arXiv:1609.04612v1 [astro-ph.CO] 15 Sep 2016 uioiydsac nfltcosmology flat in distance luminosity Keywords pta itiuino galaxies of distribution spatial naayia ouini h ope ln o the for plane complex the in solution analytical An omlg;Osrainlcsooy itne,rdhfs ra redshifts, Distances, cosmology; Observational Cosmology; : ftelmnst itnealw nigtetoparameters two the finding part imaginary allows negligible distance a and luminosity part the cons real a of cosmological of the made and is solution matter analytical pressureless with cosmology flat Abstract. E-mail: Turin,Italy 1, I-10125 P.Giuria via , Department Physics Zaninetti Lorenzo xrsinfrtedsac ouu o N ftp ai reported is Ia type of approximation. SNs minimax for the modulus distance the for expression [email protected] epeeta nltclslto o h uioiydsac nspatia in distance luminosity the for solution analytical an present We H 0 ntefaeokof framework the in n Ω and at h complex The tant. ilvelocities, dial h elpart real The . M simple A . lly An analytical solution in the complex plane for the luminosity distance in flat cosmology 2
1. Introduction
The luminosity distance in flat cosmology has been recently investigated using different approaches. A fitting formula which has a maximum relative error of 4% in the case of common cosmological parameters has been introduced by [1]. An approximate solution in terms of Pad´eapproximants has been presented by [2]. The integral of the luminosity distance has been found in terms of elliptical integrals of the first kind by [3].
2. Flat cosmology
Following Eq. (2.1) in [2], the luminosity distance dL is c 1 da L 0 M d (z; c,H , Ω )= (1 + z) 4 , (1) H0 1 ΩMa + (1 ΩM)a Z 1+z − −1 −1 where H0 is the Hubble constant expressed inp km s Mpc , c is the speed of light −1 expressed in km s , z is the redshift, a is the scale-factor, and ΩM is 8π G ρ0 ΩM = 2 , (2) 3 H0
where G is the Newtonian gravitational constant and ρ0 is the mass density at the present time. We now introduce the indefinite integral da Φ(a)= 4 . (3) ΩMa + (1 ΩM)a Z − The solution is in terms ofpF , the Legendre integral or incomplete elliptic integral of the first kind 4 F (b1, b2) b3b4b6b1b5 Φ(a)= − , (4) b7b8√b9b10 where the incomplete elliptic integral of the first kind is x dt F (x, k)= 2 2 2 , (5) 0 √1 t √1 k t Z − − see formula (19.2.4) in [4], and
a (ΩM 1) i√3+3 b1 = − , (6) v− 3 2 u ΩM a + ΩM (ΩM 1) + a i√3+1 u − − u t q i√3+1 i√3 3 b2 = − , (7) s i√3+3 i√3 1 −