Gauge-independent gravitational waves from a minimal dark $U(1)$ sector with viable dark matter candidates

Abstract

Searches for stochastic gravitational wave backgrounds from first-order phase transitions offer a powerful probe of hidden sectors, but quantitative predictions in gauge theories are obstructed by the gauge dependence of the finite-temperature effective potential and the associated tunneling action. We study a minimal gauged $U(1)$ dark sector containing a dark Higgs and a dark photon, optionally supplemented by a vectorlike dark fermion, coupled to the Standard Model through Higgs portal or kinetic mixing. Using the Nielsen identity together with a controlled derivative expansion and power counting, we construct a gauge-independent effective action in the high- and low-temperature limits, enabling model-intrinsic nucleation dynamics and robust gravitational wave predictions. We perform dedicated Monte Carlo scans in both limits and map viable microscopic parameters to detector-facing peak frequencies and amplitudes, spanning bands relevant to pulsar timing arrays and planned space-based interferometers. In our scans, supercooled transitions typically produce the strongest signals, whereas parametrically high-temperature transitions are comparatively rare and tend to be weak. We further connect the phase transition phenomenology to viable dark matter candidates within the same minimal field content, providing benchmark targets for dark photon dark matter and dark fermion dark matter, and highlighting their complementarity with gravitational wave observables. Overall, our results provide an end-to-end, gauge-independent pipeline from a minimal hidden sector Lagrangian to gravitational wave spectra and cosmologically viable dark matter benchmarks, yielding the most reliable and concrete predictions to date for a minimal gauged $U(1)$ dark sector.

Figures (7)

• Figure 1: Gauge-dependent gravitational wave signals from a Monte Carlo scan over the parameter ranges in Eq. (\ref{['eq:MCrange_GD']}) using the finite-temperature effective potential in Landau gauge ($\xi=0$). Scatter points show the peak frequency and peak amplitude of the predicted stochastic background from the first-order phase transition. Panel (a) is color-coded by the ratio $T_p/m_{A^\prime}$, and panel (b) by the $U(1)_x$ symmetry breaking scale $v_x\in[10^{-3},10^{3}]~\mathrm{GeV}$. The colored regions denote the power-law integrated sensitivities of current and future detectors, including Taiji Ruan:2018tsw, TianQin TianQin:2015yph, LISA LISA:2017pwj, $\mu$Ares Sesana:2019vho, BBO Grojean:2006bp, U$\_$DECIGO Kuroyanagi:2014qaa, ET Punturo:2010zz, and CE LIGOScientific:2016wof. Box plots show the PTA signals from NANOGrav NANOGrav:2023gor, PPTA Reardon:2023gzh, and EPTA EPTA:2023fyk, with dataset and plotting style following Athron:2020sbe.
• Figure 2: Pie chart showing the composition of phase transition regimes in the gauge-dependent scan (Landau gauge, $\xi=0$) over the parameter ranges in Eq. (\ref{['eq:MCrange_GD']}). The classification is based on the ratio $T_p/m_{A^{\prime}}$ as defined in Table \ref{['TableFrac']}, which also lists the corresponding event counts and fractions.
• Figure 3: Gauge-independent gravitational wave signals in the high-temperature regime. Each scatter point shows the peak frequency and peak amplitude of the stochastic background sourced by a first-order phase transition. In panel (a), points are color-coded by $T_p/m_{A^\prime}$. In panel (b), points are color-coded by the $U(1)_x$ symmetry breaking scale $v_x$, scanned over $v_x\in[10^{-3},10^{2}]~\mathrm{GeV}$. All points with $T_p/m_{A^\prime}<2$ violate the high-temperature assumption and should be discarded.
• Figure 4: Gauge-independent gravitational wave signals in the low-temperature regime. Each scatter point shows the peak frequency and peak amplitude of the stochastic background sourced by a first-order phase transition. In panel (a), points are color-coded by $T_p/m_{A^\prime}$. In panel (b), points are color-coded by the $U(1)_x$ symmetry breaking scale $v_x$, scanned over $v_x\in[10^{-3},10^{2}]~\mathrm{GeV}$. All points with $T_p/m_{A^\prime}>0.5$ violate the low-temperature assumption and should be discarded.
• Figure 5: Concrete gauge-independent gravitational wave predictions for the minimal gauged $U(1)_x$ dark sector, obtained by combining the physically consistent points selected from Fig. \ref{['Fig:HT']} ($T_p/m_{A^\prime} \gtrsim 2$) and Fig. \ref{['Fig:LT']} ($T_p/m_{A^\prime} \lesssim 0.5$). The gap corresponds to the intermediate regime $0.5 < T_p/m_{A^\prime} < 2$, where neither the high- nor low-temperature gauge-independent treatment is applicable. Low-temperature points are shown in blue, with darker shades corresponding to smaller $T_p/m_{A^\prime}$ (stronger supercooling). High-temperature points are shown in red, with darker shades corresponding to larger $T_p/m_{A^\prime}$. Black crosses mark three benchmark models for Case 2, for which the dark matter evolution is computed explicitly (see Table \ref{['TableCase2']}), providing benchmark targets for multi-messenger searches of the corresponding $U(1)$ model. In particular, benchmark models (e) and (f) lie within the projected sensitivity of Taiji, TianQin, and LISA.