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Getting more out of V/Vm than just the mean
by
Dilip G Banhatti
School of Physics, Madurai Kamaraj University, Madurai 625021, India
[Email: dilip.g.banhatti@gmail.com]
Abstract / Summary
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 ≡ V/Vm (0 ≤ x ≤ 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 ρ(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
Sample of quasars used
Drinkwater et al (1997) define the survey & list the properties of 323 quasars from which
286 can be used for calculating x ≡ V/Vm. The sample used is thus 89% complete relative
to the survey, which covers 3.90 sr in the sky. Using the limiting flux density S0 = 0.5 Jy
at 2.7 GHz, the limiting redshift zm is calculated for each quasar from its redshift z, ν =
2.7 GHz flux density Sν, & spectral index α (defined by α ≡ – d(log Sν)/d(log ν), or
equivalently, Sν ~ ν–α). The world model with the parameters (q0, σ0, k, λ0) = (1, 1, 1, 0)
as defined by von Hoerner (1974) is used for the functions of z needed, viz, the
luminosity distance ℓν(z) and volume v(z). These functions are:
(H0/c)2 ℓν2(α, z) = z2/(1+z)(1–α), and
(H0/c)3 v(z) = (3/2){sin–1(z/(1+z)) – (z/(1+z))√[1–(z/(1+z))2]}.
Here, c/H0 ≡ speed of light / Hubble constant, defines the linear scale.
Deriving n(z) from p(V/Vm) ≡ p(x)
Binning the zm values.
The quasars are first sorted out in increasing order of zm. The limiting redshift is
numerically calculated for each quasar using Newton-Raphson iteration (Rajarevathi
2007). The zm bins are then decided, having roughly equal numbers of sources (about 30),
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which is good enough to derive the differential distribution of x ≡ V/Vm (0 ≤ x ≤ 1) for
each of the bins. Details of this binning are given in Table 1. Also listed are numbers
proportional to the cosmological number densities n(zj) corresponding to the bin mid-
points zj. The procedure for calculating n(zj) is given later in this paper. Table 2 presents,
for comparison, the same results for the smaller sample of 76 from Wills & Lynds (1978)
used by Banhatti (2009), although the world model used for those calculations is von
Hoerner’s (1974) (q0, σ0, k, λ0) = (1/2, 1/2, 0, 0), for which the functions ℓν(z) and v(z)
are different (see Banhatti 2009).
Table 1. Limiting redshifts, their bins, mid-points & populations plus derived
cosmological number densities using 286 quasars over 3.90 sr in the sky.
zm-bin 0 to
0.3
0.3 to
0.7
0.7 to
1.2
1.2 to
1.5
1.5 to
1.8
1.8 to
2.2
2.2 to
2.8
2.8 to
4.0
> 4.0
zj (bin
mid-pt)
0.15
0.5
0.95
1.35
1.65
2.0
2.5
3.4
300*
Bin pop. 29 34 34 32 33 32 30 31 31
j (bin
no.)
1
2
3
4
5
6
7
8
9
n(zj) ~ 48770. 4717. 1560. 1167. 865. 642. 464. 194. 39.
log[n(zj)]
~
4.69
3.67
3.19
3.07
2.94
2.81
2.67
2.29
1.59
* z9 = 300 is the geometric mean of 4 & the largest zm-value ≈ 23000.
Table 2. Results of earlier similar calculation for a sample of 76 quasars.
zm-bin 0 to 0.8 0.8 to 1.6 1.6 to 2.4 2.4 to 3.2
zj (bin mid-pt) 0.4 1.2 2.0 2.8
Bin population 19 31 16 10
j (bin no.) 1 2 3 4
N(zj) ~ 1307. 255. 67. 22.
log [n(zj)] ~ 3.12 2.41 1.83 1.34
Differential distributions pi(x) of x ≡ V/Vm for the 9 zm-bins.
For each of the 9 bins, indexed by j = 1 to 9; pi(x) histograms are plotted with ∆x = 0.2
from x = 0 to 1, making 5 x-bins over [0, 1]. For p1(x), a curve is drawn by eye. For all
other pi(x), i = 2 to 9; the frequency polygon, extrapolated, with slightly higher slope than
the last segment, to x = 1, is used. Cosmological number density n(zj) is then calculated
from the formula (Banhatti 2009):
(Ω/3)(c/H0)3n(zj) = Σi=j9(Ni/v(zi))pi(xij);
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where xij = v(zj)/v(xi), and Ω is the survey solid angle.
In this formula, Ni are the bin populations of the 9 bins.
Details of n(zj) calculation are shown in Table 3. Examples are given below.
n(z7) ~ Σi=79(Ni/v(zi))pi(xi7) = (N7/v(z7))p7(x77) + (N8/v(z8))p7(x87) + (N9/v(z9))p7(x97).
The p7(x77), p7(x87) & p7(x97) values are interpolated from the p7(x) frequency polygon.
Thus, for n(z1) calculation, there are 9 terms to sum (many of which happen to be 0 due
to p1(xi1) being 0). For n(z2) there are 8 terms, & so on. Finally, for n(z9) there is only one
term: n(z9) ~ (N9/v(z9))p9(x99) = (31/2.112)2.65 = 38.9 ≈ 39.
Table 3. Calculation of n(zj). Values in rows labeled i = 1, i = 2 & so on are xij.
j 1 2 3 4 5 6 7 8 9
Nj 29 34 34 32 33 32 30 31 31
zj 0.15 0.50 0.95 1.35 1.65 2.00 2.50 3.40 300*
v(zj)
~
0.002231
0.03835
0.1251
0.2126
0.2773
0.3492
0.4436
0.5890
2.112
i = 1 1
i = 2 0.058 1
i = 3 0.018 0.307 1
i = 4 0.010 0.180 0.588 1
i = 5 0.008 0.138 0.451 0.767 1
i = 6 0.006 0.110 0.358 0.609 0.794 1
i = 7 0.005 0.086 0.282 0.479 0.625 0.787 1
i = 8 0.004 0.065 0.212 0.361 0.471 0.593 0.753 1
i = 9 0.001 0.018 0.059 0.101 0.131 0.165 0.210 0.279 1
n(zj)
~
48770.
4717.
1560.
1167.
865.
642.
464.
194.
39.
Results, Discussion & Conclusion
Plots of numbers proportional to n(zj) & log[n(zj)] listed in Tables 1 & 2 against zj show
the following broad trends. The linear plot falls steeply towards large z from a high value
at the lowest z (0.15 for the sample of 286 & 0.4 for the 76). The log-linear plot brings
out the variation at larger z values more clearly. For the smaller sample of 76, a straight
line of falling (i.e., negative) constant slope is a very good approximation. The log-linear
plot for the larger sample of 286, which starts at a significantly lower z value, falls more
steeply than the smaller sample initially, and then the slope becomes shallower (less
negative) than the smaller sample.
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Using ln(zm) in place of zm in the whole analysis leads essentially to the same results.
To note in passing, n(z) may be interpreted as the integral of the radio luminosity
function ρ(P, z) over all luminosities present in the sample as determined by the flux
density limit. Thus,
n(z; S0) = Int{S0ℓν2(α, z) to ∞}ρ(P, z)dP.
This aspect of the interpretation of the cosmological number density n(z) derived from
the differential distribution p(x) ≡ p(V/Vm) needs to be explored further and utilized in
delineating the cosmological evolution of the source population used (here quasars).
References
Banhatti, Dilip, G (2009) arXiv 0902.1139, 0903.1903, 0903.2442, 0903.2549. The last
& brifest version was presented at 27th Meeting of Astronomical Society of India at
Indian Institute of Astrophysics, Bengaluru.
Drinkwater, M J et al (1997) MNRaS 284 85-125: The PKS half-Jy flat-spectrum sample.
Rajarevathi, M (2007) MPhil Thesis Madurai Kamaraj University: The luminosity-
volume test for a sample of active galaxies.
von Hoerner, S (1974) Cosmology : Chap 13 in Kellermann, K I & Verschuur, G L (eds)
(1974 Springer) Galactic & Extragalactic Radio Astronomy pp353-92.
Wills, D & Lynds, R (1978) ApJSuppl 36 317-58: Studies of new complete samples of
quasars from 4C & PKS.
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