Use and abuse of the Fisher information matrix in the assessment of gravitational-wave parameter-estimation prospects

The Fisher-matrix formalism is used routinely in the literature on gravitational-wave detection to characterize the parameter-estimation performance of gravitational-wave measurements, given parametrized models of the waveforms, and assuming detector noise of known colored Gaussian distribution. Unfortunately, the Fisher matrix can be a poor predictor of the amount of information obtained from typical observations, especially for waveforms with several parameters and relatively low expected signal-to-noise ratios (SNR), or for waveforms depending weakly on one or more parameters, when their priors are not taken into proper consideration. In this paper I discuss these pitfalls; show how they occur, even for relatively strong signals, with a commonly used template family for binary-inspiral waveforms; and describe practical recipes to recognize them and cope with them. Specifically, I answer the following questions: (i) What is the significance of (quasi-)singular Fisher matrices, and how must we deal with them? (ii) When is it necessary to take into account prior probability distributions for the source parameters? (iii) When is the signal-to-noise ratio high enough to believe the Fisher-matrix result? In addition, I provide general expressions for the higher-order, beyond-Fisher-matrix terms in the $1/\mathrm{SNR}$ expansions for the expected parameter accuracies.

Figure 1

Consistency criterion for the simple waveform model $h({\varphi }_0)=A\cos (2\pi ft+{\varphi }_0)$ in Gaussian stationary noise, with fixed (known) $A$ and $f$. The curve plots $|\log r|$ as a function of the $1\sigma$ error ${\varphi }_0^{1\sigma }=\pm 1/A$, with specific values of $A$ called out by the circles. For the threshold $|\log r|=0.1$, consistency is achieved for $A\gtrsim 1.09$. (To generate this graph, the integration time was set to $1000/f$, and the variance of noise adjusted so that $(\overline{h}({\varphi }_0),\overline{h}({\varphi }_0))=1$.)

Figure 2

Cumulative distribution function for $|\log r|$ on the $1\sigma$ surface at various SNRs for our reference SPA model with $m_1=m_2=10M_{\odot }$. The SNR required to have 90% of the $1\sigma$ points at $|\log r|=0.1$ (dashed lines) increases considerably (in fact, to unrealistic values) as we move from the 4pp to 5pp and 6pp. This figure was produced without imposing any priors on the source parameters.

Figure 3

SNR values at which $|\log r|=0.1$ in the 4pp for our reference SPA model with $m_1=m_2=10M_{\odot }$. The six subplots display sections of parameter space corresponding to all distinct pairs of eigenvectors of the Fisher matrix; the polar radius of the curves shows the required SNR (plotted logarithmically from $\log \mathrm{SNR}=0$), while the polar angle is computed between pairs of renormalized eigenvector coefficients ${\tilde{c}}^{(i)}$. The graph at the bottom right shows the composition of the four Fisher-matrix eigenvectors in terms of the source parameters, as well as their respective eigenvalues. This figure was produced without imposing any priors on the source parameters.