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14 The Dynamical Equations of Cosmology
Thus the density throughout time is simply related to the present densities and the
scale factor; relations (14.16) are very elegant and useful. Recall also that the scale
factor may be taken as unity at the present time, making (14.16) yet simpler looking.
Equations (14.16) are key relations in obtaining the master equation for the
evolution of the scale factor that we will derive in the next section.
14.5 The Friedmann Master Equation
In this section we will obtain an equation that allows direct calculation of the scale
factor in a form especially well-suited to some of the most physically interesting
situations. We first rearrange the fundamental Einstein equation (14.5a) to give
a
2
−
8π G
3c 2
ρa
2
−
c
2
3
a
2
+ kc
2
= 0.
(14.17)
Then we substitute the density expression for matter and radiation from (14.16b) to
get
a
2
−
8π G
3c 2
ρ m0
a 0
a
3 + ρ r 0
a 0
a
4
a
2
−
c
2
3
a
2
+ kc
2
= 0.
(14.18)
This is a rather simple first order differential equation for the scale factor. It can
be put into more beautiful form by using the definition of the critical density, the
vacuum density and the curvature density in (14.7) and (14.8). Using these we may
put (14.18) into the form,
a
2
a 2 −
m0
a 0
a
3 + r 0
a 0
a
4 + V 0 + k0
a 0
a
2
H
2
0 = 0,
V 0 ≡
c
2
3H
2
0
, , k0 ≡ −
kc
2
a
2
0 H
2
0
, , m0 ≡
8π Gρ m0
3c 2 H
2
0
, , r 0 ≡
8π Gρ r 0
3c 2 H
2
0
. (14.19)
This is quite useful and elegant: it is a first order differential equation for the scale
factor in terms of powers of the scale factor. Moreover the various coefficients
denoted by a zero subscript can all be determined by observations of the present
universe.
We will refer to (14.19) as the Friedmann master equation; however Friedmann’s
name has also been attached to various related equations, including (14.5).
There is one useful feature of the Friedmann master equation that is worth noting
at this point. The various epochs in the evolution of the universe involve the scale
factor going from very small to very large values. From (14.19) we see this means
that over time, roughly speaking, the most important ingredients are radiation, then
14 The Dynamical Equations of Cosmology
Thus the density throughout time is simply related to the present densities and the
scale factor; relations (14.16) are very elegant and useful. Recall also that the scale
factor may be taken as unity at the present time, making (14.16) yet simpler looking.
Equations (14.16) are key relations in obtaining the master equation for the
evolution of the scale factor that we will derive in the next section.
14.5 The Friedmann Master Equation
In this section we will obtain an equation that allows direct calculation of the scale
factor in a form especially well-suited to some of the most physically interesting
situations. We first rearrange the fundamental Einstein equation (14.5a) to give
a
2
−
8π G
3c 2
ρa
2
−
c
2
3
a
2
+ kc
2
= 0.
(14.17)
Then we substitute the density expression for matter and radiation from (14.16b) to
get
a
2
−
8π G
3c 2
ρ m0
a 0
a
3 + ρ r 0
a 0
a
4
a
2
−
c
2
3
a
2
+ kc
2
= 0.
(14.18)
This is a rather simple first order differential equation for the scale factor. It can
be put into more beautiful form by using the definition of the critical density, the
vacuum density and the curvature density in (14.7) and (14.8). Using these we may
put (14.18) into the form,
a
2
a 2 −
m0
a 0
a
3 + r 0
a 0
a
4 + V 0 + k0
a 0
a
2
H
2
0 = 0,
V 0 ≡
c
2
3H
2
0
, , k0 ≡ −
kc
2
a
2
0 H
2
0
, , m0 ≡
8π Gρ m0
3c 2 H
2
0
, , r 0 ≡
8π Gρ r 0
3c 2 H
2
0
. (14.19)
This is quite useful and elegant: it is a first order differential equation for the scale
factor in terms of powers of the scale factor. Moreover the various coefficients
denoted by a zero subscript can all be determined by observations of the present
universe.
We will refer to (14.19) as the Friedmann master equation; however Friedmann’s
name has also been attached to various related equations, including (14.5).
There is one useful feature of the Friedmann master equation that is worth noting
at this point. The various epochs in the evolution of the universe involve the scale
factor going from very small to very large values. From (14.19) we see this means
that over time, roughly speaking, the most important ingredients are radiation, then