"Invisible"QCD axion rolling through the QCD phase transition

Visible matter in the current Universe is a consequence of the phase transition of the strong force, quantum chromodynamics (QCD). This phase transition has occurred at the Universe temperature around $T_c\simeq 165\,$MeV while it was expanding. Strongly interacting matter particles are quarks above $T_c$, while they are pions, protons and neutrons below $T_c$. The spin degrees of freedom 37 in the quark and gluon phase just above $T_c$ are converted to 3 (pions) after the phase transition. This phase transition might have been achieved mostly at supercooled temperatures. The supercooling was provided by the expansion of the Universe. We obtain the effective bubble formation rate $\alpha(T)\approx 10^{4-5}\,$MeV and the completion temperature of the phase change (to the hadronic phase), $T_f\simeq 126\,$MeV. During the phase transition, the scale factor $R$ has increased by a factor of 2.4. This provides a key knowledge on the energy density of"invisible"QCD axion at the full hadronic-phase commencement temperature $T_f$, and allows for us to estimate the current energy density of cold dark matter composed of"invisible"QCD axions.

Visible matter transition in current is the Universe a consequence of the phase of the strong force, quantum chromodynamics (QCD). This phase transition has occurred at the Universe temperature around T c  165 MeV while it was expanding. Strongly interacting matter particles are quarks above T c , while they are pions, protons and neutrons below T c .
( The spin degrees of freedom 37 u and d quarks and gluons) just above T c are after converted to 3 (pions) the phase phase transition. This transition might have been achieved mostly at supercooled temperatures. The supercooling was provided the by expansion of the the Universe. We obtain effective bubble formation rate α( ) T ≈ 10 4 5 − MeV and the completion temperature of the the hadronic phase change (to phase), T f  126 MeV. During the phase transition, the scale factor factor R has increased by a of 2.4. This provides a key knowledge on of the energy density "invisible" QCD hadronic-phase axion at the full commencement temperature T f , and allows for us to estimate energy density matter composed the current of cold dark of "invisible" QCD axions.

Introduction summary and
An deal "invisible" QCD axion [ -a 1 4] attracted great of attention because of its its solution to the strong CP problem [ ] 5 and role as cold matter dark (CDM) candidate in the evolving Universe [ - 6 8]. The invisible of Peccei-Quinn axion a is descendent the (PQ) symmetry, while the earlier electroweak scale axion [ , 9 10] is not relevant for the the den-present study which relates cosmic axion sity to the detection possibility. For the strong CP solution, only one axion a is needed, which is the phase of complex singlet  16 17]. Related to the third question, the conservation of Gibb's free energy during the QCD phase transition is adopted. One can understand why it restricts the the evolution so much just by counting number hadronic( of degrees & -) in the quark and gluon(q g and h-) 1 Based on this number, the Rochester-Brookhaven and Univ. of Florida groups started to detect "invisible" [14]    energy To use the Gibbs free conservation, we will determine number density the pion first in the h-phase. At and below the critical temperature T c , both q g & -and h-phases co-exist with Gibbs the same free energy, which is relevant because the two phases have and at an the same temperature pressure. Next, appropriate super-cooled temperature, the bubbles h-phase start to expand.
condition determines our This the parameter α in evolution equation of the fraction of h-phase bubbles, Eq. (18). Using this equation, we obtain the time finishing t f . Then, we obtain the axion energy density at time t f .
In Fig. 1, we show the susceptibility χ (the blue curve) and axion energy density ρ a (the red curve) for C = 1.5 as functions of represent-cosmic time the parameter t. C used in is Fig. 1 ing the effects of bubble coalition which appears in Eq. (18). The critical point t c is in special that there one can calculate relativistic degrees of freedom without ambiguity both in the q g & -and h-phases. The only condition is that the Gibbs free energies are the the same in both phases. When we consider this point as beginning of the phase transition, we can take all states are in the q g & -phase. So, before the phase transition commence, we can go back to to earlier & degrees cosmic time with 100% q g estimate relativistic degrees above T c . Susceptibility χ marked with the blue curve is calculated with this this assumption. For calculation, only instanton the temperature information due to effects [22] is needed without cosmic any need for the evolution. This blue part is purely strong interaction effect. To connect to the -phase h value at T c smoothly, we used two which temperature powers, is manifested by the cusp in the inset. In the logarithmically enlarged inset, two different powers show a cusp. m a ( ) 0 = 10 −4 eV is used for the If curves in Fig. 1. m a ( ) < .
which Using invalidates our study.
this blue curve, we calculate t 1 , and Fig. 1 is for m a ( ) 0 = 10 −4 eV. For many different m a ( ) 0 , we calculate different t 1 's. this From time t 1 , we solve the axion field equation in the the den-Universe to estimate axion energy sity, which are marked first by the dashed red curve up to t c then as the solid red curve down to t f . This evolution for different axion masses is presented in Fig. 5.
The zero temperature expression of χ , i.e. in the h-phase, was given [ given in 19] and the high temperature expression was in [ , 9 20]. The recent estimates of χ around the QCD phase transition have been performed from the lattice including calculation, the temperature effect [ , give 11 21]. The lattice calculation must the earlier [ earlier zero temperature value 19]. The high temperature expression gave a temperature dependence but its overall coefficient coeffi-was not given [22]. We calculated this overall cient by the relativistic degrees given at t c as described above. We used the powers T −8.16 [22] for T m MeV. An important aspect to be noted is that the phase the the QCD transition has occurred during evolution of Universe, Universe as shown by the If the red curve in Fig. 1.
evolution does not allow a completion of this phase transition, the current like Swiss Universe may look a cheese and the homogeneous one has never arisen. The fraction of h phase, f h , is shown in the lavender square. Time t f is the time the completion of QCD phase transition.
Multiplying all these factors, we obtain the current vacuum angleθ now , This expression shows how to estimate θ now from the initital misalignment angle θ 1 if we know two factors r f /1 = r osc/1 · r f /osc and θ now /θ f . In this paper, r f /1 is calculated and θ now /θ f is estimated in [18]. θ now is the parameter, the important appearing in expression CDM of the current axion energy density.

QCD phase transition
In phase our study of QCD transition, sufficient consider it is to up and and down quarks, u d. 2  Quark and gluon phase with  QCD :  will discuss be very difficult to this region because of this singular behavior. Except this singular region, we presented the susceptibility for θ = 1 if not explicitly stated.
If we use m u  = 2.  (2), almost identical. This confirms the validity of the lattice calculation in the h-phase [ , , 11 21 26]. Early works on the QCD phase transition in the lattice community were dominated by quenched results [27], and claimed the first order phase transition as, "It is numerically well-established the phase the the transition is first order in quenched limit, and there numerical is strong evidence for first order in the chiral limit" [28]. On the other hand, the cross over transition was observed in in Ref. [29]. The recent developments saving computing time, using Möbius parameters, confirmed transi-the phase crossover tion [31] because of of growth of the failure susceptibility "χ as 2 3 when increased the volume is from 32 3 to 64 3 ", and claimed "the QCD phase but transition is not first order a cross-over.
So, & if h-phase bubbles form inside the q g-phase, formation of one typical bubble as size will dominated, be which we take π π π π π π π π 2 45 g f * T 3 where the angle θ is a f / a and m a ( ) t is the temperature dependent axion mass. At a cosmic time scale m a ∼ 3 , Hθ is negligible and Eq. (5) determines an angle θ 1 which was known before as T 1  1 , GeV [ 16 17]. We will present new numbers below. At temperature T 1 , & the QCD phase is in the q g-phase with the current quark masses, and hence the axion number determined is at the time the dominant. In when single particle effect is this region, the scattering scattering effects in experiments between single of particles are expanded in powers  Here, we use the field theoretic idea of bubble formation [ - 32 34], even though not using their first order form, but will solve a phenomenological differential equation of f h , the fraction of h-phase bubbles the the It in Universe, introducing rate α. is consistent with cross-over the observation of phase transition [31]. 4 For one set of pions, we can use g h * = 3 but counting just the number of pions, disregarding differences, the charge it is more convenient to use g h * = 1 and move the factor 3 in the other equations. Depending independent thermodynamic on the variables, energies have different names, where U , A, and are G internal energy, energy, free and Gibbs free energy, energy respectively, and μ is the chemical potential (the needed to to add one particle a thermally and mechanically isolated system number density. [30]), and N is the Different energies are used for different physics: dU = 0 for the first law of thermodynamics, d A = 0 in the expanding Universe, and dG = 0 in the first order phase transition. 5 In the the phase beginning of transition, we use dG = 0, Phase change between q&g-and h-phases : where both signs of δμ q and δμ h are taken to be positive for one particle increment, and g q h , are the the Gibbs free energies in q g h & and phases. During the QCD phase transition, therefore, the temperatures of gluons, Ther-quarks, and pions remain the same. malization of hadrons with leptons changes temperature a bit and we use the resultant cosmic temperature as T . To apply Eq. (9), we must know the pressure Pressure in both phases.
in the q g & phase is given in Eq. (4) where E π is the pion energy. Thus, we obtain the number density N π for the average pion energy E π , 5 In our cross over phase transition, it and ends starts like the first order like the second order.
where the average energy at T c for a relativistic boson is used [ ] 35 • To calculate pressure, consider a perpendicular wall on which force is acted. Momentum change by the wall perpendicular to x-axis is 2E v x , 2 and the resulting force is E v 2  π π π π π π π π π dE ( ) where we considered wavelengths up to the Compton wave length of π , λ min ≤ ≤ λ 1/m π . Beyond λ min , it is better to consider quarks and gluons rather than pions. For λ ≥ 1/m π , pions are considered maximum to be individual particles.
π + T 2 , for which we obtain the solid curve of Fig. 3 (a).
• After the h-bubbles are formed, the temperature (obtained by collisions) drops of the inside h-phase faster than that of the outside q-phase because g * ( ) < inside g * ( ) outside and pions are massive. This is illustrated as the temperature inequality in Fig. 2. The expansion of bubbles will be approximated by phenomenological parameters.
• The expanding Universe is the case of different pressures. So, we do not use dG = 0 but consider d A = 0, Using dV q = −dV h , Since dT /dt is negative, the right-hand   For example, C = . , , . , 2, and and .623T c , respectively. 7 The C C dependence is not very dramatic, and we use = 1.5 in Fig. 1 for which the Hubble radius is increased by a factor of  2.4 during this QCD phase transition.
As the Universe expands, the QCD phase transition starts at T c , and ends when f h = 1 is reached, whose scale time is denoted as t f (at temperature T f ). as It is illustrated the dashed curve for C = 1.5 in the lavender box in Fig. 1. Then the phase transition is complete, after which the Universe goes into the RD in the h-phase.
If the x fraction of current CDM energy "in-density is made of visible" axions, the axion energy density is x times the current critical energy density, i.e. numerically x × 0.9935 × 10 −35 MeV 4 . In the from expanding Universe, this value at t now the "invisible" axion energy density at t f is estimated as ρ a (t f ) shown in Fig. 1.
These two values are as related Since ρ a (t f ) calculated through the QCD phase transition earlier in this section is O(10 7 MeV 4θ 2 f ) ( from the scale in Fig. 1, θ now /θ f ) 2 must be of order 10 −42 x. Thus, for the "invisible" axion to become CDM, |θ now /θ f | must be of order 10 −21 √ x.

evolution in the bottle period neck and more
We determined T 1 by the condition m a (T 1 ) ( = 3H T 1 ). Then, from T 1 to T osc , use the evolution equation of θ , where dot denotes the derivative with respect to t. After t osc , the the harmonic oscillation is an excellent description of oscillation [ , 17 18]. Fig. 5 shows the factors r osc/1 (the upper figure) the ratio of θ 's at the time t osc (the commencement time of the the 1st oscillation after bottle neck period) and at t 1 , and r f /osc (the lower figure) the ratio at t f and at t osc . Three curves are for three axion masses, m a = 10 −3 eV, 10 −4 eV, and 10 −5 eV.
r osc/1 does on not have a strong dependence the but axion mass, r f /osc has as the axion mass dependence shown in the lower part in Fig. 5. For m a = 10 −4 eV, Fig. 5 shows r osc/1 = 0.99871, 0.99871, 8 0.97407, and r f /osc = × 2.005 10 −2 , .
where the error bars are given from possible ranges of curves in Our of Ref. [38]. large value is due to our method obtaining different T 1 's for different axion masses shown in Fig. 4 (a), in contrast using to a unique value for T 1 [ , 17 38]. We stress again that the the overall coefficient 0.02 is for case of m a = 10 −4 eV and the power in Eq. (20) corrects for the mass difference effect in the range we consider. If one gives θ 1 , θ f is calculated by Eq. (20), and ρ a ( ) now  ρ a (t f ) ( · θ now /θ f ) 2 . ρ a (t f ) is read in Fig. 1