Measurement of the small-scale 3D Lyman-$\alpha$ forest power spectrum

Small-scale correlations measured in the Lyman-$\alpha$ (Ly$\alpha$) forest encode information about the intergalactic medium and the primordial matter power spectrum. In this article, we present and implement a simple method to measure the 3-dimensional power spectrum, $P_{\rm 3D}$, of the Ly$\alpha$ forest at wavenumbers $k$ corresponding to small, $\sim$ Mpc scales. In order to estimate $P_{\rm 3D}$ from sparsely and unevenly distributed data samples, we rely on averaging 1-dimensional Fourier Transforms, as previously carried out to estimate the 1-dimensional power spectrum of the Ly$\alpha$ forest, $P_{\rm 1D}$. This methodology exhibits a very low computational cost. We confirm the validity of this approach through its application to Nyx cosmological hydrodynamical simulations. Subsequently, we apply our method to the eBOSS DR16 Ly$\alpha$ forest sample, providing as a proof of principle, a first $P_{\rm 3D}$ measurement averaged over two redshift bins $z=2.2$ and $z=2.4$. This work highlights the potential for forthcoming $P_{\rm 3D}$ measurements, from upcoming large spectroscopic surveys, to untangle degeneracies in the cosmological interpretation of $P_{\rm 1D}$.