Scalar damping in cosmological phase transitions

TL;DR

The paper addresses accurate modeling of scalar damping during cosmological phase transitions and assesses the validity of phenomenological friction in hydrodynamic simulations. It derives the damping term $η_φ$ from kinetic theory by solving Boltzmann equations for SM-like plasma and via the WallGo method, focusing on top quarks and weak gauge bosons. Key findings include convergence challenges from bosonic soft modes, a marginally justified local damping description for SM content, and the result that runaway-wall pressure generally upper-bounds the local friction within the regime of validity, complemented by explicit next-to-leading corrections to the Bödeker–Moore friction. These results guide when hydrodynamic simulations can safely employ a local friction term and when nonlocal kinetic treatments are necessary, with implications for predictions of gravitational-wave signals from first-order phase transitions.

Abstract

We outline how to calculate the scalar damping term during a cosmological phase transition from kinetic theory. We determine the scalar damping rate from top quarks and weak gauge bosons in a Standard Model-like theory. We find that the convergence of the bosonic contributions hinges on how the soft modes are treated. We discuss the validity of the phenomenological friction term employed in hydrodynamical simulations. We find that for a Standard Model particle content, this approximation is (marginally) justified. We also test the hypothesis that the pressure from a runaway wall acts as an upper bound on the pressure from the local friction term. We find that next-to-leading order contributions in terms of velocity and mass are negative and that in the regime of validity, the local damping term indeed cannot surpass the pressure from runaway bubbles.

Figures (4)

• Figure 1: Left: Friction on the wall in the Standard Model. The graph shows the result for an imposed wall velocity of $v_w = 0.1$, rescaled by the phenomenological friction term, as a function of the compactified coordinate $\chi$. Right: Friction coefficient for the $W$ for different choices of momentum grid size. The pink line shows the result for the vacuum mass in the Boltzmann equation, whereas the blue dashed line includes the full asymptotic mass.
• Figure 2: Left: Friction for different values of $Q$, the factor multiplying the collision terms, as a function of $\xi = - \bar{u}^\mu_w x_\mu$, with $\bar{u}^\mu_w = \gamma_w(v_w,0,0,1)$. The solid lines show the results from WallGo, and the dashed lines show the friction term in the local approximation (normalized by $\mathcal{N}_Q$ to $\Omega^Q_{\delta,t_L}$ at the maximum). Right: Integrated friction as a function of $Q$, for the left- and right-handed top quarks and the $W$ bosons. The dotted lines show fits to $\propto 1/Q$, which is the expected behavior in the local limit. The dotted vertical lines indicate where the numerical result is more than a factor 2 smaller than the expectation in the local limit.
• Figure 3: The figures exemplify the leading corrections to the runaway pressure in terms of $m_X^2/T^2$ and $1/\gamma_w$. The left shows corrections for bosons and the right for fermions.
• Figure 4: The figures show the pressure from the equilibrium contribution, the pressure from runaway and the combination for equilibrium and local friction. Notice that equilibrium neglects the contribution from the temperature change in the wall. We also display the largest friction we found outside of the validity of the local approximation. The left figure demonstrates the contribution from bosons, the right figure shows the contribution from fermions.