Introduction to Cosmology

40 Special Relativity

Thus when the particle horizon grows with time, bodies which were at spacelike dis-

tances at earlier times enter into the light cone.

The integrands in Equations (2.48) and (2.50) are obviously the same, only the

integration limits are different, showing that the two horizons correspond to different

conformal times. If푡min=0, the integral in Equation (2.48) may well diverge, in which

case there is no particle horizon. Depending on the future behavior of푎(푡), an event

horizon may or may not exist. If the integral diverges as푡→∞, every event will sooner

or later enter the event horizon. The event horizon is then a function of waiting time

only, but there exists no event horizon at푡=∞.Butif푎(푡)accelerates, so that distant

parts of the Universe recede faster than light, then there will be an ultimate event

horizon. We shall see later that푎(푡)indeed appears to accelerate.

Redshift and Proper Distance. In Equation (1.20) in the previous chapter we
parametrized the rate of expansion푎̇by the Hubble constant퐻 0. It actually appeared
as a dynamical parameter in the lowest-order Taylor expansion of푎(푡), Equation (1.18).
If we allow퐻(푡)to have some mild time dependence, that would correspond to intro-
ducing another dynamical parameter along with the next term in the Taylor expan-
sion. Thus adding the second-order term to Equation (1.18), we have for푎(푡),

푎(푡)≈ 1 −푎̇ 0 (푡 0 −푡)+^1

2

푎̈ 0 (푡 0 −푡)^2. (2.53)

Making use of the definition in Equation (2.47), the second-order expansion for the

dimensionless scale factor is

푎(푡)≈ 1 −퐻 0 (푡 0 −푡)+

1

2

퐻̇ 0 (푡 0 −푡)^2. (2.54)

As long as the observational information is limited to the first time derivative푎̇ 0 ,

no further terms can be added to these expansions. To account for푎̈ 0 , we shall now

make use of the present value of the deceleration parameter in Equation (2.51),푞 0.

Then the lowest-order expression for the cosmological redshift, Equation (1.18), can

be replaced by

푧(푡)=(푎(푡)−^1 − 1 )

=

[

1 −퐻 0 (푡−푡 0 )−^1

2

푞 0 퐻^20 (푡−푡 0 )^2

]− 1

− 1

≈−퐻 0 (푡−푡 0 )+

(

1 +^1

2

푞 0

)

퐻 02 (푡−푡 0 )^2.

This expression can further be inverted to express퐻 0 (푡−푡 0 )as a function of the red-

shift to second order:

퐻 0 (푡 0 −푡)≈푧−( 1 +^1

2

푞 0 )푧^2. (2.55)

Let us now find the proper distance푑Pto an object at redshift푧in this approxima-

tion. Eliminating휒in Equations (2.38) and (2.39) we have

푑P=푐

∫

푡 0

푡

d푡

푎(푡)

.