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4.6 Number of templates

To search for gravitational-wave signals we evaluate the ℱ-statistic on a grid in parameter space. The grid has to be sufficiently fine such that the loss of signals is minimized. In order to estimate the number of points of the grid, or in other words the number of templates that we need to search for a signal, the natural quantity to study is the expectation value of the ℱ-statistic when the signal is present. We have
1 ( ) E [ℱ ] = -- 4 + aT ⋅ QT ⋅ M ′−1 ⋅ Q ⋅ a . (63 ) 2
The components of the matrix Q are given in Equation (52View Equation). Let us rewrite the expectation value (63View Equation) in the following form,
1 ( aT ⋅ QT ⋅ M ′− 1 ⋅ Q ⋅ a) E [ℱ ] = -- 4 + ρ2-------T------------ , (64 ) 2 a ⋅ M ⋅ a
where ρ is the signal-to-noise ratio. Let us also define the normalized correlation function
aT-⋅ QT-⋅-M-′−1-⋅ Q-⋅ a 𝒞n := aT ⋅ M ⋅ a . (65 )
From the Rayleigh principle [67] it follows that the minimum of the normalized correlation function is equal to the smallest eigenvalue of the normalized matrix QT ⋅ M ′−1 ⋅ Q ⋅ M −1, whereas the maximum is given by its largest eigenvalue. We define the reduced correlation function as
1 ( ) 𝒞(ξ,ξ ′) := --tr QT ⋅ M − 1 ⋅ Q ⋅ M ′−1 . (66 ) 4
As the trace of a matrix equals the sum of its eigenvalues, the reduced correlation function 𝒞 is equal to the average of the eigenvalues of the normalized correlation function 𝒞n. In this sense we can think of the reduced correlation function as an “average” of the normalized correlation function. The advantage of the reduced correlation function is that it depends only on the intrinsic parameters ξ, and thus it is suitable for studying the number of grid points on which the ℱ-statistic needs to be evaluated. We also note that the normalized correlation function 𝒞 precisely coincides with the autocovariance function 𝒞 of the ℱ-statistic given by Equation (51View Equation).

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

Gij Δ ξiΔ ξj = 1 − C0, (67 )
where G is given by Equation (56View Equation). 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:
∫ -----1------Γ (K-∕2-+-1) √ ------ Nt = (1 − C0 )K ∕2 πK ∕2 detG dV. (68 ) V
When C0 = 1∕2 the above formula coincides with the formula for the number Nc of cells, Equation (60View Equation). Here we would like to place the templates sufficiently closely so that the loss of signals is minimized. Thus 1 − C0 needs to be chosen sufficiently small. The formula (68View Equation) 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 reads
K∕2 ∫ √ ------ Nt = -----1------K----- detG dV. (69 ) (1 − C0 )K ∕2 2K V

For the case of the signal given by Equation (16View Equation) our formula for number of templates is equivalent to the original formula derived by Owen [74Jump To The Next Citation Point]. 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 [843519Jump To The Next Citation Point]. Applications of the geometric approach of Owen to the case of spinning neutron stars and supernova bursts are given in [2411].

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 R. An important class of coverings are lattice coverings. We define a lattice in K-dimensional Euclidean space ℝK to be the set of points including 0 such that if u and v are lattice points, then also u + v and u − v are lattice points. The basic building block of a lattice is called the fundamental region. A lattice covering is a covering of ℝK by spheres of covering radius R, where the centers of the spheres form a lattice. The most important quantity of a covering is its thickness Θ defined as

Θ := volume--of-one-K--dimensional-sphere. (70 ) volume of the fundamental region
In the case of a two-dimensional Euclidean space the best covering is the hexagonal covering and its thickness ≃ 1.21. For dimensions higher than 2 the best covering is not known. We know however the best lattice covering for dimensions K ≤ 23. These are so-called A∗K lattices which have a thickness ΘA∗K equal to
√ ------( K (K + 2))K ∕2 ΘA ∗K = VK K + 1 ---------- , (71 ) 12(K + 1)
where VK is the volume of the K-dimensional sphere of unit radius.

For the case of spinning neutron stars a 3-dimensional grid was constructed consisting of prisms with hexagonal bases [16Jump To The Next Citation Point]. 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 A ∗K lattice over the hypercubic lattice grows exponentially with the number of dimensions.

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