Expansion law and age of the Universe

2.3 Expansion law and age of the Universe

Equation (18

) has the following analytic solutions in several simple but practically important cases.

Einstein–de Sitter model ( $Ωm = 1,ΩΛ = 0$ ):
Open model with vanishing cosmological constant ( $Ωm < 1, ΩΛ = 0$ ):
Spatially-flat model with cosmological constant ( $Ωm < 1,Ω Λ = 1 − Ωm$ ):

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

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 1

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

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: $(3.0,0.0 )$ .