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 $r_{\mathrm{d}}$. 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 $z_0=-0.4_{-0.9}^{+0.8}\mathrm{\%}$, which rises to $-0.1\pm 0.7\mathrm{\%}$ 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 $z_0=0.84\mathrm{\%}$ fits the data almost as well as $\mathrm{\Lambda }$CDM, with $\mathrm{\Delta }{\chi }^2<2$. We find that $\mathrm{\Lambda }$CDM faces a $3.07\sigma$ BAO anomaly in the standard $(H_0{r}_{\mathrm{d}},{\mathrm{\Omega }}_{\mathrm{m}})$ parameter space, where different regions are preferred by BAO and non-BAO datasets from $z>0.29$. Fixing $z_0=0.84\mathrm{\%}$ reduces this to $2.79\sigma$. This suggests that a local void large enough to solve the Hubble tension cannot be ruled out by higher-redshift datasets despite its novel impacts on them.