An improved cosmic crystallography method to detect holonomies in flat spaces

A new, improved version of a cosmic crystallography method for constraining cosmic topology is introduced. Like the circles-in-the-sky method using CMB data, we work in a thin, shell-like region containing plenty of objects. Two pairs of objects (quadruplet) linked by a holonomy show a specific distribution pattern, and three filters of \emph{separation, vectorial condition}, and \emph{lifetime of objects} extract these quadruplets. Each object $P_i$ is assigned an integer $s_i$, which is the number of candidate quadruplets including $P_i$ as their members. Then an additional device of $s_i$-histogram is used to extract topological ghosts, which tend to have high values of $s_i$. In this paper we consider flat spaces with Euclidean geometry, and the filters are designed to constrain their holonomies. As the second filter, we prepared five types that are specialized for constraining specific holonomies: one for translation, one for half-turn corkscrew motion and glide reflection, and three for $n$-th turn corkscrew motion for $n=4, 3,$ and 6. {Every multiconnected space has holonomies that are detected by at least one of these five filters.} Our method is applied to the catalogs of toy quasars in flat $\Lambda$-CDM universes whose typical sizes correspond to $z\sim 5$. With these simulations our method is found to work quite well. {These are the situations in which type-II pair crystallography methods are insensitive because of the tiny number of ghosts. Moreover, in the flat cases, our method should be more sensitive than the type-I pair (or, in general, $n$-tuplet) methods because of its multifilter construction and its independence from $n$.}