Friction on ALP domain walls and gravitational waves Simone BlasiaAlberto MariottiaA aron RaseaAlexander SevrinbcKevin Turbangac aTheoretische Natuurkunde and IIHEELEM Vrije Universiteit Brussel The International

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Friction on ALP domain walls and gravitational waves

Simone Blasi,aAlberto Mariotti,aA¨aron Rase,aAlexander Sevrin,b,c Kevin Turbanga,c

aTheoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel, & The International

Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium

bTheoretische Natuurkunde, Vrije Universiteit Brussel, & The International Solvay Institutes,

Pleinlaan 2, B-1050 Brussels, Belgium

cUniversiteit Antwerpen, Prinsstraat 13, 2000 Antwerpen, Belgium

E-mail: simone.blasi@vub.be,alberto.mariotti@vub.be,

aaron.rase@vub.be,alexandre.sevrin@vub.be,kevin.turbang@vub.be

We study the early Universe evolution of axion–like particle (ALP) domain walls taking into

account the eﬀect of friction from particles in the surrounding plasma, including the case of

particles in thermal equilibrium and frozen out species. We characterize the friction force

from interactions within the ALP eﬀective theory, providing new results for the fermion

contribution as well as identifying simple conditions for friction to be relevant during the

domain wall life time. When friction dominates, the domain wall network departs from the

standard scaling regime and the corresponding gravitational wave emission is aﬀected. As

a relevant example, we show how this can be the case for ALP domain walls emitting at

the typical frequencies of Pulsar Timing Array experiments, when the ALP couples to the

SM leptons. We then move to a general exploration of the gravitational wave prospects

in the ALP parameter space. We ﬁnally illustrate how the gravitational wave signal from

ALP domain walls is correlated with the quality of the underlying U(1) symmetry.

arXiv:2210.14246v2 [hep-ph] 10 May 2024

Contents

1 Introduction 2

2 Domain walls from ALPs 4

2.1 Gravitational waves from domain wall dynamics 7

3 On the importance of friction 8

3.1 Domain wall equation of motion and friction length 9

3.2 Pressure from particle reﬂection 11

3.2.1 Fermions in thermal equilibrium 12

3.2.2 Pressure from fermion dark matter after freeze out 18

3.2.3 ALPs scattering oﬀ ALP domain walls 19

3.3 Friction from SM leptons at Pulsar Timing Arrays 20

4 Domain wall evolution with the Velocity One Scale model 21

4.1 Friction domination 21

4.2 Gravitational wave emission during friction 23

5 Exploring the ALP parameter space with gravitational waves 26

5.1 Axion quality and gravitational waves 27

5.2 Bias from Planck suppressed operators 30

6 Conclusions and outlook 31

Appendices 33

A Dark QCD ALP domain walls 33

B Reﬂection coeﬃcient 35

B.1 ALP self–reﬂection 35

B.2 Fermion reﬂection 36

C Strong friction regime 39

D Matter domination 40

– 1 –

1 Introduction

The discovery of gravitational waves by the LIGO-Virgo collaboration [1] has opened a

new window of exploration of our Universe. In this context, the detection of a stochastic

gravitational wave background (SGWB) of cosmological origin would represent a milestone

in understanding the early stages of the Universe (see e.g. [2] for a review). It is therefore

essential to study the possible sources of a SGWB of cosmological origin and how they can

impart information about fundamental physics.

In particular, strong ﬁrst–order phase transitions in the early Universe can generate a

SGWB during the dynamics associated to bubble nucleation. Furthermore, the symmetry–

breaking pattern involved in the phase transition (independently of the strength) can lead to

the formation of defects through the Kibble mechanism [3] according to the topology of the

vacuum manifold. While the SM predicts no stable defects, they can be formed in beyond

the SM (BSM) scenarios due to the breaking of new continuous or discrete symmetries.

Topological defects are known to be a strong source of SGWB, the most studied example

being cosmic strings (see e.g. [4]) arising from a non–trivial ﬁrst homotopy group.

In this paper we will instead focus on domain walls (DWs), two–dimensional defects

originating in scenarios with a spontaneously broken discrete symmetry leading to a dis-

connected vacuum manifold [3–5]. DWs can be powerful sources of a SGWB, as conﬁrmed

by numerical simulations [6–8], and their signatures in connection with BSM models have

been subject of several recent investigations [9–20].

DWs emerge naturally in models addressing the strong CP problem by the Peccei–

Quinn (PQ) mechanism [21–24]. The reason is that, taking into account the anomaly of

the global U(1)PQ under the color group, the axion potential exhibits a discrete symmetry

spontaneously broken in the vacuum. This implies the formation of DWs attached to the

strings forming at the PQ–breaking scale, which can be topologically stable depending on

the anomaly coeﬃcient, or the so–called domain wall number. More generally, one can

consider axion–like particles (ALPs) whose mass is not tied to the QCD conﬁnement scale.

These particles can arise in String Theory [25,26], or as heavy QCD axions (see e.g. [27–

37]), and they can be dark matter candidate [38–42] and have interesting phenomenology

(see e.g. [43–51] and also [52,53] for recent reviews). In ALP models one can similarly

expect the formation of a string–wall network.

If absolutely stable, DWs can come to dominate the energy density of the Universe [54],

potentially constituting a cosmological problem. However, this is not a real issue here as

global symmetries are not expected to be exact and can be explicitly broken for instance by

higher-dimensional operators [55–63]. This naturally induces a bias for the axion potential,

leading to the decay of the DW network1.

In this paper we will consider ALP DWs as our physics case, and investigate the

impact of particle friction on the DW evolution. Friction will generically slow down the

average wall velocity in the network, potentially leading to signiﬁcant departure from the

1Notice that in the case of the QCD axion, such operators can spoil the solution to the strong CP

problem leading to a tension with experimental constraints.

– 2 –

standard scaling regime, with several phenomenological implications for GWs, see e.g. [64],

and particle production.

Unlike the case of bubble walls during ﬁrst order phase transitions, particles in the

plasma will have approximately the same mass on the two sides of the DW. This is because

of the deﬁning DW symmetry relating the disconnected vacua, which is only broken by a

small bias term. In addition, as the DW motion is not accelerated by vacuum pressure (up

to small bias corrections) but rather by its tension force, DWs are only mildly relativistic

with γ∼1. Therefore, the leading–order contribution to friction arises as a consequence

of plasma particles reﬂecting on the wall surface, see e.g. [65].

For ALP models, the coupling structure between the background DW and the particles

in the plasma is ﬁxed by symmetry arguments within the ALP eﬀective ﬁeld theory (EFT).

Furthermore, the DW formation in ALP models typically occurs at temperatures much

larger than the ALP mass (which sets the DW width). This means that particles scattering

oﬀ the wall can have a very large momentum compared to the DW width, opening a new

kinematic regime for friction. These qualitative diﬀerences compared to generic scalar

theories will lead to new features in the friction force with respect to previous studies [4,66],

such as a diﬀerent scaling with the temperature. Exploration of friction in QCD axion

models was initiated in [67]. Here we present a new, detailed calculation for the friction

force in the context of ALPs by considering reﬂection from fermions (possibly being dark

matter), and comment on ALP self–reﬂection.

We will then determine the parameter space where friction can be relevant depending

on the size of the eﬀective couplings, ﬁnding that friction can aﬀect both the early and the

late evolution of the DW network when the GW emission is maximal. As a relevant example

of the latter, we show how friction from SM fermions can aﬀect the DW interpretation of

the signal observed at Pulsar Timing Array (PTA) experiments [68–70].

As a byproduct of our study, we will also point out that a certain quality is required

for the global ALP U(1) in order to generate a large SGWB signal. For instance, a naive

dimension ﬁve Planck suppressed operator makes the ALP DWs so short-lived that the

resulting GWs are undetectable, even at future experiments. On the other hand, observable

GWs are compatible with dimension six operators or larger.

The paper is organized as follows: in Section 2we review basic aspects about domain

walls (their dynamics, the bias, the gravitational wave spectrum), specializing to the DWs

arising in ALP models. In Section 3we explore the importance of friction for ALP domain

walls. We review the formalism to compute the pressure from particle reﬂection oﬀ the

wall, and then we investigate in detail the case of friction from a fermion coupled to the

ALP, and from the ALP itself. We conclude this section by showing that friction from SM

particles (speciﬁcally from the leptons) can play a signiﬁcant role in ALP domain walls

whose gravitational wave signal peaks at the frequency relevant for PTA experiments such

as NANOGrav. In Section 4we employ velocity-dependent one-scale (VOS) equations to

estimate the gravitational wave signal during friction and we explore in detail the param-

eter space for ALP models, showing which portion can be probed by current and future

gravitational wave experiments. During this analysis, we will highlight the fact that a cer-

tain quality is required for the symmetry underlying the ALP model for the gravitational

– 3 –

wave signal to be detectable.

2 Domain walls from ALPs

Domain walls are topological defects that arise in models where a discrete symmetry is

spontaneously broken (see [4,71] for comprehensive reviews). More precisely, they appear

when the vacuum manifold Mhas a non-trivial homotopy group π0(M). 2

Assuming that the discrete symmetry is restored at high temperatures, domain walls

get formed at the discrete symmetry breaking through the Kibble mechanism [3]. When

the Universe cools down below the critical temperature, uncorrelated patches in space will

randomly choose one of the degenerate vacua. Once thermal ﬂuctuations become suﬃ-

ciently suppressed, this choice cannot be undone and domains can be considered formed.

At the boundary between diﬀerent domains, the ﬁeld will be trapped at the maximum of

the potential leading to a large energy density localized in a two–dimensional surface, the

domain wall.

A very appealing way in which discrete symmetries can emerge is the case of anoma-

lous global symmetries as for DWs arising in axion models [54,72]. In general, we can

deﬁne an axion–like particle (ALP) as the pseudo Nambu–Goldstone boson arising from

the spontaneous breaking of an anomalous U(1) symmetry, the best motivated example

being the Peccei–Quinn axion. Such mechanism can be understood by considering the

following Lagrangian:

L=∂µΦ†∂µΦ−λΦ†Φ−v2

22

−V(a),(2.1)

where Φ = ρexp(ia/va)/√2 and ais the axion. The ﬁrst potential term implies that

the U(1) symmetry is spontaneously broken with ⟨Φ⟩=va/√2 and the axion domain is

[0,2πva). The last potential term in (2.1) is induced by the anomaly of the U(1) group

under a strongly coupled gauge theory whose dynamical scale is Λ, and explicitly breaks

the U(1) symmetry to a Z2Ndiscrete symmetry, where Nis the anomaly coeﬃcient. The

typical form of such explicit breaking at zero temperature is

V(a)=Λ41−cos aNDW

va (2.2)

where NDW ≡2Nis the domain wall number, and the axion decay constant is fa≡

va/NDW. At temperatures Taround the conﬁnement scale Λ and above, thermal correc-

tions to the ALP potential become important, see e.g. [44,73] and references therein for

further details. In particular, for T≫Λ the overall magnitude of the potential V(a) is

suppressed by a factor ∝(Λ/T )n, where n > 0 is a O(few) number depending on the

matter content of the theory.

The ALP potential possesses NDW discrete vacua connected by a shift symmetry,

ZNDW :a

va7−→ a

+2πk

NDW

(2.3)

2In fact, depending on the non-trivial homotopy group one can create domain walls (π0(M)), strings

(π1(M)), monopoles (π2(M)).

– 4 –

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FrictiononALPdomainwallsandgravitationalwavesSimoneBlasi,aAlbertoMariotti,aA¨aronRase,aAlexanderSevrin,b,cKevinTurbanga,caTheoretischeNatuurkundeandIIHE/ELEM,VrijeUniversiteitBrussel,&TheInternationalSolvayInstitutes,Pleinlaan2,B-1050Brussels,BelgiumbTheoretischeNatuurkunde,VrijeUniversiteitBrussel,...

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Friction on ALP domain walls and gravitational waves Simone BlasiaAlberto MariottiaA aron RaseaAlexander SevrinbcKevin Turbangac aTheoretische Natuurkunde and IIHEELEM Vrije Universiteit Brussel The International

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