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 (55
),
For the case of the signal given by
Equation (14) our formula for number of templates is equivalent to the
original formula derived by Owen
[65
]
. Owen
[65]
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
[72, 29, 16
]
. Applications of the geometric approach of Owen to the case of
spinning neutron stars and supernova bursts are given in
[20, 9]
.
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
[24]
. 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
For the case of spinning neutron stars a
3-dimensional grid was constructed consisting of prisms with
hexagonal bases
[13]
. 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.
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