Putting the Brakes on Axion Strings: Friction and Its Impact on the QCD Axion Abundance

Abstract

A compelling production mechanism for QCD axion dark matter is from the scaling dynamics of early universe axion strings. We show that in DFSZ-like models containing tree-level interactions between fermions and the axion, friction between the thermal bath and the axion string drastically changes the behavior of the axion string network for lower $f_a$ values. Friction delays the onset of scaling and increases the energy density of axions. Once the effects of friction are included, we argue that in addition to the standard value of $m_a \sim$ meV, $m_a \sim 0.1$ eV also reproduces the dark matter energy density.

Figures (4)

• Figure 1: Impact of friction on the ratio of axion abundance as computed by the loop-based estimator Eq. \ref{['eq:n_radiation']}. The shaded band shows the variation in the nuisance parameter $\alpha$ that sets the friction-limited critical loop size $\ell^K_{\rm crit} = \alpha\,\ell_f$, with $\alpha \in [1,10]$, while the solid curve corresponds to $\alpha = 2\pi$. We fix the loop emission coefficient to $\Gamma_a = 50$ and the loop fraction to $f_{\rm loop} = 0.25$. Different colors correspond to the infrared spectral cutoff $x_{\rm IR} = 30,50,100$. In the friction-limited regime at small $f_a$, the predicted axion abundance is essentially independent of $x_{\rm IR}$, whereas in the relativistic scaling regime at large $f_a$ the choice of $x_{\rm IR}$ sets the location of the $\Omega_r \simeq 1$ crossing, which lies in the range $f_a \sim 3\times 10^{10}\text{--}10^{11}\,\mathrm{GeV}$ in agreement with Refs. Buschmann:2021sdqBenabou:2024msj. The dashed lines show the corresponding predictions when friction is neglected; including friction enhances the axion abundance by a factor of $\sim 10^1\text{--}10^4$, depending on the value of $f_a$ within the friction-dominated regime.
• Figure 2: Dependence of the relic abundance ratio $\Omega_r$ (defined as Eq. \ref{['eq:OmegaR_DM_here']}) on the spectral index $q$ for four benchmark decay constants chosen to sample the scaling regime $f_a = 10^{10},\,10^{11}\,\mathrm{GeV}$ and the friction-dominated regime $f_a = 10^{7},\,10^{8}\,\mathrm{GeV}$. The solid lines correspond to the reference choice $\alpha=4$, while the shaded bands show the variation over $\alpha\in[1,10]$. The horizontal dashed line marks $\Omega_r=1$. The curves quickly approach a plateau for $q\gtrsim 1$, indicating that once the spectrum is mildly IR-weighted the abundance is set primarily by the IR cutoff rather than the detailed tilt. For larger $f_a$, the emission is in the scaling regime, where the infrared cutoff saturates to $x_{\rm IR}$ as in Eq. \ref{['eq:xmin_scaling']}, so the result is independent of $\alpha$.
• Figure 3: The relic abundance ratio $\Omega_r$ (defined in Eq. \ref{['eq:OmegaR_DM_here']}) as a function of $f_a$, computed using the spectrum-based estimator for an IR-dominated spectrum ($q \gg$1). The solid curve shows the benchmark value $\alpha=4$, while the shaded band shows the variation over $\alpha \in [1,10]$. The three curves labeled $x_{\rm IR} = 5,10,30$ illustrate that in the friction-limited Kibble regime at low $f_a$, the result is essentially independent of $x_{\rm IR}$, while $x_{\rm IR}$ controls the intercept of the high-$f_a$ scaling branch. The phenomenologically relevant information is the location of the $\Omega_r \simeq 1$ crossings: the high-$f_a$ solution at $f_a \sim \mathrm{few}\times 10^{9}\,\mathrm{GeV}$ in agreement with Ref. Gorghetto:2018mykGorghetto:2020qws (with $q>2$) and novel second crossing at $f_a \sim \mathrm{few}\times 10^{8}\,\mathrm{GeV}$, arising from the friction-enhanced axion yield. The dashed lines show the estimate obtained when friction is neglected, while including friction enhances $\Omega_r$ by a factor of order $10$ over much of the friction-dominated regime. For such IR-dominated spectra, the subsequent nonlinearity of the axion potential (discussed in Sec. \ref{['sec:dilution']}) will further modify the final abundance (cf. Fig. \ref{['fig:dilution']}).
• Figure 4: Impact of the post-oscillation dilution on the relic abundance ratio $\Omega_r$ for the IR-dominated spectral estimator with $q\gtrsim2$. The orange band show the result obtained without applying the dilution between $t_{\rm osc}$ and $t_\ell$, while the red band include this effect. The central lines correspond to the reference choice $\alpha = 4$, and the shaded bands span $\alpha \in [1,10]$. Including the dilution suppresses the abundance at smaller $f_a$, reducing the low-$f_a$ branch from $\Omega_r > 1$ to a value of order unity, and shifts the high-$f_a$$\Omega_r \simeq 1$ crossing to larger $f_a \lesssim 10^{10}\,\mathrm{GeV}$, in agreement with Ref. Gorghetto:2020qws. For the numerical coefficients entering the dilution, we use the simulation-based values of Ref. Gorghetto:2020qws, $c_m = 2.08$, $c_V = 0.13$, and $c_n = 1.35$.