# Gravitational production of dark matter during reheating

Yann Mambrinia<EMAIL_ADDRESS>Keith A. Oliveb
<EMAIL_ADDRESS>a Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  b William I. Fine Theoretical Physics Institute, School of
Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

###### Abstract

We consider the direct $s$-channel gravitational production of dark matter
during the reheating process. Independent of the identity of the dark matter
candidate or its non-gravitational interactions, the gravitational process is
always present and provides a minimal production mechanism. During reheating,
a thermal bath is quickly generated with a maximum temperature $T_{\rm max}$,
and the temperature decreases as the inflaton continues to decay until the
energy densities of radiation and inflaton oscillations are equal, at $T_{\rm
RH}$. During these oscillations, $s$-channel gravitational production of dark
matter occurs. We show that the abundance of dark matter (fermionic or scalar)
depends primarily on the combination $T_{\rm max}^{4}/T_{\rm RH}M_{P}^{3}$. We
find that a sufficient density of dark matter can be produced over a wide
range of dark matter masses: from a GeV to a ZeV.

††preprint: UMN–TH–4008/21††preprint: FTPI–MINN–21/01

##  I Introduction

While we have considerable certainty in the existence of dark matter (DM), its
identity and interactions with the Standard Model are entirely unknown. The
lack of a signal in direct detection experiments XENON ; LUX ; PANDAX  sets
strong limits on the DM-proton cross section. Furthermore the lack of
detection at the LHC nosusy  also seems also to point towards more massive
candidates and perhaps a more massive beyond-the-Standard Model sector than
was originally envisioned hut ; gunn ; ehnos .

The mechanism by which dark matter particles populate the Universe is also
unknown. Commonly, GeV-TeV DM candidates are assumed to exist in equilibrium
as part of the thermal bath. As the temperature falls below the DM mass, they
drop out of equilibrium, and their relic density freezes out hut ; swo ; gg ;
gs ; Arcadi:2017kky . However, it is quite possible that DM particles never
attain thermal equilibrium. They may either be too massive, or their
interactions with the Standard Model may be too weak. For example, particles
which interact with the Standard Model primarily through gravity, such as the
gravitino never achieve equilibrium, though they are produced by the thermal
bath at reheating after inflation nos ; ehnos ; kl ; oss . Very roughly, their
abundance $Y\sim n_{3/2}/n_{\gamma}$ can be estimated from their production
rate, $Y\sim\Gamma_{p}/H\sim T_{\rm RH}/M_{P}$ where $H$ is the Hubble
parameter, $T_{\rm RH}$ is the reheating temperature after inflation, and
$M_{P}=1/\sqrt{8\pi G_{N}}$ is the (reduced) Planck mass. This mechanism, now
generally referred to as freeze-in, applies to a wider class of dark matter
candidates known as feebly interacting massive particles or FIMPs fimp ;
Bernal ; Bernal:2017kxu ; Bernal:2019mhf ; Barman:2020plp . Other examples
include dark matter particles produced by the exchange of a massive
$Z^{\prime}$ Bhattacharyya:2018evo  or massive spin-2 Bernal:2018qlk .

It is also possible that DM is produced in the decay of the inflaton, either
directly egnop ; grav2 ; Garcia:2017tuj ; GKMO1  or radiatively Kaneta:2019zgw
. It has also been observed that annihilation-like processes such as
$\phi\phi\rightarrow SS$, where $\phi$ is the inflaton and $S$ is a dark
matter scalar, mediated by gravity can produce a sufficient abundance of dark
matter ema . Indeed, the production of dark matter solely mediated by gravity
is a minimal contribution which is nearly model independent as we discuss in
more detail below. The production of dark matter mediated by gravity from the
thermal bath is subdominant Garny:2015sjg ; Tang:2017hvq ; Bernal:2018qlk ;
Chianese:2020yjo ; Redi:2020ffc .

Often, reheating is characterized by a single temperature, $T_{\rm RH}$, which
may be defined when the energy density in the newly produced thermal bath
becomes equal to the energy density still stored in inflaton oscillations.
However, when one drops the approximation of instantaneous reheating, one
finds that initially, the Universe reheats to a potentially much higher
temperature, $T_{\rm max}$, though very little of the total energy density of
the Universe is in the form of radiation Giudice:2000ex ; grav2 ;
Garcia:2017tuj ; Chen:2017kvz ; GKMO1 ; Bernal:2020gzm . For all models in
which dark matter is produced during the reheating process after inflation,
the dark matter abundance may be sensitive to the evolution of the temperature
between $T_{\rm max}$ and $T_{\rm RH}$. The purpose of this paper is to
compute the minimal dark matter abundance produced solely through gravity
during reheating taking into account the sensitivity of the result to details
of the reheating process. We will show that the bulk of the production of dark
matter occurs at temperatures higher than the reheating temperature (when
radiation begins to dominate the energy density) while the radiation bath
begins to form. We stress that this process is always present, though other
forms of production, including preheating, may also play a role and affect the
final dark matter abundance. Here, we will only consider the universal
gravitational production channel.

The paper is organized as follows. In the next section, we lay out the
framework of our calculation. Starting with the universal coupling of gravity
to the stress-energy tensor for either scalars or fermions, we compute the
annihilation-like rate for $\phi\phi\rightarrow SS$ (for scalar dark matter)
and $\phi\phi\rightarrow\chi\chi$ in the case of fermionic dark matter. In
section III, we compute the dark matter abundance based on the detailed
process of reheating between $T_{\rm max}$ and $T_{\rm RH}$. Then in section
IV, we relate these results to possible inflaton couplings to the Standard
Model (which are responsible for reheating), and we draw our conclusions in V.

##  II The framework

The universal interaction that surely exists between the inflaton and any dark
sector is gravity. In particular, the $s$-channel exchange of a graviton shown
in Fig. 1 can be obtained from the Lagrangian (see e.g., hol )

${\cal
L}=\frac{1}{2M_{P}}h_{\mu\nu}T^{\mu\nu}_{\phi}+\frac{1}{2M_{P}}h_{\mu\nu}T^{\mu\nu}_{S/\chi}$
(1)

where $h_{\mu\nu}$ is the metric perturbation corresponding to the graviton
and we consider either a scalar111We will consider a real scalar or Dirac
fermionic as dark matter. Generalization to a complex scalar or vectorial dark
matter is straightforward. $S$ or a fermion $\chi$ as dark matter, whose
stress-energy tensors are given by

$\displaystyle T^{\mu\nu}_{X=\phi,S}$ $\displaystyle=$
$\displaystyle\partial^{\mu}X\partial^{\nu}X-g^{\mu\nu}\left[\frac{1}{2}\partial^{\alpha}X\partial_{\alpha}X-V(X)\right]$
(2) $\displaystyle T^{\mu\nu}_{\chi}$ $\displaystyle=$
$\displaystyle\frac{i}{4}\left[\bar{\chi}\gamma^{\mu}\overset{\leftrightarrow}{\partial^{\nu}}\chi+\bar{\chi}\gamma^{\nu}\overset{\leftrightarrow}{\partial^{\mu}}\chi\right]$
(3)
$\displaystyle-g^{\mu\nu}\left[\frac{i}{2}\bar{\chi}\gamma^{\alpha}\overset{\leftrightarrow}{\partial_{\alpha}}\chi-
m_{\chi}\bar{\chi}\chi\right],$

where $V(X)$ is the scalar potential for either the inflaton or scalar dark
matter.

Figure 1: Illustration of the production of dark matter through the
gravitational scattering of the inflaton.

The amplitudes relevant for the computation of the processes
$\phi(p_{1})+\phi(p_{2})\rightarrow\text{DM}^{j}(p_{3})+\text{DM}^{j}(p_{4})$
can be parametrized by

$\mathcal{M}^{\phi j}\propto
M_{\mu\nu}^{\phi}\Pi^{\mu\nu\rho\sigma}M_{\rho\sigma}^{j}\;,$ (4)

where $j$ denotes the spin of the DM involved in the process and $j=0,1/2$.
$\Pi^{\mu\nu\rho\sigma}$ denotes the propagator of the graviton with momentum
$k=p_{1}+p_{2}$,

$\Pi^{\mu\nu\rho\sigma}(k)=\frac{\frac{1}{2}\eta^{\rho\nu}\eta^{\sigma\mu}+\frac{1}{2}\eta^{\rho\mu}\eta^{\sigma\nu}-\frac{1}{2}\eta^{\rho\sigma}\eta^{\mu\nu}}{k^{2}}\,.$
(5)

The partial amplitudes, $M_{\mu\nu}^{a}$, can be expressed as

$\displaystyle M_{\mu\nu}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}-g_{\mu\nu}p_{1}\cdot
p_{2}-g_{\mu\nu}m_{\phi}^{2})\,,$ $\displaystyle M_{\mu\nu}^{1/2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}{\bar{v}}(p_{4})\left[\gamma_{\mu}(p_{3}-p_{4})_{\nu}+\gamma_{\nu}(p_{3}-p_{4})_{\mu}\right]u(p_{3})\;,$
(6)

with a similar expression for scalar dark matter in terms of the dark matter
momenta, $p_{3},p_{4}$, and mass $m_{S}^{2}$.

It is then straightforward to compute the production rate of the dark matter.
Given the inflaton number density, $n_{\phi}=\rho_{\phi}/m_{\phi}$, the rate
for the process depicted in Fig. 1 is

$\displaystyle\Gamma_{\phi\phi\rightarrow
SS}=\frac{\rho_{\phi}m_{\phi}}{1024\pi
M_{P}^{4}}\left(1+\frac{m_{s}^{2}}{2m_{\phi}^{2}}\right)^{2}\sqrt{1-\frac{m_{s}^{2}}{m_{\phi}^{2}}}$
(7)
$\displaystyle\Gamma_{\phi\phi\rightarrow\chi\chi}=\frac{\rho_{\phi}m_{\chi}^{2}}{4096\pi
M_{P}^{4}m_{\phi}}\left(1-\frac{m_{\chi}^{2}}{m_{\phi}^{2}}\right)^{3/2}$ (8)

Note the difference in behavior in the expressions for fermionic and scalar
dark matter, especially in the mass dependence. On dimensional grounds, both
are proportional to
$n_{\phi}m_{\phi}^{2}/M_{P}^{4}=\rho_{\phi}m_{\phi}/M_{P}^{4}$. However, our
rates in Eqs. (7) and (8) correspond to s-wave scattering within the
condensate. As a result, in the case of a Dirac fermion-antifermion pair in
the final state, we require a spin flip leading to a suppression by a factor
$(m_{\chi}/m_{\phi})^{2}$, making the rate proportional to
$\rho_{\phi}m_{\chi}^{2}/m_{\phi}M_{P}^{4}$. A similar expression for the rate
producing scalars was found in ema .

##  III Dark matter production

In many models of inflation, after the period of exponential expansion, the
inflaton begins a series of oscillations about a minimum. During the initial
stages of oscillations, the Universe expands as if it were dominated by non-
relativistic matter. Inflaton decays begin the process of reheating dl ; nos .
While we assume that decay products thermalize rapidly rapid , we do not
assume that the decay is instantaneous. Instead, we include the effects due to
the evolution of the temperature of the radiation bath from its initial
creation to a temperature $T_{\rm max}$ until it begins to dominate the
expansion at $T_{\rm RH}$.

Let us start by considering the evolution of the radiation bath using the
conservation equation for radiation (produced by the decay of the inflaton),

$\frac{d\rho_{R}}{dt}+4H\rho_{R}=\Gamma_{\phi}\rho_{\phi}\,,$ (9)

where $\Gamma_{\phi}$ is the inflaton decay rate, and $\rho_{R}$ and
$\rho_{\phi}$ are the energy densities of radiation and the inflaton
respectively. This equation can be rewritten as

$\frac{dX}{da}=\frac{\Gamma_{\phi}\rho_{\phi}}{H}a^{3}\,,$ (10)

where $X=\rho_{R}a^{4}$, $a$ is the cosmological scale factor and
$H\equiv{\dot{a}}/a\approx\sqrt{\rho_{\phi}}/\sqrt{3}M_{P}$ up to the epoch of
radiation domination. Assuming inflaton oscillations about a quadratic
potential222For similar results about a more general potential see GKMO2 .,
$\rho_{\phi}=\rho_{e}(a_{e}/a)^{3}$, where $\rho_{e}$ is the energy density
stored by the inflaton at the end of the inflationary phase, and Eq. (10) is
easily integrated to give

$\rho_{R}=\frac{2\sqrt{3}}{5}\Gamma_{\phi}M_{P}\sqrt{\rho_{e}}\left(\frac{a_{e}}{a}\right)^{3/2}\left[1-\left(\frac{a_{e}}{a}\right)^{5/2}\right]=\alpha
T^{4},$ (11)

where $\alpha=g_{T}\pi^{2}/30$ and $g_{T}$ is the number of relativistic
degrees of freedom, at temperature $T$. Thus, in the case of non-instantaneous
reheating, when $a\gg a_{e}$, and as long as the inflaton dominates the energy
density, the temperature of the radiation bath falls as $T\propto a^{-3/8}$ as
energy is injected into the bath from inflaton decay Giudice:2000ex ;
Garcia:2017tuj ; Kaneta:2019zgw . For $a\gg a_{e}$, with the help of Eq.(11)
we can then write

$H(T)\simeq\frac{\sqrt{\rho_{\phi}}}{\sqrt{3}M_{P}}=\frac{5}{6}\frac{\alpha}{\Gamma_{\phi}M_{P}^{2}}T^{4}\,.$
(12)

Within the same limit we can parametrize the reheating temperature by

$T_{\rm RH}=\left(\frac{4}{3\alpha
c^{2}}\right)^{1/4}\left(\Gamma_{\phi}M_{P}\right)^{1/2}$ (13)

where $c=1$ if we define the reheat temperature by $t_{\rm
RH}^{-1}=\frac{3}{2}H=\Gamma_{\phi}$ and $c=5/3$ if we define $T_{\rm RH}$ by
$\rho_{\phi}(T_{\rm RH})=\rho_{R}(T_{\rm RH})$. Here, we have chosen the
latter which can easily be derived by rewriting Eq.(11) in the limit $a\gg
a_{e}$

$\rho_{R}\simeq\frac{2\sqrt{3}}{5}\Gamma_{\phi}M_{P}\sqrt{\rho_{\phi}(T_{\rm
RH})}\left(\frac{a_{\rm RH}}{a}\right)^{3/2}\,,$ (14)

so that $\rho_{\phi}(T_{\rm RH})=\rho_{R}(T_{\rm RH})$ implies that

$\rho_{R}(T_{\rm RH})=\alpha T_{\rm
RH}^{4}\simeq\frac{12}{25}\Gamma_{\phi}^{2}M_{P}^{2}.$ (15)

Then the expression for $H(T)$ in a Universe dominated by the inflaton is
given by Kaneta:2019zgw ; GKMO2

$H(T)=\sqrt{\frac{\alpha}{3}}\frac{T^{4}}{T_{\rm RH}^{2}M_{P}}\,,$ (16)

and

$\rho_{\phi}=\rho_{R}(T_{\rm RH})\left(\frac{a_{\rm
RH}}{a}\right)^{3}=\alpha\frac{T^{8}}{T_{\rm RH}^{4}}\,.$ (17)

When the limit $a\gg a_{e}$ cannot be applied, the reheating temperature is
the solution of the equation

$\sqrt{\rho_{\rm RH}}=\alpha^{1/2}T_{\rm
RH}^{2}=\frac{2\sqrt{3}}{5}\Gamma_{\phi}M_{P}\left[1-\left(\frac{\rho_{\rm
RH}}{\rho_{e}}\right)^{5/6}\right]\,,$ (18)

where $\rho_{\rm RH}=\rho_{\phi}(T_{RH})=\rho_{R}(T_{\rm RH})$.

Note that the maximum temperature attained, $T_{\rm max}$, can be found from
the extremum of Eq. (11) with respect to $a$. We obtain

$\frac{a_{\rm max}}{a_{e}}=\left(\frac{8}{3}\right)^{2/5}$ (19)

implying

$\rho_{R}(a_{\rm
max})=\frac{\sqrt{3}}{4}\Gamma_{\phi}M_{P}\sqrt{\rho_{e}}\left(\frac{3}{8}\right)^{3/5}=\alpha
T_{\rm max}^{4}\,.$ (20)

Combining Eqs. (18) and (20) gives

$\left(\frac{T_{\rm max}}{T_{\rm
RH}}\right)^{4}=\frac{25\sqrt{\rho_{e}}}{16\sqrt{3}\Gamma_{\phi}M_{P}}\left(\frac{3}{8}\right)^{3/5}\left[1-\left(\frac{a_{e}}{a_{\rm
RH}}\right)^{5/2}\right]^{-2}\,.$ (21)

The relic abundance of dark matter is obtained by solving the Boltzmann
equation for the number density of dark matter particles

$\frac{dn_{j}}{dt}+3Hn_{j}=R_{j}(T)\,.$ (22)

Writing $Y_{j}=n_{j}a^{3}$, Eq. (22) can be simplified to

$\frac{dY_{j}}{da}=\frac{R_{j}(T)a^{2}}{H}$ (23)

where $R_{j}(T)=\frac{\rho_{\phi}}{m_{\phi}}\Gamma_{\phi\phi\rightarrow jj}$
is the production rate (per unit volume and unit time) and the yield, $Y_{j}$
is proportional to the number of dark matter quanta produced in the comoving
frame. It is important to note that because
$R_{j}\propto\rho_{\phi}^{2}\propto T^{16}$, there is a very strong
temperature dependence in Eq. (23), making the result sensitive to the maximum
temperature attained.

Using Eq. (17), it is straightforward to integrate Eq. (23) between $a_{e}$
and $a_{\rm RH}$ We find

$Y_{j}(T_{\rm
RH})=\frac{2\gamma_{j}M_{P}}{\sqrt{3}m_{\phi}}\rho_{e}^{3/2}a_{e}^{3}\left[1-\left(\frac{a_{e}}{a_{\rm
RH}}\right)^{3/2}\right]\,,$ (24)

where $\gamma_{j}=\Gamma_{\phi\phi\rightarrow jj}/\rho_{\phi}$. The yield can
also be written in terms of $\rho_{R}$, and the number density becomes

$n_{j}(T_{\rm RH})=\frac{2\gamma_{j}M_{P}}{\sqrt{3}m_{\phi}}\alpha^{3/2}T_{\rm
RH}^{6}\left[\left(\frac{a_{\rm RH}}{a_{e}}\right)^{3/2}-1\right]\,.$ (25)

Noting that we can write $(a_{e}/a_{\rm RH})=(a_{e}/a_{\rm max})(a_{\rm
max}/a_{\rm RH})=(3/8)^{2/5}(T_{\rm max}/T_{\rm RH})^{8/3}$, the density can
be expressed as

$n_{j}(T_{\rm RH})=\frac{2\gamma_{j}M_{P}}{\sqrt{3}m_{\phi}}\alpha^{3/2}T_{\rm
RH}^{2}\left[\left(\frac{8}{3}\right)^{3/5}T_{\rm max}^{4}-T_{\rm
RH}^{4}\right]\,.$ (26)

Equation (26) is valid so long as $a_{\rm RH}>a_{\rm max}$. For sufficiently
large $\Gamma_{\phi}$, the radiation energy density will equal the inflaton
oscillation energy density when $T=T_{\rm max}$, and thus $T_{\rm RH}=T_{\rm
max}$. At larger $\Gamma_{\phi}$, we can write

$n_{j}(T_{\rm RH})=\frac{2\gamma_{j}M_{P}}{\sqrt{3}m_{\phi}}\alpha^{3/2}T_{\rm
RH}^{6}\left[\left(\frac{\sqrt{\rho_{e}}}{\alpha^{1/2}T_{\rm
RH}^{2}}\right)-1\right]\,,$ (27)

as the temperature $T_{\rm max}$ is never attained.

The fraction of critical density in dark matter can be written as

$\Omega_{j}h^{2}=1.6\times 10^{8}\left(\frac{g_{0}}{g_{\rm
RH}}\right)\left(\frac{n_{j}(T_{\rm RH})}{T_{\rm
RH}^{3}}\right)\left(\frac{m_{j}}{1~{}\mathrm{GeV}}\right)\,,$ (28)

where the numerical factor is $\pi^{2}n_{\gamma}(T_{0})/2\zeta(3)\rho_{c}$,
$g_{0}=43/11$, $g_{\rm RH}=427/4$, and $n_{j}(T_{\rm RH})$ is obtained from
either Eq. (26) or Eq. (27) for $a_{\rm RH}>a_{\rm max}$ or $a_{\rm RH}<a_{\rm
max}$, respectively. We finally obtain

$\displaystyle\frac{\Omega_{s}h^{2}}{0.1}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{T_{\rm RH}}{10^{10}}\right)^{3}\left(\frac{T_{\rm
max}/T_{\rm RH}}{100}\right)^{4}\left(\frac{m_{S}}{1.75\times 10^{10}}\right)$
(29) $\displaystyle\times$
$\displaystyle\left(1+\frac{m_{s}^{2}}{2m_{\phi}^{2}}\right)^{2}\sqrt{1-\frac{m_{s}^{2}}{m_{\phi}^{2}}};\;a_{\rm
RH}>a_{\rm max}\,,$ $\displaystyle\frac{\Omega_{s}h^{2}}{0.1}$
$\displaystyle\simeq$ $\displaystyle\left(\frac{T_{\rm
RH}}{10^{10}}\right)\sqrt{\frac{\rho_{e}}{10^{60}}}\left(\frac{m_{S}}{1.9\times
10^{9}}\right)$ (30) $\displaystyle\times$
$\displaystyle\left(1+\frac{m_{s}^{2}}{2m_{\phi}^{2}}\right)^{2}\sqrt{1-\frac{m_{s}^{2}}{m_{\phi}^{2}}};\;a_{\rm
RH}<a_{\rm max}\,,$

for scalar dark matter and

$\displaystyle\frac{\Omega_{\chi}h^{2}}{0.1}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{T_{\rm RH}}{10^{10}}\right)^{3}\left(\frac{T_{\rm
max}/T_{\rm RH}}{100}\right)^{4}\left(\frac{m_{\chi}}{4.0\times
10^{12}}\right)^{3}$ (31) $\displaystyle\times$
$\displaystyle\left(\frac{3\times
10^{13}}{m_{\phi}}\right)^{2}\left(1-\frac{m_{\chi}^{2}}{m_{\phi}^{2}}\right)^{3/2};\;a_{\rm
RH}>a_{\rm max}\,,$ $\displaystyle\frac{\Omega_{\chi}h^{2}}{0.1}$
$\displaystyle\simeq$ $\displaystyle\left(\frac{T_{\rm
RH}}{10^{10}}\right)\sqrt{\frac{\rho_{e}}{10^{60}}}\left(\frac{m_{\chi}}{1.9\times
10^{12}}\right)^{3}$ (32) $\displaystyle\times$
$\displaystyle\left(\frac{3\times
10^{13}}{m_{\phi}}\right)^{2}\left(1-\frac{m_{\chi}^{2}}{m_{\phi}^{2}}\right)^{3/2};\;a_{\rm
RH}<a_{\rm max}\,,$

for fermionic dark matter. Units of energy are given in GeV (and GeV4 for
$\rho_{e}$). Note that these expressions are approximations which are valid so
long as $a_{\rm RH}$ is not very close to either $a_{\rm max}$ or $a_{e}$.
More generally, the relic density is obtained from Eq. (28) with either Eq.
(26) or Eq. (27). As expected, there is a stronger dependence of the relic
abundance on the mass of the dark matter in the fermionic case due to its
production rate.

We show in Fig. 2 the allowed parameter space in the ($m_{S,\chi}$, $T_{\rm
RH}$) plane for scalar (blue, dashed) and fermionic (red, solid) dark matter
for several values of $T_{\rm max}/T_{\rm RH}$. The lines correspond to values
of $\Omega h^{2}=0.12$ Planck  for each choice of $T_{\rm max}/T_{\rm RH}$. In
each case, $a_{\rm RH}>a_{\rm max}$. All points above the lines are excluded
because they lead to an overabundance. We see, for example, that GeV-ZeV dark
matter can be obtained with reasonable values of $T_{\rm RH}$ and $T_{\rm
max}$ by pure gravitational production through inflaton scattering. We also
see that due to the additional mass suppression $(m_{\chi}/m_{\phi})^{2}$, for
fermionic DM, it is necessary to consider higher fermionic masses ($\gtrsim
100$ PeV for $T_{\rm max}/T_{\rm RH}<1000$) to achieve
$\Omega_{\chi}h^{2}=0.12$. Note that we have assumed $m_{\phi}=3\times
10^{13}$ GeV in Fig. 2 and the kinematic suppression when $m_{s}\geq m_{\phi}$
would appear to the right of the region plotted.

Figure 2: Points yielding the Planck relic density $\Omega_{S,\chi}h^{2}=0.12$
for scalar (blue, dashed) and fermionic (red, solid) dark matter, in the
($m_{S,\chi}$, $T_{\rm RH}$) plane for several values of ${T_{\rm
max}}/{T_{\rm RH}}$ as labeled.

##  IV Coupling to the Standard Model

In the previous section, we concentrated on results for a given reheating and
maximum temperature reached by the thermal bath during the reheating process.
In any concrete model of inflation, these temperatures can be traced to two
quantities: the coupling of the inflaton to the Standard Model, which
ultimately determines the reheating temperature, and the initial energy
density of the inflaton at the end of the inflationary phase, which allows one
to determine the amount of energy available for the thermal bath, and hence
the maximum temperature. If one supposes a simple effective coupling of the
inflaton to the Standard Model fermions of the type

${\cal L}_{\phi-SM}^{y}=y~{}\phi\overline{f}f,$ (33)

leading to

$\Gamma_{\phi}=\frac{y^{2}}{8\pi}m_{\phi}\,,$ (34)

and using Eq. (21) we can write the ratio

$\left(\frac{T_{\rm max}}{T_{\rm
RH}}\right)^{4}=\frac{25\pi\sqrt{\rho_{e}}}{2\sqrt{3}y^{2}m_{\phi}M_{P}}\left(\frac{3}{8}\right)^{3/5}\left[1-\left(\frac{a_{e}}{a_{\rm
RH}}\right)^{5/2}\right]^{-2}.$ (35)

In models which are dominated by a quadratic term after inflation, we expect
$\rho_{e}\sim m_{\phi}^{2}M_{P}^{2}$ and for example, in the Starobinsky model
of inflation staro , $\rho_{e}^{1/4}\sim
0.65m_{\phi}^{1/2}M_{P}^{1/2}=5.5\times 10^{15}$ GeV egno5 . For sufficiently
large coupling, $y\simeq.67(\rho_{e}^{1/4}/10^{15}{\rm GeV})$, $T_{\rm
max}/T_{\rm RH}=1$, and reheating is nearly instantaneous. If reheating is
primarily due to a coupling of the inflaton to bosons, ${\cal L}_{\phi-
SM}^{\mu}=\mu~{}\phi bb$, equivalent results are obtained by substituting
$y\rightarrow\mu/m_{\phi}$.

Similarly, we can also compute $T_{\rm RH}$ by combining Eqs. (18) and (33):

$T_{\rm
RH}^{4}=\left(\frac{3y^{4}}{400\alpha\pi^{2}}\right)m_{\phi}^{2}M_{P}^{2}\left[1-\left(\frac{\alpha
T_{\rm RH}^{4}}{\rho_{e}}\right)^{5/6}\right]^{2}\,.$ (36)

For sufficiently large $\rho_{e}$, the second term in the bracket can be
neglected, otherwise, $T_{\rm RH}$ can be solved for iteratively. We can now
express the relic density in terms of the coupling $y$ using Eqs. (35) and
(36) in Eqs. (26) and (27). When $a_{\rm RH}\gg a_{\rm max}$, we can write
relatively compact expressions by combining Eqs.(29), (31), (35) and (36) and
one obtains for $m_{S,\chi}\ll m_{\phi}$

$\displaystyle\frac{\Omega_{S}h^{2}}{0.1}$ $\displaystyle=$
$\displaystyle\frac{y}{10^{-7}}\sqrt{\frac{\rho_{e}}{10^{64}}}\sqrt{\frac{3\times
10^{13}}{m_{\phi}}}\frac{m_{S}}{5.2\times 10^{9}}$ (37)
$\displaystyle\frac{\Omega_{\chi}h^{2}}{0.1}$ $\displaystyle=$
$\displaystyle\frac{y}{10^{-7}}\sqrt{\frac{\rho_{e}}{10^{64}}}\left(\frac{m_{\chi}}{2.6\times
10^{12}}\right)^{3}\left(\frac{3\times 10^{13}}{m_{\phi}}\right)^{3/2}\,,$
(38)

where as before dimensions for mass are in GeV, with GeV4 for $\rho_{e}$. We
see that for a given Standard Model coupling ($y=10^{-7}$ for example), the
dark matter mass needed to reach a reasonable relic abundance is much higher
in the fermionic case than in the scalar case, bearing in mind the minimality
of the model being considered.

We plot in Fig. 3 the required relation between $m_{S,\chi}$ and $y$ to obtain
$\Omega_{S,\chi}h^{2}=0.12$.

Values of $\Omega h^{2}>0.12$ are obtained for points to the right of the
negatively sloped lines and below the horizontal lines for
$\rho_{e}^{1/4}=10^{14}$ GeV. Note that for $\rho_{e}^{1/4}=10^{14}$ GeV, the
curves begin to bend when the second term in the bracket in Eq. (35)
approaches 1. For small values of $y$, $T_{\rm max}\gg T_{\rm RH}$, and the
relic density in Eq. (29) scales as $T_{\rm RH}^{3}(T_{\rm max}/T_{\rm
RH})^{4}m_{s}$. Since $(T_{\rm RH})^{3}\sim y^{3}$, $(T_{\rm max}/T_{\rm
RH})^{4}\sim y^{-2}$, we see that $y\sim m_{S}^{-1}$ for fixed relic density.
At larger values of $y$, when $a_{\rm RH}<a_{\rm max}$, the density in Eq.
(30) scales as $T_{\rm RH}m_{s}$. However, we can see from Eq. (36), that
$T_{\rm RH}$ reaches its maximal value when $\alpha T_{\rm
RH}^{4}\lesssim\rho_{e}$. At these values of $y$ (which are not much larger
than $.67(\rho_{e}^{1/4}/10^{15}~{}{\rm GeV})$ \- the value needed to make
$T_{\rm RH}=T_{\rm max}$), the number density obtained from Eq. (27) begins to
decrease. Thus a fixed energy density which is proportional to $n_{j}m_{j}$,
requires a rapidly larger value of the dark matter mass. This accounts for the
curve bending to the horizontal at large values of $m_{j}$. Physically, at
these values of $y$, the rate of inflaton decay exceeds the rate of dark
matter production from scattering, and production through the gravitational
process ceases. For $\rho_{e}^{1/4}>1.5\times 10^{15}$ GeV, this behavior
occurs only at $y>1$. Once again, the region with kinematic suppression lies
to the right of the region shown in the figure.

Figure 3: Points in the ($m_{S,\chi}$, $y$) plane yielding the Planck-
determined relic density $\Omega_{S,\chi}h^{2}=0.12$ in the case of a scalar
dark matter (blue,dashed) and a fermionic dark matter (red, solid) for two
values of $\rho_{e}$ as labeled.

It is also possible that inflaton scattering into Standard Model scalars
affects the reheating process, and the value of $T_{\rm max}$ in particular
GKMO2 . For example, a Lagrangian contribution of

${\cal L}_{\phi-SM}^{\sigma}=\sigma\phi^{2}b^{2}$ (39)

where $b$ represents a Standard Model boson. In this case, one still requires
the Yukawa coupling (33) to complete the reheating process (so that the energy
density in radiation comes to dominate over the energy density in $\phi$)
GKMO2 . The interaction generated by (39) can alter the maximum temperature
$T_{\rm max}$ for sufficiently large values of $\sigma$. In this case,

$\left(\frac{T_{\rm max}}{T_{\rm
RH}}\right)^{4}=\frac{6400\pi\sigma^{2}\rho_{e}^{3/2}}{\sqrt{3}y^{4}m_{\phi}^{5}M_{P}}\left(\frac{2}{3}\right)^{18}$
(40)

giving

$\displaystyle\frac{\Omega_{S}h^{2}}{0.1}$ $\displaystyle=$
$\displaystyle\frac{10^{-6}}{y}\left(\frac{\sigma}{10^{-9}}\right)^{2}\left(\frac{\rho_{e}}{(10^{16})^{4}}\right)^{3/2}\left(\frac{3\times
10^{13}}{m_{\phi}}\right)^{7/2}$ (41)
$\displaystyle\times\left(\frac{m_{S}}{1.2\times 10^{5}}\right)$

and

$\displaystyle\frac{\Omega_{\chi}h^{2}}{0.1}$ $\displaystyle=$
$\displaystyle\frac{10^{-6}}{y}\left(\frac{\sigma}{10^{-9}}\right)^{2}\left(\frac{\rho_{e}}{(10^{16})^{4}}\right)^{3/2}\left(\frac{m_{\chi}}{7.6\times
10^{10}}\right)^{3}$ (42) $\displaystyle\times\left(\frac{3\times
10^{13}}{m_{\phi}}\right)^{11/2}$

Note that for a given value of $\sigma$, the relic density increases with
decreasing $y$. This is valid only so long as $T_{\rm max}>T_{\rm RH}$. The
limiting value of $y$ is found by setting Eq. (40) equal to one.

We show in Fig.(4) the parameter space allowed in the plane ($y$, $\sigma$) in
the case of scalar dark matter for different values of its mass $m_{\chi}$.
Note that it is possible to have quite low values of $y$ (and thus $T_{\rm
RH}$) while still being able to produce dark matter in sufficient amounts due
to the value of $\sigma$ generating a high maximum temperature $T_{\rm max}$,
and thus a large production rate. The maximum value for $y$ occurs when
$T_{\rm max}=T_{\rm RH}$ in Eq. (40) where the relic abundance depends only on
$T_{\rm RH}$ (and is thus independent of $\sigma$) as we can see in the plot.
The same curves for fermionic dark matter are obtained for $m_{\chi}=1.5\times
10^{9}(m_{S}/{\rm GeV})^{1/3}$.

Figure 4: Points respecting Planck constraint $\Omega_{S}h^{2}=0.12$ in the
case of scalar dark matter, in the plane ($y$, $\sigma$) for different values
of $m_{S}$.

##  V Conclusions

We have derived the conditions for producing sufficient dark matter from
inflaton scattering during reheating by $s$-channel graviton exchange. This
process is always present independent of the model of inflation. We have shown
that the final abundance of dark matter depends not only on the reheating
temperature, but also on the maximum temperature and hence on the detailed
evolution of the reheating process. During the exit from exponential
expansion, many models of inflation begin a period of oscillations leading to
reheating. At the onset of these oscillations, the inflaton density is high
and the leading contribution to dark matter production occurs at the start of
reheating at $T_{\rm max}$. This represents an absolute minimal amount of dark
matter production and it contributes independent of any interactions the dark
matter may have with the Standard Model (or another dark sector if present).

Acknowledgements.  This work was made possible by Institut Pascal at
Université Paris-Saclay with the support of the P2I research departments and
the P2IO Laboratory of Excellence (program “Investissements d’avenir”
ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038). This project has
received support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sk  $-$ ${\rm l}$ odowska-Curie grant
agreement No 860881-HIDDeN and the CNRS PICS MicroDark. The work of K.A.O. was
supported in part by DOE grant DE-SC0011842 at the University of Minnesota.

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