Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 31

χ= const

t= const

Fig. 1.8.

These coordinates cover the entire hyperboloid for+∞>t>−∞and 2π≥
χ≥0 (Figure 1.8), and metric (1.92) becomes

ds^2 =dt^2 −H−^2 cosh^2 (Ht)dχ^2. (1.94)

In the four-dimensional case, this form of the metric corresponds to a closed universe
withk=+1.
Another choice of coordinates, namely,

x=H−^1 cosh(H ̃t), y=H−^1 sinh(H ̃t) sinh ̃χ, (1.95)

reduces (1.92) to a form corresponding to an open de Sitter universe:

ds^2 =d ̃t^2 −H−^2 sinh^2 (Ht ̃)dχ ̃^2. (1.96)

The range of these coordinates is+∞>t ̃≥0 and+∞>χ> ̃ −∞, covering
only the part of de Sitter spacetime wherex≥H−^1 andz>0 (Figure 1.9). More-
over, the coordinates are singular at ̃t=0.
Finally, we consider the coordinate system defined via

x=H−^1

[

cosh(H ̄t)−

1

2

exp(H ̄t) ̄χ^2

]

, y=H−^1 exp(H ̄t) ̄χ, (1.97)