40 Propagation of light and horizons
order the Hubble scale, 1/H. Consequently, the terms “Hubble scale” and “particle
horizon” are sometimes used interchangeably. Some authors even conjoin the terms
and refer to a “Hubble horizon.” However, the Hubble scale,H−^1 , is conceptually
distinct from a horizon. Whereas the particle horizon is a scale set by kinematical
considerations, the curvature scale is a dynamical scale that characterizes the rate
of expansion and enters the equations describing, for instance, the evolution of
cosmological perturbations. BecauseH−^1 is of order the 4-curvature scale, it also
characterizes the “size” of the local inertial frame.
Although the Hubble scale and particle horizon are of similar magnitudes for
some models, they can differ by a large factor when the strong energy condition
is violated,ε+ 3 p<0. In this case, from (1.66),a ̈> 0 ,that is, the expansion is
accelerating. Then, the integral in the expression
dp(t)=a(t)∫t
dt
a=a(t)∫a
da
aa ̇(2.10)
converges ast→∞anda→∞. At larget, the particle horizon is proportional to
a(t), but the curvature scale,H−^1 =a/a ̇, grows more slowly sincea ̇also increases
during accelerated expansion. For instance, the particle horizon in a flat de Sitter
universe, wherea(t)∝exp (Ht), is
dp(t)=exp(Ht)∫ttiexp(−Ht)dt=H−^1 (exp(H(t−ti))−1). (2.11)Fort−tiH−^1 , the size of the causally connected region grows exponentially
fast, whereas the curvature scale,H−^1 ,is constant. Formally, asti→−∞,the
particle horizon diverges, and hence all points were in causal contact. However,
this has limited significance since the flat slicing of de Sitter spacetime is geodesi-
cally incomplete (see next section). Moreover, when applied as an approximation
for inflation, we use only a part of the whole de Sitter spacetime. The beginning
of inflation corresponds to a finite initial timeti and, consequently, the particle
horizon is finite.
Despite the fact that the curvature scale is not, if properly considered, a horizon,
the use of the term “Hubble horizon” has become so widespread that we will
occasionally follow the “traditional terminology.” However, the reader is strongly
advised to keep in mind the distinction between thedynamicalcurvature scale and
thekinematicalhorizon.
Event horizonTheevent horizonis the complement of the particle horizon. The
event horizon encloses the set of points from which signals sentat a given moment
of timeηwill never be received by an observer in thefuture. These points have