Gravitational waves from supercooled phase transitions and pulsar timing array signals

Abstract

The recent detection of a gravitational wave background in the nano-Hertz frequency range by Pulsar Timing Array (PTA) collaborations, including NANOGrav, EPTA, and PPTA, has opened a new avenue for exploring fundamental physics in the early universe. In this work, we analyze a supercooled first-order phase transition in a hidden sector with a spontaneously broken $U(1)_X$ gauge symmetry as a source for this signal. We demonstrate that the thermal history of the hidden and visible sectors plays a crucial role in the gravitational wave power spectrum analysis. Our analysis shows that supercooled phase transitions can generate gravitational waves strong enough to explain the PTA observations while satisfying cosmological constraints from Big Bang Nucleosynthesis.

Figures (5)

• Figure 1: Illustration of the first-order phase transition showing the temperature-dependent effective potential. At the critical temperature $T_c$, the false and true vacua are degenerate. As the temperature decreases below $T_c$, the barrier between false and true vacua enables bubble nucleation and the supercooled phase transition proceeds.
• Figure 2: Evolution of the temperature ratio $\xi \equiv T_h/T$ as a function of visible sector temperature $T$ for different values of kinetic mixing $\delta$, with $m_q = 2$ GeV, $m_A = 2$ MeV, $f_x = 0.015$, and $\xi_0 = 0.001$. The black dashed line shows the separate entropy conservation approximation. For $\delta = 0$ and $10^{-10}$, the Boltzmann solution closely follows entropy conservation. For $\delta = 10^{-9}$ and $10^{-8}$, energy transfer between sectors becomes significant, driving $\xi$ toward unity (thermalization). This demonstrates that separate entropy conservation is a good approximation only for $\delta \lesssim 10^{-10}$. Figure adapted from ref. Li:2023nez.
• Figure 3: Comparison of a constant vs an evolving temperature ratio $\zeta(T_h) = T_v/T_h$ for BP1. Left: Hubble parameter $H(T_h)$ as a function of hidden sector temperature. The dashed vertical line marks the percolation temperature $T_{h,p}$ and the solid line marks the critical temperature $T_{h,c}$. Middle: False vacuum survival probability $P_f(T_h)$; the horizontal dashed line marks $P_f = 0.71$. Right: Gravitational wave power spectrum with NANOGrav 15yr data. The red curves use the evolving $\zeta_2 = \zeta(T_h) \in [1.71,\, 4.09]$ from entropy conservation, while the blue curves use a constant $\zeta_1 = 1/\xi_p = 1.71$. Figure adapted from ref. Li:2025nja.
• Figure 4: Left panel: The gravitational wave power spectrum $\Omega_{\rm GW} h^2$ for supercooled phase transitions as a function of frequency for five benchmark points BP1--BP5 in the range $[10^{-11}\text{--}10^{-5}]$ Hz. The shaded violins represent the NANOGrav 15yr free-spectrum posterior NANOGrav:2023gor. Right panel: Projected sensitivity regions of future space-based GW detectors including $\mu$Ares Sesana:2019vho, Taiji Ruan:2018tsw, LISA LISA:2017pwj, TianQin TianQin:2015yph, BBO Grojean:2006bp, and U-DECIGO Kawamura:2006up. The benchmark spectra fall within multiple detector sensitivity regions, providing further tests of the supercooled phase transition scenario. Figure adapted from ref. Li:2025nja.
• Figure 5: The three primary contributions to the gravitational wave signal for BP1: bubble collisions (red), sound waves (blue), and turbulence (green). The bubble collision contribution dominates by several orders of magnitude in the nano-Hz PTA region, while sound waves and turbulence are negligible. The shaded regions indicate projected sensitivities of future space-based detectors. Figure adapted from ref. Li:2025nja.