*Comparison of Observations with Theory*

Now let us return to the surveys, and reduce them
all to the epoch, *now*, in accordance with the principles
of relativistic cosmology. We wish to know the relative
numbers of nebulae which an observer, in an expanding
universe, would count to successive limits of apparent
faintness. The problem is intricate but it has been
thoroughly investigated, and the necessary formula is
available in quite simple terms.
^{4} Actually, the expression is
just that previously derived for uniform distribution in a stationary
universe, plus two extra terms.
One of the terms represents the recession factor, the
other represents effects of spatial curvature.

If the use of a logarithm is permitted, the situation may be clearly represented by a pair of equations. If nebulae are uniformly distributed through a non-expanding universe in which red-shifts are not primarily velocity-shifts, then the numbers should be proportional to the volumes, and the surveys should conform (and actually do conform) with the relation

where *m*_{c} is the limiting faintness expressed as a
magnitude,
corrected for local obscuration and for the
energy-effects required by the mere presence of redshifts.
^{5} The corresponding relation
for a homogeneous, expanding universe, obeying the relativistic laws of
gravitation, is

where *d* /
is the recession
factor and *C*_{v} is
the effect of spatial curvature. We wish to know whether or not
the surveys can be fitted into the latter expression.

If both of the extra terms (for recession and for curvature) were absent, the surveys would clearly fit the formula because the situation would be precisely that in a stationary universe. Now suppose we introduce only one of the extra terms, namely, the recession factor. In this way we pass from a stationary universe to an expanding universe with negligible curvature, but we destroy the agreement with the observations. The distribution is no longer uniform. The recession factors introduce departures from uniformity in the law of distribution, just as they introduced departures from linearity in the law of red-shifts.

^{4} The requisite formulae have been
derived by various
investigators. Those used in the present discussion were developed by
R. C. Tolman
(*Relativity, Thermodynamics, and Cosmology*,
Clarendon Press, (1934) and adapted to the specific problem of the
surveys by Hubble and Tolman (Mt. Wilson Contr., No. 527;
*Astrophysical Journal*, 82, 302, 1935)
Back.

^{5} The derivation is as follows. For
uniform distribution numbers
of nebulae *N* are proportional to volumes of space, and,
consequently, to
the cubes of the limiting distances r to which the counts are
carried. Hence log_{10} *N* = 3 log_{10} *r* +
constant. Among objects of the
same candle-power the distances are proportional to the inverse square
of the apparent luminosity *l*. Hence
log_{10} *r* = constant - 0.5 log_{10} *l*.
Apparent magnitudes *m* measure apparent luminosities on a logarithmic
scale. By definition,
*m* = constant - 2.5 log_{10} *l*. Hence

and