Cosmological Malmquist bias in the Hubble diagram at high redshifts

The Malmquist bias in luminosity distances for gaussian standard candles is discussed within cosmological models where the Euclidean r3-law for volumes and r−2-law for fluxes is not valid. Furthermore, the influence of K-corrections and luminosity evolution are analyzed. It is noted that the usual way of comparing theoretical predictions and data points in the Hubble diagram (log z vs. m) should be modified in view of the cosmological Malmquist bias. When the space distribution of galaxies is uniform, the classical Malmquist bias is constant at all apparent magnitudes, which is no more generally true within uniform cosmological models. Especially, calculations are made in Friedmann models for standard candles with different gaussian dispersions σ around average absolute magnitude M0. The usual log z vs. m (or Mattig) relations are deformed by amounts depending on the Friedmann model itself, on σ, and on the apparent magnitude of the standard candle. The implications on estimations of q are shown to be significant when σ ≥ 0.3 mag. It is concluded that the cosmological Malmquist bias is a necessary part of the theory of gaussian standard candles at high redshifts. It is also emphasized that one should always consider two complementary aspects of the Hubble diagram as a cosmological test, i.e. the log z vs. m and m vs. log z approaches, the first one influenced by the bias here discussed, while the second one is plagued by the magnitude limit (Malmquist bias of the 2nd kind). For example, with σ = 0.5 mag, in the case of bolometric magnitudes, the traditional log z vs. m procedure in the brighter part (〈z〉 less than about 1.5) of the Hubble diagram, would make one believe that q0 = 0.25 when it actually is 0.5. Without evolution, but in the presence of K-effect typical for V-band and E-galaxies, one would derive q0 ≈ 0.1 in the case of q0 = 1.0 when the K-effect is simply put into the zero-dispersion theoretical curve. With a good standard candle having σ = 0.3, these results would change to q0 = 0.4 (instead of 0.5) and = 0.5 (instead of 1.0). We also discuss the bias in angular size distance, which is shown to work in a different sense than the bias in luminosity distance, and the deviation from the classical bias is large already well below the distance maximum in Friedmann models.