Stability Analysis of Cosmological Perturbations in the Bumblebee Model: Parameter Constraints and Gravitational Waves

TL;DR

This work analyzes cosmological perturbations in the Bumblebee model with a perfect fluid, deriving the full second-order actions for tensor, vector, and scalar modes and enforcing ghost, Laplacian, and tachyon stability. By incorporating the observed cosmic acceleration and GW-speed bounds, the authors bound the Lorentz-violating parameter via $c_t^2=rac{1}{1-\xi b_t^2}$ and $-6 imes10^{-15}\lesssim \xi b_t^2\, ext{(≤0)}$, finding a constrained parameter space with $\\dot{H}<0$ and $\xi\, le 0$. Within this space, the theory supports six GW polarizations: two tensor ($P_+$, $P_×$), two vector ($P_x$, $P_y$), and a mixed massive scalar ($P_b$/$P_l$); tensor modes propagate subluminally, while vector and scalar modes are superluminal (unless $\xi b_t^2=0$, in which case vector/scalar modes vanish). The gauge-independence of the results across three gauge choices reinforces the robustness of the predictions, offering distinct observational signatures for testing Lorentz invariance with future gravitational-wave detectors.

Abstract

We constrain the parameter space of the Bumblebee model within a cosmological background and investigate the properties of gravitational waves under the constrained parameter space. Specifically, we derive the conditions for the absence of ghost instability, Laplacian instability, and tachyonic instability for perturbations in a cosmological background. By incorporating the observed accelerated expansion of the universe and the observational constraints on tensor gravitational waves, we derive bounds on the parameter space of the Bumblebee model. We then examine the polarization modes, the propagation speeds, and the amplitude relations of gravitational waves within this constrained framework. Our results indicate that the non-minimal coupling parameter $ξ$ must be non-positive, and the Lorentz-violating parameter $ξb^2$ has a lower bound on the order of $10^{-15}$. The propagation modes of gravitational waves in the Bumblebee model consist of two tensor modes, two vector modes, and a combination of two scalar modes. Notably, the tensor modes travel at subluminal speeds, whereas the vector and scalar modes propagate at superluminal speeds. These results provide a concrete theoretical framework and specific observational signatures for testing Lorentz invariance in the gravitational sector with future gravitational-wave detectors.

Figures (1)

• Figure 1: This figure illustrates the six polarization modes of GWs Eardley:1973zuo. Assuming the propagation direction of GWs is along the "$+z$" direction, these panels show the variation in the relative positions of test particles distributed on a spherical surface. No relative displacement occurs for test particles located along the direction perpendicular to the plane of each figure. The solid and dashed lines represent a phase difference of $\pi$, where the symbol "$\rightarrow$" denotes wave propagation to the right, and "$\odot$" indicates the wave propagation outward from the page.