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6.5. Spreading

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 rho, and the Lorentz factor, gamma, tends to a constant value gammaf for each shell. We may therefore neglect the term - (1/3)(partiale / partialr) in Eq. 18 and treat gamma and u in Eqs. 16-18 as constants. We then find that the flow moves strictly along the characteristic, betaf t - r = constant, so that each fluid shell coasts at a constant radial speed, betaf = uf / gammaf. We label the baryonic shells in the fireball by a Lagrangian coordinate R, moving with a fixed Lorentz factor gammaf(R), 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

Equation 24 (24)

The separation between two neighboring shells separated by a Lagrangian distance Delta R varies during the coasting phase as

Equation 25 (25)

Thus the width of the pulse at time t is Deltar(t) approx Deltarc + Delta gammaf(t - tc) / bar{gamma}f3 approx Ri + (t - tc) / gammaf2, where Deltarc ~ Ri is the width of the fireball when it begins coasting, bar{gamma}f is the mean gammaf in the pulse, and Delta gammaf ~ bar{gamma}f is the spread of gammaf 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 < bar{gamma}f2Ri, we have a frozen-coasting phase in which Deltar is approximately constant and the frozen pulse approximation is valid. In this regime the scalings in Eq. 23 are satisfied. However, when t - tc > bar{gamma}f2 Ri, the fireball switches to an expanding-coasting phase where Deltar propto t - tc and the pulse width grows linearly with time. In this regime the scaling of rho reverts to rho propto r-3, and, if the radiation is still coupled to the matter, e propto r-4.

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