Future null energy violation and minimal dark energy solutions to the Hubble tension

Abstract

Few phenomenological extensions of ΛCDM favour higher values of the Hubble parameter without introducing additional degrees of freedom, often at the cost of invoking phantom dynamics. We investigate ─ Phenomenological Emergent Dark Energy (PEDE) and Granda-Oliveros Holographic Dark Energy (GOHDE) ─ and examine how parameter correlations and dataset selection affect their ability to alleviate the Hubble tension. At the background level, minimally extended versions of both models prefer lower values of H0. Excluding BAO Lyα-H(z) measurements at z ≃ 2.3 yields values of H0 in significant tension with SH0ES, while their inclusion favours higher H0 due to the implied lower matter density within the fiducial cosmology. Recent DESI DR1 and DR2 results exhibit mild tension with SDSS BAO-H(z) estimates, further influencing the inferred current expansion rate in both cases. We show that stringent geometric constraints, particularly the CMB shift parameter combined with Pantheon+, substantially affect the estimation of H0 in PEDE and GOHDE models. A full CMB likelihood (low ℓ TT and EE, and NPIPE high ℓ TTTEEE) along with geometric datasets (DESI-BAO and Union3 SNe) analysis confirms that these conclusions persist beyond the background level. Finally, interpreting both scenarios as interacting dark-sector models with Q=3HγΛρ~m indicates that resolving the Hubble tension requires a dynamically evolving coupling exhibiting a singular sign reversal. This behaviour corresponds to a future violation of the null energy condition, a feature that appears as a condition for minimal models aiming to reconcile H0 tension.