5.1 The role of data in cosmology
The cosmological principle permits only global actions of uniformly distributed matter in order that a unique scale factor R (t) of the universe may exist (Sect. 3.1.1). Einstein's equations, derived under these assumptions, include general expansion and combined gravitational attraction, i.e: coordinated motion and acceleration. According to Lemaître, the cosmological constant is linked to expansion; acceleration is governed by the presence of matter. Einstein's equations, in the form first presented by Friedmann, include only the first and second time derivatives of R (t).
Severe problems arise for the comparision of theory with observational data. The simple theory based on the cosmological principle (Friedmann-Lemaître-Robertson-Walker metric) does not offer an algorithm which provides the general inclusion of local effects; and observers have largely contented themselves with simple corrections to data obtained from comparatively simple structures (such as rich clusters of galaxies). Now, we begin again to realize, how difficult it is, to derive global information using data which are (or at least may be) coarsely affected by local phenomena on all scales reached so far.
The situation is well summarized by Ehlers (1988):
``The actual universe is inhomogeneous. In theory, one tries to account for this by adding perturbations to the homogeneous isotropic background and matter variables (density, velocity, . . .).
Observational data refer, of course, to the real inhomogeneous universe. In order to derive from these data the global parameters of cosmological models, one has to apply corrections to account for local effects. This procedure is perhaps not yet well understood.
For a detailed discussion see, e.g., `Relativistic cosmology: its nature, aims and problems' (Ellis 1984).''
In practice, it requires assumptions to be made of the numerical values of the cosmological parameters, i.e. of the world model, which is to be derived from the observations.
Strict recurrence to the data is advocated by Milne (1934):
``Now what Dr. Hubble, Dr. Shapley and their co-workers actually observe may be described as follows. A certain area on a photographic plate is taken, representing a certain solid angle in the sky, and attention is fixed on a number of small nebulous patches and their spectra. For each patch its Doppler shift s and apparent brightness b are measured, and the patches are counted. Independent of all conventions as to distance, velocity, etc., the observations give firstly the behaviour of b as a function of s: secondly the number of patches n (s) ds with Doppler shifts lying between s and s + ds: thirdly, some idea of the stage of evolution of the patch of Doppler shift s - all all at a given epoch of observation. Fourthly, in principle, if observations could be carried out through long intervals of time, they would give the variation of s with epoch of observation t for a given nebulous patch, s = f (s0, t), and the functional dependence of b (s) and n (s) on t, which we may accordingly write as b (s, t) and n (s, t). Every solution of the cosmological problem, every world-model, predicts in principle the smoothed-out values to be expected for f (s0, t) for a given patch, and the brightness and distribution functions b (s, t) and n (s, t) for different patches. Two theories differ when their predictions of these functions differ. This method of comparison avoids all reference to distance-assignments, world-geometry, schemes of projection or the like.''