On Axion Thermalization in the Early Universe

We reanalyze the conditions under which we have a primordial thermal population of axions. Compared with previous studies, we find other processes, involving gluons and quarks, that dominate at high temperatures. We conclude that if the Peccei-Quinn scale fulfills $F_a<1.2 \times 10^{12} \GeV$ there is thermal axion production. In this case, a period in the early universe exists where axions interact with the QCD plasma and we point out that non-thermal axions produced before the end of this period will thermalize. This could lead to a reduction of the expected density of axions from string decay in models with $F_a<1.2 \times 10^{12} \GeV$.


I. INTRODUCTION
The Peccei-Quinn solution [1] to the strong CP-problem introduces a new U(1) P Q chiral symmetry, the spontaneous breaking of which leads to a new spinless particle, the axion [2]. If it exists, the axion has quite well defined properties: its mass and couplings are inversely proportional to F a , the energy scale of the U(1) P Q breaking. We will be particularly concerned with the coupling of the axion field a to gluons where α s = g 2 s /(4π) with g s the gauge coupling of color SU(3) c . In (1), is the dual of the gluon field G b µν , and there is a sum over the color index b. Much theoretical study and experimental effort have been devoted to this particle (for recent reviews see [3]). The laboratory, astrophysical, and cosmological constraints imply that F a must be a high energy scale, at least as large as [4] F a > 6 × 10 8 GeV (3) which in turn implies m a < 0.01 eV (4) Thus, the axion has to be a very light particle and very weakly coupled.
One of the most attractive features of the axion is that it may be one of the constituents of the dark matter. Indeed, axions are copiously produced in the early universe. These primordial axions are generated mainly non-thermally. We know three non-thermal sources at work in the evolution of the universe: the "misalignment mechanism" [5], string decay [6] and domain wall decay [7]. When the Peccei-Quinn symmetry breaks down at a temperature T ≃ F a , the vacuum angle has no preferred value; but when the temperature reaches T ≃ Λ QCD , the vacuum angle sets where the vacuum energy is minimized, which is the CP-conserving value. Relaxation from one angle to the other produces a coherent, cold condensate of axions. This is the "misalignment mechanism". On the other hand, the U(1) P Q breaking at T ≃ F a originates axion strings. If inflation takes place just after that, these strings dilute away. However, if this is not the case, we have a second mechanism of production of axions when strings decay. Also, if the axion field is not homogeneized by inflation, there are models where axion domain walls appear soon after T ≃ Λ QCD . The decay of these walls into axions is a third mechanism of non-thermal primordial axions.
Thermal production of axions was studied by Turner [8]. His main interest was to study the possibility that thermal axions have a density greater than the corresponding to nonthermal axions, and then to analyze the potential detection of photons from axion decay.
Turner considered the Primakoff process, a + q ⇄ γ + q, where q is a light quark, and photoproduction, a + Q ⇄ γ + Q, where Q is a heavy quark, and showed that when axions were once in thermal equilibrium. Evaluating the axion density he found that for F a < 4 × 10 8 GeV (6) the thermal population is greater than the non-thermal.
In [8], it was mentioned that other axion processes contribute to axion thermalization and consequently would increase the value in (5). However, these other processes were not evaluated since the range (5) includes (6) and this was enough for the purposes of [8]. We note that the experimental bound in (3) excludes the range in (6) and leaves a small interval of the condition (5).
In this paper we extend the work of [8] and investigate the contribution to axion thermalization of other processes than Primakoff and photoproduction. As we will show, the processes we discuss lead to a much larger range for F a than (5).
The reactions we consider all involve the gga vertex appearing in (1), 2) a + q ⇄ g + q and a +q ⇄ g +q 3) a + g ⇄ g + g We show the corresponding Feynman diagrams in Figs. 1, 2, and 3.
The coupling of the axion to gluons (1) is an essential ingredient of the effective theory once the Peccei-Quinn symmetry is broken, since it reproduces the chiral anomaly of the theory. In this sense our results are model independent in contrast with the result (5) arising from Primakoff and photoproduction processes that involve the aγγ coupling which is model dependent and might be small in some models.

II. EVOLUTION OF THE AXION DENSITY
In the cooling of the universe, as soon as the temperature T ≃ F a is reached, massless axions can be produced by means of reactions 1), 2), and 3). The production rate has to compete with the expansion of the universe, characterized by the Hubble expansion rate, We will assume that, in the range of temperatures where we will apply the evolution equation for interating axions, the particle content corresponds to SU ( with 3 families and one Higgs boson, with all particles at the same temperature. Then, the relativistic degrees of freedom appearing in (7) are given by [9] g * = g boson + 7 8 g f ermion = 106.75 (8) The behaviour of the axion density n a is given in terms of the Boltzmann equation [8,9,10], Here, Γ is the thermal averaged interaction rate for the process a + i ⇄ 1 + 2 with dp ≡ d 3 p/(2π) 3 . The phase space occupancy in kinetic equilibrium f eq is given by the Fermi-Dirac or Bose-Einstein distributions, where we will take the relativistic limit where the chemical potential µ goes to zero. In eq (11), |M| 2 is summed over all the degrees of freedom. The equilibrium densities appearing in (10) are given by n eq = g dp f eq where g is the degrees of freedom of the particle. In the period of interest, thermal axions are massless and have a density where ζ(3) = 1.20206 . . . is the Riemann zeta function of 3.
We follow the convenient procedure [8,9,10] of normalizing particle densities to the entropy density, During the epoch in which we will apply the Boltzmann equation (9), we can use g * S = g * = constant, and given by (8). Then we have that is constant. We can write the Boltzmann equation (9) as where we have defined We will now show that equation (18) can be integrated in our case. First, we advance that Γ ∝ T 3 (see sec. III) so that Γ/H ∝ 1/x and we can define the constant Second, we use the fact that Y eq is independent of x. We define and, finally, we write equation (18) in the form that has the solution Since we start at x = 1 (T = F a ) with no axions, in (23) we have specified the initial condition η(x = 1) = 0.
From the solution (23) we see that η starts to grow at x = 1. At x = k (that is, Γ = H) axions decouple from the QCD plasma. We call Y d the value of Y at decoupling. For x > k the value of η remains constant, so we have Y = Y d . We are interested in the situation that axions have in practice a thermal spectrum, so we will ask that Y d differs from Y eq in less than 5%, which implies When k satisfies this inequality, a thermal population of axions is born in the early universe.
In the next section we calculate Γ, which, as we see from (20), will give us the clue of which is the corresponding range of F a .

III. CALCULATION OF THERMALLY AVERAGED INTERACTION RATES
As expression (11) is Lorentz invariant, we can evaluate it in any reference system. For convenience, we choose the center of mass system and will consider that all the particles are massless because we are at very high energies. So, we will write (10) as Γ = 1 n eq a dp a 2E a dp i 2E i f eq a f eq i · (2 sσ CM ) whereσ CM is the usual total cross section in the center of mass system, but with no average over the initial degrees of freedom, and s is the Mandelstam invariant. From the Lagrangian piece (1) one can easily find the Feynman rules for the couplings between axions and gluons that appear in the diagrams in Figs.1, 2, and 3, which are necessary to calculate the cross sections for the considered processes. Some of these cross sections diverge logarithmically in the t and u channel. The processes we consider take place in a plasma with vanishing global color and so color is effectively screened for distances bigger than m −1 D , with m D the QCD Debye mass given by [11] where N c = 3 is the number of colors and N f = 6 is the number of flavors. We cutt-off the divergences using the Debye mass. Cross sections have the form where A and B are constants, which for the considered processes have the numerical value Putting (28) in (26) we find integrals of the type that can be done analitically. We have where γ = 0.5772 . . . is the Euler's Gamma constant and ζ ′ (3) = −0.1981 . . .. With these results, the integral (26) gives where the sign ± depends on whether the particle that goes whith the axion is a gluon (−) or a quark (+). Thus, for each of the processes 1), 2), and 3) we get, respectively, Γ is the sum of all Γ i , and turns out to be where we have introduced α s ≃ 1/35, corresponding to energies E ≃ 10 12 GeV.

IV. CONCLUSIONS
Using (39), (7) and the definition (20) we have The inequality (25) translates into This is our main result. Models with F a satisfying (41) have axions thermalizing in the early universe and predict that today there is a thermal population of axions.
We get a much higher value than the one in (5). One of the reasons is that the interaction rate (39) increases as T 3 at high energies, while the rate coming from the photoproduction process considered in [8] goes as T since the axion is attached to the fermion line with the usual derivative coupling. The other reason is that color and flavor factors make the interaction rate larger.
Our result in (41) is in fact conservative. This is due to the fact that the effective theory below the scale F a has couplings of the axion to the electroweak gauge bosons W ± , W 0 , and B, similar to (1). This generates processes like 1), 2), and 3) with gluons replaced by appropiate combinations of W ± , W 0 , and B. These new processes contain in general quarks and also leptons. The form of the couplings is fixed but their magnitude is model dependent since it is different for different PQ-charge assignments to the matter fields. Given this model dependence, we do not include them in our analysis. They would lead to an increase of the numerical value in (39) by a factor of ∼ 2 and consequently would make the range of F a in (41) larger by the same amount.
Once we assume that axions thermalize, it is interesting to know the density of this thermal axion population. After decoupling, axions redshift freely until today. Their density today, n a0 , can be calculated using the fact that after decoupling, the value of Y stays constant. We need the entropy density today, s 0 , which has the contribution of photons, neutrinos and axions. Even in the case that axions are relativistic today, the contribution to s 0 can be neglected since the axion temperature is much colder than the cosmic microwave background temperature, T 0 = 2.75 K. Indeed, since the time of axion decoupling, the decoupling of other particles have heated the photon bath but not the axion one. One gets that the entropy density today corresponds to g * S = 43/11 = 3.91 in (15).
Then, we use (8) This small density was also found by Turner [8]. At the view of the small mass (4) one can conclude that it is very unlikely that it can play a direct role in the efforts to detect axions [3].
It is also interesting to determine the range of temperatures where axions interact with the QCD plasma. For a given F a satisfying (41), it is not difficult to see that the thermalization range is For example, for the highest possible value of F a where still a thermal population arises, F a = 1.2 × 10 12 GeV, we get a temperature range For the lowest value of F a allowed by observation (see eq (3)), the range is Our final comment refers to a related effect also happening in the thermalization period.
If there are non-thermal axions produced at this period, they would thermalize since they would interact with the QCD plasma through reactions 1), 2) and 3). Thus, no matter which was the energy spectrum and how high was the production rate of the originally non-thermal axions, they would end up with the thermal spectrum (12) and density (14). There is no effect on axions from the misalignment mechanism and domain wall decay, since they are generated at T ≃ Λ QCD , that cannot be in the range (43). Strings start to decay into axions after their formation at T ≃ F a . Current models [6] have the bulk of axions produced also at T ≃ Λ QCD . If this is the case, then thermalization effects on string axions are small.
However, the physics of axion strings is still not completely settled and we would like to stress that if a string model or any other source of non-thermal axions have substantial axion production in the range (43) then those axions would thermalize and finally we would have a single thermal population that dilute away with the expansion of the universe as we have studied in this paper.