Theoretical aspects of the inverse Tully-Fisher relation as a distance indicator: incompleteness in log VMax, the relevant slope, and the calibrator sample bias

We study the influence of the assumption behind the use of the inverse Tully-Fisher relation: that there should be no observational cutoffs in the TF parameter log VM. It is noted how lower and upper cutoffs would be seen in a logH0 vs. “normalized distance” diagram. Analytical expressions, under the simplifying assumption of a normal distribution and the use of the correct TF slope, are derived for the resulting biases, especially the average bias which log VM cutoffs produce in the derived value of H0. This bias is shown to be relatively weak, and as such cannot explain the large differences in the reported values of H0 derived from direct and inverse TF relations. Some problems of slope and calibration are shown to be more serious. In particular, one consequence of fitting through the calibrators either the slope relevant for field galaxies or the steeper slope followed by calibrators is that the derived value of the Hubble constant comes to depend on the nature of the calibrator sample. If the calibrator sample is not representative of the cosmic distribution of log VM, large errors in the derived value of H0 are possible. Analytical expressions are given for this error that we term the calibrator sample bias.