Physical Foundations of Cosmology

2.3 Conformal diagrams 43

Note that the diagrams of different spacetimes are exactly the same if their metrics
are related by anonsingularconformal transformation:g ̃μν=a^2 (x)gμν.
In addition to the shape of the diagram, we must pay attention to the location of
singularities. Singularities, as well as the boundaries of the diagram, are determined
by the behavior of the scale factora(η,χ) and the function (η,χ) in (2.17). We
will see that it is possible to have two spacetimes whose conformal diagrams have
the same shape but different singular boundaries.

Closed radiation- and dust-dominated universesFor a closed universe filled with
radiation or dust, the conformal diagram can be immediately drawn based on the
solutions fora(η) found in Section 1.3.4. Metric (2.2) becomes

ds^2 =a^2 (η)(dη^2 −dχ^2 −sin^2 χd 

2 ), (2.19)

where

a=amsinη (2.20)

in a radiation-dominated universe and

a=am(1−cosη) (2.21)

in a dust-dominated universe (see (1.73) and (1.77)). In both cases,χandηhave
finite ranges and cover the whole spacetime:

π≥χ≥ 0 ,π>η> 0 , (2.22)

for a radiation-dominated universe and

π≥χ≥ 0 , 2 π>η> 0 , (2.23)

for a dust-dominated universe. The conformal diagrams are a square and rectangle
respectively, and are shown in Figures 2.1 and 2.2. Horizontal and vertical lines
represent hypersurfaces of constantηandχ. The lower and upper boundaries
correspond to physical singularities where the scale factor vanishes and the energy
density and curvature diverge. In both cases, the lower half of the diagram describes
an expanding universe and the upper half corresponds to a contracting phase. The
scale factor reaches its maximum value atη=π/2 in the radiation-dominated
universe and atη=πin the dust-dominated universe.
The essential difference between the diagrams is the comparative ranges ofη
andχ: for the dust-dominated universeηhas twice the range ofχ, whileηand
χhave the same range for the radiation-dominated universe. This has important
consequences for the particle and event horizons. In both cases, we can setηi= 0
at the lower boundary of the diagram. Then the particle horizon for the observer at