The equations for the evolution of a homogeneous brane world that emits gravitons at early times, and into a non-Z_2 symmetric bulk, are derived using an AdS-Vaidya spacetime approximation. The behaviour of the black hole mass parameters either side of the brane is analysed, and it is found that in general graviton emission leads to a decrease in the non-Z_2 symmetry. However, the behaviour of the dark radiation term in the Friedmann equation is more complex: it is shown that this term can increase or decrease due to the non-Z_2 symmetry, and can become negative in some cases, leading to H=0 and the brane universe collapsing. Constraints on the initial (nonzero) sizes of the mass parameters are therefore derived.
Vernon, I. R., & Jennings, D. (2005). Graviton Emission into Non-Z_2 Symmetric Brane World Spacetimes. Journal of Cosmology and Astroparticle Physics, 07, https://doi.org/10.1088/1475-7516/2005/07/011