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

1. Phenomenology of
large scale structure
Ravi K Sheth (ICTP/UPenn)

2. • The Halo Model
– Observations/Motivation
• The Excursion Set approach
– The ansatz
– Correlated steps
– Large scale bias
– Ensemble averages: Halos and peaks
– Modified gravity

5. Galaxy spatial
distribution
depends on
galaxy type
(luminosity,
color, etc.)
Not all galaxies
can be unbiased
tracers of the
underlying Mass
distribution

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

7. How to describe different point
processes which are all built from
the same underlying distribution?
THE HALO MODEL

8. 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)

9. 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

10. The halo-model of clustering
• Two types of particles: central + ‘satellite’
• ξobs(r) = ξ1h(r) + ξ2h(r)
• ξ1h(r) = ξcs(r) + ξss(r)

11. To implement
Neyman-Scott program
need model for
cluster abundance
and cluster clustering
Next generation of surveys will
have 109 galaxies in105 clusters

12. In Cold Dark Matter models, mass doesn’t move very far

13. The Cosmic
Background
Radiation
Cold: 2.725 K
Smooth: 10-5
Simple physics
Gaussian fluctuations
= seeds of subsequent
structure formation
= simple(r) math
Logic which follows
is general

14. Birkhoff’s theorem important

15. Use initial conditions (CMB)
+
model of nonlinear
gravitational clustering
to make inferences about
late-time, nonlinear structures

16. Hierarchical clustering in GR
= the persistence of memory

17. THE EXCURSION SET
APPROACH
(Epstein 1983; Bond et al. 1991; Lacey & Cole 1993;
Sheth 1998; Sheth & van de Weygaert 2004; Shen et al. 2006)

19. N-body
simulations
of
gravitational
clustering
in an
expanding
universe

20. Assume a spherical cow …
(Gunn & Gott 1972)

21. 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

22. The Nonlinear PDF
δcrit
δ0(M/V)
Linear
theory
over- This patch Halo of mass
density of volume m<M within
v contains this patch
mass M (M,v)
V v
MASS

23. • 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

24. The Random Walk Model
Higher
Redshift
Critical
over-
density
This smaller mass
patch patch within
forms more massive
halo of region
mass M
MASS

25. 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

26. Random Walk = Accretion history
eti h’
High-z
cr ot
on
ac o
‘sm
Major merger
Low-z
over-
density
larger small mass
Time mass at at high-z
evolution of low z Mapping between σ2
barrier and M depends on P(k)
depends on MASS
cosmology σ2(M)

27. 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

28. •Exact analytic solution
only for sharp-k filter
(for Gaussian fields k-
modes are independent)
•Numerical solution
easy to implement for
any filter, P(k)
•Simple analytic
bounds/approximations
available
First crossing distribution ~ independent of
smoothing filter (TH,G) and P(k) when
expressed as function of δc/σ(m)

29. 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

30. 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.

31. Completely correlated steps
Critical
over-
density
Typical mass smaller
at early times:
hierarchical clustering
MASS

32. Correlations with environment
Critical
Easier to get here
from over-dense
over-dense environment
initial
over-
density This
patch
forms ‘Top-heavy’
halo of mass function in
mass M dense regions
under-dense MASS

33. Peacock-Heavens
approximation to
correlated steps
problem is most
accurate analytic
approximation to
date
Noise around
complete Paranjape et al. 2011
correlation also
mass
quite good

34. 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),

35. • 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)

36. ν10 =
δ1 – (Sx/S0) δ0
----------------------------------------
√S1- (Sx/S0) S0 2
N.B.: Sx/S0 ≠ 1

37. 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)

38. Real and Fourier space
measures of bias
should differ,
especially for most
massive halos Tophat
Basis for getting scale
dependent bias from sharp-k
excursion set approach
Paranjape & Sheth 2011
Assembly bias from
<δ0, S0 | δc, S1, δf, Sf>

39. Most of apparent
difference is due to
difference between
real and Fourier space
bias
Real difference in
Fourier space bias is
small – but significant
at small masses

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

41. 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

42. 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)

43. Success to date motivates
search for more accurate
(better than 10%)
models of halo
abundance/clustering:
Quest for Halo Grail?

44. 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)?

45. Model the entire cosmic web,
not just the knots

46. Study of random walks with
correlated steps
(i.e. path integral methods)
=
Cosmological constraints from
large scale structures

47. ‘Modified’ gravity theories
Martino & Sheth 2009
weaker gravity on large scales stronger gravity
Voids/clusters/clustering are useful indicators

48. 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

50. Modified force modifies
(spherical) collapse
(Martino & Sheth 2009; Parfrey et al. 2011)

51. Halo density
stronger gravity
profiles in
modified
gravity
theories: weaker gravity
X-ray
observations
useful?
Martino & Sheth 2009

52. Modified halo abundances …
… potentially from two effects

53. 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.

54. Critical
overdensity
required for
collapse at present
becomes mass
dependent: δc(m)
This is generic.

55. 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