  We obtain novel closed form solutions to the Friedmann equation for
cosmological models containing a component whose equation of state is that of
radiation $(w=1/3)$ at early times and that of cold pressureless matter $(w=0)$
at late times. The equation of state smoothly transitions from the early to
late-time behavior and exactly describes the evolution of a species with a
Dirac Delta function distribution in momentum magnitudes $|\vec{p}_0|$ (i.e.
all particles have the same $|\vec{p}_0|$). Such a component, here termed "hot
matter", is an approximate model for both neutrinos and warm dark matter. We
consider it alone and in combination with cold matter and with radiation, also
obtaining closed-form solutions for the growth of super-horizon perturbations
in each case. The idealized model recovers $t(a)$ to better than $1.5\%$
accuracy for all $a$ relative to a Fermi-Dirac distribution (as describes
neutrinos). We conclude by adding the second moment of the distribution to our
exact solution and then generalizing to include all moments of an arbitrary
momentum distribution in a closed form solution.
