Gravity Waves from a Cosmological Phase Transition: Gauge Artifacts and Daisy Resummations

TL;DR

This work shows that predictions for gravitational waves from first-order cosmological phase transitions are highly sensitive to the gauge choice when using perturbative finite-temperature effective potentials. By studying the Abelian Higgs model in $R_xi$ gauges and comparing with a gauge-invariant Hamiltonian formulation, the authors demonstrate that gauge artifacts can dramatically alter the transition parameters and the resulting GW spectrum, with Landau gauge often reproducing gauge-invariant results. Daisy resummations in gauge-fixed calculations can further suppress GW amplitudes, underscoring the need for gauge-invariant resummation schemes, especially for non-Abelian theories. The paper discusses possible paths forward, including Monte Carlo approaches and higher-order Nielsen-identities-based methods, to obtain robust GW predictions in more realistic models such as the Standard Model or its extensions.

Abstract

The finite-temperature effective potential customarily employed to describe the physics of cosmological phase transitions often relies on specific gauge choices, and is manifestly not gauge-invariant at finite order in its perturbative expansion. As a result, quantities relevant for the calculation of the spectrum of stochastic gravity waves resulting from bubble collisions in first-order phase transitions are also not gauge-invariant. We assess the quantitative impact of this gauge-dependence on key quantities entering predictions for gravity waves from first order cosmological phase transitions. We resort to a simple abelian Higgs model, and discuss the case of R_xi gauges. By comparing with results obtained using a gauge-invariant Hamiltonian formalism, we show that the choice of gauge can have a dramatic effect on theoretical predictions for the normalization and shape of the expected gravity wave spectrum. We also analyze the impact of resumming higher-order contributions as needed to maintain the validity of the perturbative expansion, and show that doing so can suppress the amplitude of the spectrum by an order of magnitude or more. We comment on open issues and possible strategies for carrying out "daisy resummed" gauge invariant computations in non-Abelian models for which a gauge-invariant Hamiltonian formalism is not presently available.

Figures (10)

• Figure 1: An example of multiple phase transitions in the same model. Here, $m_h = 35$ GeV, $e^2 = \frac{2m_h}{3v}$, and $\xi = 3$. Since the existence of the secondary phase transition is gauge-dependent, it is clearly non-physical.
• Figure 2: Calculated gauge dependence of phase transition parameters for a low-mass Higgs boson. In all panels, black (grey) lines denote models with (without) resummation. The arrows denote values corresponding to the solid lines, but calculated in the gauge-invariant Hamiltonian formalism. All quantities along the y-axes are in units of GeV, except for $\beta/H$ which is unitless. In the first panel, solid, dashed and dotted lines denote the transition temperature $T_*$, the critical temperature $T_c$, and the minimum temperature at which the hot phase exists. In the second panel, solid and dashed lines denote the minima of the cold and hot phases. The third panel shows the relative difference in energy densities at both the critical temperature (dashed line) and the actual transition temperature (solid line). The final panel gives $\beta/H_*$, where $\beta$ is the approximate inverse phase transition duration and $H_*$ is the Hubble constant at the transition temperature.
• Figure 3: Calculated gauge dependence of phase transition parameters for a medium-mass Higgs boson. See fig. \ref{['fig:10mh']} for a thorough explanation of the different lines.
• Figure 4: Calculated gauge dependence of phase transition parameters for a high-mass Higgs boson. See fig. \ref{['fig:10mh']} for a thorough explanation of the different lines.
• Figure 5: Expected gravitational wave spectrum for a Higgs mass of 10 GeV, calculated in Landau gauge ($\xi=0$), two high-$\xi$ gauges ($\xi=1,5$), and the gauge-invariant Hamiltonian formalism.