Analytical Waveform Approximations and Parameter Sensitivity in Gravitational-Wave Models

How do the total mass and the mass ratio shape the mathematical form of the gravitational wave emitted by an inspiralling compact binary? We answer this question within the post-Newtonian (PN) approximation of general relativity, using the chirp time per unit frequency \(dt/df\), a quantity unaffected by the arbitrary time and phase offsets of the signal. In the variables \((\mathcal{M},\eta)\), the chirp mass fixes the leading law \(dt/df\propto\mathcal{M}^{-5/3}f^{-11/3}\), and the mass ratio enters through sensitivity coefficients \(s_k(\eta)\), which we give in closed form to 2.5PN. Their signs alternate between the 1PN correction and the radiative tail. As a result, the mass-ratio sensitivity of the chirp changes sign at \(v_\star=(743-1386\eta)/(2016\pi)\) to leading order, that is at \(f_\star\simeq16.6\text{--}111\ \mathrm{Hz}\ (M_\odot/M)\). This relation is stable under truncation to within 7% and lies below the observed band for essentially all known sources. The same coefficients govern a closed-form cycle budget. They also explain why the tail term reverses the sign of the chirp--mass--mass-ratio correlation in a Fisher-matrix analysis and reduces \(\sigma_\eta\) by factors of 2--4. A phase-coherence bound shows that the Newtonian chirp is adequate only in the last 2--26% of the frequency range below the innermost stable circular orbit.