# Gravitational portals in the early Universe

Simon Clérya<EMAIL_ADDRESS>Yann Mambrinia,b
<EMAIL_ADDRESS>Keith A. Olivec<EMAIL_ADDRESS>Sarunas
Vernerc<EMAIL_ADDRESS>a Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  b CERN, Theoretical Physics Department, Geneva, Switzerland  c
William I. Fine Theoretical Physics Institute, School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

###### Abstract

We consider the production of matter and radiation during reheating after
inflation, restricting our attention solely to gravitational interactions.
Processes considered are the exchange of a graviton, $h_{\mu\nu}$, involved in
the scattering of the inflaton or particles in the newly created radiation
bath. In particular, we consider the gravitational production of dark matter
(scalar or fermionic) from the thermal bath as well as from scattering of the
inflaton condensate. We also consider the gravitational production of
radiation from inflaton scattering. In the latter case, we also derive a lower
bound on the maximal temperature of order of $10^{12}$ GeV for a typical
$\alpha-$attractor scenario from $\phi\phi\rightarrow h_{\mu\nu}\rightarrow$
Standard Model fields (dominated by the production of Higgs bosons). This
lower gravitational bound becomes the effective maximal temperature for
reheating temperatures, $T_{\rm RH}\lesssim 10^{9}$ GeV. The processes we
consider are all minimal in the sense that they are present in any non-minimal
extension of the Standard Model theory based on Einstein gravity and can not
be neglected. We compare each of these processes to determine their relative
importance in the production of both radiation and dark matter.

††preprint: UMN–TH–4110/22††preprint: FTPI–MINN–22/02††preprint: CERN-
TH-2021-222

##  I Introduction

Despite more than 80 years since the first indication of dark matter
Zwicky:1933gu , its nature and identity still remains a mystery book . The
hypothesis for a weakly interacting massive particle (WIMP) as a dark matter
candidate is being challenged by an obvious lack of signal in dedicated direct
detection experiments such as XENON1T XENON , LUX LUX  or PANDAX PANDAX  (see
Arcadi:2017kky  for a detailed review). These experiments exclude de facto a
large part of the parameter space in models where dark matter communicates
with the Standard Model via the Higgs Burgess:2000yq ; Higgsportal , the $Z$
Zportal  or even an electroweak extension introducing a massive $Z^{\prime}$
mediator Zpportal . However, an alternative exists in the form of particles
interacting very weakly with the thermal bath, and never having reached
thermal equilibrium fimp . The seclusion can be justified by the weakness of a
coupling (gravitational in the case of the gravitino nos ; ehnos ; kl ; ekn ;
oss ) or by the exchange of very heavy mediators (generated by an extra $U(1)$
Bhattacharyya:2018evo , moduli field fimpmoduli  or massive spin-2 field
Bernal:2018qlk  as examples). A complete review can be found in Bernal:2017kxu
as well as related studies in Bernal ; Bernal:2019mhf ; Barman:2020plp ;
Chen:2017kvz .

The minimal coupling one can imagine between dark matter and the Standard
Model is gravitational mediated through a graviton ema ; Redi:2020ffc . As
this coupling is unavoidable, any process invoking graviton exchange provides
a lower limit on the amount of dark matter produced either via the thermal
bath Garny:2015sjg ; Tang:2017hvq ; Chianese:2020yjo ; Bernal:2018qlk ;
Redi:2020ffc ; Bernal:2021kaj  or directly through the scattering of the
inflaton MO ; Barman:2021ugy . The energy available in both cases partly
compensates for the strong reduction in coupling by the Planck
mass111Throughout the paper, we will consider the reduced Planck mass
$M_{P}=1/\sqrt{8\pi G_{N}}\simeq 2.4\times 10^{18}$ GeV., $M_{P}$. This is not
too surprising. Indeed, we know that in the case of a FIMP, a coupling of the
order of $\sim 10^{-11}$ is needed to produce dark matter in sufficient
quantities. This corresponds to an effective coupling of the order of
$\frac{E^{2}}{M_{P}^{2}}$, with $E\sim 10^{13}$ GeV representing the available
energy in the interaction. This energy corresponds, roughly, to the mass of
the inflaton. It is therefore at the end of inflation, during the transition
period between an inflaton-dominated universe and the radiative universe,
called reheating, that the available energy is sufficient for the efficient
gravitational production of dark matter.

The reheating process is not instantaneous Giudice:2000ex ; Bernal:2020gzm ;
GMOP . The radiation bath may be produced by inflaton decays or scattering
which require a coupling of the inflaton to the Standard Model, or as we show
below through the gravitational production of radiation. As the radiation
begins to appear, the Universe rapidly achieves a maximum temperature, $T_{\rm
max}$ and the reheating process continues until radiation domination is
achieved at $T_{\rm RH}$. The evolution of the radiation density depends on
GKMO1 ; GKMO2  1) how it is produced, that is, through decays, or scatterings,
2) the dominant final state particle spin, and 3) the form of the inflaton
potential about its minimum, which we take as
$V(\phi)\simeq\lambda\phi^{k}M_{P}^{4-k}$. This approximation is appropriate
for the Starobinsky model Staro  (leading to $k=2$), as well as more general
$\alpha$-attractor type models Kallosh:2013hoa ; Garcia:2021gsy . Once the
reheating is achieved, $T>T_{\rm RH}$, the inflaton disappears from the energy
budget and the temperature evolves isentropically $T\propto a^{-1}$, where $a$
is the scale factor of the Universe. As we show below, the evolution of the
radiation density can be modified by the gravitational production of Standard
Model quanta which induces a lower bound on the maximum temperature of the
Universe. We show that it is of the order of $10^{12}$ GeV, and is one of the
main results of our work.

If the production of dark matter occurs during reheating, it is intimately
linked to the behaviour of the inflaton and the evolution of the thermal bath.
Often it is assumed that the either the dark matter is directly coupled to the
inflaton, in which case, it can be produced directly from inflaton decays
egnop ; grav2 ; GMOP ; GKMO1  or it is coupled to the Standard Model, and thus
produced thermal as the gravitino or other super-weakly interacting particles.
In the latter case, it has also been shown that radiative decay of the
inflaton KMO  could be the dominant process to populate the dark Universe.

While reheating requires some coupling of the inflaton to the Standard Model
(as will see gravitation interactions alone will not lead to radiation
domination), the mechanism for producing dark matter may in fact be dominated
solely by gravity. In this paper, we analyze all processes involving a
gravitational interactions, comparing the modes of production via the thermal
bath, the scattering of the inflaton, and gravitational production of
particles from the thermal bath which subsequently produce dark matter through
gravity as well. In this sense, each of the physical quantities we consider,
such as the relic density or maximum temperature, must be considered as lower
bounds as the gravitational process we compute are inevitable in any theory
based on Einstein gravity. As a result, these lower bounds must be taken into
account in any kind of extension of the Standard Model, and can be thought of
as a gravitational “background noise". We do not consider preheating via
parametric or stochastic resonances as we did in GKMOV , because we want to
compute the minimal unavoidable amount of dark matter, and thus derive the
strongest model-independent constraints on the dark matter mass, supposing
that it only couples gravitationally.

The only non-gravitational coupling we consider, is a coupling of the inflaton
to SM fields to achieve reheating. Thus, we consider a generic Yukawa-like
coupling of the form, $y\phi{\bar{f}}f$, where $f$ is some Standard Model
fermion. We assume rapid thermalization, and these decays are (partially)
responsible for the growing thermal bath. However the production of dark
matter from the thermal bath is entirely gravitational.

The paper is organized as follows. The framework for our computation is
outlined in Section II. We consider both scalar and fermionic dark matter
coupled to the Standard Model and the inflaton only through gravity. We
compute the rates for the production of dark matter either through thermal
scattering (mediated by gravity alone) or from the inflaton condensate. We
choose an attractor form for the inflaton potential which when expanded about
its minimum, take the form $\phi^{k}$. Our results are sensitive to $k$.
Reheating takes place as the inflaton oscillates about this minimum. In
Section III we consider three distinct gravitational process. The gravitation
production of dark matter from the thermal bath; the gravitational production
of dark matter from the condensate; and the gravitational production of the
thermal bath from the condensate. We then compare each modes in Section IV,
before concluding in Section V.

##  II The framework

We study universal gravitational interactions that must exist between the
inflationary and dark sectors. If the space-time metric is expanded around
flat space using $g_{\mu\nu}\simeq\eta_{\mu\nu}+{\tilde{h}}_{\mu\nu}$ the
gravitational Lagrangian in the transverse-traceless gauge at second order can
be written as

${\cal
L}=\frac{M_{P}^{2}}{2}R\ni\frac{M_{P}^{2}}{8}(\partial^{\alpha}{\tilde{h}}^{\mu\nu})(\partial_{\alpha}{\tilde{h}}_{\mu\nu})=\frac{1}{2}(\partial^{\alpha}h^{\mu\nu})(\partial_{\alpha}h_{\mu\nu})$
(1)

where $h_{\mu\nu}=(M_{P}/2){\tilde{h}}_{\mu\nu}$ is the canonically normalized
perturbation and and $M_{P}=(8\pi G)^{-1/2}\simeq 2.4\times 10^{18}\,\rm{GeV}$
is the reduced Planck mass. Gravitational interactions are described by the
Lagrangian (see e.g., hol )

$\sqrt{-g}{\cal L}_{\rm
int}=-\frac{1}{M_{P}}h_{\mu\nu}\left(T^{\mu\nu}_{SM}+T^{\mu\nu}_{\phi}+T^{\mu\nu}_{X}\right)\,.$
(2)

Here SM represents Standard Model fields, $\phi$ is the inflaton and $X$ is a
dark matter candidate. The form of the stress-energy tensor $T^{\mu\nu}_{i}$
depends on the spin of the field, $i=0,\,1/2,\,1$,222In this work we consider
dark matter candidates which are either real scalars or a Dirac fermion. and
is given by

$\displaystyle T^{\mu\nu}_{0}$ $\displaystyle=$
$\displaystyle\partial^{\mu}S\partial^{\nu}S-g^{\mu\nu}\left[\frac{1}{2}\partial^{\alpha}S\partial_{\alpha}S-V(S)\right]\,,$
(3) $\displaystyle T^{\mu\nu}_{1/2}$ $\displaystyle=$
$\displaystyle\frac{i}{4}\left[\bar{\chi}\gamma^{\mu}\overset{\leftrightarrow}{\partial^{\nu}}\chi+\bar{\chi}\gamma^{\nu}\overset{\leftrightarrow}{\partial^{\mu}}\chi\right]$
(4)
$\displaystyle-g^{\mu\nu}\left[\frac{i}{2}\bar{\chi}\gamma^{\alpha}\overset{\leftrightarrow}{\partial_{\alpha}}\chi-
m_{\chi}\bar{\chi}\chi\right]\,,$ $\displaystyle T_{1}^{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[F^{\mu}_{\alpha}F^{\nu\alpha}+F^{\nu}_{\alpha}F^{\mu\alpha}-\frac{1}{2}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\right]\,,$
(5)

where $V(S)$ is the scalar potential for either the scalar dark matter or
inflaton, with $S\;=\;X,\phi$, and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
for a vector field, $A_{\mu}$. In Fig. 1, we show the $s$-channel exchange of
a graviton obtained from the Lagrangian (2) for the production of dark matter
from either the inflaton condensate or Standard Model fields. In addition, a
similar diagram exists for the production of Standard Model fields (during the
reheat process) from the inflaton condensate in the initial state.

Figure 1: Feynman diagram for the production of dark matter through the
gravitational scattering of the Standard Model particle bath or inflaton
condensate.

Although the direct coupling to the massless graviton appears to be feeble due
to Planck suppression, the energy available in the thermal bath during the
initial stage of reheating is large enough to make the gravitational
production rates significant.

The scattering amplitudes related to the production rate of the processes
$\phi/{\rm{SM}}^{i}(p_{1})+\phi/{\rm{SM}}^{i}(p_{2})\rightarrow{\rm{SM}}^{i}/X^{j}(p_{3})+{\rm{SM}}^{i}/X^{j}(p_{4})$
can be parametrized by

$\mathcal{M}^{ij}\propto
M_{\mu\nu}^{j}\Pi^{\mu\nu\rho\sigma}M_{\rho\sigma}^{i}\;,$ (6)

where ($i$, $j$) denotes the spin of the (initial,final) state involved in the
scattering process and $i,j=0,1/2,1$. $\Pi^{\mu\nu\rho\sigma}$ is the graviton
propagator for the canonical field $h$ with momentum $k=p_{1}+p_{2}$,

$\Pi^{\mu\nu\rho\sigma}(k)=\frac{\eta^{\rho\nu}\eta^{\sigma\mu}+\eta^{\rho\mu}\eta^{\sigma\nu}-\eta^{\rho\sigma}\eta^{\mu\nu}}{2k^{2}}\,.$
(7)

The partial amplitudes, $M_{\mu\nu}^{i}$, are given by

$\displaystyle M_{\mu\nu}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}-\eta_{\mu\nu}p_{1}\cdot
p_{2}-\eta_{\mu\nu}V^{\prime\prime}(S)\right]\,,$ (8) $\displaystyle
M_{\mu\nu}^{1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}{\bar{v}}(p_{2})\left[\gamma_{\mu}(p_{1}-p_{2})_{\nu}+\gamma_{\nu}(p_{1}-p_{2})_{\mu}\right]u(p_{1})\,,$
(9) $\displaystyle M_{\mu\nu}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\epsilon_{2}^{*}\cdot\epsilon_{1}\left(p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}\right)$
(10) $\displaystyle-$ $\displaystyle\epsilon_{2}^{*}\cdot
p_{1}\left(p_{2\mu}\epsilon_{1\nu}+\epsilon_{1\mu}p_{2\nu}\right)-\epsilon_{1}\cdot
p_{2}\left(p_{1\nu}\epsilon_{2\mu}^{*}+p_{1\mu}\epsilon_{2\nu}^{*}\right)$
$\displaystyle+$ $\displaystyle p_{1}\cdot
p_{2}\left(\epsilon_{1\mu}\epsilon_{2\nu}^{*}+\epsilon_{1\nu}\epsilon_{2\mu}^{*}\right)$
$\displaystyle+$ $\displaystyle g_{\mu\nu}\left(\epsilon_{2}^{*}\cdot
p_{1}\epsilon_{1}\cdot p_{2}-p_{1}\cdot
p_{2}\,\epsilon_{2}^{*}\cdot\epsilon_{1}\right)\bigg{]}\,,$

with analogous expressions for dark matter in terms of the dark matter
momenta, $p_{3},p_{4}$, and potential $V(X)$, if $X$ is a scalar. For an
initial state inflaton with $S=\phi$, we replace $M^{0}_{\mu\nu}$ with
$T^{0}_{\mu\nu}$ from Eq. (3). As we only consider vectors in the Standard
Model, their masses have been neglected in Eq. (10).

In what follows, we consider three distinct processes based on the diagram in
Fig. 1: for the production of dark matter, A) SM + SM $\rightarrow X+X$; B)
$\phi+\phi\rightarrow X+X$, where the latter involves the inflaton condensate
(zero mode) in the initial state rather than an initial state particle with
momentum $p_{1,2}$ (see below for more detail), and C) $\phi+\phi\rightarrow$
SM + SM, as a minimal and unavoidable contribution to the reheating process.

The dark matter production rate from SM fields can be readily calculated by
assuming that the initial particle states are massless. This assumption can be
justified by the fact that the energy associated with the momenta,
$p_{1}\,,p_{2}$ is extremely large at the end of inflation and dominates over
electroweak scale quantities.

The dark matter production rate $R(T)$ for the SM+SM $\rightarrow$ $X+X$
process with amplitude $\cal{M}$ 333It should be noted that we include the
symmetry factors associated with identical initial and final states in the
squared amplitude, $|{\cal{M}}|^{2}$. is

$R(T)=\frac{1}{1024\pi^{6}}\int
f_{1}f_{2}E_{1}\mathop{}\\!\mathrm{d}E_{1}E_{2}\mathop{}\\!\mathrm{d}E_{2}\mathop{}\\!\mathrm{d}\cos\theta_{12}\int|{\cal
M}|^{2}\mathop{}\\!\mathrm{d}\Omega_{13}\,,$ (11)

where $E_{i}$ denotes the energy of particle $i=1,2,3,4$, $\theta_{13}$ and
$\theta_{12}$ are the angles formed by momenta ${\bf{p}}_{1,3}$ and
${\bf{p}}_{1,2}$, respectively, and

$f_{i}=\frac{1}{e^{E_{i}/T}\pm 1}\,,$ (12)

represent the assumed thermal distributions of the incoming SM particles.

The total amplitude squared for the gravitational scattering process SM+SM
$\rightarrow$ $X_{j}+X_{j}$ is given by a sum of the three amplitudes
associated with different initial state spins,

$|\mathcal{M}|^{2}=4|\mathcal{M}^{0}|^{2}+45|\mathcal{M}^{1/2}|^{2}+12|\mathcal{M}^{1}|^{2}\,.$
(13)

These were calculated in Bernal:2018qlk  and it was found that the dark matter
production rate is given by

$R^{T}_{j}=R_{j}(T)=\beta_{j}\frac{T^{8}}{M_{P}^{4}}\,,$ (14)

where $j$ refers to the spin of $X$ (either 0 or 1/2), the constants
$\beta_{j}$ and details related to the computation of dark matter production
rate and the amplitude squared are given in Appendix A.

For the production of dark matter through the scattering of the inflaton
condensate we consider the time-dependent oscillation of a classical inflaton
field $\phi(t)$. Since our computation depends explicitly on inflaton
potential, we consider the $\alpha$-attractor T-model Kallosh:2013hoa  as a
specific example,

$V(\phi)\;=\;\lambda
M_{P}^{4}\left|\sqrt{6}\tanh\left(\frac{\phi}{\sqrt{6}M_{P}}\right)\right|^{k}\,,$
(15)

which can be expanded about the origin444It should be noted that our
discussion is general and not limited to T-models of inflation.

$V(\phi)=\lambda\frac{\phi^{k}}{M_{P}^{k-4}},\quad\phi\ll M_{P}\,.$ (16)

The time-dependent oscillating inflaton field can be parametrized as

$\phi(t)\;=\;\phi_{0}(t)\cdot\mathcal{P}(t)\,,$ (17)

where $\phi_{0}(t)$ is the time-dependent amplitude that includes the effects
of redshift and $\mathcal{P}(t)$ describes the periodicity of the oscillation.

To calculate the dark matter production rate, we combine the potential (16)
with Eq. (17), which leads to $V(\phi)=V(\phi_{0})\cdot\mathcal{P}(t)^{k}$. We
next expand the potential energy in terms of the Fourier modes Ichikawa:2008ne
; Kainulainen:2016vzv ; GKMO2

$V(\phi)=V(\phi_{0})\sum_{n=-\infty}^{\infty}{\cal P}_{n}^{k}e^{-in\omega
t}=\rho_{\phi}\sum_{n=-\infty}^{\infty}{\cal P}_{n}^{k}e^{-in\omega t}\,,$
(18)

where $\omega$ is the frequency of oscillation of $\phi$, given by GKMO2

$\omega=m_{\phi}\sqrt{\frac{\pi
k}{2(k-1)}}\frac{\Gamma(\frac{1}{2}+\frac{1}{k})}{\Gamma(\frac{1}{k})}.$ (19)

For scalar dark matter, we find that the particle production rate per unit
volume and unit time for an arbitrary value of $k$ is given by

$R^{\phi^{k}}_{0}=\frac{2\times\rho_{\phi}^{2}}{16\pi
M_{P}^{4}}\Sigma_{0}^{k}\,,$ (20)

where the factor of two accounts for the fact we produce two dark matter
particles per scattering, with

$\Sigma_{0}^{k}=\sum_{n=1}^{\infty}|{\cal
P}^{k}_{n}|^{2}\left[1+\frac{2m^{2}_{X}}{E_{n}^{2}}\right]^{2}\sqrt{1-\frac{4m_{X}^{2}}{E_{n}^{2}}}\,,$
(21)

where $E_{n}=n\omega$ is the energy of the $n$-th inflaton oscillation mode
and $m_{X}$ is the mass of the produced dark matter. A detailed calculation of
this rate is presented in Appendix B.

For the case $k=2$, we find that the particle production rate is given by

$R^{\phi^{2}}_{0}=\frac{2\times\rho_{\phi}^{2}}{256\pi
M_{P}^{4}}\left(1+\frac{m_{X}^{2}}{2m^{2}_{\phi}}\right)^{2}\sqrt{1-\frac{m_{X}^{2}}{m_{\phi}^{2}}}\,,$
(22)

where $m_{\phi}^{2}=V^{\prime\prime}(\phi_{0})$, and since
$\sum\mathcal{P}^{2}_{n}=\cos^{2}(m_{\phi}t)$, we find that only the second
Fourier mode in the sum contributes, with
$\sum|\mathcal{P}^{2}_{n}|^{2}=\frac{1}{16}$ and $E_{2}=2m_{\phi}$.555We note
that the rate calculated here differs from MO  by a factor of 8, because in
the latter the inflaton was treated as a particle and not a condensate
resulting in a difference by a factor of 2 in the applied symmetry factors. In
addition, the interaction considered there did not use a properly normalized
graviton resulting in a factor of 2 in the vertex and 16 in the rate.

For a fermionic dark matter candidate, we find the following rate

$R^{\phi^{k}}_{1/2}=\frac{2\times\rho_{\phi}^{2}}{4\pi
M_{P}^{4}}\frac{m_{X}^{2}}{m_{\phi}^{2}}\Sigma_{1/2}^{k}\,,$ (23)

where the factor of two accounts for the sum over the particle and
antiparticle final states, with

$\Sigma_{1/2}^{k}=\sum_{n=1}^{+\infty}|{\cal
P}_{n}^{k}|^{2}\frac{m_{\phi}^{2}}{E_{n}^{2}}\left[1-\frac{4m_{X}^{2}}{E_{n}^{2}}\right]^{3/2}\,.$
(24)

For the case $k=2$, we obtain

$R^{\phi^{2}}_{1/2}\simeq\frac{2\times\rho_{\phi}^{2}}{256\pi
M_{P}^{4}}\frac{m_{X}^{2}}{m^{2}_{\phi}}\left[1-\frac{m_{X}^{2}}{m_{\phi}^{2}}\right]^{3/2}\,.$
(25)

A detailed discussion related to the dark matter production rates through the
inflaton condensate scattering is given in Appendix B.

For the production of SM fields from inflaton oscillations, we follow the same
procedure, but replace the partial amplitude, $M_{\mu\nu}^{j}$, for dark
matter with the appropriate amplitude involving SM fields. Below, we consider
only the example of producing Higgs bosons, namely $\phi+\phi\rightarrow H+H$.

##  III Gravitational production of quanta

As we discussed in the previous section, the graviton can act as a portal
between the inflaton, SM fields and a potential dark matter candidate. As
outlined above we here consider three cases in detail:

  1. A.

The graviton portal between a thermal bath and dark matter. This is
essentially a gravitational freeze-in mechanism for the production of dark
matter.

  2. B.

The graviton portal between the inflaton and dark matter. In this case, the
inflaton directly populates the dark matter without the need of either the
thermal bath or a mediator between the SM and the dark matter candidate.

  3. C.

The graviton portal between the inflaton and the Standard Model sector to
produce a radiative bath at the start of reheating.

###  III.1 ${\rm SM~{}SM}\rightarrow h_{\mu\nu}\rightarrow{\rm DM~{}DM}$

The spin-2 portal for the production of dark matter was considered recently in
Bernal:2018qlk  for both massive and massless spin-2 fields. Here we restrict
our attention to the massless (graviton) portal. For an inflaton potential
with $k=2$, the scattering cross section between SM fields and dark matter is
proportional to $T^{2}/M_{P}^{4}$, and we expect the resulting dark matter
abundance to be primarily sensitive to the reheating temperature (rather than
the maximum temperature attained during the reheating process). Sensitivity to
$T_{\rm max}$ requires a cross section with a steep dependence on temperature,
$\sigma\propto T^{n}$, with $n\geq 6$. When $k>2$, sensitivity to $T_{\rm
max}$ requires only $n>(10-2k)/(k-1)$ when the primary reheating mechanism is
determined by inflaton decays as discussed below. Then, for example, when
$k=4$, when $n>2/3$, the dark matter abundance becomes sensitive to $T_{\rm
max}$. For the graviton portal, then, this occurs when $k\geq 3$.

The gravitational scattering of particles in the primordial plasma can produce
massive particles playing the role of a viable dark matter candidate $X$.
Then, the matter density $n_{X}$ obeys the classical Boltzmann equation666We
note that we include the relevant factors of 2 associated with identical
initial states in the definition of the particle production rate.

$\frac{dn_{X}}{dt}+3Hn_{X}=R^{T}_{X}\,,$ (26)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter. It is more convenient to
work with $a$ as dynamical parameter, rather than $t$ or $T$. Eq.(26) can then
be rewritten

$\frac{dn_{X}}{da}+3\frac{n_{X}}{a}=\frac{R^{T}_{X}(a)}{Ha}\,.$ (27)

Since the production rate $R^{T}_{X}$ is dependent on the initial state
energies, $i.e.$ of the temperature of the thermal bath, one needs the
expression of $T(a)$ to solve the Boltzmann equation in terms of the scale
factor. We explain the functional dependence of $R^{T}_{X}$ on $a$ below.
Defining the comoving number $Y_{X}=na^{3}$, we obtain

$\frac{dY_{X}}{da}=\frac{a^{2}R^{i}_{X}(a)}{H}\,.$ (28)

We assume an inflaton potential of the form given in Eq. (16). We next apply
the expressions for energy conservation for the inflaton density $\rho_{\phi}$
and the radiation density $\rho_{R}$

$\displaystyle\frac{d\rho_{\phi}}{dt}+3H(1+w_{\phi})\rho_{\phi}\simeq-(1+w_{\phi})\Gamma_{\phi}\rho_{\phi}$
(29)
$\displaystyle\frac{d\rho_{R}}{dt}+4H\rho_{R}\simeq(1+w_{\phi})\Gamma_{\phi}\rho_{\phi}\,.$
(30)

where $w_{\phi}=\frac{P_{\phi}}{\rho_{\phi}}=\frac{k-2}{k+2}$ GKMO2  is the
equation of state parameter. Here we assume that reheating primarily occurs
due to the inflaton effective coupling to the Standard Model fermions, given
by the Lagrangian

$\displaystyle\mathcal{L}^{y}_{\phi-SM}=-y\phi{\bar{f}}f\,,$ (31)

where $y$ is a Yukawa-like coupling and $f$ is a Standard Model fermion. The
width of $\phi$ is easily determined from the coupling (31)

$\Gamma_{\phi}=\frac{y^{2}}{8\pi}m_{\phi}\,.$ (32)

Note that for $k>2$, $m_{\phi}$ depends on $\phi$ and hence on the scale
factor $a$. We defined the inflaton energy density and pressure as

$\rho_{\phi}=\frac{1}{2}\langle\dot{\phi}^{2}\rangle+\langle
V(\phi)\rangle,~{}~{}~{}~{}P_{\phi}=\frac{1}{2}\langle\dot{\phi}^{2}\rangle-\langle
V(\phi)\rangle.$ (33)

We can solve Eqs. (29, 30) and obtain GKMO1 ; GKMO2

$\rho_{\phi}(a)=\rho_{\rm end}\left(\frac{a_{\rm
end}}{a}\right)^{\frac{6k}{k+2}}$ (34)

and

$\rho_{R}(a)=\rho_{\rm RH}\left(\frac{a_{\rm
RH}}{a}\right)^{\frac{6k-6}{k+2}}\frac{1-\left(\frac{a_{\rm
end}}{a}\right)^{\frac{14-2k}{k+2}}}{1-\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{14-2k}{k+2}}}\,,$ (35)

where these relations hold for $a_{\rm end}\ll a\ll a_{\rm{RH}}$. $a_{\rm
end}$ is a reference point marking the end of inflation. $\rho_{\phi}(a_{\rm
end})$ corresponds to the total energy density (there is virtually no
radiation at this point) when the slow-roll parameter $\epsilon=1$. At this
moment, $\rho_{\rm end}=\frac{3}{2}V(\phi_{\rm end})$ egno5 . Note that this
solution possesses a maximum for $\rho_{R}(a)$ (at $a=a_{\rm max}$). We have
also defined $\rho_{\rm RH}$ and $a_{\rm RH}$ such that $\rho_{R}(a_{\rm
RH})=\rho_{\phi}(a_{\rm RH})$. Since

$\rho_{R}=\frac{g_{T}\pi^{2}}{30}T^{4}\equiv\alpha T^{4}\,,$ (36)

where $g_{T}$ is the number of relativistic degrees of freedom at the
temperature, $T$. Thus, we have $\rho_{R}(a_{\rm max})=\alpha T_{\rm max}^{4}$
and $\rho_{R}(a_{\rm RH})=\alpha T_{\rm RH}^{4}$. The ratio of $a_{\rm max}$
to $a_{\rm end}$ is fixed and depends only on $k$ GKMO1

$\frac{a_{\rm max}}{a_{\rm
end}}=\left(\frac{2k+4}{3k-3}\right)^{\frac{k+2}{14-2k}}\,.$ (37)

Since we can express $T$ as function of the scale factor, $a$, with Eq. (35),
we can implement that relation in Eq. (14) to obtain $R_{X}^{T}$ as function
of $a$,

$R_{X}^{T}(a)=\beta_{X}\frac{\rho_{\rm
RH}^{2}}{\alpha^{2}M_{P}^{4}}\left(\frac{a_{\rm
RH}}{a}\right)^{\frac{12k-12}{k+2}}\left[\frac{1-\left(\frac{a_{\rm
end}}{a}\right)^{\frac{14-2k}{k+2}}}{1-\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{14-2k}{k+2}}}\right]^{2}.$ (38)

Using $H\simeq\frac{\sqrt{\rho_{\phi}(a)}}{\sqrt{3}M_{P}}$, which is valid for
$a\ll a_{\rm RH}$, Eq. (28) becomes

$\frac{dY_{X}}{da}=\frac{\sqrt{3}M_{P}}{\sqrt{\rho_{\rm
RH}}}a^{2}\left(\frac{a}{a_{\rm RH}}\right)^{\frac{3k}{k+2}}R_{X}^{T}(a)\,.$
(39)

The solution to this equation is

$\displaystyle n^{T}_{X}(a_{\rm RH})$ $\displaystyle=$
$\displaystyle\frac{\beta_{X}\sqrt{3}}{\alpha^{2}M_{P}^{3}}\frac{\rho_{\rm
RH}^{3/2}}{(1-(a_{\rm end}/a_{\rm
RH})^{\frac{14-2k}{k+2}})^{2}}\begin{cases}\frac{k+2}{6}\left(\frac{1}{3-k}+\frac{3}{k-1}\left(\frac{a_{\rm
end}}{a_{\rm
RH}}\right)^{\frac{14-2k}{k+2}}+\frac{(k-7)^{2}}{k^{3}+k^{2}-17k+15}\left(\frac{a_{\rm
end}}{a_{\rm RH}}\right)^{\frac{18-6k}{k+2}}-\frac{3}{k+5}\left(\frac{a_{\rm
end}}{a_{\rm RH}}\right)^{\frac{28-4k}{k+2}}\right)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad[k\neq 3]\\\
\\\ \\\ \ln\left(\frac{a_{\rm RH}}{a_{\rm
end}}\right)-\frac{5}{16}\left(3-4\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{8}{5}}+\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{16}{5}}\right)\qquad\qquad[k=3]\end{cases}$ (40)

where we integrated Eq. (39) between the values of the scale factor
corresponding to the end of inflation, $a_{\rm end}$, and the reheating
temperature (reached at $a_{\rm RH}$).

Writing the relic abundance book

$\Omega_{X}h^{2}=1.6\times 10^{8}\frac{g_{0}}{g_{\rm RH}}\frac{n(T_{\rm
RH})}{T_{\rm RH}^{3}}\frac{m_{X}}{1~{}{\rm GeV}},$ (41)

and inserting Eq. (40), we obtain

$\displaystyle\Omega^{T}_{X}h^{2}=\Omega_{k}\times\begin{cases}\frac{k+2}{6}\left(\frac{1}{3-k}+\frac{3}{k-1}\left(\frac{\rho_{\rm
RH}}{\rho_{\rm
end}}\right)^{\frac{7-k}{3k}}+\frac{(k-7)^{2}}{k^{3}+k^{2}-17k+15}\left(\frac{\rho_{\rm
RH}}{\rho_{\rm end}}\right)^{\frac{3-k}{k}}-\frac{3}{k+5}\left(\frac{\rho_{\rm
RH}}{\rho_{\rm
end}}\right)^{\frac{14-2k}{3k}}\right)~{}~{}~{}~{}~{}~{}~{}[k\neq 3]\\\ \\\
\frac{5}{18}\ln\left(\frac{\rho_{\rm end}}{\rho_{\rm
RH}}\right)-\frac{5}{16}\left(3-4\left(\frac{\rho_{\rm RH}}{\rho_{\rm
end}}\right)^{\frac{4}{9}}+\left(\frac{\rho_{\rm RH}}{\rho_{\rm
end}}\right)^{\frac{8}{9}}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[k=3]\par\end{cases}\,$
(42)

with

$\Omega_{k}=1.6\times 10^{8}\frac{g_{0}}{g_{\rm
RH}}\frac{\beta_{X}\sqrt{3}}{\sqrt{\alpha}}\frac{m_{X}}{1~{}\rm{GeV}}\frac{T_{\rm
RH}^{3}}{M_{P}^{3}}\left[1-\left(\frac{\rho_{\rm RH}}{\rho_{\rm
end}}\right)^{\frac{7-k}{3k}}\right]^{-2}\,,$ (43)

where $g_{0}=43/11$ and we take $g_{\rm RH}=427/4$ as the Standard Model
value.

We observe that, for a given reheating temperature, the relic abundance
decreases with $k$. Furthermore, whereas $\Omega_{X}^{T}h^{2}\propto T_{\rm
RH}^{3}$ for a quadratic potential, it becomes $\propto T_{\rm RH}^{2}$ for a
quartic potential, and even $\propto T_{\rm RH}$ for $k=6$. This comes from
the fact that the Hubble parameter, dominated by the evolution of the
inflaton, has a greater dependence on $T$ for larger values of $k$, slowing
down the production mechanism for large temperatures.

###  III.2 $\phi~{}\phi\rightarrow h_{\mu\nu}\rightarrow{\rm DM~{}DM}$

As noted earlier, it is also possible that the inflaton condensate can lead to
the direct production of dark matter through single graviton exchange MO .
Here, we generalize that result for $k\geq 2$. Having computed the production
rate in Eqs. (20) and (23) for scalar and fermionic dark matter respectively,
we can replace $R^{T}_{X}$ with the rates in Eq. (39). Then integrating

$\frac{dY_{X}}{da}=\frac{\sqrt{3}M_{P}}{\sqrt{\rho_{\rm
RH}}}a^{2}\left(\frac{a}{a_{\rm
RH}}\right)^{\frac{3k}{k+2}}R_{X}^{\phi^{k}}(a)$ (44)

between $a_{\rm end}$ and $a_{\rm RH}$ gives for scalar dark matter

$n_{0}^{\phi}(a_{\rm RH})=\frac{\sqrt{3}\rho_{\rm RH}^{3/2}}{8\pi
M_{P}^{3}}\frac{k+2}{6k-6}\left[\left(\frac{a_{\rm RH}}{a_{\rm
end}}\right)^{\frac{6k-6}{k+2}}-1\right]\Sigma^{k}_{0}$ (45)

which can be expressed as function of $\rho_{\rm end}$ using Eq. (34):

$n_{0}^{\phi}(a_{\rm RH})\simeq\frac{\sqrt{3}\rho_{\rm RH}^{3/2}}{8\pi
M_{P}^{3}}\frac{k+2}{6k-6}\left(\frac{\rho_{\rm end}}{\rho_{\rm
RH}}\right)^{1-\frac{1}{k}}\Sigma^{k}_{0},$ (46)

or

$\displaystyle\frac{\Omega_{0}^{\phi}h^{2}}{0.1}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{\rho_{\rm end}}{10^{64}{\rm
GeV}^{4}}\right)^{1-\frac{1}{k}}\left(\frac{10^{40}{\rm GeV}^{4}}{\rho_{\rm
RH}}\right)^{\frac{1}{4}-\frac{1}{k}}\left(\frac{k+2}{6k-6}\right)$ (47)
$\displaystyle\times$ $\displaystyle\Sigma_{0}^{k}\times\frac{m_{X}}{2.4\times
10^{\frac{24}{k}-7}{\rm GeV}}$

where we assumed $a_{\rm RH}\gg a_{\rm end}$. Note that the dependence on
$\rho_{\phi}$ used in Eq. (44) hides the fact that we considered a decaying
inflaton during the reheating.

For fermionic dark matter we obtained

$\displaystyle n_{1/2}^{\phi}(a_{\rm
RH})=\frac{m_{X}^{2}\sqrt{3}(k+2)\rho_{\rm
RH}^{\frac{1}{2}+\frac{2}{k}}}{12\pi
k(k-1)\lambda^{\frac{2}{k}}M_{P}^{1+\frac{8}{k}}}\left[\left(\frac{a_{\rm
RH}}{a_{\rm end}}\right)^{\frac{6}{k+2}}-1\right]\Sigma_{\frac{1}{2}}^{k}$
$\displaystyle\simeq\frac{m_{X}^{2}\sqrt{3}(k+2)\rho_{\rm
RH}^{\frac{1}{2}+\frac{2}{k}}}{12\pi
k(k-1)\lambda^{\frac{2}{k}}M_{P}^{1+\frac{8}{k}}}\left(\frac{\rho_{\rm
end}}{\rho_{\rm RH}}\right)^{\frac{1}{k}}\Sigma_{\frac{1}{2}}^{k}$ (48)

where we used

$m_{\phi}^{2}=V^{\prime\prime}(\phi_{0})=k(k-1)\lambda^{\frac{2}{k}}M_{P}^{2}\left(\frac{\rho_{\phi}}{M_{P}^{4}}\right)^{1-\frac{2}{k}}.$
(49)

We can simplify the expression to write

$\displaystyle\frac{\Omega_{1/2}^{\phi}h^{2}}{0.1}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{1/2}^{k}}{2.4^{\frac{8}{k}}}\frac{k+2}{k(k-1)}\left(\frac{10^{-11}}{\lambda}\right)^{\frac{2}{k}}\left(\frac{10^{40}{\rm
GeV}^{4}}{\rho_{\rm RH}}\right)^{\frac{1}{4}-\frac{1}{k}}$ (50)
$\displaystyle\times$ $\displaystyle\left(\frac{\rho_{\rm end}}{10^{64}{\rm
GeV}^{4}}\right)^{\frac{1}{k}}\left(\frac{m_{X}}{8.3\times
10^{6+\frac{6}{k}}{\rm GeV}}\right)^{3}$

Up until now, we have assumed that the thermal bath was produced via inflaton
decays. However, for low reheat temperatures, and hence small values of the
Yukawa-like inflaton coupling, $y$, it is possible that radiation, in the form
of Higgs bosons, is produced directly from the condensate via gravitational
interactions. This is considered in the next subsection.

###  III.3 $\phi~{}\phi\rightarrow h_{\mu\nu}\rightarrow{\rm SM~{}SM}$

The calculation for the production of SM fields produced by the scattering of
the inflaton via gravity is similar to the preceding calculation for dark
matter. As was shown in MO  and GKMO2 , there exists the possibility that the
thermal bath is produced not by inflaton decay but rather by inflaton
scattering after inflation. This occurs for instance for low values of $y$. In
this case, the maximum temperature is not given by the inflaton width, but by
the scattering process, whereas the final reheating (and thus $T_{\rm RH}$) is
still dominated by the decay. This is illustrated in Fig. 2 below. In fact,
the gravitational scattering $\phi\phi\rightarrow h_{\mu\nu}\rightarrow HH$ is
always present and can never be eliminated. Such a process generates an
effective coupling

${\cal L}_{h}=\sigma_{h}\phi^{2}H^{2}\,.$ (51)

From Eq. (A.23) of GKMO2 , we can write the left-hand side of Eq. (30) as

$(1+w)\Gamma_{\phi}\rho_{\phi}=N\frac{\sigma_{h}^{2}}{4\pi}\phi_{0}^{4}\omega\sum_{n=1}^{\infty}n|{\cal
P}_{n}^{k}|^{2}.$ (52)

where $N=4$ is the number of real scalars in the Standard Model, when we
neglect the Higgs mass. Identifying this rate with that in Eq. (20), and
$(1+w)\Gamma_{\phi}\rho_{\phi}=\omega R_{0}^{\phi^{k}}$, we deduce that

$\sigma_{h}=\frac{\rho_{\phi}}{8M_{P}^{2}\phi_{0}^{2}},$ (53)

for each real scalar. Thus for the Standard Model Higgs, and in the case $k=2$
we have

$\sigma_{h}=\frac{m_{\phi}^{2}}{16M_{P}^{2}}\simeq 9.8\times
10^{-12}\left(\frac{m_{\phi}}{3\times 10^{13}~{}{\rm GeV}}\right)^{2}.$ (54)

$\sigma_{h}$ can be considered as the lowest possible and inevitable value for
a quartic coupling between the inflaton and scalars. This may be important and
even dominate the reheating process at its earliest stages. We note that in a
theory with additional weak scale scalars such as the minimal supersymmetric
Standard Model (MSSM), the gravitational production is increased due to the
large number of scalars, $N=98$ in the MSSM. Note also that there is a minimal
gravitational production rate for the production of SM fermions and gauge
bosons though this is completely negligible due to the mass suppression (see
e.g. Eq. (23) for fermions). Thus if we restrict our attention to the Standard
Model, we take $N=4$ corresponding to the four real scalar degrees of freedom.

We now recompute the evolution of the radiation density using Eq. (30) and
(52),

$\frac{d\rho^{h}_{R}}{dt}+4H\rho^{h}_{R}=N\frac{\rho_{\phi}^{2}\omega}{16\pi
M_{P}^{4}}\sum_{n=1}^{\infty}n|{\cal P}_{n}^{k}|^{2}\,.$ (55)

The solution of (55) is

$\displaystyle\rho^{h}_{R}$ $\displaystyle=$ $\displaystyle
N\frac{\sqrt{3}M_{P}^{4}\gamma_{k}\Sigma^{h}_{k}}{16\pi}\left(\frac{\rho_{e}}{M_{P}^{4}}\right)^{\frac{2k-1}{k}}\frac{k+2}{8k-14}$
(56)
$\displaystyle\times\left[\left(\frac{a_{e}}{a}\right)^{4}-\left(\frac{a_{e}}{a}\right)^{\frac{12k-6}{k+2}}\right]$

with

$\gamma_{k}=\sqrt{\frac{\pi}{2}}k\frac{\Gamma(\frac{1}{2}+\frac{1}{k})}{\Gamma(\frac{1}{k})}\lambda^{\frac{1}{k}}$
(57)

and

$\Sigma^{h}_{k}=\sum_{n=1}^{\infty}n|{\cal P}_{n}^{k}|^{2}\,.$ (58)

Once again, there is a maximum temperature which can be determined by from the
value of $a_{\rm end}/a$ which maximizes Eq. (56),

$\frac{a_{\rm end}}{a_{\rm
max}}=\left(\frac{2k+4}{6k-3}\right)^{\frac{k+2}{8k-14}},$ (59)

and hence a maximum radiation density,

$\rho^{h}_{\rm
max}=N\frac{\sqrt{3}M_{P}^{4}\gamma_{k}\Sigma^{h}_{k}}{16\pi}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{2k-1}{k}}\frac{k+2}{12k-6}\left(\frac{2k+4}{6k-3}\right)^{\frac{2k+4}{4k-7}}$
(60)

For $k=2$ we have

$T_{\rm max}^{h}\simeq 3.0\times 10^{12}\left(\frac{\rho_{\rm
end}}{10^{64}~{}{\rm
GeV}^{4}}\right)^{\frac{3}{8}}\left(\frac{m_{\phi}}{3\times 10^{13}~{}{\rm
GeV}}\right)^{\frac{1}{4}}{\rm GeV}\,,$ (61)

where we have taken $N=4$ and scales as $N^{1/4}$. Furthermore, the sum
$\Sigma^{h}_{k}$ (58) begins at $n=2$, because 2 modes scatter, and the
initial mode has an energy of $2\omega$, which implies for $k=2$,

$\Sigma^{h}_{2}=2\times|{\cal
P}_{2}^{2}|^{2}=2\times\frac{1}{16}=\frac{1}{8}\,.$ (62)

It is important to stress the importance of Eqs. (60) and (61). These
correspond to an absolute lower bound on the maximal temperature of the
Universe. We have not made any assumption other than the existence of a
complex Higgs doublet and the inflaton coupled only through gravity. Our
calculation implies that the Universe must have passed through this (or a
higher) temperature during the early stages of reheating.

For $k=2$, the radiation density produced by inflaton scattering as computed
above never comes to dominate the energy density and can not lead to
reheating. Although scattering can lead to reheating if $k\geq 4$ GKMO2 .
Gravitational scattering is less efficient. The ‘quartic’ coupling defined in
Eq. (53) is only constant if $k=2$. In general, it scales as $\phi_{0}^{k-2}$.
Nevertheless, for $k>4$ reheating from gravitational scattering is possible,
though very inefficient. For example, for $k=6$, $T_{\rm RH}\lesssim 1$ eV. As
a result it is usually necessary to include a decay channel for the inflaton
as in Eq. (31).777Note that even including non-perturbative effects including
preheating, does not lead to reheating in the absence of a decay channel for
$k=2$ GKMOV . For a sufficiently large coupling, $y$, the radiation produced
by decay will always dominate over that produced by scattering as computed
above. In addition, the maximum temperature may be greater than the lower
bound in Eq. (61). However, there is a critical value of $y$, such that at
smaller couplings, the gravitational scattering process (52) dominates at some
point during the reheating process. This gives us the reheating temperature
below which the maximal temperature is fixed by (60), and is independent of
additional couplings beyond gravity between the inflaton and the standard
model sector. To determine the value of this critical coupling (and hence
reheating temperature), it is useful to rewrite Eq. (35) as

$\displaystyle\rho^{y}_{R}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}M_{P}^{4}\gamma_{k}^{3}y^{2}\Sigma^{y}_{k}}{8\pi}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{k-1}{k}}\lambda^{-\frac{2}{k}}\frac{k+2}{7-k}$
(63) $\displaystyle\times\left[\left(\frac{a_{\rm
end}}{a}\right)^{\frac{6k-6}{k+2}}-\left(\frac{a_{\rm
end}}{a}\right)^{4}\right]$

After some algebra, we found that the maximum of $\rho_{R}^{y}$ when evaluated
at $a_{\rm max}$ given by Eq. (37) is

$\displaystyle\rho_{\rm max}^{y}$ $\displaystyle=$
$\displaystyle\frac{y^{2}\gamma_{k}^{3}\sqrt{3}}{16\pi}\lambda^{\frac{-2}{k}}M_{P}^{4}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{1-\frac{1}{k}}\left(\frac{3k-3}{2k+4}\right)^{\frac{3k-3}{7-k}}$
(64) $\displaystyle\times\Sigma_{k}^{y}$

where

$\Sigma_{k}^{y}=\sum_{n=1}^{\infty}n^{3}|{\cal P}_{n}^{k}|^{2}\,.$ (65)

For $k=2$, the dominant mode is the first mode ($n=1$) which gives

$\Sigma_{2}^{y}=1^{3}\times|{\cal P}_{1}^{2}|^{2}=\frac{1}{4}$ (66)

The critical value for $y$ such that the maximum radiation density and
temperature are determined from the scattering of the inflaton condensate is
given by $\rho_{\rm max}^{y}<\rho_{\rm max}^{h}$ which leads to

$\displaystyle y^{2}$ $\displaystyle\lesssim$ $\displaystyle
N\frac{\lambda^{\frac{2}{k}}}{\gamma_{k}^{2}}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)\frac{\Sigma^{h}_{k}}{\Sigma^{y}_{k}}$ (67)
$\displaystyle\times\left(\frac{k+2}{12k-6}\right)\left(\frac{2k+4}{6k-3}\right)^{\frac{2k+4}{4k-7}}\left(\frac{2k+4}{3k-3}\right)^{\frac{3k-3}{7-k}}$

which gives for $k=2$ and $N=4$,

$y\lesssim 0.4\sqrt{\frac{\rho_{\rm end}}{M_{P}^{4}}}\simeq 6.9\times
10^{-6}\left(\frac{\rho_{\rm end}}{10^{64}{\rm GeV}^{4}}\right)^{\frac{1}{2}}$
(68)

or

$T_{\rm RH}\lesssim 3.0\times 10^{9}\left(\frac{\rho_{\rm end}}{10^{64}{\rm
GeV}^{4}}\right)^{1/2}\left(\frac{\lambda}{2.5\times
10^{-11}}\right)^{1/4}\,\rm{GeV}\,,$ (69)

where $T_{\rm RH}$ is defined by GKMO2

$\rho_{\phi}(a_{\rm RH})=\alpha T_{\rm
RH}^{4}=M_{P}^{4}\left(\frac{\sqrt{3}\gamma_{k}^{3}y^{2}\Sigma_{k}^{y}\lambda^{-\frac{2}{k}}(k+2)}{8\pi(7-k)}\right)^{k}\,.$
(70)

Thus for all models with a reheat temperature due to decays, which is less
than that given in Eq. (69), the maximum temperature during the reheat process
is determined by scattering (mediated by gravity) and thus can not be ignored.
Note also that for such small values of $y$, the kinetic effects due to the
effective mass induced by the coupling $y\phi\bar{f}f$ are non-existent, as
shown in GKMO2 .

We show in Fig. 2 the evolution of the energy densities of the inflaton
(blue), the radiation produced by inflaton decays (orange dashed), the
radiation produced by inflaton scattering mediated by gravity (green dashed),
and the total radiation density (red) as function of the scaling parameter
$a/a_{\rm end}$ for a Yukawa-like coupling $y=10^{-8}$ with $k=2$ and
$\rho_{\rm end}=10^{64}~{}\rm{GeV^{4}}$. We clearly see that the beginning of
the evolution of the radiation density is dominated by the scattering of the
inflaton via graviton exchange (orange line), which determines the maximum
temperature. For $k=2$, the radiation density from scattering falls as
$a^{-4}$ GKMO2 , whereas the density from decays falls more slowly as
$a^{-3/2}$ so that eventually the latter begins to dominate the population of
the thermal bath when $a=a_{\rm int}$, until the reheating is complete when
$\rho_{\phi}=\rho_{R}$ at $a=a_{\rm RH}$. For $a_{\rm int}\gg a_{\rm end}$, we
can approximate the cross-over point from Eqs. (56) and (63) using the
equality $\rho_{R}^{y}=\rho_{R}^{h}$. For sufficiently small $y$ and for
$k=2$, we find

$\frac{a_{\rm int}}{a_{\rm
end}}\simeq\left(\frac{8y^{2}\Sigma^{y}_{2}}{5N\Sigma^{h}_{2}}\frac{M_{P}^{4}}{\rho_{\rm
end}}\right)^{-\frac{2}{5}}\,,$ (71)

which gives $a_{\rm int}\simeq 430~{}a_{\rm end}$ in good agreement with the
numerical solution for the parameter choices used in Fig. 2. We stress that
the maximum temperature attained $T_{\rm max}\simeq 10^{12}$ GeV is
independent of any beyond the Standard Model physics, and is purely
gravitational and can not be ignored when production rates are highly
dependent on the ratio $T_{\rm max}/T_{\rm RH}$.

Figure 2: Evolution of the radiation density (red) and inflaton density (blue)
as a function of $a/a_{\rm end}$ for a Yukawa-like coupling $y=10^{-8}$ with
$\rho_{\rm end}=10^{64}$ GeV4 and $k=2$. This plot is obtained by solving
numerically equations (29), (30) and (55). The evolution of the radiation
density produced from inflaton decays (orange-dashed) and scattering mediated
by gravity (green-dashed) are also shown.

We can finally apply our result to the dark matter production through a
graviton exchange while the bath is also dominated by scattering of $\phi$
through graviton exchange. For $T_{\rm RH}\lesssim 10^{9}$ GeV, the Boltzmann
equation one needs to consider is

$\frac{dY_{X}^{h}}{da}=\frac{\sqrt{3}M_{P}}{\sqrt{\rho_{\rm
end}}}a^{2}\left(\frac{a}{a_{\rm end}}\right)^{\frac{3k}{k+2}}R_{X}^{h}(a)$
(72)

with

$R_{X}^{h}=\beta_{X}\frac{\rho_{\rm
max}^{2}}{\alpha^{2}M_{P}^{4}}\left(\frac{a_{\rm max}}{a}\right)^{8}\,.$ (73)

The result of the integration gives

$\displaystyle Y_{X}^{h}(a_{\rm int})$
$\displaystyle=\frac{N^{2}3\sqrt{3}M_{P}^{3}\beta_{X}\gamma_{k}^{2}(\Sigma_{k}^{h})^{2}}{\alpha^{2}65536\pi^{2}}\left(\frac{k+2}{8k-14}\right)^{2}$
$\displaystyle\times$ $\displaystyle\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{7k-4}{2k}}a_{\rm
end}^{3}\left[\left(\frac{k+2}{2k+10}\right)\left(1-\left(\frac{a_{\rm
end}}{a_{\rm int}}\right)^{\frac{2k+10}{k+2}}\right)\right.$ (74)
$\displaystyle+\left(\frac{k+2}{18k-18}\right)\left(1-\left(\frac{a_{\rm
end}}{a_{\rm int}}\right)^{\frac{18k-18}{k+2}}\right)$
$\displaystyle-\left.\left(\frac{k+2}{5k-2}\right)\left(1-\left(\frac{a_{\rm
end}}{a_{\rm int}}\right)^{\frac{10k-4}{k+2}}\right)\right]$

where $a_{\rm int}$ corresponds to the value of the scale factor when the
radiation density produced by inflaton decays begins dominate over that
produced by gravitational inflaton scattering (this only occurs if $y$
satisfies the bound in Eq. (67)). For $a>a_{\rm int}$, the slope of the
radiation energy density curve as a function of $a$ changes as seen in Fig. 2
and any thermal contribution to the production of dark matter originates from
inflaton decay.

For sufficiently small $y$, $a_{\rm int}\gg a_{\rm end}$, and Eq. (74) can be
simplified and we see that the dark matter yield does not depend on this
intermediate scale factor, but only on $a_{\rm end}$ and $\rho_{\rm end}$.
Thus for small $y$, we can also use Eq. (74) to evaluate the dark matter
density at $a=a_{\rm RH}$,

$\displaystyle n_{X}^{h}(a_{RH})$
$\displaystyle\simeq\frac{N^{2}\sqrt{3}\beta_{X}\gamma_{k}^{2}(\Sigma_{k}^{h})^{2}M_{P}^{3}}{196608\pi^{2}\alpha^{2}}\left(\frac{k+2}{8k-14}\right)^{2}$
$\displaystyle\times\frac{(k+2)(4k-7)^{2}}{(k-1)(k+5)(5k-2)}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{7k-4}{2k}}\left(\frac{\rho_{\rm RH}}{\rho_{\rm
end}}\right)^{\frac{k+2}{2k}}$

and the relic abundance

$\displaystyle\Omega^{h}_{X}h^{2}$ $\displaystyle=1.6\times
10^{8}\frac{g_{0}}{g_{\rm
RH}}\frac{m_{X}}{1~{}\rm{GeV}}\frac{\sqrt{3}\beta_{X}\gamma_{k}^{2}(\Sigma_{k}^{h})^{2}M_{P}^{3}}{196608\pi^{2}\alpha^{2}T_{\rm
RH}^{3}}\left(\frac{k+2}{8k-14}\right)^{2}$ (76)
$\displaystyle\times\frac{(k+2)(4k-7)^{2}}{(k-1)(k+5)(5k-2)}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{6k-6}{2k}}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{\frac{k+2}{2k}}$

Because the radiation produced by gravitational scattering dominates near
$a_{\rm max}$ only when $T_{\rm RH}$ satisfies Eq. (69), the relic density in
Eq. (76) is suppressed by $(\rho_{\rm RH}/M_{P})^{(k+2)/2k}$, and it never
dominates the gravitational production of dark matter given in Eq. (42),
though it can lead to important effects when non-gravitational production
modes with a strong dependence on temperature are considered.

##  IV Results

In the results presented below, we choose a class of inflation models, called
T-models given by Eq. (15) which take the form of Eq. (16) when expanded about
the origin. Given a specific potential, we can determined $\lambda$ from the
normalization of the CMB quadrupole anisotropy and $\rho_{\rm end}$ from the
condition $\epsilon=1$, as discussed earlier. Setting $y=10^{-7}$, for $k=2$,
we have $\lambda=2.5\times 10^{-11}$ and $\rho_{\rm end}^{1/4}=5.2\times
10^{15}$ GeV888Different values of $y$ give differences (at most) of 20% on
$\lambda$, and 5% on $\rho_{\rm end}^{1/4}$.. For $k=4$, $\lambda=3.3\times
10^{-12}$ and $\rho_{\rm end}^{1/4}=4.8\times 10^{15}\,{\rm GeV}$ whereas for
$k=6$, $\lambda=4.6\times 10^{-13}$ and $\rho_{\rm end}^{1/4}=4.6\times
10^{15}\,{\rm GeV}$. For more on the determination of these parameters, see
GKMO2 .

Given these (model-dependent) parameter values for $k=2,4,6$, we list in Table
1, the values for $T_{\rm max}$ which we obtain from $\rho^{h}_{\rm max}$ in
Eq. (60); the maximum coupling $y$ from Eq. (67) for which the gravitational
produced radiation with temperature $T_{\rm max}$ dominates over that produced
by decays; and the corresponding reheating temperature, ${T_{\rm RH}}_{\rm
max}$ obtained when $y=y_{\rm max}$ using Eq. (70) for $\rho_{\rm RH}$.
$T_{\rm max}\propto\lambda^{1/4k}$ depends weakly on the inflaton coupling,
and thus varies little for different values of $k$. The coupling $y_{\rm max}$
is independent of $\lambda$ and also varies little with k. However, the final
reheat temperature (which is not a result of purely gravitational
interactions) is very sensitive to $k$ as it scales as $y^{k/2}$ resulting in
very small reheating temperatures when $k=4$ or 6 for the small values of $y$
considered.

| $k=2$ | $k=4$ | $k=6$
---|---|---|---
$T_{\rm max}$ |  $1.0\times 10^{12}$ GeV |  $7.5\times 10^{11}$ GeV |  $6.5\times 10^{11}$ GeV
$y_{\rm max}$ | $1.8\times 10^{-6}$ | $1.4\times 10^{-6}$ | $1.1\times 10^{-6}$
${T_{\rm RH}}_{\rm max}$ | $7.9\times 10^{8}~{}{\rm GeV}$ | 470 GeV |  $9.7\times 10^{-4}$ GeV
Table 1: Lower bound on $T_{\rm max}$ generated by the process
$\phi\phi\rightarrow h_{\mu\nu}\rightarrow HH$ for different values of $k$.
The radiation from this gravitational scattering dominates when $y<y_{\rm
max}$ and we also list the corresponding reheating temperature $T_{\rm RH}$
when $y=y_{\rm max}$.

We show in Figs.(3) and (4) (for scalar and fermionic dark matter
respectively) the region in the parameter space defined by the ($m_{X}$,
$T_{\rm RH}$) plane for which we are able to obtain a relic abundance
consistent with the Planck CMB determination of the cold dark matter density,
$\Omega_{X}h^{2}=0.12$ Planck . We combine the dark matter density originating
from thermal production as given in Eq. (42) with that from scattering of the
condensate to scalars given in Eq. (47) or fermions in Eq. (50).

Figure 3: Points respecting Planck constraint $\Omega h^{2}=0.12$ in the case
of a scalar dark matter, in the plane ($m_{X}$, $T_{\rm RH}$) for different
values of $k$.
Figure 4: Points respecting Planck constraint $\Omega h^{2}=0.12$ in the case
of a fermionic dark matter, in the plane ($m_{X}$, $T_{\rm RH}$) for different
values of $k$.

For scalar dark matter, scattering in the condensate dominates the production
of dark matter and we see from Eq. (47) that an isodensity contour should obey
a simple power law, corresponding to $m_{X}\propto(T_{\rm
RH})^{\frac{4}{k}-1}$. Indeed, thermal production is not an efficient
mechanism for the scalar dark matter, and the unique mechanism which populates
the dark matter density is inflaton scattering (barring any beyond the
Standard Model contribution). To better understand this, we can compute the
ratio of the rates when $a=a_{\rm max}$, where the thermal production is
maximum. Comparing the rates in Eqs. (20) and (38)

$\displaystyle\frac{R^{\phi^{k}}_{0}(a_{\rm max})}{R^{T}_{0}(a_{\rm
max})}=\frac{\alpha^{2}\Sigma^{k}_{0}}{8\pi\beta_{0}}\left(\frac{3k-3}{2k+4}\right)^{\frac{6}{7-k}}\left(\frac{\rho_{\rm
end}}{\rho_{\rm RH}}\right)^{\frac{2}{k}}$ (77) $\displaystyle=g_{\rm
max}^{2}\frac{5760\Sigma^{k}_{0}}{3997}\left(\frac{3k-3}{2k+4}\right)^{\frac{6}{7-k}}\left(\frac{\rho_{\rm
end}}{\rho_{\rm RH}}\right)^{\frac{2}{k}}\gg 1\,,$

where $g_{\rm max}=427/4$ is the number of degrees of freedom at $a_{\rm max}$
in the Standard Model. Since $\rho_{\rm end}\gg\rho_{\rm RH}$, we clearly see
that the ratio is much greater than one. This implies that the gravitational
production is always dominated by the scattering of the inflaton zero modes.

Restricting our attention to Eq. (47) for the production of dark matter
scalars, we see that for $k=4$, something interesting happens. The relic
abundance is independent of the reheating process, and depends only on the
energy density at the end of inflation. This comes from the fact that for
increasing values of $k$, the production of dark matter is less efficient, and
from Eq. (46), we see that $n_{0}^{\phi}\propto T_{\rm RH}^{3}$ for $k=4$.
Dilution effects thus render the present abundance independent of $T_{\rm RH}$
and there is a unique universal limit of $m_{X}\lesssim 120$ GeV for scalar
dark matter and $\lesssim 1.7\times 10^{9}$ GeV for fermionic dark matter
(when inflaton scattering dominates). For $k>4$, the slope of $T_{\rm RH}$ vs
$m_{X}$ changes sign, and the required reheating temperature grows with the
dark matter mass. In this case, even sub-GeV dark matter candidates are
allowed for low reheating temperatures, whereas for $k=2$ and $k=4$ the
production process is too weak to produce MeV dark matter in sufficient
quantities to account for the cold dark matter density as determined by Planck
Planck .

The ($m_{X}$, $T_{\rm RH}$) plane for fermionic dark matter is shown in Fig.
4. In this case both the scattering from a condensate and thermal
gravitational contributions must be considered. Notice that there is a change
in slope between the required reheating temperature and dark matter mass. For
higher masses, the scattering from the condensate dominates as in the case of
scalar dark matter and we require $m_{X}\propto T_{\rm RH}^{\frac{(k-4)}{3k}}$
as can be seen from Eq. (50). However, at lower masses, because of the mass
suppression in the rate in Eq. (23) and hence the abundance of dark matter in
Eq.(48), there is a region where the thermal production dominates over
$\phi-\phi$ scattering. In this case, $m_{X}\propto T_{\rm RH}^{-3},T_{\rm
RH}^{-2},T_{\rm RH}^{-1}$, for $k=2,4$ and 6 respectively, as can be seen from
Eq. (42). The origin of this suppression is simply a helicity argument; the
scattering of two scalars generates rates where a spin-flip is required making
it proportional to the mass of the fermion in the final state. Thus the rate
vanishes for a massless fermion. This is not the case for thermal production,
because Standard Model particles in the thermal bath are relativistic and then
can still produce fermionic dark matter through scattering without being
affected by a helicity suppression. To be more quantitative, we again compare
the production rates in Eqs. (23) and (38) at $a=a_{\rm max}$

$\displaystyle\frac{R^{\phi^{k}}_{1/2}(a_{\rm max})}{R^{T}_{1/2}(a_{\rm
max})}=\frac{\alpha^{2}\Sigma^{k}_{1/2}}{2\pi\beta_{1/2}}\frac{m_{X}^{2}}{m_{\phi}^{2}}\left(\frac{3k-3}{2k+4}\right)^{\frac{6}{7-k}}\left(\frac{\rho_{\rm
end}}{\rho_{\rm RH}}\right)^{\frac{2}{k}}$ $\displaystyle=g_{\rm
RH}^{2}\frac{11520~{}\Sigma^{k}_{1/2}}{11351}\frac{m_{X}^{2}}{m_{\phi}^{2}}\left(\frac{3k-3}{2k+4}\right)^{\frac{6}{7-k}}\left(\frac{\rho_{\rm
end}}{\rho_{\rm RH}}\right)^{\frac{2}{k}}\,.$ (78)

In contrast to the scalar case, we see that there exists a value of
$m_{X}\lesssim(.13,.050,.036)(\rho_{\rm RH}/\rho_{\rm end})^{1/k}m_{\phi}$,
for $k=2,4$ and 6 respectively for which the relic abundance is dominated by
the thermal production.

##  V Conclusions

We have considered the production of matter and radiation interacting only
gravitationally with the inflaton through the exchange of a graviton
$h_{\mu\nu}$. We compared the production of dark matter from inflaton
scattering and from the thermal bath (mediated only by gravity). The former
tends to dominate the production in a large part of the parameter space.
However, we noticed a notable difference in the case of fermionic dark matter,
because the production through $\phi\phi$ scattering is suppressed by a mass
flip proportional to the dark matter mass $m^{2}_{X}$. We have also seen that
it is possible to produce radiation from inflaton scattering in the condensate
during the earlier stages of reheating. As a result, we have derived a lower
bound on the maximal temperature is expected from $\phi\phi\rightarrow
h_{\mu\nu}\rightarrow HH$ of the order of $10^{12}$ GeV for a typical chaotic
or $\alpha-$attractor scenario. This lower gravitational bound becomes the
effective maximal temperature for $T_{\rm RH}\lesssim 10^{9}$ GeV (for $k=2$.
As a conclusion, gravitational effects gives lower bounds on maximal
temperature and relic abundance that cannot be neglected and should be
considered as the minimal ingredients to add to any non-minimal extension of
the Standard Model. During the final phase of our work, a paper conducting a
similar analysis appeared Haque:2021mab . The results obtained are largely in
agreement with our own.

Acknowledgements.  The authors thank Marcos García and Kunio Kaneta for useful
discussions. This work was made possible by with the support of the Institut
Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle
Symposium 2021, with the support of the P2IO Laboratory of Excellence (program
“Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and
ANR-10-LABX-0038), the P2I axis of the Graduate School Physics of Université
Paris-Saclay, as well as IJCLab, CEA, IPhT, APPEC, the IN2P3 master projet
UCMN and 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.

## Appendix

###  A THERMAL PRODUCTION

In this appendix, we describe our calculation of the production rate for
scalar and fermionic dark matter, and include the amplitude squared for the
relevant processes. If we ignore the masses of Standard Model particles, the
rate $R(T)$ for the processes
${\rm{SM}}+{\rm{SM}}\rightarrow{\rm{DM}}_{j}+{\rm{DM}}_{j}$ can be computed
from

$R_{j}^{T}=\sum_{i=0,1/2,1}N_{i}R_{ij}=\sum_{i=0,1/2,1}\frac{N_{i}}{1024\pi^{6}}\int
f_{i}(E_{1})f_{i}(E_{2})E_{1}\mathop{}\\!\mathrm{d}E_{1}E_{2}\mathop{}\\!\mathrm{d}E_{2}\mathop{}\\!\mathrm{d}\cos\theta_{12}\int|{\cal
M}^{ij}|^{2}\mathop{}\\!\mathrm{d}\Omega_{13}=4R_{0j}+45R_{\frac{1}{2}j}+12R_{1j}\,,$
(79)

where $N_{i}$ denotes the number of each SM species of spin $i$: $N_{0}=4$ for
1 complex Higgs doublet, $N_{1}=12$ for 8 gluons and 4 electroweak bosons, and
$N_{1/2}=45$ for 6 (anti)quarks with 3 colors, 3 (anti)charged leptons and 3
neutrinos, c.f., Eq. (13). The infinitesimal solid angle is defined as

$\mathop{}\\!\mathrm{d}\Omega_{13}=2\pi\mathop{}\\!\mathrm{d}\cos\theta_{13}\,,$
(80)

with $\theta_{13}$ and $\theta_{12}$ being the angle formed by momenta
${\bf{p}}_{1,3}$ and ${\bf{p}}_{1,2}$, respectively. In the massless limit,
one can express the amplitude squared in terms of Mandelstam variables, $s$
and $t$, which are related to the angles $\theta_{13}$ and $\theta_{12}$ by
the expressions

$\displaystyle t\,=$
$\displaystyle\,\dfrac{s}{2}\left(\cos\theta_{13}-1\right)\,,$ (81)
$\displaystyle s\,=$
$\displaystyle\,2E_{1}E_{2}\left(1-\cos\theta_{12}\right)\,.$ (82)

The amplitudes and rates for scalar and fermionic dark matter are given in the
following subsections.

### Scalar dark matter

We note that we include the symmetry factors of the initial and final states
in the squared amplitudes, and indicate it with an overbar:

$\displaystyle|\overline{{\cal
M}}^{00}|^{2}=\frac{1}{4M_{P}^{4}}\frac{t^{2}(s+t)^{2}}{s^{2}}\,,$ (83)
$\displaystyle|\overline{{\cal
M}}^{\frac{1}{2}0}|^{2}=\frac{1}{4M_{P}^{4}}\frac{(-t(s+t))(s+2t)^{2}}{s^{2}}\,,$
(84) $\displaystyle|\overline{{\cal
M}}^{10}|^{2}=\frac{1}{2M_{P}^{4}}\frac{t^{2}(s+t)^{2}}{s^{2}}\,.$ (85)

Using these amplitudes in Eq. (79), we obtain Bernal:2018qlk

$\displaystyle
R^{T}_{0}=\frac{3997\pi^{3}}{41472000}\frac{T^{8}}{M^{4}_{P}}\equiv\beta_{0}\frac{T^{8}}{M^{4}_{P}}\,.$
(86)

### Fermionic dark matter

The corresponding amplitudes for fermionic dark matter are given by:

$|\overline{{\cal
M}}^{0\frac{1}{2}}|^{2}=\frac{(-t(s+t))(s+2t)^{2}}{4M_{P}^{4}s^{2}}\,,$ (87)
$|\overline{{\cal
M}}^{\frac{1}{2}\frac{1}{2}}|^{2}=\frac{s^{4}+10s^{3}t+42s^{2}t^{2}+64st^{3}+32t^{4}}{8M_{P}^{4}s^{2}}\,,$
(88) $|\overline{{\cal
M}}^{1\frac{1}{2}}|^{2}=\frac{(-t(s+t))(s^{2}+2t(s+t))}{M_{P}^{4}s^{2}}\,,$
(89)

which leads to the following rate Bernal:2018qlk

$\begin{split}&R^{T}_{\frac{1}{2}}=\frac{11351\pi^{3}}{20736000}\frac{T^{8}}{M_{P}^{4}}\equiv\beta_{\frac{1}{2}}\frac{T^{8}}{M^{4}_{P}}\,.\end{split}$
(90)

###  B INFLATON CONDENSATE SCATTERING

In this appendix, we describe our calculation of the particle production rate
of dark matter from the scattering of the inflaton condensate. If we consider
the gravitational scattering process $\phi(p_{1})+\phi({p_{2}})\rightarrow
X^{i}(p_{3})+X^{i}(p_{4})$, with $i=0,1/2$, illustrated by the Feynman diagram
in Fig. 1, the Boltzmann equation for the number density of produced dark
matter particles is given by Nurmi:2015ema ; GKMO2

$\frac{dn_{X}}{dt}+3Hn_{X}\;=\;R_{i}^{\phi^{k}}\,,$ (91)

where the rate is given by

$\displaystyle R_{i}^{\phi^{k}}\;\equiv\;g_{X}\int
d\Psi_{1,2,3,4}(2\pi)^{4}\delta^{(4)}\left(p_{1}+p_{2}-p_{3}-p_{4}\right)$
$\displaystyle~{}\times\Big{[}|\mathcal{M}|^{2}_{12\rightarrow{34}}f_{1}f_{2}(1\pm
f_{3})\left(1\pm f_{4}\right)-\left(34\leftrightarrow 12\right)\Big{]}\,$ (92)

where
$d\Psi_{1,2,3,4}=\Pi_{i=1}^{4}d^{3}{\bf{p}}_{i}/\left((2\pi^{3})2p^{0}_{i}\right)$
denotes the phase space distribution of particles $1,2,3$ and $4$,
$\mathcal{M}$ is the transition amplitude, $f_{i}$ is the phase space density
of species $i$, and $g_{X}$ denotes the number of produced dark matter
particles. If we ignore the Bose enhancement and Pauli blocking effects, the
above rate can be approximated as

$\displaystyle R_{i}^{\phi^{k}}$ $\displaystyle=$ $\displaystyle
g_{X}\int\frac{d^{3}{\bf{p}}_{3}}{(2\pi)^{3}2p_{3}^{0}}\frac{d^{3}{\bf{p}}_{4}}{(2\pi)^{3}2p_{4}^{0}}$
(93) $\displaystyle\times$
$\displaystyle(2\pi)^{4}\delta^{(4)}\left(p_{1}+p_{2}-p_{3}-p_{4}\right)|\mathcal{M}|^{2}_{12\rightarrow{34}}\,.$

For the inflaton condensate we can use the transition amplitude
$\mathcal{M}_{n}$ for each oscillating field mode of $\phi$. In this case, the
four-momentum of the $n$-th oscillation mode is given by
$p_{1}+p_{2}=p_{n}=\sqrt{s}=(E_{n},\bf{0})$ with $E_{n}$ the energy of the
$n$-th oscillation mode. Since the transition amplitude $\mathcal{M}_{n}$ of
the $n$-th oscillation does not depend on the final particle momenta
${\bf{p}}_{3,4}$, we can approximate the rate as

$R_{i}^{\phi^{k}}\;=\;\frac{g_{X}}{l!}\frac{1}{8\pi}\sum_{n=1}^{\infty}|\mathcal{M}_{n}|^{2}\sqrt{1-\frac{4m_{X}^{2}}{s}}$
(94)

where $l$ is associated with the number of identical particles in the final
state.

For the production of scalar dark matter, we find that the scattering
amplitude squared is given by

$|{\mathcal{M}}^{0\phi^{k}}_{n}|^{2}=\frac{\rho_{\phi}^{2}}{M_{P}^{4}}\left[1+\frac{2m_{X}^{2}}{s}\right]^{2}|({\mathcal{P}^{k}})_{n}|^{2}\,,$
(95)

where $s=E_{n}^{2}=n^{2}\omega^{2}$, and we used
$\rho_{\phi}=\frac{\lambda\phi^{k}}{M_{P}^{k-4}}$. We find that the inflaton
scattering rate is given by

$R^{\phi^{k}}_{0}=\frac{2\times\rho_{\phi}^{2}}{16\pi
M_{P}^{4}}\sum_{n=1}^{\infty}\left[1+\frac{2m_{X}^{2}}{E_{n}^{2}}\right]^{2}|({\mathcal{P}^{k}})_{n}|^{2}\langle\beta_{n}\left(m_{X}\,,m_{X}\right)\rangle\,,$
(96)

where

$\beta_{n}\left(m_{A},m_{B}\right)\equiv\sqrt{\left(1-\frac{\left(m_{A}+m_{B}\right)^{2}}{E_{n}^{2}}\right)\left(1-\frac{\left(m_{A}-m_{B}\right)^{2}}{E_{n}^{2}}\right)}\,,$
(97)

and we used $g_{X}=2$. For the case $k=2$, we find that the rate is given by
Eq. (22).

Similarly, for fermionic dark matter we find that the scattering amplitude
squared is,

$|{\mathcal{M}}^{1/2\,\phi^{k}}_{n}|^{2}=\frac{2\rho_{\phi}^{2}}{M_{P}^{4}}\frac{m_{X}^{2}}{s}\left[1-\frac{4m_{X}^{2}}{s}\right]|({\mathcal{P}^{k}})_{n}|^{2}\,,$
(98)

and the rate is given by Eq. (24).

The rates as defined in the text depend on various summations over the Fourier
modes of the periodicity function $\mathcal{P}(t)$. In Table 2, the numerical
values of these quantities are given for $k=2,4,6$. Values are given in the
limit of vanishing dark matter mass.

| $k=2$ | $k=4$ | $k=6$
---|---|---|---
$\Sigma_{0}^{k}$ (Eq. (21)) | $\frac{1}{16}$ | 0.063 | 0.056
$\Sigma_{1/2}^{k}$ (Eq. (24)) | $\frac{1}{64}$ | 0.061 | 0.101
$\Sigma_{k}^{h}$ (Eq. (58)) | $\frac{1}{8}$ | 0.126 | 0.124
$\Sigma_{k}^{y}$ (Eq. (65)) | $\frac{1}{4}$ | 0.241 | 0.244
Table 2: Numerical values of the various summations of the Fourier modes of
the periodicity functions used in the text. The dark matter mass has been
neglected in producing the numerical values.

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