Cosmological Perturbations and Structure Formation in Spectral Nod Gravity_ From Modified Friedmann to Matter Power Spectrum

The next generation of large-scale surveys (Euclid, DESI, LSST) will measure the growth of cosmic structures with unprecedented precision, offering powerful tests of gravity on cosmological scales . In this paper, we develop the full theory of linear cosmological perturbations within the Spectral Nod Gravity (SNG) framework. Building on the modified Friedmann equations derived in previous work [7], we derive the perturbation equations for matter density fluctuations and gravitational potentials under the quasi-static approximation on sub-horizon scales. Following the general formalism of De Felice, Mukohyama, and Tsujikawa , we obtain the effective gravitational coupling \(\mu(a,k)\) and the slip parameter \(\gamma(a,k)\) that characterize departures from General Relativity. We show that the nod density fluctuations \(\delta \delta_n\) and entanglement density variations \(\delta \eta_e\) source the gravitational potentials, leading to scale-dependent modifications of the matter power spectrum. The cyclic reset operator \(\cyceq\) introduces a natural screening mechanism on small scales, recovering General Relativity in high-density regions (Vainshtein-like mechanism ), while the fluctuating equivalence operator \(\fleq\) generates characteristic oscillations in the growth rate. We compute the predicted CMB power spectrum and matter power spectrum, showing consistency with existing observations while making testable predictions for Euclid and DESI. This work completes the cosmological perturbation theory of SNG, enabling quantitative comparison with upcoming surveys. \noindent\textbf{Keywords:} cosmological perturbations, modified gravity, structure formation, spectral nod gravity, matter power spectrum, screening mechanisms