Abstract
In this paper, the iterative method suggested by Daftardar-Gejji and Jafari hereafter called Daftardar-Jafari method (DJM) is applied for the approximate analytical representation of the luminosity distance in a homogenous Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology. We obtain the analytical expressions of the luminosity distance using the approximate solutions of the differential equation to which the luminosity distance satisfies, subject to the corresponding initial conditions. With the help 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 for the general integral formula by the Maple software. Subsequent comparison of the obtained approximate analytical formula with the corresponding numerical solution for the and quintessential models is provided and showed a high accuracy of the DJM approximations, at least for the certain values of parameters of the models. This comparison demonstrates the efficiency and simplicity of this approach to the problem of calculating the luminosity distance in theoretical cosmology.
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
The theoretical distance modulus is given by μ(z) = 5 log10(dL/M pc) + 25, where dL is the luminosity distance
The analytical calculation of the luminosity distance dL versus cosmological redshift z seems to be a very important issue in theoretical cosmology
Summary
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. It would be useful to note a new approach to perturbation method, such as the optimal perturbation iteration method (OPIM) for solving the nonlinear differential equations (see, for example, [24]-[26] and references therein ) The results of these papers reveal that the new approximate solutions obtained via OPIM applied to several problems are even more accurate and impressive than many other techniques in literature. 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 [13] Solving this equation in a spatially flat FLRW universe by means of DJM, we obtain the approximate analytical expressions for the luminosity distance in terms of redshift. For more details about the convergence of the DJM, we refer the reader to Ref. [21]
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