Simple-graduated dark energy and spatial curvature

In this work, we first discuss the possibility that dark energy models with negative density values in past can alleviate $H_0$ tension, as well discrepancy baryon acoustic oscillation (BAO) Lyman-$\alpha$ data, both which prevail within $\Lambda$CDM model. We then investigate whether two minimal extensions of model, together or separately, successfully realize such a scenario: (i) spatial curvature, which, case spatially closed universe, mimics source and (ii) simple-graduated (gDE), promotes null inertial mass usual vacuum to an arbitrary constant--if negative, corresponding decreases redshift similar phantom models, but unlike them crosses below zero at certain redshift. find that, when Planck data are not included observational analysis, simple-gDE predict interesting some significant deviations from particular, universe along positive density, work contrast each other, results minor improvement tension. The joint dataset, including presents no evidence for deviation flatness almost same cosmological constant order $\mathcal{O}(10^{-12})\,\rm eV^4$. latter predicts model up until today--so it regarding BAO Ly-$\alpha$ data--except slightly aggravates also study via dynamical analysis history Universe futures different than de Sitter future