The Halo Model

• Structure formation: cosmic capitalism

• Halos: abundances, clustering and evolution

• Galaxies: a nonlinear biased view of dark matters

• Marked correlations: There’s more to the points

Ravi K. Sheth (UPitt/UPenn)

Galaxy Surveys

Galaxy clustering depends on type

Large samples now available to quantify this

Light is a biased tracer

To use galaxies as probes of underlying dark matter distribution, must understand ‘bias’

Center-satellite process requires knowledge of 1) 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!)

Neyman & Scott

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

•Clustering model (Neyman & Scott)

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: influences from within halo, versus from outside (Sheth 1996)

Gaussian fluctuations as seeds of subsequent structure formation

Gaussianity simplifies mathematics: logic which follows is general

N-body simulations of

gravitational clustering

in an expanding universe

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

It’s a capitalist’s 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

Spherical evolution model• ‘Collapse’ depends on initial over-density i; 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 z~2 which, at z~0, are in a cluster)

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 – Hi2Ri

3/2GM

= 1 – (3Hi2 /8G) (4Ri

3/3)/M

= 1 – 1/(1+i) = i/(1+i) ≈ i

Virialization• Final object virializes: −W = 2K• Evir = W+K = 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)

Virial Motions• (Ri/rvir) ~ f(i): ratio of initial and final

sizes depends on initial overdensity• Mass M ~ (1+i)Ri

3 ~ Ri3 (since initial

overdensity « 1)• So final virial density ~ M/rvir

3 ~ (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

Spherical evolution model• ‘Collapse’ depends on initial over-density i; 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 z~2 which, at z~0, are in a cluster)

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)

The Halo Mass

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

(Reed et al. 2003)

(current parameterizations by Sheth & Tormen 1999; Jenkins et al. 2001)

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

Most massive

halos populate densest regions

over-dense

under-dense

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

n(m|) = [1 + b(m)] n(m)

Halo clustering

• Massive halos more strongly clustered

• Clustering of halos different from clustering of mass

Percival et al. 2003

massive halos

dark matter

linear theory

non-

Halo clustering Halo abundances

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

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: influences from within halo, versus from outside

The dark-matter correlation function

ξdm(r) = ξ1h(r) + ξ2h(r)

• ξ1h(r) ~ ∫dm n(m) m2 ξdm(m|r)/2

• n(m): number density of halos• m2: total number of pairs

• ξdm(m|r): 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)

• ξ2h(r) ~ larger scales, depends on halo clustering

Clustering in simulations

ξdm(r) = ξ1h(r) + ξ2h(r)

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

•Feature in data?

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

(Cole et al. 2000)

Kauffmann, Diaferio, Colberg & White 1999

Also Cole et al., Benson et al., Somerville & Primack,Colin et al.

Colors indicate age

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

• ξdm(r) = ξ1h(r) + ξ2h(r)

• Spatial distribution within halos is small-scale detail

The galaxy correlation function

ξdm(r) = ξ1h(r) + ξ2h(r)

• ξ1h(r) ~ ∫dm n(m) g2(m) ξdm(m|r)/2

• n(m): number density of halos

• g2(m): total number of galaxy pairs

• ξdm(m|r): 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)

• ξ2h(r) ~ larger scales, depends on halo clustering

Type-dependent clustering: Why?

populate massive halos

populate lower mass halos

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

Comparison with simulations

• Halo model calculation of (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

1h›2h

1h‹2h →

Sheth et al. 2001

steeper

shallower

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

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

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- forest

Marked correlation functions

Weight galaxies by some observable (e.g. luminosity, color, SFR) when computing clustering statistics (standard analysis weights by zero or one)

There’s 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…

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

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?

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

Halo-model of marked correlations

Again, write in terms of two components:

W1gal(r) ~ ∫dm n(m) g2(m) ‹W|m›2 ξdm(m|r)/gal2

W2gal(r) ≈ [∫dm n(m) g1(m) ‹W|m› b(m)/gal]2 ξdm(r)

So, on large scales, expect

M(r) = 1+W(r)1+ξ(r)

= 1 + BW ξdm(r) 1 + bgal ξdm(r)

Conclusions (mark these words!)

• Marked correlations represent efficient use of information in new high-quality multi-band datasets (there’s 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

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