Transcript No Slide Title

The Halo Model
• Observations of galaxy clustering
• The Halo Model: A nonlinear and biased view
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Real vs. Redshift space
Substructure
Weighted or Mark correlations
Correlations with environment
• A simple parametrization of galaxy formation
– A complete, observationally calibrated prescription for
higher order moments
• Beyond galaxies:
– mass, momentum and pressure fields (or statistics of
weak lensing, kSZ and tSZ power spectra)
Large Scale Structure
• 2-point and n-point statistics
• Counts in cells/void probability function
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Fractal/multi-fractal behaviour
Percolation/Minimal spanning tree
Minkowski functionals
Shape statistics
Number of data pairs with separation r
Number of random pairs with separation r
= 1 + x(r)
Power-law: x(r) = (r0/r)g
slope g = -1.8
Galaxy
clustering
depends on
galaxy type:
color,
luminosity,
etc.
Zehavi et al. 2005
SDSS
Galaxy Clustering
varies with Galaxy
Type
How is each galaxy
population related to
the underlying Mass
distribution?
Bias depends upon
Galaxy Color and
Luminosity
Need large, carefully
selected samples to
study this (e.g. Norberg
et al. 2002 2dFGRS)
Light is a biased tracer
Understanding bias important for understanding mass
Gaussian fluctuations as seeds of subsequent structure formation
Gaussianity simplifies mathematics: logic which follows is general
Hierarchical
models
Dark matter ‘haloes’ are
basic building blocks of
‘nonlinear’structure
Models of halo formation
suggest haloes have same
density whatever their
mass (Gunn & Gott 1974)
Springel et al. 2005
Cold Dark Matter
• Cold: speeds are non-relativistic
• To illustrate, 1000 km/s ×10Gyr ≈ 10Mpc;
from z~1000 to present, nothing (except
photons!) travels more than ~ 10Mpc
• Dark: no idea (yet) when/where the stars
light-up
• Matter: gravity the dominant interaction
Initial spatial distribution within patch (at z~1000)...
…stochastic (initial
conditions Gaussian
random field); study
`forest’ of merger
history ‘trees’.
…encodes information about
subsequent ‘merger history’
of object
(Mo & White 1996; Sheth 1996)
Galaxy formation
• Gas cools in
virialized dark matter
‘halos’. Physics of
halos is nonlinear, but
primarily gravitational.
• Complicated
gastrophysics (star
formation, supernovae
enrichment, etc.)
mainly determined by
local environment (i.e.,
by parent halo), not by
surrounding halos.
How to describe different point
processes which are all built from
the same underlying distribution?
Center-satellite process requires knowledge of how
1) halo abundance;
2) halo clustering;
3) halo profiles;
4) number of galaxies per halo;
all depend on halo mass.
(Revived, then discarded in 1970s by Peebles, McClelland & Silk)
(Reed et al. 2003)
The Halo
Mass
Function
•Small halos
collapse/virialize
first
•Can also model
halo spatial
distribution
•Massive halos
more strongly
clustered
Sheth & Tormen 1999; Jenkins et al. 2001; Warren et al. 2006
Still the best connection between the Excursion Set
model and the Peaks model (making this airtight is a
nice open problem)
Most
massive
halos
populate
densest
regions
Key to understand
galaxy biasing
over-dense
underdense
(Mo & White 1996;
Sheth & Tormen 2002)
n(m|d) = [1 + b(m)d] n(m) ≠ [1 + d] n(m)
Massive halos
more strongly
clustered
‘linear’ bias
factor on large
scales increases
monotonically
with halo mass
Hamana et al. 2002
Theory predicts…
• Can rescale halo abundances to ‘universal’
form, independent of P(k), z, cosmology
– Greatly simplifies likelihood analyses
• Intimate connection between abundance and
clustering of dark halos (Sheth & Tormen 1999)
– Can use cluster clustering as check that cluster
mass-observable relation correctly calibrated
• Important to test if these fortunate
simplifications also hold at 1% precision
Halo
Profiles
• Not quite
isothermal
• Depend on halo
mass, formation
time; massive
halos less
concentrated
• Distribution of
shapes (axisratios) known
(Jing & Suto 2001)
Navarro,
Frenk &
White
(1996)
The halo-model of clustering
• Two types of pairs: both particles in same halo, or
particles in different halos
• ξdm(r) = ξ1h(r) + ξ2h(r)
• All physics can be decomposed similarly: ‘nonlinear’
effects from within halo, ‘linear’ from outside
Halo Model is simplistic …
• Nonlinear physics on small scales from
virial theorem
• Linear perturbation theory on scales larger
than virial radius (exploits 20 years of hard
work between 1970-1990)
…but quite accurate!
The halo-model of galaxy clustering
• Again, write in terms of two components
– ξ1gal(r) ~ ∫dm n(m) g2(m) ξdm(m|r)/rgal2
– ξ2gal(r) ≈ [∫dm n(m)g1(m)b(m)/rgal]2 ξdm(r)
– rgal = ∫dm n(m) g1 (m): number density of galaxies
– ξdm(m|r): fraction of pairs in m-halos at separation r
• Think of number of galaxies, g1(m), as a weight applied
to each dark matter halo - galaxies ‘biased’ if g1(m) not
proportional to m
(Jing, Mo & Boerner 1998; Benson et al. 2000; Peacock & Smith 2000;
Seljak 2000; Scoccimarro et al. 2001)
Type-dependent clustering: Why?
populate massive halos
populate
lower mass
halos
Sheth & Diaferio 2001
Spatial distribution within halos second order effect (on >100 kpc)
Comparison with
simulations
• Halo model
calculation of x(r)
• Red galaxies
• Dark matter
• Blue galaxies
• Note inflection at scale
of transition from
1halo term to 2-halo
term
• Bias constant at large r
Sheth et al. 2001
x1h›x2h
x1h‹x2h →
Similarly …
• Model clustering of thermal SZ effect as a weight
proportional to pressure, applied to halos/clusters
• Model clustering of kinetic SZ signal as a weight,
proportional to halo/cluster momentum
• Model weak gravitational galaxy-galaxy lensing as
cross-correlation between galaxies and mass in halos
• (see review article Cooray & Sheth 2002)
Satellite galaxy counts ~ Poisson
• Write g1(m) ≡ ‹g(m)› = 1 + ‹gs(m)›
• Think of ‹gs(m)› as mean number of satellite
galaxies per m halo
• Minimal model sets number of satellites as
simple as possible ~ Poisson:
• So g2(m) ≡ ‹g(g-1)› = ‹gs (1+gs)› = ‹gs› +
‹gs2› = 2‹gs› + ‹gs›2 = (1+‹gs›)2 - 1
• Simulations show this ‘sub-Poisson’ model
works well (Kravtsov et al. 2004)
Luminosity dependent clustering
Zehavi et al. 2005
SDSS
• Deviation from power-law statistically significant
• Centre plus Poisson satellite model (two free parameters)
provides good description
Two approaches
• Halo Occupation Distribution
(Jing et al., Benson et al.; Seljak; Scoccimarro et al.)
– Model Ngal(>L|Mhalo) for range of L (Zehavi et al.;
Zheng et al.; Berlind et al.; Kravtsov et al.; Conroy et al.; Porciani,
Magliochetti; Collister, Lahav)
– Differentiating gives LF as function of Mhalo
(Tinker et al., Skibba et al.):
• Conditional Luminosity Function (Peacock, Smith):
– Model LF as function of Mhalo , and infer HOD
(Yang, Mo, van den Bosch; Cooray)
Why is …
• Luminosity dependence of SDSS clustering
well described by halo model with
g1(m|L) ≈ 1 + m/[23 m1(L)]
• g1(m|L) nonzero only if m>m1, where m1(L)
adjusted to match decrease of number
density with increasing L
• (Assume Poisson distribution, with mean g1,
for non-central, ‘satellite’ galaxies)
• Number density and clustering as function
of luminosity now measured in 2dF,SDSS
• Assuming there are NO large scale
environmental effects, halo model provides
estimates of luminosity distribution as
function of halo mass (interesting, relatively
unexplored connection to cluster LFs)
• Suggests BCGs are special population
(another interesting, unexplored connection
to clusters!)
Predicted
correlation
between
luminosity
and mass
Prediction based on
halo-model
interpretation of
clustering in SDSS
for galaxy samples
with various L cuts
(Zehavi et al. 2005)
total
satellite
central
<Lcen|M> ~ ln(1 + M/Mcrit)
<Lsat|M> ~ independent of M
Skibba, Sheth, Connolly, Scranton 2006
Halo model fits of Zehavi et al. (SDSS) imply
mean satellite luminosity only weak function of
halo mass
Provides good description of group catalog of
Berlind et al. (SDSS)
Worth reconsidering Scott (1957) effect, but for satellites?
BCG LF: Peebles; Tremaine & Richstone; Bhavsar & Barrow;
Lauer & Postman
Redshift space effects
Halos and Fingers-of-God
• Virial equilibrium:
• V2 = GM/r = GM/(3M/4p200r)1/3
• Since halos have same density, massive halos have
larger random internal velocities: V2 ~ M2/3
• V2 = GM/r = (G/H2) (M/r3) (Hr)2
= (8pG/3H2) (3M/4pr3) (Hr)2/2
= 200 r/rc (Hr)2/2 = W (10 Hr)2
• Halos should appear ~ten times longer along line of
sight than perpendicular to it: ‘Fingers-of-God’
• Think of V2 as Temperature; then Pressure ~ V2r
Two contributions to velocities
~ mass1/3
• Virial motions
(i.e., nonlinear
theory terms)
dominate for
particles in
massive halos
• Halo motions
(linear theory)
dominate for
particles in low
mass halos
Growth rate of halo motions ~ consistent with linear theory
Higher order moments
• In centre + Poisson satellite model, these
are all completely specified
• On large scales, higher order moments
come from suitably weighting perturbation
theory results
• Incorporating halo shapes matters on small
scales (Smith, Watts & Sheth 2006)
Three-point statistics
Wang et al. 2004
Equilateral triangles in the dark matter distribution…
…and for
galaxies…
…in real
space…
Wang et al. 2004
…and in redshift space
Wang et al. 2004
…as well as
dependence on
configuration
and on
luminosity
and on color
Wang et al. 2004
Halo Substructure = Galaxies
• Halo substructure = galaxies is good model
(Klypin et al. 1999; Kravtsov et al. 2005)
• Agrees with semi-analytic models and SPH
(Berlind et al. 2004; Zheng et al. 2005; Croton et al. 2006)
• Setting n(>L) = n(>Vcirc) works well for
all clustering analyses to date, including z~3
(Conroy et al. 2006)
Wrong subclump
model won’t work
Change in power due to
substructure; model by
Sheth & Jain (2003)
Hagan et al 2005
Weighted correlations
or mark statistics:
There’s more to the points
Luminosity as mark in SDSS
Large scale signal
consistent with halo
bias prediction; no
large scale
environmental trends
Small scale signal
suggests centre
special; model with
gradual threshold
(rather than step) is
better
Unweighted signal
centre not special
centre special
Skibba, Sheth, Connolly, Scranton 2006
Environmental effects
• Gastrophysics determined by formation
history of parent halo
• Formation history correlates primarily
with mass (massive objects form later)
• Current halo model implementations
implicitly assume that all environmental
trends come from fact that massive halos
populate densest regions
Why does this matter?
• Greatly simplifies galaxy formation models
and interpretation of galaxy clustering:
– Some semi-analytic galaxy formation models
assume this explicitly (when use semi-analytic
merger trees rather than trees from simulation)
– Implicit assumption in both HOD and CLF
implementations of halo model
Cracks in the standard model
• Sheth &Tormen (2004) measure correlation
between formation time and environment:
– At fixed mass, close pairs form earlier
– Point out relevance to halo model description
– Measurement repeated and confirmed by Gao et al.
(2005), Harker et al. (2006), Wechsler et al. (2006)
• Does this matter for surveys which use clustering
of (primarily) luminous galaxies for cosmology
(Abbas & Sheth 2005, 2006; Croton et al. 2006)?
Close pairs
form at
higher
redshifts
Sheth & Tormen
2004
Environmental
dependence of
clustering in
the SDSS
1/3 objects in densest regions
1/3 in least dense regions
•Density is number of
galaxies within 8 Mpc/h
•Statistical effect
accounts for most (all?)
of the environmental
dependence
Abbas & Sheth 2006
A Nonlinear and Biased View
• Observations of galaxy clustering on large
scales are expected to provide information
about cosmology (because clustering on
large scales is still in the ‘linear’ regime)
• Observations of small scale galaxy
clustering provide a nonlinear, biased view
of the dark matter density field, but they do
contain a wealth of information about
galaxy formation
• How to characterize this information and so
inform galaxy formation models?
Halo Model
• Describes spatial statistics well
• Describes velocity statistics well
• Since Momentum ~ mv, Temperature ~ v2,
and Pressure ~ Density ×Temperature,
Halo Model useful framework for
describing Kinematic and Thermal SZ
effects, and various secondary contributions
to CMB
• Open problem: Describe Ly-a forest
The Halo Grail
(phrase coined by Jasjeet Bagla)
Halo model
provides natural
framework
within which to
discuss, interpret
most measures
of clustering; it
is the natural
language of
galaxy ‘bias’
The Holy Grail
THE Cup!