Old high-redshift galaxies and primordial density fluctuation spectra

9 Pages·0.25 MB·English

by  J. A. Peacock

Accepted for publication in Monthly Notices of the R.A.S.
Mon.Not.R.Astron.Soc.000,000–000 (1994) Printed1February2008 (MNplainTEXmacrosv1.6)
Old high-redshift galaxies and primordial density
fluctuation spectra
J.A. Peacock1, R. Jimenez1, J.S. Dunlop2, I. Waddington2, H. Spinrad3, D.
Stern3, A. Dey4, R.A. Windhorst5
1Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ
2Institute for Astronomy, Department of Physics and Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ
8 3Department of Astronomy, Universityof California, Berkeley, Ca 94720, USA
9 4National Optical Astronomy Observatories, 950 North Cherry Avenue, Tucson, Az85726, USA
9 5Department of Physics and Astronomy, Arizona State University,Tempe, Az85287-1504, USA
1

n
ABSTRACT
a
J We have discovered a population of extremely red galaxies at z ≃ 1.5 which have
>
apparentstellar ages of 3 Gyr, based on detailed spectroscopyin the rest-frame ul-
0 ∼
traviolet. In order for galaxies to have existed at the high collapse redshifts indicated
2
by these ages, there must be a minimum level of power in the density fluctuation

1 spectrumongalaxyscales.Thispapercomparestherequiredpowerwiththatinferred
v from other high-redshift populations: damped Lyman-α absorbers and Lyman-limit
4 galaxies at z ≃ 3.2. If the collapse redshifts for the old red galaxies are in the range
8 zc ≃6 – 8, there is generalagreement between the various tracers on the required in-
1 homogeneityon1-Mpcscales.Thislevelofsmall-scalepowerrequirestheLyman-limit
1 galaxiesto be approximatelyν ≃3.0 fluctuations, implying a very largebias parame-
0 terb≃6.Ifthecollapseredshiftsoftheredgalaxiesareindeedintherangezc =6−8
8 required for power spectrum consistency, their implied ages at z ≃1.5 are between 3
9
and3.8Gyrforessentiallyanymodeluniverseofcurrentage14Gyr.Theageofthese
/
h objects as deduced from gravitationalcollapse thus provides independent support for
p theagesestimatedfromtheirstellarpopulations.Suchearly-forminggalaxiesarerare,
-
andtheircontributiontothe cosmologicalstellardensityisconsistentwithanextrap-
o
r olation to higher redshifts of the star-formation rate measured at z < 5; there is no
t evidence for a general era of spheroid formation at extreme redshifts.
s
a
Key words: galaxies: clustering – cosmology: theory – large-scale structure of Uni-
:
v verse.
i
X
r
a
1 INTRODUCTION More recently, we have obtained the deep absorption-line
spectroscopy needed in order to prove that these colours
Itiswidelybelievedthatthesequenceofcosmologicalstruc- result from a well-evolved stellar population. The mini-
ture formation was hierarchical, originating in a density mumageofthestarscanbeinferredrobustlyfromspectral
powerspectrumwithincreasingfluctuationsonsmallscales. breaks, and gives ages of 3.5 Gyr for 53W091 at z = 1.55
The large-wavelength portion of this spectrum is accessible (Dunlop et al. 1996; Spinrad et al. 1997), and 4.0 Gyr for
toobservationtodaythroughstudiesofgalaxyclusteringin 53W069 at z = 1.43 (Dunlop 1998; Dey et al. 1998). Such
the linear and quasilinear regimes. However, nonlinear evo- ages push the formation era for these galaxies back to ex-
lution has effectively erased any information on the initial tremelyhighredshifts,anditisofinteresttoaskwhatlevel
spectrum for wavelengths below about 1 Mpc. The most ofsmall-scalepowerisneededinordertoallowthisearlyfor-
sensitivewayofmeasuringthespectrumonsmallerscalesis mation. However, the dating of stellar populations rests on
via the abundances of high-redshift objects; the amplitude complexmodelling,andsoitisdesirabletohaveanindepen-
of fluctuations on scales of individual galaxies governs the dent way of checking whether these high collapse redshifts
redshift at which these objects first undergo gravitational are correct. We have carried out such a test, using the fact
collapse. that the abundances of early-forming galaxies are sensitive
Theaimofthispaperistoapplytheseargumentsabout totheamplitudeofthesmall-scalepowerspectrum.Requir-
thesmall-scalespectrumtoaparticularlyinterestingclassof ingalevelofsmall-scale powerconsistentwiththatimplied
galaxy which we have recently discovered. It has long been byotherhigh-redshiftobjectspredictsacollapseredshiftfor
apparentthatasignificant fraction oftheopticalidentifica- our red galaxies. From this, we can predict an age – which
tions of 1-mJy radio galaxies are red and inactive (Wind- can then be compared with theage results obtained by an-
horst, Kron & Koo 1984; Kron, Koo & Windhorst 1985). alyzing stellar populations.
2 J.A. Peacock et al.
We shall adopt a standard framework for interpret- (h H0/100kms−1Mpc−1).Here,zc istheredshiftofviri-
≡
ing the abundances of high-redshift objects in terms of alization; Ω is thepresent valueof thematter density pa-
m
structure-formation models, as outlined by Efstathiou & rameter; f is the density contrast at virialization of the
c
Rees(1988).Undertheassumptionthatthegrowthofstruc- newly-collapsed object relative to the background, which is
ture proceeds as a gravitational hierarchy with Gaussian adequately approximated by
primordial statistics, the abundance of objects of a given
f =178/Ω0.6(z ), (4)
mass is related directly to the rms density fluctuations on c m c
that mass scale. In Section 2, we summarize the necessary
withonlyaslight sensitivitytowhetherΛisnon-zero(Eke,
elements of this Press-Schechter theory. It will be impor-
Cole & Frenk 1996).
tanttoachieveaconsistent pictureinthisanalysis between
For isothermal-sphere haloes, thevelocity dispersion is
these observations of high-redshift density fluctuations and
thefluctuationspectrumatthepresentdeducedfromgalaxy σv =Vc/√2. (5)
clustering;Section3summarizesourknowledgeofthelarge-
Givenaformationredshiftofinterest,andavelocitydisper-
scale spectrum. We then assemble the data on masses and
sion, there is then a direct route to the Lagrangian radius
abundanceofhigh-redshiftgalaxies inSection 4,summariz-
from which the proto-object collapsed. It is sometimes ar-
ing both ourown results and those of other classes of high-
gued thattheobserved stellar velocity dispersion should be
redshift objects. The implied density fluctuation spectrum
alittle‘cooler’thanthatofthedark-matterhalohostingthe
is then discussed in Section 5, where we note that these re-
galaxy (byafactor 3/2forar−3 stellardensityprofilein
sultsrequireahighlevelofbiasforrarehigh-redshiftgalax-
an isothermal sphere, for example). However, this assumes
ies. Finally,wereturninSection 6tothequestion ofstellar p
thatthedarkmattertotallydominatesthegravity,whereas
ages in our red radio galaxies, in the light of the collapse
realgalaxiesarebaryondominatedinthecentre.Anyveloc-
redshifts implied by the constraints on small-scale density
itycorrection is thereforelikely tobesmall in practice, and
fluctuations.
we ignore the effect.
ThePress-Schechtercollapsedfractioncannowbecon-
2 PRESS-SCHECHTER APPARATUS
verted to a differential number density of objects, n(M),
The formalism of Press & Schechter (1974) gives a way of using
calculatingthefractionF ofthemassintheuniversewhich
c dF
hascollapsedintoobjectsmoremassivethansomelimitM: Mn(M)=ρb dMc. (6)
F (>M,z)=1 erf δc . (1) In practice, however, one is more likely to measure an inte-
c
− √2σ(M) grated number density N of objects which lie above above
(cid:20) (cid:21)
some mass threshold, in which case
Here, σ(M) is the rms fractional density contrast obtained
by filtering the linear-theory density field on the required MN 4πR3
scale. In practice, this filtering is usually performed with a Fc ∼ ρ =N 3 . (7)
b
spherical ‘top hat’ filter of radius R, with a corresponding
massof 4πρ R3/3,whereρ isthebackgrounddensity.The This just says that thecollapsed fraction of themass is the
b b
numberδ isthelinear-theorycriticaloverdensity,whichfor fraction of the volume contained in the Lagrangian spheres
c
a‘top-hat’overdensityundergoingsphericalcollapseis1.686 around each object. As argued above, even quite large un-
– virtually independent of Ω. This form describes numeri- certainties in Fc can have little effect on the implied value
calsimulationsverywell(seee.g.Ma&Bertschinger1994). ofσ(R).This allows ustoneglect theuncertain constantof
The main assumption is that the density field obeys Gaus- proportionality in theabove relation, which is
sian statistics, which is true in most inflationary models. ǫ+2 MN
Given some estimate of Fc, the number σ(R) can then be Fc = ǫ ρ , (8)
b
inferred. Note that for rare objects this is a pleasingly ro-
bust process: alarge error in Fc will giveonly a small error for Fc ∝M−ǫ;similarly, theuncertaintiesin estimating the
inσ(R),becausetheabundanceisexponentiallysensitiveto appropriatevalueof R from theobserved circular velocities
σ. are often unimportant. Strictly, the observations are lower
Total masses are of course ill-defined both for real as- limits: wemust make at least sufficient collapsed objects to
tronomical objectsandclumpsofparticlesinsimulations; a sitethegalaxies understudy,butsomeobjects of thismass
betterquantitytouseisthevelocitydispersion.Virialequi- maygiverisetogalaxiesofadifferenttype.However,there-
librium for a halo of mass M and proper radius r demands sultsarehighlyrobusttosubstantialchangesintheassumed
a circular orbital velocity of abundance,so we shall treat them as measurements.
Thenumberdensitiesrequireacosmologicalmodel,and
2 GM
Vc = r (2) we quote figures assuming Ω = 1. For specific calculations,
we scale to othervalues of Ω using
For a spherically collapsed object this velocity can be con-
2 2
verted directly into a Lagrangian comoving radius which nD dr=n1 D1 dr1, (9)
contains the mass of the object within the virialization ra-
where dr is increment of comoving distance and D is
dius (e.g. White, Efstathiou & Frenk 1993)
angular-diameter distance (D = R0Sk(r)/[1+z]; see the
R/h−1Mpc= 21/2[Vc/100kms−1]. (3) Appendix). The scaling of Fc with model is different, be-
Ω1/2(1+z )1/2f1/6 cause the inferred density of baryonic material depends on
m c c
Old high-redshift galaxies and primordial fluctuation spectra 3
theelement of radial distance only: thesmall-scale power spectrum at high redshift.
Fc dr=Fc1 dr1. (10)
4.1 Damped Lyman-α systems
3 POWER SPECTRA FROM GALAXY
Damped Lyman-α absorbers are systems with HI column
CLUSTERING densitiesgreaterthan 2 1024 m−2(Lanzettaetal.1991).
∼ ×
Thesmall-scaleσ(R)datahavetoberelatedtothepresent- If the fraction of baryons in the virialized dark matter ha-
dayobservationsoflarge-scalefluctuationsinordertomake los equals the global value ΩB, then data on these systems
aconsistentpicture.Thepresentlinearfluctuationspectrum can be used to infer the total fraction of matter that has
is known independent of uncertainties about bias for R >∼ collapsed into bound structures at high redshifts (Ma &
3h−1Mpc (Peacock 1997). We summarize here the results Bertschinger 1994, Mo & Miralda-Escud´e 1994; Kaufmann
of these studies. & Charlot 1994; Klypin et al. 1995). The highest measure-
We use a dimensionless notation for the power spec- ment at z 3.2 implies ΩHI 0.0025h−1 (Lanzetta et al.
trum: ∆2 is the contribution to the fractional density vari- 1991; Stohrriie≃-Lombardi, McMa≃hon & Irwin 1996). We take
ance perunit lnk. In theconvention of Peebles (1980), this ΩBh2 = 0.02 as a compromise between the lower Walker
is et al. (1991) nucleosynthesis estimate and the more recent
2 dσ2 V 3 2 estimate of 0.025 from Tytler et al. (1996), giving
∆ (k) = 4πk δ (11)
≡ dlnk (2π)3 | k| ΩHI
F = 0.12h (15)
(V being a normalization volume), and the relation to the c ΩB ≃
correlation function is
for these systems. In this case alone, an explicit value of h
2 dk sinkr isrequiredinordertoobtainthecollapsedfraction;wetake
ξ(r)= ∆ (k) . (12)
k kr h=0.65.
Z
The photoionizing background prevents virialized
Similarly, the variance in fractional density contrast aver-
gaseous systems with circular velocities of less than about
aged over spheres of radius R is
50 kms−1 from cooling efficiently, so that they cannot
2 2 dk 2 contract to the high density contrasts characteristic of
σ (R)= ∆ (k) W , (13)
k k galaxies (e.g. Efstathiou 1992). We follow Mo & Miralda-
Z
where W =3(siny ycosy)/y3; y=kR. Escud´e (1994) and use the circular velocity range 50 –
Thektwin probl−ems with galaxy clustering are (i) the 100 kms−1 (σv = 35 – 70 kms−1) to model the damped
Lyman alpha systems. Reinforcing the photoionization
power-spectrum measurements are nonlinear, rather than
argument, detailed hydrodynamic simulations imply that
the linear-theory power spectrum required by the Press-
the absorbers are not expected to be associated with
Schechtermethod;(ii) thenormalization and even shape of
very massive dark-matter haloes (Haehnelt, Steinmetz &
the galaxy spectrum is biased relative to that of the mass.
Rauch1997). This assumption isconsistent with therather
Thefirst problem can bedealt with bycalibrating thenon-
low luminosity galaxies detected in association with the
lineareffectsusingN-bodysimulations.Thesecondismore
absorbers in a numberof cases (Le Brun et al. 1996).
difficult, but soluble in a number of limits. First, the over-
all normalization can bedeterminedbythePress-Schechter
methodofSection2,asappliedtorichclusters.Thisgivesa 4.2 Lyman-limit galaxies
measurement of the rms in spheres of radius 8h−1Mpc, on
Steidelet al.(1996) observedstar-forming galaxies between
which thereis general agreement:
z = 3 and 3.5 by looking for objects with a spectral break
σ8 =[0.5 0.6]Ω−0.56 (14) redwardsoftheU band.OurtreatmentoftheseLyman-limit
−
(Henry & Arnaud 1989; White, Efstathiou & Frenk 1993; galaxies is similar to that of Mo & Fukugita (1996), who
Viana & Liddle 1996; Eke,Cole & Frenk 1996). On smaller compared the abundances of these objects to predictions
scales, bias is expected to steepen the galaxy correlations, fromvariousmodels.Steideletal.givethecomovingdensity
but this effect operates on the nonlinear data, and so has of their galaxies as
a small effect on the inferred linear spectrum for R >∼ N(Ω=1) 10−2.54 (h−1Mpc)−3. (16)
3h−1Mpc (Peacock 1997). ≃
The resulting spectrum shape appears to be inconsis- This is a high number density, comparable to that of L∗
tentwithanyvariantofpureColdDarkMatter,andisbet- galaxies in the present Universe. The mass of L∗ galaxies
ter described by Mixed Dark Matter with roughly a 30 per corresponds to collapse of a Lagrangian region of volume
cent admixture of light neutrinos (e.g. Klypin et al. 1993; 1Mpc3, so the collapsed fraction would be a few tenths
∼
Peacock 1997; Smith et al. 1997). We are now interested in of a per cent if theLyman-limit galaxies had these masses.
seeinghowwellthisspectrummatchesontothesmaller-scale Direct dynamical determinations of these masses are
data obtained from abundancesof high-redshift galaxies. still lacking in most cases. Steidel et al. attempt to infer a
velocity width by looking at the equivalent width of the C
4 DATA ON HIGH-REDSHIFT GALAXY and Si absorption lines. These are saturated lines, and so
ABUNDANCES the equivalent width is sensitive to the velocity dispersion;
values in therange
In addition to our red mJy galaxies, two classes of high-
redshiftobjecthavebeenusedrecentlytosetconstraintson σ 180 320kms−1 (17)
v
≃ −
4 J.A. Peacock et al.
are implied. These numbers may measure velocities which Gonz´alez & Gorgas 1996; Buzzoni 1996). This means that
are not due to bound material, in which case they would the use of Solar-metallicity models in estimating the age of
give an upper limit to V /√2 for the dark halo. A more thestellar populations in these galaxies is consistent.
c
recent measurement of the velocity width of the Hα emis- Havingestablishedanabundanceandanequivalentcir-
sion line in one of these objects gives a dispersion of closer cular velocity for thesegalaxies, ourtreatment of them will
to 100 kms−1 (Pettini, private communication), consistent differinonecriticalwayfromtheLyman-αandLyman-limit
with themedian velocitywidth for Lyαof 140kms−1 mea- galaxies. For these, we take the normal Press-Schechter ap-
suredinsimilargalaxiesintheHDF(Lowenthaletal.1997). proach, in which the systems under study are assumed to
Ofcourse,thesefigurescouldunderestimatethetotalveloc- benewlyborn.FortheLyman-αandLyman-limitgalaxies,
ity dispersion, since they are dominated by emission from thismaynotbeabadapproximation,sincetheyareevolving
the central regions only. For the present, we consider the rapidlyand/ordisplayhighlevelsofstar-formationactivity.
range of values σv =100 to 320kms−1, and the sensitivity Fortheradiogalaxies,conversely,thiswouldbeaverypoor
to the assumed velocity will be indicated. In practice, this assumption, since the evidence is that they existed as dis-
uncertainty in the velocity does not produce an important crete systems at redshifts much higher than the z 1.5
≃
uncertainty in the conclusions. where we see them today. Our strategy will therefore be
to apply the Press-Schechter machinery at some unknown
4.3 Red radio galaxies formation redshift, and see what range of redshift gives a
consistent degree of inhomogeneity.
Wehaveobservedtwogalaxiesatz=1.43and1.55,overan
area 1.68 10−3 sr, so a minimal comoving density is from
×
one galaxy in thisredshift range:
N(Ω=1)>10−5.87 (h−1Mpc)−3. (18) 5 THE SMALL-SCALE FLUCTUATION
∼
SPECTRUM
ThisfigureiscomparabletothedensityoftherichestAbell
clusters,andisthusinreasonableagreementwiththediscov-
5.1 The empirical spectrum
ery that rich high-redshift clusters appear to contain radio-
quiet examples of similarly red galaxies (Dickinson 1995). Fig. 1 shows the σ(R) data which result from the Press-
Since the velocity dispersions of these galaxies are not Schechteranalysis,forthreecosmologies.Theσ(R)numbers
observed, they must be inferred indirectly. This is possible measured at various high redshifts have been translated to
becauseoftheknownpresent-dayFaber-Jacksonrelationfor z = 0 using the appropriate linear growth law for density
ellipticals. For 53W091, the large-aperture absolute magni- perturbations (see Appendix).
tudeis The open symbols give the results for the Lyman-limit
(largestR)andLyman-α(smallestR)systems.Theapprox-
M (z=1.55 Ω=1) 21.62 5log h (19)
V 10
| ≃− − imately horizontal error bars show the effect of the quoted
(measured direct in the rest frame). From our Solar- rangeofvelocitydispersionsfor afixedabundance;thever-
metallicitymodels,thiswouldbeexpectedtofadebyabout tical errors show the effect of changing the abundance by a
0.9 mag. between z = 1.55 and the present, for an Ω = 1 factor 2 at fixed velocity dispersion. The locus implied by
model of present age 14 Gyr (note that Bender et al. 1996 the red radio galaxies sits in between. The different points
have observed a shift in the zero-point of the M −σv re- showtheeffectsofvaryingcollapseredshift:zc =2,4,...,12
lation out to z = 0.37 of about the expected size). If we [lowestredshiftgiveslowestσ(R)].Clearly,collapseredshifts
comparethesenumberswiththeσv –MV relationforComa of6–8arefavoured forconsistency with theotherdataon
(m M =34.3 for h=1) taken from Dressler (1984), this high-redshiftgalaxies,independentoftheoreticalpreconcep-
−
gives velocity dispersions in the range tionsandindependentoftheageofthesegalaxies.Thislevel
σv =222to292 kms−1. (20) of power (σ[R]≃2 for R≃1h−1Mpc) is also in very close
agreement with the level of power required to produce the
This is a very reasonable range for a giant elliptical, and observed structure in the Lyman alpha forest (Croft et al.
we adopt it hereafter. Assuming low-density models would 1997), so there is a good case to be made that the fluctua-
increase thesefigures by an amount smaller than the above tionspectrumhasnowbeenmeasuredinaconsistentfashion
range, so we ignore thisadditional uncertainty. down to R 0.5h−1Mpc.
≃
We note in passing that these figures also make a pre- The shaded region at larger R shows the results de-
diction for themetallicity: ducedfrom clustering data (Peacock 1997). The 1σ confi-
±
denceregion wasobtainedbyanapproximation tothefrac-
Mg =0.32to0.35, (21)
2 tional error in ∆2(k) at k 1/R. It is clear an Ω = 1
≃
(Dressler 1984) corresponding to universe requires the power spectrum at small scales to be
higherthanwouldbeexpectedonthebasisofanextrapola-
[Fe/H]=0.11to0.39, (22)
tion from the large-scale spectrum. Depending on assump-
or a metallicity of between 1.3 and 2.5 times Solar. Care tionsaboutthescale-dependenceofbias,sucha‘feature’in
is needed here, however, because this figure refers to the the linear spectrum may also be required in order to sat-
nuclear metallicity, whereas our spectra are effectively to- isfy the small-scale present-day nonlinear galaxy clustering
tal. Given the metallicity gradients in low-redshift ellipti- (Peacock1997).Conversely,forlow-densitymodels,theem-
cals, such a slightly super-Solar nuclear metallicity would pirical small-scale spectrum appears to match reasonably
resultinanintegratedmeanmetallicityofSolaratbest(e.g. smoothly onto thelarge-scale data.
Old high-redshift galaxies and primordial fluctuation spectra 5
5.2 Comparison with CDM & MDM
Fig.1alsocomparestheempiricaldatawithvariousphysical
powerspectra.ACDMmodel(usingthetransferfunctionof
Bardeenetal.1986)withshapeparameterΓ=Ωh=0.25is
shownasareferenceforallmodels.Thishasapproximately
thecorrectlevelofsmall-scalepower,butsignificantlyover-
predicts intermediate-scale clustering, as discussed in Pea-
cock (1997). The empirical shape is better described by
MDMwithΩh 0.4andΩ 0.3.Thisisthelowest curve
ν
≃ ≃
in Fig. 1c, reproduced from thefittingformula of Pogosyan
&Starobinsky(1995;seealsoMa1996).However,thiscurve
fails to supply the required small-scale power, by about a
factor3inσ;loweringΩ to0.2stillleavesaverylargedis-
ν
crepancy. This conclusion is in agreement with e.g. Mo &
Miralda-Escud´e(1994),Ma&Bertschinger(1994),butcon-
flicts slightly with Klypin et al. (1995), who claimed that
the Ω = 0.2 model was acceptable. This difference arises
ν
partlybecauseKlypinetal.adoptalowervalueforδ (1.33
c
asagainst1.686here),andalsobecausetheyadoptthehigh
normalization of σ8 = 0.7; the net effect of these changes
is to boost the model relative to the small-scale data by a
factor of 1.6, which would allow marginal consistency for
the Ω = 0.2 model. MDM models do allow a higher nor-
ν
malization than theconventionalfigureof σ8 =0.55, partly
because of the very flat small-scale spectrum, and also be-
cause of theeffects of random neutrino velocities. However,
suchshiftsareatthe10percentlevel(Borganietal.1997a,
1997b), and σ8 = 0.7 would probably still give a cluster
abundance in excess of observation. The consensus of more
recent modelling is that even Ω =0.2 MDM is deficient in
ν
small-scale power (Ma et al. 1997; Gardner et al. 1997).
All the models in Fig. 1 assume n = 1; in fact, consis-
tency with the COBE results for this choice of σ8 requires
a significant tilt for flat models, n 0.8 – 0.9. Over the
≃
range of scales probed by large-scale structure, changes in
narelargely degenerate withchangesin Ωh, butthesmall-
scale power is more sensitive to tilt than to Ωh. Tilting the
Ω = 1 models is not attractive, since it increases the ten-
dencyformodelpredictionstoliebelowthedata.However,
atiltedlow-ΩflatCDMmodelwouldagreemoderatelywell
withthedataonallscales,withtheexceptionofthe‘bump’
around R 30h−1Mpc. Testing the reality of this feature
≃
willthereforebeanimportanttaskforfuturegenerationsof
redshift survey.
Figure 1.The present-day linear fluctuation spectrum required
5.3 Limits on high-redshift clustering
in various cosmologies. This is expressed as σ(R): the fractional
rms fluctuation in density averaged in spheres of radius R. The An interesting aspect of these results is that the level of
data points areLyman-αgalaxies (open cross)andLyman-limit poweron 1-Mpcscales isonly moderate: σ(1h−1Mpc) 2.
galaxies(opencircles)Thediagonalbandwithsolidpointsshows ≃
At z 3, the corresponding figure would have been much
redradiogalaxieswithassumedcollapseredshifts2,4,...12.The ≃
lower, making systems like the Lyman-limit galaxies rather
vertical error bars show the effect of a change in abundance by
rare. For Gaussian fluctuations, as assumed in the Press-
a factor 2. The horizontal errors correspond to different choices
Schechteranalysis,suchsystemswillbeexpectedtodisplay
forthecircularvelocitiesofthedark-matterhaloesthathostthe
galaxies (R scales linearly with velocity). The shaded region at spatialcorrelationswhicharestronglybiasedwithrespectto
largeRgivestheresultsinferredfromgalaxyclustering.Thesolid theunderlyingmass. Thelinearbias parameterdependson
lines show Γ = 0.25 CDM predictions; for Ω = 1 MDM models therarenessofthefluctuationandthermsoftheunderlying
withh=0.4andΩν =0.2and0.3(lowestatleft)arealsoshown. field as
The large-scale normalization is σ8 = 0.55 for Ω = 1 or σ8 = 1 ν2 1 ν2 1
forthelow-densitymodels. b=1+ − =1+ − (23)
νσ δ
c
(Kaiser1984;Cole&Kaiser1989;Mo&White1996),where
ν = δ /σ, and σ2 is the fractional mass variance at the
c
6 J.A. Peacock et al.
Conveniently, this has a volume equivalent to a sphere of
radius 7.5h−1Mpc, so it is easy to measure the bias di-
rectly by reference to the known value of σ8. Since the de-
greeofbiasislarge,redshift-spacedistortionsfromcoherent
infall are small; the cell is also large enough that the dis-
tortionsofsmall-scalerandomvelocitiesatthefewhundred
kms−1levelarealsosmall.Usingthemodelofequation(11)
of Peacock (1997) for the anisotropic redshift-space power
spectrumandintegratingovertheexactanisotropicwindow
function,weconfirmthattheabovesimplevolumeargument
should be accurate to a few per cent for reasonable power
spectra:
σcell b(z=3)σ7.5(z=3), (25)
≃
where we define the bias factor at this scale. The results of
Mo & White (1996) suggest that the scale-dependence of
bias should beweak.
In order to estimate σcell, we havemade simulations of
syntheticredshift histograms, usingthemethodof Poisson-
sampled lognormal realizations described by Broadhurst,
Figure 2. The bias parameter at z = 3.2 predicted for the Ly- Taylor & Peacock (1995). We use a χ2 statistic to quan-
man-limitgalaxies,asafunctionoftheirassumedcircularveloc- tify the nonuniformity of the redshift histogram, and find
ity. Dotted lineshows Ω=0.3open; dashed lineis Ω=0.3flat;
solid line is Ω = 1. A substantial bias in the region of b ≃ 6 is that σcell ≃ 0.9 is required in order for the field of Steidel
et al. (1997) to be typical. It is then straightforward to ob-
predictedratherrobustly.
tain the bias parameter since, for a present-day correlation
function ξ(r) r−1.8,
∝
redshInifttohfisinatnearelysts.is, δc = 1.686 is assumed. Variations in σ7.5(z=3)=σ8×[8/7.5]1.8/2×1/4≃0.146, (26)
thisnumberoforder10percenthavebeensuggestedbyau- implying
thorswhohavestudiedthefitofthePress-Schechtermodel
b(z=3 Ω=1) 0.9/0.146 6.2. (27)
tonumericaldata.Thesechangeswouldmerelyscaleb 1by
| ≃ ≃
−
asmallamount;thekeyparameterisν,whichissetentirely
Steidel et al. (1997) use a rather different analysis which
bythecollapsedfraction.FortheLyman-limitgalaxies,typ-
concentrates on the highest peak alone, and obtain a min-
ical values of this parameter are ν 3, and it is clear that
≃ imum bias of 6, with a preferred value of 8. They use the
verysubstantialvaluesofbiasareexpected,asillustratedin
Eke et al. (1996) value of σ8 = 0.52, which is on the low
Figure 2.
side of the published range of estimates. Using σ8 = 0.55
This diagram shows how the predicted bias parameter
would lower their preferred b to 7.6, which is satisfyingly
varieswiththeassumedcircularvelocity,foranumberden-
close to our estimate. Note that, with both these methods,
sity of galaxies fixed at the level observed by Steidel et al.
itismucheasiertoruleoutalowvalueofbthanahighone;
(1996). The sensitivity to cosmological parameter is only
given asingle field,it ispossible that arelatively ‘quiet’ re-
moderate; at V = 200 kms−1, we have b 4.6, 5.5, 5.8
c gionofspacehasbeensampled,andthatmuchlargerspikes
≃
for the open, flat and critical models respectively. These
remain to be found elsewhere. Henceforth, we assume that
numbers scale approximately as V−0.4, and b is within 20
c the Steidel et al. (1997) field is typical, since there is evi-
per cent of 6 for most plausible parameter combinations.
dence that other fields have a similar appearance (Steidel,
Strictly, the bias values determined here are upper limits,
privatecommunication).
since the numbers of collapsed haloes of this circular ve-
HavingarrivedatafigureforbiasifΩ=1,itiseasyto
locity could in principle greatly exceed the numbers of ob-
translate to other models, since σcell is observed, indepen-
served Lyman-limit galaxies. However, the undercounting
dentofcosmology.ForlowΩmodels,thecellvolumewillin-
would have to be substantial: increasing the collapsed frac- crease by a factor [D2dr]/[D12dr1]; comparing with present-
tion by a factor 10 reduces the implied bias by a factor of
day fluctuations on this larger scale will tend to increase
about 2. A substantial bias seems difficult to avoid, as has
the bias. However, for low Ω, two other effects increase the
been pointed out in thecontext of CDM models by Baugh,
predicteddensityfluctuationatz=3:theclusterconstraint
Cole & Frenk (1997). increasesthepresent-dayfluctuationbyafactorΩ−0.56,and
We now compare these calculations to the recent de-
the growth between redshift 3 and the present will be less
tection by Steidel et al. (1997) of strong clustering in the
than a factor of 4. Using the Appendix to calculate these
population of Lyman-limit galaxies at z 3. The evidence
corrections, we get
≃
takes the form of a redshift histogram binned at ∆z=0.04
resolution over a field 8.7′ 17.6′ in extent. For Ω=1 and b(z=3 Ω=0.3) 0.42 (open)
× | = , (28)
z=3, thisprobes the densityfield using a cell with dimen- b(z=3 Ω=1) 0.60 (flat)
| (cid:26)
sions
which suggests an approximate scaling as b Ω0.72 (open)
cell=15.4 7.6 15.0[h−1Mpc]3. (24) or Ω0.42 (flat). Multiplying the Ω=1 figure∝of 6.2 by these
× ×
Old high-redshift galaxies and primordial fluctuation spectra 7
factorsgivesbiasvaluesof2.6(Ω=0.3open)or3.7(Ω=0.3 Convertingtheagesforthegalaxiestoanapparentcol-
flat). The significance of this observation is thus to provide lapse redshift depends on the cosmological model, but par-
the first convincing proof for the reality of galaxy bias: for ticularlyonH0.Wecancircumventsomeofthisuncertainty
Ω 0.3, bias is not required in the present universe, but byfixingtheageoftheuniverse.Afterall,itisofnointerest
≃
we now see that b>1 is needed at z =3 for all reasonable toaskaboutformationredshiftsinamodelwithe.g.Ω=1,
values of Ω. h=0.7whenthewholeuniversethenhasanageofonly9.5
Comparing these bias values with Fig. 2, we see that Gyr.IfΩ=1istobetenabletheneitherh<0.5againstall
theobservedvalueofbisquiteclosetothepredictioninthe the evidence or there must be an error in the stellar evolu-
case of Ω= 1 – suggesting that the simplest interpretation tiontimescale. Ifthestellar timescales arewrongbyafixed
ofthesesystemsascollapsedrarepeaksmaywellberoughly factor, then these two possibilities are degenerate. It there-
correct.Indeed,forhighcircularvelocitiesthereisadanger fore makes sense to measure galaxy ages only in units of
of exceedingthepredictions,and itwould create something the age of the universe – or, equivalently, to choose freely
of a difficulty for high-density models if a velocity as high an apparent Hubble constant which gives the universe an
as V 300kms−1 were to be established as typical of the agecomparabletothatinferredforglobularclusters.Inthis
c
≃
Lyman-limitgalaxies.ForlowΩ,the‘observed’biasislower spirit, Fig. 3 gives apparent ages as a function of effective
than the predictions, so there is no immediate conflict. For collapseredshiftformodelsinwhichtheageoftheuniverse
acircularvelocityof200kms−1,wewouldneedtosaythat is forced to be 14 Gyr (e.g. Jimenez et al. 1996).
thecollapsed fraction wasunderestimatedbyroughlyafac- This plot shows that the ages of the red radio galax-
tor10toclosethegap inthecaseofanopenuniverse.This ies are not permitted very much freedom. We have argued
changeincollapsed fractionincreasesthevaluesofσ inFig. for a consistent formation redshift in the range 6 to 8 on
1 bya factor of about 1.5, increasing the‘observed’ biasby abundancegrounds,andthisclearlypredictsanageofclose
the same factor. At the same time, this makes ν smaller, to 3.0 Gyr for Ω = 1, or 3.7 Gyr for low-density models,
reducingthepredictedbiasbyaboutafactor2andproduc- irrespective of whether Λ is nonzero. The age-z relation
c
ing agreement on a bias factor of between 3 and 4. Such a is rather flat, and this gives a robust estimate of age once
changeinF couldcomeabouteitherbypostulatingthatthe wehavesomeideaof z throughtheabundancearguments.
c c
conversion from velocity to R is systematically in error, or Conversely,itisalmostimpossibletodeterminethecollapse
bysuggestingthattheremaybemanyhaloeswhicharenot redshift reliably from the spectral data, since a very high
detected bytheLyman-limit search technique.It ishard to precision would be required both in the age of the galaxy
argue that either of these possibilities are completely ruled and in theage of theuniverse.
out. Nevertheless, we have reached the paradoxical conclu- What conclusions can then be reached about allowed
sion that large-amplitude clustering in the early universe is cosmological models? If we take an apparent z = 8 from
c
morenaturally understood inan Ω=1model, whereas one thepower-spectrumarguments,thentheapparentminimum
might haveexpected theopposite conclusion. age of > 4 Gyr for 53W069 can very nearly be satisfied in
bothlow-densitymodels(acurrentageof14.5Gyrwouldbe
required),butisunattainableforΩ=1.Inthehigh-density
case, a current age of 17.6 Gyr would be required to attain
6 AGES AND COLLAPSE REDSHIFTS
therequiredage for z =8; thisrequiresa Hubbleconstant
c
We now return to the red radio galaxies, and ask if the of h = 0.38. As argued above, this conclusion is highly in-
collapse redshiftsinferred aboveareconsistent withtheage sensitive to the assumed value of zc. If the true value of h
dataontheseobjects.Firstbearinmindthatinahierarchy does turn out to be close to 0.5, then it might be argued
someofthestarsinagalaxywillinevitablyformbeforethe thatΩ=1isconsistentwiththedata,givenrealisticuncer-
epochofcollapse.Indeed,somedirectobservationalevidence tainties. The ages for the low-density models would in this
for the assembly of galaxies from sub-galactic clumps may casebelargebycomparison with theobservedradio-galaxy
now be starting to emerge (Pascarelle et al. 1996). At the ages.However,theagesobtainedbymodellingspectrawith
time of final collapse, the typical stellar age will be some a single burst can only be lower limits to the true age for
fraction α of the age of theuniverseat that time: the bulk of the stars; we could easily be observing an even
older burst which is made bluer by a little recent star for-
age=t(zobs) t(zc)+αt(zc). (29) mation. A low h measurement would therefore not rule out
−
low-density models.
We can rule out α = 1 (i.e. all stars forming in small sub-
The main conclusion of this paper is thus that the ex-
unitsjustafterthebigbang).Forpresent-dayellipticals,the
istence of old radio galaxies at z = 1.5 poses two serious
tightcolour-magnituderelation onlyallows anapproximate
difficultiesfor an Ω=1Universe:(i) aconsistent pictureof
doublingofthemassthroughmergerssincethetermination
structure formation through gravitational instability from
of star formation (Bower at al. 1992). This corresponds to
Gaussian initial conditions requires a high formation red-
α 0.3 (Peacock 1991). A non-zero α just corresponds to
≃ shift for these objects, leading to an old Universe and, par-
scaling thecollapse redshift as
ticularly, a very small Hubbleconstant if the stellar ages of
apparent (1+z ) (1 α)−2/3, (30) theseobjectsareaccepted;(ii)theshapeofthepowerspec-
c
∝ − trum is complicated, with a large change in power between
since t (1+z)−3/2 at high redshifts for all cosmologies. smoothing scales of 0.5h−1Mpc and 5h−1Mpc; no known
∝
For example, a galaxy which collapsed at z=6 would have modelpredictsaspectrumwiththisshape.Theseconddiffi-
an apparent age corresponding to a collapse redshift of 7.9 cultymightbeavoidedthroughnon-Gaussianstatistics,but
for α=0.3. thefirstwouldrequireourageestimatesfortheradiogalax-
8 J.A. Peacock et al.
by the prediction of Pei & Fall (1995), based on high-z ele-
mentabundances.Ifwescalebyafactor3,andintegrateto
findthetotal densityin starsproducedat z>6,thisyields
ρ∗(zf >6) 106.2M⊙Mpc−3. (32)
≃
Since the mJy galaxies have a density of 10−5.87h3Mpc−3
andstellarmassesoforder1011M⊙,thereisclearlynocon-
flict with the idea that these galaxies are the first stellar
systemsofL∗ sizewhichform enroutetothegeneraleraof
star and galaxy formation.
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APPENDIX: FORMULAE FOR GENERAL
COSMOLOGIES
If a nonzero cosmological constant is allowed, not all of the
important cosmological formulae exist as analytical expres-
sions, but in many cases accurate approximation formulae
may be used: see Carroll, Press & Turner (1992). For con-
venience, we summarize the necessary expressions here.
Ingeneral,itisnecessarytodistinguishmatter(m)and
vacuum (v) contributions to the total density parameter.
BoththeseparametersandtheHubbleparametervarywith
scale factor a=1/(1+z):
H[a]=H0 Ωv(1 a−2)+Ωm(a−3 a−2)+a−2. (33)
− −
p