[Paper Review] Observational prospects for phase transitions at LISA: Fisher matrix analysis

Chloe Gowling, Mark Hindmarsh|arXiv (Cornell University)|Jun 10, 2021

Pulsars and Gravitational Waves Research97 references66 citations

TL;DR

This paper uses Fisher matrix analysis to assess LISA's sensitivity to gravitational wave signals from first-order cosmological phase transitions, focusing on the sound shell model for the power spectrum. It finds that with signal-to-noise ratios above 20, peak frequency and amplitude can be measured to ~10% accuracy, and wall speed is the best-constrained thermodynamic parameter, with principal components indicating sub-3% uncertainty for key parameter combinations.

ABSTRACT

A first order phase transition at the electroweak scale would lead to the production of gravitational waves that may be observable at upcoming space-based gravitational wave (GW) detectors such as LISA (Laser Interferometer Space Antenna). As the Standard Model has no phase transition, LISA can be used to search for new physics by searching for a stochastic gravitational wave background. In this work we investigate LISA's sensitivity to the thermodynamic parameters encoded in the stochastic background produced by a phase transition, using the sound shell model to characterise the gravitational wave power spectrum, and the Fisher matrix to estimate uncertainties. We explore a parameter space with transition strengths $\alpha < 0.5$ and phase boundary speeds $0.4 < v_ ext{w} < 0.9$, for transitions nucleating at $T_{ ext{N}} = 100$ GeV, with mean bubble spacings $0.1$ and $0.01$ of the Hubble length, and sound speed $c/\sqrt{3}$. We show that the power spectrum in the sound shell model can be well approximated by a four-parameter double broken power law, and find that the peak power and frequency can be measured to approximately 10% accuracy for signal-to-noise ratios (SNRs) above 20. Determinations of the underlying thermodynamic parameters are complicated by degeneracies, but in all cases the phase boundary speed will be the best constrained parameter. Turning to the principal components of the Fisher matrix, a signal-to-noise ratio above 20 produces a relative uncertainty less than 3% in the two highest-order principal components, indicating good prospects for combinations of parameters. The highest-order principal component is dominated by the wall speed. These estimates of parameter sensitivity provide a preliminary accuracy target for theoretical calculations of thermodynamic parameters.

Motivation & Objective

• Assess LISA's observational prospects for detecting gravitational waves from first-order cosmological phase transitions.
• Quantify the sensitivity of LISA to thermodynamic parameters—transition strength α, wall speed vw, nucleation temperature Tn, and mean bubble spacing—using the sound shell model.
• Evaluate the impact of astrophysical foregrounds (galactic and extragalactic compact binaries) on parameter estimation accuracy.
• Use principal component analysis of the Fisher matrix to identify the most measurable combinations of parameters and set accuracy targets for theoretical models.
• Compare the double broken power law fit to the sound shell model with the single broken power law used in prior LISA working group studies.

Proposed method

• Model the gravitational wave power spectrum using the sound shell model (SSM), which relates GW emission to fluid velocity power spectra from expanding bubble walls.
• Approximate the SSM power spectrum with a four-parameter double broken power law to enable efficient Fisher matrix analysis.
• Construct a realistic LISA noise model including contributions from unresolved galactic and extragalactic compact binaries.
• Compute the Fisher information matrix to estimate parameter uncertainties, accounting for signal-to-noise ratio (SNR) and foreground contamination.
• Perform principal component analysis (PCA) on the Fisher matrix to identify the most informative combinations of parameters and assess their relative uncertainties.
• Apply a kinetic energy suppression factor derived from numerical simulations to improve agreement between the SSM and simulation results, particularly at low wall speeds and high α.

Experimental results

Research questions

• RQ1 How accurately can LISA measure the peak frequency and amplitude of the gravitational wave spectrum from a first-order phase transition?
• RQ2 What is the impact of astrophysical foregrounds (galactic and extragalactic binaries) on the precision of parameter estimation for phase transition signals?
• RQ3 Which thermodynamic parameters—transition strength α, wall speed vw, nucleation temperature Tn, or bubble spacing—are best constrained by LISA observations?
• RQ4 How do the principal components of the Fisher matrix reveal the most measurable combinations of parameters, and what does this imply for theoretical modeling?
• RQ5 How well does the double broken power law fit the sound shell model spectrum compared to the single broken power law used in previous LISA studies?

Key findings

• For signal-to-noise ratios above 20, the peak frequency and peak amplitude of the gravitational wave spectrum can be measured with approximately 10% relative uncertainty.
• The phase boundary speed (vw) is the best-constrained thermodynamic parameter, with the highest-order principal component of the Fisher matrix dominated by wall speed.
• With SNR > 20, the relative uncertainties in the two highest-order principal components are less than 3%, indicating strong prospects for measuring key parameter combinations.
• The double broken power law provides a significantly better fit to the sound shell model spectrum than the single broken power law, with mean squared relative deviation below 0.05 across most of the parameter space.
• The inclusion of a kinetic energy suppression factor derived from numerical simulations improves the agreement between the sound shell model and simulation data, especially at low wall speeds and high transition strengths.
• Despite Fisher matrix limitations in handling degeneracies, the results suggest that theoretical calculations of thermodynamic parameters must achieve sub-3% accuracy to match LISA’s expected sensitivity for the most measurable parameter combinations.