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
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).
the subscript 0 indicates a fixed time.