Physical Foundations of Cosmology

54 Propagation of light and horizons

i+

i^0

I+

χ= const

h= const

initial singularity

̃

̃

Fig. 2.8.

Problem 2.11Draw the conformal diagram for open and flat universes where
the scale factor changes asa(t)∝tp,p> 1 .This is the situation for power-law
inflation. Note that the strong energy condition is violated in this case. Indicate the
particle and event horizons and the types of infinities. Draw the conformal diagram
for a flat universe filled by matter with equation of statep=−ε/3. Compare this
case with the Milne universe.

Problem 2.12The metric of an eternal black hole in the Kruskal–Szekeres coor-
dinate system takes the form

ds^2 =a^2 (v,u)

(

dv^2 −du^2 −^2 (v,u)d 

2

)

. (2.44)

The only extra information we need to draw the conformal diagram is that the space-
like coordinateuranges from−∞to+∞and that there is a physical singularity
located at

v^2 −u^2 = 1.

The existence of a singularity means that, for everyu,the spacetime cannot be
extended outside the interval

−

√

1 +u^2 <v<+

√

1 +u^2.