We constrain the parameter space of the bumblebee model in a cosmological background and then investigate the properties of gravitational waves within the constrained parameter space. Our analysis reveals seven perturbative degrees of freedom in the cosmological background: two tensor, two vector, and two scalar modes, along with an additional mode from the matter sector. The stability conditions for all these modes are derived. 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. Our results indicate that the nonminimal coupling parameter $\xi$ must be nonpositive, a constant background value $b_t$ of the bumblebee field implies $\sigma \ne -\frac{1}{2}\xi$, and the Lorentz-violating parameter $\xi b^2$ has a lower bound on the order of ${10}^{-15}$. We then investigate the propagation characteristics and polarization modes of gravitational waves in both the small-scale and Minkowski limits. The propagation modes of gravitational waves in the bumblebee model consist of two tensor modes, two vector modes, and one scalar mode. Notably, the tensor modes travel at subluminal speeds, whereas the vector and scalar modes propagate at superluminal speeds, when $\xi b_t^2\ne 0$. These results provide a concrete theoretical framework and specific observational signatures for testing Lorentz invariance in the gravitational sector with future gravitational-wave detectors.

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• Alternative gravity theories
• Cosmology
• Dark energy
• General relativity
• Gravitational waves