Optimal Arm Length of Gravitational-Wave Interferometers: Resonant Detection of Post-Merger Echo Structures in LIGO Data

This work investigates the relationship between the temporal delaystructure observed in gravitational-wave post-merger residuals andthe optical geometry of interferometric detectors. Previous analyses of publicly available LIGO data have reportedthe presence of a stable delay scale in the residual autocorrelationstructure of compact binary merger events. The characteristic delayis approximately \begin{equation}\Delta t \approx 2.3 \times 10^{-2} \, s .\end{equation} A signal containing a fixed delay necessarily produces a periodicspectral structure with characteristic spacing \begin{equation}f_{\mathrm{echo}} = \frac{1}{\Delta t}.\end{equation} Using the observed delay, this corresponds to a frequency scale ofapproximately $43 \, \mathrm{Hz}$. The optical response of Fabry--Perot interferometer cavities isdetermined by the free spectral range \begin{equation}f_{\mathrm{FSR}} = \frac{c}{2L}.\end{equation} Matching the cavity free spectral range to the echo frequency yieldsa geometric resonance condition for the interferometer arm length \begin{equation}L = \frac{c \Delta t}{2}.\end{equation} Substituting the observed delay leads to an optimal arm length \begin{equation}L \approx 3.46 \, \mathrm{km}.\end{equation} The current LIGO configuration employs $4 \, \mathrm{km}$ arms,which produces a free spectral range of approximately$37.5 \, \mathrm{kHz}$ and therefore lies far from resonance withthe echo carrier frequency. The present analysis shows that if coherent post-merger echostructures exist in gravitational-wave signals, an interferometerwith arm length near $3.46 \, \mathrm{km}$ would operate inresonance with the associated delay scale and could enhance thedetectability of such structures. The result follows directly from the combination of the empiricallyobserved delay scale and the optical geometry of Fabry--Perotinterferometer cavities, without relying on any specific theoreticalmodel for the origin of the echo mechanism.