Physical Foundations of Cosmology

2.2 Horizons 39

whereηi(orti) corresponds to the beginning of the universe. At timeη,the infor-
mation about events atχ>χp(η) is inaccessible to an observer located atχ=0.
In a universe with an initial singularity, we can always setηi=ti= 0 ,but in some
nonsingular spacetimes, for example, the de Sitter universe, it is more convenient
to takeηi=0. Multiplyingχpby the scale factor, we obtain the physical size of
the particle horizon:

dp(t)=a(t)χp=a(t)

∫t

ti

dt

a

. (2.7)

Until hydrogen recombination (see Section 3.6), which occurred when the uni-
verse was 1000 times smaller than now, the universe was opaque to photons. There-
fore, in practice, our view is limited to the maximum distance light can travel since
recombination. This is called the “optical” horizon:

dopt=a(η)(η−ηr)=a(t)

∫t

tr

dt

a

. (2.8)

Problem 2.2Calculateηr/η 0 in a dust-dominated universe and verify that the
present optical horizon is less than the particle horizon by only a small percentage.

Although the optical horizon is not very different from the particle horizon, it
unfortunately obscures information about the most interesting stages of the evolu-
tion of the early universe. Primordial neutrinos and gravitational waves decouple
from matter before photons, and so could, in principle, bring us this information.
Sadly, the short-term prospects of detecting primordial neutrinos or cosmological
gravitational waves are not very promising.
Let us calculate the size of the particle horizon in flat matter-dominated and
radiation-dominated universes. Substitutinga(t)∝t^2 /^3 into (2.7), we find that in
a matter-dominated universedp(t)= 3 t(c=1). If the universe is dominated by
radiation, thena(t)∝t^1 /^2 and, correspondingly,dp(t)= 2 t.

Problem 2.3Calculate the size of the particle horizon in a dust-dominated universe
with an arbitrary value of the current cosmological parameter
0 and show that

(

χp

)

=

2

a 0 H 0

0

, (2.9)

where the function is defined in (2.3).

Curvature scale (“Hubble horizon”) vs. particle horizonWhen matter satisfies the
strong energy dominance condition,ε+ 3 p> 0 ,the particle horizon is usually of