Tuning the violins: dark sector phase transition models for the PTA signal

TL;DR

This work investigates whether a dark-sector first-order phase transition can explain the nanohertz stochastic gravitational wave background seen by pulsar timing arrays. It first updates a model-independent PTA fit for the transition parameters $(α,β/H,T_ ext{reh})$ and then analyzes three concrete DSPT realizations: a thermally induced barrier in a dark Abelian Higgs sector, a two-step flip-flop transition with scalar singlets, and a loop-induced barrier in a near-conformal dark sector. The authors find that while a DSPT can fit the data in principle, only the conformal scenario achieves this with modest tuning; thermally induced and flip-flop models require substantial parameter tuning and face cosmological constraints, particularly on $oldsymbol{ΔN_ ext{eff}}$ and decay channels. They discuss viable decay portals, dark-sector spectra, and the implications for upcoming PTA and collider probes, arguing that near-conformal dynamics provide the most robust route to explain the PTA signal. The work offers a framework to scrutinize DSPT hypotheses as data improve and helps guide experimental tests of dark-sector phase transitions.

Abstract

First-order phase transitions in a dark sector have been invoked as an intriguing possibility to explain the observed stochastic gravitational wave background at nanohertz frequencies. Here we perform a comprehensive study of the generic requirements for such a phase transition to explain the observed signal while being consistent with all relevant constraints. We consider three broad model classes for strong first-order transitions, realised by an Abelian dark Higgs boson, a two-step phase transition involving two scalar singlets, and a conformal scalar field with loop-induced symmetry breaking, respectively. We discuss the tuning that is required to successfully explain the Pulsar Timing Array (PTA) signal in each of these cases, and highlight the underlying physical mechanisms. We conclude that all three scenarios can in principle describe the data, but that conformal models stand out as the most generic, and least tuned, explanation. Future observations by the PTA collaborations and collider experiments will be crucial to test the viability of this hypothesis, and to further narrow in on the model parameters, if the PTA signal is indeed due to a strong first-order phase transition.

Figures (12)

• Figure 1: Left panel:$1\sigma$ and $2\sigma$ best-fit regions of current PTA data, interpreted in terms of strength $\alpha$ and speed $\beta/H$ of a generic dark sector FOPT (blue lines), with gravitational waves dominantly produced from sound waves due to the expanding bubbles. The colour code shows the corresponding reheating temperature $T_\text{reh}$, and the hatched grey region indicates the combination of parameters that would generate a total energy density in GWs in conflict with $\Delta N_{\rm eff}$ constraints from BBN and CMB Yeh:2022heq. The dashed grey line indicates below which $\beta/H$ values the computation of the GW signal starts to become unreliable Jinno:2022mie as it is not backed up by simulations. The purple lines indicate the $1\sigma$ and $2\sigma$ regions when adding an astrophysical contribution from merging SMBH binaries to the GW signal. Right panel: Gravitational wave spectra and NANOGrav 15 yr spectrograms NANOGrav:2023gor for the best-fit point when allowing both a phase transition and merging SMBH contribution (marked with a blue star in the left panel). The grey area indicates the effect of varying amplitude and slope of the SMBH contribution within their $1\sigma$ expectation NANOGrav:2023hvm.
• Figure 2: Correlation between transition strength $\alpha$ and speed $\beta/H$ in the thin-wall approximation for $S_3(T_\text{p})/T_\text{p} = 200$ and different choices of $\alpha_\text{c}$ as introduced in eq. \ref{['eq:alpha_T_dependence']}. See text for details.
• Figure 3: Left panel. Prior choices for the Abelian dark sector model, targeted at a FOPT at the MeV scale. Right panel. The blue lines show the values of $\alpha$ and $\beta/H$ for the produced GW signal that correspond -- at the $1\sigma$ and $2\sigma$ level -- to the model parameter ranges in the left panel. For each point in these contours, the colour code indicates the ratio of mean reheating temperature $T_\text{reh}$ and vev $v$. For comparison, the grey-shaded area shows the result of the model-independent fit to the PTA data, as displayed in figure \ref{['fig:model-independent']}.
• Figure 4: Upper row: Effective potential of the Abelian dark sector model at different temperatures, choosing for illustration model parameters $g=1$, $v = 100 \, \text{MeV}$ and three different values of the quartic coupling $\lambda$ as indicated. Lower row, left: Corresponding tunnelling action $S_3$ as a function of temperature. Lower row, right: Resulting inverse transition timescale $\beta/H$ as a function of the quartic coupling $\lambda$.
• Figure 5: Left panels: Field values of $\phi_{1}$ and $\phi_{2}$ as a function of temperature (upper panel) and the corresponding potential values (lower panel). At high temperatures $T$ the global minimum is at the origin (orange line). At intermediate $T$ the global potential minimum smoothly shifts from $(\phi_1,\phi_2)=(0,0)$ to $\phi_1 = 0, \phi_2 \neq 0$ with decreasing $T$ (blue line). At low temperatures, a local minimum at $\phi_1 \neq 0, \phi_2 = 0$ appears (red line), which eventually becomes the global potential minimum. Right panel: Effective potential at the nucleation temperature $T_{\mathrm{nuc}} = 0.28\,$GeV. The blue dotted arrow shows the evolution of the $\phi_{2}$ minimum until $T_{\mathrm{nuc}}$, the red arrow shows the tunnelling path from the local minimum to the release point of the field. The parameter values for the benchmark point shown are stated in table \ref{['tab:benchmarks']} in the appendix.