R ESPETO A LA DIGNIDAD HUMANA

2.5

Summary

In this chapter, I have given a brief description of the complete samples currently available at low frequency and summarized their observational properties, e.g. radio flux, redshift and angular size. Figure 2.4 present the luminosity distribution of all the sources in 3CRR, 6CE and 7CRS samples along the redshift, and the flux limits and the selection effects are clearly shown in the diagram. I have also classified the radio morphologies for 7C-I and 7C-II samples. In the next chaper, I will use these complete samples to study how radio galaxies evolve throughout cosmological time.

Chapter 3

The cosmological evolution of the

FRII source population

Having constructed the complete samples and obtained their morphology classifications as described above, these samples could be used to investigate the cosmological evolution of the FR II source population. In this chapter, I will perform multi-dimensional Monte- Carlo simulations to generate large artificial samples of FR II sources based on analytical models. I will also compare these artificial samples with the 3CRR, 6CE and 7CRS samples in order to find the best fitting model describing the cosmological evolution of the FR II radio galaxy population.

The purpose of the work in this chapter is to use an existing model for the evolution of individual radio sources together with the redshift-dependent radio luminosity function to generate artificial samples containing a large number of sources. From these artificial samples I can find the best fitting parameters describing the radio sources and their environments, how the jet properties are distributed and how they evolve over cosmological time scales. This approach differs from that of Kaiser & Alexander (1999, hereafter KA99) who assume a birth function to describe the probability of the radio source progenitors becoming active and turning into radio sources. In this birth function approach, they simply assume that the more powerful sources are much rarer than weaker ones. More

3.1. The observed radio luminosity function at 151 MHz –25–

specifically, KA99 assume that radio sources with certain jet powers follow a power-law probability distribution in jet power:

p(Q0)dQ0 ∝ Q0dQ0 if Qmin< Q0< Qmax,

0 if Q0 ≥ Qmax or Q0 ≤ Qmin. (3.1)

KA99 argued that the intrinsic luminosity evolution of radio sources is determined by the properties of their jets and the environments which the progenitors are located in at some cosmological epoch. I use a different approach in this chapter by directly using the RLF from W01 instead of the birth function. Thus, in my approach, the radio luminosity function is guaranteed to find the right number counts for sources with different luminosi- ties at different redshift, and it is the size distribution and its evolution that ultimately constrain the model parameters.

Blundell & Rawlings (1999) also investigated the trends of radio galaxy properties with redshift. The main difference between their work and mine is that they use BRW model to describe the evolution of individual radio source while I am more concentrating on KDA model in this paper. BRW model differs from KDA model a lot and leads to steeper tracks in the P-D diagram. The main differences between the two models will be discussed later in Section 3.7. They also assume a power-law distributed birth function to investigate the distribution of the whole population. Barai & Wiita (2006, hereafter BW06) and Barai & Wiita (2007, hereafter BW07) tested the same three evolutionary models for FRII sources I use here. They showed that none of them fit the observational data, but again they only took into account the birth function instead of the RLF. I will consider this point in more detail in Section 3.1 and compare my results with those of BW06 and Blundell & Rawlings (1999) in Section 3.8.

3.1

The observed radio luminosity function at 151 MHz

In order to construct my artificial samples I need to know the relative number of objects with a given radio luminosity at a given redshift. The radio luminosity function (RLF) which has been developed based on the observational samples described in the last chapter

meet this requirement. The RLF, ρ(P, z) is defined as the number of radio sources per unit co-moving volume and per unit logarithm to base ten of luminosity at a given redshift. Several determinations of ρ(P, z) at various observing frequencies are available in the literature. To minimize the effect of the hotspot emission. I use the RLF at 151 MHz compiled by W01 on the basis of 3CRR, 6CE and 7CRS samples.

W01 model the RLF as the sum of two distinct populations that are allowed to evolve independently with redshift. The low-luminosity population, whose number density is ρl,

contains a mixture of FRI-type sources and the lowest luminosity FRII-type objects. The high luminosity population, whose number density is ρh, contains only FRII-type sources.

The total RLF is then ρ(P, z) = ρl+ ρh.

The low-luminosity population is modelled as a Schechter function, ρl= fl(z)ρl0  P P1⋆ −α1 exp  −P P1⋆  , (3.2)

where ρl0 is a normalization term. At luminosities P below the break luminosity, Pl⋆, the

RLF approximates a power-law with slope −αl. The low-luminosity population decreases

exponentially above Pl⋆. The normalization of ρlis taken to evolve with redshift through

fl(z) = (1 + z)k1 (3.3)

up to a maximum redshift zl0 beyond which fl remains constant. Here, I use model C

of W01, which gives the best fitting result to the observations, and so I adopt log ρl0 =

−7.523, αl= 0.586, log Pl⋆= 26.48, kl= 3.48 and zl0= 0.710.

The high-luminosity population is parameterized in a similar way as ρh = fh(z)ρh0  P Ph⋆ −αh exp  −Ph⋆ P  , (3.4)

where the exponential cut-off is now located below the break luminosity, Ph⋆, and the

power-law with slope −αhextends above Ph⋆. The number density of sources in the high-

luminosity population is modelled as rising up to z = zh0 and then decreasing at higher

redshifts as fh = exp " −1 2  z − zh0 zh1 2# for z < zh0 fh = exp " −1 2  z − zh0 zh2 2# for z ≥ zh0. (3.5)

3.1. The observed radio luminosity function at 151 MHz –27–

Figure 3.1: The adopted radio luminosity function corresponding to model C of W01 for ΩM = 0.3, ΩΛ = 0.7 and Ωk = 0. Dashed and dotted lines show the low-luminosity and

high-luminosity population respectively. The solid lines show the sum of both components.

The relevant constants introduced by W01 are log ρh0 = −6.757, αh = 2.42, log Ph⋆ =

27.39, zh0= 2.03, zh1 = 0.568 and zh2 = 0.956.

In this chapter I only model sources of type FRII. However, W01 do not distinguish between the FR classes in their determination of the RLF, and while ρh contains only

FRII-type objects, the exact composition of ρl in terms of FR class is not known. Here, I

simply assume that 40% of the sources contributing to the low luminosity part of the RLF are of type FRII. I find below that this assumption allows for a good fit of the properties of my artificial samples to those of the observed samples. However, the fraction of FRII-type sources in ρl may be a function of redshift and/or luminosity. In fact, in Section 8.4.2

below I show that the observed sample with the lowest flux limit I use in this paper, the 7CRS sample, is more easily modelled with an evolving FR II fraction. The birth functions used by BRW, BW06 and BW07 simply set a cutoff at low-power end, but the distinction

between FR I and FR II sources is more due to their luminosity. Thus the RLF approach may include more proper FR IIs in the final artificial samples.

W01 compute the RLF for a cosmological model with H0= 50 km s−1Mpc−1, ΩM= 0,

ΩΛ = 0 and Ωk = 1. I adopt the cosmological parameters consistent with the WMAP

results, H0 = 71 km s−1Mpc−1, ΩM= 0.3, ΩΛ= 0.7 and Ωk= 0. Hence I need to convert

the RLF, ρ, to the correct cosmological model, using the relation (Peacock, 1985): ρ1(P1, z)

dV1

dz = ρ2(P2, z) dV2

dz , (3.6)

where P is the luminosity derived for a measured flux and redshift z in a specific cosmo- logical model, while V is the comoving volume. The indices here refer to the two different cosmological models.

The comoving distance in a given cosmological model is (e.g. Hogg, 1999) DM(z) = c H0 Z z 0 dz′ p ΩM(1 + z′)3+ Ωk(1 + z′)2+ ΩΛ . (3.7)

The luminosity distance is given by DL = (1 + z)DM, so the measured flux of a source

at redshift z corresponds to different luminosities in different cosmological models, which are related by

P1D−2M,1= P2DM,2−2 . (3.8)

The comoving volume of a spherical shell at DM is:

dV = 4πD2_MdDM. (3.9)

I use the above relations to translate the RLF of W01 into my adopted cosmological model. The resulting RLF is shown in Figure 3.1.