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Cluster abundances
and clustering
Can theory step up
to precision cosmology?
Ravi K Sheth (and Marcello Musso)
ICTP/Penn (...
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 · ·...
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 ...
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...
Moving barrier: b = δc[1 + (s/δc)2/4]
Summary
It’s always good to
step up!
Collapse happens around special positions.
Can write Excursion Set Peaks model by
noting that distribution of slopes v for...
Cluster abundances and clustering Can theory step up to precision cosmology?
Cluster abundances and clustering Can theory step up to precision cosmology?
Cluster abundances and clustering Can theory step up to precision cosmology?
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Cluster abundances and clustering Can theory step up to precision cosmology?

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Review talk by Prof. Ravi Sheth at the SuperJEDI Conference, July 2013

Published in: Technology, Self Improvement
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Cluster abundances and clustering Can theory step up to precision cosmology?

  1. 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. 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. 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. 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.
  5. 5. Moving barrier: b = δc[1 + (s/δc)2/4]
  6. 6. Summary It’s always good to step up!
  7. 7. 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.

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