Introduction to Cosmology

Relativistic Distance Measures 39

Particle and Event Horizons. In Equation (1.14) we defined the Hubble radius푟H
as the distance reached in one Hubble time,휏H, by a light signal propagating along a
straight line in flat, static space. Let us define theparticle horizon휎phor휒ph(alsoobject
horizon) as the largest comoving spatial distance from which a light signal could have
reached us if it was emitted at time푡=푡min<푡 0. Thus it delimits the size of that part
of the Universe that has come into causal contact since time푡min. If the past time푡is
set equal to the last scattering time (the time when the Universe became transparent
to light, and thus the earliest time anything was visible, as we will discuss in a later
chapter) the particle horizon delimits the visible Universe. From Equation (2.38),

휒ph=푐

∫

푡 0

푡min

d푡

푎(푡)

, (2.48)

and from the notation in Equation (2.37),

휎ph=푆푘(휒ph).

A particle horizon exists if푡minis in the finite past. Clearly the value of휎phdepends
sensitively on the behavior of the scale of the Universe at that time,푎(푡ph).
If푘⩾0, the proper distance (subscript ‘P’) to the particle horizon (subscript ‘ph’)
at time푡is

푑P,ph=푎(푡)휒ph. (2.49)

Note that푑Pequals the Hubble radius푟H=푐∕퐻 0 when푘=0andthescaleisacon-
stant,푎(푡)=푎.When푘=−1 the Universe is open, and푑P,phcannot be interpreted as a
measure of its size.
In an analogous way, the comoving distance휎ehto theevent horizonis defined as
the spatially most distant present event from which a world line can ever reach our
world line. By ‘ever’ we mean a finite future time,푡max:

휒eh≡푐

∫

푡max

푡 0

d푡

푎(푡)

. (2.50)

Theparticlehorizon휎phat time푡 0 lies on our past light cone, but with time our particle
horizon will broaden so that the light cone at푡 0 will move inside the light cone at푡>푡 0
(see Figure 2.1). The event horizon at this moment can only be specified given the time
distance to the ultimate future,푡max.Onlyat푡maxwill our past light cone encompass
the present event horizon. Thus the event horizon is our ultimate particle horizon.
Comoving bodies at the Hubble radius recede with velocityc, but the particle horizon
itself recedes even faster. From

d(퐻푑P,ph)∕d푡=퐻푑̇ P,ph+퐻푑̇P,ph= 0 ,

and making use of thedeceleration parameter푞,definedby

푞=−푎̈푎

푎̇^2

=−푎̈

푎퐻^2

, (2.51)

one finds

푑̇P,ph=푐(푞+ 1 ). (2.52)