(ed.). Gravitational waves (IOP, 2001)(422s).

342 Post-Newtonian computation of binary inspiral waveformswhere A denotes some amplitude parameter. The function ˜k depends only onthe other parameters, collectively denoted by λ a where the label a ranges on thevalues 1,...,N. The family of matched filters (or ‘templates’) we consider isdefined by˜q(ω; t λ a ) = γ ′ ˜k(ω; t λ a ), (17.11)S n (ω)where t λ a is a set of test parameters, assumed to be all independent, and γ ′ isarbitrary. By substituting (17.11) into (17.5) and choosing t = 0, we get, with thenotation of (17.9),〈o, k( t λ)〉σ( t λ) =. (17.12)〈k( t λ), k( t λ)〉 1/2(Note that σ is in fact a function of both the parameters λ a and t λ a .) Nowthe experimenters choose as their best estimate of the source parametersλ a the measured parameters m λ a which among all the test parameters t λ a(independently) maximize (17.12), i.e. which satisfy∂σ( m λ) = 0, a = 1,...,N. (17.13)∂ t λ aAssuming that the signal to noise is high enough, we can work out (17.13) upto the first order in the difference between the actual source parameters and themeasured ones,δλ a = λ a − m λ a . (17.14)As a result, we obtain{δλ a = ab −〈n, ∂h}〈n, h〉 ∂h〉+ 〈h, 〉 , (17.15)∂λ b 〈h, h〉 ∂λ bwhere a summation is understood on the dummy label b, and where the matrix ab (with a, b = 1,...,N) is the inverse of the Fisher information matrix〈 ∂h ab = , ∂h 〉− 1 〈h, ∂h 〉〈h, ∂h 〉(17.16)∂λ a ∂λ b 〈h, h〉 ∂λ a ∂λ b(we have ab bc = δ ac ). On the right-hand sides of (17.15) and (17.16), thesignal is equally (with this approximation) parametrized by the measured or actualparameters. Since the noise is Gaussian, so are, by (17.15), the variables δλ a(indeed, δλ a result from a linear operation on the noise variable). The expectationvalue and quadratic moments of the distribution of these variables are readilyobtained from the facts that 〈n, f 〉 = 0 and 〈n, f 〉〈n, g〉 = 〈f, g〉 for anydeterministic functions f and g (see (17.2) and (17.3)). We then obtainδλ a = 0,δλ a δλ b = ab . (17.17)