Phase transition during inflation and the gravitational wave signal at pulsar timing arrays

Abstract

The gravitational wave (GW) signal offers a promising window into the dynamics of the early universe. The recent results from the pulsar timing arrays (PTAs) could be the first glimpse of such new physics. In particular, they could point to new details during inflation, which can not be probed by other means. We explore the possibility that the new results could come from the secondary GWs sourced by curvature perturbations, generated by a first-order phase transition during inflation. Based on the results of a field-theoretic lattice simulation of the phase transition process, we show that the GW signal generated through this mechanism can account for the new results from the PTAs. We analyze the spectral shape of the signal in detail. Future observations can use such information to distinguish the GW signal considered here from other possible sources.

Figures (9)

• Figure 1: Power spectrum of the induced curvature perturbation, $\Delta^2_\zeta$ for different choices of parameters. The solid curves are the results from numerical simulation, and the dashed curves are based on the empirical formula Eq. (\ref{['eq:DeltaFormula']}). The wiggles in the curves for $\beta/H_{\mathrm{inf}} = 10$ and 20 are the remnants of the oscillatory pattern in the integrand of (\ref{['eq:zetaq']}), which also leads to the oscillatory pattern in the primary GW spectrum as discussed in An:2020fffAn:2022cce. Large curvature perturbations can result in primordial black hole (PBH) production, and the relevant constraint is indicated by the dash-dotted line Byrnes:2018txb.
• Figure 2: The differential spectra of secondary GWs induced by first-order phase transitions during inflation for different parameters as shown in the plot. The gray violins show the periodogram for an HD-correlated free spectral process from NANOGrav:2023hvm. Four sets of the model parameters are shown as examples, which match the data collected by the NANOGrav collaboration in terms of the amplitude and spectral shape, particularly in the region $f< 3 \times 10^{-8}$ Hz. The corresponding primary GW spectra are represented by the dashed curves. In the parameter region of interest, the magnitude of the primary GWs is smaller than the secondary GWs by a few orders of magnitude.
• Figure 3: The Bayes factors for the model comparisons between the new-physics interpretations of the signal and the interpretation based solely on SMBHBs. The stars denote the Bayes factors for the model considered in this work. Other results labeled as "SIGW DELTA", "SIGW GAUSS", "SIGW BOX", "PT BUBBLE", and "PT SOUND" are the benchmark new physics models studied in NANOGrav:2023hvm. SIGW and PT stand for scalar-induced GWs and phase transition, respectively.
• Figure 4: The reconstructed posterior distributions for the parameters, $A_{\rm ref}$ and $f_{\rm ref}$ for $\beta=5H_{\rm inf}$. The upper left and lower right panels display the 1D marginalized distributions with the 68% Bayesian credible intervals, while the lower left panel depicts the 68% (darker) and the 95% (lighter) Bayesian credible regions in the 2D posterior distribution. The two benchmark stars correspond to the two curves with same color shown in Fig. \ref{['fig:GW2']}. The PBH bound on the curvature perturbation is also shown by the dash-dotted curve.
• Figure 5: Same as in Fig. \ref{['fig:posteriors1']} but for $\beta=10,\ 20H_{\rm inf}$.