  We propose an efficient analytical method for estimating the luminosity
distance in a homogenous Friedmann-Lema\^itre-Robertson-Walker (FLRW) model of
the Universe. This method is based on the homotopy perturbation method (HPM),
which has high accuracy in many nonlinear problems, and can be easily
implemented. For analytical calculation of the luminosity distance, we offer to
proceed not from the computation of the integral, which determines it, but from
the solution of a certain differential equation with corresponding initial
conditions. Solving this equation by means of HPM, we obtain the approximate
analytical expressions for the luminosity distance as a function of redshift
for two different types of homotopy. Possible extension of this method to other
cosmological models is also discussed.

General Relativity and Quantum Cosmology;Computational Physics