Gravitational-Wave Signatures of Compact Extra Dimensions and Their Detectability with Third-Generation Interferometers

We investigate gravitational-wave dynamics in theories of gravity formulated in $D=4+N$ dimensions, with matter localized on a four-dimensional brane or coupled to the effective four-dimensional metric obtained after compactification. We consider four representative geometries, namely toroidal and Calabi–Yau compactifications, and the warped Randall–Sundrum I and II models, which cover the principal discrete and continuous Kaluza–Klein spectra. We derive the modifications of binary dynamics, propagation, quasinormal modes and polarization content, and use them as a feasibility screening for third-generation interferometers, with particular attention to the Einstein Telescope. The screening shows that the direct effects of the heavy Kaluza–Klein towers are strongly suppressed and are expected to be inaccessible to third-generation detectors. Indeed, the effective-field-theory corrections scale as ${(\hslash \omega /m_1{c}^2)}^2$∼${10}^{-23}$–${10}^{-22}$, while the explicit conservative inspiral-phase correction of RS2 remains below the ET measurability threshold over the region allowed by laboratory bounds. Nevertheless, the screening identifies additional phenomenological channels that may remain testable: a tidal-charge deformation of the black-hole background, which shifts the QNM frequencies and damping times; a continuum-inspired phase deformation; and a radion-induced scalar polarization. We treat these channels as phenomenological benchmarks and formulate a Fisher-matrix Monte Carlo forecast based on the Earth-fixed geometry of the candidate sites, comparing the triangular and double-L Einstein Telescope configurations within an idealized common-noise setting.