At very late times in the matter-dominated phase the frozen pulse approximation begins to break down. In this stage the radiation density e is much smaller than the matter density , and the Lorentz factor, , tends to a constant value f for each shell. We may therefore neglect the term - (1/3)(e / r) in Eq. 18 and treat and u in Eqs. 16-18 as constants. We then find that the flow moves strictly along the characteristic, f t - r = constant, so that each fluid shell coasts at a constant radial speed, f = uf / f. We label the baryonic shells in the fireball by a Lagrangian coordinate , moving with a fixed Lorentz factor f(), and let tc and rc represent the time and radius at which the coasting phase begins, which corresponds essentially to the point at which the fluid makes the transition from being radiation dominated to matter dominated. We then find
The separation between two neighboring shells separated by a Lagrangian distance varies during the coasting phase as
Thus the width of the pulse at time t is r(t) rc + f(t - tc) / f3 Ri + (t - tc) / f2, where rc ~ Ri is the width of the fireball when it begins coasting, f is the mean f in the pulse, and f ~ f is the spread of f across the pulse. From this result we see that within the matter dominated coasting phase there are two separate regimes. So long as t - tc < f2Ri, we have a frozen-coasting phase in which r is approximately constant and the frozen pulse approximation is valid. In this regime the scalings in Eq. 23 are satisfied. However, when t - tc > f2 Ri, the fireball switches to an expanding-coasting phase where r t - tc and the pulse width grows linearly with time. In this regime the scaling of reverts to r-3, and, if the radiation is still coupled to the matter, e r-4.