**4.2. The horizon problem**

There is another odd fact about the observed universe: the apparent high
degree of uniformity
and thermal equilibrium of the CMB. While it might at first seem quite
natural for the
hot gas of the early universe to be in good thermal equilibrium, this is
in fact quite
*unnatural* in the standard Big Bang picture. This is because of
the presence of a
cosmological horizon. We recall that the horizon size of the universe is
just how far a photon can have traveled since the Big Bang,
*d*_{H} ~ *t* in units where *c* = 1. This
defines how large a "patch" of the universe can be in causal
contact. Two points in the
universe separated by more than a horizon size have no way to reach
thermal equilibrium,
since they cannot have *ever* been in causal contact. Consider
two points
comoving with respect to the cosmological expansion. The physical
distance *d*
between the two points then just increases linearly with the scale factor:

(63) |

The horizon size is just proportional to the time, or equivalently to the inverse of the Hubble parameter,

(64) |

It is straightforward to show that for any FRW space with constant equation of state, there is a conserved quantity given by

(65) |

Proof of this is also left as an exercise for the student.
If we again consider the case of "ordinary" matter, with equation of state
1 + 3*w* > 0,
we see from Eqs. (61) and (65) that the horizon
expands faster than a comoving length:

(66) |

This means that any two points at rest relative to the expansion of the
universe will
be causally disconnected at early times and causally connected at late
times. This can be visualized on a conformal space-time diagram
(Fig. 10). On a conformal diagram, we re-write
the metric in terms of *conformal time*
*d*
*dt*/*a*:

(67) |

The advantage of this choice of coordinates is that light always travels at
45° angles *d***x** =
*d* on the diagram,
regardless of the behavior of the scale factor. In a matter dominated
universe, the scale factor evolves as

(68) |

so the Big Bang, *a* = 0, is a surface
**x** = *const*. at
= 0. Two points on a given
= *const*. surface are
in causal contact if their past light cones intersect at
the Big Bang, = 0.

So the natural expectation for the very early universe is that there should be a large number of small, causally disconnected regions that will be in poor thermal equilibrium. The central question is: just how large was the horizon when the CMB was emitted? Since the universe was about 300,000 years old at recombination, the horizon size then was about 300,000 light years. Each atom at the surface of last scattering could only be in causal contact (and therefore in thermal equilibrium) with with other atoms within a radius of about 300,000 light years. As seen from earth, the horizon size at the surface of last scattering subtends an angle on the sky of about a degree.

Therefore, two points on the surface of last scattering separated by an
angle of more than a degree were out of causal contact at the time the
CMB was emitted. However, the CMB is uniform (and therefore in thermal
equilibrium) over the *entire sky* to one part in
10^{5}. How did all of these disconnected regions reach such a
high degree of thermal equilibrium?