Validation of the DESI 2024 Ly$\alpha$ forest BAO analysis using synthetic datasets

The first year of data from the Dark Energy Spectroscopic Instrument (DESI) contains the largest set of Lyman-$\alpha$ (Ly$\alpha$) forest spectra ever observed. This data, collected in the DESI Data Release 1 (DR1) sample, has been used to measure the Baryon Acoustic Oscillation (BAO) feature at redshift $z=2.33$. In this work, we use a set of 150 synthetic realizations of DESI DR1 to validate the DESI 2024 Ly$\alpha$ forest BAO measurement. The synthetic data sets are based on Gaussian random fields using the log-normal approximation. We produce realistic synthetic DESI spectra that include all major contaminants affecting the Ly$\alpha$ forest. The synthetic data sets span a redshift range $1.8<z<3.8$, and are analysed using the same framework and pipeline used for the DESI 2024 Ly$\alpha$ forest BAO measurement. To measure BAO, we use both the Ly$\alpha$ auto-correlation and its cross-correlation with quasar positions. We use the mean of correlation functions from the set of DESI DR1 realizations to show that our model is able to recover unbiased measurements of the BAO position. We also fit each mock individually and study the population of BAO fits in order to validate BAO uncertainties and test our method for estimating the covariance matrix of the Ly$\alpha$ forest correlation functions. Finally, we discuss the implications of our results and identify the needs for the next generation of Ly$\alpha$ forest synthetic data sets, with the top priority being to simulate the effect of BAO broadening due to non-linear evolution.