Measurements of the $z>5$ Lyman-$\alpha$ forest flux auto-correlation functions from the extended XQR-30 data set

Recently, the Lyman-$\alpha$ (Ly$\alpha$) forest flux auto-correlation function has been shown to be sensitive to the mean free path of hydrogen-ionizing photons, $\lambda_{\text{mfp}}$, for simulations at $z \geq 5.4$. Measuring $\lambda_{\text{mfp}}$ at these redshifts will give vital information on the ending of reionization. Here we present the first observational measurements of the Ly$\alpha$ forest flux auto-correlation functions in ten redshift bins from $5.1 \leq z \leq 6.0$. We use a sample of 35 quasar sightlines at $z>5.7$ from the extended XQR-30 data set, this data has signal-to-noise ratios of $>20$ per spectral pixel. We carefully account for systematic errors in continuum reconstruction, instrumentation, and contamination by damped Ly$\alpha$ systems. With these measurements, we introduce software tools to generate auto-correlation function measurements from any simulation. For an initial comparison, we show our auto-correlation measurements with simulation models for recently measured $\lambda_{\text{mfp}}$ values and find good agreements. Further work in modeling and understanding the covariance matrices of the data is necessary to get robust measurements of $\lambda_{\text{mfp}}$ from this data.