Resonant excitation of the axion field during the QCD phase transition

We find that the adiabatic fluctuations produced in the primordial plasma by cosmological inflation resonantly excite the axion field during the QCD phase transition by pumping axions from low momentum modes to modes with momentum up to of order $\sqrt{3} m$ where $m$ is the axion mass. We derive the momentum distribution of the excited axions. The fraction of cold axions that get excited is of order one if the axion mass is larger than a few $\mu$eV. The effect occurs whether inflation happens before or after the Peccei-Quinn phase transition.


I. INTRODUCTION
The axion is a hypothetical particle postulated to solve the Strong CP Problem, i.e. to explain why the strong interactions conserve the discrete symmetries P and CP in spite of the fact that these symmetries are broken in the Standard Model of particle physics as a whole [1][2][3]. Axions are independently motivated by cosmology because a population of cold axions is produced during the QCD phase transition, when the temperature is of order 1 GeV (k B = = c = 1) [4]. In the simplest scenarios, axions account for the cold dark matter of the Universe if the axion mass is of order 10 −5 eV, with large uncertainties [5].
The axion [2] is the quasi-Nambu-Golstone boson resulting from the spontaneous breaking of the U PQ (1) anomalous global symmetry that Peccei and Quinn [1] postulated. The properties of the axion depend for the most part on a single parameter f a called the axion decay constant. In particular the axion mass is m ≃ 6 µeV 10 12 GeV f a (1.1) at temperatures below 100 MeV. At temperatures T of order 1 GeV and higher, the axion mass is estimated to be [4] m(T ) ≃ 4 · 10 −9 eV 10 12 GeV f a GeV T 4 (1.2) using the dilute instanton gas approximation [6]. At intermediate temperatures, between 1 GeV and 100 MeV, the axion mass may be estimated from QCD lattice simulations [7,8].
The precise manner in which the axion mass turns on during the QCD phase transition affects the quantitative results derived in the present paper. An important source of uncertainty when estimating the abundance of cold axions, is whether the U PQ (1) symmetry is spontaneously broken before or after the epoch of inflationary expansion [9,10] that homogenizes the very early universe. U PQ (1) becomes spontaneously broken when the temperature falls below some critical value of order f a , during the so-called PQ phase transition. If this phase transition occurs before inflation, the axion field gets homogenized by the inflationary expansion, so that at the start of the QCD phase transition the axion field oscillations begin with almost the same amplitude everywhere. Unless this initial amplitude happens to be zero, which would be an unexpected accident, axions are produced in a process of "vacuum realignment" [4]. If the PQ phase transition occurs after inflation, the axion field enters the QCD phase transition with random initial amplitudes, uncorrelated from one QCD horizon volume to the next. In that case cold axions are produced by vacuum realignment and by the decay of axion strings and domain walls [5].
In this paper we are interested in the momentum spectrum of cold axions at the end of the QCD phase transition. If inflation occurs after the PQ phase transition the axion field is homogenized by the inflationary expansion. However, as is well known, the axion fluid has perturbations which originate as quantum mechanical fluctuations in the axion field during inflation [11]. These perturbations have been studied extensively because they produce isocurvature density perturbations that may be inconsistent with the observed cosmic microwave background (CMB) anisotropies. The observed anisotropies are consistent with purely adiabatic density perturbations. The absence of isocurvature perturbations qualitatively requires H I 10 −5 f a where H I is the Hubble constant during inflation.
We are interested in length scales much smaller than those constrained by the CMB observations since the QCD horizon stretched to the present epoch is approximately 10 17 cm, much smaller than the length scales probed in the CMB. On the other hand, since the power spectrum of density perturbations predicted by inflation is very nearly flat, observations or theoretical considerations at one scale are potentially relevant to the other scales as well.
The quantum mechanical fluctuations in the axion field during inflation are not the only source of perturbations in the axion fluid in case inflation occurs after the PQ phase transition. A second source is the effect on the axion fluid of the adiabatic perturbations in the primordial plasma [12]. For the wavevectors of interest to us, this second source of axion fluid perturbations dominates. Indeed we find that the adiabatic perturbations in the primordial plasma resonantly excite the axion field. As far as we know, this effect has not been anticipated or commented on until now.
The resonant excitation can be thought of as the processγ( k) + φ( 0) → φ( k), wherẽ γ( k) is a plasmon and φ( k) an axion, both with momentum k. It is shown in Section IV that for every comoving wavevector k there is a time interval during which this process conserves energy exactly or sufficiently well that it occurs efficiently. During that time, axions are transferred from the very low momentum modes that they occupy initially to modes of momentum up to of order √ 3m. The process transfers energy from the plasma oscillations to the axion fluid. The process is qualitatively the same whether inflation occurs before or after the PQ phase transition because in either case the initial momenta of the axions are much less than the transferred momenta. However, it is possible to derive the momentum spectrum of cold axions with the greatest precision in case inflation occurs after the PQ phase transition, and this is one of our goals here. Our treatment is complete only for modes that enter the horizon well before the QCD phase transition. The analysis of modes that enter the horizon near the start of and during the QCD phase transition is left for future work.
The outline of our paper is as follows. In Section II, we state the equation of motion for the axion field, the spectrum of initial perturbations in the axion field, and the spectrum of adiabatic perturbations in the primordial plasma. In Section III we evolve the initial perturbations in the axion field and derive the resulting axion momentum spectrum. In Section IV we derive the perturbations in the axion field induced by the adiabatic fluctuations in the primoridial plasma and the resulting axion momentum spectrum. Section V summarizes our results.

II. EQUATION OF MOTION AND INITIAL CONDITIONS
Ignoring λφ 4 self-interactions, the axion field φ( x, t) satisfies a linear equation of motion: where m(T ) is the temperature-dependent axion mass. We are interested in scalar perturbations only. So we may adopt the so-called conformal-Newtonian gauge for the space-time metric: During the QCD phase transition, ignoring small corrections due to changes in the number of thermal degrees of freedom as a function of temperature, the scale factor a(t) ∝ √ t and the Hubble expansion rate H(t) = 1 2t . When only zeroth and first order terms in Ψ( x, t), Φ( x, t) and the temperature fluctuations δT ( x, t) are included, Eq.(2.1) becomes where j = 1, 2, 3 labels the space directions. At temperatures of order 1 GeV, the axion mass is given by Eq. (1.2). It implies the time dependence: at which time the temperature is At lower temperatures, the axion mass may be obtained by QCD lattice simulations. The results of Sz. Borsanyi et al. [7] and C. Bonati et al. [8] are consistent with one another and with the temperature dependence given in Eq. (1.2) down to T 2 ≃ 160 MeV, where m(T ) turns over and becomes its zero temperature value given in Eq. (1.1). The age of the universe at temperature T 2 is 2 · 10 −5 sec [13] when the thinning out of thermal degrees of freedom during the QCD phase transition is taken ito account. In our estimates below, to simplify the calculations somewhat, we will ignore the thinning out of thermal degrees of freedom and assume This requires t 2 = mt 3 1 ≃ 10 −5 sec. In principle, our approach can be used for any smooth turn-on of the axion mass. It fails if the axion mass has a discontinuity as a function of temperature, as may occur if the QCD phase transition is first order.
The zeroth order equation for the axion field is For t << t 1 , when the axion mass is negligible, Eq. (2.8) is solved by where A and D are constants. We set D = 0 since d dt φ (0) = 0 at the end of inflation. For t >> t 1 , the change in the axion mass is adiabatic ( dm dt << m 2 ) so that The amplitude B and phase δ can be obtained in terms of A by solving Eq. (2.8) numerically. Fig. 1 shows the function φ (0) (t) for A = 1. After the QCD phase transition, the energy density in the zero mode φ (0) is which is indeed the standard result [4]. The first order equation for the perturbations φ (1) The quantum mechanical fluctuations in the inflaton field produce adiabatic fluctuations. Provided anisotropic stresses in the primordial plasma are small, as is normally the case, Ψ( k, t) = −Φ( k, t) [14]. We will assume this to be true henceforth. The power spectrum of Φ fluctuations when they reenter the horizon (k = a(t) H(t)) is where G is Newton's gravitational constant, H I is the expansion rate during inflation, and ǫ ≡ d where H 0 is the present expansion rate, n ≃ 0.96 is the scalar power index, δ H ≃ 4.6 · 10 −5 is the amplitude of adiabatic density perturbations when they reenter the horizon, Ω m ≃ 0.3 is the fraction of the critical energy density in matter today, and D 1 (t 0 ) ≃ 0.8 is the growth function. Eq. (2.16) and the various parameter estimates are taken from ref. [14]. Eq. (2.16) applies on wavevector scales of order H −1 0 . We extrapolate it, over approximately 13 orders of magnitude, to modes that enter the horizon shortly before the QCD phase transition. This yields 8πGH 2 I 9ǫ ≡ C ≃ 4.9 · 10 −9 . (2.17) Our results in Section IV are expressed in terms of C, with the value given in Eq. (2.17) substituted only at the end to make final estimates, in case the assumptions made above about the inflationary model need to be modified. After they have entered the horizon, the Φ fluctuations oscillate as follows [14] Φ We have normalized the scale factor so that a(t 1 ) = 1. k is thus the physical wavevector at time t 1 . The Φ fluctuations are accompanied by fluctuations in the temperature [14] δT ( k, t) The quantum mechanical fluctuations in the axion field, generated during inflation, have power spectrum when they reenter the horizon. There are therefore two kinds of perturbations in the axion field δφ perturbations originate as quantum mechanical fluctuations in the axion field during inflation and the φ Φ perturbations are induced in the axion field by the adiabatic fluctuations in the primordial plasma. The two kinds of perturbations are assumed to be uncorrelated, i.e.
The fluid density in non-zero mode axions is then a simple sum: δφ and ρ Φ are the contributions from φ (1) δφ and φ Φ respectively. They are evaluated in Sections III and IV respectively.

III. QUANTUM-MECHANICAL FLUCTUATIONS OF THE AXION FIELD
In this section we set j( x, t) = 0 and derive how the axion field perturbations evolve in the absence of adiabatic perturbations in the primordial plasma. The equation of motion is It may also be written (∂ 2 t + Ω 2 (k, t))B (1) ( k, t) = 0 (3.2) where B (1) ( k, t) is defined by and Ω(k, t) = 3 16t 2 + k 2 t 1 t + m 2 (t) .

5)
φ p ( k) being the initial value of the perturbation. For k >> 1 t 1 the turn on of the axion mass is adiabatic and therefore The RHS of Eq. (3.5) equals that of Eq. (3.6) for t << t 1 provided C( k) = φ p ( k)/(2 √ kt 1 ). We have therefore for all t when j( k, t) = 0 and k >> 1 t 1 . Eq. (3.8) was compared with the result of numerically solving Eq. (3.1), and found to be practically indistinguishable from it for k ≥ 2/t 1 . Fig. 2 shows the comparison for k = 2/t 1 .
Eq. (3.8) implies for t >> t 1 3.9) including the contribution of all modes that enter the horizon before t 1 . ρ (1) δφ is very small compared to ρ (0) since the observed absence of isocurvature perturbations implies H I 10 −5 f a .

IV. RESONANT EXCITATION OF THE AXION FIELD
We now turn to the effects that result from the adiabatic fluctuations in the plasma. Fig.  3 summarizes our conclusions. It shows the spectrum of excited axions. The horizontal axis is divided into three regions. The region k * < k < k c is dominated by axions which are resonantly excited at a time intermediate between t 1 and t 2 . The region k < k * is dominated by axions which are excited by a near resonance effect which occurs from the time they are resonantly excited till time t 2 . The axions that are resonantly excited after time t 2 have wavevectors k > k c .

A. General remarks
Eq. (2.12) shows that j( k, t) acts as a source for φ (1) ( k, t). This source is turned off for t << t 1 , because ∂ t φ (0) and ∂m 2 ∂T are negligible then. We rewrite Eqs. (2.12) and (2.13) in terms of B (0) (t) defined by and B (1) ( k, t) defined by Eq. (3.3), and use Eq. (2.8) to eliminate ∂ 2 t φ (0) . This yields We substitute the late time (k √ tt 1 >> 1) behaviors of Φ( k, t) and δT ( k, t), given in Eqs. (2.18) and (2.19), and that of B (0) (t) implied by Eq. (2.10). This yields N e a rre s o n a n c e R e so n a n ce k>k c with Resonant excitation of the axion field occurs because f ( k, t) includes terms that oscillate briefly with frequency ω(k, t). Indeed the frequencies that appear in f ( k, t) are Henceforth we assume a specific time dependence for the axion mass: Eq. (2.4) for t < t 2 and Eq. (1.1) for t > t 2 . Let us call k c the comoving wavevector magnitude of modes that are resonantly excited at time t 2 . Eq. (4.6) with t k = t 2 and m(t k ) = m implies Modes with comoving wavevector magnitude k c have physical wavevector magnitude equal to √ 3m at time t 2 . Modes with k > k c are resonantly excited at time stay close to each other. ω − (k, t) soon comes close ω(k, t) as well. A near resonance excitation of the axion field occurs from approximately time t k till t 2 , as discussed in Subsection IV.C. The resonance proper near time t k is discussed in Subsection IV.B. We write solves the equation of motion in the absence of source. We substitute Eq. (4.10) into Eq. (4.3), neglect d 2 a dt 2 relative to ω da dt , and keep only terms in f ( k, t) that produce resonance or near resonance. This yields (4.13) and (4.14) The initial value of Φ p ( k)a(k, t) is φp( k) ; see Eq. (3.8). Its final value is therefore Combining Eqs. (3.3), (4.10), (4.11) and (4.15-4.16), we obtain 4.18) and δ ′ (k, t) = δ(k, t) + arg(I(k)) . (4.19) δ(k, t) and δ ′ (k, t) vary only logarithmically with time after t 2 .

B. Resonant excitation for k < k c
Modes with k < k c are resonantly excited at time t k given in Eq. (4.9). The resonance proper appears only in I + (k). We evaluate that integral using the saddle point approximation. Near time t k h + (k, t) = h + (k, t k ) + 1 2 Neglecting the higher order terms in Eq. (4.24) and the slow variation of g + (k, t) and ω(k, t) inside the integral for I + (k) in Eq. (4.16), we obtain for the contribution to I(k) from the resonance proper. The resonance proper occurs over an interval of time For k < k c , we have [12m 2 k k c 6 5 + 9 2t 2 2 k c k 6 5 ] (4.27) and . (4.28) Hence the contribution of the resonance proper to the spectrum of excited axions is (4.29) For k >> 1 t 1 , when our calculation is valid, the second term in the last parenthesis in Eq. (4.29) is always much less than 1, because 4 (4.30) for all values of m. We neglect henceforth the second term in the last parenthesis in Eq. (4.29). We have then for the density of axions with k < k c that are excited by the resonance proper. As a fraction of the density of axions in the zero mode [n (0) = B 2 2t 1 ; see Eq. (2.11)] this is (4.32) Fig. 3 shows a log-log plot of the spectrum of excited axions. The axions excited by the resonance proper dominate in the region k * < k < k c . The region t −1 1 < k < k * is dominated by axions excited by the near resonance effect, as discussed in the next subsection. For k > k c the density of excited axions is very small, as discussed in Subsection IV.D.
C. Near resonance excitation for k < k c Fig. 4 shows that the frequencies ω ± (k, t) approach ω(k, t) after the resonance proper has occurred. This produces a near resonance excitation of the axion field whose strength is estimated in this subsection. We will see that the near resonance effect is strongest at times shortly before t 2 when the axion mass is still turning on and ω ± (k, t) are both closest to ω(k, t). Let us assume k is much less than k c so that the time t k of the resonance proper occurs well before t 2 . We are not so interested in k near k c because there is little time for the near resonance effect to occur in that case. The resonance proper dominates for k near k c . We choose a timet long after t k but long before t 2 . The precise value oft will be irrelevant. The contribution of the near resonance effect to the integrals I ± (k) is I ±,nr (k) = ∞ t dt g ± (k, t) ω(k, t) e i(h ± (k,t)+δ) . (4.33) conserves axion number and transfers energy from the primordial plasma to the axion fluid. The spectrum of axions that are resonantly excited is given in Eqs. (4.29), (4.38) and (4.44), and illustrated in Fig. 3. The fraction of cold axions that get excited is of order one if the axion mass is larger than a few µeV. The process of axion field excitation occurs in qualitatively the same way whether inflation occurs before or after the PQ phase transition since the initial cold axion momenta are in either case of order 1/t 1 or less, much smaller than the momenta to which they get resonantly excited.
Resonant excitation of the axion fluid during the QCD phase transition does not change the status of the axion as a cold dark matter candidate since the excited axions are nonrelativistic again long before the onset of large scale structure formation. However, the excitation spectrum may have implications for observation. In future work, we intend to study its implications for the formation of axion mini-clusters [15] and for the cosmic time at which cold axion dark matter first Bose-Einstein condenses by gravitational self-interactions [16].