Gravitational wave oscillations in Multi-Proca dark energy models

TL;DR

The paper addresses gravitational wave oscillations arising from energy exchange between metric perturbations and tensor modes in Multi-Proca dark energy models. By solving the cosmological background self-consistently and deriving the coupled tensor perturbation equations, it quantifies the GW amplitude modulation and identifies the mixing scale $m_g$ set by the vector-field mass $\mu_A$. The main finding is that $m_g \sim \mu_A \sim H_0$, making detectable oscillations require $m_g \gg H_0$, which is incompatible with the ultra-light masses needed for late-time acceleration, hence no observable modulation is expected for LIGO–Virgo or LISA within these models. The work also discusses potential early-Universe signatures in the stochastic GW background and outlines how non-minimal couplings could modify these conclusions in future studies.

Abstract

Gravitational wave oscillations arise from the exchange of energy between the metric perturbations and additional tensor modes. This phenomenon can occur even when the extra degrees of freedom consist of a triplet of massive Abelian vector fields, as in Multi-Proca dark energy models. In this work, we study gravitational wave oscillations in this class of models minimally coupled to gravity with a general potential, allowing also for a kinetic coupling between the vector field and dark matter that can, in principle, enhance the modulation of gravitational wave amplitudes. After consistently solving the background dynamics, requiring the model parameters to reproduce a phase of late-time accelerated expansion, we assess the accuracy of commonly used analytical approximations and quantify the impact of gravitational wave amplitude modulation for current detectors (LIGO--Virgo) and future missions such as LISA. Although oscillations are present in these scenarios, we find that the effective mass scale (the mixing mass) governing the phenomenon is $m_g \sim μ_A$, where $μ_A$ is the (time-dependent) effective mass of the vector dark-energy field. Detectability of gravitational wave oscillations, however, requires $m_g \gg H_0$, which is in tension with the ultra-light masses typically needed to drive accelerated expansion $μ_A \sim H_0 \sim 10^{-33}\,\mathrm{eV}$. Therefore, if gravitational wave oscillations were to be detected at the corresponding frequencies, they could not be attributed to these classes of dark-energy models.

Figures (3)

• Figure 1: Evolution of the effective EoS $w_{\rm eff}$ and the vector dark energy EoS $w_{A}$ as a function of conformal time $\eta$ for model 1 (strongly coupled exponential potential) in the left panel and model 2 (weakly coupled power-law potential) in the right panel. At late times, both models approach the de Sitter solution $w_{A} = -1$, while at intermediate stage, corresponding to the onset of the evolution, slight differences appear before reaching the radiation-like behavior $w_{A} = 1/3$. For model 1, this deviation is induced by the strong vector–matter coupling, which produces a sharp rise followed by a small bump. For model 1, we set the parameters $\alpha = 0.65$, $n = -0.3$, $\lambda = 0.34$, and $\hat{V}_{01} = 0.68001$. For model 2, we use $\alpha = -0.09$, $n = -0.009$, $\beta = -0.09$, and $\hat{V}_{02} = 0.690$. For both models we set $\chi_{0} = 5 \times 10^{-4}$, which uniquely determines the amplitude $\psi_{0}$ via Eqs. (\ref{['eq:exp_phi_0']}) and (\ref{['eq:power_phi_0']}), respectively.
• Figure 2: Evolution of the auxiliary tensor mode $y$ (left) and the GW amplitude $|h|^{2}$ (right) for model 1 (top) and model 2 (bottom). Here, $\theta=2\pi(\eta-\eta_{e}/\Delta\eta)$, with $\Delta\eta=\eta_{0}-\eta_{e}$. We show the full numerical solution, the WKB approximation, and the result obtained by neglecting the friction term. The WKB method overestimates the growth of $y$, leading to an artificial suppression of the GW amplitude, while the full solution exhibits only mild damping. The no inclusion of the friction term prevents significant growth of $y$, but keeping the GW amplitude essentially unchanged. In model 2, the sign flip of $\mu_{H}^{2}$ produces an apparent reversal of mixing in the WKB approximation, but this effect does not occur in the numerical evolution, confirming that this feature is a methodological artefact. We have taken the following initial conditions at the emission time: $h_{0}=1$, $y_{0}=0$, $h_{0}' = 10^{-3}$ and $y_{0}' = 10^{-3}$ and $k=100$.
• Figure 3: Cumulative phase shift $\Phi(z;f)$ as a function of redshift for representative GW frequency bands (PTA: $f=10^{-9}\,\mathrm{Hz}$, LISA: $f=10^{-3}\,\mathrm{Hz}$, and LIGO/Virgo: $f=10^{2}\,\mathrm{Hz}$). The phase is computed using the interpolated functions $\mu_{A}^{2}(z)$ and $H(z)$ for the model 1. At low redshift the phase grows linearly but remains extremely small, while at high redshift the integral saturates, producing a plateau consistent with the weak redshift dependence of $\mu_{A}$. In all cases, the accumulated phase remains far below unity over the full range $z\le 200$.