Suppose that the universe contained a significant component in the form of isotropic but noninteracting particles with momentum p and rest mass m. Suppose that they were created with a distribution function f(p, a) ∝ p^−q, with 4 < q < 5 extending from p ≪ m to p ≫ m.

(a) Show that the pressure and the internal energy density (not including rest mass) of these particles are both well defined.

(b) Show that the adiabatic index of this component is q/3andinterpret your answer.

(c) Explain how P and ρ for this component should vary with the scale factor a.