the halo model
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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: There’s 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 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!)
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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
• ξdm(r) = ξ1h(r) + ξ2h(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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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
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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)
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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 – Hi2Ri
3/2GM
= 1 – (3Hi2 /8G) (4Ri
3/3)/M
= 1 – 1/(1+i) = i/(1+i) ≈ i
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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)
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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
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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)
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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)
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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)
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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(m|) = [1 + b(m)] n(m)
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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-
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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
• ξdm(r) = ξ1h(r) + ξ2h(r)
• All physics can be decomposed similarly: influences from within halo, versus from outside
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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
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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?
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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
• ξdm(r) = ξ1h(r) + ξ2h(r)
• Spatial distribution within halos is small-scale detail
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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
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Type-dependent clustering: Why?
populate massive halos
populate lower mass halos
Spatial distribution within halos second order effect (on >100 kpc)
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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
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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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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- 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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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…
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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) ‹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)
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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
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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