Question

Integrate to prove equations (5.90) and (5.91) for a matter positive curvature universe.

Model Universes Table 5.1 Curved, matter-dominated universes. Density 20 < I Do > I Curvature Ultimate fate Big Chill (a α 1) Big Chill (a α t2/3) Big Crunch density, the universe will end in a fiery Big Crunch: if the density is less than or equal to the critical density, the universe will end in an icy Big Chill. In a curved universe containing only matter, the scale factor at) can be comtd The Friedmann equationcn be witten in the form a 20 (5.88) ca so the age t of the universe at a given scale factor a is given by the integral da (5.89) when S2。96 1, the solution to this integral is most compactly written in a para- metric form. The solution when Ao > 1 is a(9) = ( 1-cos θ) (5.90) and (5.91) where the parameter θ runs from 0 to 2r. Given this parametric form, the time that elapses between the Big Bang at 6-0 and the Big Crunch at θ 2π can be computed as (5.92) A plot of a versus t in the case Ω = 1.1 is shown as the dotted line in Figure The solution of Equation 5.89 for the case < 1 can be written in paramet- erunchHo (20-)32 5.4. The a α pls behavior of an Ω0 = 1 universe is shown as the solid line. ric form as 1編 21 -2o a(n) = (cosh η-1) (5.93) and 1 20 (5.94)

Answer 1