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Function: qn_template_grid()

void qn_template_grid(float dl, struct qnScope *grid)
This function is responsible for allocating the memory for a grid of templates on the parameter space and for choosing the location of the templates. The arguments are:
dl: Input. The length of the `sides' of the square templates. This quantity should be set to  $d\ell=\surd(2ds^2_{\mathrm{\scriptstyle threshold}})$ (see the discussion below).
grid: Input/Output. The grid of templates of type struct qnScope. On input, the fields that relate to parameter ranges should be set. On output, the field n_tmplt is set to the number of templates generated, and these templates are put into the array field templates[0..n_tmplt-1] (which is allocated by the function).

The function qn_template_grid() attempts to create a set of templates, $\{u_i(t)\}$, which ``cover'' parameter space finely enough that the distance between an arbitrary point on the parameter space and one of the templates is small. A precise statement of this goal, and how it is achieved, can be found in the paper by Owen [5]. We hilight the relevant parts of reference [5] here.

The templates $\{u_i(t)\}$ are damped sinusoids with a set of frequency and quality parameters $\{(f,Q)_i\}$. They are normalized so that $(u_i\vert u_i)=1$ where $(\cdot\vert\cdot)$ is the inner product defined by Cutler and Flanagan [21]. Since we are most interested in the high quality region of parameter space, it is a good approximation that the value of the one-sided noise power spectrum is approximately constant, $S_h(f)\approx S_h(f_i)$, over the frequency band of the template. This approximation simplifies the form of the inner product as the noise power spectrum appears in the inner product as a weighting function.

In order to estimate how close together the templates must be, we define the distance function  $ds^2_{ij}=1-(u_i\vert u_j)$ corresponding to the mismatch between the two templates $u_i$ and $u_j$. This interval can be expressed in terms of a metric as  $ds^2=g_{\alpha\beta}dx^\alpha dx^\beta$ where $x^\alpha=(f,Q)^\alpha$ are coordinates on the two dimensional parameter space. Such an expression is only valid for sufficiently close points on parameter space. In the limit of a continuum of templates over parameter space, the metric can be evaluated by  $g_{\alpha\beta}=-\frac{1}{2}(u\vert\partial_\alpha\partial_\beta u)$ where $\partial_\alpha$ is a partial derivative with respect to the coordinate $x^\alpha$. We find that the mismatch between templates that differ in frequency by $df$ and in quality by $dQ$ is given by

$\displaystyle ds^2$ $\textstyle =$ $\displaystyle \frac{1}{8} \biggl\{ \frac{3+16Q^4}{Q^2(1+4Q^2)^2}\,dQ^2
- 2\frac{3+4Q^2}{fQ(1+4Q^2)}\,dQ\,df + \frac{3+8Q^2}{f^2}\,df^2 \biggr\}$ (8.18.185)
  $\textstyle \approx$ $\displaystyle \frac{1}{8}\frac{dQ^2}{Q^2} - \frac{1}{4}\frac{dQ}{Q}\frac{df}{f}
+ Q^2\frac{df^2}{f^2}.$ (8.18.186)

In the approximate metric of equation ([*]), we have kept only the dominant term in the limit of high quality. The minimum number of templates, $\mathcal{N}$, required to span the parameter space such that there is no point on parameter space that is a distance larger than $ds^2_{\mathrm{\scriptstyle threshold}}$ from the nearest template can be found by integrating the volume element  $\surd\det g_{\alpha\beta}$ over the parameter space. Using the approximate metric and the parameter ranges  $Q\le Q_{\mathrm{\scriptstyle max}}$ and  $f\in[f_{\mathrm{\scriptstyle min}},f_{\mathrm{\scriptstyle max}}]$, we find that the number of templates required is
$\displaystyle {\mathcal{N}}$ $\textstyle \approx$ $\displaystyle \frac{1}{4\surd2}
(ds^2_{\mathrm{\scriptstyle threshold}})^{-1}
...ptstyle max}}
\log(f_{\mathrm{\scriptstyle max}}/f_{\mathrm{\scriptstyle min}})$  
  $\textstyle \simeq$ $\displaystyle 2700\,
\biggl(\frac{ds^2_{\mathrm{\scriptstyle threshold}}}{0.03}...
...\frac{f_{\mathrm{\scriptstyle min}}}{100\,{\mathrm{Hz}}}\Bigr)
\biggr]\biggr\}.$ (8.18.187)

The issue of template placement is more difficult than computing the number of templates required. Fortunately, for the problem of quasinormal ringdown template placement, the metric is reasonably simple. By using the coordinate $\phi=\log f$ rather than $f$, we see that the metric components depend on $Q$ alone. We can exploit this property for the task of template placement as follows: First, choose a ``surface'' of constant  $Q=Q_{\mathrm{\scriptstyle min}}$, and on this surface place templates at intervals in $\phi$ of  $d\phi=d\ell/g_{\phi\phi}$ for the entire range of $\phi$. Here, $d\ell=\surd(2ds^2_{\mathrm{\scriptstyle threshold}})$. Then choose the next surface of constant $Q$ with  $dQ=d\ell/g_{QQ}$ and repeat the placement of templates on this surface. This can be iterated until the entire range of $Q$ has been covered; the collection of templates should now cover the entire parameter region with no point in the region being farther than  $ds^2_{\mathrm{\scriptstyle threshold}}$ from the nearest template.

Author: Jolien Creighton,

next up previous contents
Next: The close-limit approximation and Up: GRASP Routines: Black hole Previous: Structure: struct qnScope   Contents
Bruce Allen 2000-11-19