Introduction to Cosmology

92 Cosmological Models

Let us now return to the general expression [Equation (5.14)] for the normalized
age푡(푧)∕푡 0 or푡(푎)∕푡 0 of a universe characterized by푘and energy density components
훺m,훺rand훺휆. Inserting the훺components into Equations (5.13) and (5.14) we have

푎̇^2

푎^2

=퐻^2 (푡)=퐻 02 [( 1 −훺 0 )푎−^2 +훺m(푎)+훺r(푎)+훺휆(푎)],

or

푡(푧)=

1

퐻 0 ∫

1 ∕( 1 +푧)

0

d푎[( 1 −훺 0 )+훺m푎−^1 +훺r푎−^2 +훺휆푎^2 ]−^1 ∕^2. (5.55)

The integral can easily be carried out analytically when훺휆=0. But this is now of
only academic interest, since we know today that훺휆≈ 0 .7 as we shall see later. Thus
the integral is best solved numerically (or analytically in terms of hypergeometric
functions or the Weierstrass modular functions [3,4]).
The lookback time is given by the same integral with the lower integration limit at
1 ∕( 1 +푧)and the upper limit at 1. The proper distance [Equation (2.39)] is then

푑P(푧)=휒(푧)=ct(푧). (5.56)

In Figure 5.2 we plot the lookback time푡(푧)∕푡 0 and the age of the Universe 1−푡(푧)∕푡 0
in units of푡 0 as functions of redshift for the parameter values훺m= 0 .27,훺휆= 1 −훺m.
At infinite redshift the lookback time is unity and the age of the Universe is zero.
Another important piece of information is that훺 0 ≈ 1 .0 (Table A.6). The vacuum
term [Equation (5.8)] (almost) vanishes, in which case we can conclude that the geom-
etry of our Universe is (almost) flat. With훺 0 = 1 .0and훺rwell known, the integral
[Equation (5.55)] really depends on only one unknown parameter,훺m= 1 −훺휆.
From the values훺휆≈^0 .7and훺m≈^1 −^0.^7 =^0 .3, one can conclude that the cosmo-
logical constant has already been dominating the expansion for some time. We shall
come back to this later.

1.0

0.8

0.6

0.4

0.2

024

lookback time

age

6810

Figure 5.2The lookback time and the age of the Universe normalized to푡 0 as functions of
redshift for the parameter values훺m= 0. 27 훺휆= 1 −훺m.