Intertwined constraints in extended cosmologies: Dark energy, curvature, neutrinos, and inflation

We present a systematic reassessment of cosmological constraints beyond ΛCDM by progressively relaxing the assumptions underlying Dark Energy (DE), Curvature, Neutrinos, and Inflation. Using the latest CMB data together with DESI BAO and different SN catalogues, we show that the preference for dynamical DE persists across all the extended cosmologies considered. Ω k remains compatible with flatness, despite a mild 2.2 σ preference for Ω k > 0 that is substantially degraded in dynamical DE extensions. Constraints on N eff are broadly consistent with N e f f = 3.04 , while cosmological upper limits on the total neutrino mass vary substantially across the cosmologies explored, ranging from ∑ m ν ≲ 0.06 eV to ≲ 0.2 eV. We quantify both the preference for the mass ordering and the apparent tension between cosmology and oscillation experiments, showing that they are strongly framework dependent. We find no evidence for inflationary tensor modes, with r ≲ 0.035. Constraints on the spectral index n s show significant model dependence. Allowing for the scalar runnings produces a mild shift toward α s > 0 and β s > 0 that can reabsorb the preference for larger n s found in small-scale CMB data, although both α s and β s remain consistent with zero at ∼ 1.5 σ . We highlight the implications for slow-roll inflation and benchmark models. None of the extensions considered here can resolve the H 0 tension. We discuss the implications for Ω m and S 8 . Overall, dynamical DE is the only significant deviation from ΛCDM and has the strongest impact on the inferred conclusions in the other sectors of the model.