4. Calculation of at
When one looks at the Hubble diagram as against m, and tries to compare the observed distribution of data points with some theoretical scheme, the basic task is calculation of the average value of at constant value of m. Already the classical Malmquist bias makes one anticipate that at m is not the same as , where has been solved from the Mattig equation (corresponding to an ideal standard candle with a zero-dispersion LF).
4.1. Basic formula
In this section we study the case of bolometric magnitude, ignoring any complications due to K-effect. It is simply assumed that in the observed sample, the galaxies at constant m, are found in space distributed according to z as expected on the basis of the co-moving volumes corresponding to dz and the space density of galaxies having absolute magnitude . It is useful to emphasize that a completeness in m is not required. It is, of course, not necessary that the sample contains all the existing standard candles in the sky having apparent magnitude . However, one must assume that the available sample galaxies at m faithfully follow the underlying z-distribution so that no selection effect distorts the distribution of z.
Denoting by µ, the average value of for a fixed m can be calculated as:
In this expression, , and the derivative was given by Eq. (9).
In the displayed examples, we use two values of the dispersion for a standard candle: 0.3 mag and 0.5 mag. Of these, 0.3 mag may be regarded in practice as a very good one. It is assumed that and km s-1 Mpc-1, and some results of calculation have been given for Friedmann models (of zero cosmological constant) with 0.25, 0.5 and 0.75.
Figs. 4-5 show the difference between the actual and the predicted by the Mattig vs. m relation which implicitly presupposes that the dispersion is zero. At zero redshift the difference is normalized to be zero by adding to the Mattig curve the classical Malmquist term : at small z the calculation correctly reproduces the classical Malmquist bias. Note that in practice, when one is not interested in the volume-limited average absolute magnitude of the standard candle, this adjustment has been left for Nature to do.
Conversely, one may read from Figs. 4-5 (by changing the sign) the correction which must be added to at a given m if one wishes to compare a Hubble diagram for bolometric magnitudes with the predictions made by the Mattig curves. The correction depends on the correct Friedmann model itself (, , and more generally, ) and on the correct parameters of the standard candle (, ). In fact, because and does not appear in the volume ratios, the correction at m is the same for any such choice of and giving fixed constant.
Let us consider only the case where standard candles may be observed in sufficient numbers already at small redshifts. There , and the zero point is obtained without a separate determination of the distance scale () and the parameter. At small redshifts also the dispersion may be determined. Hence, in such a case an important feature of the standard classical Hubble diagram test is preserved: one does not need to know the value of .
If one takes another value of , e.g. another standard candle class (keeping and fixed) which is one magnitude brighter, then the corrections from Figs. 4-5 are valid for this class when one reads them at instead of m (the intrinsically brighter class extends as deep into the redshift space at brighter apparent magnitude). Because the Malmquist correction to depends on the Friedmann model, one needs a procedure where the corrections are calculated for a range of if one wishes to derive the value of from a high-z Hubble diagram.
It is illuminating to show in Figs. 4-5 also the curve for the apparent, erroneous value of which roughly "predicts" the run of at brighter magnitudes, again on the basis of the zero-dispersion Mattig curve.
The results are such that the apparent value of goes down from the true one, and this effect is stronger for larger dispersion . With a very good standard candle with , one would derive by the normal procedure from the Hubble diagram up to about , that when it is actually 0.75. With , the inferred value would drop down to 0.45 - 0.5, while would give 0.2 - 0.3 (not displayed). If we are actually living in the Eistein-de Sitter universe (), then would give that apparently , while would perhaps make one believe that . If , then would produce an apparent (not displayed).
We emphasize that these results are concerned with observations below bolometric (where or with the adopted and ). At fainter magnitudes (for such a standard candle) the behaviour of may not be described by a unique apparent . In the next section it is shown that an improper treatment of the K-effect may substantially still enhance this source of error.
© European Southern Observatory (ESO) 1998
Online publication: October 22, 1998