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R. Sheth - The Phenomenology of Large Scale Structure


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The SEENET-MTP Workshop BW2011
Particle Physics from TeV to Plank Scale
28 August – 1 September 2011, Donji Milanovac, Serbia

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R. Sheth - The Phenomenology of Large Scale Structure

  1. 1. Phenomenology oflarge scale structure Ravi K Sheth (ICTP/UPenn)
  2. 2. • The Halo Model – Observations/Motivation• The Excursion Set approach – The ansatz – Correlated steps – Large scale bias – Ensemble averages: Halos and peaks – Modified gravity
  3. 3. Galaxy spatialdistributiondepends ongalaxy type(luminosity,color, etc.)Not all galaxiescan be unbiasedtracers of theunderlying Massdistribution
  4. 4. Light is a biased tracerUnderstanding bias important for understanding mass
  5. 5. How to describe different pointprocesses which are all built fromthe same underlying distribution? THE HALO MODEL
  6. 6. Center-satellite process requires knowledge of how1) 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)
  7. 7. The halo-model of clustering• Two types of pairs: both particles in same halo, or particles in different halos• ξobs(r) = ξ1h(r) + ξ2h(r)• All physics can be decomposed similarly: ‘nonlinear’ effects from within halo, ‘linear’ from outside
  8. 8. The halo-model of clustering• Two types of particles: central + ‘satellite’• ξobs(r) = ξ1h(r) + ξ2h(r)• ξ1h(r) = ξcs(r) + ξss(r)
  9. 9. To implement Neyman-Scott program need model for cluster abundance and cluster clusteringNext generation of surveys willhave 109 galaxies in105 clusters
  10. 10. In Cold Dark Matter models, mass doesn’t move very far
  11. 11. The Cosmic Background RadiationCold: 2.725 KSmooth: 10-5Simple physicsGaussian fluctuations= seeds of subsequentstructure formation= simple(r) mathLogic which followsis general
  12. 12. Birkhoff’s theorem important
  13. 13. Use initial conditions (CMB) + model of nonlinear gravitational clustering to make inferences aboutlate-time, nonlinear structures
  14. 14. Hierarchical clustering in GR= the persistence of memory
  15. 15. THE EXCURSION SET APPROACH (Epstein 1983; Bond et al. 1991; Lacey & Cole 1993;Sheth 1998; Sheth & van de Weygaert 2004; Shen et al. 2006)
  16. 16. N-bodysimulationsofgravitationalclusteringin anexpandinguniverse
  17. 17. Assume a spherical cow … (Gunn & Gott 1972)
  18. 18. Spherical evolution very well approximated by ‘deterministic’ mapping …(Rinitial/R)3 = Mass/(ρcomVolume) = 1 + δ ≈ (1 – δ0/δsc) −δsc …which can be inverted:(δ0/δsc) ≈ 1 – (1 + δ) −1/δsc
  19. 19. The Nonlinear PDFδcritδ0(M/V)Lineartheoryover- This patch Halo of massdensity of volume m<M within v contains this patch mass M (M,v) V v MASS
  20. 20. • Halo mass function is distribution of counts in cells of size v→0 that are not empty.• Fraction f of walks which first cross barrier associated with cell size V at mass scale M, f(M|V) dM = (M/V) p(M|V) dM where p(M|V) dM is probability randomly placed cell V contains mass M.• Note: all other crossings irrelevant → stochasticity in mapping between initial and final density
  21. 21. The Random Walk ModelHigherRedshiftCriticalover-density This smaller mass patch patch within forms more massive halo of region mass M MASS
  22. 22. From Walks to Halos: Ansätze• f(δc,s)ds = fraction of walks which first cross δc(z) at s ≈ fraction of initial volume in patches of comoving volume V(s) which were just dense enough to collapse at z ≈ fraction of initial mass in regions which each initially contained m =ρV(1+δc) ≈ ρV(s) and which were just dense enough to collapse at z (ρ is comoving density of background) ≈ dm m n(m,δc)/ρ
  23. 23. Random Walk = Accretion history eti h’ High-z cr ot on ac o ‘sm Major merger Low-z over- density larger small massTime mass at at high-zevolution of low z Mapping between σ2barrier and M depends on P(k)depends on MASScosmology σ2(M)
  24. 24. Simplification because…• Everything local• Evolution determined by cosmology (competition between gravity and expansion)• Statistics determined by initial fluctuation field: since Gaussian, statistics specified by initial power-spectrum P(k)• Fact that only very fat cows are spherical is a detail (crucial for precision cosmology); in excursion set approach, mass-dependent barrier height increases with distance along walk
  25. 25. •Exact analytic solutiononly for sharp-k filter(for Gaussian fields k-modes are independent)•Numerical solutioneasy to implement forany filter, P(k)•Simple analyticbounds/approximationsavailableFirst crossing distribution ~ independent ofsmoothing filter (TH,G) and P(k) whenexpressed as function of δc/σ(m)
  26. 26. Assumptions of convenience• Uncorrelated steps – Correlated steps (Peacock & Heavens 1990; Bond et al. 1991; Maggiore & Riotto 2010; Paranjape et al. 2011) – Non-Gaussian mass function (Matarrese et al. 2001) ~ given by changing PDF but keeping uncorrelated steps (Lam & Sheth 2009)• Barrier is sharp – Fuzzy/porous barriers (Sheth, Mo, Tormen 2001; Sheth &Tormen 2002 ; Maggiore & Riotto 2010; Paranjape et al 2011)• Ensemble of walks is also random – The real cloud-in-cloud problem
  27. 27. Peacock-Heavens approximation• Steps are correlated – correlation length scale depends on smoothing window and Pk – correlation length on scale s is Γs with Γ ~ 5 – steps further apart than this length are ~ independent• Survival probability (to have not crossed) is P (sn) ≈ ∏i p(<δc|si) n s = p(<δc|s) exp[0∫ ds/sΓ ln p(<δc|s)] First crossing distribution is -∂P /∂s.
  28. 28. Completely correlated stepsCriticalover-density Typical mass smaller at early times: hierarchical clustering MASS
  29. 29. Correlations with environmentCritical Easier to get here from over-dense over-dense environmentinitialover-density This patch forms ‘Top-heavy’ halo of mass function in mass M dense regions under-dense MASS
  30. 30. Peacock-Heavensapproximation tocorrelated stepsproblem is mostaccurate analyticapproximation todateNoise aroundcomplete Paranjape et al. 2011correlation also massquite good
  31. 31. Extensions (Paranjape, Lam, Sheth 2011)– conditional distributions (environmental trends)– large-scale bias– can both be understood as arising from fact that p(x|y) = Gaussian with <x|y> = y <xy>/<yy>– so simply make this replacement in PH expressions– moving barriers (ellipsoidal collapse, PDF),
  32. 32. • Generalization of PH to constrained walks: p(<δc|s) → p(<δc,s|δ0, S0) is trivial and accurate δ1 – (Sx/S0) δ0 mass ν10 = ------------------ √S1- (Sx/S0)2 S0• Read fine print (and correct the typos!) in Bond et al. (1991)
  33. 33. ν10 = δ1 – (Sx/S0) δ0----------------------------------------√S1- (Sx/S0) S0 2 N.B.: Sx/S0 ≠ 1
  34. 34. Large scale bias• f(m|δ0)/f(m) – 1 ~ f(ν10)/f(ν) – 1 ~ (Sx/S) b(ν) δ0 where b is usual (Mo-White) bias factor• So, real space bias should differ from Fourier space bias by factor (Sx/S)• N.B. exactly same effect seen for peaks (Desjacques et al. 2010); in essence, this is same calculation as the ‘peak’ density profile (BBKS 1986)
  35. 35. Real and Fourier space measures of bias should differ, especially for most massive halos TophatBasis for getting scale dependent bias from sharp-kexcursion set approach Paranjape & Sheth 2011Assembly bias from<δ0, S0 | δc, S1, δf, Sf>
  36. 36. Most of apparentdifference is due todifference betweenreal and Fourier spacebiasReal difference inFourier space bias issmall – but significantat small masses
  37. 37. Hamana et al. 2002Massive halosmore strongly clustered ‘linear’ bias factor on largescales increases monotonicallywith halo mass
  38. 38. Luminosity dependent clustering from mass dependence of halo bias Zehavi et al. 2005 SDSS• Deviation from power-law statistically significant• Centre plus Poisson satellite model (two free parameters)provides good description
  39. 39. Summary• Intimate connection between abundance and clustering of dark halos – Can use cluster clustering as check that cluster mass- observable relation correctly calibrated – Almost all correlations with environment arise through halo bias (massive halos populate densest regions)• Next step in random walk ~ independent of previous – History of object ~ independent of environment – Not true for correlated steps• Description quite detailed; sufficiently accurate to complete Neyman-Scott program (The Halo Model)
  40. 40. Success to date motivatessearch for more accurate (better than 10%) models of halo abundance/clustering: Quest for Halo Grail?
  41. 41. Work in progress• Correlated steps – Assembly bias• Correlated walks – What is correct ensemble? – Generically increase high-m abundance (ST99 parameter ‘a’)• Dependence on parameters other than δ – Higher dimensional walks – What is correct basis set? (e,p)? (I2,I3)?
  42. 42. Model the entire cosmic web, not just the knots
  43. 43. Study of random walks with correlated steps (i.e. path integral methods) =Cosmological constraints from large scale structures
  44. 44. ‘Modified’ gravity theories Martino & Sheth 2009weaker gravity on large scales stronger gravity Voids/clusters/clustering are useful indicators
  45. 45. In many modifications to GR,scale-dependent linear theory is generic: δ(k,t) = D(k,t) δ(k,ti) N.B. Peaks in ICs are not peaks in linearly evolved field
  46. 46. Modified force modifies (spherical) collapse (Martino & Sheth 2009; Parfrey et al. 2011)
  47. 47. Halo density stronger gravity profiles in modified gravity theories: weaker gravity X-rayobservations useful? Martino & Sheth 2009
  48. 48. Modified halo abundances …… potentially from two effects
  49. 49. δc(m) and σ(m)• In standard GR, σ(m) is an integral over initial P(k), evolved using linear theory to later (collapse) time.• In modified theories, D(k,t) means there is a difference between calculating in ICs vs linearly evolved field.• Energy conservation arguments suggest better to work in ICs.
  50. 50. Critical overdensity required forcollapse at present becomes massdependent: δc(m) This is generic.
  51. 51. Implication for halo bias• Halos are linearly biased tracers of initial field• D(k) means they are not linearly biased tracers of linearly-evolved field (Parfrey et al. 2011)• Simulations in hand too small to test, but if real, this is an important systematic