Hubble’s Law 13Or, to put it a different way, according to Hubble’s nonrelativistic law, objects at this
distance would be expected to attain the speed of light, which is an absolute limit in
the theory of special relativity.
Combining Equation (1.12) with Equation (1.11), one obtains
푧=퐻 0 푟
푐. (1.15)
In the section on Special Relativity we will see limitations to this formula when푣
approaches푐. The redshift푧is in fact infinite for objects at distance푟Hreceding
with the speed of light and thus physically meaningless. Therefore no information
can reach us from farther away, all radiation is redshifted to infinite wavelengths, and
no particle emitted within the Universe can exceed this distance.
The Cosmic Scale. The size of the Universe is unknown and unmeasurable, but if it
undergoes expansion or contraction it is convenient to express distances at different
epochs in terms of acosmic scale푅(푡), and denote its present value푅 0 ≡푅(푡 0 ).The
value of푅(푡)can be chosen arbitrarily, so it is often more convenient to normalized
it to its present value, and thereby define a dimensionless quantity, thecosmic scale
factor:
푎(푡)≡푅(푡)∕푅 0. (1.16)The cosmic scale factor affects all distances: for instance the wavelength휆of light
emitted at one time푡and observed as휆 0 at another time푡 0 :
휆 0
푅 0= 휆
푅(푡)
. (1.17)
Let us find an approximation for푎(푡)at times푡<푡 0 by expanding it to first-order time
differences,
푎(푡)≈ 1 −푎̇ 0 (푡 0 −푡), (1.18)using the notation푎̇ 0 for푎̇(푡 0 ),and푟=푐(푡 0 −푡)for the distance to the source. Thecos-
mological redshiftcan be approximated by
푧=휆 0
휆
− 1 =푎−^1 − 1 ≈푎̇ 0 푟
푐
. (1.19)
Thus 1∕ 1 +푧is a measure of the scale factor푎(푡)at the time when a source emitted
the now-redshifted radiation. Identifying the expressions for푧in Equations (1.18) and
(1.15) we find the important relation
푎̇ 0 =
푅̇ 0
푅 0
=퐻 0. (1.20)
The Hubble Constant. The value of this constant initially found by Hubble was
퐻 0 =550kms−^1 Mpc−^1 : an order of magnitude too large because his distance mea-
surements were badly wrong. To establish the linear law and to determine the global
value of퐻 0 one needs to be able to measure distances and expansion velocities well