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Register a new userCosmological number density n(z) in depth z from p(V/Vm) distribution
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Using distribution p(V/Vm) of V/Vm rather than just mean <V/Vm> in V/Vm-test leads directly to cosmological number density n(z). Calculation of n(z) from p(V/Vm) is illustrated using best sample (of 76 quasars) available in 1981, when method was developed. This is only illustrative, sample being too small for any meaningful results. Keywords: V/Vm . luminosity volume . cosmological number density . V/Vm distribution
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Using distribution p(V/Vm) of V/Vm rather than just mean <V/Vm> in V/Vm-test leads directly to cosmological number density n(z). Calculation of n(z) from p(V/Vm) is illustrated using best sample (of 76 quasars) available in 1981, when method was developed. This is only illustrative, sample being too small for any meaningful results. Keywords: V/Vm . luminosity volume . cosmological number density . 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
The classical cosmological V/Vm-test is introduced and elaborated. 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
Banhatti (2009) set down the procedure to derive cosmological number density n(z) from the differential distribution p(x) of the fractional luminosity volume relative to the maximum volume, x equiv V/Vm (0 leq x leq 1), using a small sample of 76 quasars for illustrative purposes. This procedure is here applied to a bigger sample of 286 quasars selected from Parkes half-Jansky flat-spectrum survey at 2.7 GHz (Drinkwater et al 1997). The values of n(z) are obtained for 8 values of redshift z from 0 to 3.5. The function n(z) can be interpreted in terms of redshift distribution obtained by integrating the radio luminosity function {rho}(P, z) over luminosities P for the survey limiting flux density S0 = 0.5 Jy. Keywords. V/Vm - luminosity-volume - cosmological number density - redshift distribution - luminosity function - quasars [Note: This somewhat modified version was submitted to MNRaS on 14 July 2016. It was (almost) rejected, except if thoroughly revised.]
The scientific impact of current and upcoming photometric galaxy surveys is contingent on our ability to obtain redshift estimates for large numbers of faint galaxies. In the absence of spectroscopically confirmed redshifts, broad-band photometric redshift point estimates (photo-$z$s) have been superseded by photo-$z$ probability density functions (PDFs) that encapsulate their nontrivial uncertainties. Initial applications of photo-$z$ PDFs in weak gravitational lensing studies of cosmology have obtained the redshift distribution function $mathcal{N}(z)$ by employing computationally straightforward stacking methodologies that violate the laws of probability. In response, mathematically self-consistent models of varying complexity have been proposed in an effort to answer the question, What is the right way to obtain the redshift distribution function $mathcal{N}(z)$ from a catalog of photo-$z$ PDFs? This letter aims to motivate adoption of such principled methods by addressing the contrapositive of the more common presentation of such models, answering the question, Under what conditions do traditional stacking methods successfully recover the true redshift distribution function $mathcal{N}(z)$? By placing stacking in a rigorous mathematical environment, we identify two such conditions: those of perfectly informative data and perfectly informative prior information. Stacking has maintained its foothold in the astronomical community for so long because the conditions in question were only weakly violated in the past. These conditions, however, will be strongly violated by future galaxy surveys. We therefore conclude that stacking must be abandoned in favor of mathematically supported methods in order to advance observational cosmology.
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