Thermal evolution of dark matter and gravitational-wave production in the early universe from a symplectic glueball model

TL;DR

This paper nonperturbatively determines the equation of state for a confining $Sp(2)$ gauge theory using lattice simulations, revealing a first-order confinement/deconfinement transition with latent heat $L_{ m h}/T_c^4\approx1.69(12)$ and EOS quantities that approach but remain below the Stefan–Boltzmann limit near $T_c$. It then analyzes the cosmological evolution of a dark $Sp(2)$ sector as a DM candidate, including potential reheating, relic scenarios (freeze-out and cannibalism), and the generation of gravitational waves from the phase transition. The GW spectrum is modeled with standard sources (bubble collisions, sound waves, turbulence), with semi-quantitative predictions suggesting a dominant sound-wave contribution and peak frequencies in the millihertz range, potentially accessible to space-based detectors like LISA/DECIGO/BBO. The study highlights both the predictive power of a lattice-determined EOS for a dark gauge sector and the need for further work to refine non-equilibrium dynamics and extend the analysis to broader gauge groups and matter content. Overall, the work provides a concrete, nonperturbative framework linking a dark glueball DM scenario to a potentially observable gravitational-wave signature from a finite-temperature phase transition.

Abstract

The hypothesis that dark matter could be a bound state of a strongly coupled non-Abelian gauge theory is theoretically appealing and has a variety of interesting phenomenological implications. In particular, an interpretation of dark matter as the lightest glueball state in the spectrum of a dark Yang-Mills theory, possibly coupled to the visible sector only through gravitational interactions, has been discussed quite extensively in the literature, but most of previous work has been focused on dark SU(N) gauge theories. In this article, we consider an alternative model, based on a symplectic gauge group, which has a first-order confinement/deconfinement phase transition at a finite critical temperature. We first determine the equation of state of this theory, focusing on temperatures close to the transition, and evaluating the associated latent heat. Then we discuss the evolution of this dark-matter model in the early universe, commenting on the mechanisms by which it could indirectly interact with the visible sector, on the spectrum of gravitational waves it could produce, and on the relic abundances it would lead to. Our discussion includes an extensive review of relevant literature, a number of comments on similarities and differences between our model and dark SU(N) gauge theories, as well as some possible future extensions of the present study.

Figures (10)

• Figure 1: Binder cumulant associated with the Polyakov loop, defined in eq. \ref{['Binder_cumulant']}, extracted from simulations on lattices with $N_\tau=8$ sites in the Euclidean-time direction and for different spatial volumes $L^3$ (displayed by symbols of different shapes and colors), and shown as a function of $\beta=8/g_0^2$. The continuous curves are splines to guide the eye.
• Figure 2: Main plot: fit of the data reported in table \ref{['tab:scale-setting_values']} to eq. \ref{['scale-setting_fit']}. Inset plot: comparison of our non-perturbative scale setting (green points) with the mean-field improved perturbative scale setting (magenta line) discussed in the text.
• Figure 3: Trace of the energy-momentum tensor $\Delta$ in the $\mathop{\mathrm{\mathrm{Sp}}}\nolimits(2)$ theory, expressed in units of the fourth power of the temperature, as obtained from our simulations at finite values of the lattice spacing, equal to one fifth (black squares), one sixth (red triangles pointing up), and one seventh (green circles) of the lattice extent in the Euclidean-time direction. In addition, the figure also includes the results obtained from simulations on lattices with values of the lattice spacing equal to one eighth (blue triangles pointing down) and one ninth (orange diamonds) of the lattice extent in the Euclidean-time direction, for a more limited set of temperatures ($T/T_{\hbox{\tiny{c}}}=0.9$, $0.95$, $1$, $1.05$, and $1.1$).
• Figure 4: Continuum extrapolation for $\Delta/T^4$ at $T/T_{\hbox{\tiny{c}}}=1.5$: the data obtained from the infinite-volume extrapolation of the results of simulations at $T/T_{\hbox{\tiny{c}}}=1.5$ on lattices with $N_\tau=5$, $N_\tau=6$, and $N_\tau=7$ points along the Euclidean-time direction (black circles) are fitted as a function of $1/N_\tau^2$, including a constant and linear term. The fit result is denoted by the dashed black line. The red square denotes the extrapolation to the continuum limit.
• Figure 5: Results for thermodynamic-equilibrium quantities extrapolated to the continuum limit, and plotted as a function of the $T/T_{\hbox{\tiny{c}}}$ ratio: the main plot shows the entropy density in units of $T^3$ (magenta symbols), the energy density (blue symbols), the trace of the energy-momentum tensor (brown symbols), and the pressure (red symbols) in units of $T^4$. The horizontal bars on the right-hand side of the figure show, from top to bottom, the values corresponding to the Stefan--Boltzmann limit for $s/T^3$, for $\epsilon/T^4$, and for $p/T^4$. In the inset plot, the same quantities (except for $\Delta/T^4$, whose value in the free limit is zero) are normalized by their values in the Stefan--Boltzmann limit, with the same colors as in the main plot.