Cosmic implications of a low-scale solution to the axion domain wall problem

The post-inflationary breaking of Peccei-Quinn (PQ) symmetry can lead to the cosmic domain wall catastrophe. In this Letter we show how to avoid domain walls implementing the Instanton Interference Effect (IIE) with a new interaction which itself breaks PQ symmetry and confines at an energy scale smaller than $\Lambda_{QCD}$. We give a general description of the mechanism and consider its cosmological implications and constraints within a minimal model. Contrary to other mechanisms we do not require an inverse phase transition neither fine-tuned bias terms. Incidentally, the mechanism leads to the introduction of new self-interacting dark matter candidates and the possibility of producing gravitational waves in the frequency range of SKA. Unless a fine-tuned hidden sector is introduced, the mechanism predicts a QCD axion in the mass range $1\text{ meV}-15\text{ meV}$.


INTRODUCTION AND MOTIVATION
The axion solution to the strong CP problem [1][2][3][4][5][6] is a well-known paradigm where domain walls bounded by strings 1 emerge [9,10]. This is because as the Universe cools down two different phase transitions occur. In the first one the PQ scalar where t R is the Dynkin index and q R are the PQ charge of the fermions. Notice that the axion decay constant and the VEV, | Φ | = v, are related as It is well-known that for N QCD = 1 the string-wall system is not stable because the tension of the walls make strings to collapse to a point and the network decays fastly into non-relativistic axions [11]. However, for N QCD > 1 each string gets attached to N QCD walls and the network cannot decay: the walls are topologically protected and stable. Their evolution with cosmic expansion is slower than that of matter or radiation and will eventually dominate the energy density of the Universe.
To avoid such a cosmological catastrophe one needs to make them disappear or evade their formation. Actually, as showed in Ref. [12], simple KSVZ axion models with different representations for the exotic quarks generating the PQ anomaly generically have N QCD > 1 and, therefore, do suffer from DW problem. Therefore, simply neglecting the problem and assume N DW = 1 sounds too simplistic.
A straightforward way to solve the domain wall problem is to invoque cosmic inflation [13][14][15]. Domain walls are, in this case, pushed beyond the horizon and will not harm our cosmic evolution. This is the case of a PQ symmetry spontaneously broken before inflation and never restored after reheating. This case, however, can be constrained by isocurvature fluctuations in the CMB [16].
Here instead we consider a scenario in which PQ symmetry is broken after inflation. Different post-inflation mechanisms have been proposed since the '80s. In the Lazarides-Shafi mechanism [17], for example, one asso-  [19] or the Witten effect have been explored more recently [20].
As a guideline, to make the walls unstable one needs to remove their topological protection by explicitly breaking the discrete symmetry that relates the degenerated set of vacua. The simplest solution following this philosophy is almost as old as the DW problem and is known as the bias term solution [10]. It consists in adding to the scalar potential an ad-hoc term This term breaks explicitly PQ symmetry and produces a potential for the axion field that generates a effective theta term which is constrained to be θ ≤ 10 −10 . Then, if one assumes natural values for the phase, δ ∼ O(1), the dimensionless parameter Ξ is constrained to be [21] Ξ ≤ 2 · 10 −45 N −2

(5)
In this article we will consider a natural (and not finetuned) realization of the bias term by using a low scale version of the IIE. The introduction of a new confining interaction and its associated instantons will generate the explicit breaking of the Z N DW symmetry, unbroken by QCD instantons.
The paper is organized as follows. We first introduce the IIE and explain how it solves the domain wall problem. Then, we explore the constraints on the confinement scale of the HC sector we introduce. Once we set the scale of the new sector, we specify to a minimal SU (N ) model and explore its cosmological implications. Finally, we conclude and comment on future directions to follow.

THE INSTANTON INTERFERENCE EFFECT
The Instanton Interference Effect (IIE) is a compelling mechanism to avoid the cosmic domain wall problem in a inflation independent way (it applies in both, pre and post-inflationary PQ breaking scenarios) [22,23]. In this 2 See [18] for a different realization in the context of family symmetries.
mechanism one adds to the invisible QCD axion model a new non-abelian gauge group HC which is also anomalous under PQ with SM the usual Standard Model gauge symmetry the new group will in general break the PQ symmetry down to a Z N HC subgroup. On the other hand, QCD instantons break the same symmetry to a Z N QCD discrete symmetry. The full axion potential reads with κ ≈ 1.2 × 10 −3 , in which we recognize the QCD term, with anomaly coefficient N QCD , and the new term due to the HC interaction which confines at some scale Λ HC and has its own anomaly coefficient, N HC . When N QCD and N HC are co-prime numbers, the PQ symmetry is completely broken by the combination of explicit symmetry breaking by instantons and the domain wall problem is solved. This is the Instanton Interference Effect (IIE). We notice that a similar periodic bias term with N HC = 1 has been considered in [24]; there the authors focus on the generation of primordial black holes and did not consider neither the origin nor its cosmological implications (and constraints)which are intriguing and diverse.
In its high-scale realization [22,23], the IIE mechanism requires a confinement scale much larger than the QCD scale. In fact, if PQ is spontaneously broken by hypercolor HC condensates, as in [23], f a and Λ HC scales coincide. Therefore one needs to turn off the interaction below a critical temperature, once PQ is spontaneously broken and the axion field sits on the same vacuum everywhere, in order to allow the standard PQ mechanism to work and drive dynamically the axion field to a CP conserving minimum. To allow this, a new fermion in the HC sector and a new scalar with an inverted phase transition are needed.
It is attractive to explore the possibility of a low-scale version of the IIE with Λ G Λ QCD , which is indeed the goal of the present work. As we will show, this sce- relaxed and the only requirement is that they must disappear before matter-radiation equality, given roughly by T eq ∼ 1 eV. The reason is that since these domain walls can only decay into axions or gravitational waves (gravitons) the interaction of their decay products with the SM is not efficient and they do not spoil BBN. In general, for N DW > 1, domain walls will dissappear when the discrete symmetry Z N QCD unbroken by QCD instantons starts to feel the effects of HC instantons. Then, due to the IIE the Z N QCD symmetry gets explicitly broken and the domain walls are not topologically protected.
This occurs because the initially degenerated vacua get a small splitting from the bias term. Therefore the new term in the potential causes an energy difference between the false and the true minima ∆ρ ∼ Λ 4 HC leading to a pressure, p V ∼ ∆ρ, which acts against the domain wall. We remark that, in our case, the bias term comes from the potential V HC (a) (generated by HC instantons). The condition that the vacuum pressure generated from the energy difference exceeds the wall tension reads p V > p tension (8) and therefore with σ = 8m a f 2 a . This allow us to write the Hubble parameter when the walls decay as: We can impose that the domain walls decay before matter domination, obtaining the lower bound to the confinement scale A different, and actually more stringent, bound can be obtained by the requirement that DW never dominate the energy density of the Universe. The contribution of domain walls to the energy density of the Universe is The Universe becomes dominated by domain walls when they reach relativistic speeds at the time t c ∼ (Gσ) −1 .
Consequently one has to impose that the time associated to the decay is much shorter: Using equation (10) we get which looks more stringent than Eq. (11) for reasonable values of f a .

Upper bound to ΛHC
The axion potential (see Eq. (7)) presents two contributions from QCD and HC instantons. The QCD contribution is the one that turns on first. Then, below a critical temperature, the HC potential also turns on and a small θ ef f will be generated. We can minimize the above potential to get and impose the solution of the Strong CP problem is not spoiled.
We see that in the limit Λ HC Λ QCD , the effective CP violating phase behaves as θ ef f ∝ Requiring |a| 10 −10 implies, as already stated, an up- the smaller (and fine tuned) is the shift δ, the larger will be the upper limit (see yellow curve in Fig.1).

AXION DARK MATTER ABUNDANCE
One of the most attractive feature of the PQ solution of the Strong CP problem is that it provides a new dark matter candidate, the axion, to which a great experimental and theoretical effort has been devoted .
In usual models, when DW are short lived the three non-thermal contributions (decay of DW, coherent oscillations and cosmic string radiation) are comparable [21]. where is the energy of the domain walls, then converted into axion, at the time of their decay. The relic density of cold axions from DW is given by It is useful to write the ratio and, then, use the relations The Hubble parameter at matter-radiation equality and the critical density are given by Finally we find Ω a,w = 0.12 × f a 1.8 × 10 9 GeV where we used m a ∼ 5.7 10 9 GeV f a meV .
In Fig.2 we show the relation between the scale Λ HC and the decay constant f a for Ω a,w h 2 = 0.12 in a generic model with N HC = 2. Axions from coherent oscillations and radiated by strings can be estimated as [21] Ω coh h 2 ∼ 0.0009 × f a 1.8 × 10 9 GeV 7/6 , One can easily see that the axions coming from the decay of the walls will dominate the relic density today unless we have a large value of N 2 DW . In this case axions radiated from strings become important. 3 Then, to obtain a lower bound for Λ HC from DM abundance one has to distinguish between different axion models. This is because different lower bounds or constraints for the axion decay constant f a will pose different lower bounds for Λ HC . If one considers the DFSZ, processes involving the axion coupling to electrons contribute to fast stellar cooling. From the constrain g ae ≤ 2.6 × 10 −13 [46] one gets f a ≥ 6.5 sin 2 β × 10 8 GeV , where tan β = v u /v d is the ratio of up-type and downtype Higgs doublets in the DFSZ model. For hadronic KSVZ models one has the constraints coming from SN1987A, where processes like N N → N N a generate a more efficient energy-loss channel, resulting in a reduced neutrino burst duration. This constraints the axion decay constant to be f a ≥ 4 × 10 8 GeV [36]. For the sake of clarity, in the rest of the paper we will consider a general KSVZ-like model with N QCD = 1 and heavy, vector-like quarks which we denote as Q.

ON THE HC SECTOR
The dark HC sector can have a rich structure and many groups can lead to the desired interference effect, making this solution quite generic (up to simple model building). 3 We warn the reader that the subject of axions from topological defects is of course a technically complicated one, to which a lot of effort has been dedicated (for an incomplete list of works see [48][49][50][51][52][53][54][55]), therefore it follows that any estimate has to be taken with a grain of salt. Here, for definiteness, we limit our analysis to the case of a dark SU (N ) HC with a dark fermion, F , also charged under PQ. Guided by minimality we assume that the fermions F with HC charge take their mass from the same scalar which breaks PQ symmetry, Φ, and gives also mass to the colored fermions Q of a generic KSVZ model.
We also stress that the situation is quite different from other models with a non-abelian dark sector where the new degrees of freedom carry also other SM charges (see for example [56][57][58][59][60][61]). For us all the particles of the dark HC sector are singlets under SU (3) C × SU (2) L × U (1) Y .
Moreover, we expect the mass of the dark sector fermion, where Y 1 is a yukawa coupling and v = N QCD f a is the scale of the PQ spontaneous symmetry breaking. Requiring the Yukawas not to bee too small (in order not to introduce again a small parameter as the θ-term) it is easy to see that we will be always in the situation where the mass of the fermions is much larger than the confinement scale In this case one can have stable, heavy bound states com- which not necessarily coincides with the SM temperature T . The relic density of glueballs is given by [62] We can easily see that this relic density becomes larger than the DM relic density unless is small. This is known as the Dark Glueball Problem [63,64]. In Fig.3 we show the glueballs relic density for two diffent values of the confinement scale in the range of interest. We see that the temperature ratio has to be 10 −3 − 10 −2 in order not to overclose the Universe.
We notice that if the dark fermion is in the fundamental representation of SU (N ) HC , it will be N HC = 1.
While this of course leads to the IIE and the resolution of the domain wall problem, it could seem as arbitrary as taking N QCD = 1 from the beginning. As an example we explore a more general situation in which the fermion is in the adjoint representation, F ∼ (Adj, 1, 1, 0). In such a case the spectrum will be composed of HC glueballs and glueballinos as in [62] and the HC anomaly  [62,65]: where we take also σv ∼ α 2 HC (m F )/m 2 F . This expression neglects the effect of re-annihilation considered in [56]. A more sophisticated expression should take this effect into account, investigating both de-excitation and dissociation processes. However, in the present setup glueballs do not decay and therefore we expect re-annihilation to be inefficient. Consequently, we will neglect its effects.
Concerning the glueballino relic density, we note that, as usually happens for stable relics that were in thermal equilibrium, they can overclose the Universe if its mass is larger than O(100) TeV [66]. In addition, our scenario also suffers from the glueball problem and, therefore, one needs 1. In such a case, when m F 100 TeV and 1, the relic density of dark matter will be QCD axion dominated. In a following section we will identify the regions of the parameter space where dark matter is composed by axions and those where dark matter is made of glueballs and glueballino.
Finally, let us comment that the suppression of the dark sector temperature respect to the Standard Model also allows to evade bounds from the number of relativistic degrees of freedom N ef f . Results from Planck give N ef f = 2.99 ± 0.17 [67]. We have then to worry if our framework implies non-standard values of N ef f . The temperature of the Standard Model when the dark sector confines is given by Therefore, for 10 −2 there are no relativistic, massless species to act as the hidden-sector bath during BBN because HC confinement occurs well before BBN and structure formation. Thermal equilibrium between the HC sector and the standard model In the previous section we introduced a new parameter = T HC /T . Glueball and/or glueballino relic abundance in the considered model turn out to be an important problem unless is small. This means that the HC sector needs to have a smaller temperature than the SM thermal bath. We make the assumption that the two sectors start with different equilibrium temperatures 4 . Still, we have then to make sure that the two sectors never enter thermal equilibrium. The strongest constraint on the 4 This might occur due to their coupling with the inflaton sector.
parameters of the theory comes from the scattering pro- mediated by an heavy fermion, F . The rate for this process at high temperatures is given by which has to be compared with the Hubble rate In the above equation G is the Newton constant, T is the temperature of the standard model and g * the number of relativistic degrees of freedom, which is given by The comparison can be made for T HC = m F = T , that is to say when T = m F / . Below this temperature the number density of heavy fermions, F , is rapidly suppressed by Boltzmann factor. Expressing m F = Y 1 f a we find that in order not to enter thermal equilibrium GeV , axions from domain wall decay overclose the Universe (see Eq. (23)). On the other hand, SN1987A gives a lower limit f a 4 · 10 8 GeV [36]. It follows that the value of the small-scale formation woes in astrophysics [68,69].
As showed in [62], it is also possible to arrange the parameters of the model, just by slightly increasing , to transition can lead to a GW signal in the frequency range of future detectors [73]. The situation under consideration, m F Λ HC , is analogous to a pure Yang-Mills theory, which is well known to exhibit a strong first order phase transition [74]. However, a GW signal is proportional to the energy density of the dark sector, which is proportional to the fourth power of the temperature T h . As we have seen, the latter is smaller than the temperature of the visible world and the GW signal will be consequently suppressed. Using the same input parameters and results of [73], properly taking into account the suppression of energy density due to the low HC sector temperature, we find that the signal frequency will fall in the range of frequency of SKA [75] (thank to the difference in temperatures between dark and visible sectors) f peak ≈ 3.33 · 10 −9 Hz Λ HC 1M eV 10 −2 (38) but just ouside its sensitivity for the considered values of . Nevertheless this is an interesting situation that deserves further study and will be presented elsewhere.
Finally, we also checked the possible GW production from domain wall decay into gravitons [76], which unfortunately turns out to be subdominant and always negligible in the explored parameter space. for the phase δ, the decay constant, f a , is constrained to be 4 · 10 8 GeV ≤ f a 2 · 10 9 GeV . This corresponds to an axion mass range of 1 meV − 15 meV. It is remarkable that this lies close to the expected sensitivity of axion antennas [77], dielectric haloscopes [78] or ARI-ADNE [79]. Axion experiments will therefore be able to test this scenario in the near future.