The luminosity-volume method : Derivation of the cosmological number density in depth from V/Vm distribution [Number density in depth from luminosity-volume]

The luminosity-volume method : Derivation of the cosmological number density in
depth from V/Vm distribution
[Number density in depth from luminosity-volume]

by

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

Abstract / Summary. 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

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, {σ<x>2 = 1 / (12.N) for a sample of size N,} and

<(x - <x>)2.r> = 1 / {22.r.(2.r + 1)}, <(x - <x>)2.r +1> = 0, for r = 1, 2, ….

The mean <x> of a sample of size N 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..

Mid-range as an unbiased estimate of mean
Another unbiased estimate of <x> is xmid ≡ (1/2).(xleast + xgreatest), whose variance is

σx-mid2 = 1 / {2.(N + 1).(N + 2)}for a sample of size N.

Clearly, σx-mid2 < σ<x>2 for N > 2 and σx-mid2 << σ<x>2 for N >> 2.

So maybe (V/Vm)mid rather than <V/Vm> should be used as an estimate of <V/Vm>pop,
instead of the usual practice of using <V/Vm>. However, an advantage of <V/Vm> is that
it is distributed normally (with variance 1 / (12.N)) to a very good approximation for any
respectable N (say, N > 10). To consider (V/Vm)mid instead, since its variance is less, we
must find its (approximate) distribution, that is, asymptotic distribution as N → ∞
(Banhatti 2009c, in preparation).

Median
The above suggestion of preference for the mid-point of the range as an estimate of
central tendency of a (random) variable rather than the mean finds resonance in the trend
of plotting median values of quantities rather than their means to bring out correlations
between them more clearly (Swarup 1975, Kapahi 1975, Swarup, Subrahmanya &
Venkatakrishna 1982, Banhatti 1985, Banhatti & Ananthakrishnan 1989).

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
For example, Kulkarni & Banhatti (1983) and Banhatti (1985) applied the luminosity-
volume test to the somewhat unusual cosmology implied by Hoyle-Narlikar (1972)
conformal gravity. 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. But some cosmological models
may involve refocusing of light in curved, possibly closed and finite, space, so that there
may be a minimum flux density at infinity, or even increasing flux density beyond a
critical redshift (Kulkarni & Banhatti 1983). A model can, in principle, even have

multiple minima, maxima and / or poles in its ℓν(z) relation. In all such cases, the
luminosity-volume test can be applied provide the volume “occupied” by a source and
the maximum volume “available” to the source are reckoned appropriately (Longair &
Scheuer 1970, Lynds & Wills 1972, Avni & Bahcall 1980, Kulkarni & Banhatti 1983,
Banhatti 1985).

Generalities
The statistics of the V/Vm-variable has been considered in some detail by Van Waerbeke
et al (1996) in the context of standard cosmologies and cosmological luminosity
evolution of the quasar population, using m-z Hubble diagrams and the spread of the
V/Vm-variable relative to the cosmological parameters. Our purpose here is to relate the
distribution of the V/Vm-variable to the number density as a function of the redshift z, as
briefly indicated by Kulkarni & Banhatti (1983) and Banhatti (1985), putting in any
models only when it is necessary. Of course, to calculate the occupied and available
volumes V and Vm via the volume function v(z) and luminosity distance ℓ(z) or ℓν(z), a
cosmological or world model must be assumed. For a general treatment where ℓν(z) is not
monotonic (for a given source luminosity Lν), see Longair & Scheuer (1970), Kulkarni &
Banhatti (1983), Banhatti (1985) and Banhatti (2009e, in preparation). Here, we use a
model with monotonic ℓν(z) to illustrate the derivation of the number density n(z) from
the distribution p(x) of x = V/Vm. To be fully general, one must also use a free-form
source spectrum rather than parametrizing it, e.g., by a single parameter α, as is usually
done. (See Banhatti 2009f (in preparation) for an attempt at such a free-form treatment.)
One could take a more complex parametrized form for the spectrum (say, spectral indices
αthin, αlow, αhigh and low- and high-frequency cutoffs νlow and νhigh), since it is the next best
approximation after a single spectral index α ≡ αthin. (See Banhatti 2009g (in preparation)
for such an attempt.) However, our main aim here is to illustrate the relation between the
number density n(z) and the (differential) distribution p(x) of x = V/Vm.

Calculation of the Limiting Redshift zm
For a (radio) source of (monochromatic radio) luminosity Lν, monochromatic flux density
Sν, (radio) spectral index α (≡ - dlog Sν / dlog ν so that Sν is proportional to ν-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) proportional to Lν1/2. Then, for different values of
α, this amounts to choosing different Lν(α).

Explicitly, for Hoyle-Narlikar (1972) conformal gravity,

(H0 / c)2.ℓν2(α, z) = z2 / (1 + z)1 – α and (H0 / c)3.v(z) = {z / (1 + z)}3
: HN model (Kulkarni & Banhatti 1983).

The function ℓν(α, z) is the same as the model q0 = σ0 = k = 1, λ0 = 0 in von Hoerner’s
(1974) notation, which may be called the (1, 1, 1, 0) model. The other function v(z),
relevant for luminosity-volume test, is not the same (cf. von Hoerner 1974).

For the (1, 1, 1, 0) model, which also applies to Hoyle-Narlikar (1972) conformal gravity,
as shown by Kulkarni & Banhatti (1983), who complement Canuto & Narlikar (1980) in
cosmological tests of HN model,

┌ is monotonic increasing from 0 to ∞ for α > -1 ┐
│ │
ℓν(α, z) ┼ is monotonic increasing from 0 to 1 for α = -1 ┤
│ │
└ has a maximum at z0 = -2 / (1 + α ) = 2 / (|α | -1) │
& goes to 0 at z = 0 and ∞ for α < -1 ┘

Observationally, α > -1 for any radio frequency of interest and also at optical frequencies.
Hence, for the cosmological samples of interest, ℓν(α, z) is monotonic increasing from 0
(at z = 0) to ∞ (at z = ∞), to which case we have limited ourselves for simplicity.

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) (Banhatti 2009h, in
preparation). 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.
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

zi = (i – ½).∆z = {(i – ½) / k}.zmax. Calculate n(z) at these points: n(zi).

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 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 used by Kulkarni & Banhatti (1983) for testing mean <V/Vm> against ½ in a
model partially indicated by (1, 1, 1, 0) in von Hoerner’s (1974) notation (see above).

Here we use this (small) sample only to illustrate derivation of n(z) from p(x) ≡ p(V/Vm).
We use the 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 (½, ½, 0, 0) model (i.e., E-de S)

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.

In the second table, (a) 5th to 8th columns are 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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