Addressing the DESI DR2 Phantom-Crossing Anomaly and Enhanced $H_0$ Tension with Reconstructed Scalar-Tensor Gravity

Recent cosmological data, including DESI DR2, highlight significant tensions within the $\mathrm{\Lambda }$CDM paradigm. When analyzed in the context of General Relativity (GR), the latest DESI data favor a dynamical dark energy (DDE) equation of state, $w(z)$, that crosses the phantom divide line $w=-1$. However, this framework prefers a lower Hubble constant, $H_0$, than Planck 2018, thereby worsening the tension with local measurements. This phantom crossing is a key feature that cannot be achieved by minimally coupled scalar fields (quintessence) within GR. This suggests the need for a new degree of freedom that can simultaneously: (A) increase the best-fit value of $H_0$ in the context of the DESI DR2 data, and (B) allow the crossing of the $w=-1$ line within a new theoretical approach. We argue that both of these goals may be achieved in the context of Modified Gravity (MG), and in particular, Scalar-Tensor (ST) theories, where phantom crossing is a natural and viable feature. We demonstrate these facts by analyzing a joint dataset including DESI DR2, Pantheon+, CMB, and growth-rate (RSD) data in the context of simple parametrizations for the effective gravitational constant, ${\mu }_G(z)\equiv G_{eff}\mathrm{/}{G}_N$, and the DDE equation of state, $w(z)$. This MG framework significantly alleviates the tension, leading to a higher inferred value of $H_0=70.6\pm 1.7{\text{km s}}^{-1}{\text{Mpc}}^{-1}$. We also present a systematic, data-driven reconstruction of the required underlying ST Lagrangian and provide simple, generic analytical expressions for both the non-minimal coupling $F(\mathrm{\Phi })=1+\xi {\mathrm{\Phi }}^2{e}^{n\mathrm{\Phi }}$ and the scalar potential $U(\mathrm{\Phi })=U_0+a{e}^{b{\mathrm{\Phi }}^2}$, which well-describe the reconstructed functions.