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 .