**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
, 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}
= *u*_{f} /
_{f}.
We label the baryonic shells in the fireball by a Lagrangian coordinate
, moving with a fixed
Lorentz factor _{f}(), and let *t*_{c} and *r*_{c}
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

(24) |

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

(25) |

Thus the width of the pulse at time *t* is
*r*(*t*)
*r*_{c}
+
_{f}(*t* - *t*_{c}) /
_{f}^{3}
*R*_{i} + (*t* - *t*_{c}) /
_{f}^{2}, where
*r*_{c}
~ *R*_{i}
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* - *t*_{c} <
_{f}^{2}*R*_{i}, 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* - *t*_{c} >
_{f}^{2} *R*_{i}, the fireball
switches to an expanding-coasting phase where
*r*
*t* -
*t*_{c} 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}.