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

• Received 8 April 2021
• Accepted 3 June 2021

DOI:https://doi.org/10.1103/PhysRevD.104.023505