Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 33

χ = const

t– = const

Fig. 1.10.

it is recast as

ds^2 =−dl^23 d=−dτ^2 −H−^2 sin^2 (Hτ)(dχ^2 +sin^2 χdθ^2 ). (1.100)

Then, analytically continuingτ→it+π/2, we obtain a three-dimensional de
Sitter spacetime in the form of a closed Friedmann universe:

ds^2 =dt^2 −H−^2 cosh^2 (Ht)(dχ^2 +sin^2 χdθ^2 ). (1.101)

Note that the coordinateχvaries only from 0 toπ, covering the entire space. The
same construction works for a four-dimensional closed de Sitter universe.
To obtain an open de Sitter metric we must analytically continuetwocoordinates
in (1.100) simultaneously:τ→i ̃tandχ→iχ ̃, giving

ds^2 =d ̃t^2 −H−^2 sinh^2 (H ̃t)(dχ ̃^2 +sinh^2 χ ̃dθ^2 ). (1.102)

Generalizing the procedure to four dimensions is again straightforward.

De Sitter universe as a solution of Friedmann equations with cosmological constant
(four-dimensional case) A cosmological constant is equivalent to a “perfect fluid”
with equation of statep=−ε.It follows from ( 1.64) that

dεV=− 3 (εV+pV)dlna= 0 ,