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First-order cosmological phase transitions in the radiation dominated era

Ariel Megevand

TL;DR

This work tackles the problem of how first-order cosmological phase transitions proceed in a radiation-dominated universe and how they imprint relics. It develops a fully analytical treatment of the two-stage evolution—an initial reheating phase followed by a phase-equilibrium stage—using thin-wall and linearized free-energy approximations, while incorporating back-reaction on the cosmic expansion. The main contributions are closed-form expressions for key quantities such as the bubble number density $n_b$, interface velocity $v_w$, and the duration of the stages, all expressed in terms of thermodynamic parameters like the latent heat $L$, surface tension $\sigma$, and friction $\eta$, and it distinguishes impurity-mediated from homogeneous nucleation. The results offer a practical toolkit to connect microphysical model inputs to cosmological outcomes (e.g., baryogenesis efficiency, defect production, and magnetic-field generation) and to assess how changes in particle content influence phase-transition dynamics and relics.

Abstract

We consider first-order phase transitions of the Universe in the radiation-dominated era. We argue that in general the velocity of interfaces is non-relativistic due to the interaction with the plasma and the release of latent heat. We study the general evolution of such slow phase transitions, which comprise essentially a short reheating stage and a longer phase equilibrium stage. We perform a completely analytical description of both stages. Some rough approximations are needed for the first stage, due to the non-trivial relations between the quantities that determine the variation of temperature with time. The second stage, instead, is considerably simplified by the fact that it develops at a constant temperature, close to the critical one. Indeed, in this case the equations can be solved exactly, including back-reaction on the expansion of the Universe. This treatment also applies to phase transitions mediated by impurities. We also investigate the relations between the different parameters that govern the characteristics of the phase transition and its cosmological consequences, and discuss the dependence of these parameters with the particle content of the theory.

First-order cosmological phase transitions in the radiation dominated era

TL;DR

This work tackles the problem of how first-order cosmological phase transitions proceed in a radiation-dominated universe and how they imprint relics. It develops a fully analytical treatment of the two-stage evolution—an initial reheating phase followed by a phase-equilibrium stage—using thin-wall and linearized free-energy approximations, while incorporating back-reaction on the cosmic expansion. The main contributions are closed-form expressions for key quantities such as the bubble number density , interface velocity , and the duration of the stages, all expressed in terms of thermodynamic parameters like the latent heat , surface tension , and friction , and it distinguishes impurity-mediated from homogeneous nucleation. The results offer a practical toolkit to connect microphysical model inputs to cosmological outcomes (e.g., baryogenesis efficiency, defect production, and magnetic-field generation) and to assess how changes in particle content influence phase-transition dynamics and relics.

Abstract

We consider first-order phase transitions of the Universe in the radiation-dominated era. We argue that in general the velocity of interfaces is non-relativistic due to the interaction with the plasma and the release of latent heat. We study the general evolution of such slow phase transitions, which comprise essentially a short reheating stage and a longer phase equilibrium stage. We perform a completely analytical description of both stages. Some rough approximations are needed for the first stage, due to the non-trivial relations between the quantities that determine the variation of temperature with time. The second stage, instead, is considerably simplified by the fact that it develops at a constant temperature, close to the critical one. Indeed, in this case the equations can be solved exactly, including back-reaction on the expansion of the Universe. This treatment also applies to phase transitions mediated by impurities. We also investigate the relations between the different parameters that govern the characteristics of the phase transition and its cosmological consequences, and discuss the dependence of these parameters with the particle content of the theory.

Paper Structure

This paper contains 33 sections, 124 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cosmological consequences of phase transitions
  3. Electroweak baryogenesis.
  4. Baryon inhomogeneities.
  5. Topological defects and magnetic fields.
  6. Phase equilibrium
  7. Supercooling
  8. Bubble nucleation
  9. Phase transition dynamics
  10. Bubble coalescence
  11. Bubble number and interface velocity
  12. Thin wall approximation and linearization of $V$
  13. The first stage: nucleation and reheating
  14. The second stage: phase equilibrium
  15. Inhomogeneous nucleation
  16. ...and 18 more sections

Figures (3)

  • Figure 1: The expansion rate of the Universe for a phase transition at constant $T=T_c$, in the case $\rho _{u} =\pi ^{2}g_{\ast }T^{4}/30$ (negative cosmological constant).
  • Figure 2: The expansion rate of the Universe for a phase transition at constant $T=T_c$, in the case $\rho _{u} =\pi ^{2}g_{\ast }T^{4}/30+L$.
  • Figure 3: Typical evolution of the temperature during a phase transition with supercooling.