15.5 The LCDM Universe
239
For this model our task is to evaluate the integral in (15.3b) without the radiation
term, that is
a
0
da
V 0 a
2
+
m0
a
+ k0
−1/2
= H 0 t.
(15.14)
This is an implicit solution, but this form of solution is not particularly useful since the
integral does not involve elementary functions; it can of course be solved numerically.
But for the case of zero curvature the integration can be done in terms of elementary
functions. It now appears that curvature of the real universe is either zero or quite
small so we will focus on this favored case, and since the result is important we will
do the integration in explicit detail.
To do the integral in (15.14) we first simplify it by introducing a new dimensionless
time τ and scale factor y. The substitutions and the resulting equation are
τ =
V 0 H 0 t =
/3 ct, a =
m0
V 0
1/3
y, τ =
y
0
dy
y 2 + 1/y
. (15.15)
Then we make another substitution for the variable of integration y
3
= x
2 and obtain
τ =
2
3
x
0
dx
√
1 + x 2
=
2
3
log(x +
1 + x 2 ), x
2
= y
3
.
(15.16)
Fortunately this may be easily inverted to give x(τ ) and thus y(τ ) from (15.15),
1 + x 2 = e
3τ/2
− x, x(τ ) = sinh(3τ /2), y(τ ) =
sinh
3
2
τ
2/3
. (15.17)
Finally, in terms of the original functions and parameters, the scale function is
a(t) =
m0
V 0
1/3
sinh
√
3
2
ct
2/3
.
(15.18)
Figure 15.4 shows the behavior of the scale factor as well as the asymptotic forms
for small and large times.
Note that setting the scale factor equal to 1 at the present time gives the age of the
universe for the LCDM model; we will return to this in Chap. 16. See also Exercise
15.7.
Equation (15.18) is a remarkable result. It is the exact solution of the dynamical
equations for the currently favored model of the real universe, the flat LCDM model.
239
For this model our task is to evaluate the integral in (15.3b) without the radiation
term, that is
a
0
da
V 0 a
2
+
m0
a
+ k0
−1/2
= H 0 t.
(15.14)
This is an implicit solution, but this form of solution is not particularly useful since the
integral does not involve elementary functions; it can of course be solved numerically.
But for the case of zero curvature the integration can be done in terms of elementary
functions. It now appears that curvature of the real universe is either zero or quite
small so we will focus on this favored case, and since the result is important we will
do the integration in explicit detail.
To do the integral in (15.14) we first simplify it by introducing a new dimensionless
time τ and scale factor y. The substitutions and the resulting equation are
τ =
V 0 H 0 t =
/3 ct, a =
m0
V 0
1/3
y, τ =
y
0
dy
y 2 + 1/y
. (15.15)
Then we make another substitution for the variable of integration y
3
= x
2 and obtain
τ =
2
3
x
0
dx
√
1 + x 2
=
2
3
log(x +
1 + x 2 ), x
2
= y
3
.
(15.16)
Fortunately this may be easily inverted to give x(τ ) and thus y(τ ) from (15.15),
1 + x 2 = e
3τ/2
− x, x(τ ) = sinh(3τ /2), y(τ ) =
sinh
3
2
τ
2/3
. (15.17)
Finally, in terms of the original functions and parameters, the scale function is
a(t) =
m0
V 0
1/3
sinh
√
3
2
ct
2/3
.
(15.18)
Figure 15.4 shows the behavior of the scale factor as well as the asymptotic forms
for small and large times.
Note that setting the scale factor equal to 1 at the present time gives the age of the
universe for the LCDM model; we will return to this in Chap. 16. See also Exercise
15.7.
Equation (15.18) is a remarkable result. It is the exact solution of the dynamical
equations for the currently favored model of the real universe, the flat LCDM model.