Bardeen, Bond, Kaiser

1
Bardeen, Bond, Kaiser Szalay (1986)The
Statistics of Peaks in Gaussian Random Fields

• Edwin Sirko
• 2004-11-22

2
(No Transcript)
3
(No Transcript)
4
Outline
5
(No Transcript)
6
Gaussian random fields what they are

• Central limit theorem
• Random-phase assumption of independent Fourier
  modes
• White noise field
• Convolved with square root of correlation
  function

Bertschinger (2001) ApJS 137, 1
7
Gaussian random fields useful things

• Randomly selected point has a Gaussian
  distribution
• Derivatives, integrals, linear functions of F are
  also Gaussian
• Characterized completely by power spectrum P(k)
• Isotropy makes this P(k)
• Rigorous multivariate definition

8
Gaussian fields why they are important

• Predicted by inflation
• The density field is our Gaussian random field

Gaussian fields what they are not

• Topological defect models
• Anything with a nonzero three-point correlation
  function (bispectrum) the nonlinear universe

9
Gaussian fields what else they are

• CMB
• Ocean waves
• Quasar light curves
• Accuracy in clocks
• Flow of Nile over last 2000 years
• Music

Press (1978) ComAp 7, 103 http//map.gsfc.nasa.gov
/
10
More comments on noise

• Their noise is our signal
• f0 white noise, Johnson noise in electrical
  circuits
• f-1 pink noise, flicker noise, 1/f noise,
  scale-invariant
• f-2 brown noise, random walk
• f-3

http//astronomy.swin.edu.au/pbourke/fractals/noi
se/
11
Gaussian random fields definitions
12
Smoothing

• Physical
• Silk damping, free streaming
• Artificial
• To study difference between clusters and galaxies

13
The spectral parameters

• g
• depends on
• P(k) which depends on cosmology
• RF smoothing
• Approaches 1 if the power spectrum is a shell in
  k-space
• Less than 1 if the power spectrum is broad
• R
• Measure of coherence scale

14
Peak density

• Strategy evaluate
• This will depend on spectral parameters g and R

15
Biasing

• Bias the mass correlation function and galaxy
  (or cluster) correlation function differ
• In other words, galaxies dont trace mass
• Explained naturally if bright galaxies form
  preferentially at high peaks

16
Peak enhancement by background field

• Assume galaxies form at peaks with F gt Ft
• Superimpose field Fb
• Enhancement factor in local density of peaks
• In other words, a modest overdensity on some
  large mass scale can lead to a strong enhancement
  in the local density of galaxies.

17
Correlation functions of peaks
18
Profiles
http//mathworld.wolfram.com/Spheroid.html
19
Borgani et al. 1992
Naselsky et al. 2004
Thoul Weinberg 1996
Pudritz 2002
Van de Weygaert Icke 1989
Turner et al. 1993
McDonald Miralda-Escude 1999
Zhang et al 1997
Kaufmann Straumann 2000
Castro 2003
Suginohara Suto 1991
Ma Shu 2001
Theuns et al. 1998
20
Transfer function / Power spectrum
21
Conclusions

• Inflation predicts the density perturbation field
  to be a Gaussian random field
• Gaussianity is also simple because it can be
  described by just the power spectrum
• BBKS derived peak density, correlation function,
  and profiles
• These things depend only on two parameters of the
  power spectrum
• BBKS is mostly cited because of their fit to the
  transfer function

22
(No Transcript)
23
(No Transcript)