Redshift distances in curved Friedmann-Lemaître-Robertson-Walker spacetime
Authors: Haase Steffen 
Published: June 2023

In the present paper we use the curved Friedmann-Lemaitre-Robertson-Walker metric describing a spatially homogeneous and isotropic universe to derive the cosmological redshift distance in a way which differs from that which can be found in the general astrophysical literature.
Using the curved Friedmann-Lemaitre-Robertson-Walker metric the radial physical distance is described by R(t) = a(t)χ(r) with χ(r) = arcsin(r) for the curvature parameter ε = (+1) and χ(r) = arsinh(r) for ε = (-1), respectively. In this equation the radial co-moving coordinate is named r and a(t) means the time-depending scale parameter. We use the co-moving coordinate r_e (the subscript e indicates emission) describing the place of a galaxy which is emitting photons and r_a (the subscript a indicates absorption) describing the place of an observer within a different galaxy on which the photons - which were traveling thru the universe - are absorbed. Therefore the physical distance - the real way of light - is calculated by D = a(t_0)χ(r_a) - a(t_e)χ(r_e) ≡ R_0a - R_ee. Here means a(t_0) the today’s (t_0) scale parameter and a(t_e) the scale parameter at the time t_e of emission of the photons. The physical distance D is therefore a difference of two different physical distances from an origin of coordinates being on r = 0.
Nobody can doubt this real travel way of light: The photons are emitted on the co-moving coordinate place r_e and are than traveling to the co-moving coordinate place r_a. During this traveling the time is moving from t_e to t_0 (t_e ≤ t_0) and therefore the scale parameter is changing in the meantime from a(t_e) to a(t_0).
Using this right physical distance D we calculate the redshift distance and some relevant classical cosmological equations (effects) for both possible values of ε = (±1) and compare these theoretical results with some measurements of astrophysics (quasars, SN Ia and galaxy containing a black hole).
We get the today’s Hubble parameter H_0a,ε=(+1) ≈ 65.117 km/(s Mpc) for ε = (+1) and H_0a,ε=(-1) ≈ 65.189 km/(s Mpc) for ε = (-1), respectively, as a main result. This values are a little smaller than the Hubble parameter H_0,Planck ≈ 67.66 km/(s Mpc) resulting from Planck data 2018.
Furthermore, we find for the radius of the by us so-called Friedmann sphere R_0a,ε=(+1) ≈ 2,697.62 Mpc and R_0a,ε=(-1) ≈ 3,011.07 Mpc. This radius corresponds to a maximum possible distance of seeing within an expanding universe. Photons emitted at this distance are infinite red shifted.
The today’s mass density of the Friedmann sphere results in ρ_0m,ε=(+1) ≈ 1.037 x 10^-30 g/cm3 and ρ_0m,ε=(-1) ≈ 9.24 x 10^-32 g/cm3, respectively. For the mass of the Friedmann sphere we get M_Fs,ε=(+1) ≈ 2.506 x 10^54 g and M_Fs,ε=(-1) ≈ 3.10 x 10^53 g, respectively.
The mass of black hole within the galaxy M87 has the value M_BH,M87,ε=(+1) ≈ 4.1469 x 10^43g and M_BH,M87,ε=(-1) ≈ 4.1468 x 10^43 g, respectively. The redshift distance of this object is D_ε=(+1) ≈ 19.60 Mpc and D_ε=(-1) ≈ 19.60 Mpc, respectively, but its today’s distance is only D_0,ε=(+1) ≈ 8.13 Mpc and D_0,ε=(-1) ≈ 6.78 Mpc, respectively.