Fisher matrix has been extensively used to assess the accuracy of estimation of astrophysically-interesting
parameters of different gravitational-wave signals. For ground-based interferometric detectors, the first
calculations of the Fisher matrix concerned gravitational-wave signals from inspiralling compact binaries
(made of neutron stars or black holes) in the leading-order quadrupole approximation [50, 76, 63] and from
quasi-normal modes of Kerr black hole [48].
Cutler and Flanagan [41] initiated the study of the implications of the higher-order post-Newtonian
(PN) phasing formula as applied to the parameter estimation of inspiralling binary signals. They used the
1.5PN phasing formula to investigate the problem of parameter estimation, both for spinning and
non-spinning binaries, and examined the effect of the spin-orbit coupling on the estimation of parameters.
The effect of the 2PN phasing formula was analyzed independently by Poisson and Will [106
] and Królak,
Kokkotas and Schäfer [75
]. In both cases the focus was to understand the leading-order spin-spin coupling
term appearing at the 2PN level when the spins were aligned perpendicularly to the orbital plane.
Compared to [75], [106] also included a priori information about the magnitude of the spin parameters,
which then leads to a reduction in the rms errors in the estimation of mass parameters. The case of a 3.5PN
phasing formula was studied in detail by Arun et al. [17]. Inclusion of 3.5PN effects leads to an
improved estimate of the binary parameters. Improvements are relatively smaller for lighter
binaries. More recently the Fisher matrix was employed to assess the errors in estimating the
parameters of nonspinning black-hole binaries using the complete inspiral-merger-ring-down
waveforms [7].
Various authors have investigated the accuracy with which the LISA detector can determine binary parameters including spin effects. Cutler [40] determined LISA’s angular resolution and evaluated the errors of the binary masses and distance considering spins aligned or anti-aligned with the orbital angular momentum. Hughes [60] investigated the accuracy with which the redshift can be estimated (if the cosmological parameters are derived independently), and considered the black-hole ring-down phase in addition to the inspiralling signal. Seto [128] included the effect of finite armlength (going beyond the long wavelength approximation) and found that the accuracy of the distance determination and angular resolution improve. This happens because the response of the instrument when the armlength is finite depends strongly on the location of the source, which is tightly correlated with the distance and the direction of the orbital angular momentum. Vecchio [140] provided the first estimate of parameters for precessing binaries when only one of the two supermassive black holes carries spin. He showed that modulational effects decorrelate the binary parameters to some extent, resulting in a better estimation of the parameters compared to the case when spins are aligned or antialigned with orbital angular momentum. Hughes and Menou [61] studied a class of binaries, which they called “golden binaries,” for which the inspiral and ring-down phases could be observed with good enough precision to carry out valuable tests of strong-field gravity. Berti, Buonanno and Will [29] have shown that inclusion of non-precessing spin-orbit and spin-spin terms in the gravitational-wave phasing generally reduces the accuracy with which the parameters of the binary can be estimated. This is not surprising, since the parameters are highly correlated, and adding parameters effectively dilutes the available information.
Extensive study of accuracy of parameter estimation for continuous gravitational-wave signals from spinning neutron stars was performed in [64]. In [129] Seto used the Fisher matrix to study the possibility of determining distances to rapidly rotating isolated neutron stars by measuring the curvature of the wave fronts.
In order to test the performance of the maximization method of the -statistic it is useful to perform
Monte Carlo simulations of the parameter estimation and compare the simulated variances of the estimators
with the variances calculated from the Fisher matrix. Such simulations were performed for various
gravitational-wave signals [73, 26, 65
, 36]. In these simulations one observes that, above a certain
signal-to-noise ratio, called the threshold signal-to-noise ratio, the results of the Monte Carlo simulations
agree very well with the calculations of the rms errors from the inverse of the Fisher matrix. However, below
the threshold signal-to-noise ratio they differ by a large factor. This threshold effect is well known in signal
processing [139]. There exist more refined theoretical bounds on the rms errors that explain this
effect, and they were studied in the context of the gravitational-wave signals from coalescing
binaries [98
].
Use of the Fisher matrix in the assessment of accuracy of the parameter estimation has been critically
examined in [138], where a criterion has been established for the signal-to-noise ratio above which
the inverse of the Fisher matrix approximates well covariances of the parameter estimators.
In [148, 142] the errors of ML estimators of parameters of gravitational-wave signals from
nonspinning black-hole binaries were calculated analytically using a power expansion of the bias and
the covariance matrix in inverse powers of the signal-to-noise ratio. The first-order term in
this covariance matrix expansion is the inverse of the Fisher information matrix. The use of
higher-order derivatives of the likelihood function in these expansions makes the errors prediction
sensitive to the secondary lobes of the pdf of the ML estimators. Conditions for the validity of the
Cramèr–Rao lower bound are discussed in [142] as well, and some new features in regions of the
parameter space so far not explored are predicted (e.g., that the bias can become the most
important contributor to the parameters errors for high-mass systems with masses
and
above).
There exists a simple model that explains the deviations from the covariance matrix and reproduces well the results of the Monte Carlo simulations (see also [25]). The model makes use of the concept of the elementary cell of the parameter space that we introduced in Section 4.3.2. The calculation given below is a generalization of the calculation of the rms error for the case of a monochromatic signal given by Rife and Boorstyn [116].
When the values of parameters of the template that correspond to the maximum of the functional
fall within the cell in the parameter space where the signal is present, the rms error is satisfactorily
approximated by the inverse of the Fisher matrix. However, sometimes, as a result of noise,
the global maximum is in the cell where there is no signal. We then say that an outlier has
occurred. In the simplest case we can assume that the probability density of the values of the
outliers is uniform over the search interval of a parameter, and then the rms error is given by
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