... the geometry of the Universe in the standard model is specified by a set of cosmological parameters. The values of these cosmological parameters can therefore be estimated by measuring the geometrical properties of the Universe. The starting point is to find two observables that are related to each other only through the geometrical properties of the Universe. The most important example here is the redshift–distance relation , two types of distances can be defined through observational quantities. One is the luminosity distance
, which relates the luminosity of an object, 
, to its flux, 
, according to 
f. The other is the angular-diameter distance, 
, which relates the physical size of an object, 
, to its angular size, 
. In general, the redshift–distance relation can formally be written as
where d stands for either 
, and by definition 
. For redshifts much smaller than 1, the redshift–distance relation reduces to the Hubble expansion law 
and so the Hubble constant 
can be obtained by measuring the redshift and distance of an object (ignoring, for the moment, that objects can have peculiar velocities). Redshifts are relatively easy to obtain from the spectra of objects, and in §2.1.3 we have seen how to measure the distances of a few classes of astronomical objects. The best estimate of the Hubble constant at the present comes from Cepheids observed by the HST, and the result is
(Freedman et al., 2001). In order to measure other cosmological parameters, one has to determine the nonlinear terms in the redshift–distance relation, which typically requires objects at 
. For example, measuring the light curves of Type Ia supernovae out to 
has yielded the following constraints:
(e.g. Perlmutter et al., 1999). Using Eq. (2.63) and neglecting 
because it is small, the above relation gives 
. Since 
, we have 
, i.e. the expansion of the Universe is speeding up at the present time. Important constraints on cosmological parameters can also be obtained from the angular spectrum of the CMB temperature fluctuations. As shown in Fig. 1, the observed angular spectrum 
contains peaks and valleys, which are believed to be produced by acoustic waves in the baryon–photon fluid at the time of photon–matter decoupling. As we will see in §6.7, the heights/depths and positions of these peaks/valleys depend not only on the density of baryonic matter in the Universe, but also on the total mean density, Hubble's constant and other cosmological parameters. In particular, the position of the first peak is sensitive to the total density parameter 
(or the curvature K). Based on the observational results shown in Fig. 1, one obtains
where 
and 
are the density parameters of total matter and of baryonic matter, respectively . Note that this implies that the Universe has an almost flat geometry, that matter accounts for only about a quarter of its total energy density, and that baryons account for only ~ 17% of the matter.
Fig.1. The angular power spectrum, C, of the CMB temperature fluctuations in the WMAP full-sky map. This shows the relative brightness of the ‘spots’ in the CMB temperature map vs. the size of the spots. The shape of this curve contains a wealth of information about the geometry and matter content of the Universe. The curve is the model prediction for the best-fit ΛCDM cosmology. [Adapted from Hinshaw et al. (2007) by permission of AAS]
