Axion domain wall network evolution in the early universe

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An attractive candidate dark matter (DM) particle is the axion. This is the pseudo Nambu-Goldstone boson from the spontaneous symmetry breaking of a chiral U(1)_PQ introduced by Peccei & Quinn to solve the 'strong CP' problem, viz. that the strong interactions in the Standard Model (SM) do not violate CP symmetry as is evidenced by the vanishing electric dipole moment of the neutron. While the original 'Weinberg-Wilczek' axion corresponding to PQ symmetry breaking at the electroweak scale was never found, it was soon realised that the relevant scale can be much higher, thus yielding an 'invisible axion' with very suppressed (derivative) couplings to SM fields. Nevertheless collective oscillations of the relic axion field has the same equation of state as non-relativistic particles so relic axions can account for the DM of the universe for a PQ scale around 10^{10-11} GeV. The cosmological evolution of such an axion is interesting because of a sequence of symmetry breaking which produces potentially stable topological defects. Below the Peccei-Quinn scale f_PQ the vacuum manifold is not simply connected. This implies the existence of closed paths in physical space which get mapped onto non-trivial paths in field space winding around the origin. Such field configurations correspond to cosmic strings. When the temperature drops to A_QCD ~ 300 MeV, non-perturbative QCD effects generate a mass for the axion which breaks the symmetry to a Z(N) with N the number of quarks charged under U(1)_PQ. The vacuum manifold of Z(N) is disconnected which implies the existence of paths in physical space which map onto paths interpolating between two vacuum states in field space. Such paths necessarily leave the vacuum manifold and the resulting structure is a domain wall.Topologically each string must be connected by N domain walls once the axion gets a mass. Due to the surface energy of domain walls, a network of strings and domain walls with N= 1 would be unstable and collapse. However networks with N >1 are stable in principle and can lead to a cosmological catastrophe if they come to dominate the energy density of the universe after they form [. The standard argument for evading the 'domain wall problem' when N >1 is to invoke non-renormalisable Plank-scale suppressed operators reflecting quantum gravity effects which explicitly break the initial U(1)PQ and lift the degeneracy of the vacuum states, resulting in an effective pressure term which causes the domain of true vacuum to grow. Such explicit breaking of U(1)PQ is however experimentally constrained as it reintroduces the CP violation which is required by experimental upper limits on the neutron electromagnetic dipole moment (EDM) to be negligibly small. By requiring axion domain walls to disappear in time to avoid cosmological catastrophe one can thus infer a lower bound on the neutron EDM. Therefore the `tilt' solution to the domain wall problem is potentially testable by future neutron EDM experiments. We propose an alternative mechanism to render domain walls unstable by instead introducing a statistical `bias' in the population of the vacuum states. Such bias is naturally generated by the evolution of the axion field during inflation, under the assumption that inflation lasts long enough for the universe to cool down below A_QCD during inflation so the axion acquires a mass. This possibility has recently been exploited to open up previously excluded axion parameter space. It has been demonstrated that such bias leads for Z(2) models to exponential decay of the co-moving domain wall energy density. The impact of this on the most likely mass for axions to make up dark matter will be studied, as this is particularly relevant to ongoing searches for axions including in the UK 'Quantum Sensors for the Hidden Sector' programme.