Cosmic topology. Part IIc. Detectability with non-standard primordial power spectrum

Non-trivial spatial topology of the Universe can imprint potentially observable signatures on the cosmic microwave background (CMB). In this study, we investigate how deviations from the standard nearly-scale-free primordial power spectrum impact observables for the fully compact, orientable Euclidean topologies ($E_1$--$E_6$). We examine how such deviations modify the detectability of the underlying topology, depending on whether they are an intrinsic consequence of non-trivial topology or independent of it. We compute CMB temperature correlation matrices across a range of topologies, fundamental domain sizes, and observer locations for both standard and modified primordial power spectra. The impact of these modifications on the detectability of topology is quantified using the Kullback-Leibler divergence, providing an estimate of the distinguishability of non-trivial and simply-connected topologies based solely on CMB temperature observations. In addition, we employ the CatBoost machine learning algorithm to classify harmonic-space realizations of CMB temperature maps and thereby assess the observational prospects for topology detection. Signatures of non-trivial topology are encoded in the off-diagonal structure of the CMB temperature correlation matrices and are most prominent on the largest angular scales. Deviations from the simple power-law primordial spectrum at these scales can substantially alter the detectability of topology, either enhancing its characteristic CMB imprints or suppressing them below observational sensitivity. Our results demonstrate that uncertainties in the primordial power spectrum must be carefully accounted for in robust searches for cosmic topology using the CMB.