Spectrum of radiation from global strings and the relic axion density

TL;DR

This paper addresses how the radiation spectrum from global strings influences the relic axion density, introducing a parametric framework that separates loop and long-string contributions and encodes spectrum effects in the functions $G_1$ and $G_2$. It demonstrates, via numerical simulations of perturbed straight strings, that rigorous removal of the string self-field is essential to recover the propagating axion spectrum, which is predicted to be exponential rather than hard. Depending on the spectrum and energy partition between loops and long strings, the inferred axion mass $m_a$ can range from a few microelectronvolts to around $160\,\mu\text{eV}$, with corresponding detection frequencies spanning GHz to tens of GHz, and often exceeds simple initial misalignment predictions. The results highlight large theoretical uncertainties and call for careful, controlled simulations to reliably constrain axion properties from string networks, bearing on experiments and cosmology alike.

Abstract

We discuss key aspects of the nature of radiation from global strings and its impact on the relic axion density. Using a simple model we demonstrate the dependence on the spectrum of radiation emitted by strings. We then study the radiation emitted by perturbed straight strings paying particular attention to the difference between the overall phase of the field and the small perturbations about the string solution which are the axions. We find that a significant correction is required to be sure that one is analyzing the axions and not the self-field of the string. Typically this requires one to excise a sizeable region around the string - something which is not usually done in the case of numerical field theory simulations of string networks. We have measured the spectrum of radiation from these strings and find that it is compatible with an exponential, as predicted by the Nambu-like Kalb-Ramond action, and in particular is not a ``hard'' spectrum often found in string network simulations. We conclude by attempting to assess the uncertainties on relic density and find that this leads to a range of possible axion masses when compared to the measured density from the Cosmic Microwave Background, albeit that they are typically higher than what is predicted by the Initial Misalignment Mechanism. If the decay is via a ``soft spectrum'' from loops produced close to the backreaction scale we find that $m_{\rm a}\approx 160\,μ{\rm eV}$ and a detection frequency $f\approx 38\,{\rm GHz}$. If axions are emitted directly by the string network, and we use emission spectra reported in field theory simulations, then $m_{\rm a}\approx 4\,μ{\rm eV}$ and $f\approx 1\,{\rm GHz}$, however this increases to $m_a \approx 125\,μ{\rm eV}$ and $f\approx 30\,{\rm GHz}$ using our spectra for the case of an oscillating string. In all scenarios there are significant remaining uncertainties that we delineate.

Figures (16)

• Figure 1: The function $G_2(q,n*)$ as a function of $q$ for various values of $n_*$. The clear features are an asymptote for $q\rightarrow\infty$ and a strong dependence on $n_*$ for large negative $q$. We have also included as an inset the dependence on $n_*$ for values of $q$ close to one. Remembering that the axion mass at fixed $\Omega_{\rm a}h^2$ is proportional to $G_2$, this makes it clear that the predicted value depends very strongly on the spectrum unless $q\gg 1$. Note also that the contributions from both loop and long strings will have a similar functional form, but possibly very different parameters.
• Figure 2: Illustrations of the perturbed string configurations for $n_{x}=n_{y}=n_{z}=101,201,401,801$. The positions presented are those which are computed by the string position finding algorithm. The four setups correspond to the same wavelength of $L = 50\Delta x=35$ and $\varepsilon_0=0.5$. Note that the string amplitude is significantly smaller than the box size in all four cases.
• Figure 3: Relative amplitude, $\varepsilon(t)=2\pi A/L$ for the four simulations discussed in the text with $n_x=101, 201, 401$ and 801 for $L=35$ and $\varepsilon_0=0.5$. They are similar for the first few oscillations but deviate after this due the effects of the boundary implying that even with the absorbing boundary conditions there are some reflections of radiation. Included also is a linear fit to $\varepsilon^{-2}$ versus time to the simulation with $n_x=801$.
• Figure 4: The axion radiation field, $\Delta \alpha$, in the $x-y$ plane for $n_x=801$, $z=412\Delta x$ and $\varepsilon_0=0.5$ for three different time steps (from the left): $t=199\Delta t$, $t=209\Delta t$ and $t=219\Delta t$. We have zoomed in the region in the $x-y$ plane where $-100\le x /\Delta x,y/\Delta x\le 100$. It is clear that there is a quadrupole pattern near the center of the string. The string was initially displaced in the x-direction.
• Figure 5: Comparison of $\hat{r^i} D_i \alpha$ (left) and $\phi \hat{r^i} \partial_i \Delta \alpha$ (right) for a single string simulation with $n_x=801$, $\varepsilon_0= 0.5$ for the slice where $z = 412\Delta x$ taken at time $t = 678\,\Delta t$. We have zoomed in the region in the $x-y$ plane where $-300\le x /\Delta x,y/\Delta x\le 300$. As can be clearly seen the two quantities are almost identical, with small differences very close to the string. After the application of a mask of radius $r = 10\,\Delta x$ (bottom row) the two quantities become identical.