Measuring Gravitational Wave Spectrum from Electroweak Phase Transition and Higgs Self-Couplings

TL;DR

The paper addresses how to extract information about a first-order EWPT and Higgs self-couplings from a stochastic SGWB measured by space-based detectors, focusing on Taiji-like missions. It integrates a frequency-domain detector model with realistic noise and astrophysical foregrounds and performs parameter inference using both the Fisher information matrix and Bayesian MCMC to recover the spectral parameters $( ext{Ω}_0, ext{f}_p)$. These spectral constraints are then mapped to the xSM parameter space $(v_s,m_{h_2}, heta,b_3,b_4)$ and its thermodynamics $(T_n,eta/H_n,K)$, enabling predictions for the Higgs self-couplings $( ext{Δ} extκ}_3, ext{Δ} extκ}_4)$. The study highlights parameter degeneracies that limit precision but demonstrates that GW observations can substantially constrain the Higgs potential in regions complementary to collider probes, underscoring the synergistic potential of GW astronomy and collider physics for electroweak symmetry breaking.

Abstract

In this work, we demonstrate the complete process of using space-based gravitational wave detectors to measure properties of the stochastic gravitational wave background resulting from a first order electroweak phase transition, to infer the parameters governing the phase transition dynamics as well as that of the underlying particle physics model, and eventually to make predictions for important physical observables such as the Higgs cubic and quartic self-couplings which are difficult to measure at colliders. This pipeline is based on a frequency domain simulation of the space-based gravitational wave detector Taiji, taking into account dominant instrumental noises and astrophysical background, where the data analysis is carried out using both the Fisher information matrix and Bayesian inference with Markov-Chain Monte Carlo numerical sampling. We have applied this framework to the simplest extension of the Standard Model, the singlet extension, and show the measured uncertainties of the parameters at various levels of inference, and show that the Higgs cubic and also the quartic coupling can be highly constrained from gravitational wave measurement. We also show the impact from the problem of parameter degeneracy, highlighting the corresponding limitation on parameter inference and on making predictions.

Figures (11)

• Figure 1: Comparison of the response functions (left) for the $A$, $E$, and $T$ channels of Taiji (solid curves) and LISA (dashed curves), together with their respective noise spectral densities (right).
• Figure 2: This figure shows the theoretical PSD curve (blue) and the simulated data in channels $A$ and $T$ (red points). The curve is computed using Eqs. \ref{['udd']} and \ref{['esc']}, while the red points are generated from Eq. \ref{['hdm']}. The analysis focuses on the frequency range $\left[3 \times 10^{-5}\,\text{Hz},\,0.5\,\text{Hz}\right]$, with $\Omega_\text{ast} = 10^{-8}$ and $\varepsilon = 2/3$.
• Figure 3: Uncertainty analysis for the parameters $k$ (left) and $b$ (right) in the linear model $y = kx + b$ based on the FIM. The figure compares the relative uncertainties for three different values of the data size $n$. As expected, a larger number of data points leads to smaller parameter uncertainties.
• Figure 4: Comparison between the parameter constraints obtained from MCMC sampling and the confidence ellipses predicted by the FIM. The injected parameter values are $k = 2$ and $b = 1$, as indicated by the dashed lines. The darker and lighter shaded regions correspond to the 68% and 95% confidence intervals, respectively. In the left panel, the confidence ellipses (blue contours) are generated by drawing samples from a multivariate Gaussian distribution using the covariance matrix derived from the FIM. Superimposed red and green curves denote the analytical 68% and 95% Mahalanobis distance contours, respectively, and show excellent agreement with the sampled ellipses, thereby validating the correctness of the numerical Fisher-based approach. In the right panel, we present a direct comparison between the MCMC posterior distributions (in red) and the FIM forecasts (in blue). The overlap and slight deviations highlight both the validity and the limitations of the Gaussian assumption in the Fisher formalism. In this example, the minor discrepancy is likely caused by statistical fluctuations, while in more complex models with non-Gaussian posteriors, such deviations could become more significant.
• Figure 5: Comparison of parameter estimation uncertainties obtained from the FIM (blue) and MCMC sampling (red) for four representative parameters: two instrumental noise parameters, $N_{\mathrm{acc}}$ and $\delta x$, and two astrophysical parameters, $\Omega_{\mathrm{ast}}$ and $\varepsilon$. The injected values, $N_{\mathrm{acc}} = 3 \times 10^{-15}$, $\delta x = 8 \times 10^{-12}$, $\Omega_{\mathrm{ast}} = 1 \times 10^{-8}$, and $\varepsilon = 2/3$, are indicated by solid blue lines, while the MCMC-recovered best-fit values are shown as solid red lines. The dark and light blue shaded regions denote the 68% and 95% confidence contours predicted by the FIM, respectively, whereas the red contours represent the corresponding credible regions from MCMC sampling. The diagonal panels display the marginalized one-dimensional posterior distributions from both approaches, with dashed vertical lines marking the $1\sigma$ intervals (blue for FIM and red for MCMC). The relative uncertainties predicted by the FIM are approximately 0.088%, 0.0064%, 0.24%, and 0.78%, while those obtained from the MCMC posteriors are nearly identical: 0.088%, 0.0065%, 0.24%, and 0.79%. The shape and orientation of the ellipses illustrate the correlations between parameters: elongated and tilted contours indicate strong degeneracies, whereas more circular contours suggest weaker coupling.