Cluster abundancesand clustering

Can theory step upto precision cosmology?

Ravi K Sheth (and Marcello Musso)

ICTP/Penn (CP3-IRMP, Louvain)

• Motivation: Solving Press-Schechter

• The importance of stepping up

• One step beyond

• Stepping up is good where ever you are

Press-Schechter: Want δ ≥ δc

Bond, Cole, Efstathiou, Kaiser:δ(s) ≥ δc and δ(S) ≤ δc for all S ≤ s:

f(s)∆s =∫ δc

−∞dδ1 · · ·

∫ δc

−∞dδn−1

∫ ∞

δcdδn p(δ1, . . . , δn)

Since s = n∆s this requires n-pointdistribution in limit as n → ∞ and ∆s → 0.(Best solved by Monte-Carlo methods.)

Musso-Sheth: δ ≥ δc while ‘stepping up’δ(S) ≥ δc and δ(S − ∆S) ≤ δc

δ(S) ≥ δc and δ(S) − ∆Sdδ/dS ≤ δcδ(S) ≥ δc and δ(S) ≤ δc + ∆Sv

so

f(s)∆s =∫ ∞

0dv

∫ δc+∆Sv

δcdbp(b, v)

=∫ ∞

0dv∆Sv p(δc, v)

making

f(s) = p(δc|s)∫ ∞

0dv v p(v|δc)

Requires only 2-point statistics.Logic general; applies to very NG fields also

One step beyond:

Start from exact statement:

p(≥ b|s) =∫ s

0dS f(S) p(≥ b, s|first at S)

Approximate as:

p(≥ b|s) ≈∫ s

0dS f(S) p(≥ b, s|B,S)

Completely correlated: p(≥ δc, s|δc,S) = 1

(what Press-Schechter really means)

Completely uncorrelated:

p(≥ δc, s|δc,S) = 1/2

(Bond, Cole, Efstathiou, Kaiser)

Next simplest approximation (step up):

p(≥ b|s) =∫ s

0dS f(S) p(≥ b, s|first at S)

≈∫ s

0dS f(S) p(≥ b|up at S)

where

p(≥ b|up at S) =

∫∞0 dVVp(≥ b,V|B)∫∞

0 dVVp(V|B)

Requires only 3-point statistics.

Works for all smoothing filters and

(monotonic) barriers.

Moving barrier: b = δc[1 + (s/δc)2/4]

Summary

It’s always good tostep up!

Collapse happens around special positions.

Can write Excursion Set Peaks model by

noting that distribution of slopes v for

peaks is different from that for random

positions (of same height):

f(s) = p(b|s)∫ ∞

b′dv (v − b′) p(v|b)Cpk(v)

Including this extra factor is necessary for

matching halo counts.

Other things than initial overdensity may

also matter (e.g. external shear, alignment

of initial shape and shear, etc.)

These lead to models with more than one

walk, sometimes called stochastic barrier

models, which generically exhibit ‘nonlocal’

stochastic bias.