Malmquist Bias - Applications

Applications

Anytime a magnitude-limited sample is used, one of the methods described above should be used to correct for the Malmquist bias. For instance, when trying to obtain a luminosity function, calibrate the Tully-Fisher relation, or obtain the value of the Hubble constant, the Malmquist bias can strongly change the results.

The luminosity function gives the number of stars or galaxies per luminosity or absolute magnitude bin. When using a magnitude-limited sample, the number of faint objects is underrepresented as discussed above. This shifts the peak of the luminosity function from the faint end to a brighter luminosity and changes the shape of the luminosity function. Typically, the volume weighting method is used to correct the Malmquist bias so that the survey is equivalent to a distance-limited survey rather than a magnitude-limited survey. The figure to the right shows two luminosity functions for an example population of stars that is magnitude-limited. The dashed luminosity function shows the effect of the Malmquist bias, while the solid line shows the corrected luminosity function. Malmquist bias drastically changes the shape of the luminosity function.

Another application that is affected by the Malmquist bias is the Tully-Fisher relation, which relates the luminosity of spiral galaxies to their respective velocity width. If a nearby cluster of galaxies is used to calibrate the Tully-Fisher relation, and then that relation is applied to a distant cluster, the distance to the farther cluster will be systematically underestimated. By underestimating the distance to clusters, anything found using those clusters will be incorrect; for example, when finding the value of the Hubble constant.

These are just a few examples where the Malmquist bias can strongly affect results. As mentioned above, anytime a magnitude-limited sample is used, the Malmquist bias needs to be corrected for. A correction is not limited to just the examples above.