Axion domain wall baryogenesis

We propose a new scenario of baryogenesis, in which annihilation of axion domain walls generates a sizable baryon asymmetry. Successful baryogenesis is possible for a wide range of the axion mass and decay constant, $m \simeq 10^8 -10^{13}$ GeV and $f \simeq 10^{13} - 10^{16}$ GeV. Baryonic isocurvature perturbations are significantly suppressed in our model, in contrast to various spontaneous baryogenesis scenarios in the slow-roll regime. In particular, the axion domain wall baryogenesis is consistent with high-scale inflation which generates a large tensor-to-scalar ratio within the reach of future CMB B-mode experiments. We also discuss the gravitational waves produced by the domain wall annihilation and its implications for the future gravitational wave experiments.


I. INTRODUCTION
Axions may be ubiquitous in nature. Indeed, there appear many axions through compactification of the extra dimensions in the string theory [1,2]. Some of them may remain relatively light and play an important role in cosmology such as inflation, dark matter and dark energy. In this paper we shall present a new scenario of baryogenesis, in which axions play a key role.
The axion exhibits a shift symmetry, where C is a real transformation parameter. While the shift symmetry keeps the axion potential flat at the perturbative level, non-perturbative effects break the symmetry to a remnant discrete one.
Let us suppose that one of the non-perturbative effects gives the dominant contribution to the axion potential, which is expressed as where m is the axion mass and f is the decay constant. Then, the axion potential has a series of N (approximately) degenerate vacua, where the precise value of N depends on the details of the UV theory. 1 If the axion is lighter than the Hubble parameter during inflation, it acquires quantum fluctuations which extend beyond the Hubble horizon. For sufficiently large quantum fluctuations, some of the N vacua might be populated, which results in domain wall formation after inflation. The domain walls are cosmologically problematic, and so, they must annihilate before dominating the Universe. This is possible if the degeneracy between different vacua is lifted by other non-perturbative effects [7][8][9][10].
The domain wall annihilation and the emitted gravitational waves have been extensively studied in the literature [11][12][13][14].
In this paper we point out that the annihilation of domain walls also induces the baryon asymmetry of the Universe. Suppose that the axion is derivatively coupled to the standard model (SM) quarks and/or leptons, where c i is a coupling constant. The time derivative of the axion plays a role of the effective chemical potential, which spontaneously breaks the CPT symmetry. 2 This enables the generation of the baryon or lepton asymmetry in thermal plasma if the baryon or lepton number is broken, and this is the so-called spontaneous baryogenesis scenario [16][17][18].
The current to which the axion is coupled does not have to coincide exactly with the baryon or lepton current; for instance, it could be a U(1) hypercharge current [18] or a Peccei-Quinn current [19]. Such derivative couplings to the baryon and lepton currents can also be induced if the axion has an anomalous coupling to the SU(2) gauge fields [17].
In this case the chemical potential is induced by sphalerons [20,21], because a non-zero time derivative of the axion generates energy difference between the states with different winding number and B + L number. Therefore, the chemical potential is expected to be suppressed at T 10 12 GeV where sphalerons decouple from the cosmic expansion. Note that there is no such suppression of the effective chemical potential if one starts with the derivative couplings with baryon and/or lepton current (more precisely, B − L current), as we shall do below. We shall see that, if the axion has such derivative couplings, a sizable baryon asymmetry can be generated when the axion domain walls annihilate.
Before going into details, let us give a rough sketch of our scenario. For simplicity, we assume that only two vacua, a 1 and a 2 with a 1 < a 2 , are populated during inflation, leading to formation of domain walls separating the two vacua. Generalization to the case of multiple vacua is straightforward. After formation, domain walls randomly move around at relativistic speed, collide and annihilate continuously, so that the domain wall network show the dynamical scaling behavior [22][23][24][25]. Every time a domain wall goes through some point in space, the field value of the axion changes either from a 1 to a 2 or from a 2 to a 1 . Such transition induces a temporal and local chemical potential for baryons or leptons. No net baryon asymmetry is generated by the domain wall dynamics in the scaling regime, however, because both transitions occur with an equal probability and there is no preference of baryons over anti-baryons. The asymmetry between the two vacua becomes important when the domain walls annihilate because of the energy bias. Suppose that one of the vacua is energetically preferred, e.g., V (a 1 ) < V (a 2 ). When domain walls annihilate, the value of a then decreases from a 2 to a 1 in a region of the false vacuum, which gives a preference to baryons over anti-baryons for a certain choice of the couplings. Thus, the axion domain wall annihilation can generate the baryon asymmetry of the Universe.
Our scenario has several advantages. First, it is known that the spontaneous baryogenesis in the slow-roll regime generically leads to baryonic isocurvature perturbations [26], which makes the scenario incompatible with high-scale inflation. 3 In our scenario, however, the baryonic isocurvature perturbations can be significantly suppressed, because of the scaling property of the domain wall network. In particular, our scenario is consistent with large-field inflation, and therefore, the required high reheating temperature can be realized more easily. Secondly, the axion field value is kept large inside domain walls, which enables a large effective chemical potential even when the axion mass m becomes larger than the Hubble parameter. Without domain walls, the spontaneous baryogenesis would become inefficient when the axion starts to oscillate about the minimum [28].
Therefore, the axion domain wall baryogenesis scenario works for a wide range of the axion mass and the inflation scale.
Lastly let us comment on differences of our scenario from other works. In the thickwall regime of the electroweak baryogenesis, the passage of an expanding bubble wall generates a non-zero chemical potential, which leaves net baryon asymmetry in thermal plasma based on the spontaneous baryogenesis [17,18] (see also Ref. [29]). The bubble 3 It is possible to give the axion a mass of order the Hubble parameter in the spontaneous baryogenesis using a flat direction [19], thus avoiding the isocurvature constraint. Also, no isocurvature perturbation is induced in the gravitational baryogenesis [27].
walls play a similar role to that of domain walls in our scenario. The difference is that the electroweak spontaneous baryogenesis relies on the first order phase transition of two (or more) Higgs fields, and the sphaleron process is exponentially suppressed in the symmetry breaking vacuum. As a result, the estimate of the final baryon asymmetry requires a precise determination of the critical field value as well as detailed analysis of the diffusion process during the phase transition [30]. In our scenario, on the other hand, the baryon (or lepton) number violation is operative equally in the two minima.
Also it relies on the domain wall dynamics of a single axion field, whose behavior is well studied with numerical simulations. This makes our scenario relatively simple and robust.
Recently, the authors of Ref. [31] proposed a scenario where the axion has only anomalous coupling to SU(2) L gauge fields. They studied a spatially homogeneous axion field in the slow-roll regime, and explored the parameter space of the axion mass and decay constant preferred by the string axions. The parameter ranges have an overlap with our scenario.
One difference is that we start with derivative couplings of the axion with baryon and/or lepton currents. Another is that our scenario relies on the domain wall dynamics, while Ref. [31] focused on the homogeneous axion field.
The rest of this paper is organized as follows. In Sec. II, we briefly review the evolution of axion domain walls. We estimate the baryon asymmetry induced by the domain wall annihilation in Sec. III. The last section is devoted to discussion and conclusions.

II. AXION DOMAIN WALLS
Let us consider an axion whose potential is given by where m and f are the mass and the decay constant of the axion a. We assume that two adjacent minima, a 1 = 0 and a 2 = 2πf , are populated with more or less equal probability during inflation, and that domain walls separating the two minima are formed after inflation. This is the case if the quantum fluctuations of the axion, δa ∼ H inf /2π, is comparable to the decay constant, or if the initial position of the axion is sufficiently close to the local maximum, a max = πf . Our scenario can be straightforwardly applied to the case in which more than two minima are populated.
The domain wall solution in a flat spacetime is given by where x is the spatial coordinate perpendicular to the domain wall, v is the domain wall velocity and γ is the relativistic factor defined by for the potential (4).
The domain walls are formed when H m. According to the numerical and analytic calculations [22][23][24][25], within a few Hubble time after the formation, the domain walls quickly follow the scaling law, i.e., where there are only one or a few domain walls in each Hubble horizon. The domain walls must annihilate and disappear before they start to dominate the Universe, since otherwise the Universe would be too inhomogeneous. We assume that there is another shift-symmetry breaking term which generates a bias between the two minima, . Then domain walls annihilate rapidly when the energy density of domain walls becomes comparable to the energy bias [8][9][10], Marginally relativistic axion particles with a typical momentum, k ∼ m, are copiously produced through the axion domain wall annihilation. Those axion particles soon become non-relativistic due to the cosmic expansion [12][13][14]. In addition, axion coherent oscillations are produced at the domain wall formation, and we shall discuss their cosmological impact later in this paper.
The axion particles eventually decay into SM particles through their couplings with the SM sector. In general, the axion can have derivative couplings to fermions like (3), which are allowed by the shift symmetry (1). Specifically we focus on the case in which the axion has derivative couplings only to the SM left-handed lepton currents, 4 Our results remain practically unchanged even if one adds additional derivative couplings to other SM fermions. If the axion is coupled to the SM sector only through the above interaction (9), it mainly decays into a pair of SU(2) L gauge bosons and hypercharge gauge bosons through its anomalous couplings [16]. The decay width into a pair of gauge bosons is approximately given by where α 2 and α are respectively SU(2) L and U(1) Y gauge coupling constants and N f is the number of of generation, and we will set N f = 3 in the following. Approximating that this is the main decay channel, the axion decay temperature is where we have defined the decay temperature by 3H(T a ) = Γ a . If those axion particles dominate the Universe before the decay, there will be an extra entropy production by the axion decay, which dilutes pre-existing baryon asymmetry by some amount. As we shall see, the entropy dilution becomes important for a large decay constant and a small axion mass.

III. BARYOGENESIS BY DOMAIN WALL ANNIHILATION
A. Analytical estimate of the asymmetry Now let us discuss baryogenesis by the axion domain walls under the existence of the derivative coupling to the lepton current given by (9). As previously noted, ifȧ is non-vanishing, the derivative couplings behave like an effective chemical potential, where µ eff =ȧ/f is the effective chemical potential for the lepton number (L).
The axion domain walls can generate the effective chemical potential because of the large spatial gradient of the axion field inside the wall. Since domain walls are moving at nearly the speed of light, the time derivative of the axion field at some fixed spatial point becomes large while domain walls are passing through. The effect of the gradient term is negligible if the domain wall is sufficiently thick compared to the diffusion length.
If the L-number violating operator is in equilibrium, and if the chemical potential is spatially homogeneous, the difference of number densities between lepton and anti-leptons would be produced as n eq − n eq 2µ eff T 2 for µ eff T , where we have taken into account the spin degrees of freedom and the number of generation. It depends on the rate of the L-number violating process as well as the domain wall dynamics whether the lepton asymmetry reaches the equilibrium value in the expanding Universe. One needs to solve the Boltzmann equation for the lepton asymmetry, n L = n − n¯ , where Γ is the interaction rate for the L-violating processes. Note here that the chemical potential in n eq L depends on the position and velocity of domain walls. As the L-number violating operator, we consider ∆L = 2 scattering processes, ↔ HH, H ↔¯ H , which are mediated by heavy right-handed Majorana neutrinos in the seesaw mechanism [32][33][34][35].
Here and in what follows we assume that the right-handed neutrinos are so heavy that they can be integrated out in our analysis. The interaction rate for the ∆L = 2 processes is roughly given by [36] where v EW = 174 GeV and m i with i = 1, 2, 3 denotes the mass of three active neutrinos. The decoupling temperature of the L-violating process in the radiation dominated Universe is where we have assumed the normal ordering for the neutrino mass differences and used the experimental value, For the reheating temperature T R lower than T dec , the L-violating process remains decoupled from the cosmic expansion.
As we shall see below, even in this case, a non-zero lepton asymmetry is induced by the domain wall annihilation.
Let us first consider an ideal situation where a domain wall passes through the origin x = 0 at t = t DW . Using Eq. (5), the effective chemical potential at the origin evolves with time as It takes roughly ∆t ∼ (mγv) −1 for the domain wall to pass through the origin, and so, the induced lepton asymmetry by passage of the domain wall is estimated as Note that the lepton asymmetry becomes independent of the velocity of the domain walls.
As the domain wall passes through, a similar amount of the lepton number density will be induced inside the Hubble horizon.
In the scaling regime, domain walls randomly move around inside the Hubble horizon so as to collide and annihilate continuously. In particular, since there is no preference for either of the vacua, the effective chemical potential can be positive or negative with equal probability. Therefore there will be no net lepton asymmetry left, even though some amount of the lepton asymmetry with either positive or negative sign is induced each time a domain wall passes through. Such lepton asymmetry has fluctuations of order unity inside the Hubble horizon, but it has no sizable fluctuations at superhorizon scales, because of the scaling property of the domain-wall network.
A non-zero net lepton asymmetry is induced when domain walls annihilate and disappear owing to the energy bias. This is because one of the two vacua is energetically annihilation occurs after the decoupling, the induced asymmetry tends to be suppressed because the L-violating process is inefficient. The maximum asymmetry is therefore where s and g * s are respectively the entropy density and the relativistic degrees of freedom.
We have substituted g * s = 106.75 and the decoupling temperature (15) in the second equality, assuming the radiation-dominated Universe. The negative sign is inserted in the second equality to obtain positive baryon asymmetry through sphalerons.
If the reheating temperature T R is lower than the decoupling temperature T dec , the interaction rate for the L-violating processes never exceeds the expansion rate of the Universe. One can see this by noting that Γ/H reaches the maximal value (smaller than unity) at the reheating as long as the temperature of the dilution plasma obeys T ∼ (HT 2 R M P ) 1/4 before the reheating. Hence the maximal asymmetry in this case is obtained if the domain wall annihilation occurs at the reheating, and it is roughly given We shall see later in this section that the maximal asymmetry is indeed generated if the domain wall annihilation takes place at T = min[T dec , T R ].
at the domain wall formation (H form ∼ m), where n L is given by (17).
Secondly, the domain wall must be sufficiently thick to justify our analysis where we have neglected dissipation of the asymmetry. The thickness of the wall is roughly m −1 and the typical mean free path of the particle in plasma is of order T −1 . Thus, the thick-wall condition is given by where T ann denotes the temperature at the domain wall annihilation.
Thirdly, we have assumed that the domain wall annihilation takes place well after the domain wall network start to follow the scaling law. It takes a few Hubble time after the formation to reach the scaling regime, and therefore we conservatively require where H form and H ann are the Hubble parameter at the domain wall formation and annihilation, respectively.
Fourthly, we require that the decay constant is larger than the quantum fluctuations of the axion to ensure the validity of analysis using the potential (4). Specifically, we impose a lower bound on f as where H inf is the Hubble parameter during inflation. If this bound is not satisfied, the corresponding U(1) symmetry may be restored, or the saxion field may be destabilized.
Finally we assume that there is (effectively) only single path connecting the two vacua a 1 and a 2 . Apparently this is not satisfied if a U(1) symmetry is explicitly broken down to In the numerical calculations we impose the above conditions to ensure successful domain wall baryogenesis. It turns out that all the conditions are easily satisfied for the parameters of our interest.

C. Numerical calculations
The net lepton asymmetry is effectively induced by the domain wall annihilation, during which domain walls sweep typically about a half of the space. To model the domain wall dynamics during the annihilation, we approximated the situation by a single domain wall passing through the origin, where we numerically solve the Boltzmann equation (13), combined with the evolution equations for the energy density of the inflaton (ρ I ) and radiation (ρ r ),ρ where Γ I is the decay rate of the inflation, and we define the reheating temperature in our analysis by 3H(T R ) = Γ I . This approximation is valid because no net asymmetry is induced during the scaling regime, and so, we can focus on the domain wall dynamics during the one or a few Hubble time before the annihilation.
In Fig. 1 we show the induced lepton asymmetry as a function of the domain wall annihilation temperature for various values of the reheating temperature. In the top and bottom panels, we have set the axion mass to be m = 10 11 GeV and 10 12 GeV, respectively.
Here we have not taken into account the entropy production by the subsequent axion decay, which we shall return to in a moment. As expected, the maximal asymmetry is obtained when T ann min(T dec , T R ), in good agreement with the analytic estimate (18).
In the bottom panel, one can see that the lepton asymmetry is highly suppressed in the case of e.g. T ann > T dec and T R = 10 14 GeV. This is because the asymmetry induced by the domain wall annihilation is subsequently washed out by the L-number violating processes in equilibrium. In general, we expect that the wash-out process is efficient when At the domain wall annihilation, marginally relativistic axions are copiously produced, and they may come to dominate the Universe before they decay into gauge bosons. Once the axion dominates the Universe, its subsequent decay produces a large entropy, diluting pre-existing asymmetry. Thus, the final baryon asymmetry is fixed after the axion decay, if there is entropy dilution. Taking into account the sphaleron process 5 , the resultant baryon asymmetry is estimated as where ∆ is the dilution factor by the axion decay given by .
The numerical factor 1/2 comes from the fact that the transition from the false vacuum to the true vacuum takes place in about half of the whole space.
In Figs. 2 and 3 we show the contours of the final baryon asymmetry, n B /s, in the m-f plane for various values of T R . Here we have set T ann = min(T dec , T R ) so that the baryon asymmetry takes the largest possible value for a given reheating temperature. The baryon asymmetry can be suppressed by either increasing or decreasing T ann (see Fig. 1(a)). One can see that a sufficient amount of baryon asymmetry, n B /s 10 −10 , can be generated for T R 2 × 10 11 GeV. In the lower shaded (magenta) region, there is no entropy dilution, i.e., ∆ 1, and so, n B /s takes a constant value. As f becomes large, n B /s decreases owing to the entropy dilution factor ∆ 1. This is because, as f increases, the energy density of the axion particles increases and the lifetime of the axions becomes longer. The horizontal dashed (green) lines and dash-dotted (cyan) lines represent the lower bound on the axion decay constant, f H inf /2π, for H inf = 10 14 GeV and σH > µ eff n L | H=m , respectively (cf. (20) and (23)). The yellow-shaded region in upper right corner in Fig. 3 is ruled out from the domain wall domination at annihilation. Below the dotted (blue) line, baryonic isocurvature perturbations and their non-Gaussianity would exceed the observational bound, if the L-number violating rate (14) is valid at the domain wall formation. In other words, in the region slightly below the dotted (blue) line, baryonic isocurvature perturbations and their non-Gaussianity may be found in the near future observations. We will discuss this issue in the next subsection.  Finally, the domain-wall dynamics toward the scaling regime will also induce the baryon isocurvature perturbations. For domain walls to be formed, or more precisely, for infinitely long domain walls to be formed, the probabilities to realize the two vacua must be comparable, but they do not have to be exactly equal to each other. It implies that, when domain walls are formed, the spatial volume of one of the vacua is generically larger (or smaller) than that of the other by (at most) a few tens of percent. The ratio of the two volumes will quickly converge to unity as the domain-wall network approaches the scaling evolution. This is because the two vacua are degenerate in energy and there is no preference to one over the other once the scaling regime is reached. In this process toward the scaling regime, there is an overall transition from one of the vacua to the other, which similarly induces the baryon asymmetry. Let us denote the asymmetry by Y DW,form . As the bias of the spatial volumes is induced by the quantum fluctuations of the axion, Y DW,form has isocurvature fluctuations at large scales. The magnitude of Y DW,form is expected to be comparable to Y osc , and the sign is opposite. So, there is a partial cancellation, but in general, there is no exact cancellation. For our scenario to work, both Y osc and Y DW,form must be sufficiently suppressed, since otherwise the baryonic isocurvature perturbations and their non-Gaussianity, would be too large to be consistent with observations.
The baryon asymmetry generated at the domain-wall formation can be suppressed as follows. If the lepton-number violation processes are in equilibrium between the formation and annihilation of domain walls, the initial asymmetry Y osc and Y DW,form can be washed out. This is the case if the reheating temperature is higher than ∼ 10 13 GeV. The current constraint on the matter isocurvature perturbation S from the Planck observation reads P S < 8.7 × 10 −11 [37]. Using the fact that baryon isocurvature perturbation is written as P

IV. DISCUSSION AND CONCLUSIONS
Collapsing domain walls are cosmological sources of gravitational waves [11][12][13]. The gravitational wave spectrum is peaked at a frequency, f peak 160 kHz ξ −1/2 g * 106.75 corresponding to the Hubble horizon scale at the domain wall annihilation [14]. Here ξ and T X are defined as ξ = min 1, For successful baryogenesis, T X must be higher than 2 × 10 11 GeV, and so, the peak frequency is at O(100) kHz or higher, which is too high to be detected by near future observations. We note however that there have been proposed several new detection techniques with the sensitive frequency region around MHz [39,40], which may be able to probe gravitational waves produced in our scenario.
So far, we have considered the L-number violating processes mediated by heavy righthanded neutrinos in the seesaw mechanism. Other types of the baryon/lepton violating operator is also possible and the corresponding decoupling temperature for the baryon/lepton violating processes could be lowered. One of the examples is the R-parity violating operator, in the supersymmetric Standard Model. In this case, the interaction rate for the Lviolating processes scales as Γ ∝ T 5 for T m˜ and Γ ∝ T for T m˜ , where m˜ is the slepton mass. For instance, if we take λ ∼ 10 −8 and m˜ 10 9 GeV, the L-violating process marginally reaches equilibrium and soon decouples at T dec ∼ 10 9 GeV. Since the maximal possible value of lepton asymmetry is roughly given by n L /s ∼ 0.1T dec /M P from the first equality in (18), successful baryogenesis is possible with T ann ∼ 10 9 GeV. In this case, the peak frequency of the gravitational waves from the domain wall annihilation can be within the sensitivity range of the ground-based detector such as advanced-LIGO [41] and KAGRA [42,43]. For instance, if we take T R ∼ m ∼ 10 9 GeV and f ∼ 10 13 GeV, domain walls dominate the Universe at the annihilation and the peak frequency falls in the sensitivity range of these experiments. A naive order-of-magnitude estimate suggests, however, that the signal strength is a few orders of magnitude smaller than the predicted sensitivity, and either some deviation from the scaling regime or further improvement of the sensitivity would be necessary to directly probe such signals.
In this paper we have proposed a baryogenesis scenario using axion domain walls. Axion domain walls are produced if the axion acquires sufficiently large quantum fluctuations during inflation or if it initially stays sufficiently close to the local maximum. While no net baryon asymmetry is produced in the scaling regime, collapsing axion domain walls produce a large enough baryon asymmetry to explain the observed value. This is because the energy bias between the two vacua, and therefore between baryons and anti-baryons, becomes relevant only when domain walls annihilate. In particular, baryon isocurvature perturbations can be significantly suppressed in our scenario, either because the asymmetry produced by the initial field configurations is washed out by the L-number violating interactions in equilibrium, or because the L-number violating interaction is simply suppressed at the domain wall formation. In some parameter region, baryon isocurvature perturbations and their non-Gaussianity are suppressed, but non-negligible, which may be detected by future observations. Our scenario works together with high-scale inflation which predicts a large tensor-to-scalar ratio within the reach of future B-mode observations. The required relatively high reheating temperature can be realized in high-scale inflation more easily. This should be contrasted to other spontaneous baryogenesis scenarios in which the inflation scale is severely constrained by the isocurvature perturbations.
Although we have focused on the axion domain wall throughout this paper, our analysis can also be straightforwardly applied to a wide class of domain walls such as the Standard Model Higgs domain wall [44,45].