The cosmological constant
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Figure 1.1. Hubble diagram giving the effective magnitude versus redshift for the
supernovae in the primary low-extinction subset. The full line is the best-fit flat-universe
cosmology from the low-extinction subset. the broken and dotted lines represent the
indicated cosmologies.
Hence. for nearby supernovae the luminosity distance is proportional to the
redshift of the source.
Astronomers measure the apparent magnitude m of the various supernovae
sources. The difference m - M. where M .... -19.5. is the (assumed constant)
intrinsic magnitude of the source. is just the logarithm of the luminosity distance.
So the apparent magnitude is predicted to be linear in In;z for small;z. This is
consistent with the data for z ;S 0.1. see figure 1.1 taken from [2]. For more distant
supernovae the linear relationship between DL and ;z is distorted by quadratic
terms depending on the present deceleration parameter qO of the universe. The
data for 0.7 ;S ;z ;S 1 do display such a distortion. see figure 1.1 [2].
For an FRW universe. it follows from (1.36) and the definition (1.22) of qO
that. in general. the deceleration may be written as
qO = ! L(I + 3Wi)S'2i
(1.89)
for a universe with components labelled by ; having energy density Pi and
pressure Pi == WjPi; here S'2j == Pi/Pc where Pc == 3HJ/87rGN is the
critical density. In particular. for a universe with just (pressureless) matter and