The spectrum of axions in a scaling string network

TL;DR

This work analyzes axion production from a post-inflation Peccei–Quinn string network in the radiation era. By decomposing the axion current into propagating and string-sourced components and employing an Unconnected Segment Model, the authors demonstrate that the axion emission spectrum approaches a scale-invariant form with a logarithmic dependence ${\mathcal P}_{\text{ax}}(k,\tau) \propto p_{\text{ax}}\ln(k\tau)$ (with $p_{\text{ax}} \approx 10$) and that the total axion number density converges to $n_{\text{ax}} = 1.66(17) f_{a}^2 H$, providing a robust, extrapolated prediction for the late-time spectrum. They introduce current-based observables $J_0$, $J_s$ and $J_\pm$ to separate string and propagating axion contributions, and compute unequal-time correlators to probe decoherence and the relative string contribution, finding that about 30% of previous estimates based on $J_0$ alone arise from string fields. The study yields a practical axion-mass-equivalent scale of $m_{\text{ax}}^* \sim 4 \mu\text{eV}$ for a scaling network and points to future work on domain-wall annihilation and matter-era dynamics to complete the cosmological axion mass prediction.

Abstract

Cosmic strings formed when the Peccei-Quinn symmetry breaks post-inflation are expected to emit axions throughout their lifetime. The details of the evolution of this network and the associated spectrum of axions are crucial for obtaining an accurate axion mass prediction, thus guiding searches at haloscopes. In a previous publication, we obtained evidence for the standard scaling of axion string networks, showing that the number of horizon lengths of string per horizon volume asymptotes to an $\mathcal{O}(1)$ constant. In this article, we turn our attention to the axion spectra, studying spectra of all components of the axion current and their unequal time correlators. With the new information we are better able to distinguish the contributions from propagating axions from the field carried by the strings, and show that previous measurements of the axion energy spectrum based only on the timelike component of the current are approximately 30\% derived from the string fields. We introduce a simple model based on an ensemble of string segments, which accounts for the general features of the spectra and time correlations. We conclude that axion emission from a scaling string network is close to scale-invariant ($q \approx 1$), and that the energy spectrum of sub-horizon modes behaves as $p_\text{ax}\ln( k τ)$, where $k$ is the comoving wavenumber, $τ$ the conformal time and $p_\text{ax} \simeq 10$. The number density spectrum evolves towards a single curve for $kτ\lesssim 10^2$, with higher wavenumber deviations arising from initial conditions and resonant axion production at the string width scale. The total number density of axions produced from strings is $n_\text{ax}=1.66(17) f_\text{a}^2 H$, where $f_\text{a}$ is the axion decay constant and $H$ the Hubble rate. We report on axion production from the final collapse of the network in a future work.

Figures (17)

• Figure 1: Top: string local rest-frame length density parameter, against conformal time in code units. Middle: ratio of the comoving universe-frame mean string separation $\xi^\text{c}_\text{w}$ to comoving horizon length $\tau$. Bottom: RMS velocity. Times between the end of diffusive time evolution $\tau_{\rm dif}$ and the start of the physical evolution $\tau_{\rm cg}$ are plotted with dotted lines. The fixed point values obtained in Ref. Correia:2024cpk are shown as the horizontal olive dashed line with 1$\sigma$ uncertainty denoted by the band. The times in which power spectra and UETCs are calculated are shown as grey vertical lines. The legend gives the initial field correlation length in code units, in which the string width at the initial time is $w_\text{str} = 0.5$.
• Figure 2: Decoherence function for the transverse currents $J_\pm$ in the Unconnected Segment model \ref{['e:DpmUSM']}, with the speed distribution given in Eq. \ref{['e:vDist']}, evaluated at $x_1 = 25$, where $x_1 \equiv k\tau_1$. Also shown is the unaveraged expression \ref{['e:DpmUSMnoav']}, evaluated at the RMS velocity $\bar{v} = 0.52$.
• Figure 3: Wavenumbers for the analysis of power spectra derived from significant length scales, in units of the final time at which power spectra are recorded $\tau_{\rm fin}$: lattice spacing $\Delta x$, string width $w_\text{str}$ and the simulation box side length $N\Delta x$. The coloured solid lines are mean universe frame string separation $\xi_{\rm w}^\text{c}$, with the same colour key for initial field correlation lengths as Fig. \ref{['f:zetaw']}. The coloured dashed lines are their values at $\tau_{\rm dif}$ (see Table \ref{['simparams']}).
• Figure 4: Final $\tau$-scaled power spectra. Top left: ${\mathcal{P}}_0$, the power spectrum of $J_0$. Top right: ${\mathcal{P}}_s$, the power spectrum of $J_s$. Bottom left: the sum ${\mathcal{P}}_+ + {\mathcal{P}}_-$ (solid) and the difference of ${\mathcal{P}}_0 - {\mathcal{P}}_s$ (dashed). Bottom right: the difference ${\mathcal{P}}_+ - {\mathcal{P}}_-$. The power spectra are plotted against comoving wavenumber $k$ in units of the comoving horizon $\tau$ (bottom axis) and in units of the comoving string width (top axis). The top left plot is annotated with wavenumbers corresponding to several relevant length scales. Vertical dashed lines mark the horizon at the start of the second order time evolution $\tau_{\rm dif}$, the horizon at the start of physical evolution $\tau_{\rm cg}$, half the comoving string width at $\tau_{\rm cg}$, and half the comoving string width at $\tau_{\rm dif}$. Coloured dots mark the wavenumbers corresponding to the universe-frame mean string separation at the start of physical evolution, $\xi(\tau_{\rm cg})$, in each set of simulations. The black dashed line in the inset in the top right figure is the fit to the mean of the $l_\phi = 20$ power spectra, using Eq. \ref{['e:PowSpeFitFun']}, in the range $25 < k\tau < 100$.
• Figure 5: Emission spectra for (top to bottom) $J_0$ and $J_s$, between times $\tau = [2000, 2340]$ (left) and $\tau = [2360, 2801]$ (right). The spectra have been smoothed with a Savitsky-Golay filter savitzky1964smoothing, with window length 16 and polynomial order 3. The vertical dashed lines show the comoving wavenumber corresponding to half the physical saxion mass at a conformal time half way between the end-points.