}^{2}}_{\phi\phi\to
SS}=\frac{1}{64M_{P}^{4}}\left(m_{S}^{2}+2m_{\phi}^{2}\right)^{2}$ (D.8)

which on substitution in Eq. (D.7) gives rise to

$\Bigl{(}\sigma v_{\text{rel}}\Bigr{)}_{\phi\phi\to
SS}=\frac{m_{\phi}^{2}}{4096\pi
M_{P}^{4}}\left(x^{2}+2\right)^{2}\sqrt{1-x^{2}}$ (D.9)

with $x=m_{S}/m_{\phi}$. Corresponding DM production rate is then given by

$\Gamma_{\phi\phi\to SS}=n_{\phi}\Bigl{(}\sigma
v_{\text{rel}}\Bigr{)}_{\phi\phi\to SS}=\frac{\rho_{\phi}m_{\phi}}{4096\pi
M_{P}^{4}}\left(x^{2}+2\right)^{2}\sqrt{1-x^{2}}\,.$ (D.10)

###  D.2 Fermionic Dark Matter

For the Dirac fermionic DM case $\psi$, using Eq. (D.1) , we obtain

$\overline{\mathcal{M}^{2}}_{\phi\phi\to\psi\psi}=\frac{m_{\chi}^{2}}{128M_{P}^{4}}\left(m_{\phi}^{2}-m_{\chi}^{2}\right)$
(D.11)

which in turn, following Eq. (D.7), gives rise to

$\Bigl{(}\sigma
v_{\text{rel}}\Bigr{)}_{\phi\phi\to\psi\psi}=\frac{m_{\phi}^{2}}{4096\pi
M_{P}^{4}}x^{2}\left(1-x^{2}\right)^{3/2}$ (D.12)

where $x=m_{\psi}/m_{\phi}$. Then the rate of fermionic DM turns out to be

$\Gamma_{\phi\phi\to\psi\psi}=n_{\phi}\Bigl{(}\sigma
v_{\text{rel}}\Bigr{)}_{\phi\phi\to\psi\psi}=\frac{\rho_{\phi}m_{\phi}}{4096\pi
M_{P}^{4}}x^{2}\left(1-x^{2}\right)^{3/2}.$ (D.13)

###  D.3 Vector Dark Matter

The spin-averaged amplitude squared for the case of massive vector DM $X$
reads

$\overline{\mathcal{M}^{2}}_{\phi\phi\to
XX}=\frac{m_{\phi}^{4}}{1024M_{P}^{4}}\,\Bigl{(}4+4x^{2}+19x^{4}\Bigr{)}$
(D.14)

Following the CM frame kinematics as before we get the 2-to-2 annihilation
cross-section as

$\Bigl{(}\sigma v_{\text{rel}}\Bigr{)}_{\phi\phi\to
XX}=\frac{m_{\phi}^{2}}{32768\pi
M_{P}^{4}}\sqrt{1-x^{2}}\,\Bigl{(}4+4x^{2}+19x^{4}\Bigr{)},$ (D.15)

where $x=m_{X}/m_{\phi}$. The rate of DM production is given as

$\Gamma_{\phi\phi\to XX}=\frac{\rho_{\phi}m_{\phi}}{32768\pi
M_{P}^{4}}\sqrt{1-x^{2}}\,\Bigl{(}4+4x^{2}+19x^{4}\Bigr{)}.$ (D.16)

##  Appendix E Rates for Dark Matter Self-interaction

In this section, we compute the 2-to-2 and 2-to-4 DM interaction rates for the
case of the SSDM model, in the limit where $\lambda_{\mathcal{H}s}=0$.

###  E.1 2-to-2 Processes

The thermally averaged 2-to-2 elastic-scattering cross-section is given by
[167]

$\langle\sigma v\rangle_{2\to
2}(T^{\prime})=\frac{1}{8m_{\text{DM}}^{4}\,T^{\prime}\,K_{2}\left(m_{\text{DM}}/T^{\prime}\right)^{2}}\int_{4m_{\text{DM}}^{2}}^{\infty}ds\,\sigma\left(s\right)\left(s-4m_{\text{DM}}^{2}\right)\sqrt{s}\,K_{1}\left(\frac{\sqrt{s}}{T^{\prime}}\right),$
(E.1)

where $\sigma\left(s\right)=\lambda_{s}^{2}/32\pi s$. This leads to the 2-to-2
interaction rate

$\Gamma_{2\to 2}(T^{\prime})=n_{\text{eq}}\,\langle\sigma v\rangle_{2\to
2}=\frac{\lambda_{s}^{2}}{64\pi^{3}}\frac{{T^{\prime}}^{2}}{m_{\text{DM}}}\frac{K_{1}(2\,m_{\text{DM}}/T^{\prime})}{K_{2}(m_{\text{DM}}/T^{\prime})}\,.$
(E.2)

###  E.2 2-to-4 Processes

In the nonrelativistic limit, the thermally averaged 4-to-2 cross-section can
be expressed as [74]

$\langle\sigma v^{3}\rangle_{4\to 2}\simeq\frac{\sqrt{3}}{256\pi
m_{\text{DM}}^{4}}\left|\mathcal{M}\right|_{4\to
2}^{2}\simeq\frac{27\sqrt{3}}{8\pi}\frac{\lambda_{s}^{4}}{m_{\text{DM}}^{8}}\,.$
(E.3)

Thus, the nonrelativistic 2-to-4 equilibrium rate turns out to be

$\Gamma_{2\to 4}(T^{\prime})=n_{\text{eq}}^{3}\,\langle\sigma
v^{3}\rangle_{4\to
2}\simeq\frac{27}{128\,\pi^{11/2}}\sqrt{\frac{3}{2}}\,\lambda_{s}^{4}\,T^{\prime}\left(\frac{T^{\prime}}{m_{\text{DM}}}\right)^{7/2}e^{-3\frac{m_{\text{DM}}}{T^{\prime}}}.$
(E.4)

Alternatively, in the ultrarelativistic limit, the 2-to-4 cross-section
behaves like

$\sigma_{2\to 4}\left(s\right)\simeq\mathcal{K}\,\frac{\lambda_{s}^{4}}{s}\,,$
(E.5)

where $\mathcal{K}\simeq 2\times 10^{-4}$ is a dimensionless prefactor
numerically computed with Calchep [168]. The interaction rate in the
ultrarelativistic limit can be written as

$\Gamma_{2\to 4}(T^{\prime})=n_{\text{eq}}\,\langle\sigma v\rangle_{2\to
4}\simeq\frac{\zeta(3)}{\pi^{2}}\,\mathcal{K}\,\lambda_{s}^{4}\,T^{\prime}\,.$
(E.6)

Equations (E.4) and (E.6) match when $T^{\prime}\simeq m_{\text{DM}}$.

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