Exam (elaborations) TEST BANK FOR Fundamentals of Cosmology, 2nd ed By J. (James) Rich (Solution Manual)-Converted

1.1 If ΩM = ΩT = 1, then ΩM(a) = ΩT(a) = 1 for all a(t). Structure formation never ceases as larger and larger regions of negative Newtonian energy detach from the expansion. The values of the Ω’s for the ΩM = ΩT = 0.3 and the ΩM = 0.3,ΩΛ = 0.7 models are shown in Fig. 1. Fig. 1 The solid lines show ΩM(a) and ΩΛ(a) for (ΩM = 0.3,ΩΛ = 0.7). The dashed line shows ΩM(a) for (ΩM = ΩT = 0.3). The universe remains matter dominated for a longer period in the first case 1 2 Solutions 1.2 The time that has passed since the universe became vacuum dominated is (including only the vacuum energy density) t0 − tm=v H−1 0 ∼  da a √ ΩΛ = 1 3 √ ΩΛ ln(ΩΛ/ΩM) = 0.39 . (1) Numerical integration including both matter and vacuum gives 0.32. The duration of the matter-dominated epoch is (including only the matter density) tm=v − tr=m H−1 0 ∼ a−3/2 0  da a  ΩMa−3 ∼ (2/3) 1 Ω 1/2 Λ = 0.78 . (2) Numerical integration including matter and vacuum gives 0.69. The duration of the radiation-dominated epoch is (including only the radiation density) tr=m − tinf H−1 0 ∼ a−2 0  da a  ΩRa−4 ∼ (1/2) Ω 3/2 R Ω2M = 5.4 × 10−6 (3) for ΩR = 1.68Ωγ ∼ 8.5 × 10−5 (three massless neutrino species). Numerical integration including both radiation and matter gives 4.2 × 10−6. The time would not change by much if you had taken ainf = 0. The time when the first nuclei formed: tnuc − tinf H−1 0 ∼ a−2 0  da a  ΩRa−4 ∼ (3 × 10−9)2 2 √ ΩΛ ∼ 4.9 × 10−16, (4) i.e. 3.4 min. 1.3 The universe is expanding today because it was expanding yesterday (see (1.58)). It was expanding yesterday because..... It will be difficult to get an ultimate explanation since it will require knowledge of the physics that was in charge of things before the expansion began. Chapter 2 2.1 The flux from a typical galaxy of redshift z  1 is φ ∼ 2 × 1010L/(2eV/photon) 4π(zdH)2 ∼ 100m−2s−1/z2 . (5) Solutions 3 The ratio of the flux of