A new analytical approximation of luminosity distance by optimal HPM-Padé technique
Introduction
Recent astronomical observations clearly indicate that the universe is currently expanding with an increasing speed, and is spatially flat and vacuum dominated [1], [2]. In order to explain this mysterious phenomenon, many cosmological models [3] have been proposed. Since the relation between cosmological distances and redshift depends on the parameters of underlying cosmological models, the accurate and efficient analytical computation of these cosmological distances becomes an important issue for the comparison of different cosmological models with observation data in modern precision cosmology.
As is well known to all, in the general lambda cold dark matter (CDM), luminosity distance which is the most important distance from an observational point of view, can only be expressed in the term of integrals over the cosmological redshift, and computation pressure of the integral of luminosity distance is usual very large. Therefore, for the purpose of avoiding numerical quadrature, several analytical approaches [4], [5], [6], [7], [8], [9] to approximate luminosity distance in flat universe have been proposed for decades. Among these methods, one of the most widely used approaches is to apply Taylor expansion to approximate luminosity distance [10], [11]. Obviously, Taylor polynomial expanding at may have divergence problem caused by cosmological observations that exceed the limits of it. In fact, the supernova data that we obtain now is at least back to data available [12]. Thus several works suggested [11], [13], [14] that Padé rational polynomial has the ability to approximate luminosity distance, due to its good convergence property in a relatively larger redshift range.
In addition, by solving the differential equation of luminosity distance, Shchigolev and Yu obtained two formulae for approximating luminosity distance with smaller error over a relatively small redshift interval based on homotopy perturbation method (HPM) [9] (hereafter Shch17) and optimal homotopy perturbation method (OHPM) [15], respectively. The HPM was first put forward by He [16] to solve nonlinear differential equations, which yields a very accurate solution via one or two iterations. After that various modifications of HPM [17], [18], [19] were given by various investigators, such as the OHPM coupled with the least squares method [19], [20], optimal homotopy asymptotic method [21], and so forth. In a word, we can obtain more accurate approximations for luminosity distance over a relatively small redshift interval, based on the use of HPM technique (or modifications of HPM). Thus, to reach a compromise between accuracy and redshift convergence interval, the combination of padé approximant and HPM are therefore adequate candidates to carry out this goal. In fact, homotopy perturbation method-Padé technique (HPM-Padé) has been recognized as a good one to apply the series solution to improve the accuracy and enlarge the convergence interval [22], [23] in the study of nonlinear differential equations.
Therefore, in this paper, we will apply HPM-Padé to obtain a more accurate approximate analytical expression for luminosity distance in a relatively larger redshift range, based on solving the differential equation of luminosity distance in a flat universe. The rest of this paper is as follows. In Section 2, we briefly review the differential equation of luminosity distance in a spatially flat universe. The HPM-Padé rational approximation of luminosity distance is given in Section 3. In Section 4, comparison of our rational approximation polynomial for computing luminosity distance is made with the results obtained other existing methods. Then we confront the analytical approximate expression of luminosity distance that was obtained by HPM-Padé technique with the observational data, for the purpose of checking whether it works well. Note that Markov Chain Monte Carlo (MCMC) code emcee [24] is used in the data fitting. Finally, some brief conclusions are given in Section 5.
Section snippets
Differential equation of luminosity distance in a flat universe
The general expression of theoretical modulus in a flat universe is defined as follows [1], [2] where is the luminosity distance. In order to verify theoretical calculation, we define the luminosity distance of SNe Ia in a flat CDM universe as follows [25] where and are the energy densities corresponding to matter and cosmological constant, respectively: , is the speed of light, is cosmology redshift, and is the
Solution of homotopy perturbation method
For the homotopy perturbation technique has already become standard and concise, its basic idea can be referred to [16], [17]. In order to solve Eq. (9) by homotopy perturbation technique, we build the homotopy as follows Let us assume that the solution of Eq. (10) in the form of a series in : Substituting Eq. (11) into Eq. (10), and collecting coefficients with the same power of , we get a set of differential equations:
Performance of the HPM-Padé rational approximation
In this section, the performance of HPM-Padé rational approximation in Section 3 is assessed, which includes two steps. Firstly, comparison of our proposed approach for computing CDM model luminosity distance is made with the results obtained by other methods. Then we confront the analytical expression of luminosity distance that was obtained by the HPM-Padé technique with the observational data, for the purpose of checking whether it works well.
Conclusion and discussion
In this paper, based on use of homotopy perturbation method-Padé (HPM-Padé) technique, a new analytical approximation of luminosity distance in the flat universe is proposed. The numerical results clearly indicate that HPM-Padé technique has obvious advantages in improving accuracy of approximating luminosity distance over relative larger cosmological redshift range interval. It is worthy noting that the choice of the orders () for luminosity distance rational polynomial is very important,
CRediT authorship contribution statement
Bo Yu: Writing - original draft, Methodology, Visualization. Jian-Chen Zhang: Writing - review & editing, Data curation. Tong-Jie Zhang: Conceptualization, Methodology, Supervision, Visualization. Tingting Zhang: Editing, Data curation, Methodology.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank Anonymous Referees for their valuable comments for revising and improving earlier draft of our manuscript. We are grateful to Prof. Jin-Yu He for his kind help. We are grateful to Kang Jiao for useful discussions. This work was supported by National Key R&D Program of China (2017YFA0402600), and National Science Foundation of China (Grants No. 11929301, 61802428,11573006).
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