Annu. Rev. Astron. Astrophys. 1997. 35:
101-136
Copyright © 1997 by . All rights reserved |

**3.2. Situations Where the First or General Malmquist Bias Appears**

Let us consider a class of galaxies having a gaussian luminosity
function *G*(M_{p},
) acting in the role of
a standard candle. Such a class may be defined, e.g. via a fixed value
of the TF parameter *p* = log V_{max}, and it is assumed
that the calibration has been made, i.e. the volume-limited value of
M_{p}
is known. A sample of these galaxies with magnitudes measured up to some
limit gives us a collection of derived distance moduli *µ* = m
- M_{p}. If in addition the redshifts are available, one may
construct the *µ*-log V Hubble diagram, and try to determine
the Hubble constant H from the following expected relation:

(7) |

If the error in the distance modulus has a gaussian distribution
*G*(0,
), one can take simple
averages <log V> and <*µ*>, and solve for <log
H>. In reality, there is a systematic error in each subaverage
<*µ*_{i}> (*µ*_{i} = constant)
and this systematic error generally depends on the value of
*µ*_{i} if the space distribution of galaxies is not
uniform. The error
*µ*(*µ*_{i}) is given by
Malmquist's formula (Equation 3). One point should be especially
emphasized: The ratio *N*(*µ* +
b^{2}) /
*N*(*µ*) refers to the distance moduli in the ideal case
when the sample is not influenced by a magnitude limit, i.e. when
*N*(*µ*) contains information on the real space
distribution of the galaxies.

One implication of the general Malmquist bias is that the Hubble diagram does not necessarily show a linear Hubble law, because the bias depends on the real distribution of galaxies and is different for different (derived) distances. Also, in different directions of the sky, one may derive by the mentioned method different values of the Hubble constant, possibly interpreted, but incorrectly, to be caused by our own peculiar motion or by large scale streams of galaxies. Such explanations have been applied to the Rubin-Ford effect (Sandage & Tammann 1975a, b, Fall & Jones 1976, James et al 1991), the supergalactic anisotropy (Teerikorpi & Jaakkola 1977), and the Great Attractor velocity field (Landy & Szalay 1992).

The general Malmquist correction has in recent years
been discussed particularly in connection with the attempts to derive maps
of peculiar velocities or galaxy streams. The radial component of a peculiar
velocity for a galaxy with true distance r and observed correct radial
velocity V_{obs} is as follows:

(8) |

Let us look at galaxies in some direction, with their derived distances
within a narrow range r_{der} ± 1/2 *dr*.
Perhaps these galaxies are also in the real space so close by that they have
a common peculiar velocity (bulk flow or stream), though different
cosmological components. Then <V_{obs}> - H_{o}
^{.} <r_{der}> does not give the correct
peculiar velocity, unless a proper correction for
<r_{der}> is made, i.e. the Malmquist correction of the
first kind, which gives the correct average distance for the galaxies in
the r_{der} ± 1/2 *dr* range.

Lynden-Bell et al (1988) used such a correction in their Great Attractor paper, though they assumed a uniform distribution of galaxies. This means that for their diameter (vs velocity dispersion) indicator for elliptical galaxies, the bias formula is analogous to the classical Malmquist formula:

(9) |