Analytic Approximation of Luminosity Distance in Cosmology via Variational Iteration Method

V. K. Shchigolev

doi:10.13189/ujcmj.2017.050303

Abstract

This work deals with a new approach in the approximate analytical representation of the luminosity distance in a homogenous Friedmann-Lemaitre-Robertson-Walker (FLRW) model of the Universes by means of the variational iteration method (VIM). For the analytical calculation of the luminosity distance, we obtain the approximate solution of the differential equation which the luminosity distance obeys, using the corresponding initial conditions. On the basis of this approximate solution, a simple analytic formula for the luminosity distance as a function of redshift is obtained and compared with a numerical solution from the general integral formula.

Highlights

The database, obtained in the observational astronomy on Supernovae of type Ia as one of the best cosmological distance indicators [1]-[3], encourages theoretical cosmologists to a strong restriction of the essential parameters in their cosmological models
In order to test any cosmological models with the help of the SNIa Union2 database, we have to use the maximumlikelihood approach minimizing χ2 which measures the deviations of the theoretical predictions from the observations
The theoretical distance modulus is given by μ(z) = 5 log10(dL/M pc) + 25, where dL is the luminosity distance depending on the cosmological redshift z and the model parameters

Summary

Introduction

The database, obtained in the observational astronomy on Supernovae of type Ia as one of the best cosmological distance indicators [1]-[3], encourages theoretical cosmologists to a strong restriction of the essential parameters in their cosmological models. An analytical approximation of the luminosity distance as a function of the redshift z can be a rather useful in cosmological modeling. We use the idea of approximate analytical calculation of the luminosity distance by virtue of solving the corresponding differential equation with certain initial conditions, proposed in [11]. Solving this equation in a spatially flat FLRW universe by means of VIM, we obtain the approximate analytical expressions for the luminosity distance in terms of redshift for the different content of cosmological models. We show that by using the VIM, the expression for dL(z) in arbitrary accuracy can be obtained by implementing a simple procedure for the governing equation

The main idea of VIM

Differential Equation for the Luminosity Distance

Luminosity Distance in VIM Approximation

Application Examples

Findings

Conclusions