This document discusses improvements to the Press-Schechter theory for modeling the abundances and clustering of dark matter halos. It proposes that modeling halo collapse as requiring the density to "step up" above a critical density threshold at progressively larger spatial scales provides a better approximation than assuming fully correlated or uncorrelated densities. This "stepping up" approach requires only 2-point statistics and can be applied to non-Gaussian fields. The document also suggests that modeling the distribution of density slopes at peak positions provides a way to match halo counts through an Excursion Set Peaks model.

1. Cluster abundances
and clustering
Can theory step up
to 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

2. 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
∞
δc
dδn p(δ1, . . . , δn)
Since s = n∆s this requires n-point
distribution 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 ≤ δc
δ(S) ≥ δc and δ(S) ≤ δc + ∆S v
so
f(s) ∆s =
∞
0
dv
δc+∆S v
δc
db p(b, v)
=
∞
0
dv ∆S v p(δc, v)
making
f(s) = p(δc|s)
∞
0
dv v p(v|δc)
Requires only 2-point statistics.
Logic general; applies to very NG fields also

3. One step beyond:
Start from exact statement:
p(≥ b|s) =
s
0
dS f(S) p(≥ b, s|first at S)
Approximate as:
p(≥ b|s) ≈
s
0
dS 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)

4. Next simplest approximation (step up):
p(≥ b|s) =
s
0
dS f(S) p(≥ b, s|first at S)
≈
s
0
dS f(S) p(≥ b|up at S)
where
p(≥ b|up at S) =
∞
0 dV V p(≥ b, V|B)
∞
0 dV V p(V|B)
Requires only 3-point statistics.
Works for all smoothing filters and
(monotonic) barriers.

8. Moving barrier: b = δc[1 + (s/δc)2/4]

9. Summary
It’s always good to
step up!

10. 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.