Amplification of Cluster Correlations
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**Session 115 -- Cosmology and Dark Matter**
*Oral presentation, Saturday, January 15, 10:15-11:45, Salon III Room (Crystal Gateway)*

## [115.06] Amplification of Cluster Correlations

*Istv\'an Szapudi (JHU), Alexander S. Szalay (JHU, E\"otv\"os University)*

\def\xiav{\avrg{\xi}}
\def\xiN{\avrg{\xi^{(N)}}}
\def\avrg#1{\langle{#1}
\rangle}

We study the higher order correlation properties of the
distribution of galaxies in the Lick, IRAS, and APM catalogs.
We show, using a new technique, how the much debated cluster
correlation function is directly related to the higher order
galaxy correlations. Mathematical techniques from
biased galaxy formation are adopted: here the biasing simulates the
detection of the clusters, rather than their dynamical
formation.
For statistical purposes we define the factorial moment
correlators as
$$
w_{kl}(r_{12}) = { \avrg{(N_1)_k (N_2)_l}-\avrg{(N)_k}
\avrg{(N)_l}
\over\avrg{N}^{k+l} },
$$
where we used the notatation $(N)_k = N(N-1)..(N-1+k)$ for the factorial
moments. Since these correspond
to the continuum moments after the transition to discrete
we essentially calculate the correlation functions in a
catalog in which the density is raised to a power. This is a
`soft clipping' because it enhances the density contrast
similarly to `sharp clipping', where
clusters are identified as cells with density exceeding a certain
threshold.
The advantage of `soft clipping' is that
the $w_{kl}$'s are obtained as a finite
sum of higher order correlation functions.
The resulting `cluster' catalogs have amplifications
similar to rich clusters, and the galaxy-`cluster'
cross-correlation is close to the geometric mean of the galaxy
and cluster autocorrelations.
We also present a novel
method to determine the amplitudes of the irreducible N-point
correlation functions in the gravitational clustering hierarchy.
The case of `sharp clipping' is investigated theoretically with
the method of generating functions.
It is shown that the shape of the
cluster correlation function is always identical to the galaxy
correlation function and we calculate the amplification for some
special cases. The theoretical results agree well
with known observations.

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