**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

sand apparent brightnessbare measured, and the patches are counted. Independent of all conventions as to distance, velocity, etc., the observations give firstly the behaviour ofbas a function ofs: secondly the number of patchesn (s) dswith Doppler shifts lying betweensands + ds: thirdly, some idea of the stage of evolution of the patch of Doppler shifts- 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 ofswith epoch of observationtfor a given nebulous patch,s = f (s_{0},t), and the functional dependence ofb (s)andn (s)ont, which we may accordingly write asb (s, t)andn (s, t). Every solution of the cosmological problem, every world-model, predicts in principle the smoothed-out values to be expected forf (s_{0},t) for a given patch, and the brightness and distribution functionsb (s, t)andn (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.''