1.3 From Newtonian to relativistic cosmology 35k= 1k= 0 k= − 1ta(t)Fig. 1.11.then reaches its minimum value, and subsequently increases. In flat and open co-
ordinates,a(t) always increases astgrows but vanishes ast→−∞andt=0,
respectively. However, the vanishing of the scale factor does not represent a real
physical singularity but simply signals that the coordinates become singular. For
tH−^1 , the expansion is nearly the same in all coordinate systems, namely, ex-
ponential witha∝exp(Ht).
Problem 1.19CalculateH(t) and (t) in open and closed de Sitter universes.
Verify thatH(t)→Hand (t)→1ast→∞in both cases.
In a pure de Sitter universe, there is no real evolution. In this sense, de Sitter
spacetime is similar to Minkowski spacetime. As in the case of the Milne universe,
the apparent expansion reflects the nonstatic character of the chosen coordinate
systems. However, unlike Minkowski spacetime, there exists nostaticcoordinate
system which can cover de Sitter spacetime on scales exceedingH−^1. We will see
later that only a de Sitter solution withslightly brokentime-translational symmetry
plays an important role in physical applications. The notion of de Sitter expan-
sion is still useful in the presence of perturbations that break the exact symmetry,
and the coordinate systems (1.106) are well suited to study the behavior of these
perturbations and the subsequent exit from the de Sitter stage.
Problem 1.20Verify that for a flat universe filled with radiation and cosmological
constant, the scale factor grows as
a(t)=a 0 (sinh 2Ht)^1 /^2 , (1.107)