3.2 Basic parameters

3.2.1 R (t) - the scale factor, and its time derivatives

Friedmann (1922) has shown that in a non-stationary world Einstein's field equations, which tie geometry and matter together, relate the time-dependent scale factor R (t) and its first and second derivative to the mass density in the universe, the constant of gravity, the velocity of light and the cosmological constant. His equations

with

assume implicitly that the pressure p is zero. Later, these equations were extended to include negative and null curvature (Sect. 3.1.2).

Lemaître (1927) formulated the time dependence of the field:

``Space is homogeneous and has constant positive curvature; space-time is also homogeneous, for all events are perfectly equivalent. But the partition of space-time into space and time disturbs the homogeneity. The co-ordinates used introduce a centre. A particle at rest at the centre of space described a geodesic in the universe; a particle at rest otherwhere than at the centre does not describe a geodesic. The co-ordinates chosen destroy the homogeneity of the universe and produce the paradoxical results which appear at the so-called `horizon' of the centre. When we use co-ordinates and corresponding partition of space and time of such a kind as to preserve the homogeneity of the universe, the field is found to be no longer static: the universe becomes of the same form as that of Einstein, with a radius no longer constant but varying with the time according to a particular law.''

After introducing R as a function of time and its first and second derivatives, Lemaître arrives at the same expression for the field equations as Friedmann, including, however, the pressure term

where is again given by Einstein's expression. Lemaître continued:

``The four identities giving the expression of the conservation of momentum and of energy reduce to

which is the energy equation. This equation can replace [the above equation]. As V = ² R³, it can be written as

showing that the variation of total energy plus the work done by radiation-pressure in the dilatation of the universe is equal to zero.''

He summarized:

``It remains to find the cause of the expansion of the universe. We have seen that the pressure of radiation does work during the expansion. This seems to suggest that the expansion has been set up by the radiation itself. In a static universe light emitted by matter travels round space, comes back to its starting-point, and accumulates indefinitely. It seems that this may be the origin of the velocity of expansion / R which Einstein assumed to be zero and which in our interpretation is observed as the radial velocity of extra-galactic nebulae.''

The search for a relation between the expansion of the universe and energy available was continued by Lemaître (1931b). Another three years later, he offered a new physical explanation for the expansion (Sect. 4.1).

In his 1927 paper he also introduced the relation between redshift and variation of the scale factor:

``The relation between radial velocity (redshift) is also given:

is the apparent Doppler effect due to variation of the radius of the universe. It equals the ratio of the radii of the universe at the instants of observation and emission diminished by unity.''

Later it became evident that the first derivative measured in units of R is the increment H in the Hubble diagram or the Hubble parameter while the second derivative measured in units of H² is the acceleration parameter q. The symbols

were introduced by Robertson (1955) and Hoyle and Sandage (1956), respectively. The present values ⁽⁶⁾ are H₀ and q₀.

An attempt to measure the curvature radius of space was first made by Schwarzschild (1900):

``The question shall be discussed how large the curvature radius must at least be chosen . . . For both cases the elliptical and the hyperbolical space we will now discuss the problem of parallax determination.''

The method using parallaxes was employed not only by Schwarzschild but also, in the early-days of relativistic cosmology, by de Sitter (1917). In a more general way

distance/scale factor = f (distance measure, metric)

the method is still the only one available. It is essentially the measurement of a three-dimensional geometrical figure projected onto the surface of four-dimensional space. The area of the figure relates to the volume of a geodesic sphere and thus to its radius of curvature. The result is obtained through model fitting, so that the figure must be determined theoretically for comparison.

For space with a time-dependent scale factor, the relation between observational and theoretical quantities has been discussed in detail by Heckmann (1942). His photometric distance is

with R     = scale factor at the time of observation
z       = redshift
S () = distance r.

For small redshifts D is equal to the usual photometric distance. Distance r should be compared to the spatial distance defined by Whittaker (1931):

``The spatial distance of two material particles in a general Riemannian space-time may then be thought of as a relation between two world-points which are on the same null geodesic. It is obviously right that 'spatial distance' should exist only between points which are on the same null geodesic; for it is only then that the particles are in direct physical relation with each other. This statement brings out into sharp relief the contrast between `spatial distance and `interval' defined by

for between points on the same null geodesic, the `interval' is always zero. Thus 'spatial distance exists when, and only when, the `interval' is zero.''

A symbolic presentation is given in Fig. 25 taken from Heckmann (1942).

Figure 25. Light path on a geodesic (Heckmann 1942). ``Concerning the dependence of the apparent magnitude of a light source from its position on the space geodesic which connects it to the observer.''

z is directly related to the Hubble parameter H, the deceleration parameter q and the cosmological constant . m_N is related to the `angular' correction of magnitude m for distant objects. m_z is the luminosity correction.

For models with = 0 the dependence of apparent magnitude m on H₀, q₀ and z was given by Mattig (1958, Fig. 26).

Figure 26. Mattig's (1958) presentation of magnitude correction m as function of z and q0.

A recent revival of the idea of measuring the Riemann curvature k / R² in a universe with = 0 or is due to Ehlers and Rindler (1987). Through a combination of m (z) and N (m) measurements, the numerical values of the cosmological parameters can be determined, as is apparent from Heckmann's equations.

⁷ Series expansions were used at least since 1931 (Lemaître) to relate observable quantities and cosmological parameters. Heckmann's presentation is particularly transparent because H and q appear explicitly. Back.