86 Cosmological Models
This equation does not contain푘and휆, but that is not a consequence of having started
from Equations (5.4) and (5.5). If, instead, we had started from Equations (5.17)
and (5.18), we would have obtained the same equation.
Note that all terms here have dimension of energy density per time. In other words,
Equation (5.24) states that the change of energy density per time is zero, so we can
interpret it as thelocal energy conservation law. In a volume element d푉,휌푐^2 d푉 rep-
resents the local decrease of gravitating energy due to the expansion, whereas푝d푉is
the work done by the expansion. Energy does not have a global meaning in the curved
spacetime of general relativity, whereas work does. If different forms of energy do not
transform into one another, each form obeys Equation (5.24) separately. The Einstein
equation needs to be extended in some way to be able to serve as a global energy con-
servation law. There is no “correct” way to do it, but many suggestions. We shall not
pursue that search here.
As we have seen, Equation (5.24) follows directly from Friedmann’s equations with-
out any further assumptions. But it can also be derived in another way, perhaps more
transparently. Let the total energy content in a comoving volume푎^3 be
퐸=(휌푐^2 +푝)푎^3.The expansion isadiabaticif there is no net inflow or outflow of energy so that
d퐸
d푡= d
d푡[(휌푐^2 +푝)푎^3 ]= 0. (5.25)
If푝does not vary with time, changes in휌and푎compensate and Equation (5.24)
immediately follows.
Equation (5.24) can easily be integrated,
∫
휌̇(푡)푐^2
휌(푡)푐^2 +푝(푡)
d푡=− 3
∫푎̇(푡)
푎(푡)
d푡, (5.26)if we know the relation between energy density and pressure—theequation of stateof
the Universe.
Entropy Conservation and the Equation of State. In contrast, the law ofconser-
vation of entropy푆is not implied by Friedmann’s equations, it has to be assumed
specifically, as we shall demonstrate in Section 5.2,
푆̇= 0. (5.27)Then we can make an ansatz for the equation of state: let푝be proportional to휌푐^2 with
some proportionality factor푤which is a constant in time,
푝=푤휌푐^2. (5.28)Inserting this ansatz into the integral in Equation (5.26) we find that the relation
between energy density and scale is
휌(푎)∝푎−^3 (^1 +푤)=( 1 +푧)^3 (^1 +푤). (5.29)Here we use푧as well as푎because astronomers prefer푧since it is an observable. In
cosmology, however, it is better to use푎for two reasons. Firstly, redshift is a property