Projected_clustering

Projected clustering

The projected clustering signal,
$w_p(r_p)$
is defined as
$w_p(r_p) = \int \xi(r_p,r_\sigma) d r_\sigma \,.$
Note that this implies that
$w_p(r_p)$
has dimensions of length.

Alternatively, one can define the quantity
$\tilde{w}_p(r_p) = \frac{ \int_{0}^{r_\sigma^{\rm max}} DD_{\rm RSD}(r_p,r_\sigma) d r_\sigma}{ \int_{0}^{r_\sigma^{\rm max} } RR(r_p,r_\sigma) d r_\sigma } - 1$
where
$DD_{\rm RSD}$
is the data pairs in redshift space. In contrast to
$w_p(r_p),$
the quantity
$\tilde{w}_p(r_p),$
is dimensionless. It is super clear that because redshift space distortions scatter the pairs along the line of sight direction, as
$r_\sigma^{\rm max} \rightarrow \infty$
the redshift space distortion effect should be mitigated and in this limit
$\tilde{w}_p(r_p) = \frac{ \int_{0}^{r_\sigma^{\rm max}} DD(r_p,r_\sigma) d r_\sigma}{ \int_{0}^{r_\sigma^{\rm max} } RR(r_p,r_\sigma) d r_\sigma } - 1$
where
$DD$
are the data pairs in real space.