Axions withma∼>10−2eV are thermally produced in the standard radiation-dominated cosmology.
We now show that in LTR models, these axions have suppressed abundances. We consider standard hadronic axions, which do not couple to standard-model leptons but do couple to pions and photons through higher order terms [56, 57]. We do not consider flaton models, or other scenarios in which PQ symmetry breaking is related to supersymmetry breaking [211–215]. For temperatures T ∼< 150 MeV, the dominant channels for axion production areπ++π−→a+π0, π++π0
→π++a,
and π−+π0
→ a+π− [93]. Axion scattering rates are suppressed relative to particle-number-
changing interactions by factors of T2/f2
PQ and thus decouple very early. Thus, axions stay in
kinetic equilibrium because ofπ+π↔a+π, and kinetically decouple when they chemically freeze out.
Nucleonic channels are negligible at these temperatures [93], as we verify below. If pions are in chemical equilibrium and Bose enhancement can be neglected, the axion production rate is obtained
Figure 3.2 The top panel shows the Hubble parameter as a function of temperature in an LTR cosmology with Trh = 20 MeV and HI = 1 MeV. Initially H ∝ T4g∗(T), but at temperatures
cooler than T ∼ Trh, H ∝ T2
p
g∗(T).The second panel shows the comoving radiation energy densityR=ρRT0−4a4and scalar energy density Φ =ρφT0−3Trh−1a3as a function of temperature. At
T ∼Trh,Rflattens out to a constant and Φ drops off to zero, indicating the conclusion of reheating.
from the Lagrangian in Eq. (2.15) to be [6, 50, 93, 216]
Γ = 3ζ(3)T 5C2 aπ 1024π7f2 afπ2 Z dx1dx2 x2 1x22 y1y2 f(y1)f(y2) Z 1 −1 dµ s−m2 π 3 5s−2m2 π s2T4 , Caπ= 1−r 3 (1 +r), s= 2 m2π+T2(y1y2−x1x2µ), (3.11)
where xi = pi/T is the dimensionless pion momentum, yi = px2
i +m2π/T2 is the dimensionless pion energy, f(yi) = 1/[exp (yi)−1] is the pion distribution function, Caπ is the dimensionless axion-pion coupling constant, andζ(x) is the Riemannζ-function. The PQ energy scale isfa. The
neutral pion mass ismπ '135 MeV [91]. The PQ scale can be expressed in terms of the axion mass [87]: fa' √r 1 +r fπmπ ma . (3.12)
Here r ≡ mu/md ∼ 0.56 is the up/down quark mass ratio and fπ ' 93 MeV is the pion decay
constant [50, 91].
freeze-out temperature using the condition Γ (TF)∼H(TF). As the reheating temperature is low-
ered, axions freeze out at higher temperatures due to the higher value of H, as shown in Fig. 3.3. As the reheating temperature is increased, the T ∝ a−3/8 epoch becomes increasingly irrelevant,
and the freeze-out temperature of the axion asymptotes to its standard radiation-dominated value. Examining Eq. (3.11), we see that Γ∝f−2
a ∝m2a, so higher-mass axions keep up with the Hubble
expansion for longer and generally decouple at lower temperatures. Thus, for higher ma, a more
radical change to the thermal history (even lowerTrh), is needed to driveTFto a fixed higher value.
Figure 3.3 Freeze-out temperature of the reactions π++π− ↔ π0+a, π++π0 ↔ π+ +a, and π−+π0
↔ π−+a, shown as a function of the reheating temperature T
rh, for 4 different axion
masses. More massive axions are coupled more strongly, leading to later freeze-out than for lighter axions.
As axions are spin-0 relativistic bosons, their number density at freeze-out isna(Tf) =ζ(3)TF3/π2.
If we assume that axion production ceases at freeze-out, the density of axions at any subsequent time is just na(Tf) [aF/a0]3, whereaF is the value of the cosmological scale factor at axion freeze-out.
The reheating time scale,trh'1/Γ, is much shorter than the Hubble time forT ∼< Trh, and so it is a
good approximation to treat the break between theT ∝a−3/8andT
∝a−1epochs as instantaneous
atT =Trh. Doing so, we apply Eq. (3.9) prior to the completion of reheating anda∝T−1g−∗S1/3(T)
afterwards, to obtain Ωah2=ma,eV 130 10 g∗S,F γ(Trh/TF),
γ(β)∼ β5g∗,rh g∗,F 2g ∗S,F g∗S,rh ifβ 1, 1 ifβ 1, (3.13)
wherema is the axion mass in units of eV.
Low reheating temperatures drive up the freeze-out temperature. When Trh ∼< TF, the present
mass density in axions is severely suppressed, because of the sharper dependence of the scale factor a onT during reheating. This is a result of entropy generation. Using the numerical solution for a(T) from Sec. 3.2, we obtain Ωa, accounting for the smooth transition between theT ∝a−3/8and
T ∝a−1 regimes. In Fig. 3.4, we show Ω
a normalized by its standard value, Ω0a, as a function of
Trh. At reheat temperatures just a factor of a few below the usual axion freeze-out temperature for
a given axion mass, the axion abundance is suppressed by a factor of 10−4
−10−3. ForT
rh TF,
the axion abundance asymptotes towards its standard value. We see that the constraints on axion decay from Chapter 2.5.3 then imply a very strict limitTrh 10 MeV ifE/N = 2 is fixed there,
and uncertainties inmu/mdare ignored.
In the case of kination, axion freeze-out temperatures are still raised, but there is no additional entropy production. Axion abundances are given by Eq. (3.1), but with the higher g∗S,F values
appropriate at higher values ofTF.
For the LTR case, our results do not depend on the initial value HI of the Hubble parameter.
As seen in Fig. 3.1, changes to HI determine the moment of the fast rise to Tmax, but have little
influence on the expansion history forT < Tmax. For convenience, we chooseHI= 1 MeV for our
calculations, corresponding toTmax'20 GeV.
Our calculation is valid only if axions are produced in equilibrium by thermal pions. We check that pions are in chemical equilibrium using the chiral Lagrangian density term for pion-photon interactions, which scales as [217]
Lππγγ∼
απ0π0FµνFµν f2
π
. (3.14)
We use standard expressions for the number density of pions in the non-relativistic regime, and estimate the cross-section for theππ ↔γγ reaction accurate to order-of-magnitude level assuming that the photon momentum k ∼T. Using the usual Γ (TF)∼ H(TF) criterion, we estimate that
pions stay in equilibrium from the QCD phase transition until long after nucleosynthesis (down to T ∼ keV, yielding exponentially suppressed abundances today), and so we may use equilibrium expressions for the number density of pions.
For axion freeze-out to occur during the era where pions have condensed out of the quark-gluon plasma, we must have Trh ∼> 10 MeV. Outside this range, our assumptions break down. For
sufficiently low values of ma andTrh, pionic cross sections lead to TF∼>200 MeV, earlier than the
production cross sections.
Furthermore, forTrh∼<10 MeV, pions will decay before they can come into equilibrium. In both
cases, axion abundances are suppressed relative to our calculation. For axion masses saturating our upper limits and Trh ∼>10 MeV, we have checked that we are well within the equilibrium regime.
We restrict ourselves to this range, noting that forTrh∼<10 MeV, more suppressed abundances will
lead to an even more dramatic relaxation to cosmological axion limits. Coherent oscillations of the axion field produce a condensate that behaves as cold dark matter [76], but the resulting additional abundance is negligible forma∼>10−2eV at all values ofTrh under consideration here [76].
It is instructive to verify that production off the ample thermal bath of protons and neutrons for T ∼<ΛQCD is indeed negligible in our regime of interest. In low-energy chiral perturbation theory,
the axion-nucleon interaction Lagrangian density is [93]
LaN=∂µA
fa
CaNnγµγ5n+Cappγµγ5p, (3.15)
whereAis the axion field,CaN is the axion-neutron coupling constant, andCapis the axion-proton
coupling constant. Both coefficients are of order unity. The lowest-order axion-nucleon reactions are NN→NN a and Nπ →N a. Using an equilibrium value for the total nucleon density (adequate because the baryon asymmetry is not very relevant until T ∼ 22 MeV when nucleons freeze out [50]), we obtain the following expression for the ratio of the relevant chemical reaction rate to the Hubble expansion Γ H = 611m2 a,eV pmN T e− mN T ifT < Trh, 6.11×10−4m2 a,eV MeVTrh 2qm5N T5e− mN T ifT > Trh. (3.16)
Numerically, we find that for the range of ma and Trh considered in this paper, the freeze-out
temperatureTFof the nucleon-axion chemical reactions lies in the range 100 MeV∼< TF∼<200 MeV.
Thus, our assumption that the pionic reactions dominate over the nucleonic ones is solid.