Smearing scale in Laguerre reconstructions of the correlation function

To a good approximation, on large scales, the evolved two-point correlation function of biased tracers is related to the initial one by a convolution with a smearing kernel. For Gaussian initial conditions, the smearing kernel is Gaussian, so if the initial correlation function is parametrized using simple polynomials, then the evolved correlation function is a sum of generalized Laguerre functions of half-integer order. This motivates an analytic "Laguerre reconstruction" algorithm which previous work has shown is fast and accurate. This reconstruction requires as input the width of the smearing kernel. We show that the method can be extended to estimate the width of the smearing kernel from the same dataset. This estimate, and associated uncertainties, can then be used to marginalize over the distribution of reconstructed shapes and hence provide error estimates on the value of the distance scale. This procedure is not tied to a particular cosmological model. We also show that if, instead, we parametrize the evolved correlation function using simple polynomials, then the initial one is a sum of Hermite polynomials, again enabling fast and accurate deconvolution. If one is willing to use constraints on the smearing scale from other datasets, then marginalizing over its value is simpler for this latter, "Hermite" reconstruction, potentially providing further speed-ups in cosmological analyses.