**6.4. The Matter-Dominated Phase**

The radiation dominated regime extends out to a radius
*r* ~ (*e*_{0} /
_{0})
*r*_{0}. At larger radii, the first and last terms in
Eq. 20 become comparable and
tends to
its asymptotic value of
_{f}
= (4*e*_{0} /
3_{0} +
1)_{0}.
This is the matter
dominated regime. (The transition occurs when 4*e* / 3 =
, which
happens when
=
_{f}
/ 2.) In this regime, *D*
*r*^{2/3}, leading to the scalings:

(23) |

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.