Synopsis

   #include <lal/Ring.h>

Routines for generating waveforms for black hole ringdown.

The ringdown waveform is an exponentially-damped sinusoid

$$q(t) = \left\{ \begin{array}{ll} (2\pi)^{1/2}e^{-\pi ft/Q... ...x{for $t\ge0$} \0 & \mbox{for $t<0$} \end{array} \right.$$ (39.1)

where $f$ is the central frequency of the ringdown waveform and $Q$ is the quality factor.

For a black hole ringdown, the gravitational waveform produced, averaged over the various angles, is

$$h(t) = Aq(t)$$ (39.2)

where the central frequency and quality of the ringdown are determined from the mass and spin of the black holes. An analytic approximation yields [1,2]

$$f \simeq 32 \textrm{kHz}\times[1-0.63(1-{\hat{a}})^{3/10}](M_\odot/M)$$ (39.3)

and

$$Q \simeq 2(1-{\hat{a}})^{-9/20}$$ (39.4)

with the black hole mass given by $M$ and its spin by $S={\hat{a}}GM^2/c$ (where $G$ is Newton's constant and $c$ is the speed of light). The dimensionless spin parameter ${\hat{a}}$ lies between zero (for a Schwarzschild black hole) and unity (for an extreme Kerr black hole). The amplitude of the waveform depends on these quantities as well as the distance $r$ to the source and the fractional mass loss $\epsilon$ radiated in gravitational waves [4]:

$$A = 2.415\times10^{-21}Q^{-1/2}[1-0.63(1-{\hat{a}})^{3/10}]... ...M}{M_\odot}\right) \left(\frac{\epsilon}{0.01}\right)^{1/2}.$$ (39.5)

The mismatch between two nearby templates is given by $ds^2$, which can be thought of as the line interval for a mismatch-based metric on the $(f,Q)$ parameter space [3,4]:

$$ds^2 = \frac{1}{8} \biggl\{ \frac{3+16Q^4}{Q^2(1+4Q^2)^2} ... ...Q^2}{fQ(1+4Q^2)} dQ df + \frac{3+8Q^2}{f^2} df^2 \biggr\}.$$ (39.6)

When expressed in terms of $\log f$ rather than $f$, the metric coefficients depend on $Q$ alone. We can exploit this property for the task of template placement. The method is the following: First, choose a ``surface'' of constant  $Q=Q_{\mathrm{\scriptstyle min}}$, and on this surface place templates at intervals in $\phi=\log f$ of  $d\phi=d\ell/\surd g_{\phi\phi}$ for the entire range of $\phi$. Here, $d\ell=\surd(2ds^2_{\mathrm{\scriptstyle threshold}})$. Then choose the next surface of constant $Q$ with  $dQ=d\ell/\surd g_{QQ}$ 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.