Modelling the luminosities and sizes of radio galaxies: radio luminosity function at z = 6
MMNRAS , 1–13 (2015) Preprint 8 October 2018 Compiled using MNRAS L A TEX style file v3.0
Modelling the luminosities and sizes of radio sources: radioluminosity function at z = 6
A. Saxena (cid:63) , H. J. A. R¨ottgering and E. E. Rigby Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
ABSTRACT
We present a model to predict the luminosity function for radio galaxies and theirlinear size distribution at any redshift. The model takes a black hole mass functionand Eddington ratio distribution as input and tracks the evolution of radio sources,taking into account synchrotron, adiabatic and inverse Compton energy losses. Wefirst test the model at z = 2 where plenty of radio data is available and show thatthe radio luminosity function (RLF) is consistent with observations. We are able toreproduce the break in luminosity function that separates locally the FRI and FRIIradio sources. Our prediction for linear size distribution at z = 2 matches the observeddistribution too. We then use our model to predict a RLF and linear size distributionat z = 6, as this is the epoch when radio galaxies can be used as probes of reionisation.We demonstrate that higher inverse Compton losses lead to shorter source lifetimesand smaller sizes at high redshifts. The predicted sizes are consistent with the generallyobserved trend with redshift. We evolve the z = 2 RLF based on observed quasar spacedensities at high redshifts, and show that our RLF prediction at z = 6 is consistent.Finally, we predict the detection of 0.63, 0.092 and 0.0025 z (cid:62) z (cid:62) Key words: radio luminosity function, radio galaxies, high redshift, reionisation
Powerful high redshift radio galaxies (HzRGs) are found toreside in massive galaxies, which are thought to be progen-itors of the massive ellipticals that we observe today (Bestet al. 1998; McLure et al. 2004). These host galaxies con-tain huge amounts of dust and gas, and are observed tobe forming stars intensively (Willott et al. 2003). HzRGsare also associated with cosmological over-densities such asgalaxy clusters and proto-clusters (R¨ottgering et al. 2003;Stevens et al. 2003; Kodama et al. 2007; Venemans et al.2007; Galametz et al. 2012; Mayo et al. 2012). These prop-erties make HzRGs important tools to study the formationand evolution of massive galaxies and large-scale structurein the universe. For a review about the nature and proper-ties of HzRGs, their hosts and their environments, we referthe reader to Miley & De Breuck (2008).Radio galaxies at the highest redshifts, particularly inthe Epoch of Reionisation (EoR), have the potential to beimportant probes of cosmology. Constraining when and howthe universe made a phase transition from neutral to com- (cid:63)
E-mail: [email protected] pletely ionised is one of the most exciting challenges in cos-mology today and luminous radio galaxies at z (cid:62) z (cid:62) z (cid:62)
6, this line fallsin the low-frequency radio regime ( ν <
200 MHz) and maybe detected as an absorption feature in the radio spectra of z (cid:62) α forest) in the continuum of a z > z (cid:62) c (cid:13) a r X i v : . [ a s t r o - ph . GA ] M a y A. Saxena et al. can be used to constrain black hole growth models. Under-standing the growth of SMBHs over cosmic time using ob-servations of active galactic nuclei (AGN) has been a subjectof intense study. The growth of the black hole mass functionhas been studied using AGN and quasar luminosity func-tions and varying key parameters associated with AGN suchas SMBH accretion rate, AGN duty cycles and radiative effi-ciencies (Shankar et al. 2009; Kelly et al. 2010; Shankar et al.2013). Willott et al. (2010b) managed to extend studies outto z = 6. However, most observations used to study SMBHgrowth are at optical, infrared and Xray wavelengths. Notmany AGN at high redshifts have been observed to be lu-minous at radio wavelengths and this could partly be dueto a lack of deep, all-sky radio surveys. The general under-standing, however, is that radio emission is powered by thesame mechanism that is also responsible for the optical/IRand Xray emission from AGN, the accretion of material onto the central SMBH.Many attempts have been made to measure the evo-lution of the radio luminosity function (RLF). Dunlop &Peacock (1990) reported evidence for the existence of a red-shift cut-off in the space density of quasars and radio galax-ies over the redshift range 2 −
4. Subsequent studies alsoindicated that space densities of radio sources undergo con-tinued decline between z (cid:39) . . z > . z ∼ z ∼ . z peak ,is higher, thereby demonstrating that z peak is a function ofradio luminosity. Rigby et al. (2015) extended their earlierstudy to even lower radio luminosities and found consistentresults.It is largely agreed, however, that the RLF at anyepoch is dominated by two distinct classes of objects –star-forming systems at the lowest luminosities, which aregenerally hosted by late-type galaxies, and radio-loud AGNat higher luminosities, generally found to reside in massiveearly-type galaxies and powered by accretion on to the cen-tral SMBH (Jackson & Wall 1999). Radio-loud AGN can fur-ther be divided into two categories based on their radio mor-phologies (Fanaroff & Riley 1974). The Fanaroff-Riley classI (FRI) objects are brightest at their centres and have inter-mediate radio luminosities, whereas the FR class II (FRII)objects are brightest at the edges, away from the central re-gions and are some of the most luminous radio sources ob-served. Dunlop & Peacock (1990) showed that the dividingline between the FR classes, generally thought to be around10 W Hz − at 150 −
200 MHz, is remarkably close to thebreak in the local RLF.Observations have been complemented by theoreticalstudies aiming to understand the evolution of the radio lu- minosities and sizes of both FRI and FRII objects. Theseinclude modelling the dynamics of a radio source poweredby relativistic jets resulting from the accretion on to theSMBH (Kaiser & Alexander 1997) and modelling the dif-ferent physical processes through which a radio source maylose energy (Kaiser et al. 1997; Blundell et al. 1999; Alexan-der 2002). Some studies have also explored the link betweenthe growth of the two FR classes (Alexander 2000; Kaiser &Best 2007). However, not a lot of work has gone into paint-ing a complete picture that incorporates the growth of blackholes with the evolution of individual radio sources.In this work, we attempt to model the radio luminosityfunction based on black hole mass functions. Such an ap-proach naturally establishes a link between the the growthof black holes and the resulting radio luminosity function,which can be tested using existing and upcoming radio sur-veys. We begin by testing our model at redshift 2, for whichthere are sufficient observations available. We then extendour model to z = 6 with the ultimate goal of predicting aradio luminosity function close to or even into the EoR. Fi-nally we use the modelled RLF to make predictions aboutthe number of radio sources that could be observed in thecurrent and upcoming state-of-the-art low-frequency radiosurveys.The layout of this paper is as follows. In Section 2 weconstruct and describe our model for the growth and evo-lution of luminosities and linear sizes of radio sources. InSection 3 we present luminosity and size predictions fromour model at z = 2 and test them using data available inthe literature. We extend our model to z = 6 in Section4 and present our predicted luminosity function and linearsize distribution. We then test the predictions at z = 6 bycomparing with a RLF obtained using a pure density evo-lution model extrapolated from lower redshifts. We presentthe number of expected sources in current and future lowfrequency radio surveys and assess the trade-off betweencoverage area and depth, determining the optimum surveyparameters to maximise detection of radio sources at highredshifts. Finally, we present a summary of our findings inSection 5.Throughout this paper we assume a flat ΛCDM cos-mology with H = 67 . m = 0 . It is generally believed that radio galaxies are powered byaccretion of material on to the supermassive black hole(SMBH). Therefore, we take black hole mass obeying theblack hole mass functions (BHMF) determined at differentepochs by Shankar et al. (2009) as one of the inputs. TheseBHMFs have been derived using AGN bolometric luminos-ity functions estimated using optical and X-ray observations,making certain assumptions about radiative efficiency. TheBHMF is generally well fit by a Schechter function of the
MNRAS , 1–13 (2015) adio luminosity function at high-z form φ ( M BH ) = φ (cid:63) (cid:18) M BH M (cid:63) (cid:19) α exp (cid:20) − M BH M (cid:63) (cid:21) (1)where φ (cid:63) and M (cid:63) are the characteristic space density andblack hole mass, respectively and α is the low-mass endslope.For an actively accreting SMBH with an accretionrate dM/dt , the bolometric luminosity can be written as L bol = (cid:15) ( dM/dt ) c , where (cid:15) is the efficiency parameter and c is the speed of light. The maximum possible luminosityachievable through this mechanism is the Eddington lumi-nosity, L Edd and the ratio of the bolometric luminosity tothe Eddington luminosity, the Eddington ratio, is writtenas λ = L bol /L Edd . This can take values between 0 and 1,and is an indicator of how ‘actively’ a SMBH is accreting.The Eddington ratio is another input in our model, whichis drawn from a log-normal distribution that Shankar et al.(2013) found to fit the observed AGN luminosity functionswell and is also supported by Willott et al. (2010b) at z ∼ The jet power is thought to be closely coupled to the blackhole mass, spin and accretion rate via the Blandford-Znajekmechanism (Blandford & Znajek 1977). Jet power can becalculated either by using the thin disk solution (Shakura &Sunyaev 1973), which typically works for black holes accret-ing at higher Eddington ratios ( λ > .
01) or by assuminga thick accretion disk with an advection dominated accre-tion flow (ADAF) for black holes accreting at lower rates(Narayan & Yi 1994). The expression for jet power in thethin disk regime can be written as (Meier 2002; Orsi et al.2016) Q jet = 2 × (cid:18) M BH M (cid:12) (cid:19) . (cid:18) λ . (cid:19) . a W (2)and in the ADAF regime can be written as Q jet = 2 . × (cid:18) M BH M (cid:12) (cid:19) (cid:18) λ . (cid:19) a W (3)where M BH is the black hole mass, λ is the Eddington ratioand a is the black hole spin. We use a Monte Carlo basedapproach where a value of λ is randomly sampled from thegiven input distribution (which we elaborate upon in thenext section) and assigned to a black hole mass to calculatejet powers for each source using Equations 2 or 3, dependingon the Eddington ratio.Although studies have indicated that AGN might expe-rience recurrent jet activity through refuelling of the centralblack hole due to a possible merger, the lifetime of a lumi-nous radio source is typically of the order of 10 yr, with theAGN duty cycle found to be close to 10 yr (Bird et al. 2008).Note that duty cycle values have been shown to strongly de-pend on the stellar mass of host galaxies (Best et al. 2005).However, the typical duty cycle timescale is longer than thetime for which we evolve our sources, which we talk about inthe following sections. Therefore, any recurrent jet activityis not taken into account in our modelling. Several studies have aimed to establish an empirically de-rived relation between the intrinsic jet power and the ob-served radio luminosity (Bˆırzan et al. 2004, 2008; Cavagnoloet al. 2010). There have also been several attempts to con-struct analytical models for the growth of jets in radiosources, especially in strong radio sources with an FRII-type morphology that are typically associated with HzRGs(Kaiser & Alexander 1997; Kaiser et al. 1997; Blundell et al.1999; Alexander 2000, 2002).Kaiser & Best (2007, hereafter KB07) studied analyti-cally the growth of radio galaxies in the context of the ra-dio luminosity function and determined that the luminosityfunction at any epoch consists of sources that are dominatedby different energy loss mechanisms, depending on their agesand sizes. Each energy loss phase affects their growth andthe evolution of their radio luminosity differently. We usethe KB07 prescriptions to track the growth of radio sourcesin our model, which contribute to the luminosity functions.We briefly describe the implemented energy loss mechanismsbelow.
Synchrotron radiation from ultra-relativistic charged par-ticles is believed to be the major source of virtually allextragalactic radio sources. In star-forming galaxies, syn-chrotron radiation originates from electrons from HII re-gions, accelerated by Type II and Ib supernovae. In radio-loud AGN, charged particles are accelerated in the jets andlobes launched by the central SMBH due to accretion ofmatter, and such systems are the focus of this model.The energy density of the magnetic field associated witha radio source decreases as a source grows. Therefore, syn-chrotron emission dominates early in the source’s lifetime.KB07 noted that the luminosity of a source growing in thisregime stays constant, with the luminosity at an observingfrequency ν given by L ν ≈ m e c f n A / ν Q jet (4)where m e is the mass of an electron, c is the speed oflight, f n and A are constants with fiducial values 1 . × s kg − m − and 4, respectively. After the early stages of synchrotron losses, the magneticfield strength declines and adiabatic losses begin to domi-nate. The gas density around a radio-loud AGN into whichthe jets and lobes grow is generally represented by a single- β model of the form ρ r = ρ [1 + ( r/d ) ] β/ (5)where ρ is the density in the inner-most regions, d is thedistance until which the gas density remains constant (andfollows a power-law decline afterwards) and r is the distancefrom the centre of the gas distribution. Such a profile hasbeen found to fit X-ray emission from hot gas in ellipticalgalaxies (that usually host radio-loud AGN), galaxy groupsand clusters (Fukazawa et al. 2004). MNRAS000
Synchrotron radiation from ultra-relativistic charged par-ticles is believed to be the major source of virtually allextragalactic radio sources. In star-forming galaxies, syn-chrotron radiation originates from electrons from HII re-gions, accelerated by Type II and Ib supernovae. In radio-loud AGN, charged particles are accelerated in the jets andlobes launched by the central SMBH due to accretion ofmatter, and such systems are the focus of this model.The energy density of the magnetic field associated witha radio source decreases as a source grows. Therefore, syn-chrotron emission dominates early in the source’s lifetime.KB07 noted that the luminosity of a source growing in thisregime stays constant, with the luminosity at an observingfrequency ν given by L ν ≈ m e c f n A / ν Q jet (4)where m e is the mass of an electron, c is the speed oflight, f n and A are constants with fiducial values 1 . × s kg − m − and 4, respectively. After the early stages of synchrotron losses, the magneticfield strength declines and adiabatic losses begin to domi-nate. The gas density around a radio-loud AGN into whichthe jets and lobes grow is generally represented by a single- β model of the form ρ r = ρ [1 + ( r/d ) ] β/ (5)where ρ is the density in the inner-most regions, d is thedistance until which the gas density remains constant (andfollows a power-law decline afterwards) and r is the distancefrom the centre of the gas distribution. Such a profile hasbeen found to fit X-ray emission from hot gas in ellipticalgalaxies (that usually host radio-loud AGN), galaxy groupsand clusters (Fukazawa et al. 2004). MNRAS000 , 1–13 (2015)
A. Saxena et al.
When the source is in the adiabatic loss phase, the lu-minosity evolves as (KB07) L ν ∝ D (8 − β ) / (6)where D is the linear size of the radio lobe and β depends onthe profile of the ambient gas surrounding the radio source. Once the size of the lobe has exceeded the extent of the x-rayhalo of the host galaxy, and the energy density of the mag-netic field in the lobe is comparable to the energy densityof the ambient cosmic microwave background (CMB) radia-tion, losses due to inverse Compton scattering against CMBphotons begin to dominate. KB07 found that the luminosityin this phase evolves as L ν ∝ D ( − − β ) / (7)The CMB energy density scales with redshift, z , as ∝ (1 + z ) . Therefore at higher redshifts, IC losses shouldbegin to dominate much earlier in the source’s lifetime. Wecalculate the distance from the centre of the galaxy afterwhich IC losses begin to dominate, D IC , by equating the lu-minosities produced by adiabatic losses and IC losses, givenby equations (3) and (7) in KB07. This depends on the jetpower of the source and can be written as (See AppendixA) D IC ∝ ( ρd β ) − / Q / jet (1 + z ) − / (8) Having set up the various phases of evolution, we can trackthe luminosity and size evolution of all our sources. Equation(A2) in KB07 describes the growth of the lobe size, D withtime D = C (cid:18) Q jet ρd β (cid:19) / (5 − β ) t / (5 − β ) (9)where C is a constant. For simplicity, we use the fiducial val-ues from KB07 and set the extent till which the gas densityremains constant to d = 2 kpc at z = 2. We then include red-shift evolution of linear sizes of galaxies. van der Wel et al.(2014, and references therein) find that over the redshiftrange 0 < z <
3, the size evolution of early-type galaxies ismuch faster than late-type galaxies. Further, Mosleh et al.(2012) find that the sizes of Lyman break selected galaxies(LBGs) with stellar masses of 10 . < M (cid:63) < . evolveas (1 + z ) − . from z = 1 to 7, which is consistent with thefindings of van der Wel et al. (2014). At higher redshifts,7 < z <
12, Ono et al. (2013) find the size evolution ofLBGs to follow (1 + z ) − . . Radio galaxies are more likelyto be hosted by early-type galaxies up to redshifts of z = 3,but at higher redshifts, HzRGs are seen to be forming starsintensively (Miley & De Breuck 2008).Owing to the faster evolution of early-type galaxies seenat moderate redshifts and the redshift evolution of LBGsat higher redshift, and assuming that early in the universeradio galaxies are expected to be hosted by galaxies withproperties similar to LBGs, we include the size evolutionwith redshift as (1 + z ) − . , which is consistent with the Figure 1.
Input black hole mass function and the Eddingtonratio at z = 2 taken from Shankar et al. (2009) and Shankaret al. (2013). Shown in the figure is the best fit Schechter functiondetermined at z = 2. The Eddington ratio distribution is a log-normal peaking at λ = 0 . majority of studies carried out at z >
2. Therefore, theparameter d takes values d = (cid:40) , if z (cid:54) × [(1 + z ) / − . kpc , if z > ρ = 10 − kg m − at all redshifts.The initial growth of the lobe would be in a region wherethe ambient gas density is roughly constant, i.e. D < d andtherefore, β = 0. This very early growth would be domi-nated by synchrotron losses as the magnetic field is influ-ential. Considering Equation 4, the luminosity would leveloff at the stage when synchrotron radiation dominates. Af-ter the lobe has grown to a size of 2 kpc, the source entersthe adiabatic loss phase. Here, we consider the gas densityprofile around the radio source to follow a power law de-cline (Equation 5), with β = 2 being used for further evolu-tion. Finally, once the source has grown to a size when theCMB energy density begins to play an important role, in-verse Compton (IC) losses begin to dominate. The values ofall model parameters used in this study are shown in TableA1 in Appendix A. We first implement our model at z = 2 to compare thepredictions of our model to data, as the radio luminosityfunction at this epoch is relatively well constrained (Dunlop& Peacock 1990; Blundell et al. 1999; Willott et al. 2001;Rigby et al. 2011, 2015). We use the Shankar et al. (2009)BHMF determined at z = 2, which is a Schechter func-tion with parameters log φ (cid:63) = − . M (cid:63) = 9 . α = − . λ = 0 .
16 with dispersion of 0.5 dex fromShankar et al. (2013) (both shown in Figure 1) as input. For
MNRAS , 1–13 (2015) adio luminosity function at high-z simplicity, we assume the Eddington ratio to be independentof the black hole mass. We randomly sample a value of theEddington ratio from the chosen distribution and assign itto each black hole.We assume a mass dependence for the spin parameter ofour black holes, following the conclusions of Volonteri et al.(2007). They note that disk galaxies harbour lower massSMBHs and weaker AGN, and grow by accreting smallerpackets of material. This would skew the black hole spin dis-tribution to lower values. Brighter radio sources, however,are found in elliptical galaxies that host massive SMBHs.These black holes must have had a major accretion episode,likely powered by a merger that was responsible for form-ing the host elliptical galaxies too. During this episode thespin must have increased significantly. Volonteri et al. (2007)show that the peak in distribution of spin parameter forblack holes with masses greater than 10 M (cid:12) lies between0 . − .
0. Therefore, black holes with masses greater than 10 M (cid:12) in our simulation are randomly assigned a spin from therange [0 . , .
0) and all other black holes are assigned a spinfrom the range [0 . , . The resulting jet power distributions for the 3 differentchoices of spin parameter distributions are shown in Figure2. Cases where smaller black holes have higher spin (Case 2)and a constant spin parameter (Case 3) are unable to pro-duce the most powerful jets, which have been observed in theliterature (see Godfrey & Shabala 2013). The widely sup-ported idea that more massive black holes are highly spunup (Case 1) accurately reproduces the bimodal distributionof jet powers, which is thought to result in the fundamentaldifference between the FRI and FRII type radio galaxies.This bimodality is likely due to the two different accretionregimes implemented in our model (ADAF and thin diskregimes), that depend on the accretion rate. Therefore, go-ing forward, we only consider results from Case 1 of spinparameter choice.
The first step at which we construct the luminosity functionis when the linear size of all sources in the simulation is2 kpc, i.e. when all sources are dominated by synchrotronlosses. Since the growth rate depends on jet power, the radiosources take different times to attain a size of 2 kpc, thusleading to an age distribution. To now track the evolutionof radio luminosities, we evolve our sources in time stepsof 0.2 Myr, calculating linear sizes at each time step. Thesize determines which phase of energy loss each source isin and we use this to calculate the evolved luminosities ateach time step according to the prescriptions described inthe previous section. The simulation is run for a total timeof 9 Myr, which is the typical lifetime of a radio galaxy.
Figure 2.
Distribution of jet powers predicted by our model at z = 2 for different choices of spin parameters. Case 1 is whenmore massive black holes have a higher spin, Case 2 is when moremassive black holes have lower spin and Case 3 is when all blackholes have a constant spin. Case 1 most accurately reproducesthe bimodality in jet power, and is able to produce sources withvery powerful jets that result in some of the largest and brightestradio galaxies in the universe. Although it has been shown that the radio-loud fractionof AGN is a function of the stellar mass of the host galaxyand the black hole mass in the local universe, these values arerelatively unconstrained at high redshifts (Best et al. 2005;Williams & R¨ottgering 2015). The physical conditions of theuniverse change dramatically going from z = 0 to z = 2, so asimple extrapolation from the local universe would not work.It has been observed however, that roughly 10% of all galax-ies are AGN (Martini et al. 2013, and references therein) andof these, only 10% are generally found to be bright in theradio or radio-loud, even out to higher redshifts (Ba˜nadoset al. 2015). Further, the correlation between stellar massand radio-loud fraction is weaker for high-excitation radiogalaxies (Janssen et al. 2012), which most radio sources athigh redshifts are expected to be. Therefore, we take a sim-plistic approach and randomly select 1% of all our sources tobe included in the final radio luminosity function. It is im-portant to note that we do not select objects to be AGN orradio-loud depending on the black hole mass, as black holemasses of (radio-loud) AGN seem to be distributed evenlyover several orders of magnitude (Woo & Urry 2002).We use the Shankar et al. (2009) black hole space densi-ties to normalise the radio luminosity function in the follow-ing way. The space densities from the analytical BHMF areused to assign a maximum volume, V max , to each simulatedblack hole, which is the volume probed by a complete sur-vey that would enable the black hole to be detected. Thesecalculated volumes are then used to normalise the resultingradio luminosity function, by summing over 1 /V max in eachluminosity bin, which is the traditional way of calculatingluminosity functions (Schmidt 1968). Luminosity distributions constructed for 3 time steps areshown in Figure 3. The bright end of the luminosity dis-
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Figure 3.
Time evolution of the radio luminosity function (RLF)at z = 2 at various time steps in our model. Each curve showsthe predicted luminosity function at a given average source age.The brightest sources lose their energy very quickly and therefore,those observed must be young. Also shown are the space densitiescalculated by Rigby et al. (2015), which have been recalculatedat 150 MHz. Additionally, two distinct radio populations are ap-parent, with the dividing luminosity lying between 10 and 10 W Hz − , which is consistent with the luminosity that is generallyconsidered as the dividing line between the FRI and FRII sources(Dunlop & Peacock 1990; Willott et al. 2001). tribution is seen to change, whereas the faint end remainsroughly constant over time. This is mainly because themost powerful sources in the simulation grow faster and thisrapid growth leads to increased adiabatic losses. Addition-ally, sources with powerful jets enter the regime of InverseCompton losses and end up losing energy much quicker. Thissuggests that the most powerful sources must be very youngand as a result, compact.We plot the space densities calculated by Rigby et al.(2015) for comparison. Since the space densities were cal-culated at an observing frequency of 1.4 GHz, we scaleit to obtain powers at 150 MHz using the z − α relation α = 0 . .
21 log(1 + z ), determined by Ker et al. (2012).Overall, the data seems to match the predictions well andis also consistent with the expectation that most powerfulsources must be younger. There is a slight disagreement inthe luminosity bin 25 < log P <
26, where the observedspace densities are higher than our prediction. However,this may be explained by the apparent presence of twodistinct populations, with the dividing luminosity between10 − W Hz − . This coincides with the dividing lumi-nosity between FRI and FRII radio sources, which is seen inobservations (Dunlop & Peacock 1990; Willott et al. 2001)and was also predicted by KB07.Another interesting feature especially prominentaround this dividing luminosity, is the transition of sourcesfrom the FRII to the FRI regime over time (purely on thebasis of radio power; within the framework of this simula-tion, we are unable to separate the sources morphologically).Such a scenario was suggested by KB07, where the develop-ment of an FRII structure in a source with a weak jet canbe disrupted by dense external gas, which starts to fill thelobe volume, eventually reaching the jet flow. At this point Figure 4.
The so-called P-D tracks, showing the evolution ofradio power (P) with linear size (D) for sources with differentradio jet powers at z = 2. Sources with stronger jets have a wellbehaved evolution, whereas sources with weaker jets have a breakin their track very early in their lifetime. This is because weakersources enter the regime of inverse Compton losses much quicker. the morphology is expected to change from FRII to FRI ac-companied by a decrease in radio luminosity, which is whatwe may be seeing in our simulation. Such a behaviour, com-bined with the apparent bimodality in the distribution of jetpowers as seen in Figure 2 suggests that the origin of the FRIand FRII populations could be purely due to the mechanismby which the central SMBH is accreting gas. Such a sce-nario was suggested by Best & Heckman (2012), where theyconstructed a large sample of radio sources from the SDSSand classified them broadly into two categories based ontheir emission line properties, low-excitation radio galaxies(LERGs) and high-excitation radio galaxies (HERGs). Theyfound LERGs to dominate at low radio luminosities andHERGs to dominate at high luminosities, with the switchin population dominance found to be at L = 10 WHz − . LERGs are powered by radiatively inefficient accre-tion, with much lower accretion rate compared to the radia-tively efficient accretion that powers HERGs. This divide inluminosity is roughly where we observe two separate popu-lations too, which we attribute to the difference in accretionmechanism. Therefore, the scenarios suggested by Best &Heckman (2012) and what we see in our simulation seem tobe consistent. The included energy-loss prescriptions in our model give riseto ‘P-D tracks’ for every source in our simulation, which havebeen historically used to study the evolution of radio lobesin FRII galaxies (Blundell et al. 1999; Alexander 2000, 2002;Kaiser & Best 2007). Figure 4 shows P-D tracks for sourceswith different jet powers for the entirety of the simulationrun (9 Myrs). Sources with low jet powers show a breakin their P-D track, which arises when there is a decline intheir radio luminosities after attaining a certain linear size,when the dominant energy loss mechanism changes from adi-abatic losses to inverse Compton (IC) losses from the CMB.Growth of the source in the IC regime is slower. Sources
MNRAS , 1–13 (2015) adio luminosity function at high-z Figure 5.
Comparison of the cumulative distribution of linearsizes of FRII radio sources predicted by our model (red line) withradio galaxies with confirmed redshifts in the range z = 1 . − . z = 2 is unable to producesources with sizes greater than 250 kpc, we have excluded theseparticular MRC sources in this comparison for clarity. with high jet powers follow, however, are more likely neverto be dominated by IC losses and follow a well-behaved pathalong the P-D diagram.Since the average linear sizes of observable sources inour model depend strongly on the average age of sourcesat a given time step, we weigh the observed sizes by age.This is done by assigning a probability of detection, whichis defined as the age of a source divided by its total lifetimein the simulation. This means that younger sources have alower probability of detection than sources that have existedin the universe for a longer time. We construct a linear sizedistribution in time steps of 1 Myr and normalise this distri-bution by assigning a detection probability to each source.To test the size predictions from our model, we usethe catalogue compiled by Singal & Laxmi Singh (2013),which contains linear sizes and redshifts for radio sourcesin the Molonglo Reference Catalogue (MRC; Large et al.1981). The original MRC contains 12,141 discrete sourceswith flux density (cid:62) . . − . >
250 kpc (263, 284 and 458 kpc at redshifts 1.54, 1.78and 1.54 respectively). Our model at z = 2 is unable to pro-duce sources with sizes greater than 250 kpc. This could bebecause we do not model recurrent jet activity, which may be responsible for the continued growth of a radio source byperiodically refuelling the jet from the SMBH. Therefore,in the comparison shown in Figure 5, we have excluded thethree largest sources from the MRC. It is safe to assume,however, that the inclusion of the largest sources should notaffect the cumulative distribution of linear sizes too much asthese sources lie well above the mean linear size. Having tested the predictions of our model at z = 2 wherethere are sufficient observations available, we now extend themodel to z = 6 to predict a radio luminosity function at thisredshift, and describe the implementation in the followingsection. We use the black hole mass function (BHMF) and Edding-ton ratio distribution at z = 6 determined by Willott et al.(2010b) using the quasar luminosity function and estima-tion of black hole masses through measurements of MgIIline widths. The luminosity function they use to derive theBHMF has been found to be consistent with more recentstudies of z ∼ M (cid:63) = 2 . × M (cid:12) , φ (cid:63) ( M BH ) = 1 . × − Mpc − dex − and faint-end slope α = − .
03. Further, Willott et al. (2010b) found the Ed-dington ratio distribution at z = 6 to be a log-normal dis-tribution, peaking at 0.6 and with a dispersion of 0.30 dex,which is consistent with the distribution used by Shankaret al. (2013) and this is what we use in our model.We also account for the redshift-evolution of linear sizesof galaxies and how that affects the parameter d , which rep-resents the extent of constant gas density around a source.This redshift evolution goes as ∝ (1 + z ) − . as discussed inSection 2.3.4. Finally, in order for the black hole masses tobe high enough at z = 6, they must have been accreting veryclose to the Eddington limit from very early times after theirformation and must be nearly maximally spun up (Shapiro2005; Daly 2009; Dubois et al. 2014). Therefore, we set thespin parameter at this redshift to a = 1. The simulation isrun for a total of 5 Myrs. The time evolution of the distribution of radio luminositiesat z = 6 is shown in Figure 6. Note that only sources withFRII-like radio powers are shown, as less luminous sourcesat z = 6 are beyond the detection capabilities of currentradio surveys. There are only a handful of sources with verypowerful radio luminosities ( > W Hz − ). This is mainlydue to the generally lower SMBH masses at higher redshiftsand the increased inverse Compton losses due to the denserCMB. The time evolution of the luminosity function is muchstronger too, which can be attributed to rapid energy losses.This is especially prevalent at the most luminous end of the MNRAS000
03. Further, Willott et al. (2010b) found the Ed-dington ratio distribution at z = 6 to be a log-normal dis-tribution, peaking at 0.6 and with a dispersion of 0.30 dex,which is consistent with the distribution used by Shankaret al. (2013) and this is what we use in our model.We also account for the redshift-evolution of linear sizesof galaxies and how that affects the parameter d , which rep-resents the extent of constant gas density around a source.This redshift evolution goes as ∝ (1 + z ) − . as discussed inSection 2.3.4. Finally, in order for the black hole masses tobe high enough at z = 6, they must have been accreting veryclose to the Eddington limit from very early times after theirformation and must be nearly maximally spun up (Shapiro2005; Daly 2009; Dubois et al. 2014). Therefore, we set thespin parameter at this redshift to a = 1. The simulation isrun for a total of 5 Myrs. The time evolution of the distribution of radio luminositiesat z = 6 is shown in Figure 6. Note that only sources withFRII-like radio powers are shown, as less luminous sourcesat z = 6 are beyond the detection capabilities of currentradio surveys. There are only a handful of sources with verypowerful radio luminosities ( > W Hz − ). This is mainlydue to the generally lower SMBH masses at higher redshiftsand the increased inverse Compton losses due to the denserCMB. The time evolution of the luminosity function is muchstronger too, which can be attributed to rapid energy losses.This is especially prevalent at the most luminous end of the MNRAS000 , 1–13 (2015)
A. Saxena et al.
Figure 6.
Luminosity functions at z = 6 at various stages ofsource evolution. The mean ages of the sources contributing tothe luminosity function are shown. At this redshift, it is clearthat sources are much younger and lose their energy much morerapidly. This increased loss of energy can be attributed to themuch stronger CMB energy density, which increases the inverseCompton losses. After ∼ z = 6 must be very young. The faint end isseen to evolve less than the bright end. Figure 7.
Comparison of the evolution of radio power with linearsize (P-D tracks) for sources with given jet powers in our simu-lation at z = 6. Clearly, the Inverse Compton losses come in toeffect much quicker at higher redshifts, as evident from the breakin the P-D tracks at much smaller linear sizes. Overall, sourcesat z = 6 find it much harder to grow to large sizes due to theincreased CMB energy density. luminosity function, which can be seen from the ‘P-D’ tracksfor sources in our simulation, shown in Figure 7. More lumi-nous sources are fuelled by powerful jets and grow faster. Asa result, they enter the regime of strong Inverse Compton(IC) losses much earlier in their lifetime. At lower luminosi-ties, evolution is slower as it consists of sources with rela-tively weaker jets that grow at a lower rate and lose theirenergy less rapidly.Overall at z = 6, sources are younger and lose energy at Figure 8.
The age-weighted probability distribution function andthe cumulative distribution (inset) of predicted linear sizes of ra-dio galaxies at z = 6. The mean linear size is expected to beroughly 25-30 kpc. Large sources at this epoch die out quickerdue to the increased energy density of the CMB. Therefore, radiosources at z = 6 are generally expected to be young and verycompact. a rate that is much higher than seen at lower redshifts. Thishas implications on the nature of sources that we can expectto observe at z (cid:62)
6. Any luminous source observed at thisepoch must be very young and compact, consistent withthe ‘inevitable youthfulness’ of radio sources in the earlyuniverse, as suggested by Blundell & Rawlings (1999).
The higher CMB energy density prevents lobes from grow-ing to large sizes (a few hundred kpc) and as a result, linearsizes at higher redshifts are generally smaller compared tolower redshifts. We apply the age-weighted normalisation in-troduced in Section 3.5, keeping in mind that the detectionprobability of younger sources is less than older ones. Theresulting linear size distribution for sources brighter than0.5 mJy at 150 MHz, which represents the lowest flux den-sities for the LOFAR Tier 1 survey (Shimwell et al. 2017),is shown in Figure 8. Also shown in the inset is the cumu-lative distribution function of the linear size distribution.We find that the median size of radio galaxies at z = 6 is20 kpc, which translates to an angular scale of roughly 3 . z = 6 seems to be in line with theobserved sizes of currently known high-redshift radio galax-ies, as reported by Blundell et al. (1999), Singal & LaxmiSingh (2013), and van Breugel et al. (1999), shown in Fig-ure 9. Also shown as reference is the redshift dependenceof the parameter D IC (see Section 2.3.3) for a constant jetpower (dashed line). It is worth noting that the samplesused to compare our results are flux limited, with the Blun-dell et al. (1999) sample containing sources with S (cid:62) . S > . S (cid:62)
180 mJy. These are likely to introduce various se-
MNRAS , 1–13 (2015) adio luminosity function at high-z Figure 9.
The median linear size of radio sources at z = 6 with S > . D IC , which we show tobe a function of redshift (Equation 8). Our predictions seem tobe in line with the generally decreasing trend of linear sizes withredshift. lection effects that may affect linear size determination thatour model does not take into account. Additionally, the fluxlimits of the above mentioned samples may have implica-tions on linear sizes being probed, as radio power and sizeseem to be correlated, as seen through the P-D tracks andas reported by Ker et al. (2012). Regardless, the sizes pre-dicted by our model seem to be in line with the observedtrend. For further analysis, we take into consideration the predicted z = 6 radio luminosity function (RLF) at an average sourceage of ∼ .
32 Myr. At this time step, the functional form ofthe RLF is well behaved and samples a broad range of lu-minosities. The resulting luminosity function is well fit witha double power law of the form φ ( P ) = φ (cid:63) × (cid:34)(cid:18) PP (cid:63) (cid:19) α + (cid:18) PP (cid:63) (cid:19) β (cid:35) − (10)with parameters log φ (cid:63) = − .
05 Mpc − dex − , log P (cid:63) =27 .
91 W Hz − , faint end slope α = 1 .
04 and bright endslope β = 2 . z ∼ φ (cid:63) = 1 . × − Mpc − (log P ) − , log P (cid:63) = 28 .
95 W Hz − , α = 0 . Figure 10.
Comparison of the radio luminosity function at z = 6predicted by our model (black points) and the RLF obtained byusing a pure density evolution model from z = 2 (solid curve).The dashed curve shows the luminosity function at z = 2 deter-mined by Rigby et al. (2015). The evolutionary parameter foundto best match our prediction at z = 6 is q = − .
49, which in-dicates stronger evolution for radio galaxies but is within theobserved evolution of optical and Xray quasar space densitiesat high redshifts. The shaded region shows RLFs expected with − . < q < − . and β = 2 .
74. We then assume an exponential declinein space density with redshift, which can be written as φ ( P , z ) = φ ( P , q ( z − . , where q is the evolution-ary parameter. Several studies of quasars at various wave-lengths have inferred the value of q to range from − .
59 to − .
43 (Fan et al. 2001; Fontanot et al. 2007; Brusa et al.2009; Willott et al. 2010a; Civano et al. 2011; Roche et al.2012). Assuming the evolution of radio galaxies to matchthe evolution of optically-selected quasars at high redshifts,it seems reasonable to scale the radio luminosity functionusing an exponentially declining model that seems to workfor quasars.We find that q = − .
49 fits our predicted RLF best,which is shown in Figure 10. The dashed region repre-sents − . < q < − .
43. The evolution we find is slightlystronger than what is observed in quasars. This is not sur-prising, as the dependence of radio luminosity on size andlifetime should introduce deviations from the evolution ob-served for quasars. Our model over-predicts space densitiesat the faintest end. This could be due to the fact that thefaint-end slope of the black hole mass function we use as in-put is poorly constrained, which is most likely affecting thefaint-end of our predicted RLF too. At the highest luminosi-ties, space densities predicted by our model are lower thanthose expected from pure density evolution by a factor of0.67 dex in the luminosity bin log P = 28 . − . P = 28 . − .
0, where our model barelypredicts any sources. This is most likely due to the increas-ing dominance of inverse Compton losses at high redshifts,which lowers the space density of the brightest sources thatgrow and lose energy quicker. This effect is not accountedfor by pure density evolution, suggesting that a luminosity-dependent density evolution (LDDE) model (Dunlop & Pea-cock 1990; Willott et al. 2001; Rigby et al. 2011, 2015) may
MNRAS000
MNRAS000 , 1–13 (2015) A. Saxena et al. be better suited to describing the evolution of space densitiesof radio sources out to high redshifts.
We now use the modelled RLF at z = 6 to make predictionsabout number counts in current and future low-frequencyradio surveys. Expected number counts are essential for de-signing surveys and observing strategies that target the iden-tification of the highest-redshift radio galaxies. To calculate expected number counts, we integrate the RLFat z = 6 down to flux limits chosen to represent varioussurveys at 150 MHz with instruments such as LOFAR ,GMRT , and MWA , which are shown in Table 1. It is clearthat the current and upcoming surveys with LOFAR willbe the way forward to detecting a large number of z (cid:62) z (cid:62)
6. Direction-dependent (DD) calibration, which takescare of effects such as varying ionospheric conditions anderrors in beam models, is currently ongoing on the LoTSSfields and will result in high-fidelity images at full resolutionand sensitivity (Shimwell et al. 2017). On completion, theLoTSS direction-dependent survey shall provide a large skycoverage (Dec > z (cid:62) −
53 (Intema et al. 2017). In this sur-vey, one could expect to detect around 92 sources at z = 6.The probability of 21-cm absorption arising from theneutral intergalactic medium at z > τ , that pervades the uni-verse. This in turn depends on a number of key parameterssuch as the temperature of the CMB, spin temperature ofneutral hydrogen and the neutral hydrogen fraction (Car-illi et al. 2002). The detection of such features in the radiocontinuum of z > z = 7 with a flux density of 50 mJy, alonga line-of-sight with τ = 0 .
12. Such a detection would re-quire an integration time of 1000 hours using a bandwidthof 5 kHz. Using a larger bandwidth may bring down the re-quired integration time. Detection of a source with a fluxdensity of 50 mJy at z >
6, however, is very unlikely butthere may be more than 30 sources with flux densities > Figure 11.
Number of high- z radio sources expected to be de-tected as a function of flux density limits reached in a fixed ob-serving time of 100 hours, using standard LOFAR configuration.The upper x-axis shows sky coverage in the given observing timeat a particular flux density limit. Also shown are the 5 σ limits ofcurrent and future low-frequency surveys. The LoTSS DI and DDsurveys seem to be ideal to detect a large number of z > (Carilli et al. 2002). The on-going direction-dependent all-sky survey with LOFAR will be extremely efficient in layingthe groundwork for future 21cm absorption studies with theSKA, when much fainter sources can be used for detectionof 21cm absorption. To compare what the ideal trade-off between depth and skycoverage would be that maximises the detection of z (cid:62) z >
5. The lu-minosity functions at z = 5 and z = 7 are obtained byassuming pure density evolution with q = − .
49, which wasfound to best-fit our z = 6 RLF prediction in the previ-ous section. There clearly lies a ‘sweet-spot’ in the coveragearea vs. depth trade-off, which covers a large enough areaand goes deep enough in a way that maximises detection ofhigh- z radio sources. Note that the 4 π sr. limit is achievedjust after a flux density limit of 1 mJy in the given observingtime. MNRAS , 1–13 (2015) adio luminosity function at high-z Table 1.
Predicted number of radio galaxies at z = 6 detected with 5 σ in current and future surveys at 150 MHz.Survey Ref. Flux lim (mJy/beam) N/deg Area (deg ) Total NLOFAR Tier 3 (planned) 0.01 7.18LOFAR Tier 2 (planned) 0.03 2.28LoTSS a direction-dependent (ongoing) Shimwell et al. (2017) 0.1 0.63 20500 12915LoTSS direction-independent Shimwell et al. (2017) 0.5 0.092 350 32TGSS ADR Intema et al. (2017) 3.5 0.0025 37000 92GLEAM Wayth et al. (2015) 5 0.001 31000 31LOFAR MSSS HBA Heald et al. (2015) 10 0.0001 20500 2 a LOFAR Two-metre Sky Survey
In this study, we have used a semi-analytical model basedon prescriptions laid out by KB07 to predict the luminos-ity and linear size distribution of radio sources. The modeltakes as input the black hole mass function and Edding-ton ratio distribution at any epoch and implements simpleenergy loss mechanisms that dominate at different phasesof a radio source’s lifetime. Radio jet powers are assignedto each active black hole depending on the black hole massand the Eddington ratio, which is randomly sampled fromthe input distribution. As the radio source grows, it initiallyloses energy predominantly by synchrotron emission whenits magnetic field is strong. Adiabatic losses take over in theintermediate phase, with inverse Compton losses due to theCMB radiation dominating in the later stages of a source’slifetime.We first implement our model at z = 2 where sufficientdata for radio luminosities and linear sizes is available. Mak-ing certain assumptions about the prevalent physical condi-tions and the black hole spin distribution, we predict a ra-dio luminosity function that is consistent with observations.We are also able to reproduce the break in luminosity thatmarks the distinction between FRI and FRII radio sourcepopulations, supported by both theory and observations inthe literature. We argue that this bi-modality in source pop-ulation may be due to the accretion mechanism (thin diskvs. ADAF) that is responsible for powering the radio jets.Further, we are able to reproduce the distribution of linearsizes observed in flux limited surveys from the literature.We then extend our modelling to z = 6 where radiosources can be unique probes of the epoch of reionisation as21cm absorption features in the radio continuum of a sourceat z > z = 6.We show that radio sources at z = 6 do not live for verylong compared to typical ages of sources in the low-redshiftuniverse. This is mainly due to radio jets being intrinsicallyweaker because of lower black hole masses and due to in-verse Compton (IC) scattering being highly dominant inthe early universe (due to a higher CMB energy density)that frustrates the jets and suppresses the growth of linearsizes of sources at high redshifts. Further, the predicted dis-tribution of linear sizes is consistent with observations ofthe currently known highest redshift radio galaxies and thegenerally decreasing trend observed between linear sizes andredshift.We compare the predicted z = 6 RLF with a pure den- sity evolution model from lower- z for radio galaxies, basedon the evolution of QSO luminosity functions. The evolutionis of the form φ ( P , z ) = φ ( P , q ( z − . . We find that q = − .
49 fits reasonably well with luminosities between10 . − W Hz − . There is some disagreement at thehighest luminosity end and we attribute this to the signifi-cantly enhanced inverse Compton losses at higher redshifts,that have the most impact on luminous sources. Such aneffect is not captured by a pure density evolution model andwe argue that luminosity-dependent density evolution wouldbetter explain the redshift evolution of the radio luminosityfunction.We finally predict the number of high redshift radiogalaxies that may be observed in current and future low-frequency surveys with LOFAR, GMRT and MWA. To bet-ter understand the trade-off between coverage area anddepth in a way that maximises detection of high- z radiosources, we calculate the total number of sources that wouldbe expected as a function of flux density limits for a fixed ob-serving time. We show that the LOFAR Two-metre Sky Sur-vey (LoTSS) direction-independent and direction-dependentcalibration surveys sit at the sweet-spot of coverage area anddepth, and should be most effective at detecting large num-bers of z > z > ACKNOWLEDGMENTS
We thank the referee for useful comments and suggestions.AS is grateful to Philip Best, George Miley and KinwahWu for fruitful discussions and comments over the course ofthis work. AS and HJR gratefully acknowledge support fromthe European Research Council under the European UnionsSeventh Framework Programme (FP/2007-2013)/ERC Ad-vanced Grant NEWCLUSTERS-321271. EER acknowledgesfinancial support from NWO (grant number: NWO-TOPLOFAR 614.001.006).This work has made extensive use of IPython (P´erez& Granger 2007). This research made use of Astropy, acommunity-developed core Python package for Astronomy(Astropy Collaboration et al. 2013). The data analysis andvisualisation were done using TOPCAT (Taylor 2005). Allfigures used in this paper were produced using matplotlib(Hunter 2007). This work would not have been possiblewithout the countless hours put in by members of the open-source community all around the world.
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MNRAS000 , 1–13 (2015) A. Saxena et al.
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APPENDIX A: CALCULATION OF INVERSECOMPTON LOSS DISTANCE
Here we show the calculation of the distance from the galaxyat which Inverse Compton (IC) losses begin to dominate.This is done by equating the predicted luminosities from the
MNRAS , 1–13 (2015) adio luminosity function at high-z Table A1.
Fiducial model parameters used in this study, takenfrom KB07.Parameter ValueA 4 ρ − kg m − d z (cid:54) × [(1 + z ) / − . kpc at z > u × − J m − f n . × s kg − m − f p . C . a / f L . × − × / √ .
15 J / m / s kg − adiabatic loss phase and the IC loss phase. The expressionsfor luminosities are adapted from Kaiser & Best (2007).For a source in the adiabatic loss phase at an observingfrequency of 150 MHz L = 3 f L Q jet p / a t (A1)where Q jet is the jet power, p is the pressure in the lobe andthe constants f L = 3 . × − (cid:0) . (cid:1) − / J / m / s kg − ,and a = 3 / p = f p ( ρd β ) / Q / D ( − − β ) / (A2)The luminosity of a source in the IC loss phase can bewritten as L = 3 m e c f n Q jet p √ Au CMB0 (1 + z )
150 MHz (A3)where m e is the mass of an electron, c is the speed of light, u CMB is the CMB energy density and the constants f n =1 . × s kg − m − and A = 4 (Kaiser & Best 2007).The present day CMB energy density, u CMB0 is ∼ × − J m − and scales with redshift z as (1 + z ) .Linear size, D is related to the age of the source, t as D ∝ t / (5 − β ) (A4)For expansion into the ambient medium, we take β = 2and therefore, there is a linear relation between source age t and linear size D , t ∝ D . We then equate equations A1and A3 using A2 to calculate the distance at which IC losseswould begin to dominate over adiabatic losses. This distancedepends on the initial jet power of the source and is givenby D IC = const × ( ρd β ) − / Q / (1 + z ) − / (A5)where the constant is given by m e c f n f / p C (3 + a )14 f L √ Au
150 MHz (A6)The values of model parameters used in this study areshown in Table A1.
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