We study the system of axion strings that forms in the early Universe if the Peccei-Quinn symmetry is restored after inflation. Using numerical simulations, we establish the existence of an asymptotic solution to which the system is attracted independently of the initial conditions. We study in detail the properties of this solution, including the average number of strings per Hubble patch, the distribution of loops and long strings, the way that different types of radiation are emitted, and the shape of the spectrum of axions produced. We find clear evidence of logarithmic violations of the scaling properties of the attractor solution. We also find that, while most of the axions are emitted with momenta of order Hubble, most of the axion energy density is contained in axions with energy of order the string core scale, at least in the parameter range available in the simulation. While such a spectrum would lead to a negligible number density of relic axions from strings when extrapolated to the physical parameter region, we show that the presence of small logarithmic corrections to the spectrum shape could completely alter such a conclusion. A detailed understanding of the evolution of the axion spectrum is therefore crucial for a reliable estimate of the relic axion abundance from strings. IntroductionThe QCD axion [1-7] is the simplest and most robust of the known solutions of the Standard Model (SM) Strong CP problem, and it also automatically forms a component of cold dark matter (DM) [8][9][10]. Consequently, a QCD axion that makes up the entire measured DM relic abundance is one of the best motivated scenarios for physics beyond the SM. In addition, numerous experiments aimed at detecting axions are currently running or in development. These will be sensitive to a substantial proportion of the relevant parameter space, and, if an axion is discovered, they could measure its mass and couplings precisely (see e.g. [11,12]).The dynamics by which QCD axion DM is produced, and its final relic abundance, depends on the cosmological history of the Universe (see e.g. [13,14]). If the Peccei-Quinn (PQ) symmetry that gives rise to the axion was broken after inflation, the axion field had initially random fluctuations over the present day observable Universe. Instead, if the PQ symmetry was broken during inflation and never subsequently restored, the axion field was initially homogeneous. In this case the axion relic abundance is incalculable because it depends on the local value of the axion field after inflation. 1 In this paper we study the class of models in which PQ breaking happens after inflation. This includes theories in which inflation happened at a scale above the axion decay constant f a , and also those with inflation at a lower scale but which were reheated to a temperature above f a . 2 In this case, assuming a standard cosmological history, the relic abundance is calculable in terms of the axion mass due to the random initial conditions. 3 . As a result, there is in principle a unique calculable predict...

We study the contribution to the QCD axion dark matter abundance that is produced by string defects during the so-called scaling regime. Clear evidence of scaling violations is found, the most conservative extrapolation of which strongly suggests a large number of axions from strings. In this regime, nonlinearities at around the QCD scale are shown to play an important role in determining the final abundance. The overall result is a lower bound on the QCD axion mass in the post-inflationary scenario that is substantially stronger than the naive one from misalignment.

We improve the precision of the topological susceptibility of QCD, and therefore of the QCD axion mass, by including O(α em ) and NNLO corrections in the chiral expansion, which amount to 0.65(21)% and -0.71(29)% respectively. Both corrections are one order of magnitude smaller than the known NLO ones, confirming the very good convergence of the chiral expansion and its reliability. Using the latest estimates for the light quark masses the current uncertainty is dominated by the one of the low-energy constant 7 . When combined with possible improvements on the light quark mass ratio and 7 from lattice QCD, our computation could allow to determine the QCD axion mass with per-mille accuracy.