
2 COLD AXIONS
1 Introduction
The firm observational evidence for some form of dark matter, approximately five times more
abundant than ordinary matter, leaves no doubt about the need for physics beyond the standard
model. Motivated particle candidates are not postulated to exist ad hoc but they tackle more than
one problem. The candidate discussed here, naturally arising from the investigation of symmetries
whose violation is small and structured, is one of the best examples.
The experimentally observed invariance of strong interactions when we flip the arrow of time
is rather surprising. The Peccei-Quinn (PQ) mechanism [1,2]is an elegant dynamical solution, and
all microscopic realizations share some common features. The standard model is extended via a
new global symmetry U(1)PQ that must satisfy two key requirements: spontaneously broken, and
anomalous under strong interactions. The PQ symmetry breaking scale is bound to be much higher
than the energies we can reach with experiments, and the only accessible low-energy residual is a
pseudo-Nambu-Goldstone boson (PNGB) arising from the spontaneous PQ breaking. This pseudo-
scalar field is known as the QCD axion [3,4], and as a consequence of the color anomaly it couples
to gluons via the dimension 5 operator
LPQ ⊃αs
8π
a
fa
GA
µν e
GAµν . (1)
Here, we have the QCD fine structure constant αs=g2
s/(4π), the gluon field strength tensor Ga
µν,
and its dual e
Gaµν. This operator defines the axion decay constant fa. Interactions with other
standard model fields are model dependent. Given its PNGB nature, all couplings are suppressed
by the scale fa. QCD non-perturbative effects generate a potential once strong interactions confine,
and this leads to an axion mass [5]
ma'5.7 µeV 1012 GeV
fa. (2)
A light and elusive particle could play a prominent role in the early universe. Famously, the
QCD axion is a viable dark matter candidate [6–8]as discussed in Sec. 2. The PQ framework
leads to other fascinating phenomena in the early universe with testable consequences today. A
hot axion population, discussed in Sec. 3, is a notable example. We focus on thermal production
in Sec. 4, and we give updated predictions for the KSVZ [9,10]and DFSZ [11,12]frameworks. We
also compare cosmological bounds on flavor-violating axions with laboratory searches and present
an updated cosmological bound on the QCD axion mass. Conclusions are given in Sec. 5.
2 Cold Axions
The axion contribution to the observed dark matter relic density can be computed by analyzing the
field evolution across the expansion history. At early times, at temperatures much higher than the
QCD confinement scale, the finite-temperature axion potential is negligible and Hubble friction
prevents any motion. The field starts evolving only when the primordial plasma temperature
reaches values around the proton mass and the non-perturbative axion potential switches on.
The resulting motion is described by damped harmonic oscillations around the minimum of the
potential, and the amplitude of such oscillations gets damped by the Hubble expansion as non-
relativistic dark matter. This is the misalignment production mechanism.
2