The Halo Model

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The Halo Model

• Structure formation cosmic capitalism
• Halos abundances, clustering and evolution
• Galaxies a nonlinear biased view of dark
  matters
• Marked correlations Theres more to the points
• Ravi K. Sheth (UPitt/UPenn)

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Galaxy Surveys
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Galaxy clustering depends on type
Large samples now available to quantify this
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Light is a biased tracer
To use galaxies as probes of underlying dark
matter distribution, must understand bias
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• Center-satellite process requires knowledge of
• halo abundance 2) halo clustering 3) halo
  profiles
• 4) how number of galaxies per halo depends on
  halo mass.
• (Also a simple model of earthquakes and
  aftershocks!)

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• Neyman Scott
• Hypothesis testing (J. Neyman)
• Model of ozone
• Model of cancer
• Model of BCGs (E. Scott)
• Clustering model (Neyman Scott)

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The halo-model of clustering

• Two types of pairs both particles in same halo,
  or particles in different halos
• ?dm(r) ?1h(r) ?2h(r)
• All physics can be decomposed similarly
  influences from within halo, versus from outside
  (Sheth 1996)

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Gaussian fluctuations as seeds of subsequent
structure formation
Gaussianity simplifies mathematics logic which
follows is general
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N-body simulations of gravitational clustering
in an expanding universe
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Cold Dark Matter

• Simulations include gravity only (no gas)
• Late-time field retains memory of initial
  conditions
• Cosmic capitalism

Co-moving volume 100 Mpc/h
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Its a capitalists life

• Most of the action is in the big cities
• Newcomers to the city are rapidly stripped of
  (almost!) all they have
• Encounters generally too high-speed to lead to
  long-lasting mergers
• Repeated harassment can lead to change
• Real interactions take place in the outskirts
• A network exists to channel resources from the
  fields to feed the cities

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Spherical evolution model

• Collapse depends on initial over-density Di
  same for all initial sizes
• Critical density depends on cosmology
• Final objects all have same density, whatever
  their initial sizes
• Collapsed objects called halos
• 200 denser than background, whatever their
  mass

(Tormen 1998)
(Figure shows particles at z2 which, at z0, are
in a cluster)
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Spherical evolution model

• Initially, Ei GM/Ri (HiRi)2/2
• Shells remain concentric as object evolves if
  denser than background, object pulls itself
  together as background expands around it
• At turnaround E GM/rmax Ei
• So GM/rmax GM/Ri (HiRi)2/2
• Hence (Ri/rmax) 1 Hi2Ri3/2GM
• 1 (3Hi2 /8pG)
  (4pRi3/3)/M
• 1 1/(1Di)
  Di/(1Di) Di

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Virialization

• Final object virializes -W 2K
• Evir WK W/2 -GM/2rvir -GM/rmax
• So rvir rmax/2 final size, density of object
  determined by initial overdensity
• To form an object at present time, must have had
  a critical overdensity initially
• To form objects at high redshift, must have been
  even more overdense initially
• At any given time, all objects have same density
  (high-z objects denser)

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Virial Motions

• (Ri/rvir) f(Di) ratio of initial and final
  sizes depends on initial overdensity
• Mass M (1Di)Ri3 Ri3 (since initial
  overdensity 1)
• So final virial density M/rvir3 (Ri/rvir)3
  function of critical density hence, at any
  given time, all virialized objects have the same
  density, whatever their mass
• V2 GM/rvir M2/3 massive objects have larger
  internal velocities/temperatures

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Spherical evolution model

• Collapse depends on initial over-density Di
  same for all initial sizes
• Critical density depends on cosmology
• Final objects all have same density, whatever
  their initial sizes
• Collapsed objects called halos
• 200 denser than background, whatever their
  mass

(Tormen 1998)
(Figure shows particles at z2 which, at z0, are
in a cluster)
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Initial spatial distribution within patch (at
z1000)...
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)
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The Halo Mass Function
(Reed et al. 2003)

• Hierarchical no massive halos at high-z
• Halo abundance evolves strongly
• Massive halos (exponentially) rare
• Observable ? mass difficult

(current parameterizations by Sheth Tormen
1999 Jenkins et al. 2001)
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Non-Maxwellian Velocities?

• v vvir vhalo
• Maxwellian/Gaussian velocity within halo
  (dispersion depends on parent halo mass)
  Gaussian velocity of parent halo (from linear
  theory independent of m)
• Hence, at fixed m, distribution of v is
  convolution of two Gaussians, i.e.,
• p(vm) is Gaussian, with dispersion
• svir2(m) sLin2 (m/m)2/3svir2(m) sLin2

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Exponential tails are generic

• p(v) ?dm mn(m) G(vm)
• F(t) ?dv eivt p(v) ?dm n(m)m
  e-t2svir2(m)/2 e-t2sLin2/2
• For P(k) k-1, mass function n(m) power-law
  times exp-(m/m)2/3/2, so integral is
• F(t) e-t2sLin2/2 1 t2svir2(m)-1/2
• Fourier transform is product of Gaussian and FT
  of K0 Bessel function, so p(v) is convolution of
  G(v) with K0(v)
• Since svir(m) sLin, p(v) Gaussian at vltsLin
  but exponential-like tails extend to large v
  (Sheth 1996)

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Comparison with simulations
Sheth Diaferio 2001
Sheth Diaferio 2001

• Gaussian core with exponential tails as expected!

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Universal form?

• Spherical evolution (Press Schechter 1974
  Bond et al. 1991)
• Ellipsoidal evolution (Sheth Tormen 1999
  Sheth, Mo Tormen 2001)
• Simplifies analysis of cluster abundances (e.g.
  ACT)

Jenkins et al. 2001
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Most massive halos populate densest regions
over-dense
under-dense
Key to understand galaxy biasing (Mo White
1996 Sheth Tormen 2002)
n(md) 1 b(m)d n(m)
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Halo clustering
massive halos

• Massive halos more strongly clustered
• Clustering of halos different from clustering of
  mass

non-
linear theory
dark matter
Percival et al. 2003
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Halo clustering ? Halo abundances

• Clustering is ideal (only?) mass calibrator
  (Sheth Tormen 1999)

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The halo-model of clustering

• Two types of pairs both particles in same halo,
  or particles in different halos
• ?dm(r) ?1h(r) ?2h(r)
• All physics can be decomposed similarly
  influences from within halo, versus from outside

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The dark-matter correlation function

• ?dm(r) ?1h(r) ?2h(r)
  
• ?1h(r) ?dm n(m) m2 ?dm(mr)/r2
• n(m) number density of halos
• m2 total number of pairs
• ?dm(mr) fraction of pairs which have separation
  r depends on density profile within m-halos
• Need not know spatial distribution of halos!
• This term only matters on scales smaller than the
  virial radius of a typical M halo ( Mpc)
• ?2h(r) larger scales, depends on halo clustering

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Clustering in simulations

• Expect (and see) feature on scale of transition
  from one- halo to two-halo
• Feature in data?

• ?dm(r) ?1h(r) ?2h(r)

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Power-law x(r) (r0/r)g slope g
-1.8
Totsuji Kihara 1969
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• 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

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(Cole et al. 2000)
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Kauffmann, Diaferio, Colberg White
1999 Also Cole et al., Benson et al.,
Somerville Primack, Colin et al.
Colors indicate age
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Halo-model of galaxy clustering

• Two types of pairs only difference from dark
  matter is that now, number of pairs in m-halo is
  not m2
• ?dm(r) ?1h(r) ?2h(r)
• Spatial distribution within halos is small-scale
  detail

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The galaxy correlation function

• ?dm(r) ?1h(r) ?2h(r)
  
• ?1h(r) ?dm n(m) g2(m) ?dm(mr)/r2
• n(m) number density of halos
• g2(m) total number of galaxy pairs
• ?dm(mr) fraction of pairs which have separation
  r depends on density profile within m-halos
• Need not know spatial distribution of halos!
• This term only matters on scales smaller than the
  virial radius of a typical M halo ( Mpc)
• ?2h(r) larger scales, depends on halo clustering

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Type-dependent clustering Why?
populate massive halos
populate lower mass halos
Spatial distribution within halos second order
effect (on gt100 kpc)
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Comparison with simulations
Sheth et al. 2001
steeper

• Halo model calculation of x(r)
• Red galaxies
• Dark matter
• Blue galaxies
• Note inflection at scale of transition from
  1-halo term to 2-halo term
• Bias constant at large r

shallower
?x1hx2h
x1hx2h ?
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Color dependent clustering
Zehavi et al. (SDSS)
steeper
shallower
Reddest galaxies (oldest stars) in most massive
halos?
Form of g(m) required to match clustering data
summarizes and constrains effects of complicated
gastrophysics.
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Galaxy formation models correctly identify the
halos in which galaxies form Galaxy halo
substructure is reasonable model
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A Nonlinear and Biased View

• Observations of galaxy clustering on large scales
  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
• g(m) characterizes this information and so can
  inform galaxy formation models

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Successes and Failures

• Distribution of sizes Lognormal seen in SDSS
• Morphology-density relation (oldest stars in
  clusters/youngest in field)
• Type-dependent clustering red galaxies have
  steep correlation function clustering strength
  increases with luminosity
• Distribution of luminosities
• Correlations between observables
  (luminosity/color, luminosity/velocity dispersion)

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Sizes of disks and bulges
Observed distribution Lognormal Distribution
of halo spins Lognormal Distribution of halo
concentrations Lognormal (Bernardi et al.
2003 Kauffmann et al. 2003 Shen et al. 2003)
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The ISW effect
Cross-correlate CMB and galaxy distributions Inte
rpretation requires understanding of galaxy
population
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Cross-correlate LRGs with CMB
Measured signal combination of ISW and SZ
effects Estimate both using halo model
(although signal dominated by linear theory)
Signal predicted to depend on b(a) D(a) d/dt
D(a)/a
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Evolution and bias
Work in progress to disentangle evolution of
bias from z dependence of signal (Scranton
et al. 2004)
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Halo clustering ? Halo abundances

• Clustering is ideal (only?) mass calibrator
  (Sheth Tormen 1999)

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Environmental effects

• Fundamental assumption all
  environmental trends come from fact that massive
  halos populate densest regions

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Summary

• Hierarchical clustering cosmic capitalism
  Many models (percolation, coagulation, random
  walks) give equivalent descriptions
• All models separate cosmology/dynamics from
  statistics P(k)
• Gastrophysics determined by mass of parent halo
• All effects of density (environment) arise
  through halo bias (massive halos populate densest
  regions)
• Description quite detailed language of halo
  model also useful for other biased observables

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Halo Model

• Describes spatial statistics well
• Describes velocity statistics well
• Since Momentum mv, Temp v2 m2/3, and
  Pressure Density Temp
• Halo Model useful language for interpreting
  Kinematic and Thermal SZ effects, various
  secondary contributions to CMB, and gravitational
  lensing (see Cooray Sheth 2002 review)
• Open problem Describe Ly-a forest

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Marked correlation functions
Weight galaxies by some observable (e.g.
luminosity, color, SFR) when computing clustering
statistics (standard analysis weights by zero or
one)
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Theres more to the point(s)

• Multi-band photometry becoming the norm
• CCDs provide accurate photometry possible to
  exploit more than just spatial information
• How to estimate clustering of observables, over
  and above correlations which are due to spatial
  clustering?
• Do galaxy properties depend on environment?
  Standard model says only dependence comes from
  parent halos

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Marked correlations
(usual correlation function analysis sets m 1
for all galaxies)
W(r) is a weighted correlation function, so
marked correlations are related to weighted ?(r)
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Luminosity as a mark

• Mr from SDSS
• BIK from semi-analytic
• model
• Little B-band light
• associated with
• close pairs more B-band
• light in field than clusters
• Vice-versa in K
• Feature at 3/h Mpc in all
• bands Same physical
• process the cause?
• e.g. galaxies form in groups
• at the outskirts of clusters

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Colors and star formation

• Close pairs tend to be redder
• Scale on which feature
• appears smaller at higher z
• clusters smaller at high-z?
• Amplitude drops at lower z
• close red pairs merged?
• Close pairs have small
• star formation rates scale
• similar to that for color even
• though curves show
• opposite trends!
• Same physics drives both
• color and SFR?

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Stellar mass

• Circles show M, crosses show LK
• Similar bumps, wiggles in both offset related to
  rms M, L
• Evolution with time M grows more rapidly in
  dense regions

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Halo-model of marked correlations
Again, write in terms of two components W1gal(r)
?dm n(m) g2(m) Wm2 ?dm(mr)/rgal2 W2gal(r)
?dm n(m) g1(m) Wm b(m)/rgal2 ?dm(r) So,
on large scales, expect
1W(r) 1?(r)
1 BW ?dm(r) 1 bgal ?dm(r)
M(r)

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Conclusions (mark these words!)

• Marked correlations represent efficient use of
  information in new high-quality multi-band
  datasets (theres more to the points)
• No ad hoc division into cluster/field,
  bright/faint, etc.
• Comparison of SDSS/SAMs ongoing
• test Ngalaxies(m)
• then test if rank ordering OK
• finally test actual values
• Halo-model is natural language to interpret/model

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Halo-model calculations

• Type-dependent (n-pt) clustering
• ISW and tracer population
• SZ effect and halo shapes/motions
• Weak gravitational lensing
• Absorption line systems
• Marked correlations

Review in Cooray Sheth 2002

Work in progress
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The Holy Grail
The Halo Grail
Halo model provides natural framework within
which to discuss, interpret most measures of
clustering it is the natural language of galaxy
bias
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Grail Bowl? Dish?
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The Cup!
India Cricket World Champions