Axions Production

The Peccei-Quinn (PQ) symmetry breaks at a high temperature, providing a well known dynamical solution to The Strong CP Problem. In this scenario, the complex axion field takes a non-vanishing vacuum expectation value, which fixes one of its components, leaving just a phase, which we call an axion.

There are various production mechanisms for axions, which can be thermal or non-thermal. Which of these mechanisms dominate, strongly depends on whether the P! temperature is higher or lower than the reheating temperature.

The axion mass arises from non-perturbative QCD effects, which are negligible at high temperatures, but become relevant at a critical time, $t_1$, that satisfies $m_a{t}_1\approx 1$, and where the temperature of the Universe is approximately $T_1\approx 1GeV$.

Thermal Production

Axions can be produced in the plasma of the early Universe, mainly by processes that involve quarks and gluons. These processes can lead to a population of hot axions if the Peccei-Quinn axion scale is $f_a<10^{12}GeV$. After colour confinement, the leading thermal production process is through coupling to pions, $\pi +\pi \leftrightarrow \pi +a$. These hot axions would contribute to the universe energy density and, like in the case of neutrinos, cosmological constraints impose upper bounds on their mass.

Thermally produced axions would contribute to the effective number of relativistic degrees of freedom, which is measured to be $N_{eff}=3.15\pm 0.23$, leading to an upper bound on the axion mass of $m_a<0.529eV$.

Misalignment Mechanism

Cold axions can be produced in the early universe through this mechanism.

Consider a model, where the potential for the complex field $\varphi (x)$ at a high temperature reads

$V(\varphi )=\frac{\lambda }{4}(|\varphi ){|}^2-v_{\mathfrak{a}}^2{)}^2$.

Then, when the Universe cools to $T_{PQ}\sim v_a$, the field $\varphi$ takes a non-vanishing vev in each Hubble volume $⟨\varphi ⟩\equiv v_{\mathfrak{a}}{e}^{i\theta (x)}$. The remaining angle $\mathfrak{a}(x)=v_{\mathfrak{a}}\theta (x)$ is related to what we will refer to as the axion.

The axion mass plays no role for high temperatures but when the Universe cools down to the confinement scale $T\sim \Lambda$, non-perturbative QCD corrections “tilt” the potential,

$\tilde{V}(\theta )=m_{\mathfrak{a}}^2(T)f_{\mathfrak{a}}^2(1-\cos \theta )$.

This forces the axion to “realign”. This also gives the axion an effective mass whose time-dependence can be calculated to be $m_a(T)=0.2m_{\mathfrak{a}}{\left(\frac{\Lambda }{f_{\mathfrak{a}}}\right)}^4$.

The eom for the axion in a FRW Universe,

$\ddot{\theta }+3\dot{H}\theta -\frac{1}{a^2(t)}{\nabla }^2\theta +m_{\mathfrak{a}}^2(T(t))\sin \theta =0$.

If $T_{PQ}>T_R$, the the axion reached the PQ maximum before reheating, taking a random value for the phase, later fixed by cosmic expansion. This means that $\theta$ is homogenous at QCD confinement. Then, the spatial derivative can be neglected and we are left with the equation of a damped harmonic oscillator (using the small angle approximation),

$\ddot{\theta }+3H\dot{\theta }+m_{\mathfrak{a}}^2\theta =0$.

The axion mass is not relevant at early times ($\theta$ is constant), but after a time $t_1=1/m_{\mathfrak{a}}(T_1(t_1))$, which is equivalent to a critical temperature $\approx 1\text{ GeV}$. At this point, the axion field responds by attempting to minimise its potential, oscillating around minimum (vacuum-realignment). The critical time and temperature are

$t_1\approx 2\times 10^{-7}\text{ s }{\left(\frac{f_{\mathfrak{a}}}{10^{12}}\text{ GeV}\right)}^{1/3},T_1\approx 1\text{ GeV}{\left(\frac{10^{12}\text{ GeV}}{f_{\mathfrak{a}}}\right)}^{1/6}$.

The momentum of the resulting quanta of the axion field can be estimated. Assuming a typical coupling of $f_{\mathfrak{a}}\sim 10^{-12}$, (axion mass $m_{\mathfrak{a}}\sim 6\times 10^{-6}\text{ eV}$),

$p_{\mathfrak{a}}(t_1)\sim \frac{1}{t_1}\sim 10^{-9}\text{ eV}$.

Thus, classically, spatially coherent oscillating fields are equivalent to a coherent state of extremely non-relativistic dark matter (cold), i.e., CDM.

Finally, the energy density of the axion field,

$\rho =\frac{f_{\mathfrak{a}}^2}{2}({\dot{\theta }}^2+m_{\mathfrak{a}}(t{)}^2{\theta }^2)$.

The initial energy of the axion, before starting to oscillate,

$\rho =\frac{f_{\mathfrak{a}}^2}{2}{m}_{\mathfrak{a}}(t_1{)}^2{\theta }_1^2$.

As a cosmological density parameter,

${\Omega }_{\mathfrak{a}}{h}^2=0.15{\left(\frac{f_{\mathfrak{a}}}{10^{12}\text{ GeV}}\right)}^2{\theta }_1^2$.

According to this equation, an axion with a scale $f_{\mathfrak{a}}\sim 10^{12}\text{ GeV}$ would reproduce the CDM relic density observed by the Planck satellite, assuming that the original misalignment is of the order of ${\theta }_1\sim 1$ (for naturalness argument). Otherwise, by abandoning naturalness and introducing a degree of fine-tuning in ${\theta }_1$, the axion scale can be raised arbitrarily. This is called “anthropic”.