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Cosmological number density in depth from V/Vm distribution
Dilip G. Banhatti
School of Physics, Madurai Kamaraj University, Madurai 625021
Abstract. 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
Luminosity-distance and volume
For cosmological populations of objects, distance is measured by (monochromatic)
luminosity-distance ℓν(z) (at frequency ν), function of redshift z of object. Similarly,
volume of sphere passing through object and centered around observer is (4.π / 3).v(z).
Both ℓν(z) and v(z) are specific known functions of z for given cosmological model.
Calculation of limiting redshift zm
For source of (monochromatic radio) luminosity Lν, flux density Sν, (radio) spectral index
α (≡ - dlog Sν / dlog ν), and redshift z, Lν = 4.π.ℓν2(α, z).Sν. For survey limit S0, value of
limiting redshift zm is given by ℓν2(α, z) / ℓν2(α, zm) = S0 / Sν ≡ s, 0 ≤ s ≤ 1, for source of
redshift z and spectral index α. For simplest case,
[ℓν(α, z) / ℓν(α, zm)]2 = s has single finite solution zm for given α, z and Sν, S0. Different
values zm correspond to different Lν(α).
Relating n(z) to p(V/Vm)
Let N(zm).dzm represent number of sources of limiting redshifts between zm and zm + dzm
in sample covering solid angle ω of sky. Then 4.π.N(zm) / ω is total number of sources of
limit zm per unit zm-interval. Since volume available to source of limit zm is
V(zm) = (4.π / 3).(c / H0)3.v(zm), (where speed of light c and Hubble constant H0 together
determine linear scale of universe,) number of sources (per unit zm-interval) per unit
volume is
{3.N(zm) / ω}.(H0 / c)3.(1 / vm), where vm ≡ v(zm). Let nm(zm, z) be 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 distribution of x ≡ V/Vm for given zm. For z > zm, nm(zm, z) = 0, since sources with
limiting redshift zm cannot have z > zm. To get 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)).
Scheme of Calculation
Any real sample has maximum zmax for zm. So, n(zmax) = 0. In fact, lifetimes of individual
sources will come into consideration, as well as structure-formation epoch at some high
redshift (say, > 10). Thus, n(z) calculation will give useful results only upto redshift
much less than zmax. Formally writing zmax instead of ∞ for 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.
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To apply to real samples, this must be converted to sum. Divide zm-range 0 to zmax into k
equal intervals, each = zmax / k = ∆z. Mid-points are
zj = (j – ½).∆z = {(j – ½) / k}.zmax. Calculate n(z) at these points: n(zj). Converting
integral to sum,
(ω / 3).(c / H0)3.n(zj) = ∑i=jk {Ni / v(zi)}.pi(xij), where xij = v(zj) / v(zi). (1)
It is useful to use Zm = ln zm as redshift variable. Integral and converted 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 population of ith zm-bin and Li that
of ith Zm-bin. There are K bins for Zm, and K and k will, in general, be different.
Illustrative Calculation in 1981
Wills & Lynds (1978) have defined carefully sample of 76 optically identified quasars.
We use this sample only to illustrate derivation of n(z) from p(x) ≡ p(V/Vm). We use
Einstein-de Sitter cosmology or q0 = σ0 = ½, k = λ0 = 0 or (½, ½, 0, 0) world model in
von Hoerner’s (1974) notation, 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 with initial guess z for zm. Values of z, zm
are then used to calculate v(z), v(zm) and hence x = V/Vm. All 76 V/Vm-values are used to
plot histogram. 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,
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 calculate and plot 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 do calculations using this approximation in addition to using
actual values. Finally we calculate (ω / 3).(c / H0)3.n(zj) using (1) and (1’). (See tables.)
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).
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Use of approximations pi(x) = 2.x in evaluating sums (1) for each row j = 1, 2, 3, 4 gives
virtually same results. Table below shows steps in evaluating 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.
Number of sources in bin j is denoted Lj for ln-scale (instead of Nj for linear scale).
Conclusion
Due to too small sample, results are only indicative. Main aim is illustrating method
fully.
Acknowledgments
Work reported evolved out of discussions with Vasant K Kulkarni in 1981. Computer
Centre of IISc, Bangaluru was used for calculations. First draft was written in 2004-2005
in Muenster, Germany. Radha D Banhatti provided, as always, unstinting material, moral
& spiritual support. Uni-Muenster is acknowledged for use of facilities & UGC, New
Delhi, India for financial support.
References
von Hoerner, S 1974 Cosmology : Chapter 13 in Kellermann, K I & Verschuur, G L (eds)
1974 Galactic & Extragalactic Radio Astronomy (Springer) 353-392.
Wills, D & Lynds, R 1978 ApJSuppl 36 317-358 : Studies of new complete samples of
quasi-stellar radio sources from the 4C and Parkes catalogs.
(See longer versions astro-ph/0903.1903 and 0902.2898 for fuller exposition and
references.)
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