Gravitational waves from first order phase transitions as a probe of an early matter domination era and its inverse problem

TL;DR

This work analyzes gravitational waves from a short-lasting first-order phase transition inside an early matter-dominated era, showing that the initial GW spectrum is RD-like but acquires a distinctive mode-dependent redshift attributable to MD. The authors derive how MD modifies the phase-transition parameters through $\beta/T$ scaling and demonstrate that the MD spectral feature can reveal the epoch of transition and the end of MD, while also addressing degeneracies in the inverse problem where multiple macroscopic parameter sets can yield similar signals. They argue that MD helps in breaking some degeneracies via the kink feature, particularly when multiple GW contributions (bubbles, sound waves, turbulence) are observed, with full parameter determination being most feasible for runaway bubbles in plasma. The results underscore the potential of future GW observations to probe the thermal history of the early universe and constrain TeV-scale phase-transition physics.

Abstract

We investigate the gravitational wave background from a first order phase transition in a matter-dominated universe, and show that it has a unique feature from which important information about the properties of the phase transition and thermal history of the universe can be easily extracted. Also, we discuss the inverse problem of such a gravitational wave background in view of the degeneracy among macroscopic parameters governing the signal.

Figures (6)

• Figure 1: GWs in RD. Top: Non-runaway bubbles with $\alpha=0.5, \ \beta/H_*=100, \ v_w=0.95, \ T_*=100 \mathinner{\mathrm{GeV}}$. Red and blue dashed lines are contributions from sound waves and turbulence, respectively. Black solid line is the sum of those two contributions. Bottom: Runaway bubbles in Plasma with $\alpha_\infty=0.1, \ \alpha=0.2, \ \beta/H_*=100, \ v_w=1, \ T_*=1 \mathinner{\mathrm{TeV}}$. Color scheme is the same as Top panel except the green line which represents the contribution from bubble collisions.
• Figure 2: $\Omega_{\rm GW}^{\rm MD}/\Omega_{\rm GW}^{\rm RD}$ in Eq. (\ref{['Omega-MD-to-RD']}) for a temperature at the peak frequency of each contribution.
• Figure 3: GWs in MD (solid lines) relative to the ones in RD (dashed lines) for the same $T_*$ as in MD. . Red lines are for $(T_*,T_{\rm d})=(10^5 \mathinner{\mathrm{GeV}}, 8 \times 10^4 \mathinner{\mathrm{GeV}})$. Blue lines are for $(T_*, T_{\rm d})=(100 \mathinner{\mathrm{GeV}}, 50 \mathinner{\mathrm{GeV}})$. Top: Non-runaway bubbles with $\alpha=0.5, \ (\beta/H_*)_{\rm RD} = 100, \ v_w = 0.95$. Bottom: Runaway bubbles in Plasma with $\alpha_\infty = 0.1, \ \alpha=0.2, \ (\beta/H_*)_{\rm RD}=100, \ v_w=1$.
• Figure 4: Degeneracy (solid lines) in GW spectra in RD for non-runaway bubbles with different choices of parameters. Solid lines are summations of the contributions from sound waves and turbulences. Each dashed line is the contribution of turbulence only for a specific set of parameters. We took $v=0.95$ for all different lines. For black lines, $\alpha = 0.5$ was taken. For the each of other color lines, $\alpha$ was reduced by a factor $2^{2.5},3^{2.5},4^{2.5}$ for red, blue, and green (solid and dashed) lines, respectively. Also, the same factors were applied to $\beta/H_*$ but the inverse of the factor applied to $T_*$ so as to keep $(\beta/H_*)T_*$ fixed.
• Figure 5: Degeneracy in GW spectra in RD for non-runaway bubbles. Top: Parameter space allowing only single contribution (from sound wave) in the reach of LISA. The labels of color caption indicate $\log(T_*/\mathinner{\mathrm{GeV}})$. Parameters were scanned, covering $(\beta/H_*)/v_w = \left[3,10^4\right], \ \kappa_{\rm sw} \alpha/(1+\alpha) = \left[ 10^{-4},0.5\right]$ and $T_*/\mathinner{\mathrm{GeV}}= \left[10,10^4\right]$. They were constrained such that $f_{\rm turb}^{\rm peak} = (2.7-3.3) \times 10^{-3} {\rm mHz}$ around the best sensitivity region of LISA and $\Omega_{\rm turb}^{\rm RD} < 2.016 \times 10^{-12} \leq \Omega_{\rm sw}^{\rm RD}$. The gray dashed diagonal lines are examples of constant $\Omega_{\rm sw}^{\rm RD}$. Bottom: $\kappa_{\rm sw} \alpha/(1+\alpha)$ as a function of $\alpha$ and $v_w$.