2
The standard model of cosmology
1.2 The Robertson-Walker metric
The standard description of the hot big bang assumes a universe which is
homogeneous and isotropic with a metric involving a single function R(t),
the 'scale factor' (or 'radius' of the universe). The appropriate metric is the
Robertson-Walker metric
ds2=dt2-R2(t)( dr
2 +r 2 d0 2 +r 2 sin 2 (}dt/>2)
(1.1)
1- kr 2
where the (time and spherical polar) coordinates (t, r, (), tP), called the 'comoving'
coordinates, are the coordinates of an observer in free fall in the gravitational
field of the universe. The parameter k takes the values -I, 0, I corresponding
to a universe which has spatial curvature which is negative, zero or positive,
respectively. (This can be seen from the curvature scalar derived from the second
equality of (1.30) with a change in sign for Euclidean rather than Minkowski
space.) Units have been chosen in which the speed of light c is I.
An immediate use of this metric is to calculate the size of regions of the
universe that have been in causal contact (in the sense that there has been the
possibility of causal influence occurring between points within the region at some
time between the big bang at I = 0 and time t). Causal influences cannot occur
over distances greater than the (proper) distance dH(I) that light has been able to
travel from the the big bang at I = 0 to the time t being studied. This distance
is called the 'particle horizon'. Without loss of generality, consider emission of
a light signal from coordinate (r, 0, tP) at I = 0 to coordinate (0, (), tP) at time t
along the (radial) geodesic with () and tP constant. (It may be checked that this is
indeed a geodesic by using the coefficients of affine connection given in the next
section (exercise I).) For a light beam, ds 2 = 0 and we have
dt 2
dr 2
(1.2)
R2(t) = l-kr2 ·
Thus, the largest value of r at t = 0 to be in causal contact with r = 0 at time I is
given implicitly by
f' dt' (' dr'
(1.3)
lo R(t') = lo JI - kri2
This equation determines the particle horizon. The proper distance to the particle
horizon at time I is
r
dr'
dH(t) = R(I) l a ~kri2
t dt'
(1.4)
= R(/) l a R(t')·
The standard model of cosmology
1.2 The Robertson-Walker metric
The standard description of the hot big bang assumes a universe which is
homogeneous and isotropic with a metric involving a single function R(t),
the 'scale factor' (or 'radius' of the universe). The appropriate metric is the
Robertson-Walker metric
ds2=dt2-R2(t)( dr
2 +r 2 d0 2 +r 2 sin 2 (}dt/>2)
(1.1)
1- kr 2
where the (time and spherical polar) coordinates (t, r, (), tP), called the 'comoving'
coordinates, are the coordinates of an observer in free fall in the gravitational
field of the universe. The parameter k takes the values -I, 0, I corresponding
to a universe which has spatial curvature which is negative, zero or positive,
respectively. (This can be seen from the curvature scalar derived from the second
equality of (1.30) with a change in sign for Euclidean rather than Minkowski
space.) Units have been chosen in which the speed of light c is I.
An immediate use of this metric is to calculate the size of regions of the
universe that have been in causal contact (in the sense that there has been the
possibility of causal influence occurring between points within the region at some
time between the big bang at I = 0 and time t). Causal influences cannot occur
over distances greater than the (proper) distance dH(I) that light has been able to
travel from the the big bang at I = 0 to the time t being studied. This distance
is called the 'particle horizon'. Without loss of generality, consider emission of
a light signal from coordinate (r, 0, tP) at I = 0 to coordinate (0, (), tP) at time t
along the (radial) geodesic with () and tP constant. (It may be checked that this is
indeed a geodesic by using the coefficients of affine connection given in the next
section (exercise I).) For a light beam, ds 2 = 0 and we have
dt 2
dr 2
(1.2)
R2(t) = l-kr2 ·
Thus, the largest value of r at t = 0 to be in causal contact with r = 0 at time I is
given implicitly by
f' dt' (' dr'
(1.3)
lo R(t') = lo JI - kri2
This equation determines the particle horizon. The proper distance to the particle
horizon at time I is
r
dr'
dH(t) = R(I) l a ~kri2
t dt'
(1.4)
= R(/) l a R(t')·