Measurement of the matter-radiation equality scale using the extended Baryon Oscillation Spectroscopic Survey Quasar Sample

The position of the peak of the matter power spectrum, the so-called turnover scale, is set by the horizon size at the epoch of matter-radiation equality. It can easily be predicted in terms of the physics of the Universe in the relativistic era, and so can be used as a standard ruler, independent of other features present in the matter power spectrum, such as baryon acoustic oscillations (BAO). We use the distribution of quasars measured by the extended Baryon Oscillation Spectroscopic Survey (eBOSS) to determine the turnover scale in a model-independent fashion statistically. We avoid modelling the BAO by down-weighting affected scales in the covariance matrix using the mode deprojection technique. We measure the wavenumber of the peak to be $k_\mathrm{TO} = \left( 17.6^{+1.9}_{-1.8} \right) \times 10^{-3}h/\mathrm{Mpc}$, corresponding to a dilation scale of $ D_\mathrm{V}(z_\mathrm{eff} = 1.48) = \left({36.2^{+4.1}_{-4.4}}\right)r_\mathrm{H}$. This is not competitive with current BAO distance measures in terms of determining the expansion history but does provide a useful cross-check. We combine this measurement with low-redshift distance measurements from type-Ia supernova data from Pantheon and BAO data from eBOSS to make a sound-horizon free estimate of the Hubble-Lema\^itre parameter and find it to be $H_0=\left({74.7\pm 9.6}\right) \ \mathrm{km/s/Mpc}$ with Pantheon, and $H_0=\left({72.9^{+10.0}_{-8.6}}\right) \ \mathrm{km/s/Mpc}$ with eBOSS BAO. We make predictions for the measurement of the turnover scale by the Dark Energy Spectroscopic Instrument (DESI) survey, the Maunakea Spectroscopic Explorer (MSE) and MegaMapper, which will make more precise and accurate distance determinations.