# Random Fields in Cosmology

The idea of the course is to get familiar with the properties of the random (mostly gaussian) fields in cosmological context. Random fields is fundamental concept in many areas of cosmology - for example development of the Large Scale Structure is understood in terms of evolution of the random density field. Rich statistical and structural properties of initial density field, viewed as a random one, are directly responsible for many aspects of the observed LSS.

If you have any questions e-mail me at pogosyan@cita.utoronto.ca

Here is the current list of problems suggested. The due date is mid April.

Dmitry Pogosyan

### Problem #1

Calculate and discuss \gamma, R_0, and R_* parameters for

1. 3-dimensional Gaussian random density field with the spectrum P(k)=k^n, n=]-3,1]
2. the corresponding to this density field of gravitational potential.
3. For the density field compare results when the high-k smoothing is done with different (commonly used) filters W(kRf), Rf is an (arbitrary) smoothing scale
1. Sharp k-cutoff --- Ws(k Rf)=step_function[1-k Rf]
2. "Top-hat" filter -- Wt(k Rf)=9 [Sin(k Rf)/(k Rf)-Cos(k Rf)]^2/(k Rf)^4
3. Gaussian filter --- Wg(k Rf)=exp[-(k Rf)^2]
4. In case some integrals do not formally converge, discuss how they behave after artifical cutoff is imposed, and then taken away
5. In discussion make connection between calculated parameters and overall shape of a field.