Phase transitions in the early universe

TL;DR

<3-5 sentence high-level summary>These notes provide a compact, self-contained introduction to first-order phase transitions in the early universe, outlining how a thermally driven Higgs potential enables bubble nucleation and subsequent hydrodynamic evolution. They connect microscopic field-theory descriptions to macroscopic fluid dynamics and quantify how phase-transition dynamics imprint gravitational waves in a spectrum that could be observed by LISA. The framework identifies key observables—nucleation temperature, transition strength, rate parameter, and wall velocity—that determine the GW signal, while noting the need for nonperturbative methods and simulations in strong transitions. Overall, the work highlights how GW observations could probe beyond-Standard-Model physics and early-universe cosmology.

Abstract

These lecture notes are based on a course given by Mark Hindmarsh at the 24th Saalburg Summer School 2018 and written up by Marvin Lüben, Johannes Lumma and Martin Pauly. The aim is to provide the necessary basics to understand first-order phase transitions in the early universe, to outline how they leave imprints in gravitational waves, and advertise how those gravitational waves could be detected in the future. A first-order phase transition at the electroweak scale is a prediction of many theories beyond the Standard Model, and is also motivated as an ingredient of some theories attempting to provide an explanation for the matter-antimatter asymmetry in our Universe. Starting from bosonic and fermionic statistics, we derive Boltzmann's equation and generalise to a fluid of particles with field dependent mass. We introduce the thermal effective potential for the field in its lowest order approximation, discuss the transition to the Higgs phase in the Standard Model and beyond, and compute the probability for the field to cross a potential barrier. After these preliminaries, we provide a hydrodynamical description of first-order phase transitions as it is appropriate for describing the early Universe. We thereby discuss the key quantities characterising a phase transition, and how they are imprinted in the gravitational wave power spectrum that might be detectable by the space-based gravitational wave detector LISA in the 2030s.

Figures (21)

• Figure 1: This figure shows the dimensionless function $J_{B}$ that is proportional to the free energy of bosons as defined in Eq. (\ref{['eq:BosFreEne']}), as a function of mass-to-temperature ratio (thick line). Also the expansions for large $T$ (dashed), Eq. \ref{['eqn:bosonic-partition-func-largeT']} and small $T$ (dotted), Eq. \ref{['eqn:bosonic-partition-func-lowT']} are shown. The large-T expansion is performed up to order four in $m/T$, being a good approximation up to $m/T\sim 1.1$.
• Figure 2: This figure shows the dimensionless function $J_{F}$ that is proportional to the free energy of fermions as defined in Eq. (\ref{['eqn:BosFreEnDim']}) as a function of mass-to-temperature ratio (thick line). Additionally, the expansion for large $T$ (dashed), Eq. \ref{['eqn:fermionic-partition-func-largeT']} and small $T$ (dotted) , in analogy to Eq. \ref{['eqn:bosonic-partition-func-lowT']} are shown. Note that in the small $T$ limit, both the fermionic and the bosonic expansions agree. The large $T$ expansion is performed up to order four in $m/T$, working well up to $m/T\sim 0.5$, hence being sligthly worse than the bosonic high-T expansion, depicted in Fig. \ref{['fig:free_energy_baryon']} .
• Figure 3: This figure shows the effective number of relativistic degrees of freedom $g_\text{eff}$ of a Standard Model plasma as a function of temperature, taking into account interactions between particles, with both perturbative and lattice methods Laine:2015kra.
• Figure 4: The figure shows the thermal effective Higgs potential $V_T(\phi)$ at different temperatures. For large temperatures $T\gg T_{\rm c}$ (red) the potential has a minimum at $\phi=0$ and the ground state is symmetric. Below the temperature $T_1>T_{\rm c}$ (dark green) a second, but higher lying minimum develops. At the critical temperature $T_c$ (green) both minima are degenerate. Below the critical temperature, the new minimum at non-zero field value is the global minimum representing the true (stable) ground state.
• Figure 5: The phase diagram of the Standard model. For Higgs masses of $m_H\lesssim 75$ GeV the Standard Model undergoes a first-order phase transition. For larger Higgs masses, there is no phase transition between the symmetric phase $\phi=0$ and the Higgs phase $\phi=v_\text{EW}$, but a cross-over. Including higher-order interactions changes the picture significantly.