A New Analytic Approximation of Luminosity Distance in Cosmology Using the Parker–Sochacki Method

Abstract

The luminosity distance dL is possibly the most important distance scale in cosmology and therefore accurate and efficient methods for its computation is paramount in modern precision cosmology. Yet in most cosmological models the luminosity distance cannot be expressed by a simple analytic function in terms of the redshift z and the cosmological parameters, and is instead represented in terms of an integral. Although one can revert to numerical integration techniques utilizing quadrature algorithms to evaluate such an integral, the high accuracy required in modern cosmology makes this a computationally demanding process. In this paper, we use the Parker–Sochacki method (PSM) to generate a series approximate solution for the luminosity distance in spatially flat ΛCDM cosmology by solving a polynomial system of nonlinear differential equations. When compared with other techniques proposed recently, which are mainly based on the Padé approximant, the expression for the luminosity distance obtained via the PSM leads to a significant improvement in the accuracy in the redshift range 0≤z≤2.5. Moreover, we show that this technique can be easily applied to other more complicated cosmological models, and its multistage approach can be used to generate analytic approximations that are valid on a wider redshift range.