How do you prove that a is exponential function for t?

1. Matter dominated universe Pm Pr, A = 0 (p = Pm at a = ao) 2 3 ASIS = 8nG Pm 3 a a t2/3 a a 2. Radiation dominated universe Pm = 1 = 0 p = Pr. 4 a 1/2 3. Cosmological constant dominated universe (early universe, small a ) 2 EVE t a= : inflation a 3 Friedmann equation H(to) = H. Pm a 3 + Pr 8G 3 -P 8TG k H2 HZ 1 k 1 Ha?" 3H + = H2 25 + + hk az + 3H Pc 3 Pc Pm Pr k 1 m 127 k = Pc Pc H2 S2r k H2 = HZ [+ + +22] a2 t = to, a(to) = 1, H(to) = H Nam + 12r + 2x + 24 = 1 80G ke2 Friedmann equation H2 = 0 .P Total density : p = Pm + Pr 2 1 + 3 a2 3 1 matter radiation (photons) For ordinary matter, p decreases with the expansion asp a For photons, p () (a) 1 redshift expansion p = pm (*)* + Pr (*)*, a(to) = ao Critical density, matter dominated Universe (at present) Pr = 1 = 0 k = 0 (a= 80G Pc and Pc = Pm 3 1. Matter dominated universe Pm Pr, A = 0 (p = Pm at a = ao) 2 3 ASIS = 8nG Pm 3 a a t2/3 a a 2. Radiation dominated universe Pm = 1 = 0 p = Pr. 4 a 1/2 3. Cosmological constant dominated universe (early universe, small a ) 2 EVE t a= : inflation a 3 Friedmann equation H(to) = H. Pm a 3 + Pr 8G 3 -P 8TG k H2 HZ 1 k 1 Ha?" 3H + = H2 25 + + hk az + 3H Pc 3 Pc Pm Pr k 1 m 127 k = Pc Pc H2 S2r k H2 = HZ [+ + +22] a2 t = to, a(to) = 1, H(to) = H Nam + 12r + 2x + 24 = 1 80G ke2 Friedmann equation H2 = 0 .P Total density : p = Pm + Pr 2 1 + 3 a2 3 1 matter radiation (photons) For ordinary matter, p decreases with the expansion asp a For photons, p () (a) 1 redshift expansion p = pm (*)* + Pr (*)*, a(to) = ao Critical density, matter dominated Universe (at present) Pr = 1 = 0 k = 0 (a= 80G Pc and Pc = Pm 3