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Is evolution of radio sources necessary?
Physics LettersA 158 (1991) 282—290 North-Holland
Is evolution of radio sources necessary? G. Bigot’, S. Rauzy
1,2
and R. Triay
Centre de Physique Theorique ~, CNRS Luminy, Case 907, F-13288 Marseille Cedex 9, France Received 31 May 1991, accepted for publication 12 July 1991 Communicated by J.P. Vigier
The null-correlation approach has been used to study the distribution of the 3CR sample of radio galaxies in the Hubble diagram, with the following results: This sample is about complete at flux density limit S 178= 10 Jy, a relatively small number of strong sources SI78> 100 Jy are missing (that correspond probably to local sources). The radio luminosity distribution function is given by a power law ccL —?~ with y(z) 2, which interprets the large scatter of the distribution in the Hubble diagram. Because of such a distribution in luminosity, this sample is not adapted for deriving the world model. The evolutionary effects are not substantial enough to be revealed within the range of observations.
1. Introduction In early days of cosmology, extragalactic radio source counts were used together with optical data in tests for discriminating between world models. Unfortunately, these approaches were destined to fail because the luminosities cover such a wide range in the Hubble diagram that no standard candle has been recognized; moreover evolutionary effects were detected, e.g., by means of the V/ Vmax test [1J, see also ref. [2]. Nowadays, these sources are discussed in terms of evolving population, e.g., see refs. [3—6],with the goal of determining the (evolving) luminosity distribution function, e.g., see refs. [7,8]. The typical model-generating procedure is to guess a form with a limited number of free parameters that is fitted to the data assuming a particular world model. However, this procedure does not disentangle the evolution from geometrical effects, since the source count is made compatible with an arbitrary world model once an appropriate evolutionary law is chosen, e.g., see ref. [9]. On the other hand, DasGupta et al. [101 derived a non-evolving luminosity function of the 3CR sample of radio galaxies satisfying the constraint of sky brightness, but the above problem still survives. It turns out that such an obstacle can be avoided by means of the null-correlation test, since the world model (in agreement with the data) is determined independently of the luminosity function. This test was formulated by fiche and Souriau [111 for quasar statistics and later adapted for the brightest cluster galaxies by Bigot et al. [12]. In addition, these authors proposed a method for obtaining the luminosity distribution function free of an a priori hypothesis upon its form, see also ref. [131. Here, we outline and apply these techniques to the 3CR radio galaxies. Compared to the usual approaches, which might be embarrassed by well-known problems in observation (noise, confusion and resolution), the null correlation has the advantage of not requiring the completeness of the sample (as long as the selection effects do not depend on redshift). The geometry is based on Friedmann—LemaItre models without as~uminga priori a null cosmological constant. This 2
Université d’Aix—Marseille II, Marseille Cedex, France. Service d’Astrophysique CEA-CEN Saclay (Allocataire MRT), Gif-sur-Yvette, France. Université d’Aix—Marseille I (Université de Provence), Marseille Cedex, France. Laboratoire Propre, LP7061.
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0375-9601/9 1/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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is motivated in part by the recent interest in the cosmological constant, e.g., see ref. [14]. The notations in statistic theory used in the text can be found in ref. [15].
2. The null-correlation approach For convenience in calculation we use the (radio) magnitude m = —2.5 log S~+ const, e.g., see ref. [16], instead of the flux density S~,(at frequency v). Let Z(z) denote the dimensionless measure of the luminosity distance at redshift z, see ref. [17], and Ks~(z)the K-correction term. By grouping these redshift dependent terms of the luminosity—distance relation into a single function, (1)
~(z)=Ks,(z)+5[log(l+z)+logZ(z)] ,
the reduced absolute magnitude (hereafter RAM) reads as follows, u=m—C(z).
(2)
This stands for the K-corrected apparent magnitude in the galaxy rest-frame of an object placed at a distance ofone Hubble radius. If the luminosity of sourcesreads L, = L~(v/v0) then the K-correction term is given by K~,(z) = 2.5 (a 1) log( 1+ z), where a denotes the spectral index. In Friedmann—LemaItre models, the comoving distance at redshift z is given by 4_1COR2+QOR’ (3) dR —
r(z)=
‘
l/(1+z)
,J~OR
the dimensionless measure of the luminosity distance by Z(z)=sin[r(z)~J~]/~J~,
ifk 0>O, ifk0=O,
=T(z),
=sinh[r(z)~fj7~j]/~,/i~J,ifk0.czO,
(4)
and the comoving volume by 3, =~,tr(z)
312{sinh[2r(z)Ikol”2]—2r(z)IkoI”2},
=x1k01
ifk0>0, ifk 0=0, ifk 0’
where k0 = —1 q0+ ~Q0denotes the reduced curvature parameter and ~o= constant, see refs. [17,181. —
(5) —
q0 the reduced cosmological
2.1. The null-correlation test In the case of a non-evolutionary scenario, ifthe selection effects depend only on radio magnitude then the
probability density (hereafter p.d.) describing the distribution of sources reads dPr~1=~ço(m)f(~.t) d~uO(z)O(z~O~—z) dv,
(6)
where A is a normalization factor, ~,(m) stands for a selection function (0 ~ ~
1) describing the selection effects,f(~u)denotes the radio-luminosity distribution function (hereafter RLF) written in terms ofthe RAM, 283
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0 denotes the Heaviside function, Zç0~the redshift of the formation epoch and V= V(z) the comoving volume. Let us define the function
/
w(z)=wp(z)/ I
N
~
cop(z)=
Wp(Z,,~),
k=I
exp[~fl(ln lO)~]
(7)
, (Jr/lit,
where fi is an arbitrary constant, the derivative 8 V/8C= (0 V/Oz) / (OC/Oz) is calculated according to (1)— (5). This function is used to generate weighting factors Wk= W(Zk) that are as equally distributed as possible by choosing the value of fi to this end. Let us denote N
fN
k=I ~=~ Wk/ik,
\l/2 2
o~(u)=(\k=I ~ Wk(~Uk(/~>w)
and similarly for m. According to the central limit theorem, it turns out that if the working hypotheses used to define the p.d. (6) are correct then the following statistic,
1
f’(q
N
0, Q0)= a~Cu)a~(m) ~ k=1
W~(/.4~— Ku> W)(mk—
(8)
has a limiting distribution about zero for the true values of cosmological parameters. This statement, which forms the basic of the null-correlation test, can be lightly established by writing eq. (6) as follows,
d Prdat acp(z)q(m) dmf(u) d~u, where p(z)=
IOV/OCIO(z)O(zfo~—z)
expresses the correlation (function) between the random variables j~and m. Hence, we understand that, for the part of the data satisfying the following inequality, lLk>ILmin=max{mk}C(zfo~)
(9)
,
k
8 V/8~ I as denominator (however, it turns out that this condition is fulfilled for all objects). In practice, the data must be numerous enough so that the correlation between >~,.i co~(~,~) and Wp(Zk) can be neglected (the correlation coefficient
this correlation is cancelled by the weighting factor (7) since one finds the term
between these random variables being equal to 1 /~J7~). Therefore, since the statistic (8) stands for the sample correlation coefficient of weighted variables, its expected value vanishes for the correct values of cosmological parameters. Thus these can be determined as roots of the equation f’(q0, Q0) = 0; which is resolved by a computer program (the accuracy of estimates depends upon the data distribution). It is interesting to note that this approach does not assume the completeness of the sample, i.e., ~,( m) = O( mum m) where mum denotes the limiting search magnitude. On the other hand, the use of weighting factors has the disadvantage of losing statistical information, the percentage of deficit can be defined as follows, —
1
N
2~=1+—~wklnwk, lnNk..~I the weaker this (one has
(10) .~
=
0 for equally distributed
Wk=
1/N) the more reliable the results.
2.2. Determination of the radio-luminosity function (RLF)
According to (6) and (7), for the true values of cosmological parameters, the distribution of weighted data is described by the following p.d., 284
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~,~(m) dmf~(~u) dp0[~(zfo~)—m+~L],
where the functions f~Cu)ccfCu)Xl0-~’~and ~,~(m)x~,(m)X
1oflm/5•
For sufficiently large N, one has N ~ k= I
where q’~(m)dm O[C(zr~)—m+~u]
is a constant function for values /~~ksatisfying eq. (9). Therefore, the form of the RLF can be inferred at abscissas u> ji,,~,,,independently of ~,( m) by means of the following statistic, ~
2~u
(11)
W~,
where the interval i~t must be chosen wide enough for minimizing random fluctuations. In the following, let
us suppose that the candidate shape is given by an exponential form, i.e., the RLF reads fCu)=exP(_
(12)
0(li,~ax~4U),
where s is a scale parameter and Pm~,a location parameter. The more likely values of these parameters are obtained by computing the statistics and a,,,(ji), namely the average and the sample standard deviation of the weighted random variable p, defined above. Indeed,the related p.d. function is given by exp[(/L—jt,,~~~)/s~] 1 ~, ~ sw~expL~..umax—umin,,swJ—f
,
br
/Lmin
where
i!_~plnlo
(13)
and the calculation yields the following equations, 2w~UTnaxlimjn lniI 1 + (~u I — —
S,,,
1~~ /Lmjn) 2 2 (/Lmaxw) 0~w(/L)J 2+< 2
W—umjfl+(l/Sw)[crw(J1) 4u>w(
w—pTfljfl)]
ILrnax/1min —
\
\
(14) (15)
l+(l/Sw)(<óU>w~/Lmjn)
/Lmax
By substituting
—
(15)
into (14), we obtain an equation involving SW only, and that is solved by the Newton
scheme; one chooses an initial value for 1 /s~by substituting ~Umax= maxk {/Lk} into (14). Once SW is determined, we obtain p,,~axfrom (15) and s from (13). The likelihood of the exponential hypothesis (12) is inspected by means of the Kolmogorov—Smirnov approach, i.e., by evaluating the statistical significance of the deviation between the step-function Fe,, Cu) = Ek OCu— ILk) Wk and its expected form F~(p)= f~f~(ji) d/L. However, it must be noted that this approach is not warranted like that because ofthe presence of weighting factors in the above
definitions of the cumulative distribution functions. We remedy (ad hoc) this weakness in measuring such a deviation as follows, 285
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sup [c(p)~F~(ji)—F~,(ji)I],
p
where c(/L)=(N~0Cu_/uk)wk)
~
is a correction factor. 2.3. The selection effects
Let N(
0( m mk) denote the count of objects of magnitude brighter than m. According to (2),
~k
—
(6), (12), one has N(
where
J
Zform
W(m)xexp(m/s)
exp[—~(z)/s]0(~u,,~+~(z)—m)dV(z).
(16)
Hence, it follows straightforwardly that, if the parameters s and 4u,,,~,,are known, then the shape of the selection function ço(m) can be estimated from the following discrete derivatives, 1
1
q~(m)cc~.—~0(Am—lm—mkl) !I’(mk)’
(17)
where W( mk) is calculated by a computer program according to (16), the width of Am is chosen wide enough to minimize the statistical fluctuations. Note that this estimation does not use the weighting factors (7). The completeness to a limiting search magnitude mljm can checked properly by means of the Kolmogorov—Smirnov test, i.e., the sample cumulative distribution function N(
3. Application to 3CR radio galaxies The latest summary of the identification content of the 3CR sample of radio sources at 178 MHz [19] was given by Spinrad et al. [20], additional redshifts were obtained by Djorgovski et al. [211.This gives a sample of 200 radio galaxies located at lb I ~ 100 down to a flux density of S178 = 3.7 Jy. A straightforward analysis shows that the distribution of spectral indices is not or wealdy correlated to flux density and to redshift (since the relatedcorrelation coefficients are found to be 0.11 ±0.19 and 0.22 ±0.19 within la-error) which suggests that the RLF can be written in terms of 4u and z only, and accredits the working hypothesis (6) for testing the non-evolutionary scenario. The statistic (8) was evaluated within the domain (0.05~Q~ 1; —1~q0~<2), while onlyasmall part should correspondto a physically acceptable solution, the results are listed in table 1. It is interesting to note that the amount of information lost, £~=3% ±1%, is negligible, which ensuresthe efficiency of this statistical approach. 286
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Table I The entire sample of 200 3CR radio galaxies up to a flux density of S
178 = 3.7 Jy is used. The results are quoted as follows: I’statistic; percentage of information lost .2i; scale parameter s, location parameter ~ Kolmogorov probability testing an exponential shape for the radio luminosity function (written in terms ofmagnitude).
—1.0
ao= —0.5
Q0=0.05
Q0=0.l
Q0=O.5
Qo=1
—0.21
—0.21
—0.20
—0.19
2%
2%
2%
2%
s
0.95 37.22
0.96 37.25
0.99 37.34
1.01 37.68
Kolmogorov probability
39% —0.22 3%
44% —0.22
75% —0.21
87% —0.20
3% 1.00 37.52 75%
—0.22 3%
—0.22 3%
3% 1.01 37.81 85% —0.21
3%
1.00 37.50 72%
1.02 37.78
1.02 37.76
f’statistic
f’statistiC
s
q0=0.0
Kolmogorov probability f’statistic
s
q0=0.5
82%
86%
90%
93%
f’statistic
—0.22 1.03
—0.22 3% 1.03
—0.22 3% 1.04
—0.21 3% 1.05
Kolmogorov probability
37.91 90%
37.95 90%
38.31 94%
38.98 94%
f’statistic
—0.22
—0.22
—0.22
—0.21
4%
4%
4%
1.04
1.04
1.05
1.06
38.04 93% —0.22 4%
38.09 93% —0.22 4%
38.50 94% —0.21 4%
39.14 94% —0.21 4%
1.05 38.21
1.05 38.25
1.06 38.63
1.07 39.27
93% —0.22
93% —0.22
94% —0.22
95% —0.21
4%
4%
4%
4%
1.06
1.06
1.07
1.07
38.31
38.35
38.72
39.34
94%
94%
95%
95%
3%
s Kolmogorov probability qo=l.5
f’statistic
s q0=2.0
1.04 38.11
90% —0.20 3% 1.05 38.70
Kohnogorov probability
s q0=l.0
3%
1.04 38.32
Kolmogorov probability Fstatistic s Kolmogorov probability
3%
We obtained f’(q0, Q0) = —0.21 ±0.01, which might suggest the presence of evolutionary effects if this discrepancy from zero is statistically significant (however, it is shown subsequently that this turns out to be indisentangable from statistical fluctuations). Fig. 1 shows the candidate shape of the RLF, as obtained from eq. (11) by assuming q0= 0.5 and £Jo= I; this results depends weakly on the world model used. The exponential shape, which is manifest, can be interpreted as a huge excess of weak sources. Since this determination is independent of selection effects, such a distribution in luminosities is not a deficiency of a relatively small number of strong sources as claimed [5,22]. By assuming that the RLF is given by (12), we derive from (13) and (14) a scale parameter s= 1.01 ±0.06 and a location parameter /L,~=38± I, see table 1. This solution is secure at 40—95% of chance as given by the Kolmogorov—Smirnov test. Written in term of luminosity, the RLF reads as a power law cxL ~ where y= 2.01±0.06does not depend on luminosity. Fig. 2 shows the selection function q,( m) as derived from (17); the dashed curve comes from a complete 287
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-
-z~ ‘t.~
5
0
25
30
35
RAM ,a Fig. 1. The natural logarithm of the radio luminosity distribution function versus the RAM, the derivation is performed assuming the standard model q 0= 0.5. It is clear that this curve lies about a straight line within statistical fluctuations, which shows that the radio luminosity distribution function has an exponentialshape.
0
tJ1~
~
~11w
~0.
-2
—4 102
S178 (Jy)
to the 3CR data and the dashed onecomes from a simulation ofa complete sample having a similar radio luminositydistribution function. By taking into account the magnitude ofstatistical fluctuations that are shown by the dashed curve, it is manifest that these curvesdeviate significantly at low flux density S178< 10 Jy and possibly also at high flux density S178> 100 Jy. Fig. 2. The natural logarithm ofthe selection function versus the flux density S178. The plain curve corresponds
sample simulation. By comparing these curves we understand that the steep crest rising from weakest flux densities up to about S178 = 10 Jy is distinct from a statistical fluctuation. This deviation can be interpreted as a
deficiency of weak sources at flux densities S178 < 10 Jy which is probably due to the well-known difficulty in estimating the flux density of nearest sources. Namely, the weakest local sources are missing whereas the other ones have undervalued estimates offlux density. Aless significant deviation is also present at high flux densities 288
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S178~100 Jy; the “local hole”, see ref. [5], might be possibly connected to this trough. Therefore this investigation shows that the sample is about complete up to S178 = 10 Jy, but certainly not at the nominal cutoff of 9 Jy. A hundred random complete samples were simulatedaccording to (6) and (12) with a similar RLF, zm= 10, q0 = 0.5 and (2~=1. By estimating the statistic (8) using these samples, we obtained a standard deviation O)-~0.24 which is of the order of the value obtained from the 3CR sample. Thus we conclude that the magnitude of evolutionary effects is not important enough to be detected by the null-correlation approach. It is clear that if these are present, see ref. [21, then they should behave as geometry to understand the null value of (8). Moreover, in such a case, the above determinations of f(IL) and 9,(m) are not so accurate as that since the formulas used above do not assume evolution. A safe speculation might be that the RLF has an exponential shape with redshift dependent parameters s(z) and ,.i,,,~,(z), the assumption that these parameters depend also
on the RAM IL should be considered (only in case of failure of this scenario) as further step. By writing the p.d. (6) in term of m and z as follows, dPrdatoc~’(m)fI(m, z) dm2(z) dz,
Jj(m, z)=exp[rn/s(z)]O(p,,,ax(z)+~(z)—m), (18) we can easily note that the variation of the cutoffp < jt,,,~(z) with redshift cannot be disentangled from number density evolution, as described by f2 (z), unless it is visible through the distribution of 3CR radio galaxies in the Hubble diagram (but this is not the case). The question of whether it exists or plays only a technical role with any physical meaning will be discussed in ref. [231.In accordance with Longair [24], who claimed that pure luminosity or pure number density evolution does not work, it is interesting to mention that in the present case, any number density evolution can be formally put into p,,,8~(z)to give pure luminosity evolution. The peculiar characteristic of such a distribution in luminosity is that the luminosity—distance relation is ineffective for determining the world model. Indeed, according to (18), the functionfj (m, z) does not depend on the distance modulus, see (1), whereas it acts as a correlation function between the variables m and z. Thus this relation does not provide us with information on cosmology but only on the variation of scale parameter with redshift, and the constant s(z) the more independent the random variables m and z. This shows in a clear cut way that the discrimination between cosmological models is not feasible, which is the reason why the statistic (8) does not depend significantly on cosmological parameters. f2(z)eXP{[Pmax(Z)+~(Z)]/S(Z)}O(z)O(ZformZ)IOV/öZI
,
Acknowledgement One of us (RT) thanks J.V. Narlikar who has motivated this work.
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