Question

Take h = 0.7. Orad,0 = 0.0m,0-0.3, and ΩΛ,0 0.7. (a) Use the formula for how the Hubble parameter H(t)-à/a changes with scale factor and express it as a function of redshift. b) How fast was the universe expanding at z-0.1, z 0.4, z 0.7, z -2, 2-5, and z = 1100? (in units of km/s/Mpc)? (c) Suppose instead that there is the same amount of matter but no dark energy (i.e. no cosmological constant). What would the expansion rates be at the above redshifts? (d) Calculate the percentage differences between the expansion rates for the above redshifts. If you wanted to distinguish between these two possible Universes, which two redshifts from the list above would provide the most useful measurements? 0.1, 0.4, z0.7, z

Answer 1

Given that h = 0.7, ohm_rad, 0 = 0 ohm_m, 0 = 0.3 ohm_A, 0 = 0.7 a we know a(t) = (t/t)^2/3 = 1/1 + x 1 H(t) = a(t)/a(t) = 2/3 t^- 1/3/t^2/3 (t/t_0)^- 2/3 = 2/3.1/t H(t) = 2/3.1/t.t_0/t_0.2/3l_0 (t_0/l) But t_0/t = (1 - z)^3/2 [from 1] because H(x) = 2/3 + 0 (1 + x)^3/2 H(x) = H_0 (+ 1 + 2)^3/2 b H(z = t^1_0 (1 + 0.1)^3/2 = 100.hlms^- 1 mpc^- 1 (1 + 0.1)^3/2 = 100 times 0.7 kms^- 1 mpc^- 1 (1 + 0.1)^3/2 H(z = 0.1) = 80.758 kms^- 1 mpc^- 1 \r\n H(z = 0.4) = 115.955 km/s/mpc H(z = 0.7) = 70 times (1 + 0.7)^3/2 = 115.157 m/s mpc H(z = 2) = 70 times (1 + 2)^3/2 = 363.73 kms/mpc H(z = 5) = 70 times(15)^3/2 = 1028.785 km/s/mpc H(z = /100) = 70 times (1 + 1100)^3/2 = 2557284.336 km/s