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Bardeen, Bond, Kaiser & Szalay (1986)“The Statistics of Peaks in Gaussian Random Fields”: 

Bardeen, Bond, Kaiser & Szalay (1986) “The Statistics of Peaks in Gaussian Random Fields” Edwin Sirko 2004-11-22

Outline: 

Outline

Gaussian random fields: what they are: 

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

Gaussian random fields: useful things: 

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:

Gaussian fields: why they are important: 

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

Gaussian fields: what else they are: 

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/

More comments on noise: 

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/noise/

Gaussian random fields: definitions: 

Gaussian random fields: definitions

Smoothing: 

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

The spectral parameters: 

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

Peak density: 

Peak density Strategy: evaluate This will depend on spectral parameters g and R*

Biasing: 

Biasing Bias: the mass correlation function and galaxy (or cluster) correlation function differ In other words, galaxies don’t trace mass Explained naturally if bright galaxies form preferentially at high peaks

Peak enhancement by background field: 

Peak enhancement by background field Assume galaxies form at peaks with F > 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.”

Correlation functions of peaks: 

Correlation functions of peaks

Profiles: 

Profiles http://mathworld.wolfram.com/Spheroid.html

Slide19: 

Borgani et al. 1992 Castro 2003 Kaufmann & Straumann 2000 Ma & Shu 2001 McDonald & Miralda-Escude 1999 Naselsky et al. 2004 Pudritz 2002 Suginohara & Suto 1991 Theuns et al. 1998 Thoul & Weinberg 1996 Van de Weygaert & Icke 1989 Zhang et al 1997 Turner et al. 1993

Transfer function / Power spectrum: 

Transfer function / Power spectrum

Conclusions: 

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