Jump Starting the Dark Sector with a Phase Transition Michele Redi Andrea Tesi

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Jump Starting the Dark Sector

with a Phase Transition

Michele Redi, Andrea Tesi

INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy

Department of Physics and Astronomy, University of Florence, Italy

Abstract

We study the possibility to populate the dark sector through a phase transition. We

will consider secluded dark sectors made of gauge theories, Randall-Sundrum sce-

narios and conformally coupled elementary particles. These sectors have in common

the fact that the action is approximately Weyl invariant, implying that particle pro-

duction due to time dependent background is strongly suppressed. In particular no

signiﬁcant production takes place during inﬂation allowing to avoid strong isocurva-

ture constraints from CMB. As we will show, if the scale of inﬂation is large compared

to the dynamical mass scale, these sectors automatically undergo a phase transition

that in the simplest cases is controlled by the Hubble parameter. If the phase transi-

tion takes place during reheating or radiation the abundance obtained can be larger

than particle production and production from the SM plasma. For phase transitions

completing during radiation domination, the DM mass is predicted in the range 108

GeV while larger values are required for phase transitions occurring during reheating.

E-mail: michele.redi@fi.infn.it,andrea.tesi@fi.infn.it

arXiv:2210.03108v3 [hep-ph] 5 Sep 2024

1 Introduction

The possibility that Dark Matter (DM) is just gravitationally coupled to the visible sector is

certainly worrisome, since it comes with no obvious experimental and observational signatures,

but it is one that we have to start embracing seriously. If gravity is all we got, we should at

least explain how dark sectors can ever be populated and the observed DM relic abundance

reproduced. It would be remarkable if gravity itself is responsible for the DM genesis.

One unavoidable contribution to the dark sector abundance is the production through tree

level graviton exchange, also known as gravitational freeze-in [1,2]. This is eﬃcient if the

reheating temperature is close to the experimental bound from inﬂation TR|max ≈5×1015 GeV

and rapidly declines for smaller values. Another generic mechanism is particle production due

the time dependent background [3,4]. This is at work during inﬂation for minimally coupled

scalars and for massive vector ﬁelds [5] leading to masses as low as 10−5eV. In the ﬁrst case

this however comes at the price of strong constraints from isocurvature perturbations [6], while

for the vector it relies on the Stueckelberg mechanism for mass generation [7].

In this paper we introduce a new mechanism that allows secluded sectors to be populated

even in absence of any coupling to the Standard Model (SM) and to the inﬂaton. We consider

interacting dark sectors with a dynamical mass scale M, arising either from conﬁnement or

from spontaneous symmetry breaking. If the scale of inﬂation HI> M the sector is in the

unbroken phase during inﬂation and undergoes a phase transition during reheating or radiation

domination. Assuming no thermal population the energy gained from the phase transition, of

order M4, populates the dark sector whose lightest state is automatically stable and constitutes

the DM candidate. In the simplest scenarios we ﬁnd,

Ωh2

0.1≈TR

1012GeV M

108GeV2

, TR<pMMPl (1)

leading to heavy DM scenarios.

Production from a phase transition is particularly transparent for Weyl invariant sectors

because inﬂation automatically prepares the system in a false vacuum empty state. This is

attractive as it avoids strong isocurvature constraints from inﬂationary production. The only

relevant scales in the evolution are Hubble and Mso that the phase transition is triggered when

H∼M, a condition realized during reheating or radiation domination. This is complementary

to particle production due to the time dependent background but as we will see typically leads

to a larger abundance. The contribution can also dominate gravitational freeze-in depending on

the reheating temperature.

As examples we consider asymptotically free dark gauge theories, strongly coupled Conformal

Field Theories (CFT) with deformations (and their holographic realization in Randall-Sundrum

scenarios) and conformally coupled elementary scalars (with a second order instability). If the

scale of inﬂation is large compared to the mass all these scenarios undergo a phase transition

after inﬂation that populates the dark sector. The lightest state of the sector is automatically

stable providing a natural DM candidate. Moreover, these theories are approximately Weyl

invariant at high energies so that no signiﬁcant particle production happens during inﬂation.

The assumption of a Weyl invariant sector is not very restrictive being only violated by minimally

coupled scalars that are naturally associated to spontaneous breaking of a global symmetry.

The paper is organized as follows. In section 3we discuss in detail the phase transition

mechanism for a conformally coupled scalar with an instability. In this case one can follow

explicitly the dynamics of the scalar during the phase transition and determine the abundance.

We show in this simple example that the phase transition dominates particle production in

the expanding universe and gravitational freeze-in if TR<1013 GeV. In section 4we consider

gauge theories arguing that the conﬁnement phase transition can also successfully populate the

dark sector. We also study inﬂationary production that is controlled by the β−functions of

the theory. For strongly coupled CFTs we use the AdS/CFT correspondence to determine the

inﬂationary production. A potentially sizable contribution exists in this case that is associated

to the explicit breaking a conformal invariance of gravity. We then turn to the contribution from

the phase transition that can be determined using the dilaton eﬀective action. We summarise in

6. In appendix Awe derive analytically the abundance of conformally coupled scalar produced

in radiation domination.

2 Phase Transition Mechanism

Approximately Weyl invariant dark sectors such as gauge theories are very diﬃcult to produce

if they are only gravitationally coupled to the SM. In fact in the limit of exact Weyl invariance

time dependence of the background drops out from the classical equations of motions eliminating

particle production during and after inﬂation. Particle production is then controlled by the

explicit breaking of Weyl symmetry that eventually induces a mass scale M.

These sectors can however be produced gravitationally from the SM thermal bath through

tree level graviton exchange (also known as, gravitational freeze-in) [1,2]. This type of thermal

production is only eﬃcient for very large SM reheating temperature. As we will discuss, another

mechanism can be relevant due to the dynamics of the mass scale. In the case where the dark

sector does not thermalize among itself, it can gain an energy density proportional to M4when

Hubble becomes comparable with M.

In the rest of this section we give an outline of the two contributions, writing the general

scaling of the energy density produced in the two cases.

2.1 Gravitational freeze-in: tree level graviton exchange

After reheating of the SM to temperature TRdark sector states are produced through s-channel

graviton exchange from the SM plasma. This production is analogous to a UV dominated freeze-

in, since it originates from SM thermal initial states. In the approximation that TRis much

larger than SM thresholds and than the mass M, it was shown in [8] that the abundance of free

particles can be written in a completely general way as1

sGR

= 6 ×10−6McDTR

MPl 3

, TR> M , (2)

where cDis the central charge (cD= 4/3,4,16 for a conformally coupled scalar, Weyl fermion,

massive gauge ﬁeld respectively) of the approximate relativistic CFT that describes the dark

sector at these energies.

If interactions allow the dark sector to thermalize in the relativistic regime, it develops a

dark sector temperature TD, given by TD= 0.25(cD/gD)1/4(TR/MPl)3/4T, where Tis the SM

temperature and gDis the number of degrees of freedom. In such a case the abundance becomes

sGR,therm.

= 1.5×10−4MgDcD

gD3/4TR

MPl 9/4

,(3)

that is larger than for free particles.

2.2 Phase transition and particle-production

Gravitational freeze-in becomes ineﬃcient when TRis small, as it stems from the formulas,

leaving a practically empty dark sector at the onset of radiation domination. In such a case, the

dark sector is only characterized by the relative size of the Hubble parameter Hand the mass

scale M.

We wish to argue that when H∼Man energy density of order ∼M4becomes available

in the dark sector, which soon after starts to redshift as matter. Under this assumption the

abundance is found to be

sPT ∼MMin "M

MPl 3

,TRM

Pl #,(4)

independently of the details of inﬂation. The above formula captures the situation where critical

condition H≈Mhappens both during radiation or during reheating.

While the formula above is simply based on the argument that an energy density of order M4

becomes dynamical at a critical time, we will show that it represents at least two well deﬁned

mechanisms by which a dark sector can be populated: phase transitions and particle production

due to the expanding universe.

•Phase Transition. Thanks to the generation of the dynamical scale Mat the phase

transition, the dark sector gains an energy density ∆V≈M4. Since the UV contributions

from gravitational freeze-in and inﬂationary production are very suppressed, it is a good

approximation to consider the phase transition to happen practically at zero temperature.

In a cosmological scenario, the critical parameter in this case can be identiﬁed as H≈M.

1We report throughout the ratio of energy to entropy densities. The energy fraction is then given by Ωh2=

0.27/eVρ/s.

•Particle Production. Thanks to the expansion of the Universe, when H≈Mthe

Fourier modes of the quantum ﬁelds undergoing the phase transition (or simply acquiring

a mass term) experience a large deviation from adiabaticity. Such condition signals the

non-thermal production of non-relativistic particles that can be determined computing

Bogoliubov coeﬃcients.

In both cases the expected scaling is the one of eq. (4). However, as we will show explicitly

in a few cases, the contribution from the phase transition can be enhanced at weak coupling and

it can be the dominant source of energy allowing for a ‘jump start’ of secluded dark sectors.

3 Elementary conformal scalar

We ﬁrst consider a scalar ﬁeld with conformal coupling to the curvature. The generic renormal-

izable theory is described by the lagrangian

L=(∂µϕ)2

2−1

2µ2ϕ2+1

12ϕ2R−λ

4ϕ4.(5)

The coupling to curvature guarantees that classically for µ= 0 the action is invariant under

a Weyl transformations, gµν →Ω(x)2gµν and ϕ→Ω(x)−1ϕso that the scale factor can be

removed from the equations of motion. A related important fact is that the coupling to curvature

generates a positive mass squared 2H2

Iduring inﬂation. Weyl invariance is explicitly broken by

the mass term and the running of the couplings (see for example [9]).

We assume that the sector is decoupled from the SM, so that it can be populated only

through gravitational eﬀects. In this work, we wish to distinguish between quantum production

due to the non-adiabatic evolution of the Bunch-Davies vacuum, and production from a phase

transition. A second order phase transition is possible if µ2=−M2/2, while µ2=M2is the

other possible branch. This parametrization is chosen to denote with Mthe mass of the scalar

both in unbroken and broken phase.

3.1 Particle production from time dependent background

We start reviewing particle production in the scenario with no interactions and positive mass

term M2=µ2. In a expanding universe particle are produced due to the time dependence

of the background. This is controlled by the explicit breaking of Weyl invariance that for the

conformally coupled scalar is the mass term.

Upon rescaling the ﬁeld by the scale factor a,ϕ=v/a, the equation of motion in conformal

time takes the form

v′′

k(η) + ω2

k(η)vk(η)=0, ω2

k(η) = k2+M2a2(η) (6)

where Mis the physical mass today.

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JumpStartingtheDarkSectorwithaPhaseTransitionMicheleRedi,AndreaTesiINFNSezionediFirenze,ViaG.Sansone1,I-50019SestoFiorentino,ItalyDepartmentofPhysicsandAstronomy,UniversityofFlorence,ItalyAbstractWestudythepossibilitytopopulatethedarksectorthroughaphasetransition.Wewillconsidersecludeddarksectorsmad...

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Jump Starting the Dark Sector with a Phase Transition Michele Redi Andrea Tesi

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