The Halo Model• Observations of galaxy clustering• The Halo Model: A nonlinear and biased view

– 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

• Fractal/multi-fractal behaviour

• Percolation/Minimal spanning tree

• Minkowski functionals

• Shape statistics

Number of data pairs with separation rNumber of random pairs with separation r = 1 + (r)

Power-law: Power-law: (r) = (r(r) = (r00/r)/r)

slope slope = -1.8= -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

Springel et al. 2005

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)

Cold Dark Matter

• Cold: speeds are non-relativistic

• To illustrate, 1000 km/s ×10Gyr ≈ pc; 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)

The Halo Mass

Function•Small halos collapse/virialize first•Can also model halo spatial distribution•Massive halos more strongly clustered

(Reed et al. 2003)

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

over-dense

under-denseKey to understand

galaxy biasing (Mo & White 1996; Sheth & Tormen 2002)

n(m|) = [1 + b(m)] n(m) ≠ [1 + ] 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 (axis-ratios) 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)/gal2

– ξ2gal(r) ≈ [∫dm n(m)g1(m)b(m)/gal]2 ξdm(r)

– gal = ∫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

Spatial distribution within halos second order effect (on >100 kpc)

Sheth & Diaferio 2001

Comparison with simulations

• Halo model calculation of (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

1h›2h

1h‹2h →

Sheth et al. 2001

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

Skibba, Sheth, Connolly, Scranton 2006

<Lcen|M> ~ ln(1 + M/Mcrit)

<Lsat|M> ~ independent of M

Prediction based on halo-model interpretation of clustering in SDSS for galaxy samples with various L cuts (Zehavi et al. 2005)

central

satellitetotal

Halo model fits of Zehavi et al. (SDSS) imply mean satellite luminosity only weak function of halo massProvides 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/4)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

= (8G/3H2) (3M/4r3) (Hr)2/2

= 200 /c (Hr)2/2 = (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 ~ V2

Two contributions to velocities• 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

~ mass1/3

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

Equilateral triangles in the dark matter distribution…

Wang et al. 2004

…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)

Hagan et al 2005

Change in power due to substructure; model by Sheth & Jain (2003)

Wrong subclump model won’t work

Weighted correlations or mark statistics:

There’s more to the points

Luminosity as mark in SDSS

Skibba, Sheth, Connolly, Scranton 2006

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

centre not special

centre special

Unweighted signal

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

Abbas & Sheth 2006

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

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- forest

The Holy Grail

Halo model provides natural framework within which to discuss, interpret most measures of clustering; it is the natural language of galaxy ‘bias’

The Halo Grail(phrase coined by Jasjeet Bagla)

THE Cup!