QCD Axion Dark Matter from a Late Time Phase Transition

We investigate the possibility that the Peccei-Quinn phase transition occurs at a temperature far below the symmetry breaking scale. Low phase transition temperatures are typical in supersymmetric theories, where symmetry breaking fields have small masses. We find that QCD axions are abundantly produced just after the phase transition. The observed dark matter abundance is reproduced even if the decay constant is much lower than $10^{11}$ GeV. The produced axions tend to be warm. For some range of the decay constant, the effect of the predicted warmness on structure formation can be confirmed by future observations of 21 cm lines. A portion of parameter space requires a mixing between the Peccei-Quinn symmetry breaking field and the Standard Model Higgs, and predicts an observable rate of rare Kaon decays.

Introduction.-The CP violation in QCD [1], expressed by the so-called θ parameter, is extremely small -θ < 10 −10 [2]. The smallness of the CP violation is elegantly explained by the Peccei-Quinn (PQ) mechanism [3,4]. One introduces a spontaneously broken global symmetry which is explicitly broken by the QCD anomaly, and predicts a pseudo-Nambu-Goldstone boson called an axion [5,6]. For a large enough symmetry breaking scale, the axion is stable and a dark matter candidate [7][8][9].
Two production mechanisms of axion dark matter in the early universe are widely recognized. One is the misalignment mechanism [7][8][9], where the displacement of the axion field from the vacuum turns into oscillations which behave as dark matter. Another is the emission of axions from the string-domain wall network produced after the spontaneous breaking of the PQ symmetry [10][11][12][13]. Both mechanisms require that the axion decay constant f a is large -f a > ∼ 10 11 GeV. (The estimation of the abundance in the latter mechanism assumes a scaling law of topological defects. See [14][15][16] for a possible violation of the scaling law and its impact on axion abundance.) In this letter, we point out a new production mechanism for axion dark matter under the assumption that the phase transition temperature of the PQ symmetry breaking is far below the symmetry breaking scale. We find that axions are abundantly produced via parametric resonance arising from oscillations of the symmetry breaking field [17][18][19][20] after the phase transition. Since the phase transition temperature is low, the axions are not thermalized and remain as dark matter. The axions produced from the late time phase transition can explain the observed dark matter abundance even if the decay constant is much smaller than 10 11 GeV.
Low phase transition temperatures are natural in supersymmetric theories. This is because the radial direction of the PQ symmetry breaking field, commonly called the saxion, is the scalar partner of the nearly-massless axion. The mass of the saxion is given by a supersymmetry breaking soft mass and is much smaller than the PQ symmetry breaking scale. This small mass in turn yields a relatively low phase transition temperature.
In contrast to the two conventional mechanisms, the produced axions are initially relativistic and red-shift sufficiently to be dark matter. In some of the parameter space, the axions are still warm enough to affect structure formation by an observable amount.
There are intensive ongoing and future experimental efforts to search for the axion with a small decay constant, such as IAXO [21,22], TASTE [23], Orpheus [24], MADMAX [25], ARIADNE [26,27], and many others [28][29][30]. Astrophysical observations and the above searches for the QCD axion could probe the dynamics of the PQ phase transition.
Late time PQ phase transition.-We consider a coupling of the PQ symmetry breaking field P to new PQ charged fermions ψ andψ, The new fermions may be identified with the hidden quarks of the KSVZ model [31,32]. This coupling gives a thermal mass to P , where m is the mass of P around the origin. As the temperature drops below the critical temperature the PQ symmetry is spontaneously broken. For m f a , the critical temperature may be significantly below the PQ symmetry breaking scale.
While small mass values are typically encountered in supersymmetric theories, PQ symmetry breaking scales much larger than the mass m are naturally obtained through the stabilization of P by a higher dimensional interaction [33], with n > 2, or through the renormalization group running of the soft mass of P [34], The parameters of these potentials have been chosen such that the saxion mass around the vacuum is m s (∼ m), the vacuum expectation value of |P | is f a / √ 2, and the vacuum energy vanishes.
In both cases, the PQ symmetry breaking field at the origin has a potential energy density ∼ m 2 s f 2 a . This is larger than the radiation energy density ∼ T 4 c if which we assume in the following. The case with smaller y can be analyzed in a similar manner. Since the potential energy dominates, a period of so-called thermal inflation [35,36] occurs with a Hubble scale H PT ∼ m s f a /M Pl before the phase transition.
Axions from inhomogeneity.-Just after the phase transition, the configuration of the PQ symmetry breaking field is inhomogeneous. The correlation length r c of the configuration is determined by the Kibble-Zurek mechanism [37,38]. As the Universe approaches the PQ phase transition, the mass of the PQ symmetry breaking field nearly vanishes and the correlation length of the field grows. The correlation length is frozen when the time before the transition is equal to the equilibrium relaxation time. This correlation length is given by [39] The gradient energy density of the inhomogeneity is f 2 a /r 2 c . Typically one cosmic string per correlation length volume, r 3 c , exists with energy density f 2 a /r 2 c . The inhomogeneous configuration is quickly homogenized until the correlation length becomes as large as the horizon size. The gradient and string energy should be emitted as axions with typical wavelength r c . The number density of axions produced from the inhomogeneity is then The potential energy density m 2 s f 2 a is converted into the oscillation energy of the saxion, which subsequently red-shifts in proportion to the inverse cube of the scale factor of the universe. We thus normalize the number density of axions by the energy density of the saxion oscillation, n a /ρ s , which does not change under red-shifting. For axions coming from the inhomogeneity, Axions from parametric resonance.-The oscillations of the saxion after the phase transition also produce axions via parametric resonance. Just after the phase transition, the saxion oscillates with an amplitude on the order of f a . The production rate of axions via parametric resonance is then as large as the frequency of the oscillations ∼ m s [20]. On the other hand, the Hubble expansion rate is as large as m s (f a /M Pl ) m s . Thus the parametric resonant production of axions is very efficient.
Parametric resonance creates axions with momenta m s /2 and continues until the newly produced axion energy density is roughly equal to the initial saxion energy density, m 2 s f 2 a . We label this second contribution to the axion density as n PR a , so that One might be concerned that the inhomogeneity caused by the phase transition could ruin the parametric resonance process, which requires coherent oscillations. We do not anticipate that this is the case since the wavelengths in the resonance band are ∼ 1/m s , which is much shorter than the length scale on which the PQ symmetry breaking field is correlated, r c . Hence the oscillations are effectively coherent for the modes in the resonance band.
Comparing Eqs. (9) and (10), we see that for f a < M Pl , the parametric resonance axions are the dominant contribution to the axion population. In what follows, we only take into account the parametric resonance axions. With this, one finds that the axion number density normalized by the entropy density s is where T RH is the reheat temperature after the thermal inflation. To obtain the axion dark matter abundance the reheat temperature T RH must be above If T RH is higher than this value, axions are overproduced, but the introduction of extra entropy production from heavy fields can generate the correct abundance. Obtaining this reheat temperature is discussed below. Axion Warmness.-The axions are produced relativistically and may behave as warm dark matter. To be concrete, we consider a specific model with the PQ potential in Eqs. (4,5). The expressions in the following depend on the potential, but a similar analysis can be performed for more general potentials.
To estimate the warmness, we first note that the ratio between the axion momentum, k a , and the cube root of the axion number density, n 1/3 a , is constant throughout the evolution of the universe, k a where n is an integer larger than 2 in Eq. (4) or 1 for the potential in Eq. (5). Here it is assumed that half of the potential energy of the saxion is transferred into axions with momenta k a = m s /2. Using the observed dark matter abundance, we obtain v a 6 × 10 −4 n 1/3 f a 10 9 GeV T eV (15) for the velocity of the axions at temperature T . We have assumed T MeV to express the entropy density in terms of T . In Fig. 1, we show contours of the axion velocity at T = 1 eV for n = 3.
The constraint on the warmness of dark matter is frequently estimated for a model where dark matter consists of a massive Weyl fermion with mass m WDM that decouples while relativistic and is later diluted. In such a model, the typical velocity of dark matter, v WDM , at temperature T is given by This result, combined with the warm dark matter mass bound, m WDM > 3.3 keV [40], yields the generic velocity bound of v < 10 −4 at T = 1 eV. This warmness bound imposes the following constraint on the saxion mass The green shaded region in Fig. 1 is disfavored by this constraint. Future observations of 21cm lines can probe m WDM < 10-20 keV [41], which corresponds to v a > ∼ 10 −5 at T = 1 eV, as indicated by the arrow in Fig. 1. For convenience, we provide the correspondence between the mass of this fermionic dark matter and the parameters of our model, Other astrophysical constraints.-In addition to the warmness bound, we consider constraints from the cooling of red giant (RG) and horizontal branch (HB) stars [42][43][44] by the emission of saxions. We follow the analysis performed in [45,46]. For RG and HB stars, one must demand that the energy transport by new particles with effective nucleon couplings not exceed 10 erg g −1 s −1 . These constraints are displayed as the blue region in Fig. 1 labeled as "RG & HB".
The orange shaded and dashed excluded parameter regions in Fig. 1 labeled "SN1987A or N eff " arise from the SN1987A constraint of [47], as well as the constraint on the effective number of relativistic degrees of freedom N eff [48]. For SN1987A, the energy loss should not exceed 10 19 erg g −1 s −1 . This leads the the boundary of the orange shaded region as well as the orange dashed curve. If one takes the SN1987A constraint on energy loss directly, the region below the orange dashed curve would be excluded. However, if one assumes a strong enough coupling between the saxion and the Standard Model Higgs, the saxion enters the so-called trapping regime through this saxion-Higgs mixing, and the region below the dashed orange curve is permitted. The orange shaded region remains constrained since the mixing keeps the saxion in thermal equilibrium with electrons even after neutrinos decouple in the early universe. Hence the depletion of the saxion energy heats up photons, resulting in N eff < 3. Assuming that neutrinos suddenly decouple at T 2 MeV, we determine a lower bound on the saxion mass of m s > ∼ 4 MeV. The purple shaded region is the bound related to axions arising from SN1987A [49][50][51][52][53].
We also note that there is at least an order of magnitude uncertainty in the SN1987A constraints [54][55][56][57][58][59]. This could lead to a larger parameter space.
The saxion-Higgs mixing results in rare decays of Kaons. The large mixing in the trapping regime can be probed by NA62 and KLEVER experiments [60][61][62].
Thermalization of saxions and axions.-The saxion should be thermalized at or above the temperature T DM in Eq. (13). We consider the case where the PQ symmetry breaking field P couples to a pair of new fermions f andf via the Yukawa coupling where µ is the mass of the fermion. For T > µ, the saxion thermalizes with a rate 0.1T µ 2 /f 2 a [63,64], leading to a reheating temperature T RH 100 GeV µ 100 GeV If the fermion is charged under the Standard Model gauge group, the mass µ must be above 100 GeV. This reheating temperature is larger than the lower bound given in Eq. (13) above, and more than enough axion dark matter is produced. T RH = T DM can be obtained through thermalization from coupling the saxion with Standard Model particles or with particles that are neutral under the Standard Model gauge group.
Axions produced in our model are never thermalized. The thermalization rate of an axion is suppressed by the decay constant and the momentum of the axion [65], where b is a constant which depends on the axion coupling. If the axion couples to gluons, b is loop-suppressed and is as small as 10 −5 . If instead the axion couples to a light fermion in the thermal bath, b may be as large as O(1). During the matter dominated era by the saxion oscillation, the momentum of axions is given by The energy density of the thermal bath never exceeds that of the saxion. Hence the thermalization rate is bounded from above, The ratio between the thermalization rate and the Hubble expansion rate is where the last inequality is saturated right after the phase transition. In the region of parameter space that produces sufficiently cold axion dark matter, the axions are never thermalized. One can see that the late-time phase transition is crucial. If the mass m s is as large as f a , the thermalization is effective. After the saxion decays and the radiation dominated era begins, the thermalization rate decreases faster than the Hubble expansion rate and the thermalization of the axions never becomes efficient.
Discussion.-We have investigated a production mechanism for QCD axion dark matter associated with PQ symmetry breaking at a low temperature. We find that axions are primarily produced by parametric resonance via oscillations of the PQ symmetry breaking field. The low phase transition temperature fits naturally in supersymmetric theories.
The axions produced by this mechanism tend to be warm. The prediction on axion warmness is shown in Fig. 1 and constrains the allowed parameter space. Future observations of 21cm lines will probe the parameter space further. Discovery of the QCD axion in laboratories and the determination of dark matter warmness by astrophysical observations will suggest that axion dark matter was produced by parametric resonance. Fig. 1 is one of the primary results of this letter and contains information beyond the warmness constraint. As outlined above, one also has bounds from energy loss in RG and HB stars and supernovae by saxion emission, as well as axion emission in the supernovae case. We note that our parameter space easily allows for rather low values of the axion decay constant, particularly if strong saxion-Higgs coupling occurs to trap saxions inside the supernova core, or if the traditional SN1987A bound is loosened. The region with large saxion-Higgs mixing can be probed by observations of rare Kaon decays.
There are several uncertainties in our estimation of the warmness. First, we have assumed that half of the energy density of the saxion oscillation is transferred into axions. In reality the transferred fraction will not be exactly half. Second, we have assumed that the momentum as well as the number density of the axions decrease only by the cosmic expansion. However, the momentum/number density can slowly increase/decrease by axion self interactions, see [66] for a related discussion. These two effects will change the prediction on axion velocity by an O(1-10) factor. Whether or not the whole parameter space can be probed depends on these uncertainties, which can be fixed by numerical computation.
We list other known mechanisms to produce axion dark matter for f a 10 11 GeV; 1)axion emission from long-lived topological defects which collapse via explicit PQ symmetry breaking [11,[67][68][69][70], 2)parametric resonant production of axions from oscillations of the saxion with a large initial field value [71], 3)a misalignment angle fine-tuned to be close to π [72][73][74], 4)dynamical mechanisms that set the misalignment angle close to π [75,76], 5)the misalignment mechanism with nonstandard cosmology [77], and 6)delayed oscillations of the axion field because of a non-zero kinetic energy of the axion field [78,79]. Among them, 2) can also produce warm axions, but the produced axions are much colder than our mechanism for a given (m s , f a ). We also note that the large field value assumed in [71] requires that the potential be flat for large field values and is therefore incompatible with the potential in Eq. (4).
Our axion production mechanism involves a PQ symmetry breaking field that is initially trapped at the origin. We may consider a generic situation where a PQ symmetry breaking field is trapped at some other point in field space and later begins to oscillate with a large amplitude.
One example is a model with the superpotential where P andP are PQ symmetry breaking fields and X is a chiral multiplet that fixes them on the moduli space PP = V 2 PQ . The moduli space is lifted by additional superpotential terms that spontaneously break supersymmetry [80][81][82], or by the soft masses of P and P . Ref. [83] investigates the trapping of the PQ symmetry breaking fields on the moduli space by a thermal potential and finds that oscillations occur in some region of the parameter space. Axions should then be produced via parametric resonance in this setup as well.