[tex]
\langle v^2 \rangle = (Haf)^2 \int_0^\infty y {\rm d}y \xi(y)
[/tex]
This can be extended (by performing the right integral of 14.9) to correlations of neighbouring velocity fields as
[tex]
\langle v(\mathbf{x}) v(\mathbf{x+r}) \rangle = (Haf)^2 \left[ \int_0^r y^2r^{-1} {\rm d}y\xi(y) + \int_r^\infty y {\rm d}y \xi(y) \right)
[/tex]
and hence
[tex]
\langle v(\mathbf{x}) v(\mathbf{x+r}) \rangle = \langle v^2 \rangle \frac{ \int_0^r y^2r^{-1} {\rm d}y \xi(y) + \int_r^\infty y {\rm d}y \xi(y)}{\int_0^\infty y {\rm d}y \xi(y)}
[/tex]
I have two problems with this and would appreciate feedback:
- I have never seen the equation above before and Peebles didn't put it in his book, although it is fairly obvious thing to do. Anybody has a ref, or is it wrong for some reason?
- It doesn't work... On simulations it predicts correlations that are much stronger than observed (i tried comparing halo centers, whose correlation function DOES match the linear correlation functions and thus they should still be in linear mode). I think that this might be a selection function, simply due to the fact that by comparing velocity of halos you calculate the object weighted rather than volume weighted quantity.