Einstein equations for a Friedmann-Robertson-Walker universe
5
leading to
qO) 2
2
Z = Ho(to - td + ( I + 2" Ho (to - t) + ....
( 1.24)
Since z is the physically measurable quantity, it is useful to invert (1.24). For
small z
10 - I) = ~o [z - (I + ~qo) z2 + ... ].
(1.25)
Then, after expanding 1/ R(t) in (1.11) in powers of t - to, we may determine rl
as a function of z. Expanding (1.11) gives
R:tO) [(to - td + ~Ho(to - t)2 + .. -] = rl + O(r~). (1.26)
Thus, in terms of the redshift,
rl=
)
[
R(to)Ho Z - 2"(1 I + qo)z2 + .... ]
(1.27)
We shall use this result in section 1.7 to calculate the 'luminosity distance' of a
(supernova) source as a function of the redshift.
1.3 Einstein equations for a Friedmann-Robertson-Walker
universe
It is straightforward to calculate the coefficients of affine connection for the metric
(1.1). The non-zero components are
o R . R
.
r ij = - /igij
rjo = /i 8ij = rOj
(1.28)
.
rJk =
1 'I
19' (Bkglj + Bjglk - Blgjk).
(1.29)
Here xi, i = 1,2, 3, denotes the (spatial) coordinates (r, 8, r/J). Equation (1.29)
is just the coefficients of affine connection for the three-dimensional subspace
(r, 8, r/J). It is also straightforward to calculate the Ricci tensor RJJ.II from the
cofficients of affine connection (exercise 2). It has non-zero components
R
R R2
Roo = -3/i and Ri}
[
2k]
= -
R + 2 R2 + R2 gij.
(1.30)
The corresponding curvature scalar is
=
R R2 k ]
IR == gJJ.1I Rj.l.1I -6 [ /i + R2 + R2 •
( 1.31)