Assume that the energy density in the Universe is a power-law in the scale factor:
ε = εoαn
where n can be positive, zero, or negative. Assume also that k = +1.
(a) Solve for the equation of state w as a function of n.
(b) Is the turning point for the scale factor (e.g. α = 0) a minimum or a maximum?
(c) At what scale factor does the turning point occur and what is the maximum or minimum size of the Universe at this time? Assume that Ro = 2c/H0 (where H0 = 67 km*s^-1Mpc^-1 is the present day Hubble), that the universe is currently expanding, and that the scale factor today is one (a0 =1).
(d) Describe what pathologies a cosmology in which the Universe has a minimum size might have.