Bayesian Analysis of the Complex Singlet Model with Phase Transition Gravitational Waves

TL;DR

This work analyzes the Complex Singlet Extension of the Standard Model as a source of a strong first-order electroweak phase transition whose stochastic gravitational-wave background could be detected by a Taiji-like space-based observatory. By building a frequency-domain likelihood that accounts for instrumental noise, galactic and astrophysical foregrounds, and the SW GW template, the authors perform both Fisher-matrix forecasts and Bayesian nested sampling to reconstruct the GW spectrum parameters and map them onto the CxSM scalar potential, including Higgs self-couplings. They demonstrate good agreement between the two inference methods and find that Taiji can yield decisive evidence for a SW signal (e.g., $\ln\mathrm{BF}\approx11.6$) with ${\Omega_0}$ and ${f_p}$ tightly constrained; these GW-derived constraints can then be propagated to the Higgs cubic and quartic deviations $\delta\kappa_3$, $\delta\kappa_4$, highlighting complementarity with collider reach. The study provides a rigorous framework linking phase-transition physics to GW data and underscores the complementary potential of future space-based GW detectors and collider experiments in probing the Higgs sector and electroweak-scale new physics.

Abstract

We explore the prospects of probing the Complex Singlet Extension of the Standard Model (CxSM) with gravitational waves from the Electroweak phase transition. The study establishes a connection of the scalar potential parameters, the thermodynamic properties of the phase transition, with the directly measured stochastic gravitational-wave background in the presence of astrophysical background and foreground. Considering the space-based gravitational wave detector Taiji, we construct a frequency-domain likelihood that incorporates instrumental and astrophysical noises, and perform both Fisher-matrix forecasts and Bayesian Nested Sampling analysis. The comparison of these two approaches demonstrates consistent parameter recovery and highlights the sensitivity of Taiji to millihertz gravitational-wave signals. We further propagate the inferred constraints on the gravitational-wave spectrum back to the underlying CxSM parameters, obtaining meaningful limits on the Higgs self-couplings. The results emphasize the complementarity between gravitational-wave observations and collider measurements, showing that future missions such as Taiji can serve as a powerful probe of electroweak-scale new physics and the dynamical origin of the Higgs sector.

Figures (6)

• Figure 1: Sensitivity curves for LISA and Taiji. Solid lines correspond to the model-based results from Eq. \ref{['hgv']}.
• Figure 2: Comparison of parameter constraints obtained from NS (red) and FIM (blue). The dark and light shaded regions denote the $68\%$ and $95\%$ confidence intervals, respectively. The analysis includes two instrumental noise parameters ($N_{\mathrm{acc}}, \delta x$), four galactic foreground parameters ($A_1, \alpha_1, A_2, \alpha_2$), two astrophysical background parameters ($\Omega_{\mathrm{ast}}, \varepsilon$), and two SW signal parameters ($\Omega_0, f_{\mathrm{p}}$). The corner plot illustrates that the NS posteriors are in good agreement with the FIM predictions, with mild deviations attributable to fluctuation in sampling and simulation, and weak non-Gaussian effects. The diagonal panels display the one-dimensional marginalized distributions, with dashed vertical lines marking the $1\sigma$ intervals (blue for FIM, red for NS). The inset figure in the upper-right corner shows the energy-density spectra of the individual components: the galactic foreground $\Omega_\mathrm{dwd}$ (gray), the astrophysical background $\Omega_{\mathrm{GW,ast}}(f)$ (green), the SW signal $\Omega_{\mathrm{sw}}(f)$ (orange), and the detector sensitivity curve (purple). The simulated total data points are shown in blue.
• Figure 3: Relative uncertainty on $\Omega_0$ obtained from NS (red points) compared with the FIM prediction (blue line). As expected, the uncertainty decreases for larger $\Omega_0$, reflecting the improvement in parameter precision with increasing SW signal strength.
• Figure 4: Constraints on the geometric parameters $\log_{10}\Omega_0$ and $\log_{10}f_{\mathrm{p}}$ obtained from NS. Blue points represent the full set of parameter combinations from the CxSM scan, while red and green points correspond to those falling within the 68% and 95% confidence regions, respectively. The left panel shows the complete distribution of sampled points, whereas the right panel provides a zoomed-in view highlighting the region favored by the data.
• Figure 5: Constraints on the CxSM parameters inferred from the NS posterior distributions. Blue points denote the full set of scanned parameter points, while red and green points correspond to those lying within the 68% and 95% credible regions, respectively, as determined from the geometric parameter posteriors. The distributions illustrate how GW observations can significantly narrow the viable regions of the scalar potential parameter space.