Chapter 14
The Dynamical Equations of Cosmology
Abstract The Einstein equations applied to the FLRW metric give basic dynamical
equations for cosmology, specifically for the scale factor. The dynamical equations
depend on the physical properties of the constituents of the cosmic fluid, which we
take to be vacuum or dark energy, cold matter, radiation, and an effective curvature.
Together with the behavior of the constituents the dynamical equations lead to what
we here call the Friedmann master equation for the scale factor of the universe; it is
remarkably useful.
14.1 The Einstein Field Equations for Cosmology
In the preceding chapters we have set up the infrastructure of cosmology, and now
we need to add dynamics via the Einstein field equations applied to the FLRW metric
and the perfect fluid energy-momentum tensor (Adler 1975; Misner 1973; Peebles
1993). We repeat here from the last chapter the FLRW metric, the cosmic perfect
fluid energy-momentum tensor, and the field equations in mixed index form.
ds
2
= c
2 dt
2
− a(t)
2
dr
2
1 − kr 2 + r
2
(dθ
2
+ sin
2
θ dϕ
2
)
,
(14.1a)
T
μ ν = ρu
μ u ν + p
u
μ u ν − g
μ ν
,
(14.1b)
G
μ ν + g
μ ν = CT
μ ν = −
8π G/c
4
T
μ ν .
(14.1c)
We use the form of the FLWR metric (13.12b) with dimensionless scale factor a(t),
radial coordinate r with the dimension of a distance, and an energy-momentum tensor
with the dimensions of energy density; here ρ and p both have the dimensions of
energy density or mass density times c
2 . The velocity u
μ is dimensionless. Either
the mixed index form of the tensors or the lower index forms are convenient to use.
Recall that the mixed metric tensor g
μ
ν is the Kronecker δ
μ
ν .
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_14
223
The Dynamical Equations of Cosmology
Abstract The Einstein equations applied to the FLRW metric give basic dynamical
equations for cosmology, specifically for the scale factor. The dynamical equations
depend on the physical properties of the constituents of the cosmic fluid, which we
take to be vacuum or dark energy, cold matter, radiation, and an effective curvature.
Together with the behavior of the constituents the dynamical equations lead to what
we here call the Friedmann master equation for the scale factor of the universe; it is
remarkably useful.
14.1 The Einstein Field Equations for Cosmology
In the preceding chapters we have set up the infrastructure of cosmology, and now
we need to add dynamics via the Einstein field equations applied to the FLRW metric
and the perfect fluid energy-momentum tensor (Adler 1975; Misner 1973; Peebles
1993). We repeat here from the last chapter the FLRW metric, the cosmic perfect
fluid energy-momentum tensor, and the field equations in mixed index form.
ds
2
= c
2 dt
2
− a(t)
2
dr
2
1 − kr 2 + r
2
(dθ
2
+ sin
2
θ dϕ
2
)
,
(14.1a)
T
μ ν = ρu
μ u ν + p
u
μ u ν − g
μ ν
,
(14.1b)
G
μ ν + g
μ ν = CT
μ ν = −
8π G/c
4
T
μ ν .
(14.1c)
We use the form of the FLWR metric (13.12b) with dimensionless scale factor a(t),
radial coordinate r with the dimension of a distance, and an energy-momentum tensor
with the dimensions of energy density; here ρ and p both have the dimensions of
energy density or mass density times c
2 . The velocity u
μ is dimensionless. Either
the mixed index form of the tensors or the lower index forms are convenient to use.
Recall that the mixed metric tensor g
μ
ν is the Kronecker δ
μ
ν .
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_14
223