Covariance matrices for the Lyman-$\alpha$ forest using the lognormal approximation

We investigate the nature of correlations in the small-scale flux statistics of the Lyman-$\alpha$ (Ly$\alpha$) forest across redshift bins. Understanding these correlations is important for unbiased cosmological and astrophysical parameter inference using the Ly$\alpha$ forest. We focus on the 1-dimensional flux power spectrum (FPS) and mean flux ($\bar F$) simulated using the semi-numerical lognormal model we developed in earlier work. The lognormal model can capture the effects of long wavelength modes with relative ease as compared to full smoothed particle hydrodynamical (SPH) simulations that are limited by box volume. For a single redshift bin of size $\Delta z\simeq 0.1$, we show that the lognormal model predicts positive cross-correlations between $k$-bins in the FPS, and a negative correlation for $\bar F\times$ FPS, in qualitative agreement with SPH simulations and theoretical expectations. For measurements across two neighbouring redshift bins of width $\Delta z$ each (obtained by 'splitting' skewers of length $2\Delta z$ in half), the lognormal model predicts an anti-correlation for FPS $\times$ FPS and a positive correlation for $\bar F\times$ FPS, caused by long wavelength modes. This is in contrast to SPH simulations which predict a negligible magnitude for cross-redshift correlations derived from such `split' skewers, and we discuss possible reasons for this difference. Finally, we perform a preliminary test of the impact of neglecting long wavelength modes on parameter inference, finding that whereas the correlation structure of neighbouring redshift bins has relatively little impact, the absence of long wavelength modes in the model can lead to $\gtrsim2-\sigma$ biases in the inference of astrophysical parameters. Our results motivate a more careful treatment of long wavelength modes in analyses that rely on the small scale Ly$\alpha$ forest for parameter inference.