Reconstructing the Long-wavelength Matter Density Fluctuation Modes from the Scalar-Type Clustering Fossils

Revealing the large-scale structure from the 21cm intensity mapping surveys is only possible after the foreground cleaning. However, most current cleaning techniques relying on the smoothness of the foreground spectrum lead to a severe side effect of removing the large-scale structure signal along the line of sight. On the other hand, the clustering fossil, a coherent variation of the small-scale clustering over large scales, allows us to recover the long-wavelength density modes from the off-diagonal correlation between short-wavelength modes. In this paper, we study the requirements for an unbiased and optimal clustering-fossil estimator and show that (A) the estimator is unbiased only when using an accurate bispectrum model for the long-short-short mode coupling and (B) including the connected four-point correlation functions is essential for characterizing the noise power spectrum of the estimated long mode. The clustering fossil estimator based upon the leading-order bispectrum yields an unbiased estimation of the long-wavelength ($k\lesssim 0.01~[h/{\rm Mpc}]$) modes with the cross-correlation coefficient of $0.7$ at redshifts $z=0$ to $3$.