QCD Axion Dark Matter with a Small Decay Constant

The QCD axion is a good dark matter candidate. The observed dark matter abundance can arise from misalignment or defect mechanisms, which generically require an axion decay constant $f_a \sim \mathcal{O}(10^{11})$ GeV (or higher). We introduce a new cosmological origin for axion dark matter, parametric resonance from oscillations of the Peccei-Quinn symmetry breaking field, that requires $f_a \sim (10^8 -10^{11})$ GeV. The axions may be warm enough to give deviations from cold dark matter in Large Scale Structure.

The QCD axion is a good dark matter candidate. The observed dark matter abundance can arise from misalignment or defect mechanisms, which generically require an axion decay constant fa ∼ O(10 11 ) GeV (or higher). We introduce a new cosmological origin for axion dark matter, parametric resonance from oscillations of the Peccei-Quinn symmetry breaking field, that requires fa ∼ (10 8 − 10 11 ) GeV. The axions may be warm enough to give deviations from cold dark matter in Large Scale Structure.
Introduction.-The absence of CP violation from QCD is a long-standing problem in particle physics [1] and is elegantly solved by the Peccei-Quinn (PQ) mechanism [2,3] involving a spontaneously broken anomalous symmetry. The scheme predicts the existence of a light boson [4,5], the axion, which is constrained by data to be extremely light and hence is stable on cosmological time scales and a dark matter candidate.
What cosmological production mechanism yields axions that account for dark matter? Axions produced from the thermal bath have too low an abundance to be dark matter, and are too hot. Two production mechanisms have been considered; both are IR mechanisms with axions produced near the QCD phase transition. While they depend on the PQ symmetry breaking scale, f a , they are insensitive to details of the UV axion theory and dynamics of the PQ phase transition. i) Initially the axion is nearly massless, and takes a generic field value misaligned by angle θ mis from the vacuum value. Around the QCD phase transition the axion obtains a mass, and the energy density in the resulting oscillations account for the dark matter abundance [6][7][8] Ω a h 2 | mis 0.01 θ 2 mis f a 10 11 GeV 1.19 . (1) ii) When PQ symmetry is broken after inflation, cosmic strings are produced [9]. Domain walls form between these strings around the QCD phase transition and, if the domain wall number is unity, the string-domain wall network is unstable and decays into axions [10], yielding a dark matter density [11,12] Ω a h 2 | string−DW 0.04-0.3 f a 10 11 GeV 1.19 . (2) These mechanisms most naturally lead to axion dark matter for f a ∼ (10 11 −10 12 ) GeV, and the misalignment mechanism could also yield dark matter for larger f a as θ mis is reduced. However, axions are not expected to be dark matter for smaller f a 1 . In this letter we introduce a mechanism for UV production of dark matter axions from the early evolution of the PQ symmetry breaking field S in a relatively flat potential. We assume S has a large initial value, S i f a , e.g. from a negative Hubble induced mass during inflation [20]. After S begins oscillating, parametric resonance [21,22] creates a huge number of axions. Assuming no subsequent entropy production where m S,i is the initial S mass. Unlike (1) and (2), the abundance grows at low f a and sufficient dark matter can be obtained for 10 8 GeV < f a < 10 11 GeV. Several experimental efforts are ongoing for axions with small f a , including IAXO [23,24] and TASTE [25] for solar axions, Orpheus [26] and MADMAX [27] for halo axions, and ARIADNE [28,29] for axion mediated CP-violating forces. See Refs. [30][31][32] for other proposals. Helioscopes will explore the lower end of this range, and it is worth noting that our mechanism allows values of f a even below 10 8 GeV, which are also consistent with the supernova bound [33][34][35][36][37] if there is a mild cancellation in the axionnucleon coupling. In our mechanism, axions are initially produced with momenta of order m S,i . After subsequent red-shifting, the axion velocity at a temperature of 1 eV is 1 The special cases for axion dark matter with lower fa are as follows. Refs. [13][14][15] study the anharmonicity effects of the axion cosine potential when θ mis is tuned to approach π, requiring a small inflation scale to avoid quantum fluctuations. Ref. [16] analyzes axion production in non-standard cosmological eras with kinetic energy domination even at temperatures below a GeV.
Refs. [11,[17][18][19] consider topological defects for a domain wall number larger than unity with an explicit PQ breaking, where a fine tuning is required to solve the strong CP problem.

arXiv:1711.10486v2 [hep-ph] 26 Jul 2018
For a sufficiently large m S,i , the axion is warm. Axion production by oscillating PQ field.-The axion mass is [4] m a = 6 meV 10 9 GeV f a .
The axion number density n a0 that explains the observed dark matter abundance, ρ DM /s 0.4 eV, is where s is the entropy density of the Universe. The required Y a is much larger than thermal for all f a ≥ 10 8 GeV, so axions must be produced non-thermally. The radial direction of the PQ symmetry breaking field, which we call the saxion, S, (following terminology of supersymmetric theories) is taken to have a large initial field value. As the Hubble scale H becomes smaller than the saxion mass, the saxion begins coherent zeromode oscillations. The mass m S (S) ≡ V (S) is generally S dependent and differs from the vacuum value. For a wide range of potentials V (S), the saxion yield is conserved with the initial value in a radiation-dominated universe, where M Pl 2.4 × 10 18 GeV. A result for the matter-dominated case can be straightforwardly derived. Suppose that this saxion yield is converted into axions, e.g. by decay. For f a S i < M Pl , the dark matter axion yield of Eq. (6) results only if m S,i S i , requiring V (S) to be flat. For a saxion to axion conversion rate Γ * , at an oscillation amplitude S * , the axion momentum p a (T ) is assuming that axions are produced with momenta ∼ m S (S * ) ≡ m S * . Axions are colder for a larger Γ * since they receive more red-shifting after production. Given the recent constraint on the warmness of dark matter (see e.g. [38,39]), we require that the axion velocity is smaller than 10 −3 at a temperature of 1 eV, placing a lower bound on Γ * . Let us assume that saxion to axion conversion occurs via a perturbative decay rate of Γ * ∼ m 3 S * /S + 2 , where S + = max(S * , f a ). The number density and momenta of the axions are given by One can verify that the bound from coldness, (9), and the abundance requirement, (6), are incompatible with each other for any (S * , m S * ), for all f a larger than the experimental lower bound of 10 8 GeV.
Parametric Resonance-In fact, the production rate is typically much larger than the perturbative decay rate. The saxion couples to the axion through the potential of the PQ breaking field, so that the axion mass oscillates. Axion modes with non-zero momenta, which we call fluctuations, grow rapidly by parametric resonance [21,22] from initial seeds set by quantum fluctuations. The axion momentum is typically of order m S . The energy of the fluctuations grows exponentially and at some point, typically soon after oscillations begin, becomes comparable to that of the saxion oscillation. At this stage, the back reaction on the saxion is non-negligible and parametric resonance ceases. Due to efficient scattering between the oscillating saxion field and the axion fluctuations, the entire zero-mode energy of the saxion is converted into comparable saxion and axion fluctuations, Y S ∼ Y a ∼ Y Si , given in Eq. (7).
The evolution of the fluctuations after this stage is model-dependent, and a model-by-model lattice simulation is needed to rigorously follow the dynamics. In this letter we consider the case where the co-moving momentum and number density of the fluctuations are approximately conserved, so that the axion yield is comparable to that of the original saxion oscillation, given by Eq. (7).
It is striking that this production mechanism, with Γ ∼ m S,i , occurs at a very early time. This allows large redshifting after production, giving axion momenta that easily satisfy the coldness constraint (9) for sufficiently small saxion masses. So far we assumed that entropy is conserved after axion production, but this depends on the fate of the saxions. To avoid overclosure, the energy of the saxions must be transferred to the thermal bath with some rate Γ s and hence thermalize at T th . However, thermalization after matter domination generates entropy, leading to an axion yield that is independent of S i where m S,0 ≡ m S (f a ) is the vacuum saxion mass.
In the following, we apply the above mechanism to a few simple models of PQ symmetry breaking, giving results with and without matter domination by saxions.
Production of axions via parametric resonance is discussed in the literature. Refs. [40][41][42] consider production of QCD axions by oscillations of the PQ symmetry breaking field, with axions identified as dark radiation. Ref. [43] investigates production of axion-like particle dark matter via derivative couplings with a scalar condensation. However, these papers do not consider production of QCD axion dark matter from parametric resonance.
Quartic potential.-We consider a potential of the PQ symmetry breaking field P , The saxion mass is field-dependent, m S √ 3λS for S f a and m S,0 = √ 2λf a . The saxion begins to oscillate when the Hubble scale is λS i / √ 3 ≡ H osc . After the saxion oscillation energy is transferred to axion and saxion fluctuations, the fluctuation momentum, initially of order λS, slowly grows via number-reducing scatterings as k ∝ R 1/7 [44] when the fluctuation amplitude is larger than f a . Here R is the scale factor of the Universe. The evolution of the momentum for a smaller amplitude is not known, but we expect that the growth of the momentum becomes ineffective since the interaction term λ 2 |P | 4 is less effective. The overall growth of the momentum is at the most O(10) in the parameter space of interest, and we ignore it.
First we discuss the case of rapid thermalization of the saxion fluctuations so that they never dominate. The axion yield and momentum are given by p a s 1/3 giving constraints on the parameter space, m S,0 400 MeV S i 2 × 10 17 GeV The axions should not be thermalized. For small f a thermalization seems inevitable, but in our mechanism it is suppressed by the large field value of S. The axions scatter with the thermal bath with a rate During the zero-mode saxion oscillation S eff may be much smaller than the amplitude of the oscillation as P coherently gets small during the oscillation [45]. However, S eff is still large enough to prevent thermalization. After the saxion oscillation energy is converted into fluctuations, S eff is of order the amplitude of the fluctuation and decreases in proportion to the temperature, until it becomes of order f a , after which S eff = f a . The ratio of the scattering rate to the Hubble scale is maximized when S eff after removing f a using the constraint Eq. (16) for the dark matter abundance. The upper panel in Fig. 1 summarizes the constraints on m S,0 and S i . On each contour, dark matter is explained by axions produced from parametric resonance for the corresponding f a . In the pink region, the axions are too warm to be dark matter; the y-axis on the right labels the axion velocity v a at T = eV. In the yellow region, the resonantly produced axions thermalize and are not dark matter. The constraint leading to the orange region is explained later.
Once the zero-mode saxion oscillation is transferred into fluctuations of axions and saxions, their energy density evolves as radiation. After S eff becomes of order f a , the saxion behaves as matter, and may dominate the energy density of the universe. The axion abundance is then given by Eq. (12), and the axion momentum is With this dilution, the dark matter abundance requires the saxion mass to be smaller than in Eq. (16). The constraint from axion thermalization is derived in a similar manner to Eq. (18), and is given by The constraints on parameters with a saxion domination era are shown in the lower panel of Fig. 1. The region to the right of each color line is excluded for the corresponding f a . In regions between the orange boundary and the colored lines, the dark matter abundance is explained with T th given by Eq. (12). Quadratic potential.-In supersymmetric QCD axion models, the saxion potential may be approximately quadratic. One example is a two-field model with superpotential and soft supersymmetry breaking terms Another example is a one-field model with a potential given by soft supersymmetry breaking and radiative corrections [46][47][48], For simplicity we approximate the dynamics of the saxion as the oscillation by an almost constant quadratic term with m S,i m S,0 ≡ m S ∼ m P , m.
In these models, the saxion starts oscillating when the Hubble scale is H osc m S /3. We parametrize the time of parametric resonance as t ≡ N p /m S . After the zeromode saxion oscillation energy is transferred to axion fluctuations, the axion yield is of order that of the original saxion in Eq. (7). The axion momentum is We have checked with a lattice simulation using LAT-TICEEASY [49] that, in the one field model of Eq. (22), the number density is conserved, the axion momentum does not grow due to number-reducing scatterings, and N p = O(100). We discuss this issue in detail in a separate publication [50]. The results below also apply to the two-field theory if the comoving number density and momentum do not vary much.
We consider first the case that saxions thermalize before they dominate. The abundance and coldness of axion dark matter then require m S 2 GeV S i 10 15 GeV The former is derived in a similar manner as for the quartic potential. When the saxion transfers its energy to the thermal bath before it dominates, S eff suddenly drops to f a , enhancing Γ a . The latter bound is derived by conservatively assuming that saxions are destroyed at the temperature of saxion domination. If saxions are destroyed at higher temperatures, the constraint becomes stronger. The bound is stronger than the one from the warmness of the axion only for f a 10 9 GeV. In the case of an era of saxion domination, the axion abundance is given by Eq. (12), and the momentum is The saxion mass must be smaller than given in Eq. (24). Axion thermalization places an upper bound on m S , The former (latter) case is for thermalization during the radiation-dominated (saxion-dominated) era. Formulae to derive the latter bound can be found in the appendix of Ref. [51]. Finally, when the saxion is destroyed S eff suddenly drops to f a , and the axion thermalization rate is enhanced, which gives an upper bound on m S , m S < 2 × 10 4 GeV f a 10 10 GeV .
The constraints on the quadratic theory are shown in Fig. 2. In comparison with the quartic theory, a large vacuum mass of the saxion is allowed, since the mass is independent of the saxion field value.
For S i smaller than 6 × 10 15 GeV, saxions begin oscillating in the thermal potential generated by the free energy of quarks and gluons [52]. (In principle, this can be avoided by giving a large mass to gluons.) This not only changes the saxion evolution, but may also lead to formation of clumpy objects such as Q-balls [53] and Iballs/oscillons [54][55][56], which may drastically change axion production. We discuss this in a future work [50].
The fate of the saxion.-Remnant saxions must be removed without causing cosmological problems. If saxions, with yield (6), do not decay or scatter with the thermal bath, they dominate the energy density below temperature T dom , In this case, Eqs. (6) and (12) imply T th > m S,0 . Without saxion domination, the saxion must be thermalized at a temperature above T dom , larger than m S,0 . In both cases, T th > m S,0 is required, and hence scattering with the thermal bath is important for destruction. One may naively expect that scattering between axions and the thermal bath is also effective, thermalizing the axion. This is, however, not always true. Consider a coupling between P and the Higgs field H, V = κ|P | 2 |H| 2 . This induces a saxion-Higgs coupling, m 2 H S|H| 2 /f a , but not an axion-Higgs coupling. Here m 2 H is a contribution to the Higgs mass squared from PQ symmetry breaking. The saxion scattering rate with the thermal bath, before the electroweak phase transition, and for T > m S , is [45] With this interaction the saxion is thermalized below a temperature T th , which may be large enough. Thermalized saxions interact with standard model fermions via mixing with the Higgs. For sufficiently large m H , saxions are still in thermal equilibrium while nonrelativistic, and hence disappear without leaving any imprint by decaying into standard model fermions. Saxion masses below O(10) MeV are however excluded, for the f a of interest, from saxion cooling of supernovae [33][34][35][36][37]57]. For small m H , on the other hand, saxions decouple from the thermal bath while relativistic, and decay dominantly to a pair of axions at temperature T dec 1 MeV m S,0 10 MeV The resulting axions are observed as dark radiation of the Universe, with an energy density, normalized to one generation of neutrinos, given by , where g * (T ) is the number of degrees of freedom of the thermal bath at temperature T , and T D is the saxion decoupling temperature. The experimental upper bound is ∆N eff < 0.6 [58]. For m S,0 < 10 MeV, there is too much dark radiation (small m H ) or too rapid supernova cooling (large m H ), as shown by the orange region in Figs. 1 and 2.
Discussion.-In a wide class of theories, a large initial value of the PQ breaking field leads to cosmological production of axions via parametric resonance. The resulting dark matter abundance scales as 1/f a and dominates at low f a . Figures 1 and 2 show that, in theories with quartic and quadratic potentials, there are large regions of parameter space allowing axion dark matter with f a ∼ (10 8 −10 11 ) GeV. The parts of these regions close to the "too warm" boundaries have Large Scale Structure that deviates from that of cold dark matter. The required flatness of the potential leads to a mass hierarchy, m S,0 f a , that can be understood with supersymmetry. Parametric resonance randomizes the axion direction so that axions are also produced from the mis-alignment mechanism with θ mis = O(1), overproducing axions for f a 10 12 GeV. In the quartic theory, PQ symmetry is restored, leading to domain wall formation [59,60]. This might also occur in the one-field quadratic model. The domain wall number should be unity so that domain walls are unstable [61]. The unstable string domain wall network overproduces axions if f a 10 12 GeV. A numerical simulation [62] suggests that domain walls are not produced in the two-field quadratic model.
If the reheating temperature of the Universe is low enough, oscillations begin during an inflaton-dominated era; this case will be discussed in a future publication, along with a detailed analysis on thermal potentials [50].