Gravity waves from a cosmological phase transition: Gauge artifacts and daisy resummations

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.

Figure 1

An example of multiple phase transitions in the same model. Here, $m_h=35\mathrm{GeV}$, $e^2=\frac{2m_h}{3v}$, and $\xi =3$. Since the existence of the secondary phase transition is gauge dependent, it is clearly nonphysical.

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. 2 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. 2 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.

Figure 6

Expected gravitational wave spectrum for a Higgs mass of 35 GeV, calculated in Landau gauge ($\xi =0$), one or two high-$\xi$ gauges ($\xi =1$, 5), and the gauge-invariant Hamiltonian formalism.

Figure 7

Expected gravitational wave spectrum for a Higgs mass of 120 GeV, calculated in Landau gauge ($\xi =0$), a high-$\xi$ gauge ($\xi =1$), and the gauge-invariant Hamiltonian formalism.

Figure 8

Comparison of gravitational wave spectra calculated without daisy resummation in Landau gauge, and with resummation (dashed lines) in Landau gauge and two other $R_{\xi }$ gauges.

Figure 9

Comparison of gravitational wave spectra calculated without daisy resummation in Landau gauge, and with resummation (dashed lines) in Landau gauge and one or two other $R_{\xi }$ gauges.

Figure 10

Comparison of gravitational wave spectra calculated without daisy resummation in Landau gauge, and with resummation (dashed lines) in Landau gauge and one other $R_{\xi }$ gauge.