Like in the calculation of the number of cells in order to estimate the number of templates we perform a Taylor expansion of up to second order terms around the true values of the parameters, and we obtain an equation analogous to Equation (57),

where is given by Equation (56). By arguments identical to those in deriving the formula for the number of cells we arrive at the following formula for the number of templates: When the above formula coincides with the formula for the number of cells, Equation (60). Here we would like to place the templates sufficiently closely so that the loss of signals is minimized. Thus needs to be chosen sufficiently small. The formula (68) for the number of templates assumes that the templates are placed in the centers of hyperspheres and that the hyperspheres fill the parameter space without holes. In order to have a tiling of the parameter space without holes we can place the templates in the centers of hypercubes which are inscribed in the hyperspheres. Then the formula for the number of templates readsFor the case of the signal given by Equation (16) our formula for number of templates is equivalent to the original formula derived by Owen [74]. Owen [74] has also introduced a geometric approach to the problem of template placement involving the identification of the Fisher matrix with a metric on the parameter space. An early study of the template placement for the case of coalescing binaries can be found in [84, 35, 19]. Applications of the geometric approach of Owen to the case of spinning neutron stars and supernova bursts are given in [24, 11].

The problem of how to cover the parameter space with the smallest possible number of templates, such that no point in the parameter space lies further away from a grid point than a certain distance, is known in mathematical literature as the covering problem [28]. The maximum distance of any point to the next grid point is called the covering radius . An important class of coverings are lattice coverings. We define a lattice in -dimensional Euclidean space to be the set of points including such that if and are lattice points, then also and are lattice points. The basic building block of a lattice is called the fundamental region. A lattice covering is a covering of by spheres of covering radius , where the centers of the spheres form a lattice. The most important quantity of a covering is its thickness defined as

In the case of a two-dimensional Euclidean space the best covering is the hexagonal covering and its thickness . For dimensions higher than 2 the best covering is not known. We know however the best lattice covering for dimensions . These are so-called lattices which have a thickness equal to where is the volume of the -dimensional sphere of unit radius.For the case of spinning neutron stars a 3-dimensional grid was constructed consisting of prisms with hexagonal bases [16]. This grid has a thickness around 1.84 which is much better than the cubic grid which has thickness of approximately 2.72. It is worse than the best lattice covering which has the thickness around 1.46. The advantage of an lattice over the hypercubic lattice grows exponentially with the number of dimensions.

http://www.livingreviews.org/lrr-2005-3 |
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |