Astron. Astrophys. 339, 647-657 (1998)

## 3. Volumes probed by galaxies at

If M for a standard candle has a gaussian distribution around , galaxies with and observed to have apparent magnitude m, must lie at z given by . Galaxies from the luminous wing of the LF and also having apparent magnitude m, originate from the volume beyond , while those from the fainter wing come from the foreground volume.

In the classical case, one has instead of redshift the usual Euclidean distance r and the distance modulus has the familiar connection with r. Then in the case of a uniform space distribution, the background and foreground volumes defined by galaxies having, say, and and falling on the magnitude interval , are different (see Fig. 1). Their ratio is a useful indicator of the Malmquist bias. Classically, it has the constant value:

In Friedmann models, such foreground and background volumes must be defined as co-moving volumes.

 Fig. 1. Classical Malmquist Bias: Symmetrical parts of the gaussian luminosity function, observed through the magnitude window [m-1/2dm, m+1/2dm], originate from different foreground and background volumes as determined by the -law of Euclidean volumes and the -law of fluxes.

In order to get at once a feeling of how the classical case is deformed when one considers a Friedmann model, we calculate the mentioned " volume ratios" for a few in the Einstein-de Sitter model and with two other models ( 0.05 and 1.0), and compare these with the classical ratio (that depends only on ).

Suppose that the dispersion of the standard candle is . Then galaxies symmetrically around , at and , are found at redshifts and () (see Fig. 2). These redshifts are determined by the requirements:

The desired ratio of the volumes is then

In Table 1 the volume ratios are given for 0.001, 0.1, 0.5, 1.0, 1.5, and 2.0, when 0.5 mag. The corresponding classical ratio for 0.5 mag is = 1.995, close to the calculated volume ratio for in all Friedmann cases. We shall return to the last columns of Table 1 in Sects. 5 and 7 where the influence of the K-correction and non-zero cosmological constant are respectively discussed.

 Fig. 2. Cosmological Malmquist Bias in Friedmann models: Symmetrical parts of the gaussian luminosity function, observed through the magnitude window , originate from different comoving volumes. Fundamental theory gives the change of M with redshift z (together with the K-term K(z)) and the value of comoving volume, allowing one to calculate the average value of logz for standard candles with apparent magnitude m.

Table 1. volume ratios

At increasing redshifts the ratios get progressively smaller which means that the Malmquist bias decreases (the volumes where the bright and faint Gaussian wings are sampled differ less). It is useful to inspect this table when one formally identifies redshift z with distance. Then the case has the same dependence of m on z as classically m has on distance r, and the strong dependence of the volume ratio on z hence reflects directly the "geometry" of such a z-space. For smaller , both dependence and geometry change, their effects working in opposite directions, and the net effect is that the volume ratios differ a little less from the classical one, in comparison with . We illustrate the relevant trends in Fig. 3, where 3a shows the luminosity distance versus z, 3b shows the angular size distance versus z, and 3c shows the volume derivative versus z, for a few Friedmann models. In all diagrams, we have also indicated the relation for the "classical" case, where Euclidean distance.

 Fig. 3. These diagrams show how luminosity distance , angular size distance (both in unit of ), and derivative of comoving volume (per unit solid angle) depend on redshift z, for Friedmann models with 0.05, 0.5, and 1.0 (). "Eucl" means the case when z is proportional to distance in an Euclidean space - it forms an instructive comparison to what happens in a Friedmann universe. E.g., is closest to Euclidean as regards the luminosity distance, but farthest from Euclidean as regards the volume derivative.

© European Southern Observatory (ESO) 1998

Online publication: October 22, 1998