PIUAN-2022-720FT UMNTH420222 FTPIMINN2226 Gravity as a Portal to Reheating Leptogenesis and Dark Matter

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PI/UAN-2022-720FT, UMN–TH–4202/22, FTPI–MINN–22/26

Gravity as a Portal to Reheating,

Leptogenesis and Dark Matter

Basabendu Barman,a,b Simon Cl´ery,cRaymond T. Co,dYann

Mambrini,cKeith A. Olived

aCentro de Investigaciones, Universidad Antonio Nari˜no

Carrera 3 este # 47A-15, Bogot´a, Colombia

bInstitute of Theoretical Physics, Faculty of Physics, University of Warsaw,

ul. Pasteura 5, 02-093 Warsaw, Poland

cUniversit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

dWilliam I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University

of Minnesota, Minneapolis, MN 55455, USA

E-mail: basabendu88barman@gmail.com,simon.clery@ijclab.in2p3.fr,

rco@umn.edu,yann.mambrini@ijclab.in2p3.fr,olive@physics.umn.edu

Abstract. We show that a minimal scenario, utilizing only the graviton as an intermediate

messenger between the inﬂaton, the dark sector and the Standard Model (SM), is able to gen-

erate simultaneously the observed relic density of dark matter (DM), the baryon asymmetry

through leptogenesis, as well as a suﬃciently hot thermal bath after inﬂation. We assume an

inﬂaton potential of the form V(φ)∝φkabout the minimum at the end of inﬂation. The pos-

sibility of reheating via minimal gravitational interactions has been excluded by constraints

on dark radiation for excessive gravitational waves produced from inﬂation. We thus extend

the minimal model in several ways: i) we consider non-minimal gravitational couplings–this

points to the parameter range of DM masses MN1'2−10 PeV, and right-handed neutrino

masses MN2'(5 −20) ×1011 GeV, and TRH .3×105GeV (for k≤20); ii) we propose

an explanation for the PeV excess observed by IceCube when the DM has a direct but small

Yukawa coupling to the SM; and iii) we also propose a novel scenario, where the gravitational

production of DM is a two-step process, ﬁrst through the production of two scalars, which

then decay to fermionic DM ﬁnal states. In this case, the absence of a helicity suppression

enhances the production of DM and baryon asymmetry, and allows a great range for the

parameters including a dark matter mass below an MeV where dark matter warmness can

be observable by cosmic 21-cm lines, even when gravitational interactions are responsible for

reheating. We also show that detectable primordial gravitational wave signals provide the

opportunity to probe this scenario for TRH .5×106GeV in future experiments, such as

BBO, DECIGO, CE and ET.

arXiv:2210.05716v2 [hep-ph] 1 Dec 2022

Contents

1 Introduction 1

2 The framework 3

3 Gravitational production of RHNs 6

3.1 Gravitational dark matter 7

3.2 Gravitational leptogenesis 11

3.3 Gravitational reheating temperature 12

3.4 Non-minimal gravitational production 14

3.5 Gravitational waves generated during inﬂation 17

4 Results and discussion 19

4.1 A stable DM candidate 19

4.2 The case for a decaying gravitational DM & IceCube events 23

5 Dark matter & leptogenesis with a Majoron 25

6 Conclusions 28

1 Introduction

Since the ﬁrst calculation of the mass of the Milky Way by Henry Poincar´e in 1906 [1],

and the conclusion that the “dark matter is present in much greater amount than luminous

matter” by Fritz Zwicky in 1933 [2], the virial method has been used frequently to compute

the amount of dark matter (DM) in the Universe. The presence of dark matter is generally

deduced from its gravitational eﬀects. The precise abundance of DM is obtained from ob-

servations of the anisotropy spectrum of the cosmological microwave background (CMB) [3].

Initially, a neutrino, or more generally a heavy neutral lepton was thought to be an ideal

dark matter candidate [4]. This candidate was assumed to interact weakly with the Stan-

dard Model (SM) and required a GeV scale mass to satisfy relic abundance constraints [5,6].

Generalizations of the heavy neutrino DM candidate are referred to as WIMPs (weakly in-

teracting massive particles). Perhaps the best studied WIMP is the lightest supersymmetric

particle in supersymmetric extensions of the SM [7]. Other well studied candidates include

those generated by a Higgs-portal [8–18] or a Z-portal [19,20]. However, these minimal con-

structions are now heavily constrained (see, for example, Refs. [21,22] for reviews), and even

extensions to Z0-portal [23–27] are in tension with the electroweak nature of dark matter.

The WIMP relic density is often determined by thermal freeze-out. WIMPs are assumed

to be in thermal equilibrium at temperatures higher than the mass, and the relic density is

determined by the equilibrium density when DM annihilations can no longer keep up with

the expansion rate of the Universe. It is also possible that particles with interactions much

weaker than electroweak interactions and are never fully in equilibrium, but are nonetheless

produced in the thermal bath after inﬂation. An example of such a candidate is the gravitino

[7,28–30]. More generally, these Feebly Interacting Massive Particles (FIMPs) have been

proposed [31–33] as an alternative to WIMPs (see [34] for a recent review). In this framework,

– 1 –

the DM candidate is never thermalized due its extremely weak coupling to the SM, so weak

that they evade the current accelerator constraints.

In the early Universe, FIMPs can be produced from either the decay or annihilation

of states in the visible sector. When the SM temperature becomes smaller than the typical

mass scale of the interaction (i.e. the maximum of the DM and the mediator mass), the

generation process becomes suppressed, leaving a constant comoving DM number density.

Such a scenario is often referred as the freeze-in mechanism [32]. In contrast to the “WIMP-

miracle” which produces the observed relic density with near weak-scale couplings and masses,

a “FIMP-miracle” occurs when one considers renormalizable couplings of order ∼ O(10−11)

independent of the mass of the DM. If a priori such couplings seem unatural, UV versions

of the freeze-in mechanism may invoke eﬀective couplings, suppressed by a large mass scale

above the temperature of the thermal bath. This can be achieved via non-renormalizable

operators [35], suppressed by a high mass scale, e.g., in models where the mediators between

the visible sector and the dark sectors are very massive. This is the case in uniﬁed theories

like SO(10) with a heavy Z0gauge boson [33,36,37], moduli ﬁelds [38], high scale SUSY [39–

43] or heavy spin-2 constructions [44]. In other examples, freeze-in of DM may proceed via

loops [45,46] or 4-body ﬁnal states [47]. All of these scenarios are particularly interesting, as

the DM yield is sensitive to the highest temperature Tmax reached by the SM plasma [48–53],

controlled by the dynamics of the inﬂaton decay.

Even feebler interactions are possible when the only eﬀective coupling at the UV scale

is gravity. Indeed, the minimal irreducible interaction that should exist between DM and

the Standard Model (SM) is mediated by graviton exchange [44,54–71] which can lead to

the observed amount of DM through the scattering of the particles in the thermal bath or

directly through the gravitational transfer of the energy stored in the inﬂaton condensate, as

already been discussed in detail Refs. [65–69].

DM requires an extension to the SM, but it is not the only reason why an extension

is necessary. As is well known, the visible or baryonic matter content of the Universe is

asymmetric. One interesting mechanism to produce the baryon asymmetry of the Universe

(BAU) via the lepton sector physics is known as leptogenesis [72], where, instead of creat-

ing a baryon asymmetry directly, a lepton asymmetry is generated ﬁrst and subsequently

gets converted into baryon asymmetry by the (B+L)-violating electroweak sphaleron tran-

sitions [73]. In thermal leptogenesis [74–77], the decaying particles, typically right-handed

neutrinos (RHNs), are produced thermally from the SM bath. However, the lower bound

on the RHN mass in such scenarios (known as the Davidson-Ibarra bound), leads to a lower

bound on the reheating temperature TRH &1010 GeV [78] so that the RHNs can be produced

from the thermal bath. One simpler alternative is the non-thermal production of RHNs [79–

83] originating from the decay of inﬂaton. This interaction is necessarily model dependent

as it depends on the Yukawa interaction between the inﬂaton and the RHNs.

In addition to providing the DM abundance, gravitational interactions can also be the

source of baryogenesis. As shown in [84], it is possible to have a model-independent theory

of non-thermal production of RHNs from inﬂation, once the inﬂaton potential is speciﬁed.1

The abundance of RHNs is calculated in the same manner as the dark matter abundance

and can lead to observed BAU from the out-of-equilibrium CP violating decay of the RHNs,

produced during the reheating epoch.

1The simultaneous generation of gravitational DM and the baryon asymmetry was also discussed in [85].

Our results diﬀer, as their choices of parameters are in conﬂict with the tensor-to-scalar ratio bound from

Planck.

– 2 –

As noted above, the DM and RHNs may be produced through gravitational interactions

emanating from the thermal bath or directly from the inﬂaton condensate. It has also

been argued that the thermal bath itself may be generated from gravitational interactions

[68,84,86]. However, reheating the Universe from graviton exchange processes alone requires

a steep inﬂaton potential during reheating, resulting in a low reheating temperature and

a massive enhancement of tensor modes after inﬂation. Hence, the minimal scenario of

gravitational reheating is excluded by an excessive generation of dark radiation in the form

of gravitational waves (GWs) during BBN, as already noted in [87]. This limitation of

minimal gravitational reheating is one motivation to introduce, as a natural generalization,

non-minimal couplings of ﬁelds with gravity.

Motivated by these arguments, we derive a simultaneous solution for the DM abun-

dance, the baryon asymmetry, and the origin of the thermal bath from purely gravitational

interactions. In this sense, our scenario can be considered as the most minimal possible,

since we do not introduce any new interactions for any process beyond the SM, except for

gravity. The only new ﬁelds required are the dark matter candidate and the RHNs (which are

anyway needed for the generation of neutrino masses). Our only model dependence comes

from the choice of the particular inﬂaton potential. However we are mostly sensitive to the

shape of the potential about the minimum after inﬂation. To be deﬁnite, we adopt the class

of inﬂationary models called T-models [88]. But, as will be shown, even this dependence

proves to be weak when it comes to combining the constraints of reheating, baryogenesis,

and the dark matter relic density. We further show that the present framework can give rise

to a detectable inﬂationary GW background, that in turn excludes the minimal gravitational

reheating scenario which leads to an excess of the present-day GW energy density, in conﬂict

with the BBN prediction. However, a large part of the parameter space still remain within

the reach of several futuristic GW detection facilities.

The paper is organized as follows. After presenting our framework in Sec. 2, we re-

view the gravitational production from inﬂaton scattering and the thermal bath in Sec. 3,

where we also discuss the eﬀect of non-minimal gravitational interactions. We derive the

set of parameters (dark matter mass, RHN mass, and reheating temperature, TRH, which

simultaneously provide the correct relic density and BAU in Sec. 4. If the dark matter is

not absolutely stable, we are able to propose an explanation for the PeV events observed at

IceCube in the case of a long-lived candidate. Finally, we propose a novel scenario where the

gravitational production is a two-step process passing through a scalar singlet which couples

with the RHN sector in Sec. 5, before concluding in Sec. 6.

2 The framework

If the metric is expanded around Minkowski space-time: gµν 'ηµν +2hµν

MP, then the gravita-

tional interactions are described by the Lagrangian [89,90]

√−gLint =−1

hµν Tµν

SM +Tµν

φ+Tµν

X,(2.1)

where φis the inﬂaton and Xis a particle which does not belong to the SM.2In the present

context we consider Xto be a spin 1/2 Majorana fermion which can be associated with the

2MP= (8πGN)−1/2'2.4×1018 GeV is the reduced Planck mass.

– 3 –

dark matter or a right-handed neutrino. The graviton propagator for momentum pis

Πµνρσ(p) = ηρν ησµ +ηρµησν −ηρσηµν

2p2.(2.2)

The form of the stress-energy tensor Tµν

idepends on the spin of the ﬁeld and, for Majorana

spin-1/2 fermions, takes the form

Tµν

1/2=i

8¯χγµ↔

∂νχ+ ¯χγν↔

∂µχ−gµν i

4¯χγα↔

∂αχ−mχ

2χcχ,(2.3)

whereas for a scalar ϕ,

Tµν

0=∂µϕ∂νϕ−gµν 1

2∂αϕ ∂αϕ−V(ϕ).(2.4)

There are of course many possible scalar potentials V(φ) which can account for inﬂation.

However, the calculations relevant in this paper are largely independent of the potential

during inﬂation and depend only on the shape of the potential about the minimum. Without

loss of generality, we will assume that V(φ) is among the class of α-attractor T-models [88]

V(φ) = λM4

P√6 tanh φ

√6MP

,(2.5)

which can be expanded about the origin3

V(φ) = λφk

Mk−4

;φMP.(2.6)

In this class of models, inﬂation occurs at large ﬁeld values (φ>MP), and after the

period of exponential expansion, the inﬂaton begins to oscillate about the minimum and the

process of reheating begins. ]The end of inﬂation may be deﬁned when ¨a= 0 where ais the

cosmological scale factor. The inﬂaton ﬁeld value at that time is given by [52,91] as

φend 'r3

8MPln 1

2+k

3k+pk2+ 3.(2.7)

It is easy to show that at the end of inﬂation, the condition ¨a= 0 is equivalent to ˙

φ2

end =

V(φend) and thus the inﬂaton energy density at φend is ρend =3

2V(φend). The overall scale

of the potential parameterized by the coupling λ, can be determined from the amplitude of

the CMB power spectrum AS,

λ'18π2AS

6k/2N2,(2.8)

where Nis the number of e-folds measured from the end of inﬂation to the time when the

pivot scale k∗= 0.05 Mpc−1exits the horizon. In our analysis, we use ln(1010AS) = 3.044

[92] and set N= 55. This leads to an inﬂaton mass of mφ'1.2×1013 GeV for k= 2. More

generically, mφ'1.2×1013 is also the inﬂaton mass at the end of inﬂation for any larger

3Our discussion is general and not limited to T-models of inﬂation as the way we express the minimum of

the potential is generic.

– 4 –

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PI/UAN-2022-720FT,UMN{TH{4202/22,FTPI{MINN{22/26GravityasaPortaltoReheating,LeptogenesisandDarkMatterBasabenduBarman,a;bSimonClery,cRaymondT.Co,dYannMambrini,cKeithA.OlivedaCentrodeInvestigaciones,UniversidadAntonioNari~noCarrera3este#47A-15,Bogota,ColombiabInstituteofTheoreticalPhysics,FacultyofP...

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