Introduction to Cosmology

Relativistic Distance Measures 41

We then insert푎(푡)from Equation (2.54) to lowest order in푡 0 −푡, obtaining

푑P≈푐

∫

푡 0

푡

[ 1 +퐻 0 (푡 0 −푡)]d푡=푐(푡 0 −푡)

[

1 +^1

2

퐻 0 (푡 0 −푡)

]

. (2.56)

Substituting the expression in Equation (2.55) into this yields the sought result:

푑P(푧)≈

푐

퐻 0

(

푧−

1

2

( 1 +푞 0 )푧^2

)

. (2.57)

The first term on the right gives Hubble’s linear law [Equation (1.15)], and thus the
second term measures deviations from linearity to lowest order. The parameter value
푞 0 =−1 obviously corresponds to no deviation. The linear law has been used to deter-
mine퐻 0 from galaxies within the Local Supercluster (LSC). On the other hand, one
also observes deceleration of the expansion in the local universe due to the lumpiness
of matter. For instance, the local group clearly feels the overdensity of the Virgo clus-
ter at a distance of about 17Mpc, falling towards it with a peculiar velocity of about
630kms−^1 [3]. It has been argued that the peculiar velocities in the LSC cannot be
understood without the pull of the neighboring Hydra–Centaurus supercluster and
perhaps a still larger overdensity in the supergalactic plane, a rich cluster (the A3627)
nicknamed the ‘Great Attractor’.
It should be clear from this that one needs to go to even greater distances, beyond
the influences of local overdensities, to determine a value for푞 0. Within the LSC it is
safe to conclude that only the linear term in Hubble’s law is necessary.
Equation (2.57) is the conventional formula, which is a good approximation for
small푧. The approximation obviously deteriorates as푧increases, so that it attains its
maximum at푧= 1 ∕ 1 +푞 0. In Figure 2.5 we plot the function푑P(푧)for small values of푧.

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5

Figure 2.5 Approximate distance measures푑to second order in푧. The solid curve shows
the linear function, Equation (1.15); the short-dashed curve the proper distance푑P,Equa-
tion (2.57); the medium-dashed curve the luminosity distance푑L, Equation (2.60); and the
long-dashed curve the angular size distance푑A, Equation (2.64). The value of푞 0 is− 0 .5.