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

1. Introduction

In the treatment of the classical Malmquist (1920) bias, the space is assumed to be Euclidean and transparent. Furthermore, the standard candles (with a gaussian distribution of absolute magnitudes) do not change with lookback time nor have any K-effect. In modern cosmological models these assumptions are not valid, when e.g. Supernovae of type Ia are detected up to and radio galaxies and quasars well beyond . As the Hubble diagram is usually looked at as vs. m and the theoretical predictions are compared with vs. m, it is important to study how the non-classical situation influences the run of the standard candle data points in a Hubble diagram and how one should compare observations and model curves.

1.1. Malmquist biases of the first and second kind

Actually, there appears in the literature two ways of looking at the Hubble diagram - both at small redshifts where one studies the Hubble constant and large scale galaxy streams, and also at high redshifts where one has in mind applications on the value of and on evolution of galaxies. These two ways, either considering vs. m or vs. for a gaussian standard candle, are respectively accompanied by two distortion effects, Malmquist bias of the 1st kind and Malmquist bias of the 2nd kind (see Teerikorpi 1997 where this terminology is suggested in order to unify the rather varied practice). The 1st bias is closely related to the classical Malmquist bias, while the 2nd bias arises due to a magnitude limit cutting away parts of the fainter wing of the luminosity function (LF) of the standard candle.

These two different biases are often called in the contemporary literature as "Malmquist bias" and "selection bias", respectively, with the understanding that the Malmquist bias is basically caused by the geometry or space distribution of the standard candles, while the selection bias refers to the influence of observational (e.g. magnitude) limits. Concrete examples of how these two types of biases are discussed and treated in the study of the local galaxy universe, one may find e.g. in Willick (1994), Freundlich et al. (1995), Gonzales & Faber (1997), and Theureau et al. (1997). Sandage (1995) gives a very illuminating discussion of how these biases are related to each other in the classical space, uniformly filled by galaxies.

It is useful to state short definitions of these two types of biases, analogous to Sect. 3.1 in Teerikorpi (1997), but here in terms of and m, directly applicable to a Hubble diagram, instead of true and derived distance moduli. Consider a standard candle with (bolometric) average absolute magnitude . Within the cosmological model in question, there is an exact relation between apparent magnitude m of the candle having and redshift z:

This may be converted to give

Assume that the candles have been selected from the sky only on the basis of apparent magnitude so that there is no selection, say, discriminating against some values of z. For the sample galaxies within there is the average .

Then the Malmquist bias of the first kind is

and the Malmquist bias of the second kind is

Note that the first bias is concerned with the average at a fixed value of apparent magnitude m, as compared with the prediction from the exact relation . The second bias is concerned with the average apparent magnitude at some constant redshift. At small redshifts, where and const, these definitions go over to the usual classical ones.

These two biases have been rather extensively discussed in connection with the Hubble diagram at small redshifts, in the local galaxy universe (see the above references), while they have received less attention at high redshifts. Perhaps one reason is that the classical Hubble magnitude-redshift cosmological test has lost much of its attraction when one has encountered formidable problems trying to use evolving galaxies as standard candles, the uncertain evolution apparently dominating over other effects. When the Hubble diagram is presented, either as vs. m or m vs. , it has been common to display, in the frame of Friedmann models, the classical Mattig (1958) curves. Such curves give the connection between redshift and apparent (bolometric) magnitude for ideal standard candles with exactly the same absolute magnitude (see Sandage (1995) for an interesting discussion of the Mattig equation). Use of the Mattig curves is justified in the m vs. approach, if there is no Malmquist bias of the 2nd kind or the selection bias, because then for a gaussian standard candle really follows the exact Mattig curve (if one can ignore any scatter in z). Sometimes, but not at all always, the selection bias has been discussed in the m vs. approach, while the bias of the 1st kind has been generally ignored in the vs. m approach where it inevitably appears. Its omission actually is equivalent to the assumption that the bias is constant at all m, as is true classically (when the space distribution of standard candles is uniform).

1.2. log z vs. m and m vs. log z approaches

The traditional vs. m approach, instead of vs. , is often preferable, because in the latter case one encounters problems with the magnitude limit, or more generally, the progressively increasing incompleteness (Malmquist bias of the 2nd kind) which may be difficult to model. This happens just there, i.e. at high redshifts, where one also expects test information about the world model. In the vs. m approach, the magnitude limit or observational incompleteness in m does not matter, and the relevant effects are directly related to either the world model or to intrinsic properties of the standard candles. Of course, there may be cases when the magnitude limit problem is not serious, as has been stated by Perlmutter et al. (1997) in their study of the Hubble diagram of Supernovae. Then the vs. m approach should give a consistent result.

It is important to note that the alternative approaches give complementary cosmological test information. As a simple example, one may have obtained the "classical" relation const. from m versus for a standard candle. Then, without any knowledge of Friedmann models or physics of redshift, one might conclude that redshift z is just proportional to Euclidean distance in a static universe. In a homogeneous universe this conclusion may be tested from versus m. In the Euclidean case, one would thus recover the expected relation const', where const' includes a constant Malmquist term. However, in a Friedmann universe, versus m would not give directly , but only if one takes into account the non-classical Malmquist bias.

When we speak in this paper about the Hubble diagram, we have in mind especially the traditional vs. m approach. Consequently, we are concerned with the Malmquist bias of the first kind, which we call the cosmological Malmquist bias, in distinction from the classical bias .

1.3. Aim and contents of the paper

We describe in the present paper the basic ingredients of the cosmological Malmquist bias (of the 1st kind) and how its amount is calculated in a given situation. Primarily, the intention is to show how much the omission of the cosmological Malmquist bias influences analysis of the Hubble diagram and encourage its consideration in future studies of observational data.

We study the cosmological generalization of the Malmquist bias step by step, starting from the pure (non-classical) volume and flux effects in the case of bolometric magnitudes, in order to see clearly the difference to the classical bias (Sects. 2 - 4).

Then it is emphasized in Sect. 5 that one cannot simply "shift the curve" in order to include the K-effect (or the redshift dependent luminosity evolution). In Sect. 6, the analogous problem with angular size is shortly discussed, and Sect. 7 illustrates how inclusion of the cosmological constant influences the situation. The present discussion is concerned with the Friedmann model, and one should note that any other model needs its own detailed treatment. However, the basic ideas are general.

© European Southern Observatory (ESO) 1998

Online publication: October 22, 1998
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