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

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

Dilip G. Banhatti
School of Physics, Madurai Kamaraj University, Madurai 625021, India

Abstract. 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

Introduction
A celestial source of isotropic luminosity L at the distance r has the observed flux density
S = L / 4.π.r2. Using a telescope of detection limit S0, this source can be observed out to a
maximum distance rm given by S0 = L / 4.π.rm2. We can associate two volumes with the
source: the volume V = 4.π.r3 / 3 actually “occupied” by the source, and the maximum
“luminosity-volume” Vm = 4.π.rm3 / 3 that the source could occupy and still be detected
by the telescope at its detection limit S0. The variable x ≡ V/Vm = (r / rm)3 characterizes
the fraction of available volume occupied by the source: 0 ≤ x ≤ 1. If the observer is
surrounded by a distribution of celestial sources which has uniform density per unit
volume relative to r, then x or V/Vm is uniformly distributed on [0, 1]. Conversely, a
uniform V/Vm-distribution implies a uniform number density (per unit volume) as a
function of the distance r from the observer. Testing this for a given sample of N celestial
sources may be called the luminosity-volume or V/Vm-test, although historically only the
mean <V/Vm> and the standard deviation σ<V/Vm> of the mean were tested against the
population mean <V/Vm>pop = ½ and σ<V/Vm> = 1 / √(12.N) (Schmidt 1968, 1978,
Schmidt et al 1988, Lynds & Wills 1972, Lynden-Bell 1971, Schmitt 1990).

The Uniform Random Variable
In general, for a continuous random variable x, uniform on [0,1], <x> = ½, σx2 = 1/12,
and σ<x>2 = 1 / (12.N) for a sample of size N. The mean <x> of a sample of N instances is
an unbiased estimate of <x>pop within σ<x> = 1 / √(12.N) with probability 68%, since <x>
is normally distributed to a very good approximation..

Luminosity-distance and Volume
For cosmological populations of objects (galaxies, galaxy clusters, radio sources, quasars,
γ-ray sources, …) the distance measure r must be replaced by the luminosity-distance
ℓ(z), and is a function of the redshift z of the object. Similarly, the volume of the sphere
passing through the object and centered around the observer is (4.π / 3).v(z) rather than
(4.π / 3).r3. Both ℓ(z) and the volume v(z) are specific known functions of z for a given
cosmological or world model.

The luminosity-volume Test
In general, the (monochromatic) luminosity-distance ℓν(z) depends on z through the
spectral shape of the (radio) source since the redshift (by definition) shifts light from
higher to lower frequencies ν. For the (radio) sources (quasars) generally used for such
cosmological investigations, the spectral shape is roughly parametrized by the negative
slope -α of a power-law between Sν and ν (α ≡ - dlog Sν / dlog ν or Sν proportional to ν-α).
In the simplest case, a (radio) source of a given (monochromatic) luminosity Lν appears
to a fixed observer to become monotonically fainter to zero flux density as it is taken
farther and farther away to infinity.

Calculation of the Limiting Redshift zm
For a (radio) source of (monochromatic radio) luminosity Lν, monochromatic flux density
Sν, (radio) spectral index α, and redshift z, Lν = 4.π.ℓν2(α, z).Sν. For a survey limit S0, the
value(s) of zm is (are) given by ℓν2(α, z) / ℓν2(α, zm) = S0 / Sν ≡ s, 0 ≤ s ≤ 1, for a source of
redshift z and spectral index α. This becomes clear on writing the Luminosity Lν in terms
of S0 and zm as Lν = 4.π.ℓν2(α, zm).S0, and comparing or identifying the two expressions
for Lν. For simplicity, restrict attention to only those cosmological models in which
[ℓν(α, z) / ℓν(α, zm)]2 = s has a single finite solution zm for given α, z and Sν, S0, for the
(radio) source under consideration. In other words, we restrict to those models for which
ℓν(α, z) is monotonic increasing with z, and ℓν(α, 0) = 0 & ℓν(α, ∞) = ∞. For a sample of
flux density limit S0, choosing sources of constant zm means, for the same α, choosing
constant ℓν(α, zm) = (Sν / S0)1/2. ℓν(α, z), which is proportional to Lν1/2. Then, for different
values of α, this amounts to choosing different Lν(α). We consider the function ℓν(α, z) in
the world model q0 = σ0 = 1/2, k = λ0 = 0 in von Hoerner’s (1974) notation, which may be
called the (1/2, 1/2, 0, 0) model, and is also known as Einstein-de Sitter cosmology.

Relating n(z) to p(V/Vm)
With a large enough flux density-limited deep sample, one may select (radio) sources
within a narrow range of zm, and still have sufficient number to determine the number
density n(z) from the differential distribution p(x) or p(V/Vm). Until such very large and
deep samples are available, sources of different zm must be combined together to get a
large enough sample to derive n(z) sensibly (Kulkarni & Banhatti 1983, Banhatti 1985).
Let N(zm).dzm represent the number of (radio) sources of limiting redshifts between zm
and zm + dzm in the sample being considered, which covers solid angle ω of the sky, so
that 4.π.N(zm) / ω is the total number of sources of limit zm per unit zm-interval. Since the
total volume available to sources of limit zm is V(zm) = (4.π / 3).(c / H0)3.v(zm), (where the
speed of light c and the Hubble constant H0 together determine the linear scale of the
universe,) the number of such sources (per unit zm-interval) per unit volume is
{3.N(zm) / ω}.(H0 / c)3.(1 / vm), where vm ≡ v(zm). Denote by nm(zm, z) the number of
sources / unit volume / unit zm-interval at redshift z. Then, n(z) ≡ ∫z∞ dzm. nm(zm, z), and
nm(zm, z) = {3.N(zm) / ω}.(H0 / c)3.(1 / v(zm)).pm(v(z) / v(zm)) for 0 ≤ z ≤ zm, where pm(x)
is the (differential) distribution of x ≡ V/Vm for a given zm. For z > zm, nm(zm, z) = 0,
since the sources with limiting redshift zm cannot have z > zm (for the type of
cosmological model we are considering, viz, with ℓν(α, z) monotonic increasing from 0
(at z = 0) to ∞ (at z = ∞)). To get the total n(z) for all zm-values, integrate over zm:
n(z) = {3 / ω}.(H0 / c)3. ∫z∞ dzm.( N(zm) / v(zm)).pm(v(z) / v(zm)).

The Scheme of Calculation
The upper limit ∞ for zm in this integral is illusory, since for a real sample, however deep
and large, there will be a maximum zmax for the limiting redshift zm. Consequently,
n(zmax) = 0. In fact, the lifetimes of individual (radio) sources will come into the
calculation, as well as the galaxy / cluster / structure-formation epoch at some high
redshift (say, > 10). Thus, any such n(z) calculation will give useful results only upto a
redshift much less than zmax. Formally writing zmax instead of ∞ for the upper limit,
n(z) = {3 / ω}.(H0 / c)3. ∫zz_max dzm.( N(zm) / v(zm)).pm(v(z) / v(zm)) for 0 ≤ z ≤ zmax.
To apply to real samples, this must be converted into a sum. To this end, divide the zm-
range 0 to zmax into k equal intervals, each = zmax / k = ∆z. The mid-points are
zj = (j – ½).∆z = {(j – ½) / k}.zmax. Calculate n(z) at these points: n(zj). Converting the
integral into a sum,
(ω / 3).(c / H0)3.n(zj) = ∑i=jk {Ni / v(zi)}.pi(xij), where xij = v(zj) / v(zi). (1)

It is often more useful / appropriate to use Zm = ln zm as the redshift variable. The
integral and the corresponding sum are then:
n(z) = {3 / ω}.(H0 / c)3. ∫zz_max dZm.(zm.N(zm) / v(zm)).pm(v(z) / v(zm)) for 0 ≤ z ≤ zmax, and
(ω / 3).(c / H0)3.n(zj) = ∑i=jK {zi.Li / v(zi)}.pi(xij), where xij = v(zj) / v(zi). (1’)

In these two forms (with zm and Zm as variables), Ni is the population of the ith zm-bin
and Li that of the ith Zm-bin. There are K bins for the ln zm ≡ Zm variable, and K and k
will, in general, be different. Further, the zm- (or Zm-) bins need not all be of the same
size. Unequal bins are also allowed / possible and may be more convenient. Since the
population M of a bin has uncertainty √M due to counting (or Poisson) statistics, it is
advantageous to choose bins so as to have roughly equal numbers of sources each. This
way, the error-bars are about the same through the range of zm (or Zm).

Illustrative Calculation Done in 1981
Wills & Lynds (1978) have listed a carefully defined sample of 76 optically identified
quasars. We use this (small) sample only to illustrate derivation of n(z) from p(x) ≡
p(V/Vm). We use Einstein-de Sitter cosmology or (½, ½, 0, 0) world model, for which
(H0 / c)2.ℓν2(α, z) = 4.(1 + z)α / {√(1 + z) – 1}2 and (H0 / c)3.v(z) = 8.{1 – 1 / √(1 + z)}3.

For each quasar, zm is calculated by iteration using Newton-Raphson method starting
with initial guess z for zm. The values of z, zm are then used to calculate v(z), v(zm) and
hence x = V/Vm. All the 76 V/Vm-values are used to plot a histogram. A good
approximation for p(x) is p(x) = 2.x, which is normalized over [0,1]. The limiting
redshifts zm range from 0 to 3.2. Dividing into four equal intervals, the bins centered at
0.4, 1.2, 2.0 and 2.8 contain 19, 31, 16 and 10 quasars. Although each of these 4 subsets
is quite small, we calculated and plotted histograms pi(x), i = 1, 2, 3, 4 for each subset for
x-intervals of width 0.2 from 0 to 1, thus with 5 intervals centered at x = 0.1, 0.3, 0.5, 0.7
and 0.9. Each normalized pi(x) is also well approximated by pi(x) = 2.x except
p4(0.2994). So we have done the calculations using this approximation in addition to
using the actual values. Finally we calculate (ω / 3).(c / H0)3.n(zj) using (1) (linear scale

for zm) and (1’) (ln, i.e., natural logarithmic, scale for zm, viz., using Zm = ln zm rather
than zm). All these calculations are tabulated below.

Table for pi(x) and p(x)
x No. p1(x) No. p2(x) No. p3(x) No. p4(x) No. p(x)
0.1 0 0 1 0.161 0 0 0 0 1 0.066
0.3 2 0.526 2 0.323 3 0.9375 1 0.5 8 0.526
0.5 3 0.789 6 0.968 2 0.625 1 0.5 12 0.789
0.7 8 2.105 8 1.290 7 2.1875 5 2.5 28 1.842
0.9 6 1.580 14 2.258 4 1.25 3 1.5 27 1.776
Totals 19 31 16 10 76

Table of n(z) calculation using linear scale for limiting redshifts
j zj Nj → v(zj) i = 1 i = 2 i = 3 i = 4 → n(zj)
1 0.4 19 2.97E-2 1 0.1074 0.0492 0.0321 1307.
2 1.2 31 0.27666 1 0.4580 0.2994 255.
3 2.0 16 0.60399 1 0.6536 67.
4 2.8 10 0.92407 1 22.

Notes for second table: (a) 5th to 8th columns list xij-values,
(b) → v(zj) ≡ (H0 / c)3.v(zj) = 8.{1 – 1 / √(1 + zj)}3, and
(c) → n(zj) ≡ (ω / 3).(c / H0)3.n(zj).

Use of approximations pi(x) = 2.x in evaluating the sums (1) for each row j = 1, 2, 3, 4
gives virtually the same results. Another table below shows steps in evaluation of n(z)
using ln-scale for limiting redshifts, and pi(x) = 2.x, so that no xij-values need be
calculated.
Table of n(z) calculation using ln-scale for limiting redshifts
j Zm-range mid-Zm zm (i.e. zj) Lj → v(zj) → n(zj)
1 -1.5to-0.9 -1.2 0.3012 7 0.015012 355.
2 -0.9to-0.3 -0.6 0.5488 11 0.060673 301.
3 -0.3to+0.3 0.0 1.0000 27 0.201010 337.
4 +0.3to+0.9 +0.6 1.8221 23 0.530388 181.
5 +0.9to+1.5 +1.2 3.3201 8 1.117620 48.

The number of sources in bin j is denoted Lj for ln-scale (instead of Nj for linear scale).

Concluding Remarks
Due to too few sources in the total sample, and even fewer in the subsamples for different
limiting redshift ranges, the results of the calculation are only indicative. No conclusion
about the distribution of quasars in redshift is warranted at this stage. For more
meaningful results, cosmological samples of size at least a few hundred is needed.
Statistical errors for the results should also be calculated. This paper details the method
for calculating n(z) from a well-defined sample, and illustrates the method fully, using
such a sample of quasars. We propose to apply the method for larger samples of different
types of cosmological populations (galaxies, galaxy clusters, radio sources, quasars, γ-ray

sources, …). Note that the V/Vm method was first developed for examining the
distribution of stars in our Milky Way Galaxy. This application of the test has recently
been revived for specific types of stars like white dwarfs.

Acknowledgments
The work reported evolved out of discussions with Vasant K Kulkarni in 1981. Computer
Centre of Indian Institute of Science was used for the calculations done in 1981. The first
draft was written up in 2004-2005 in Muenster, Germany. Radha D Banhatti provided, as
always, unstinting material, moral & spiritual support. Westfaelische-Wilhelms
University of Muenster is acknowledged for use of facilities. University Grants
Commission, New Delhi is acknowledged for financial support.

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