1. 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)

2. 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

5. Number of data pairs with separation r Number of random pairs with separation r = 1 +  (r)

6. Power-law:  (r) = (r 0 /r)  slope  = -1.8 slope  = -1.8

7. Galaxy clustering depends on galaxy type: color, luminosity, etc. Zehavi et al. 2005 SDSS

8. 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)

9. Light is a biased tracer Understanding bias important for understanding mass

10. Gaussian fluctuations as seeds of subsequent structure formation Gaussianity simplifies mathematics: logic which follows is general

11. 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 )

12. 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

14. 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)

15. Random Walks Gaussian initial fluctuation field + spherical evolution model = hierarchical growth of structure (Bond et al. 1991; Lacey & Cole 1993; Sheth 1998; Sheth & van de Weygaert 2003) ←Vv→

16. 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.

18. How to describe different point processes which are all built from the same underlying distribution?

19. 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)

20. 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

21. Still the best connection between the Excursion Set model and the Peaks model (making this airtight is a nice open problem)

22. Recall n(m|  ) = [1 + b(m)  ] n(m) Easier to get to here from here than from here Dense regions host more massive halos b(m) increases as m increases, so n(m|  ) ≠ (1+  ) n(m) Fundamental basis for models of halo bias (and hence of galaxy bias)

23. 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) ≠ [1 +  ] n(m)

24. Massive halos more strongly clustered ‘linear’ bias factor on large scales increases monotonically with halo mass Hamana et al. 2002

25. 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

26. 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)

27. The halo-model of clustering Two types of pairs: both particles in same halo, or particles in different halos ξ dm (r) = ξ 1h (r) + ξ 2h (r) All physics can be decomposed similarly: ‘nonlinear’ effects from within halo, ‘linear’ from outside

28. 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!

29. The dark-matter correlation function ξ dm (r) = ξ 1h (r) + ξ 2h (r) The 1-halo piece ξ 1h (r) ~ ∫dm n(m) m 2 ξ dm (m|r)/  2 n(m): number density of halos ξ dm (m|r): fraction of total pairs, m 2, in an m- halo 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)

30. ξ dm (r) = ξ 1h (r) + ξ 2h (r) ξ 2h (r) = ∫dm 1 m 1 n(m 1 ) ∫dm 2 m 2 n(m 2 ) ξ 2h (r|m 1,m 2 )     Two-halo term dominates on large scales, where peak-background split estimate of halo clustering should be accurate:  h ~ b(m)  dm ξ 2h (r|m 1,m 2 ) ~ ‹  h 2 › ~ b(m 1 )b(m 2 ) ‹  dm 2 › ξ 2h (r) ≈ [∫dm mn(m) b(m)/  ] 2 ξ dm (r) On large scales, linear theory is accurate: ξ dm (r) ≈ ξ Lin (r) so ξ 2h (r) ≈ b eff 2 ξ Lin (r)

31. The halo-model of galaxy clustering Again, write in terms of two components – ξ 1gal (r) ~ ∫dm n(m) g 2 (m) ξ dm (m|r) /  gal 2 – ξ 2gal (r) ≈ [∫dm n(m)g 1 (m)b(m)/  gal ] 2 ξ dm (r) –  gal = ∫dm n(m) g 1 (m): number density of galaxies –ξ dm (m|r): fraction of pairs in m-halos at separation r Think of number of galaxies, g 1 (m), as a weight applied to each dark matter halo - galaxies ‘biased’ if g 1 (m) not proportional to m (Jing, Mo & Boerner 1998; Benson et al. 2000; Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001)

32. Type-dependent clustering: Why? populate massive halos populate lower mass halos Spatial distribution within halos second order effect (on >100 kpc) Sheth & Diaferio 2001

33. 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

34. 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 )

35. Satellite galaxy counts ~ Poisson Write g 1 (m) ≡ ‹g(m)› = 1 + ‹g s (m)› Think of ‹g s (m)› as mean number of satellite galaxies per m halo Minimal model sets number of satellites as simple as possible ~ Poisson: So g 2 (m) ≡ ‹g(g-1)› = ‹g s (1+g s )› = ‹g s › + ‹g s 2 › = 2‹g s › + ‹g s › 2 = (1+‹g s ›) 2 - 1 Simulations show this ‘sub-Poisson’ model works well ( Kravtsov et al. 2004 )

36. 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

37. Two approaches Halo Occupation Distribution ( Jing et al., Benson et al.; Seljak; Scoccimarro et al. ) –Model N gal (>L|M halo ) 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 M halo ( Tinker et al., Skibba et al.): Conditional Luminosity Function ( Peacock, Smith ): –Model LF as function of M halo, and infer HOD ( Yang, Mo, van den Bosch; Cooray)

38. Why is … Luminosity dependence of SDSS clustering well described by halo model with g 1 (m|L) ≈ 1 + m/[23 m 1 (L)] g 1 (m|L) nonzero only if m>m 1, where m 1 (L) adjusted to match decrease of number density with increasing L (Assume Poisson distribution, with mean g 1, for non-central, ‘satellite’ galaxies)

39. 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!)

40. Predicted correlation between luminosity and mass Skibba, Sheth, Connolly, Scranton 2006 ~ ln(1 + M/M crit ) ~ 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 satellite total

41. 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?

42. BCG LF: Peebles; Tremaine & Richstone; Bhavsar & Barrow; Lauer & Postman

43. Luminosity dependent clustering Luminous galaxies more strongly clustered Natural if more luminous galaxies populate more massive halos Simply change g 1 (m) in formalism Zehavi et al. 2002 (SDSS) (Jing, Mo, Boerner 1998; Scoccimarro et al. 2001; Yang et al. 2003)

44. Color dependent clustering Reddest galaxies (oldest stars) in most massive halos? Form of g(m) required to match clustering data summarizes and constrains effects of complicated gastrophysics.

45. Redshift space effects

46. Halos and Fingers-of-God Virial equilibrium: V 2 = GM/r = GM/(3M/4  ) 1/3 Since halos have same density, massive halos have larger random internal velocities: V 2 ~ M 2/3 V 2 = GM/r = (G/H 2 ) (M/r 3 ) (Hr) 2 = (8  G/3H 2 ) (3M/4  r 3 ) (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 V 2 as Temperature; then Pressure ~ V 2 

47. Non-Maxwellian Velocities? v = v vir + v halo 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(v|m) is Gaussian, with dispersion  vir 2 (m) +  Lin 2 = (m/m * ) 2/3  vir 2 (m * ) +  Lin 2

48. 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 ~ mass 1/3

49. Exponential tails are generic p(v) = ∫dm mn(m) G(v|m) F (t) = ∫dv e ivt p(v) = ∫dm n(m)m e -t 2  vir 2 (m)/2 e -t 2  Lin 2 /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 -t 2  Lin 2 /2 [1 + t 2  vir 2 (m * )] −1/2 Fourier transform is product of Gaussian and FT of K 0 Bessel function, so p(v) is convolution of G(v) with K 0 (v) Since  vir (m * )~  Lin, p(v) ~ Gaussian at |v|<  Lin but exponential-like tails extend to large v (Sheth 1996)

50. Comparison with simulations Gaussian core with exponential tails as expected! Sheth & Diaferio 2001

51. 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 )

52. Three-point statistics Equilateral triangles in the dark matter distribution… Wang et al. 2004

53. …and for galaxies… …in real space… Wang et al. 2004

54. …and in redshift space Wang et al. 2004

55. …as well as dependence on configuration and on luminosity and on color Wang et al. 2004

56. 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(>V circ ) works well for all clustering analyses to date, including z~3 ( Conroy et al. 2006 )

57. Hagan et al 2005 Change in power due to substructure; model by Sheth & Jain (2003) Wrong subclump model won’t work

58. Weighted correlations or mark statistics: There’s more to the points

59. 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

60. 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

61. Recall n(m|  ) = [1 + b(m)  ] n(m) Easier to get to here from here than from here Dense regions host more massive halos b(m) increases as m increases, so n(m|  ) ≠ (1+  ) n(m) Fundamental basis for models of halo bias (and hence of galaxy bias)

62. 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

63. 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 )?

64. Close pairs form at higher redshifts Sheth & Tormen 2004

65. 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

66. 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?

67. Halo Model Describes spatial statistics well Describes velocity statistics well Since Momentum ~ mv, Temperature ~ v 2, 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

68. 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)

69. THE Cup!