Theoretical uncertainties for cosmological first-order phase transitions

TL;DR

This work systematically compares perturbative approaches to cosmological first-order phase transitions within the SMEFT, focusing on gravitational-wave predictions. It shows that conventional 4d daisy-resummed calculations suffer large renormalisation-scale uncertainties and gauge-dependence, leading to order-of-magnitude ambiguities in the predicted peak GW amplitude. By employing dimensional reduction to a 3d effective theory, the authors achieve markedly reduced scale dependence and obtain gauge-invariant results through an explicit ħ-expansion, while enabling controlled higher-loop and nucleation analyses. The study also assesses high-temperature truncations, nonperturbativity, and nucleation corrections, concluding that the 3d DR method provides a more reliable and systematically improvable framework for GW phenomenology, with important implications for LISA and beyond. Overall, the work advocates using dimensionally-reduced perturbation theory as the standard tool for predicting SGWB signals from SMEFT-type first-order phase transitions.

Abstract

We critically examine the magnitude of theoretical uncertainties in perturbative calculations of first-order phase transitions, using the Standard Model effective field theory as our guide. In the usual daisy-resummed approach, we find large uncertainties due to renormalisation scale dependence, which amount to two to three orders-of-magnitude uncertainty in the peak gravitational wave amplitude, relevant to experiments such as LISA. Alternatively, utilising dimensional reduction in a more sophisticated perturbative approach drastically reduces this scale dependence, pushing it to higher orders. Further, this approach resolves other thorny problems with daisy resummation: it is gauge invariant which is explicitly demonstrated for the Standard Model, and avoids an uncontrolled derivative expansion in the bubble nucleation rate.

Figures (13)

• Figure 1: A common method of calculating the thermal parameters of a phase transitions is very sensitive to the choice of renormalisation scale. Here we show this dependence in the popular daisy-resummed "4d approach" for a benchmark point of our SMEFT defined by Eq. \ref{['eq:O6']}, without the renormalisation group (RG) running of couplings. The LISA signal-to-noise ratios are 6 and 210 for the renormalisation scales $\bar{\mu}=T/2$ and $2\pi T$ respectively, for the calculation of which we have used PTPlotCaprini:2019egz and assumed a three year mission profile.
• Figure 2: A comparison of the dependence on the renormalisation scale, $\bar{\mu}$, in the daisy-resummed (black) and dimensionally-reduced (blue) approaches. The thermodynamic quantities are calculated for different choices of $\bar{\mu}$, with uncertainty bands indicating the envelope spanned by these choices. The optimal $\bar{\mu} = 2.2 T$ is established in Eq. \ref{['eq:bmu4:dr']}.
• Figure 3: The remormalisation scale dependence of the critical temperature, $T_{\rm c}(\bar{\mu})$, in the 4d (black) and 3d (blue) approaches at the benchmark point $M=640$ GeV. As discussed in the text, in the 3d approach the other thermodynamic parameters show a similar $\bar{\mu}$ dependence to $T_{\rm c}$.
• Figure 4: Gauge dependence of thermal parameters for the 4d approach at $\bar{\mu} = T$ (black) and the 3d approach at $\bar{\mu} = 2.2T$ (blue). In both cases the continuous lines denote $\xi_1=\xi_2=0$ and the dot-dashed lines denote $\xi_1 = \xi_2 = 3$. At $\xi_1=\xi_2=10$ the 4d approach breaks down, whereas in the 3d approach the artificial gauge dependence is largely indiscernible, even to very large values of $\xi_i$. A dashed blue line demonstrates this at $\xi_1=\xi_2=100$ for the 3d approach. Note that the residual gauge dependence in the 3d approach is an artifact of an incomplete operator basis in this EFT, and as such is not morally equivalent to the inherent gauge dependence in the 4d approach.
• Figure 5: $\alpha$, $\beta/H_p$, each with two lines: (i) full thermal functions at $\bar{\mu}=T$ (your existing results) and (ii) high-$T$ approximation (up to and including logarithms) for all thermal functions (both fermions and bosons), again at $\bar{\mu}=T$. Thermodynamic parameters in the 4d approach, with and without high-$T$ expansion of thermal functions, with a choice $\bar{\mu}=T$ for the RG-scale.