The smearing scale in Laguerre reconstructions of the correlation function

TL;DR

This paper introduces an analytic Laguerre reconstruction method for the correlation function in cosmology, which can estimate the smearing kernel width and improve error estimates without relying on specific cosmological models.

Contribution

The work extends Laguerre reconstruction to estimate the smearing kernel width from data and compares it with Hermite parametrization, enhancing the method's flexibility and speed.

Findings

The method accurately estimates the smearing kernel width from data.

Laguerre and Hermite parametrizations enable fast deconvolution of the correlation function.

Marginalizing over the smearing scale improves error estimates on cosmological distances.

Abstract

To a good approximation, on large cosmological scales the evolved two-point correlation function of biased tracers is related to the initial one by a convolution. 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 which are 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 Hermite reconstruction, potentially providing further speed-up in cosmological analyses.