3.1.4 Look-back time

Figure 24, taken from de Sitter (1930), shows an early, probably the first, diagram of the relation between world radius R, represented by the dimensionless quantity z, and time t, represented by the dimensionless quantity - ₀ (see Sect. 3.1.1). De Sitter's definitions are:

and

with = wavelength of radiation
c = velocity of light
= Lemaitre's cosmological constant in unit length^-2,
the subscript 0 indicates a fixed time.

With the appropriate choice of R₀, e.g. the present scale factor, an infinite scale factor is reached asymptotically in a finite interval - ₀, corresponding to infinite time, provided that the cosmological constant is a function of time (see Sect. 3.2.4). The curves labeled I-VII represent different world models.

Figure 24. ``Relation between z and - 0. The vertical coordinate is z, the horizontal coordinate is - 0.'' (de Sitter 1930)

With reversed scales, taking again the present for reference, one finds that R approaches 0 asymptotically in a finite interval - ₀. In this form the diagram displays the look-back time for various models. Different scaling permits to display the past behaviour of different models in the form that is frequently used today (e.g. Tinsley 1968).