Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by . All rights reserved |

**4.3. The Predicted Hubble Diagram With Correction for Luminosity
Evolution**

In an evolving universe, the mean age of galaxies decreases with
increasing redshift simply because we sample earlier times as we look
out in distance. A first estimate of the expected change of E galaxy
luminosities with look-back time, based on the change of the turnoff
luminosity in the HR diagram for an old coeval population, gave a mean
evolutionary rate of
*L* ~ *t*^{-4/3} for a flat luminosity function at the
main sequence turnoff
(Sandage 1961b).
Because the luminosity function
is not flat but rises for faint magnitudes below the turnoff, this is
an *upper* limit, overestimating the luminosity rate by about 30%.

Call the change of magnitude due to evolution
*E*(*t*). Equation 33,
transformed to heterochromatic magnitudes, then becomes

(35) |

which is a basic equation that is used extensively in the following sections.

As for the size of *E*(*t*), the simple evolution rate for an
old coeval population of
*L* ~ *t*^{-4/3} quoted above would give a magnitude
variation of
mag = - 2.5
log *t*_{0} / *t*_{1}, where
*t*_{0} and *t*_{1} are the ages of the source at
light reception at the Earth and at light emission from the source,
respectively. If *t*_{0} = 15 × 10^{9} yr and
we inquire for the case of a look-back time of 10^{9} yr, then
mag = - 2.5
log(15/14)^{4/3} = 0.10 mag for small
*t*_{0}/*t*. The sense of the
correction is that galaxies were
brighter in the past. If this rate is ~ 30% too high
(Tinsley & Gunn 1976),
the rough estimate of *E*(*t*) is then ~ 0.07 mag per
10^{9} yr.

Elaborate calculations of *E*(*t*) form the subject of galaxy
evolution via stellar population synthesis, pioneered by
Tinsley (1968,
1972a,
b,
1976,
1977a,
b,
1980,
and references therein) and her
colleagues. The exact rate depends on the various assumptions of star
formation rates over time and on the slope of the main sequence
luminosity function. However, order-of-magnitude corrections,
changing *t* to *z* via equations in the next section, give
*m*
- 2.5 log(*l* +
*z*). For small *z* the correction again is
approximately 0.07 mag per 10^{9} yr on a time scale of
*t*_{0} ~ 15 × 10^{9} yr for the
age of the Universe-nearly the same as the early, quite elementary
estimates.

For *very* large redshifts, where the look-back times are of the order
of the age of the Universe, much more elaborate evolutionary models
are required than simple main sequence burn-down rates near the
present main sequence termination point. The philosophy by which the
rates can be calculated near the beginning of galaxy formation was
first set out by
Tinsley (1968).
Modern calculations include those of
Bruzual & Kron (1980),
Bruzual (1981,
1983a,
b), and
Arimoto & Yoshii (1986,
1987).
A summary of *E(t)* over the age range of 10^{7} to
1.5 × 10^{10} yr is given by Yoshii & Takahara
(1988,
their Figure 2) for E/S0 and Sdm galaxies in the *UBVRIJK*
photometric bands.