Derivation of the cosmological number density in depth from V/Vm distribution

The classical cosmological V/Vm-test is introduced. Use of the differential distribution p(V/Vm) of the V/Vm-variable rather than just the mean <V/Vm> leads directly to the cosmological number density without any need for assumptions about the cosmological evolution of the underlying (quasar) population. Calculation of this number density n(z) from p(V/Vm) is illustrated using the best sample that was available in 1981, when this method was developed. This sample of 76 quasars is clearly too small for any meaningful results. The method will be later applied to a much larger cosmological sample to infer the cosmological number density n(z) as a function of the depth z. Keywords: V/Vm . luminosity volume . cosmological number density . V/Vm distribution

💡 Research Summary

The paper revisits the classic V/Vm test, a statistical tool originally devised to assess the spatial uniformity and evolutionary behavior of astronomical source populations such as quasars. In its traditional form the test relies on the mean value ⟨V/Vm⟩, which should be 0.5 for a non‑evolving, uniformly distributed sample. However, the mean discards the detailed shape of the V/Vm distribution, making it difficult to separate genuine cosmological evolution from selection effects.

The author proposes to use the full differential distribution p(V/Vm) instead of just its mean. By calculating, for each object, the observable volume V (the comoving volume out to its redshift) and the maximum volume Vm in which the object could still be detected given the survey’s flux limit, one obtains a set of V/Vm values. These values are then binned and smoothed to produce an empirical p(V/Vm). The key theoretical step is to relate p(V/Vm) to the cosmological number density n(z) through an integral equation that involves only the comoving volume element dV/dz and the selection function implicit in Vm. Explicitly,

n(z) = dN/dz = ∫₀¹ p(x) ·