4.4 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.

Thus, we assume that the data x contains the gravitational-wave signal h (t;𝜃 ) defined in Eq. (32View Equation), so x(t;𝜃) = h(t;𝜃) + n(t). The parameters 𝜃 = (a,ξ) of the signal consist of extrinsic parameters a and intrinsic parameters ξ. The data x will be correlated with the filters ′ hi(t;ξ ) (i = 1,...,n) parameterized by the values ′ ξ of the intrinsic parameters. The ℱ-statistic can thus be written in the form [see Eq. (62View Equation)]

ℱ [x(t;a,ξ);ξ′] = 1N [x (t;a,ξ );ξ ′]T ⋅ M (ξ ′)−1 ⋅ N[x(t;a,ξ);ξ′], (101 ) 2
where the matrices M and N are defined in Eqs. (60View Equation). The expectation value of the ℱ-statistic (101View Equation) is
[ ′] 1 ( T ′ ′−1 ′T ) E ℱ [x (t;a,ξ);ξ ] = 2- n + a ⋅ Q(ξ,ξ ) ⋅ M (ξ) ⋅ Q (ξ,ξ ) ⋅ a , (102 )
where the matrix Q is defined in Eq. (90View Equation). Let us rewrite the expectation value (102View Equation) in the following form,
[ ] 1 ( ) E ℱ [x (t;a,ξ );ξ′] = -- n + ρ(a,ξ)2𝒞n(a,ξ, ξ′) , (103 ) 2
where ρ is the signal-to-noise ratio and where we have introduced the normalized correlation function 𝒞n,
′ aT ⋅ Q(ξ,ξ ′) ⋅ M (ξ′)−1 ⋅ Q (ξ,ξ′)T ⋅ a 𝒞n(a,ξ,ξ ) := ------------T---------------------. (104 ) a ⋅ M (ξ ) ⋅ a
From the Rayleigh principle [90] it follows that the minimum value of the normalized correlation function is equal to the smallest eigenvalue of the matrix M (ξ)−1 ⋅ Q(ξ,ξ ′) ⋅ M (ξ ′)−1 ⋅ Q(ξ, ξ′)T, whereas the maximum value is given by its largest eigenvalue. We define the reduced correlation function 𝒞 as
′ 1- ( − 1 ′ ′ −1 ′T) 𝒞 (ξ, ξ) := 2 Tr M (ξ ) ⋅ Q (ξ,ξ ) ⋅ M (ξ ) ⋅ Q(ξ,ξ ) . (105 )
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 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 Eq. (89View Equation).

As 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 Eq. (95View Equation),

m ∑ Gð’œâ„¬Δ ξ𝒜 Δ ξℬ = 1 − C0, (106 ) 𝒜,ℬ=1
where G is given by Eq. (94View 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 (m ∕2 + 1)∫ ∘ -------- Nt = ------------------------ detG (ξ)dV. (107 ) (1 − C0)m∕2 πm ∕2 V
When C = 1∕2 0 the above formula coincides with the formula for the number N cells of cells, Eq. (98View 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 (107View 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
1 mm ∕2∫ ∘ -------- Nt = --------m∕2---m--- detG (ξ)dV. (108 ) (1 − C0) 2 V

For the case of the signal given by Eq. (34View Equation) our formula for the number of templates is equivalent to the original formula derived by Owen [102Jump To The Next Citation Point]. Owen [102] 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 [121, 45, 26Jump 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 [33, 16].

4.4.1 Covering problem

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 [38]. This was first studied in the context of gravitational-wave data analysis by Prix [111]. 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 m-dimensional Euclidean space ℝm 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 ℝm 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-m--dimensional-sphere. (109 ) 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. However, we know the best lattice covering for dimensions m ≤ 23. These are A ∗m lattices, which have thicknesses ΘA ∗m equal to
( )m ∕2 Θ ∗ = V √m--+--1 m-(m-+--2) , (110 ) Am m 12(m + 1)
where Vm is the volume of the m-dimensional sphere of unit radius. The advantage of an ∗ A m lattice over the hypercubic lattice grows exponentially with the number of dimensions.

For the case of gravitational-wave signals from spinning neutron stars a 3-dimensional grid was constructed [18Jump To The Next Citation Point]. It consists of prisms with hexagonal bases. Its thickness is around 1.84, which is much better than the cubic grid with a thickness of approximately 2.72. It is worse than the best 3-dimensional lattice covering, which has a thickness of around 1.46.

In [19] a grid was constructed in the 4-dimensional parameter space spanned by frequency, frequency derivative, and sky position of the source, for the case of an almost monochromatic gravitational-wave signal originating from a spinning neutron star. The starting point of the construction was an A ∗ 4 lattice of thickness ≃ 1.77. The grid was then constrained so that the nodes of the grid coincide with Fourier frequencies. This allowed the use of a fast Fourier transform (FFT) to evaluate the maximum-likelihood ℱ-statistic efficiently (see Section 4.6.2). The resulting lattice is only 20% thicker than the optimal A ∗ 4 lattice.

Efficient 2-dimensional banks of templates suitable for directed searches (in which one assumes that the position of the gravitational-wave source in the sky is known, but one does not assume that the wave’s frequency and its derivative are a priori known) were constructed in [104Jump To The Next Citation Point]. All grids found in [104] enable usage of the FFT algorithm in the computation of the ℱ-statistic; they have thicknesses 0.1 – 16% larger than the thickness of the optimal 2-dimensional hexagonal covering. In the construction of grids the dependence on the choice of the position of the observational interval with respect to the origin of time axis was employed. Also the usage of the FFT algorithms with nonstandard frequency resolutions achieved by zero padding or folding the data was discussed.

The above template placement constructions are based on a Fisher matrix with constant coefficients, i.e., they assume that the parameter manifold is flat. The generalization to curved Riemannian parameter manifolds is difficult. An interesting idea to overcome this problem is to use stochastic template banks where a grid in the parameter space is randomly generated by some algorithm [89, 57, 86, 119].

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