Ly-${\mathsf \alpha}$ forest: efficient unbiased estimation of second-order properties with missing data

Context. One important step in the statistical analysis of the Ly- α forest data is the study of their second order properties. Usually, this is accomplished by means of the two-point correlation function or, alternatively, the K -function. In the computation of these functions it is necessary to take into account the presence of strong metal line complexes and strong Ly- α lines that can hidden part of the Ly- α forest and represent a non negligible source of bias. Aims. In this work, we show quantitatively what are the effects of the gaps introduced in the spectrum by the strong lines if they are not properly accounted for in the computation of the correlation properties. Methods. We propose a geometric method which is able to solve this problem and is computationally more efficient than the Monte Carlo (MC) technique that is typically adopted in Cosmology studies. The method is implemented in two different algorithms. The first one permits to obtain exact results, whereas the second one provides approximated results but is computationally very efficient. The proposed approach can be easily extended to deal with the case of two or more lists of lines that have to be analyzed at the same time. Numerical experiments are presented that illustrate the consequences to neglect the effects due to the strong lines and the excellent performances of the proposed approach. Results. The proposed method is able to remarkably improve the estimates of both the two-point correlation function and the K -function.