Measurement of the power spectrum turnover scale from the cross-correlation between CMB lensing and Quaia

We use the projected clustering of quasars in the Gaia-unWISE quasar catalog, Quaia, and its cross-correlation with CMB lensing data from Planck, to measure the large-scale turnover of the matter power spectrum, associated with the size of the horizon at the epoch of matter-radiation equality. The turnover is detected with a significance of between $2.3$ and $3.1\sigma$, depending on the method used to quantify it. From this measurement, the equality scale is determined at the $\sim20\%$ level. Using the turnover scale as a standard ruler alone (suppressing information from the large-scale curvature of the power spectrum), in combination with supernova data through an inverse distance ladder approach, we measure the current expansion rate to be $H_0=62.7\pm17.2\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. The addition of information coming from the power spectrum curvature approximately halves the standard ruler uncertainty. Our measurement in combination with calibrated supernovae from Pantheon$+$ and SH0ES constrains the CMB temperature to be $T_{\rm CMB}=3.10^{+0.48}_{-0.36}\,{\rm K}$, independently of CMB data. Alternatively, assuming the value of $T_{\rm CMB}$ from COBE-FIRAS, we can constrain the effective number of relativistic species in the early Universe to be $N_{\rm eff}=3.0^{+5.8}_{-2.9}$.