Back to the phase space: thermal axion dark radiation via couplings to standard model fermions

TL;DR

The paper develops a momentum-space Boltzmann framework to track the phase-space distribution ${\mathcal F}_a(k,t)$ of axions produced by standard model fermions, without assuming early thermalization. It analyzes leptophilic and hadrophilic couplings, computes the resulting dark radiation in terms of ${\Delta N_{\rm eff}}$, and demonstrates that infrared production below the weak scale yields distinctive spectral and energy-density signatures. The authors show that a fully phase-space treatment can shift ${\Delta N_{\rm eff}}$ predictions beyond the reach of standard approximate methods, making spectral distortions and decoupling dynamics observable by future CMB experiments. They also quantify theoretical uncertainties from the QCD crossover and running, finding robust predictions for top-quark couplings but more cautious bounds for bottom and charm due to nonperturbative effects near confinement.

Abstract

We investigate the cosmological consequences of axion interactions with standard model fermions accurately and precisely. Our analysis is entirely based on a phase space framework that allows us to keep track of the axion distribution in momentum space across the entire expansion history. First, we consider flavor-diagonal couplings to charged leptons and quantify the expected amount of dark radiation as a function of the coupling strength. Leptophilic axions are immune from complications due to strong interactions and our predictions do not suffer from theoretical uncertainties. We then focus on flavor-diagonal interactions with the three heavier quarks whose masses are all above the scale where strong interactions become non-perturbative. The top quark case is rather safe because its mass is orders of magnitude above the confinement scale, and the consequent predictions are solid. The bottom and charm masses are in more dangerous territory because they are very close to the QCD crossover. We present a comprehensive discussion of theoretical uncertainties due to both the choice of the scale where we stop the Boltzmann evolution and the running of QCD parameters. Finally, we compute the predicted amount of dark radiation expressed as an effective number of additional neutrino species. We compare our predictions with the ones obtained via standard approximate procedures, and we find that adopting a rigorous phase space framework alters the prediction by an amount larger than the sensitivity of future CMB observatories.

Figures (9)

• Figure 1: Axion production rates as a function of the temperature for leptophilic axion (left panel) and hadrophilic axion (right panel). Colored lines, both solid and dashed, show the production rates when only the interaction with that specific standard model particle is switched on in the Lagrangian. The difference between solid and dashed lines is the chosen value for $f_a / c_i$ as explained in the legenda. The dashed-dot black line shows the temperature-dependent Hubble expansion rate.
• Figure 2: Numerical results for leptophilic axions: asymptotic phase space distributions shown in terms of comoving variables defined in the main text (left) and evolution of the ratio $T_a / T$ between axion and bath temperatures (right). Different rows contain results for tau (top), muon (middle), and electron (bottom). Colored lines correspond to different choices for $f_a$ (all dimensionless Wilson coefficients are set to $c_\psi = 1$), and thick dotted lines identify the Bose-Einstein equilibrium distribution.
• Figure 3: Numerical results for hadrophilic axions. Notation as in Fig. \ref{['fig:PSDleptons']}.
• Figure 4: Predicted value of $\Delta N_{\rm eff}$ for different choices of $T_{\rm STOP}=\{1,2,5\}$ GeV, i.e. the temperature at which we shut off the axion-quark coupling. The predictions for the bottom and charm quarks are shown in the left and right panels, respectively.
• Figure 5: Calculation of $\Delta N_{\rm eff}$ with different running of QCD parameters for bottom (left) and charm (right) quarks. We compare the results presented in this paper (solid lines) where we fix the strong gauge coupling and quark mass with cases where we let run either $\alpha_s(T)$ (dashed) or the quark mass $m_q(T)$ (dotted).