Alleviating the H₀ tension in Rastall gravity

The persistent discrepancy between local determinations of the Hubble constant $H_0$ and the Planck 2018 value ($67.4 \pm 0.5~{\rm km\,s^{-1}\,Mpc^{-1}}$) within $\Lambda$CDM remains a central challenge in precision cosmology. We investigate the Hubble tension in $\Lambda$CDM and its Rastall extension (R-$\Lambda$CDM) for flat, open, and closed geometries. We analyze three primary dataset combinations: D$_1$ (late-time probes: SN + $H(z)$ + $f\sigma_8$), D$_2$ (late-time probes combined with DESI DR2 BAO and BBN), and D$_3$ (late-time probes combined with BAO and Planck 2018 CMB distance priors). Parameters are constrained via Markov Chain Monte Carlo sampling, and tensions with SH0ES ($73.2 \pm 1.3~{\rm km\,s^{-1}\,Mpc^{-1}}$) and Planck are expressed in units of the combined uncertainty. In addition, we include a Planck-only configuration (D$_4$) as a reference baseline to isolate early-Universe constraints on $H_0$. Within $\Lambda$CDM, D$_1$ and D$_2$ yield $H_0 \simeq 70.75$-$71.43~{\rm km\,s^{-1}\,Mpc^{-1}}$, reducing the SH0ES discrepancy to $1.11\sigma$--$1.63\sigma$ while maintaining a $3.62\sigma$-$4.23\sigma$ tension with Planck. Including CMB distance priors (D$_3$) shifts the result to $H_0 \simeq 67.18$--$67.55~{\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.09\sigma$-$0.38\sigma$ but increasing the SH0ES discrepancy to $4.25\sigma$-$4.51\sigma$.