The Alcock-Paczy\'nski effect from Lyman-$\alpha$ forest correlations: Analysis validation with synthetic data

The three-dimensional distribution of the Ly$\alpha$ forest has been extensively used to constrain cosmology through measurements of the baryon acoustic oscillations (BAO) scale. However, more cosmological information could be extracted from the full shapes of the Ly$\alpha$ forest correlations through the Alcock-Paczy\'nski (AP) effect. In this work, we prepare for a cosmological analysis of the full shape of the Ly$\alpha$ forest correlations by studying synthetic data of the extended Baryon Oscillation Spectroscopic Survey (eBOSS). We use a set of one hundred eBOSS synthetic data sets in order to validate such an analysis. These mocks undergo the same analysis process as the real data. We perform a full-shape analysis on the mean of the correlation functions measured from the one hundred eBOSS realizations, and find that our model of the Ly$\alpha$ correlations performs well on current data sets. We show that we are able to obtain an unbiased full-shape measurement of $D_M/D_H(z_\mathrm{eff})$, where $D_M$ is the transverse comoving distance, $D_H$ is the Hubble distance, and $z_\mathrm{eff}$ is the effective redshift of the measurement. We test the fit over a range of scales, and decide to use a minimum separation of $r_\mathrm{min}=25\ h^{-1}\text{Mpc}$. We also study and discuss the impact of the main contaminants affecting Ly$\alpha$ forest correlations, and give recommendations on how to perform such analysis with real data. While the final eBOSS Ly$\alpha$ BAO analysis measured $D_M/D_H(z_\mathrm{eff}=2.33)$ with $4\%$ statistical precision, a full-shape fit of the same correlations could provide a $\sim2\%$ measurement.