Scalar overproduction in standard cosmology and predictivity of nonthermal dark matter

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Scalar overproduction in standard cosmology

and

predictivity of non–thermal dark matter

Oleg Lebedev

Department of Physics and Helsinki Institute of Physics,

Gustaf H¨allstr¨omin katu 2a, FI-00014 Helsinki, Finland

Abstract

Stable scalars can be copiously produced in the Early Universe even if they have no coupling

to other ﬁelds. We study production of such scalars during and after (high scale) inﬂa-

tion, and obtain strong constraints on their mass scale. Quantum gravity-induced Planck-

suppressed operators make an important impact on the abundance of dark relics. Unless

the corresponding Wilson coeﬃcients are very small, they normally lead to overproduction

of dark states. In the absence of a quantum gravity theory, such eﬀects are uncontrollable,

bringing into question predictivity of many non-thermal dark matter models. These con-

siderations may have non-trivial implications for string theory constructions, where scalar

ﬁelds are abundant.

arXiv:2210.02293v1 [hep-ph] 5 Oct 2022

Contents

1 Introduction 1

2 Decoupled scalar production during inﬂation 2

2.1 Longinﬂation................................... 3

2.2 Shortinﬂation................................... 7

3 Decoupled scalar production after inﬂation 7

3.1 Backgrounddynamics .............................. 8

3.2 Decoupled scalar production in the inﬂaton oscillation epoch . . . . . . . . . 9

3.2.1 Perturbative estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Quantum gravity eﬀects 12

5 Freeze-in production 15

6 Implications and discussion 16

6.1 Quantum gravity eﬀects on non-thermal dark matter . . . . . . . . . . . . . 16

6.2 Suppressing inﬂationary particle production . . . . . . . . . . . . . . . . . . 17

6.3 Implications for models of inﬂation: an example . . . . . . . . . . . . . . . . 18

7 Conclusion 20

1 Introduction

Inﬂationary cosmology [1, 2, 3] addresses many conceptual challenges of modern physics.

It has become an integral part of the cosmological standard model, which assumes a pe-

riod of inﬂation, followed by the inﬂaton oscillation epoch, reheating and a long period

of radiation–dominated Universe evolution. These ingredients are suﬃcient to explain the

observed structure of the Universe [4].

In this work, we study constraints on stable scalars in standard cosmology. Such ﬁelds

are produced in the Early Universe by various mechanisms, which can lead to overabundance

of dark relics. Throughout this work, we make the following assumptions:

•high scale inﬂation

•existence of a stable scalar with mass below the inﬂationary Hubble rate and the

inﬂaton mass

•the scalar is minimally coupled to gravity and has a very weak or no coupling to other

ﬁelds and itself

We consider both inﬂationary and postinﬂationary particle production. These mechanisms

are eﬃcient unless the scalar is very heavy. The scalar ﬁeld is assumed to be very weakly

coupled, possibly decoupled from the observable and inﬂaton sectors, while its self-interaction

is small enough such that it does not reach thermal equilibrium. A special case of a dark

relic of this type is non–thermal dark matter (DM).

Figure 1: Typical dark relic production mechanisms: (from left to right) via inﬂation,

inﬂaton oscillations, perturbative inﬂaton decay, thermal emission.

Very weakly interacting particles and non–thermal dark matter, in particular, have

memory in the sense that their abundance is additive and accumulates over diﬀerent stages

of the Universe evolution. As illustrated in Fig. 1, the most common production mechanisms

are provided by (1) inﬂation, (2) inﬂaton oscillations, (3) perturbative inﬂaton decay, and

(4) freeze–in type thermal emission. These are all very eﬃcient even at tiny values of the

couplings. If the scalar is completely decoupled from other ﬁelds, gravitational particle pro-

duction still takes place during and after inﬂation, leading to strong constraints on its mass

scale.

Quantum gravity eﬀects play an important role in these considerations. Such eﬀects are

thought to generate (gauge-invariant) couplings between the diﬀerent sectors of the model,

typically in the form of higher dimensional Planck–suppressed operators. We show that

these operators make an important impact on the abundance of dark relics, as long as the

inﬂaton ﬁeld value is large at the end of inﬂation. This brings into question predictivity of

non–thermal dark matter unless such operators are well under control. The latter is only

possible if a UV complete model of gravity is available.

The goal of this work is to obtain constraints on properties of dark feebly interacting

relics assuming high scale inﬂation. These constraints turn out to be quite strong, especially

taking into account quantum gravity eﬀects. While we focus mostly on scalar ﬁelds, some

aspects of our study easily generalize to fermions (Sec. 6.1).

2 Decoupled scalar production during inﬂation

Gravitational particle production has been the subject of many research works [5, 6, 7, 8, 9].

Even if a given ﬁeld has no couplings to other ﬁelds, it can be produced by gravity due to

the space-time expansion. The latter creates the necessary non-adiabaticity if the Hubble

rate is high enough. As a result, particles with sub-Hubble masses are abundantly produced,

possibly causing cosmological problems [12, 13].1

Particle creation can also be formulated in terms of the ﬁeld ﬂuctuations characteristic

of the de Sitter phase. This is done most conveniently using the corresponding equilibrium

probability distribution of Starobinsky and Yokoyama [14], which is the path we take in

1Some amount of heavy particles with masses above the Hubble rate is also generated [10, 11].

this work. In Refs. [15, 16, 17], this approach has been used to compute the generated dark

matter abundance.

Let us study inﬂationary scalar production following Ref. [17]. Consider a real scalar s

with the potential

V(s) = 1

2m2

ss2+1

4λss4,(1)

which has no couplings to other ﬁelds apart from gravity. Suppose its self-coupling is weak

and it is eﬀectively massless during inﬂation,

λs1, msH . (2)

In this case, the scalar experiences large quantum ﬂuctuations induced by the Hubble ex-

pansion. The equilibrium distribution of the ﬁeld is given by the probability density [14]

P(s)∝exp −8π2V(s)/(3H4).(3)

This equilibrium is reached on the time scale (√λsH)−1[14]. One can then distinguish two

cases depending on the weakness of the coupling: the inﬂation period is long enough for the

ﬂuctuations to equilibrate and the inﬂation period is too short such that equilibrium never

sets in.

2.1 Long inﬂation

In this case, one can read oﬀ the scalar ﬂuctuation size from the equilibrium distribution.

At the end of inﬂation, the variance of sis given by

hs2i ' 0.1×H2

end

√λs

,(4)

where Hend is the Hubble rate at the end of inﬂation. In other words, the scalar develops

a condensate whose value is at least of the order of the Hubble rate and can be far greater

than that if the self–coupling is very weak.2

After inﬂation ends, the condensate evolves through the following stages:

•it stays frozen as long as the Hubble rate is greater than the eﬀective mass of s

•starts oscillating in an s4potential when the Hubble rate decreases to the eﬀective

mass of s

•oscillates in an s2potential when the eﬀective mass of sbecomes comparable to the

bare mass ms, making it a non-relativistic dark relic

Let us now go through these stages in detail. For convenience, introduce the “average” ﬁeld

value

¯s≡phs2i.(5)

2In the limit λs= 0, the variance grows linearly in time ∝H3t.

The eﬀective mass of the scalar is given by

eﬀ =m2

s+ 3λs¯s2.(6)

Immediately after inﬂation, meﬀ H, so that the potential can be neglected and the ﬁeld is

eﬀectively “frozen”, d

dt ¯s'0. For many purposes, the ﬁeld can be treated as homogeneous,

except for the ﬂuctuations generated by inﬂation. Soon after the end of inﬂation, the Universe

becomes dominated by radiation. The precise nature of this transition (“reheating”) is

unimportant for us. In the instant reheating approximation, the reheating temperature Treh

is found via

3H2

endM2

Pl =g∗(Treh)π2

30 T4

reh ,(7)

where g∗is the number of eﬀective degrees of freedom at Treh. The resulting Hubble rate

then decreases as H∝a−2with the scale factor.

The second stage in the evolution sets in when

osc ∼m2

eﬀ .(8)

We assume msto be small enough such that m2

s3λs¯s2at this stage. The condensate

starts oscillating in a quartic potential and behaves as radiation,

¯s∝a−1.(9)

When the amplitude reduces further, the self-interaction term becomes comparable to the

mass term in the potential,

2m2

s¯s2

dust ∼1

4λs¯s4

dust ⇒¯sdust 'r2

λs

ms,(10)

which signiﬁes the onset of the third phase in the evolution of s. After that, the ﬁeld behaves

as non–relativistic collisionless dust with the energy density scaling as a−3. In summary, we

have the following stages in the evolution of s:

¯send

−→ ¯sosc

a−1

−→ ¯sdust ,(11)

The number density in the “dust” phase is given by

n(a) = ρ(a)

ms'm3

λs

dust

a3.(12)

The abundance of the squanta is conveniently expressed in terms of the conserved quantity

Y=n

sSM

, sSM =2π2

45 g∗sT3,(13)

where sSM is the entropy density of the SM thermal bath at temperature Tand g∗sis the

eﬀective number of degrees of freedom contributing to the entropy. Ycan, for instance, be

evaluated at the onset of the non–relativistic period characterized by temperature Tdust. At

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Scalaroverproductioninstandardcosmologyandpredictivityofnon{thermaldarkmatterOlegLebedevDepartmentofPhysicsandHelsinkiInstituteofPhysics,GustafHallstrominkatu2a,FI-00014Helsinki,FinlandAbstractStablescalarscanbecopiouslyproducedintheEarlyUniverseeveniftheyhavenocouplingtootherelds.Westudyproducti...

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Scalar overproduction in standard cosmology and predictivity of nonthermal dark matter

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