Seeing in the Dark: Gravitational Waves from Resonant Production of Dark Photon Dark Matter

Technical Summary: Gravitational Waves from Resonant Production of Dark Photon Dark Matter

Problem Statement
Ultralight dark photons (DPs) are compelling dark matter (DM) candidates, particularly in the regime where their mass $m_{A'} \ll \text{eV}$. Since a thermal population would be too warm, these DPs must be produced nonthermally. A specific mechanism involves the Abelian-Higgs model, where the DP mass is generated by a dark Higgs field. If the dark Higgs is initially displaced from its minimum at the end of inflation, its subsequent oscillations induce a periodically varying mass for the DP, potentially generating a relic abundance via parametric resonance.

Previous analytical studies of this mechanism (e.g., Ref. [15]) relied on linear Floquet theory. However, linear analysis fails to capture critical nonlinear phenomena, including backreaction, rescattering, and the redistribution of energy between the dark Higgs and the DPs. Consequently, the final state of the dark sector, the precise relic abundance, and the associated gravitational-wave (GW) signatures remained poorly characterized.

Methodology
The authors advance the understanding of this mechanism through dedicated three-dimensional lattice simulations of the nonlinear Abelian-Higgs dynamics.

• Model: They utilize a minimal gauged Abelian-Higgs model where a complex scalar $\Phi$ (dark Higgs) charged under $U(1)_X$ acquires a vacuum expectation value, giving mass to the DP $A_\mu$. The potential is quartic ($V \propto |\Phi|^4$) in the large-field regime relevant for the initial displacement.
• Simulation Setup: The authors developed a custom code to solve the equations of motion in the temporal gauge on a $3+1$ dimensional lattice. Unlike previous studies that assumed a prescribed background evolution, this work evolves the Hubble rate self-consistently with the dark-sector energy density and pressure via the Friedmann equation.
• Regime: The simulations focus on the "quartic regime" where the initial displacement $|\Phi_{\text{in}}| \gg v$ (the vacuum expectation value), rendering the quadratic mass term negligible. The system is evolved through the resonant production phase, the fragmentation of the condensate, and the post-resonance dynamics until the field is trapped in the true vacuum.

Key Contributions and Results

1. Nonlinear Fragmentation and Backreaction:

   • The simulations demonstrate that parametric resonance does not merely populate DPs; it fragments the coherent dark Higgs condensate into a highly inhomogeneous, phase-disordered state.
   • Backreaction is significant: the produced gauge and scalar fluctuations carry comparable amounts of energy ($\rho_{\text{Higgs}}/\rho_{\text{gauge}} = \mathcal{O}(1)$), contradicting linear predictions that suggest an exponential dominance of one mode over the other.
   • The residual homogeneous condensate is strongly depleted ($|\langle \Phi \rangle|/\sqrt{\langle |\Phi|^2 \rangle} \sim 10^{-3}$), resulting in a locally Higgsed, nonthermal state rather than a coherent condensate.

2. Dark Matter Relic Abundance and Composition:

   • The final dark sector composition is determined by the energy partition between DPs and dark Higgs excitations after resonance shuts down.
   • The authors present a formula for the relic abundance $\Omega_X h^2$ (Eq. 8) that depends on the initial dark-sector energy fraction $\alpha_{\Phi, \ast}$, the quartic coupling $\lambda_\Phi$, and a dimensionless factor $\Xi_N$ extracted directly from the lattice simulations. The detailed derivation of this yield is provided in Section II of the Supplemental Material.
   • Crucially, the dark sector is generically multicomponent, consisting of both DPs and dark Higgs bosons. The ratio depends on post-resonance number-conserving scatterings ($AA \leftrightarrow hh$). In the gauge-dominated regime ($q \gg 1$), the DP is heavier than the dark Higgs; efficient conversion can lead to a scalar-dominated relic, while inefficient conversion leaves a DP-dominated relic.

3. Stochastic Gravitational-Wave Signal:

   • The fragmentation process generates large anisotropic stress, sourcing a stochastic GW background with a peaked spectrum.
   • The peak frequency $f_{\text{peak}, 0}$ is determined by the energy scale of the resonance (Eq. 14), which correlates with the dark Higgs mass and the initial displacement.
   • The peak amplitude $\Omega_{\text{GW}, 0} h^2$ scales with the square of the initial energy fraction ($\alpha_{\Phi, \ast}^2$).
   • The paper establishes a strong correlation between the GW signal and the dark sector properties: the peak frequency probes the mass scale, while the amplitude constrains the initial energy fraction.

4. Observational Reach:

   • For benchmark parameters (e.g., $q \sim 10^3$), the predicted GW signals fall within the sensitivity ranges of current and future detectors, including Pulsar Timing Arrays (PTA), LISA, DECIGO, and ground-based interferometers (LIGO, CE, ET).
   • Specifically, the model predicts ultralight DPs in the sub-$\mu\text{eV}$ range for signals detectable by ground-based interferometers.
   • The authors note that constraints from Big Bang Nucleosynthesis (BBN) on the effective number of neutrino species ($\Delta N_{\text{eff}}$) limit the initial energy fraction $\alpha_{\Phi, \ast}$ to $\lesssim 5\%$, which in turn caps the maximum GW amplitude.

Significance and Claims
The paper claims to provide the first self-consistent determination of both the dark matter relic abundance and the associated GW signal for resonant DP production, moving beyond the limitations of linear Floquet analysis.

• Correction of Previous Models: The work demonstrates that linear analysis is insufficient for characterizing the final state, as it misses the fragmentation of the condensate and the resulting energy partition.
• Predictive Power: By linking the GW spectrum directly to the dark sector mass scale and composition, the paper offers a new, complementary probe for ultralight DP dark matter. The detection of such a GW signal would simultaneously reveal the mass of the dark sector particles and the dynamics of their production.
• Novel Cosmological History: The study highlights a generic possibility where the dark sector evolves through a fragmented, nonthermal phase before settling into the symmetry-broken vacuum, distinct from the thermal restoration scenarios often considered in Higgs models.

The authors conclude that this mechanism opens a new avenue for probing ultralight dark matter, with the GW signal serving as a direct window into the nonlinear, nonperturbative dynamics of the early universe dark sector.