infall velocity

[Anze Slosar]

I am confused, again.

The party line on correlation function in redshift space goes along the lines of squashing on large scales due to large scale infall which can be described using linear approx plus small scal elongation due to fingers-of-God.

Ok, let's forget about the FOG effect and discuss the infall only. Going from real space correlation function to redshift space correlation function involves the well known Kaiser/Hamilton recipe and involves calculating the multipoles of [tex]\xi(s)[/tex] from [tex]\xi(r), \bar{\xi}(r),\bar{\bar{\xi}}(r)[/tex] and [tex]\beta[/tex]. I would naivelly expect, that one could equivalently calculate it from

[tex]
\xi(\sigma,\pi) = \int \xi_0 (r(v)) f(v,r(v)) \frac{z(v)}{r(v)} dv,
[/tex]

where [tex]z(v)=\pi-v[/tex] and [tex]r(v)=\sqrt{\sigma^2+(\pi-v)^2
}[/tex] and [tex]f(v,r)[/tex] desribes the mean infall velocity at distance [tex]r[/tex] between two galaxies. In other words, I would like to use the same recipe as for calculating FOG, but this time [tex]f(v)[/tex] instead of being a typical exponential distribution with equal wings on both sides, it would be probably just a delta function at some mean infall velocity (as a function of radius) or at least some sensible distribution.

For [tex]\sigma=0[/tex], the Kaiser recipe reduces to:

[tex]
\xi(s) = (1+\beta)^2\xi(r)-\frac{4\beta}{r^3}\int_0^r \xi(r')r'^2 {\rm d}r' - \frac{4\beta^2}{r^5} \int_0^r \xi(r') r'^4 {\rm d}r'
[/tex]

So, changing the variables this implies (well at least roughly; note that I use H for Heaviside function, nut Hubble)):

[tex]
f(v,r) = (1+\beta)^2 \delta(v) - \frac{4\beta}{r^3}(r-v)^2 H(v)H(r-v) - \frac{4\beta^2}{r^5} (r-v)^4 H(v) H(r-v),
[/tex]

which doesn't look like a sensible distribution at all. It has nothing at negative side and is negative at the positive side, plus a delta function.
It is sensible, it increases the correlation function at a position and the puts some negative wings at larger [tex]\pi[/tex]. It is just not what an infall would do, i.e. take stuff from correlation function at a point and put it to smaller [tex]\pi[/tex] Could anybody enlighten me on this?

In fact, I tried [tex]f(v,r) = \delta(v+\alpha r) and at say [tex]\alpha=0.4[/tex] it does give a sensible looking squashed contours, so the general idea seems right... Does anybody know how this relates to Scoccimaro's criticism of kaiser formalism [astro-ph/0407214]?

[Anze Slosar]

One mistake spotted; at least the part of the problem is that in case of radial dependence on should write

[tex]
1+\xi(\sigma,\pi) = \int (1+\xi_0 (r(v))) f(v,r(v)) \frac{z(v)}{r(v)} dv
[/tex]

rather than

[tex]
\xi(\sigma,\pi) = \int \xi_0 (r(v)) f(v,r(v)) \frac{z(v)}{r(v)} dv
[/tex]

(apparently, this is a quite common mistake)
Soccimarro's paper actually derives dispersion model's PVD but his solution diverges in the limit of no dispersion.