Binary Quasars at High Redshift. II. Sub-Mpc Clustering at z ~ 3-4
ABSTRACT We present measurements of the small-scale (0.1 r 1 h –1 Mpc) quasar two-point correlation function at z>2.9, for a flux-limited (i < 21) sample of 15 binary quasars compiled by Hennawi et al. The amplitude of the small-scale clustering increases from z ~ 3 to z ~ 4. The small-scale clustering amplitude is comparable to or lower than power-law extrapolations (assuming a fixed slope γ = 2) from the large-scale correlation function of the i < 20.2 quasar sample from the Sloan Digital Sky Survey. Using simple prescriptions relating quasars to dark matter halos, we model the observed small-scale clustering with halo occupation models. We found that the level of small-scale clustering favors an active fraction of black holes in (M 1013 h –1 M ☉) satellite halos fs 0.1 at z 3.
- SourceAvailable from: Giuseppe Tormen[show abstract] [hide abstract]
ABSTRACT: Using a cosmological N-Body simulation and a sample of re-simulated cluster-like haloes, we study the mass loss rates of dark matter subhaloes, and interpret the mass function of subhaloes at redshift zero in terms of the evolution of the mass function of systems accreted by the main halo progenitor. When expressed in terms of the ratio between the mass of the subhalo at the time of accretion and the present day host mass the unevolved subhalo mass function is found to be universal. However, the subhalo mass function at redshift zero clearly depends on $M_0$, in that more massive host haloes host more subhaloes. To relate the unevolved and evolved subhalo mass functions, we measure the subhalo mass loss rate as a function of host mass and redshift. We find that the average, specific mass loss rate of dark matter subhaloes depends mainly on redshift. These results suggest a pleasingly simple picture for the evolution and mass dependence of the evolved subhalo mass function. Less massive host haloes accrete their subhaloes earlier, which are thus subjected to mass loss for a longer time. In addition, their subhaloes are typically accreted by denser hosts, which causes an additional boost of the mass loss rate. To test the self-consistency of this picture, we use a merger trees constructed using the extended Press-Schechter formalism, and evolve the subhalo populations using the average mass loss rates obtained from our simulations, finding the subhalo mass functions to be in good agreement with the simulations. [abridged] Comment: 12 pages, 12 figures; submitted to MNRASMonthly Notices of the Royal Astronomical Society 12/2007; · 5.52 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: We study the two-point correlation function of a uniformly selected sample of 4426 luminous optical quasars with redshift 2.9 ≤ z ≤ 5.4 selected over 4041 deg2 from the Fifth Data Release of the Sloan Digital Sky Survey. We fit a power-law to the projected correlation function wp(rp) to marginalize over redshift-space distortions and redshift errors. For a real-space correlation function of the form ξ(r) = (r/r0)-γ, the fitted parameters in comoving coordinates are r0 = 15.2 ± 2.7 h-1 Mpc and γ = 2.0 ± 0.3, over a scale range 4 h-1 Mpc ≤ rp ≤ 150 h-1 Mpc. Thus high-redshift quasars are appreciably more strongly clustered than their z ≈ 1.5 counterparts, which have a comoving clustering length r0 ≈ 6.5 h-1 Mpc. Dividing our sample into two redshift bins, 2.9 ≤ z ≤ 3.5 and z ≥ 3.5, and assuming a power-law index γ = 2.0, we find a correlation length of r0 = 16.9 ± 1.7 h-1 Mpc for the former and r0 = 24.3 ± 2.4 h-1 Mpc for the latter. Strong clustering at high redshift indicates that quasars are found in very massive, and therefore highly biased, halos. Following Martini & Weinberg, we relate the clustering strength and quasar number density to the quasar lifetimes and duty cycle. Using the Sheth & Tormen halo mass function, the quasar lifetime is estimated to lie in the range ~4-50 Myr for quasars with 2.9 ≤ z ≤ 3.5, and ~30-600 Myr for quasars with z ≥ 3.5. The corresponding duty cycles are ~0.004-0.05 for the lower redshift bin and ~0.03-0.6 for the higher redshift bin. The minimum mass of halos in which these quasars reside is (2-3) × 1012 h-1 M for quasars with 2.9 ≤ z ≤ 3.5 and (4-6) × 1012 h-1 M for quasars with z ≥ 3.5; the effective bias factor beff increases with redshift, e.g., beff ~ 8 at z = 3.0 and beff ~ 16 at z = 4.5.The Astronomical Journal 12/2007; 133(5):2222. · 4.97 Impact Factor
Article: The Void Phenomenon Explained[show abstract] [hide abstract]
ABSTRACT: We use high-resolution N-body simulations, combined with a halo occupation model of galaxy bias, to investigate voids in the galaxy distribution. Our goal is to address the 'void phenomenon' of Peebles (2001), which presents the observed dearth of faint galaxies in voids as a challenge to the current cosmology. In our model, galaxy luminosity is determined only as a function of dark matter halo mass. With this simple assumption, we demonstrate that large, empty voids of ~15 Mpc/h in diameter are expected even for galaxies seven magnitudes fainter than L*. The predictions of our model are in excellent agreement with several statistical measures; (i) the luminosity function of galaxies in underdense regions, (ii) nearest neighbor statistics of dwarf galaxies, (iii) the void probability function of faint galaxies. In the transition between filaments and voids in the dark matter, the halo mass function changes abruptly, causing the maximum galaxy luminosity to decrease by ~5 magnitudes over a range of ~1 Mpc/h. Thus the boundary between filaments and voids in the galaxy distribution is nearly as sharp for dwarfs as for ~L* objects. These results support a picture in which galaxy formation is driven predominantly by the mass of the host dark matter halo, and is nearly independent of the larger-scale halo environment. Further, they demonstrate that LCDM, combined with a straightforward bias model, naturally explains the existence of the void phenomenon. Comment: 8 pages, 5 figures, submitted to ApJ. citations updatedThe Astrophysical Journal 04/2008; · 6.73 Impact Factor
arXiv:0908.3908v1 [astro-ph.CO] 26 Aug 2009
DRAFT VERSION AUGUST 26, 2009
Preprint typeset using LATEX style emulateapj v. 10/09/06
BINARY QUASARS AT HIGH REDSHIFT II: SUB-MPC CLUSTERING AT Z ∼ 3−4
YUE SHEN1, JOSEPH F. HENNAWI2,3,4,5, FRANCESCO SHANKAR6, ADAM D. MYERS5,7, MICHAEL A. STRAUSS1, S. G. DJORGOVSKI8,
XIAOHUI FAN9, CARLO GIOCOLI10, ASHISH MAHABAL8, DONALD P. SCHNEIDER11, DAVID H. WEINBERG12
Draft version August 26, 2009
We present measurements of the small-scale (0.1 ? r ? 1 h−1Mpc) quasar two-point correlation function
at z > 2.9, for a flux-limited (i < 21) sample of 15 binary quasars compiled by Hennawi et al. (2009). The
amplitude of the small-scale clustering increases from z ∼ 3 to z ∼ 4. The small-scale clustering amplitude
is comparable to or lower than power-law extrapolations (with slope γ = 2) from the large-scale correlation
function of the i < 20.2 quasar sample from the Sloan Digital Sky Survey. Using simple prescriptions relating
quasars to dark matter halos, we model the observed small-scale clustering with halo occupation models. Re-
producing the large-scale clustering amplitude requires that the active fraction of the black holes in the central
galaxies of halos is near unity, but the level of small-scale clustering favors an active fraction of black holes in
satellite galaxies 0.1 ? fs? 0.5 at z ? 3.
Subject headings: black hole physics – galaxies: active – cosmology: observations – large-scale structure of
universe – quasars: general – surveys
With the rapid progress in observational and computational
cosmology in the past two decades due to dedicated sur-
veys and numerical simulations, it is now possible to study
the quasar population within the hierarchical structure forma-
tion framework (e.g., Kauffmann & Haehnelt 2000; Volon-
teri, Haardt & Madau 2003; Wyithe & Loeb 2003; Hopkins
et al. 2008; Shankar et al. 2008, 2009; Shen 2009). If lumi-
nous quasars are the progenitors of the most massive galaxies
today, then they occupy the rare peaks in the initial density
fluctuation field, i.e., they are biased tracers of the underly-
ing matter distribution (e.g., Bardeen et al. 1986; Efstathiou
& Rees 1988; Cole & Kaiser 1989; Djorgovski 1999; Djor-
govski et al. 1999). The quasar two-point correlation func-
tion has now been measured for large survey samples to un-
precedented precision (e.g., Porciani et al. 2004; Croom et
al. 2005; Myers et al. 2006, 2007a; Shen et al. 2007, 2008,
2009; da Ângela et al. 2008; Ross et al. 2009). These stud-
ies suggest that quasars live in massive dark matter halos of
Mhalo? a few×1012h−1M⊙; their bias relative to the under-
lying matter increases rapidly with redshift. However, such
1Princeton University Observatory, Princeton, NJ 08544.
2Department of Astronomy, Campbell Hall, University of California,
Berkeley, California 94720.
3NSF Astronomy and Astrophysics Postdoctoral Fellow.
4Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidel-
5Visiting Astronomer, Kitt Peak National Observatory, National Optical
Astronomy Observatory, which is operated by the Association of Universi-
ties for Research in Astronomy (AURA) under cooperative agreement with
the National Science Foundation.
6Max-Planck-Institüt für Astrophysik, Karl-Schwarzschild-Str.
85748, Garching, Germany.
7Department of Astronomy, University of Illinois at Urbana-Champaign,
Urbana, IL 61801.
8Division of Physics, Mathematics, and Astronomy, California Institute
of Technology, Pasadena, CA 91125.
9Steward Observatory, 933 North Cherry Avenue, Tucson, AZ 85721.
10Institutüt für Theoretische Astrophysik, Zentrum für Astronomie der
Universität Heidelberg Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany.
11Department of Astronomy and Astrophysics, 525 Davey Laboratory,
Pennsylvania State University, University Park, PA 16802.
12Astronomy Department, Ohio State University, Columbus, OH 43210.
studies are unableto probethe smallest scales (r ?1h−1Mpc),
where matter evolves nonlinearly and the distributions of
quasars within dark matter halos start to play a role in de-
termining their clustering properties. This is because fiber-
two targets closer than the fiber collision scale ∼ 1′.
Hennawi et al. (2006) compiled a sample of close quasar
binaries at z < 3 by spectroscopic follow-up observations
of candidates selected from the Sloan Digital Sky Survey
(SDSS; York et al. 2000) imaging data. Using this binary
sample,theymeasuredthe correlationfunctiondownto scales
as small as Rprop∼ 15kpc, where Rpropis the transverse sep-
aration in proper units; they confirmed and extended pre-
vious tentative claims (e.g., Djorgovski 1991) that quasars
exhibit excess clustering on small scales (most notably at
Rprop? 40kpc) compared with the naive power-law extrapo-
lation of the large-scale correlation function. This small-scale
in a more homogeneoussample, albeit at a lower level of “ex-
The large-scale quasar correlation function has now been
measured at high redshift (z ? 3, Shen et al. 2007), where
quasars cluster much more strongly than their low redshift
counterparts. It is natural to extend the work of Hennawi
et al. (2006) to study the small-scale quasar clustering at
z > 3. However, such investigations are challenging for two
reasons: first, the number density of the quasar population
dropsrapidlyafterthe peakofquasaractivity at z∼2−3 (e.g.,
Richards et al. 2006); second, quasar pairs on tens of kpc to
1Mpc scales are rare occurrences – only ? 0.1% of quasars
have a close quasar companion with comparable luminosity.
Hence a large search volume is needed to build up the statis-
tics. Hennawi et al. (2009,hereafterPaper I) have, for the first
time, compiled such a binary quasar sample at z ? 3, which
we use here to study the clustering of quasars on small-scales.
The rareness of close quasar pairs is not in direct contra-
diction with the major merger scenario of quasar triggering,
because the probability that two quasars are triggered and
identified simultaneously during the early stage of a major
merger (i.e., with separations on halo scales rather than on
galactic scales) is low in theoretical models (e.g., Volonteri
2SHEN ET AL.
et et al. 2003; Hopkins et al. 2008). However, even a hand-
ful of close quasar pairs will contribute significantly to the
small-scale clustering amplitude because the mean number
density of quasars is so low that the expected number of ran-
dom companions on such small scales is tiny. Also note that
although quasar pairs with comparable luminosities are rare,
there might be more fainter companions (i.e., low luminosity
AGN or fainter quasars) around luminous quasars (e.g., Djor-
govski et al. 2007), as expected from the hierarchical merger
In this paper we measure the small-scale quasar clustering
at z ?3 using a set of 15 quasar pairs in the sample of Paper I.
We adopt the same cosmology as in Paper I, with Ωm= 0.26,
ΩΛ= 0.74 and h = 0.7. Comoving units will be used unless
otherwise specified, and we use subscriptpropfor properunits.
2. THE SAMPLE
Our parent sample is the high-redshift binary quasar cata-
log presented in Paper I. This sample includes 27 quasar pairs
with relative velocity |∆v| < 2000kms−1at 2.9 < z < 4.5,
down to a limiting magnitude i < 21 after correcting for
Galactic extinction, selected over 8142 deg2of the SDSS
imaging footprint prior to DR6 (Adelman-McCarthy et al.
2008). The detailed target selection criteria, completeness
analysis, and follow-up spectroscopy can be found in Paper I.
To construct our clustering subsample, we first exclude eight
pairs that failed to pass the selection criteria (for which the
completeness cannot been quantified) described in §2 of Pa-
per I, leaving 19 pairs. Second, the follow-up spectroscopic
observations are the most complete out to an angular sepa-
ration θ ≈ 60′′, because those targets were assigned higher
priority for follow-up spectroscopy, and therefore we restrict
ourselves to pairs with angular separation θ < 60′′; this re-
strictionexcludesonepairat z<3.5andthree pairsat z>3.5.
Our final clustering subsample thus includes 15 pairs with
seven pairs at z < 3.5 and eight pairs at z > 3.5, with pro-
jected comoving separations R ∼ 0.1−1h−1Mpc and proper
separations Rprop∼ a few tens to a few hundreds of kpc (see
fig. 8 of Paper I).
The sparseness of the sample requires different techniques
for measuring the clustering strength, from the traditional
binned wp statistic (e.g., Davis & Peebles 1983). Here we
adopt the Maximum-Likelihood(ML) approach used in Shen
et al. (2009), as described below. We report our ML esti-
in §2.1; the systematic uncertainties are discussed in §2.2. To
reduce the impact of the selection incompleteness at z ∼ 3.5
due to stellar contaminants (see Paper I), and to explore red-
shift evolution, we measure the small-scale clustering in two
redshiftbins: 2.9<z<3.5(low-z)and 3.5<z<4.5(high-z).
2.1. Clustering Measurements
Here we recast the ML approach of Shen et al. (2009).
We choose a power-law model for the underlying correla-
tion function: ξ(r) ≡ (r/r0,ML)−γML. We then compute the
expected number of quasar pairs within a comoving cylindri-
cal volume with projected radius R to R+dR and half-height
∆H =20 h−1Mpc. This half-heightis chosen to reflect our ve-
locity constraint in defining a quasar pair and to minimize the
effects of redshift distortions and errors. Assuming Poisson
statistics, the likelihood function can be written as:
where µ = 2πRh(R)dR is the expected number of pairs in the
interval dR, the index i runs over all pairs in the sample and
the index j runs over all the elements dR in which there are
no pairs. The expected pair surface density h(R) is given by
where n(z) is the cumulativequasar luminosity functiondown
to a limiting magnitude (in this case i = 21), fcomp(z) is the
completeness in selecting binary candidates for follow-up
spectroscopy as quantified in Paper I (see their fig. 7), and
Vcis the comoving volume between redshifts zminand zmax
covered by the binary survey. The factor of 1/2 in eqn. (2)
removes duplicate counts of pairs.
Note we only consider the completeness in target selection
fcomp≡ ftarg(i.e., the fraction of quasar binaries that would
have been selected by the algorithm in Paper I) throughout
this section. We discuss the effects of the completeness of the
spectroscopic follow-up of survey candidates (i.e., the frac-
tion of targets that have been observed), fspec, in §2.2.
Defining the usual quantity S ≡ −2lnL we have
S ≡ −2lnL = 2
with all the model-independentadditive terms removed. Here
[Rmin,Rmax] is the range of comoving scales over which
we search for quasar pairs. To include all observed pairs
with angular separation θ < 60′′, we choose [Rmin,Rmax] =
[0.04,1] h−1Mpc for the low-z bin and [Rmin,Rmax] =
[0.1,1.3] h−1Mpc for the high-z bin. We verified that our re-
sults were not sensitive to the exact values of these limits. If
we fit both r0,MLand γMLwe found that the best-fit model
favors γML> 2.3 for both redshift bins, a value substantially
steeper than the slope on large scales, γ ∼ 2 (e.g., Shen et al.
2007). However, the spectroscopic completeness, fspec, prob-
ably depends on angularseparation, because we tended to ob-
serve the closest candidates first; this may introduce an arti-
ficially steep slope in the correlation function. Therefore we
fix the slope γML= 2 (i.e., close to the measured slope of the
large-scale correlation function, Shen et al. 2007) and mini-
mize the merit functionS with respect to r0,MLonly. A power-
lawslopeγ ∼2is also foundfortheclusteringofSDSS LRGs
to z ∼ 0.4 (e.g., Masjedi et al. 2006) and photometric SDSS
quasars (e.g., Myers et al. 2006, 2007a) over a wide range of
scales down to r = 0.01h−1Mpc .
Alternatively, we may estimate the projected correlation
function, i.e., the wpstatistic, for these pairs. Following the
definition of wp(e.g., Davis & Peebles 1983), we have
drical annulus over which we search for pairs, R is the geo-
metric mean pair separation in the bin, n(z) is the cumulative
quasar number density, Npair≡ ΣDD is the observed num-
ber of quasar pairs in the bin, and RR is the expected num-
ber of random-randompairs in the cylindrical shell with radii
(R1,R2) and height dH. Note that there are some approxima-
tions and ambiguities involved in Eqn. (4), such as the posi-
wp(R) = 2
dHξ2D(R,H) = 2
1) is the projected comoving area of the cylin-
SMALL-SCALE QUASAR CLUSTERING AT z ∼ 3−43
FIG. 1.— Measurements of the small-scale clustering for the low-z bin (left) and high-z bin (right). Filled circles are the large-scale correlation function data
from Shen et al. (2007, the all sample) and dashed lines are their power-law fits with fixed slope γ = 2. Squares are our estimate of wpusing Eqn. (4), estimated
in a large radial bin (filled) and two smaller radial bins (open). Points are placed at the logarithmic mean of pair separations in the bin, horizontal error bars show
the bin size, and vertical error bars show Poisson errors. The black hatched regions show our ML power-law fits to the small-scale pairs (§2.1; fspec= 1), with the
vertical extent enclosing the 1σ statistical uncertainty from the ML fitting. If we assume minimal spectroscopic completeness, fspec= 0.38 (0.52) for the low-z
(high-z) bin, the ML results are shown as red hatched regions (see §2.2). These estimates, however, should be considered as solid upper limits.
tion of the bin center, hence it can only be treated as a crude
estimate for wp.
For both the ML approach and the wpstatistic we need to
estimate the integral?zmax
edge of the faint end of the luminosity function (i < 21) of
quasars at redshift 2.9<z <4.5. We have searched the litera-
ture for usable LF within these redshift and luminosity ranges
(e.g., Wolf et al. 2003; Jiang et al. 2006; Richards et al. 2006;
Hopkins et al. 2007). The Jiang et al. (2006) LF data probe
sufficiently faint but do not extend to z > 3.6; the Richards
et al. (2006) data have the desired redshift coverage but do
not probe deep enough. By comparing the Richards et al. LF
with the COMBO-17 LF (Wolf et al. 2003), we found that
the COMBO-17 PDE fit gives better estimates of the LF at
z > 3.5, e.g., it agrees well with the Richards et al. LF at
the high luminosity end, and produces the expected flatten-
ing at fainter luminosities. Motivated by these comparisons,
we adopt a combination of the Jiang et al. fit (at z < 3.5) and
the COMBO-17 PDE fit (at z > 3.5) for the model LF, scaled
to our standard cosmology. We estimate an uncertainty in the
cumulative number density (i < 21) of ∼ 20%, based on the
statistical uncertainties in these LF fits and comparison be-
tween these optical LFs and the bolometric LF compiled by
Hopkinsetal. (2007),wherethefaintendLFat theseredshifts
is further constrained by X-ray data. This estimate of uncer-
tainty in the model LF is, however, conservative at z > 3.5,
since there are no direct optical LF measurements down to
i = 21 within this redshift range. We will discuss the contri-
bution of the uncertainty in the LF to the systematic errors in
our small-scale clustering measurements, in §2.2.
Our clustering measurements are summarized in Fig. 1,
where we plot for comparison the large-scale (R ? 2 h−1Mpc)
correlation function data from Shen et al. (2007, the all sam-
ple), for the low-z (left) and high-z (right) bins respectively.
TheML approachyields r0,ML=8.31+1.77
zminfcompn2dVc. This requires knowl-
−1.61h−1Mpc forthe low-
z bin, and r0,ML= 18.22+3.47
errors are 1σ statistical only; these results are shown as black
hatched regions whose horizontal and vertical extent encloses
thefittingrangeandstatistical errors. Forthebinnedwpstatis-
tic, we take all the pairs and use Eqn. (4) to estimate wpfor
the two redshift bins with Poisson errors. We then plot the wp
estimates at the (geometric)mean values of separations?R? as
filled squares in Fig. 1. To indicate the uncertainties in the bin
center, we draw horizontalerror bars which enclose the fitting
ranges in the ML approach. In both redshift bins we further
divide the pairs into two radial bins (with more or less equal
number of pairs each), with the dividing scale R = 0.34 and
0.56h−1Mpc for the low-z and high-z cases (the dividingscale
is set by the geometric mean of the maximum separation of
observed pairs in the inner bin and the minimum separation
of observed pairs in the outer bin). The wpestimates for the
ofthewpstatistic areconsistentwiththeMLresults withinthe
errors. However, due to the ambiguity of placing bin centers
when there are only a few pairs, the wpdata points cannot be
used in the power-law fit. Our ML approach is not subject
to such ambiguities, and therefore providesreliable clustering
measurements. We tabulate the ML results in Table 1.
−3.12h−1Mpc for the high-z bin, where
2.2. Systematic Uncertainties
uncertainties in our ML results. The two major systematics
come from the adopted luminosity function and the sample
completeness. Our model luminosity function is quite uncer-
tain down to i = 21, especially at z > 3.5 where no direct op-
tical LF data are available. As we described above, the uncer-
tainty in the LF is ∼20%. In addition, the relative uncertainty
in our pair target selection completeness is ? 10% (Paper I).
These taken together, introduce a systematic uncertainty in
the best-fit r0,MLof σr0= ±1.5h−1Mpc and ±3.1h−1Mpc for
the low-z and high-z bins, respectively; these values are com-
4SHEN ET AL.
FIG. 2.— HOD model predictions for a flux limit of i < 21 compared with the clustering data (notations are the same as Fig. 1). The solid lines are the HOD
predictions, where the dotted lines are the two-halo term contribution to wp. Three HOD models with satellite halo duty cycle fs= 0.1 (blue), 0.5 (cyan) and 1.0
(magenta) are presented. For clarity, we have removed the upper limits on the small-scale clustering shown in Fig. 1 (the red hatched regions).
ESTIMATES OF r0FOR FIXED POWER-LAW (γ = 2) CORRELATION
NOTE. — The second column lists our ML results assuming spec-
troscopic completeness fspec= 1 (§2.1). The third column lists ML
upper limits assuming the lowest fspec (§2.2). The fourth column
lists the large-scale correlation lengths from Shen et al. (2007, the
all sample). Uncertainties are 1σ statistical only.
parable to the statistical uncertainties reported above.
In addition, our spectroscopy is incomplete even at θ < 60′′
– only ∼38% and ∼ 52% of the high-prioritylow-z and high-
z binary targets have been observed (see Table 3 of Paper I).
Therefore we are undoubtedlymissing some quasar pairs and
our ML results are lower limits. Because targets further away
from the stellar locus were assigned higher priority (Paper I),
it is difficult to assess the effective spectroscopic complete-
ness (because the most promising candidates were observed
first); we expect that the effective spectroscopic completeness
is larger than 50%. In the extreme case fspec= 0.38 (low-z)
and 0.52 (high-z), we repeat our ML analysis in §2.1 with
fcomp= ftarg× fspec and find r0,ML= 13.81+2.82
spectively,where errors are 1σ statistical. These estimates are
shown as red hatched regions in Fig. 1 and should be consid-
ered as solid upper limits.
The ML results in §2.1 have comparable or lower cluster-
ing amplitude at 0.1 ? R ? 1 h−1Mpc than the extrapolations
from the fits for the large-scale correlation functions (Shen et
al. 2007, 2009). This does not directly contradict the results
in Hennawi et al. (2006) for z < 3 quasars since: 1) our sam-
ple barely probes scales below R ∼ 0.1h−1Mpc where most
−4.28h−1Mpc for the low-z and high-z case re-
of the excess clustering occurs for the z < 3 sample (Hen-
nawi et al. 2006), and 2) the quasar sample in Shen et al.
(2007) has i < 20.2, while our binary sample has i < 21, thus
luminosity-dependent clustering at such high redshift and lu-
minosity ranges might play a role (e.g., Shen 2009). In the
next section we show how these small-scale clustering mea-
surements can be used to constrain halo occupation models.
The small-scale clustering measurements presented above
can be used to constrain the statistical occupation of quasars
within dark matter halos at z ? 3. Given that we have a
poor understanding of the physics of quasar formation, we
use a simple phenomenological model relating quasars to ha-
los to modelthe observedclusteringresults. The details of the
model will be presented elsewhere (Shankar et al., in prepara-
tion); below we briefly describe the model assumptions.
We assume there is a monotonic relationship between
quasar luminosity and the mass of the host dark matter halo
(including subhalos), with a log-normal scatter Σ (in dex).
Therefore for a flux-limited quasar sample, the minimal halo
mass Mminand the average duty cycle f, defined as the frac-
tion of halos that host a quasar abovethe luminosity threshold
at a given time, can be jointly constrained from abundance
matching and the clustering strength (e.g., Martini & Wein-
berg 2001; Haiman & Hui 2001; Shen et al. 2007; White et
dlogM , (5)
where nQSO,i<21(z) is the cumulative quasar number density
with flux limit i < 21, M is the halo mass, Φhalo(M,z) is the
halo mass function per logM interval, and 0 < f(M,z) < 1 is
the average halo duty cycle, which may be a function of both
redshift and halo mass.
SMALL-SCALE QUASAR CLUSTERING AT z ∼ 3−45
In general the halo mass function Φhalo(M,z) includes con-
tributionsfrom bothhalos (Φc) andtheir subhalos(Φs), where
we use the Sheth & Tormen (1999) halo mass function for the
former and the unevolved subhalo mass function from Gio-
coli et al. (2008) for the latter. It is important to use the
unevolved mass (i.e., mass defined at accretion before tidal
stripping takes place) for subhalos, since subhalos will lose a
the parent halo. We denote the average duty cycles for central
and satellite halos as fcand fsrespectively. Note that we as-
sume halos and subhalos of the same mass host quasars of the
sameluminosity– ofcourse,subhaloswithina givenhalowill
be less massive and thus host quasars fainter on average than
the central quasar. The satellite duty cycle fsis the fraction
of black holes in subhalos that are active at a given time. The
fractionofluminousquasarsthataresatellites is alwayssmall,
regardlessof fs, becausethe numberof massive satellite halos
is itself small.
An importantconsequenceofthe rareness ofbinaryquasars
is that the abundance matching, i.e., Eqn. (5), can be done
using central halos only, and we have f ≈ fc, Φhalo≈ Φcin
Eqn. (5); the satellite duty cycle fswill only affect the small-
and their abundance, Shankar et al. (2008, 2009) found large
values of duty cycle fc∼ 0.5−1 are needed, as well as small
scatter for the quasar-halo correspondence, if the Sheth et al.
(2001)bias formula is used (cf. Shen et al. 2007 for the usage
of alternative bias formulae). For simplicity we fix fc= 1 and
Σ = 0.03 dex in what follows, which produces adequate fits
for the large-scale clustering and abundance matching13.
Using Eqn. (5), we determine the minimal halo mass to be
Mmin∼ 1013h−1M⊙for both redshift bins. We then use stan-
dard halo occupation distribution (HOD) models (e.g., Tinker
et al. 2005) to compute the one-halo term correlationfunction
with different values of satellite duty cycle 0 < fs< 1.
Fig. 2 shows several examples of our HOD model at z =3.1
(left panel) and z = 4 (right panel) with fs= 0.1 (blue), 0.5
(cyan) and 1.0 (magenta) for a flux limit of i = 21. Solid lines
are the total correlation while the dotted line denotes the two-
halo term contribution. As expected, the value of fshas no
effect on the large-scale clustering; it only changes the small-
scale clustering amplitude. These are not actual fits to the
data because the quality of our measurements does not allow
a reliable HOD fit. Nevertheless, it seems that some active
satellite halos are required, but only ? 50% of satellite ha-
los can be active at a given time in order not to overshoot the
small-scale clustering. This constraint is less stringent if we
consider instead the upper limits on the small-scale clustering
discussed in §2.2. One potential concern regardingour model
is that the adopted subhalo mass function has not yet been
tested against simulations for the extreme high-mass end and
redshift ranges considered here; nevertheless our model ap-
proach demonstrates how the small-scale clustering measure-
ments can be used to constrain quasar occupations within ha-
los. We defer a more detailed investigation on the uncertain-
ties and caveats of our halo models to a future paper (Shankar
et al., in preparation).
We have measured the small-scale (0.1h−1Mpc ? R ?
1h−1Mpc) clustering of quasars at high redshift (z ?3), based
on a sample of 15 close binaries from Paper I. Strong clus-
tering signals are detected, comparable to or lower than
the extrapolations from the large-scale clustering based on
SDSS quasar samples. The small-scale clustering increases
in strength from z ∼ 3 to z ∼ 4, consistent with that of the
large-scale clustering (Shen et al. 2007, 2009).
Using a simple prescription relating quasars to dark matter
halos, we constrain the average duty cycles of satellite ha-
los at z ? 3 from the small-scale clustering measurements.
We found tentative evidence that only ∼ 10%−50% of satel-
lite halos with mass ? 1013h−1M⊙can host an active quasar
(with i < 21). With the completion of our ongoing binary
quasar survey, we will have better estimates of the spectro-
scopic completeness and therefore will confirm our results.
Future surveys of fainter binary quasars at z > 3 will
increase the sample size and hence the signal-to-noise ra-
tio of the small-scale clustering measurements.
measurements, together with better understandings of the
halo/subhalo abundance and clustering at z > 3 from simula-
tions, will provideimportant clues to the formationof quasars
at high redshift.
We thank Silvia Bonoli, Charlie Conroy, Phil Hopkins and
Linhua Jiang for helpful discussions. This work was par-
tially supported by NSF grants AST-0707266 (YS and MAS)
and AST-0607634 (DPS). FS acknowledges partial support
from NASA grant NNG05GH77G and from the Alexander
von Humboldt Foundation. SGD and AM acknowledge par-
13Although the model still underpredicts the large-scale clustering a bit
for the high-z bin even with fc= 1, as noted in earlier papers (White et al.
2008; Shankar et al. 2008; Shen 2009).
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