6.4. The Matter-Dominated Phase
The radiation dominated regime extends out to a radius r ~ (e0 / 0) r0. At larger radii, the first and last terms in Eq. 20 become comparable and tends to its asymptotic value of f = (4e0 / 30 + 1)0. This is the matter dominated regime. (The transition occurs when 4e / 3 = , which happens when = f / 2.) In this regime, D r2/3, leading to the scalings:
The modified scalings of and e arise because the fireball now moves with a constant radial width in the comoving frame. (The steeper fall-off of e with r is because of the work done by the radiation through tangential expansion.) Moreover, since e << , the radiation has no important dynamical effect on the motion and produces no significant radial acceleration. Therefore, remains constant on streamlines and the fluid coasts with a constant asymptotic radial velocity. Of course, since each shell moves with a velocity that is slightly less than c and that is different from one shell to the next, the frozen pulse approximation on which Eq. 19 is based must ultimately break down at some large radius.