Thermal Damping of Mass-Modulating Scalars

TL;DR

This work develops a general Boltzmann-based framework to quantify thermal damping of scalars that modulate bath particle masses, extending previous adiabatic analyses to underdamped, sinusoidally evolving backgrounds. It derives analytic expressions for the damping rate $Υ$ in both adiabatic and underdamped limits and generalizes to multiple interacting bath species, providing a comprehensive treatment of mass-modulating scalars in cosmology. The authors apply the formalism to neutrino-coupled scalars, hadron-mass modulators including QCD axions, and WIMP-mass modulators, showing that thermal damping can compete with or exceed Hubble friction in certain temperature ranges and can alter late-time abundances and experimental targets. The results reveal new dynamical behaviors, such as attractor-like evolution and cross-damping between species, and offer practical criteria for when thermal damping must be included in cosmological scalar dynamics.

Abstract

The cosmological evolution of a scalar field is shaped by Hubble damping. Any non-gravitational couplings of the scalar with the primordial thermal bath generically contribute additional damping. Although rarely considered, such thermal damping could be the dominant dissipative effect. We derive approximate but highly general thermal-damping rates of scalar fields that modulate the masses of thermally populated particles. We extend previous results to cover cases of particular phenomenological interest where the scalar background oscillates sinusoidally but not necessarily slowly compared to the thermalization rates of the primordial bath. Based on these results, we estimate the thermal damping of scalars coupled to neutrinos linearly, to gluons quadratically, and to WIMPs linearly, and demonstrate its importance in certain parameter space of these models. We also estimate the thermal damping rates in models of QCD axion.

Figures (6)

• Figure 1: Illustration of thermal damping of $\chi$-mass modulating scalar field $\phi$. As the scalar field $\phi$ oscillates (top), it modulates $\chi$'s effective mass $M_\chi$ (middle purple) which, in turn, modulates $\chi$'s equilibrium number density $n_\chi^{\rm eq}$ (bottom purple). The $\chi$ field remains out of equilibrium as its actual number density $n_\chi$ (bottom orange) continually seeks the moving $n_\chi^{\rm eq}$ (bottom purple) and thereby backreacts dissipatively on $\phi$ (top purple).
• Figure 2: Origin of thermal damping in the toy model of Section \ref{['sec:intuition']}. Here, the model parameters are chosen to be the same as in Fig. 1. Left: $\chi$'s contribution to $\phi$'s effective potential gradient $|V'-m_\phi^2\phi|\approx |\lambda n_\chi \phi/m_\chi|$ plotted against the absolute value of the $\phi$ field. It shows that this potential-gradient contribution is non-conservative: gentler during descent ($d|\phi|/dt<0$), steeper during ascent ($d|\phi|/dt>0$). The area enclosed by the orange curve is the magnitude of the non-conservative work on the $\phi$ field in one half-oscillation, $\approx \int_{-\Phi}^{\Phi} d\phi\,V'$. Right: The $\chi$ particles are produced ($\dot{n}_\chi>0$) by the thermal bath when they are relatively light and decay/annihilate ($\dot{n}_\chi<0$) to the thermal bath when they are relatively heavy. In other words, the thermal bath "buys" $\chi$ particles when they are energetically cheap and "sells" them later when their energy costs are relatively high. The area enclosed by the orange curve, $\oint dn_\chi M_\chi$, is the net energy gain ("profit") of the thermal bath, which is equal to the dissipative energy loss of $\phi$ by energy conservation.
• Figure 3: The neutrino thermalization rate $\Gamma_{\rm th}$, scalar effective mass $M_\phi$, and oscillation-averaged thermal damping rate $\left<\Upsilon_\nu\right>_{\rm osc}$ in the neutrino-coupled scalar model of Eq. \ref{['eq:neutrinoLagrangian']}. Here, we set $g=10^{-11}$, $m_\phi=10^{-10}\mathinner{\mathrm{eV}}$, and that $\phi$ makes up the total dark matter abundance today, $\Phi=\Phi_{\rm CDM}$. In evaluating the thermal damping rate, we take this present-day value of $\Phi$ and evolve $\Phi$ backward in time accounting for the thermal mass of $\phi$ but assuming $\left<\Upsilon_\nu\right>_{\rm osc}=0$.
• Figure 4: The pion effective thermalization rate $\Gamma_{\rm th}^{\rm eff}$, scalar effective mass $M_\phi$, and oscillation-averaged thermal damping rate $\left<\Upsilon_\pi\right>$as a function of temperature in the scalar quadratically coupled to gluons model, Eq. \ref{['eq:Lquadgluon']}. Here, assume $d=10^{23}$ and $m_\phi=10^{-10}\mathinner{\mathrm{eV}}$, and fix the scalar amplitude to $\Phi=M_{\rm pl}/\sqrt{d}$. The vertical dashed line indicates the rough boundary between the adiabatic and non-adiabatic regimes. The displayed $\Gamma_{\rm th}^{\rm eff}$ is meaningful only in the adiabatic regime, where the adiabatic thermal damping formula, Eq. \ref{['eq:upsilon_adiabatic']}, applies. In the non-adiabatic regime, the thermal damping rate is given by the more general Eq. \ref{['eq:upsilon_underdamp']}, in which there is no simple notion of thermalization rate.
• Figure 5: Evolution of the scalar field amplitude $\Phi$ (left) and number density $n_\phi=M_\phi^2\Phi$ (right) in the scalar quadratically coupled to gluons model, Eq. \ref{['eq:Lquadgluon']}. Here, we assume the same model parameters as in Fig. \ref{['fig:quadratic_case']} and pick initial conditions such that $\Phi=10^6\mathinner{\mathrm{GeV}}$ (turquoise), $10^5\mathinner{\mathrm{GeV}}$ (purple), $3\times10^3\mathinner{\mathrm{GeV}}$ (orange) at $T=100$ MeV. The evolution of $\Phi$ and $n_\phi$ are shown in solid lines, which in this these figures merge almost completely to a single line. Their evolution in the absence of thermal damping is shown in dashed lines for comparison. On the left plot, the thick pink line is the scalar amplitude $\Phi_\Upsilon$ at which $\Upsilon\sim H$ and the approximate BBN limit ($\delta \Lambda_{\rm QCD}/\Lambda_{\rm QCD}\lesssim 2\times 10^{-3}$Flambaum:2002de) is shown as gray shaded region.