Causality of brane universe via the general bulk-based formalisms with the non-zero Schwarzschild mass

In brane-world scenarios, electromagnetic waves (EMWs) are confined to the brane, while gravitational waves (GWs) can propagate through the bulk spacetime. This fundamental difference has been exploited in multiple cosmological studies to address some issues, such as the well-known horizon problem. This paper reinvestigates the problem using general bulk-based formalisms, with specific focus on how the non-zero Schwarzschild mass modifies geodesic motion. Our results demonstrate that the Schwarzschild mass significantly modifies the gravitational-to-photon horizon ratio. In the low-energy regime, our analysis constrains the anti-de Sitter curvature radius, i. e. l H₀ 10^-29. Our finding agrees quantitatively with prior work. In the high-energy regime, the gravitational-to-photon horizon ratio rg/r_γ increases by thirty orders of magnitude, reaching 10^33. Subject to the nucleosynthesis constraint σ^1/4 < 1 MeV, the ratio becomes 10^40. In this region, we observe the phenomenon of graviton bouncing by the brane, a behavior that has also been documented in prior literatures. Based on the observation of approximately 5\% dark radiation during the nucleosynthesis epoch, we further derive constraints on the relevant model parameters. Our results demonstrate that the non-zero Schwarzschild mass profoundly affects brane-world gravity. Crucially, some resulting effects may provide mechanisms to address persistent challenges in standard cosmology.