Dynamical dark energy with AdS-dS transitions vs. Baryon Acoustic Oscillations at $z=$ 2.3-2.4

In this paper, written in memory of Alexei Starobinsky, we discuss the observational viability of the Ph-${\mathrm{\Lambda }}_{\mathrm{s}}$CDM model - a dynamical dark energy scenario based on a phantom scalar field undergoing an anti-de Sitter (AdS) to de Sitter (dS) transition - and revisit the Sahni-Shtanov braneworld model in light of updated BAO Ly-$\alpha$ data at $z\sim 2.3$. Both models are able to remain consistent with Planck CMB data while offering potential resolutions to the $H_0$ tension. In both cases, the expansion rate $H(z)$ is suppressed relative to Planck-$\mathrm{\Lambda }$CDM at high redshift and enhanced at low redshift, while remaining consistent with the comoving distance to recombination as estimated by Planck-$\mathrm{\Lambda }$CDM. Comparing model predictions with BAO-inferred values of $H(z)$, we find that SDSS Ly-$\alpha$ data at $z\approx 2.33$ mildly favor such dynamical models, whereas the recent DESI Ly-$\alpha$ measurements agree more closely with $\mathrm{\Lambda }$CDM. Although current high-redshift BAO data do not decisively favor one model over another, our findings illustrate how frameworks originally developed to address earlier anomalies - such as the braneworld scenario - may gain renewed relevance in confronting today’s cosmological tensions.