190 Cosmic Microwave Background
The authors of the currently best measurement then end up presenting ten different
versions rather than one recommendation because they do not want to offend anyone.
What one may consider an admissible modification of a model is a cultural ques-
tion to some extent, and probably this is true even in particle physics. However, in
cosmology there appears to be a wider diversity of what people consider to be a rea-
sonable model. For instance, the age of the Universe was quoted in Equation (1.22)
to have a finite value with a precision of 50 Myr, but in A. Linde’s eternal universe and
in the bouncing universe the age may well be infinite.
Thus I caution the reader to take recommended values with a grain of salt.
Expansion Time. The so-called timescale test compares the lookback time in Fig-
ure 5.2 at redshifts at which galaxies can be observed with푡 0 obtained from other
cosmochronometers inside our Galaxy, as discussed in Section 1.4. Thus we have to
make do with a consistency test. At moderately high redshifts where the훺mterm
dominates and훺휆can be neglected, Equation (5.55) can be written
퐻 0 푡(푧)≈^2
3√
훺m( 1 +푧)−^3 ∕^2. (8.42)
Let us multiply the퐻 0 and푡 0 values in Table A.2 and A.6 to obtain a value for the
dimensionless quantity
퐻 0 푡 0 = 0. 95. (8.43)As we already saw in Equation (5.44) this rules out the spatially flat matter-dominated
Einstein–de Sitter universe in which퐻 0 푡 0 <^23.
The Magnitude–Redshift Relation. Equation (2.61) relates the apparent magni-
tude푚of a bright source of absolute magnitude푀at redshift푧to the luminosity
distance푑L. We noted in Section 1.4 that the peak brightness of SNe Ia can serve as
remarkably precise standard candles visible from very far away; this determines푀.
Although the magnitude–redshift relation can be used in various contexts, we are only
interested in testing cosmology.
The luminosity distance푑Lin Equation 2.60 is a function of푧and the model-
dependent dynamical parameters, primarily훺m,훺휆and퐻 0. The redshift푧or the
scale푎at the emission of light can be measured in the usual way by observing the
shift of spectral lines. The supernova lightcurve shape gives supplementary informa-
tion: in the rest frame of the supernova the time dependence of light emission follows
a standard curve, but a supernova at relativistic distances exhibits a broadened light
curve due to time dilation.
Let us define a ‘Hubble-constant free’ luminosity distance
퐷L(푧, 훺m≡퐻(푧)푑L(푧, 훺m), (8.44)where퐻(푧)is a simplification of퐻(푡)in Equation 5.55. The apparent magnitude푚B
corresponding to푑L(B for the effective B-band, a standard blue filter) becomes
푚B=푀B−5log퐻 0 + 25 +5log퐷L(푧, 훺m,훺휆). (8.45)
This can be fitted to supernova data to determine best values in the space of the
matter density parameter,훺m, and the cosmological constant density parameter,훺휆),