Emergence of cosmic structures around distant radio galaxies and quasars

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quasars

Overzier, Roderik Adriaan

Citation

Overzier, R. A. (2006, May 30). Emergence of cosmic structures around distant radio

galaxies and quasars. Retrieved from https://hdl.handle.net/1887/4415

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Corrected Publisher’s Version

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Licence agreement concerning inclusion of doctoral thesis in the

_{Institutional Repository of the University of Leiden}

Downloaded from:

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Abstract. _{This c}_hapt_{er at}_t_empt_{s t}_{oprovide new c}_onst_raint_{s ont}_{he sc}_enariof_{or t}_{he f}_ormat_ionof galaxy clusters, based, inpart, onthe observationalevidence presentedinthis thesis. The chapter is structuredin3 parts. InpartI, we compile the firstoverviewofobservationalevidence for overden-sities ofgalaxies betweenz=2 andz=6. The overdensities, estimatedfrom the number densities

ofstar-forminggalaxies (Lyα_emit_t_inggal_{axies andLymanbreakgal}_axies)rel_at_{ive t}_{orandom fiel}_ds,

are∼1−7.Ifthese structures were tocollapse under the influence oftheir owngravity, their masses

wouldbe∼1014to1015 M. Because this is comparable tothe masses ofclusters ofgalaxies inthe localuniverse, we define the term ‘protocluster’ as beinganobjectthatmeets the requirements for forminga boundobjectonthe mass scale ofa cluster prior to,or at, the presentepoch, butwhich has notyetcollapsedandvirializedatthe epoch correspondingtoits observedredshift. InpartII, we use simple theoreticaldescriptions for the growth ofoverdensities ina ΛCDM universe tostudy the evolutionofthe sample ofcandidate protoclusters compiledinpartI. Usingconservative estimates ofthe overdensities, we findthatthe majority ofthe structures are likely tocollapse withina finite time. We identify severalstructures as meetingthe requirements for virializationatz≈0.5, whereas

others are expectedtohave fully collapsedby the presentepoch. We compare the predict edabun-dance ofdarkhalos as a functionoftheir (linear)overdensities, mass andredshift, tothe protocluster data. We findthatthe protoclusters lie indarkhalos with number densities of10−6

to10−5 Mpc−3

, andconclude thatthey are associatedwith clusters thatbecome virializedbetweenz≈0 for M'1015

Mandz≈1 for M'10 14 _M

. We show thatthis is inagreementwith recentresults from N-body simulations. We compare the extrapolatedbias ofdarkhalos hostingprotoclusters andradiogal ax-ies atz∼3 tothe bias ofLymanbreakgalaxies (LBGs)anddistantredgalaxies (DRGs)atz=3−6.

The bias ofprotoclusters atz∼3 (b∼8)implies thattheir present-day descendants lie indarkhalos

thatare∼5−10 times more massive thanthose hostingthe z=0 descendants ofluminous LBGs or

DRGs, although eventhe latter populations are associatedwith group-or moderate clustertype en-vironments atz=0. InpartIII, we modelthe star formationhistory ofcluster redsequence galaxies

inorder tocompare their luminosities extrapolatedtoz>_{2 t}_ot_{he prot}_oc_l_ust_{er dat}_{a. We findt}_hatt_he

totalstellar mass inthe cluster redsequence was builtupover the redshiftrange z∼10−2 with star

formationrates ofseveralhundreds toa thousandofMyr −1

assumingconstantstar formation. We show thatthere is goodagreementwith the star formationrates as measuredfor Lyα_emit_t_{ers and}

LBGs, andthe totalextrapolatedstar formationrates inprotoclusters. Summarizing,the overdensi-ties, the masses andthe star formationrates ofprotocluster candidates are ingeneralagreementwith the properties expectedfor the progenitors ofclusters inthe localuniverse.

R. A. Overzier, G. K. Miley, H. J. A. R¨ottgering,S.Mei& B. P. Venemans(2006)

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10.1 I

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The formation and evolution of structure in the universe is a fundamental research area in mod-ern cosmology. Clusters of galaxies represent the most extreme deviation from initial condi-tions in the universe, and are therefore good lab-oratories for testing evolutionary scenarios for the formation of the large-scale structure, and their properties are closely tied to the cosmo-logical parameters (Bahcall & Fan 1998). While clusters of galaxies have been studied exten-sively in the relatively nearby universe, their evolutionary history becomes obscure beyond roughly half the Hubble time. Their progenitors are extremely difficult to identify when the den-sity contrast between the forming cluster and the field becomes very subtle, and mass conden-sations on the scales of clusters are extremely rare at any epoch (Kaiser 1984).

Overdensities of galaxies have been discov-ered out to z ≈6 (e.g. Pascarelle et al. 1996;

Steidel et al. 2005;Keel et al. 1999;Francis et al. 2001;M ¨oller & Fynbo 2001;Venemans et al. 2002, 2004, 2005a,b;Shimasaku et al. 2003; Ouchi et al. 2005;Stiavelli et al. 2005). Some of these structures were found as by-products of wide field surveys using broad or narrow band imaging. Others were traced by a luminous ra-dio galaxy or quasar that pinpointed the over-dense regions (e.g. Steidel et al. 1998;Kurk et al. 2000;Pentericci et al. 2000;Venemans et al. 2002, 2004;Miley et al. 2004;Venemans et al. 2005a,b; Overzier et al. 2006a,b;Zheng et al. 2006). Al-though these structures are all overdense com-pared to the field, their derived physical prop-erties are generally highly uncertain. In gen-eral, the galaxy overdensities are on the order of a few and imply group- or clusterlike masses of 1013−15

M, projected sizes of several to sev-eral tens of (comoving) Mpc, and in some cases measured velocity dispersions of a few hundred km s−1_{determined from emission line galaxies.} Because of the variation, as well as the uncer-tainties, in their sizes, topologies and masses they have been associated with overdense re-gions within the large-scale structure, such as filaments, or more special structures that are the

progenitors of local clusters and galaxy groups. The inferred total masses of the suspected protoclusters usually relies on the assumptions that (i) the overdensity measured for one tracer population of galaxies (usually Lyα emitters

(LAEs) or Lyman break galaxies (LBGs)) is rep-resentative for the total underlying mass over-density of the structure (its various components being dark matter, gas and other types of galax-ies, besides the tracer population) and (ii) that the various other galaxy populations occupy the same volume that is occupied by the tracer pop-ulation. These assumptions are at present diffi-cult to verify without time-consuming spectro-scopic campaigns over large areas of the sky. Can we use any other signatures expected from forming clusters to establish whether the high redshift galaxy overdensities observed are in-deed the suspected sites of cluster formation? One of the tell-tale signs of clusters, at least in the relatively nearby universe, is their high lu-minosity and extended X-ray emission. How-ever, the detectability of distant clusters using X-ray observations is proportional to (1+_z)−4_,

even when it is assumed that there is no evo-lution in cluster abundance, as well as their properties, with redshift. The most distant X-ray clusters that have been found (e.g. Rosati et al. 1998;Mullis et al. 2005) are expected to lie very close to the maximum redshift achiev-able by current surveys (z∼1.5). Because the

X-ray luminosity of clusters is furthermore pro-portional to the square of the gas density within the virial radius of the cluster, and virialization is believed to have occurred by z∼1−1._{5 (for}

the most distant and massive clusters known), little X-ray emission is expected from clusters or cluster progenitors beyond z'2. X-rays are

therefore not useful as a tool for the identifica-tion of a (forming) cluster.

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Figure 10.1—Color-magnitude diagram ofCl1252–2927at z₌1

.24within the central 20ofthe cluster,

withellipti-cal galaxies indicatedby (filled) circles andS0 galaxies by squares. The lines indicate fits to the CMR for ellipticals only (solidline) andfor all early-types (dashed). The dot-dashedline represents the relation for the Coma Cluster, transformedto these bandpasses at z=1.24assuming no

evolution. The figure was taken from Blakeslee et al. (2003).

in clusters (see Fig. 10.1). The relation is formed almost exclusively by early-type galaxies. The existence of the red sequence implies that star formation ceased at a sufficiently early epoch to allow the colors redden passively up to the cluster age at the observed epoch. The early-type galaxies on the red sequence are the most massive and oldest galaxy constituents of clus-ters, even for clusters at z∼1, where masses of ∼1011 M and formation redshifts of ∼2−5 are inferred (e.g. Ellis et al. 1997; van Dokkum et al. 2000; Stanford et al. 2002, 2005; Blakeslee et al. 2003, 2006; Holden et al. 2005; Postman et al. 2005; Mullis et al. 2005; Mei et al. 2006). The epoch of cluster formation is presumed to be marked by the violent build-up of the stellar mass contained in this early-type population. Constraints on the star formation history are (i) the color-magnitude relation, (ii) galaxy mor-phologies, (iii) the metal enrichment of the in-tra cluster medium (ICM), and (iv) the Butcher-Oemler effect (the empirical evidence that dis-tant clusters have a higher fraction of relatively blue (late-type) galaxies than nearby clusters. How do thestar formationrates observedin

protoclusters correspondto predictions based ontheformationoftheseredsequencegalax -ies? Aretheamplitudes ofthegalaxyove r-densities observedconsistent withwhat struc-tureformationpredicts for theprogenitors of galaxyclusters?The answers on these impor-tant questions may shed new light on the pro-cess of structure formation in the universe.

In this chapter we will attempt to address the questions raised above. The structure of this chapter is as follows. In Part I, we review the evidence of galaxy overdensities observed be-tween z=_{2 and 6, and summarize the main}

properties of the structures used in the subse-quent parts of this chapter. In Part II, we review the theory of structure formation, and compare the evolution of overdensities associated with massive dark matter halos to the protocluster data compiled in Part I. We will investigate the abundances, bias and the likely present-day de-scendant populations of the halos hosting pro-toclusters with respect to other classes of high redshift objects. In Part III, we construct a sim-ple ‘toy model’ for the star formation history of the red sequence population of galaxy clusters, and compare the evolution of the total luminos-ity as predicted by our model against the proto-cluster data.

10.2 Pa

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At present the investigation of candidate pro-tocluster fields has resulted in the discovery of about 15 structures spanning the redshift range z=₂₋_{6 (corresponding to}_∼₂.5 Gyr of cosmic

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Table 10.1 —Observational data on protocluster candidates. Object z _Samplea _{Field size}b δc

g σdv Me References† (arcmin2 ) (km s−1 ) (M) PKS 1138–262 2.16 Lyα ₇_×₇ ₃_±₂ ₉₀₀_±₂₄₀ _3–4 _1,2,3,4,5 HS1700–FLD 2.30 BX 8×8 6.₉+2 .1 −2.1 – 14 6 MRC0052–241 2.86 Lyα ₇_×₇ ₂.₀+0.5 −0.4 980±120 3–4 7 MRC0943–242 2.92 Lyα ₇_×₇ ₂.₂+0.9 −0.7 715±105 4–5 7 SA22–FLD 3.09 LBG 9×18 3.₆+1.4 −1.2 – 10–14 8 Lyα ₉_×₉ ₅_±₂ _– _– ₉ MRC0316–257 3.13 Lyα ₇_×₇ ₂.₃+0.5 −0.4 640±195 3–5 7,10 TNJ2009–3040 3.16 Lyα ₇_×₇ ₀.₇+0.8 −0.6 515±90 – 7 TNJ1338–1942 4.11 Lyα ₇_×_{7 (}_×₂₎ ₃.₇+1.0 −0.8 265±65 6–9 7,11 LBG 3.₄_×₃.₄ _∼₂₋₁₆ _– _– _12,_13,₁₄ SDF 4.86 Lyα ₂₅_×₄₅ ₂.₀+1.0 −2.0 – >₃ ₁₅ TNJ0924–2201 5.19 Lyα ₇_×₇ ₁.₅+1.6 −1.0 305±110 4–9 7,16 LBG 3.₄_×₃.₄ ₁.₀_±₀.₅ _– _– ₁₇ SXDF-Object‘A’f _5.₇₀ _Lyα ₆₀_×₆₀ ₂.₃+0.6 −0.6 ∼180 1–3 18 SDSSJ0836+0054 5.82 LBG 3.₄_×₃.₄ ₂.₀+3.0 −1.5 – 1–6 19

a_{Method ofs}_ampl_{e s}_el_ec_ti_{on:(Lyα) narrowband Lyα, (LBG) Lymanbreaktec}_hni_{que, (BX) the ‘}_BX’c_ri_teri_{a ofAdel}_{berger et al}_{. (2005).} b_Approxi_{mate fiel}_{d s}_i_ze.

c_Ampl_i_{tude ofthe gal}_{axy overdens}_i_{ty, δ}

g= (Σ − ¯Σ)/ ¯Σ, calculated from the surface overdensity (Σ) with respect to the average field density ( ¯Σ). Except for HS1700–FLDand SA22–FLDwhere ample spectroscopicinformationwasavailable, the overdensitiesquoted here do not infer any informationonthe velocity structure ofthe systems, and are therefore likely a lower limit to the true overdensitiesinrealspace.

d_Vel_oc_i_{ty di}_s_pers_i_{on(where avai}_l_abl_e).

e_I_nf_{erred mas}_{softhe overdens}_i_{ty i}_nuni_tsof1014_M .

f_Onl_{y the ri}_c_hes_{t ofthe two z = 5.7 overdens}_i_ti_esdi_s_c_{overed i}_nthi_sfiel_{d i}_sl_i_s_ted.

†_Ref_erenc_es_{:(1)Kurket al}_{. (2000), (2)Penteri}_c_c_{iet al}_{. (2000), (3)Penteri}_c_c_{iet al}_{. (2002), (4)Kurket al}_{. (2004a), (5) Kurket al}_{. (2004b), (6)Stei}_del

et al. (2005), (7) Venemanset al. (2005a), (8)Steidelet al. (1998), (9)Steidelet al. (2000), (10)Venemanset al. (2005b), (11)Venemanset al. (2002), (12)Miley et al. (2004), (13)Zirm et al. (2005), (14)Overzier et al. (2006a), (15) Shimasakuet al. (2003), (16)Venemanset al. (2004), (17) Overzier et al. (2006b), (18)Ouchiet al. (2005), (19)Zhenget al. (2006).

10.2.1 Thetargets

•PKS1138–262 (z=2.16)This 70×70fieldc

on-tains a significantoverdensity ofspectrosc op-ically confirmedLyα emitters arounda

mas-sive radiogalaxy (Kurket al.2000;Pentericci et al.2000;Kurket al.2004a,b).Furthermore, the fieldhas beenfoundtobe relatively richin Hαemittinggalaxies havingdifferent velocity

andspatialdistributions comparedtothe Lyα

emitters, severalspectroscopically c onfirmedX-ray sources (Pentericci et al.2002;Croft et al. 2005), as wellas candidate 4000 ˚A breakob-jects at the protocluster redshift, indicatingthat different galaxy populations already exist in clusters severalGyr before virialization.Being

amongthe closest ofthe candidate protocl us-ters, this is a particularly important target for linkinghighredshift protoclusters toclusters at low andintermediate redshifts (z∼1)inorder tostudy morphologicalandkinematicalevol u-tionofgalaxy clusters andtheir star formation histories.The host galaxy ofthe radiosource has beenfoundtoconsist ofa massive c om-ponent surroundedby a large concentrationof smaller, disturbedobjects, andthe entire system inembeddedina 100 kpchaloofemissionline gas (see Fig.1.3inthis thesis).

• HS1700–FLD (z₌₂.30) A highly significant

overdensity at z=2.300±0.015was discovered

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by Steidel et al. (2005) to select star-forming galaxies at z = 2.3±0.4. The structure

cor-responds to the largest, spectroscopically con-firmed galaxy overdensity known at z>_{2, and}

it is expected to become virialized by z ∼ 0 on a mass scale of ∼ 1015 _M

. Comparison between the best-fit spectral energy distribu-tions of galaxies in the protocluster and the field suggests that the structure is relatively rich in evolved galaxies, as expected from simple theo-retical predictions for accelerated structure for-mation (Steidel et al. 2005).

• MRC 0052–241 (z ₌2._{86) andMRC 0943–}

242(z= 2.92) The 70×70 fields towards both

of these z≈2._{9 radio galaxies are significantly}

rich in Lyα_{galaxies in a narrow redshift}

inter-val centred on that of the radio sources. The sys-tems each have∼70 Lyα_{imaging candidates (of}

which>_{20 spectroscopically identified),}

corre-sponding to galaxy overdensities ofδ_g >_{2 and}

masses of ∼3−5×1014_M

(Venemans et al. 2005a).

• SA22–FLD (z = 3.09) This structure was

serendipitously discovered by Steidel et al. (1998), following the presence of a significant ‘spike’in the spectroscopic redshift distribution of a large sample of z∼3 Lyman break galaxies in this field (90

×180

). The derived number den-sity of these structures implies that they are con-sistent with being the progenitors of moderately rich galaxy clusters in their early stages of evo-lution. Subsequent study of this field has shown a similar overdensity in Lyα_{galaxies, as well as}

the presence of several large Lyα_‘

blobs’associ-ated with the protocluster (Steidel et al. 2000).

•MRC 0316–257 (z=3.13) This structure

con-sists of 31 confirmed Lyα _{emitters in a 7}0

×70 field centered on a powerful radio galaxy (Ven-emans et al. 2005a,b). Its velocity dispersion of

∼600 km s−1

, roughly halfway between the typ-ical velocity dispersion associated with the local Hubble flow at z & 3 and that of massive viri-alized clusters at z . 1, possibly indicating that the structure is at an intermediate evolutionary stage.

•TN 2009–3040 (z=3.16) This target is one of

two of the least rich of the overdensities around

radio galaxies found by Venemans et al. (2005a). Although due to its small overdensity it is not considered to be a candidate protocluster, the subclustering of Lyα _{emitters both in angular}

and velocity space suggests that they are phys-ically linked to the radio galaxy, maybe in a group. We include it in our analysis below for comparison.

• TN J1338-1942(z= 4.11) This is the richest

structure among the radio galaxy-selected pro-tocluster targets (Venemans et al. 2005a). Ven-emans et al. (2002) have estimated a mass of

∼1015 _M

based on the overdensity of Lyα emitters within a structure of∼2 Mpc in radius around the radio galaxy TN J1338–1942. The field has been shown to have a similar excess of Lyman break galaxies selected using HST/ACS over an 3.₄0

×3.₄0

area (see chapters 4, 5 & 6 of this thesis), indicating the enhanced star form-ing activity and clusterform-ing associated with the forming structure and the radio source (Miley et al. 2004; Zirm et al. 2005; Overzier et al. 2006b).

•SDF (z=4.86) Strong clustering of Lyα

emit-ters was found in a 20×50 Mpc (in co-moving units) elongated region in the Subaru Deep Field (SDF) using a narrowband centred on Lyα

at z=4.86 (Shimasaku et al. 2003). The

over-density and large structure size may signal the formation of a galaxy cluster.

•TN J0924–2201 (z= 5.19) This radio galaxy,

the most distant known, has 6 spectroscopi-cally confirmed Lyα_{emitting companion}

galax-ies, and appears to lie within an overdense re-gion (Venemans et al. 2004). Observations with HST/ACSfurther indicated an excess of V606 -break (z∼5) objects to∼99% confidence, sug-gesting that the radio galaxy lies in a relatively rich environment, possibly a protocluster.

•SXDF (z=5.70) Ouchi et al. (2005) discovered

two overdense structures of∼1 Mpc in phys-ical size of Lyα _{galaxies within the} _∼ _{1 deg}2 Subaru/XMM-Newton Deep Field (SXDF), cor-responding to a similar volume density of mod-erately rich clusters found in the present-day universe.

• SDSS J0836+0054(z =5.82) The most

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is associated with a large number of candi-date companion objects characterized by very red (1.3<i775−z850 <2.0) color (Zheng et al. 2006,see chapter 8 of this thesis). The surface density in this field is approximately six times higher than the number expected from deep ACS fields, although the relatively small num-ber statistics and lack of spectroscopic confirma-tion make it difficult to quantify the excess.

10.2.2 Derivation ofthetotalmass

The large galaxy overdensities relative to the field observed in each of these structures can be translated into an estimate of the total mass of the structure (cf. Steidel et al. 1998; Venemans et al. 2005a) as follows. The observed galaxy over-density,δg, relates to the mass overdensity,δm, through the bias parameter, b, that relates galax-ies to the underlying dark matter:

1+_b_δ

m=|C|(1+δg), (10.1) where

C=₁+_f− _{f (1}+δ_m₎1/3,_f =_Ω_m_(z)4/7 _(10.2)

is a correction for redshift-space distortion due to peculiar velocities assuming that the object is breaking away from the Hubble expansion (Steidel et al. 1998). Taking the typical bias val-ues of ∼ 2−5 as found for various popula-tions of star-forming galaxies in protoclusters estimated from the clustering of these popula-tions in large field surveys (e.g. Ouchi et al. 2004; Adelberger et al. 2005; Lee et al. 2005), typ-ically yields δm ∼0.2−2 for the targets listed in Table 10.1. Assuming that this overdensity is representative for the true mass overdensity of the structure as a whole as measured within a certain structure volume (or surface area), the total mass is given by

M'(1+_δ_m_{) ¯}_ρ_V_, _(10.3)

where ¯ρis the present-day mean density of the Universe.

The mass estimates of the protocluster candi-dates are typically on the order of 1014−15

M

(see Table 10.1), similar to that of massive clus-ters. However, given that for typical cosmo-logical surveys V & 1014−15

/ρ¯, the genuine pro-genitors of present-day clusters must be shown to have an overdensity sufficiently large in or-der for the structure to collapse and virialize by the current epoch. This will be the subject of the following section, where we shall consider the basic theory of structure formation in or-der to compare its predictions with the observed properties of protocluster candidates summa-rized above.

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The structure of this section is as follows. We first present the elements of the spherical col-lapse model, and use its predictions to inves-tigate whether the properties of our candidate ‘protoclusters’ are consistent with the evolution of the structures into bound objects. Then, we use the framework of structure formation to in-vestigate the evolution of cluster-like halo abun-dances, and compare its predictions to the avail-able protocluster data.

10.3.1 Linearsphericalcollapse

In the local Universe, clusters of galaxies are gravitationally bound objects with masses of∼

1014−16 _M

(bound objects with mass of a few times 1013

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initial perturbations at recombination as charac-terized by the matter power spectrum.

Although collapsed structures ranging from dwarf galaxies to massive galaxy clusters are highly non-linear systems, in explaining struc-ture formation we can largely rely on the linear theory of spherical collapse (see, e.g., Peacock 1999). This is because the growth of cosmolog-ical structure proceeds approximately linearly up to the point of collapse. After collapse, the structure evolves non-linearly until it is virial-ized. In the linear collapse model, a uniform (top-hat) spherical overdense region with a den-sity larger than the local critical denden-sity will behave like an isolated region that initially ex-pands but then collapses to form a bound ob-ject. The turn-around time is the time at which the outermost shell of the system has reached its maximum radius, and decouples from the cosmic expansion resulting from the collapse. In this simplified model, the collapse is com-plete at twice the turn-around time. In real-ity, the object does not collapse to a singular-ity, but stabilizes after a finite time at the radius of virialization, roughly half the radius reached at maximum expansion (see Fig. 10.2). The critical overdensity predicted by the linear col-lapse model is δc '1.686, which is almost in-dependent of cosmology. This critical overden-sity can be used as a simple criterion to study the collapse of density perturbations at differ-ent epochs and at differdiffer-ent mass scales. Once the structure virializes, the true overdensity of the structure will be∼200, at which the linear model becomes insufficient to describe the dy-namical evolution.

10.3.2 Comparison between protocluster over-densities and the requirements for spherical collapse

Here we will investigate whether the matter overdensities estimated for protoclusters at z=

2−6 are sufficient for the structures to col-lapse, and, if so, at which epoch virialization takes place. As discussed in the previous sec-tion, gravitationally bound or collapsed objects of mass M are expected to have formed when their linear matter overdensity, δL, exceeds the

Figure 10.2—Cartoonoftheuniform (top-hat)modelfor sphericalcollapse(from Kaiser 2002). Eachregioninthe universewillexpandwithtime(openline), unlessitslocal densityexceedsthecriticaldensity. Inthiscase, thepert ur-bationreachesa maximum radiusattheturn-aroundtime, andcollapsesattwicethattime(closedline). Inreality, the overdensitydoesnotcollapsetozero, butstabilizesafter a finitetimeattheradiusofvirialization, roughlyhalft hera-diusreachedatmaximum expansion(wavyline).

critical density for collapse of 1.686. In or-der to make the comparison between δL used in the theory, and the matter overdensity, δm, that were estimated based on galaxy overden-sities (δg) measured towards the protocluster targets discussed in the previous section, we use the approximation given by Mo & White (1996,see Bernardeau (1994) for a derivation) that relates the mass overdensity to the linearly extrapolated overdensity in the early stages of the (spherical) collapse:

δL = ₋1.35(1+ δm)− 2/3 +0.₇₈₇₈₅₍₁+ δm)− 0.58661 + −1.12431(1+ δm)− 1/2 +1.₆₈₆₄₇. ₍_10.₄₎ In Fig. 10.3we presentthe linearlyext rap-olatedoverdensitiesdeterminedfrom the pro-tocluster datasummarizedin Section 10.2(i n-dicatedbythe points).For simplicity, the t ar-getshave been groupedin approximate redshift where appropriate.The shadedregionsindicate the linear overdensitiesandtheir 1σuncertainty

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Figure 10.3— Linear overdensitiesasa functionofredshiftbasedonthe observationalevidence for protocluster candidates assummarizedinTable 10.1.Pointswitherror barsshow the linear overdensitiescorrespondingtothe measuredgalaxy overdensities(mostly from Lyα_emi_t_t_ers_)as_s_umi_ngs_pheri_c_alc_ol_l_aps_e,andt_{he evol}_ut_i_onoft_{he overdens}_i_t_i_eswi_t_hreds_hi_f_ti_s

shownby the shadedregions(1σ_range)_{.The hori}_zont_{albar i}_ndi_c_at_est_{he c}_ri_t_i_c_alc_ol_l_aps_{e t}_res_hol_d,δc₌1.686,for forming

boundobjects.The available data suggeststhatthe protocluster candidateshave varyingpropertieswhenevolvedtothe currentepoch:some structuresundergocollapse by z∼0._5,ot_{hersby z}_≈_0,whi_l_{e s}_{ome s}_t_ruc_t_{uresare notdens}_{e enoughf}_or

undergoingcollapse evenby z=0.See textfor details.

to δL(z2)₌ D_(z2)δL(z1) D_(z1) , (10.5) where D_(z)= _g(z)_/[g(0)(1+_z)]_, (10.6) g(z)= 5 2ΩM,z[Ω 4/7 M,z−ΩΛ,z+ +₍₁+_ΩM,z/2)(1+Ω_Λ,z/70)]− 1 , ΩM,z= ΩM(1+z) 3 /E2(z), ΩΛ,z= Ω_Λ/E 2 (z), E2 (z)= _[Ω_Λ+(1−ΩΛ−ΩM)(1+z) 2 +_ΩM(1+z) 3 ].

using the approximations tothe cosmological growthin a ΛCDM universe given byCarrollet al.(1992).

In eachofthe panels ofFig.10.3we have indicatedthe criticallinear overdensityfor col -lapse.The simple extrapolation ofthe measured overdensities tolater epochs illustrates a range ofinteresting aspects ofthese structures.Some ofthe structures (e.g.the targets HS1700–FLD atz=2.3,TN J1338–1942 atz=4.1 andSXDF

atz=5.7)have sufficientlylarge overdensities

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well before the present epoch. For these objects, collapse is predicted to occur at z∼0.5. Most of

the other structures are consistent with collapse by z∼0 (e.g., PKS 1138–262 at z=2.16,SA22–

FLD and MRC 0316–257 both at z =3.1, SDF

at z=4.9, TN J0924–2201 at z=5.2, and SDSS

J0836+0054 at z=5.8). Some of the

overdensi-ties observed seem too small for collapse even by z=0, for example for 2009–3040 at z=3.1,

in agreement with the conclusion of Venemans et al. (2005a) that the overdensity is too small to qualify as a protocluster candidate.

Taking the results at face value, we have demonstrated that the current sample of proto-clusters represents a class of objects that, gen-erally speaking, meets the requirements for col-lapse on a mass scale that is comparable to that of galaxy clusters. How do the linear overden-sities derived here, and the abundance of pro-toclusters observed, compare to the predicted halo abundances as a function of redshift and halo mass?

10.3.3 Theevolutionofmassfluctuations Here we give the necessary ingredients for describing the evolution of mass fluctuations starting from the initial perturbations to the present (see, e.g., Peebles 1980;Peacock1999; Mo & White 2002;Kaiser 2002;Tozzi2006). Be-cause the predictions for structure formation rely heavily on the cosmology, observational cosmologists have had to struggle with widely varying model predictions depending on which cosmological model was used. Currently, the-ory predictions have become considerably more reliable due to the vastly improved accuracy of the fundamental cosmological parameters. In the discussion below we will use the concor-dance ΛCDM cosmology, ΩM₌0_.3, Ωλ ₌0_.7, h₌₀

.7, H0=100h km s−

1 .

Because the cosmicdensityfieldisapproxi -matedtobe linear, atanymomentwe canrelate a givenmasstothe radiusofa sphericalvolume inwhichthatmassiscontained

R(M)= 3M

4πρ¯0 1/3

, (10.7)

where ¯ρ0 isthe currentaverage densityofthe universe. Under the assumptionofGaussian fluctuationsinthe densityfield(aspredictedby inflation), the entire massfieldcanbe charact er-izedonlybyitsvariance

σ2 (M)= 4π (2π)3 Z ∞ 0 dk kk 3 Plin(k)W 2 (kR), (10.8) whereσ(M) isthe rmsvalue ofthe densityfluc -tuationsonmassscale M whensmoothedwitha top-hatfilter ofradiusR havinga Fourier trans -formofW(x)=₍₃/_x3_)[_s_i_nx−x cosx], andP_lin(k)

isthe linear power spectrum ofdensityfluct ua-tionsextrapolatedtoz=_{0. We us}_{eda ΛCDM}

power spectrum ofthe form Plin(k)∝ knT2

k(q) withn=_{1 andt}_{he mat}_t_{er t}_rans_f_{er f}_unc_t_i_on

Tk(q) = ln(1 +₂_._34q) 2.34q (10.9) ×[1+₃_.₈_9q+₍₁₆_._1q)2+ +₍₅_.₄_6q)3+₍₆_.₇_1q)4_]−1/4_, q = _(k/_h)Γ, Γ = _e_xp(ΩB+ √ 2H0ΩB/ΩM)/ΩMH0c

from Bardeenetal. (1986);Sugiyama (1995). The power spectrum isnormalizedbyrequiring thatthe present-dayrmsmassfluctuationina sphere ofradius8 h−1

Mpcisσ8=0.9.

The above prescriptionsfor massfluctuations canbe usedtopredictthe number densityof present-daybound structuressuchasgalaxy clusters, andtheir evolutionatdifferentepochs. Thisproblem isusuallyaddressedbyst udy-ingthe massfunction n(M,z)dM, which de-scribesthe number densityofbound objects withmassesbetweenM andM+_{dM atreds}_hi_f_t

z, or N(>M,z), whichisthe number densityof objectsmore massive thanM. Althoughthe col -lapse andvirializationofoverdensitiesare non-linear innature, the processofcollapse i sun-likelytochange the totalmasscontainedbythe overdensityfilteredona scale R. Therefore, the massfunctioncanbe constructedfromthe num-ber densityofregionsthathave anoverdensity

δ > δc.

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andredshiftcanbe approximatedusingthe un-conditionalmassfunction derivedbySheth& Tor-men(1999);Shethetal. (2001);Sheth& Tormen (2002): νf (ν)=_2A 1+ 1 ν02p ν02 2π 1/2 exp −ν 02 2 , (10.10) whereν0₌√

aν, a=₀_._707,andq=₀_._{3 are bas}_ed

ona fittothe massfunctionofthe numerical GIFsimulations(Kauffmannetal. 1999), and A=₀._{322 f}_ol_l_owsf_romt_{he requi}_rementt_hatt_he

integraloff (ν) over allνshouldgive unity. The halomassM isrelatedtothe rmsdensityfluc -tuationsthroughthe parameter

ν ≡ _δ c D(z)σ(M) 2 . (10.11)

The massfunctionisaccordingly n(M,z)dM= ρ¯

Mνf (ν) dν

ν . (10.12)

The typicalhalomassthatcollapsesateac hred-shift, Mc(z), isdefinedbyν =1, i.e., σ(Mc)=

δc/D(z).

We have usedthe above formalism t orepro-duce the evolutionofhaloabundancesasshown byMo& White (2002). The resultisshown inFig. 10.4, showingthe number densityof collapsed, darkhalosasa functionofredshift and(minimum) mass. The figure illustratesthat the number densityofcollapsedhalosofmass

> 1015

M are asfrequentinthe present-day universe, ashalosof>1012

M atz∼10, and thatlocallyhalosof>1012

M are 4 ordersof magnitude more abundantthanthose of>1015 M.

10.3.4 Comparisonbetweenprotoclusterover -densitiesandhaloabundances

InFig. 10.5we plotthe predictedabundance ofthe progenitor halosof1015

M halos, asa functionoftheir linear overdensityandredshift (the predictionsfor 1014

M halosare shown asdottedlines). For your guidance, checkthat

0 5 10 15 20 z -8 -6 -4 -2 0 2 lo g N c (> M ) [( h -1 M p c) -3 ] 7 8 9 10 11 12 13 14 15

Figure 10.4—Darkhalonumber densities as a functionof redshift. Thenumberedlines indicatethecumulati veabun-danceofhalos ofmass log(M_/M).

the number densitycorrespondingtothe li n-ear collapse thresholdof1.686 (dashedline) at z=_{0 i}_s_'₁₀−6 _h3 _Mpc−3_{, c}_ons_i_s_t_entwi_t_ht_he

abundance ofcollapsedhalosofthismassplot -tedinFig. 10.4. The plotfurther showsthat atz∼2, for example, the linear overdensityof

suchhaloswas∼0.8.Assumingthatthe prot

o-clustersare the progenitorsof∼1015M halos (see Table 10.1), how dotheir overdensitiesand number densitiesfitinwiththese modelpre-dictions? Unfortunately, the abundance ofthe protoclustersislargelyunknowngiventhe very few objectsdiscoveredtodate andthe compli -catedselectioneffects. However, we canplace at leastsome constraintsfromthe observedspread in(linear) overdensitiesofδL=0.3−0.8, and

the correspondingredshiftsofthe protoclusters ofz=₂₋_{6 (poi}_nt_si_ns_i_{de t}_{he darks}

hadedre-gioninFig. 10.5). Furthermore, we indicate the approximate number densityofpowerful radiogalaxiesatz∼3 asestimated1

byVen-emansetal. (2002) (indicatedbythe medi um-darkshadedregion,allowingfor a factor 10

un-1

Basedonthenumber densityofradiogalaxies at2.7<

z_<₃_._{4 wi}_t_hl_umi_nosi_t_i_e_{s e}_xc_e_e_di_{ng 10}33

erg s−1

Hz−1

sr−1

at2.7 GHz,taking intoaccounta radiosourcelife-timeof 107

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0.5

1.0

1.5

2.0 δ

L

-8

-7

-6

-5

-4

lo

g

N

(>

δ

L

)

[(

h

-1

M

p

c)

-3

]

0.5

1.0

1.5

2.0 δ

L

-8

-7

-6

-5

-4

lo

g

N

(>

δ

L

)

[(

h

-1

M

p

c)

-3

]

z=0 z=0.5 z=1 z=2 z=3 4 5 6 7 1014 MO• 1015 MO•

Figure 10.5—The cumulative number densities of the progenitors of 1015_M

halos as a function of linear overdensity and

redshift (numbered solid lines). Dotted lines are for dark halos of mass 1014

M. The points inside the dark shaded area

indicate the range of overdensities (δL∼0.3−0.8)measuredatthe corresponding redshifts ofthe protocluster targets ofTable

10.1 andFig.10.3.Althoughthere are fewconstraints onthe abundance ofprotoclusters,we markthe approximate number densityofpowerfulradiogalaxies atz∼3 (medium shadedregion),andthe number densityofprotoclusters inferredby Steideletal.(1998) basedonthe statistics ofLBGredshiftspikes (lightshadedregion).The verticaldashedline indicates the criticalcollapse thresholdof1.686.Ifradiogalaxies,LBGspikes andthe galaxyoverdensities observedare associatedwith the progenitors of∼1015Mhalos,their number densities oflogN(> δL)∼ −(5−6) h3Mpc−3implycluster virializationat

z_∼₀.₅₋_{0 f}_{or t}_{he l}_{ower val}_ue,andnoc_ompl_et_{e c}_ol_l_{apse byz}₌_{0 f}_{or t}_{he hi}_{gher val}_{ue.The hat}_c_{hedareamarks t}_{he observed}

number densityofclusters atz. 1 withX-rayluminosities of& 5×1043

erg s−1

(Rosatietal.2002).

certainty in the number density). We al soindi-cate the number density ofz∼3 protoclusters

as inferredby Steidelet al. (1998) basedon the statistics ofLBG redshift spikes (light shaded region,witha factor 10 uncertainty). Ifradio galaxies,LBGspikes andthe range ofstructure overdensities observedalltrace the same mas-sive halos,their number densities are predicted tobe in the range from N(> δL)₌10−

6

to10−5 h3

Mpc−3_{. Fol}_l_{owing the redshif}_{t evol}_{ution at} these haloabundances using the lines in Fig. 10.5 imply cluster virialization by z≈0.5 for the

overdensities at z'4−6.Some ofthe

overden-sities at z'3 seem toosmallfor complete col

-lapse even by z=0.

We alsoindicatedthe observednumber den-sity ofclusters at z . 1 withX-ray l uminosi-ties of& 5×1043erg s−1(hatchedregion in Fig. 10.5). For a massive cluster tocollapse by z∼1,

it is predictedtohave δL _≈0.7 at z_∼4. This is higher than observedfor protoclusters at this redshift:TNJ1338–1942at z=4.1,for example,

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0 1 2 3 4 5 6 7 z 0 2 4 6 8 10 B ia s 1010 1011 1012 1013

Figure 10.6— The inferred bias of luminous radio galaxies (star) and LBG protoclusters (large square) based on their number densities derived by Venemans et al. (2002) and Steidel et al. (1998). Other populations shown are Lyman break galaxies at z=3−6 (circles, taken from Ouchi et al.

(2004) and chapter 8 of this thesis), near-infrared selected galaxies (relatively red objects (upper open square), blue ob-jects (lower open square) and all objects (small filled square), from Daddi et al. (2003)), and luminous radio sources at z_∼_{1 (bar, derived from chapter 2 of this thesis).}

10.3.5 Thebiasofprotoclustersandtheirz=

0 descendants

In the halo mass model given in Section 10.3.3, the bias of the dark halos is given by

bDH=1+ 1 δc ν02₊_b ν0(1₋c) − ν02c_/√_a ν02c₊_b(₁ −c)(1−c/2) , (10.13)

where a=0.707,b=0.5,c=0.6 (Sheth etal.

2001). We have estimatedthe biasofprotoclus -ters,bylookingupthe bDH(z₌3) corres pond-ingtothe number densitiesofluminousradio galaxiesandLBGspikesatz∼3 gi

venbyVen-emansetal. (2002) andSteideletal. (1998). We findb∼8 (see Fig. 10.6). Atthisredshift,the

biascorrespondstocollapsedhalosof∼5×1013

M_{. For c}_ompari_s_{on,we have al}_s_oi_ndi_c_at_ed inFig. 10.6 the biasvaluesfoundfor several other classesofobjects. Atz∼ 1,luminous

radiosourceshave b∼ 2−3,suggestingthat

theycorrespondtomassive halos(see chapter

2 ofthisthesis). Atz=3−5,the clustering

ofbright(L & L∗) LBGsindicate b∼3−5 and halomassesof∼1012M (circles,from Ouchi etal. (2004)). Near-infraredselectedgalaxiesat z_∼_{3 s}_{how t}_hatt_{he bi}_asmays_t_rongl_ydepend oncolour (squares,from Daddietal. (2003)). For completeness,we have alsoillustratedthe (mild) constraintsonthe biasofi775-dropoutsat z_∼_{6 deri}_vedi_nc_hapt_{er 8 oft}_hi_st_hes_i_s_.

FollowingOuchietal. (2004),we canext rap-olate the biasmeasurementsfor these objectsto z₌_{0 us}_i_ng(Shet_{h etal}_{. 2001):} b0_DH=₁+ ₍₁₀_.₁₄₎ + D(z) δc h ν02₊_b_ν0(1−c) − ν02c_/√a ν02c+b(1−c)(1−c/2) i , and byassumingthatthe biasofa given class ofobjectsisrepresentative ofthe biasofthe darkhaloshostingthem,i.e.,bg ' bDH. The resultsare shown in Fig. 10.7,where we plot the number densitiesofthe present-daydesc en-dantsofthe objectsin Fig. 10.6asa function of halomass. Aspreviouslyshown byOuchiet al. (2004),the descendantsofallthe classesof objectsshown lie in the massrange corres pond-ingtogroupsand clustersofgalaxies. Albeitby construction,Fig. 10.7illustratesthatthe num-ber densityofthe z=0 descendantsofluminous

radiogalaxiesand protoclustersatz∼3 implies

masseswellin the range for galaxyclusters, 10.3.6 Summary

There are severalconclusionsthatfollow from our analysisofthe observed properti esofpro-tocluster candidates:

(1)We calculated the linear overdensitiesc or-respondingtothe massoverdensitiesderived from the observations,findingδ_L_∼₀.₃₋₀._8.

(2)The observed overdensitiesare in approxi -mate agreementwiththe amplitudesofmassive haloprogenitorsatz=2−6.

(3)The number densitiesofmassive darkhalo progenitorsagree approximatelywiththe den-sitiesexpected for luminoushigh-redshiftradio galaxies(corrected for a radiosource life-time of

∼107yr),and the estimated number densityor

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12.5 13.0 13.5 14.0 14.5 15.0 15.5 logM [(h70 -1 MO•)] -8 -7 -6 -5 -4 -3 -2 lo g N (> M ) [( h70 -1 M p c) -3 ]

De

s

c

e

n

d

a

n

t

s

a

t

z

=0

Groups Clusters

Figure 10.7—The halomassandnumber densityofthe z=

0 descendantsofthe populationsshowninFig.10.6.The de-scendantsofallofthe highredshiftpopulationsshownare expectedtoendupinhaloswithmassesrangingfrom those ofgroupstoclusters,withthe higher cluster-like masses clearlypreferredbythe descendantsofradiogalaxiesand protoclusters.

(4) The overdensities of the protocluster candi-dates are, on average, sufficiently large to form bound objects with masses of 1015 _M

at z . 0.5. (5) The estimated number density of protoclus-ters implies b∼8 at z∼3, roughly twice as high

as the bias of luminous galaxies at similar red-shifts as measured by, e.g., Daddi et al. (2003) and Ouchi et al. (2004).

10.4 _{Part I}

_I

_:Mode

_l

_i

_ngthehi

_s

_toryof

thec

l

us

te

r re

ds

e

que

nc

e

As detailed inthe previous sections, the ampli -tudes and masses ofthe galaxyoverdensities found suggestthattheyare likelyprogenitors ofclusters. However, sofar we have ignored the baryonicmatter componentofclusters al -together. Here, we shallconstructa verysi m-ple ‘toymodel’for the star formationhistoryof clusters. Because itis anestablished factthat mostofthe stellar mass contained inthe cl us-ter red sequence populationwas formed

atred-shifts similar tothe redshifts ofour protocluster candidates (see Blakeslee etal. 2006,and ref er-ences therein), we willrestrictour analysis to modelingofthe red sequence ofclusters. The goalofour simulations is tobuild a libraryof models thatare able toreproduce the observed intrinsicscatters around the color-magnitude relation(CMR)as observed for massive hi gh-redshiftclusters, and compare the extrapolated star formationhistoryofthose models against the available protocluster data. For these si mu-lations we shalluse the observed CMRofthree massive clusters chosenfrom literature. These are Cl1358+6245atz=0.33 (vanDokkum et

al. 1998), MS1054–0321 atz=0.834(Blakeslee

etal. 2006), and RDCS0910+5422atz=1.106

(Meietal. 2006). The mainobservationaldetails for these clusters are summarized inTable 10.2. 10.4.1 Themodel

The firstpartofour procedure is similar to the method used byvanDokkum etal. (1998), Blakeslee etal. (2003, 2006)and Meietal. (2006), whosimulated the scatter around the CMRto estimate the meanages ofred sequence galaxies ina number ofmassive clusters atz=0.3−1.3

observed withHST. We have used Bruzual& Charlot(2003,BC03)stellar populationmodels tocalculate template colors givena particular star formationhistory. We onlyconsider the solar metallicitymodels witha Salpeter initial mass functionand highresolution(“bc2003 hr -m62salpssp”). We use the truncatedstarfor -mationmodelfor the star formationhistoryof cluster galaxies, whichassumes thatstars form ata constantrate betweent= t1 and t₌ t2, where t1 and t2 are randomlychosentolie at t0<t1<t2<tendwitht0the time atrecombi na-tionand tendthe age ofthe universe atthe red-shiftofthe cluster.

We simulated ‘clusters’byrandomlycreating ‘galaxies’withdifferentformationepochs and differentburstdurations. Instead ofthe st an-dard procedure employed byBlakeslee etal. (2003)and Meietal. (2006)ofcreating∼10,000

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Table10.2—_Sc_at_t_e_rsoft_hec_ol_or-_magni_t_udere_l_at_i_oni_{n3 mas}_s_i_vec_l_us_t_e_rs_. Cluster z La bol,44 T b X fEc NdE Remax Colorf σ g int Refs.† (erg s−1 ) (keV) (arcmin) Cl 1358+6245 0.33 10.65 7.0 0.24 46 4 V606–I814 0.022±0.003 1,2 MS1054–0321 0.83 16.43 8.0 0.49 46 3 V606–z850 0.080±0.015 3,4,5,6 RDCSJ0910+5422 1.11 2.14 7.2 0.37 20 2 i775–z850 0.044±0.010 6,7,8 a

Bolometric X-rayluminosityinunits of1044

erg s−1_. b_X-_rayt_e_mpe_rat_ure_.

c_El_l_i_pt_i_c_alf_rac_t_i_on.

d_Numbe_{r ofe}_l_l_i_pt_i_c_al_sont_here_{d s}_e_que_nc_e_. e_Maxi_{mum radi}_usf_{or obj}_e_c_ts_e_l_e_c_t_i_on.

f_Col_{or us}_e_{d f}_{or fit}_t_i_ngt_hec_ol_or-_magni_t_udere_l_at_i_on. g_Obs_e_rve_{d i}_nt_ri_ns_i_cs_c_at_t_e_r.

†_Re_f_e_re_nc_e_s_:(_{1)vanDokkum e}_tal_{. (}₁₉₉₈₎_,(_2)Fabri_c_ante_tal_{. (}₁₉₉₁₎_,(_3)Bl_ake_s_l_e_ee_tal_{. (}₂₀₀₆₎_,(_4)Gi_oi_ae_tal_{. (}₂₀₀₄₎_,(_5)Rome_{r e}_tal_{. (}₂₀₀₀₎_,(₆₎

Postmanetal. (2005),(7)Meietal. (2006),(8)Stanford etal. (2002).

Table10.3—_Color-_magni_t_udere_lat_i_ons_i_mulat_i_ons_.

Cluster z _Model σ td min hτLie ht1if h∆τig Cl 1358+6245 0.33 a 0.022 3.9 6.8 1.6 2.9 Cl 1358+6245 0.33 b + 2.0 5.7 2.2 4.0 Cl 1358+6245 0.33 c – 5.4 7.6 1.3 1.9 MS1054–03 0.83 a 0.080 1.3 3.9 1.4 2.3 MS1054–03 0.83 b + 0.8 3.7 1.3 3.0 MS1054–03 0.83 c – 2.3 4.3 1.2 2.0 RDCSJ0910+5422 1.11 a 0.042 1.0 3.3 1.0 2.2 RDCSJ0910+5422 1.11 b + 0.1 2.7 1.3 3.0 RDCSJ0910+5422 1.11 c – 2.1 3.8 0.9 1.7 a_Thebe_s_t_-_fitmode_l(_s_e_et_e_xtf_{or de}_t_ai_l_s₎_.

b_The‘_maxi_mal_l_{yyoung’mode}_l(_s_e_et_e_xtf_{or de}_t_ai_l_s₎_. c_The‘_maxi_mal_l_yol_d’mode_l(_s_e_et_e_xtf_{or de}_t_ai_l_s₎_. d_Themi_ni_{mum mode}_lage_.

e_Theme_anl_umi_nos_i_t_y-_we_i_ght_e_{d age}_. f_Theme_ans_t_artt_i_meoft_hes_t_{ar f}_ormat_i_on. g_Theme_andurat_i_onoft_hes_t_{ar f}_ormat_i_on.

tothe actual number ofredsequence galaxies observedonthe CMR.For eachrandomlysi m-ulatedcluster we evolve the star formationhi s-tories tothe redshift (zobs) or epoch(tend) ofob-servation,andcalculate the average color and the scatter ofthe simulatedpopulationtotest

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redse-Figure 10.8— Points showthe scatter in the simulated color-magnitude relation at z=0._33.Eac_{hsimulation c}_{onsisted of46}

model SEDs withconstant formation ofvariable,random duration (from t1to t2) between the age ofthe cluster (tend) and the

age ofrecombination (t0). In (a),we showthe scatter in V606–I814as a function ofthe minimum age at z₌0.33(tend). In (b),we

illustrate howthe increase in the minimum age ofthe models implies a decrease in the (mean) duration ofthe starburst,given the shorter time windowallowed for star formation. In (c)–(e),we respectivelyplot the mean ofthe luminosity-weighted ages at z=0._{33,the mean duration ofstar f}_{ormation (}h_∆τi = ht₂₋t₁_i_{),and the mean time ofonset ofthe star f}_{ormation (}_ht₁_i₎ versus the scatter found in eachofthe simulations. Shaded regions indicate the±1σ_{range ofthe observed intrinsicsc}_atter

from van Dokkum et al. (1998). Large solid circles indicate the best fit model (middle circle in panel (a)),as well as a maximally young and a maximallyold model that still fit the observed intrinsicscatter (left and right circles in panel (a),resp.). These three models are used in Fig. 10.11 to evaluate the star formation historyofthe red sequence galaxies at>₀._33.

quence. The formation epochs and star forma-tion histories are saved to carry out step 2 of our simulation. In this step, the BC03 models are normalised so that the total stellar mass of the model at z∼1 amounts to the typical mass of red sequence galaxies. Given the set of models that yields an appropriate red sequence when evolved to the observed cluster redshifts, and the proper mass normalisation, the full star for-mation history of the simulated cluster is now fixed. It is straightforward to compute the total star formation rate or luminosity of the simu-lated ‘clusters’ at any redshift and in any band-pass. Summarizing, our toy model depends on the following input parameters or constraints: (1) a global model form of star formation history (we assume constant star formation).

(2) Ne, the number of morphologically selected elliptical galaxies in the cluster CMR.

(3) Me, the typical mass of cluster elliptical

galaxies (∼1011_M_).

(4)σ_int, the intrinsic scatter at z₌z_obs.

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mod-0 1 2 3 4 5 6 7 Minimum age(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er 0 1 2 3 4 5 6 7 Minimum age(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er

1054

(a) 0 1 2 3 4 5 6 7 Minimum age(Gyr) 0 1 2 3 4 5 6 7 M ea n b u rs t d u ra ti o n (b) 0 1 2 3 4 5 6 7 Meanluminosityweightedage(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er 0 1 2 3 4 5 6 7 Meanluminosityweightedage(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er (c) 0 1 2 3 4 5 6 7 Meanburstduration(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er 0 1 2 3 4 5 6 7 Meanburstduration(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er (d) 0 1 2 3 4 5 6 7 Meanburststart(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er 0 1 2 3 4 5 6 7 Meanburststart(Gyr) 0.00 0.02 0.04 0.06 0.08 0.10 S ca tt er (e)

Figure 10.9—Simulated scattersinV606–z850ofthe color-magnitude relationinthe cluster MS 1054–0321 at z₌0.834.The

shaded regionindicatesthe±1σrange ofthe observed intrinsicscatter from Blakeslee et al.(2006).See the legend ofFig.10.8

for further details.

els also showthatfor massive cluster ellipticals, the number ofequal mass progenitors is quite low (∼2−3). For suchequal mass mergers to

take place, itis conceivable thatthe two merging galaxies will roughlyhave comparable f orma-tionhistories giventhattheyhave comparable mass atthe time ofthe merger. Our model is i n-dependentofthe exactmechanism thatdrives the star formation (e.g., induced by merger, monolithiccollapse, AGN or superwindf eed-back,etc.). Althoughour model uses a verysi m-plisticapproachthatshouldnotbe considered as analternative for elaborate semi-analytical models or N-bodysimulations ofcluster f orma-tion, itis sufficientfor carryingouta roughor-der ofmagnitude comparisonwiththe available data onprotocluster candidates.

10.4.2 ModelResults

For each simulation, we created Ne galaxies withrandomlychosent1, t2 (giving∆τ = t2₋ t1). ∆τ was inthe range 0.1–10.0 Gyr, withi n-crements of0.1 Gyr. Our three clusters 1358, 1054and 0910 have respectively46, 46and 20 elliptical galaxies onthe CMR. To aidthe practicality, we simulated the model

popula-tions as a functionofthe minimum age, tmin, ofgalaxies atthe cluster redshift. Figs. 10.8, 10.9and10.10 showthe scatters calculatedfrom the simulations (points), comparedto the ob-servedscatter andits 1σerror range

(shadedre-gions). Our best-fitmodel for 1358(z=0.33)

hada minimum age of3.9Gyr (z>0.9), a mean

luminosity-weightedage of6.8Gyr (z ∼2.2),

a meanburstdurationof2.9Gyr, anda mean star formationstartingtime of1.6Gyr after re-combination. For 1054(z=0.83), the minimum

age was 1.3 Gyr (z>₁._{2)andmeanlumi}_nosi_t

y-weightedage of3.9Gyr (z=2.4). For 0910 (z=

1.₁₁₎_{, t}_{he mi}_ni_{mum age was 1.0 Gyr (z}>₁.₅₎

andthe meanτL₌3.3 Gyr (z_∼3.1). The re-sults are summarizedinTable 10.3. Our results are inagreementwiththe results foundbyvan Dokkum etal. (1998), Blakeslee etal. (2006)and Meietal. (2006)for the same clusters.

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Figure 10.10 — Simulated scatters in i775–z850of the color-magnitude relation in the cluster RDCS 0910+5422 at z=1.106. The

shaded region indicates the±1σrange of the observed intrinsic scatter from Mei et al. (2006). See the legend of Fig. 10.8 for

further details.

de-evolved the star formation histories to earlier times, and calculated the total cluster luminos-ity at rest frame 1500 ˚A by adding up the lumi-nosities of each of the Ne models. The result is our final model for the star formation history of each cluster, and is shown in Fig. 10.12. Also plotted are two alternative, extreme model out-comes that reproduced the observed intrinsic scatter (dotted lines). In these models, the star formation either shut off later (on average), or started earlier, compared to the best-fit model. We also indicated the effect of dust to the total luminosity (dashed line). The dust contents was assumed to be constant, and decoupled from redshift, taking E(B−V)=0.1 and applying the

dust law of Calzetti et al. (2002).

Our model shows (depending on the assump-tions) that the massive end of the cluster red se-quence may have formed its stars at a relatively constant rate of several hundred M yr−1over the redshift range∼10−2.

10.4.3 Comparisonwithprotoclusterdata One of the main questions that motivated the analysis presented in this chapter is whether the total stellar mass observed at the brightest

∼3 mag end of z . 1 cluster red sequences is

consistent with being produced by the Lyman break galaxies (and Lyα emitters) observed in

protoclusters at z=2−6. Our models, as

illus-trated in Fig. 10.12, imply that such hypothesis would require an average SFR of several hun-dred to a thousand Myr−1over this whole red-shift range. An estimate of the measured SFR in protocluster fields due to the observed Ly-man break and Lyα_{populations can be obtained}

by looking at the typical star formation rate of these objects. Lyα emitters have a typical SFR

of a few Myr −1

(Venemans et al. 2005a), which multiplied by the typical number of∼30−60

found in protocluster regions gives about 100-200 M yr

−1

. This is still an underestimate of the total UVSFR, as field studies indicate that only about 25%of LBGs have a sufficiently large equivalent width in Lyα_{to be observed and}

de-tected as Lyα _{emitters according to the most}

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10 8 6 4 2 0 Time since big bang (Gyr) 0 10 20 30 40 Ne 10 8 6 4 2 0

Time since big bang (Gyr) 0 10 20 30 40 Ne 7 6 5 4 3 2 1 0

Time since big bang (Gyr) 0 10 20 30 40 Ne 6 5 4 3 2 1 0

Time since big bang (Gyr) 0 5 10 15 20 Ne 6 5 4 3 2 1 0

Time since big bang (Gyr) 0 5 10 15 20 Ne

Figure 10.11 —The greybars show the starting epoch and duration of constant star formation for each individual member among the set of simulated histories that best matched the observed intrinsic scatter of Cl 1358+6245 (left),Cl 1054–0321 (middle),and Cl 0910+5422 (right). The dashed vertical line indicates the age of the universe at the redshift of the clusters. See Table 10.2 for the model results.

0.1 1.0 10.0 Redshift 1026 1027 1028 1029 1030 1031 1032 P ro to -C lu st er L1 5 0 0 ( er g s -1 H z -1) 0.1 1.0 10.0 1026 1027 1028 1029 1030 1031 1032 0.1 1.0 10.0 100.0 1000.0 10000.0 T o ta l S F R ( M O • y r -1) 0.1 1.0 10.0 1026 1027 1028 1029 1030 1031 1032 0.1 1.0 10.0 1026 1027 1028 1029 1030 1031 1032 0.1 1.0 10.0 1026 1027 1028 1029 1030 1031 1032 1358 1 10 Redshift 1026 1027 1028 1029 1030 1031 1032 P ro to -C lu st er L1 5 0 0 ( er g s -1 H z -1) 1 10 1026 1027 1028 1029 1030 1031 1032 0.1 1.0 10.0 100.0 1000.0 10000.0 T o ta l S F R ( M O • y r -1) 1 10 1026 1027 1028 1029 1030 1031 1032 1 10 1026 1027 1028 1029 1030 1031 1032 1 10 1026 1027 1028 1029 1030 1031 1032 1054 1 10 Redshift 1026 1027 1028 1029 1030 1031 1032 P ro to -C lu st er L1 5 0 0 ( er g s -1 H z -1) 1 10 1026 1027 1028 1029 1030 1031 1032 0.1 1.0 10.0 100.0 1000.0 10000.0 T o ta l S F R ( M O • y r -1) 1 10 1026 1027 1028 1029 1030 1031 1032 1 10 1026 1027 1028 1029 1030 1031 1032 1 10 1026 1027 1028 1029 1030 1031 1032 0910

Figure 10.12—The total combined star formation rate historyof the red sequence populations corresponding to the star formation histories shown in Fig. 10.11. Each BC03 template was scaled so that the total stellar mass at tendamounted to 10

11

M. The modelluminosities at rest frame 1500 ˚A were traced backin time, and at eachepochthe luminosities ofthe models

were added to give an estimate ofthe totalluminosityofthe ‘protocluster’ (thicksolid line). The luminosityat rest frame 1500 ˚A was used as a proxyfor the star formation rate using a conversion factor of1 Myr

−1

=L1500(erg s−1Hz−1)/8×1027

from Madau et al. (1998). Dotted lines indicate the different evolution withredshift ofthe totalcluster luminositywhen we adopt the maximallyyoung and maximallyold models that stillmatched the observed scatter, as indicated in Figs. 10.8, 10.9and 10.10. The dashed line shows the best-fit cluster luminosityevolution after applying a constant (withredshift) dust reddening factor ofE(B−V₎₌₀_._{1 to the modell}_umi_nosi_ty_.

be close toa thousandM yr −1

. Inthisesti -mate we have notincludedthe highstar f orma-tionratesobservedinthe radiogalaxiest hem-selvesofanother severalhundredsofM yr

−1 . Hence,the propertiesofthe LBGandLyα

popu-lationsinthe distantprotoclustersare consistent withthem beingthe progenitorsofthe evolved galaxiesonthe redsequence atz . 1.

10.5 Di

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We summarize our resultsasfollows:

• The observedoverdensitiesassociatedwith

the protoclusterscandidateslistedinTable 10.1

are inroughagreementwiththose expectedfor massive haloprogenitorsatz=2−6.

• The number densitiesofsuchmassive dark

haloprogenitorsare inroughagreementwith the number density ofluminoushigh-redshift radiogalaxies(assuminga radiosource lif e-time of∼ 107 yr),andthe estimatednumber

density ofLBGoverdensities(‘redshiftspikes’).

• The overdensitiesofthe protocluster candi

-datesare,onaverage,sufficiently large for f orm-inga boundobjectwitha massof1015_M

at z_{. 0}.5.