5.2 Redshifts and distances
Light and particles (galaxies) move along (different) geodesics. When, in an expanding universe, light travels along a smooth geodesic from the source towards the observer, its frequency changes with the changing scale factor according to zcosm = / = R / R (Sect. 3.2.1). This is the global effect.
Redshift zcosm is related to the distance r of the object, given by the invariable fraction of the scale length, and the present scale factor R0. When distance is measured by a distance-dependent object property, such as apparent brightness or angular extent, the deduced distance value depends not only on the cosmological parameters: Hubble constant H0, deceleration parameter q0 and curvature constant k, but also on the measuring process. If the process is brightness measurement the resulting distance is luminosity distance rM; angular diameters give angular distances r, parallax measurements parallax distances rp, etc. The differences occur because the scale factor R0 enters differently into the measured quantities.
McCrea (1935, following Tolman, 1930, Walker, 1933, and others) gives an extensive discussion of distance determinations which he introduces:
``. . . . any specific astronomical measurement of `distance' . . . . carried out in any relativity model of space-time must lead to a result which depends on the particular operations of measurement.''
For small distances from the observer, z can be approximated by the relativistic or the classical Doppler formula, and the distance r is determined from v / H0.
Contributions to the observed redshift ztotal result from the warping of the smooth geodesic due to local mass concentrations:
Local effects on the light path contribute zl. Local effects, which can be described as peculiar motions of the emitting particle (galaxy) and the observer along particle geodesics, contribute zm. When they are sufficiently small, z and r can again be approximated by the classical or special relativistic Doppler formula. The superposition ztotal = zcosm + zl + zm makes it observationally difficult to separate the cosmological and the local contributions.