Constraints on the gravitational potential from DESI DR2 BAO and its implications for the local void scenario

We constrain the difference in gravitational potential between our location and sources at $z \gtrsim 0.3$ using datasets at those redshifts. Our motivation is that the Hubble tension might be caused by a local void, as suggested by galaxy number counts. This would increase the redshift through outflow and gravitational redshift (GR). Only the latter is important at high redshift, where a void contributes a fixed additional GR contribution of $z_0$ due to our location on a potential hill. This $z_0$ model has various subtle effects that were not previously considered, including a hotter CMB and reduced BAO $r_{\rm d}$. We test whether $z_0$ can have the previously expected value of 0.84\%, which was based on fitting void parameters to galaxy number counts and local $H_0$ measurements. Combining BBN, CMB, BAO, and CC datasets at $z > 0.5$, we find that $z_0$ = $-0.4 \pm 0.9\%$, which rises to $0.0^{+0.6}_{-0.7}\%$ when extending our analysis down to $z > 0.29$. Although the results prefer the standard value of $z_0 = 0$, the best-fitting model with $z_0 = 0.84\%$ fits the data almost as well as $\Lambda$CDM, with $\Delta \chi^2 < 2$. We find that $\Lambda$CDM faces a $2.81\sigma$ BAO anomaly in the standard $(H_0 r_{\rm d}, \Omega_{\rm m})$ parameter space, where different regions are preferred by BAO and non-BAO datasets from $z > 0.29$. Fixing $z_0 = 0.84\%$ reduces this to $2.39\sigma$. This suggests that a local void large enough to solve the Hubble tension cannot be ruled out by higher-redshift datasets despite its novel impact on them.