2.3 Expansion law and age of the Universe

Equation (18View Equation) has the following analytic solutions in several simple but practically important cases.

In the above, t0 denotes the present age of the Universe,

∫ 1 t = -1- ∘--------------xdx----------------, (25 ) 0 H0 0 Ωmx + (1 − Ωm − ΩΛ)x2 + Ω Λx4
and we adopt the initial condition that a = 0 at t = 0. The expression clearly indicates that t0 increases as Ωm decreases and/or ΩΛ increases. Figure 1View Image plots the scale factor as a function of H0 (t − t0), and Table 1 summarizes the age of the Universe.


Table 1: The present age of the Universe in units of (h∕0.7)−1 Gyr.
Ωm Open model Spatially-flat model
(ΩΛ = 0) (Ω Λ = 1 − Ωm)
1.0 9.3 9.3
0.5 10.5 11.6
0.3 11.3 13.5
0.1 12.5 17.8
0.01 13.9 28.0

View Image

Figure 1: Evolution of the cosmic scale factor as a function of H0 (t − t0). The present value of the scale factor a0 is set to unity; solid line: (Ωm, Ω Λ) = (0.3,1.7 ), dotted line: (0.3,0.7), dot-dashed line: (0.3,0.0), long dashed line: (1.0, 0.0), short dashed line: (3.0,0.0 ).

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