cosmological entropy as reflecting an observers lack of knowledge of the universe beyond
his event horizon.
2 π
Euclidean metric periodic with period H
Temperature = H
2 π
⇒
Area of event horizon = 4 π
H 2
Entropy =
π
H 2
De Sitter space is not a good model of the universe we live in because it is empty and
it is expanding exponentially. We observe that the universe contains matter and we deduce
from the microwave background and the abundances of light elements that it must have
been much hotter and denser in the past. The simplest scheme that is consistent with our
observations is called the Hot Big Bang model.
In this scenario, the universe starts at a singularity filled with radiation at an infinite temperature. As it expands, the radiation cools and its energy density goes down. Eventually
the energy density of the radiation becomes less than the density of non relativistic matter
which has dominated over the expansion by the last factor of a thousand. However we
can still observe the remains of the radiation in a background of microwave radiation at a
temperature of about 3 degrees above absolute zero.
The trouble with the Hot Big Bang model is the trouble with all cosmology without
a theory of initial conditions: it has no predictive power. Because general relativity would
50
Hot Big Bang Model
radius/
temperature
radius of
the universe
temperature
of the universe
time
break down at a singularity, anything could come out of the Big Bang. So why is the
universe so homogeneous and isotropic on a large scale yet with local irregularities like
galaxies and stars. And why is the universe so close to the dividing line between collapsing
again and expanding indefinitely. In order to be as close as we are now the rate of expansion early on had to be chosen fantastically accurately. If the rate of expansion one second after the Big Bang had been less by one part in 1010, the universe would have collapsed after
a few million years. If it had been greater by one part in 1010, the universe would have
been essentially empty after a few million years. In neither case would it have lasted long
enough for life to develop. Thus one either has to appeal to the anthropic principle or find
some physical explanation of why the universe is the way it is.
Hot Big Bang model does not explain why :
1. The universe is nearly homogeneous and isotropic but with small pertur-
bations.
2. The universe is expanding at almost exactly the critical rate to avoid col-
lapsing again.
Some people have claimed that what is called inflation removes the need for a theory
of initial conditions. The idea is that the universe could start out at the the Big Bang in
almost any state. In those parts of the universe in which conditions were suitable there
would be a period of exponential expansion called inflation. Not only could this increase
the size of the region by an enormous factor like 1030 or more, it would also leave the region homogeneous and isotropic and expanding at just the critical rate to avoid collapsing again.
The claim would be that intelligent life would develop only in regions that inflated. We
51
should not, therefore, be surprised that our region is homogeneous and isotropic and is
expanding at just the critical rate.
However, inflation alone can not explain the present state of the universe. One can
see this by taking any state for the universe now and running it back in time. Providing it
contains enough matter, the singularity theorems will imply that there was a singularity
in the past. One can choose the initial conditions of the universe at the Big Bang to be the
initial conditions of this model. In this way, one can show that arbitrary initial conditions at the Big Bang can lead to any state now. One can’t even argue that most initial states
lead to a state like we observe today: the natural measure of both the initial conditions
that do lead to a universe like ours and those that don’t is infinite. One can’t therefore