Redshift distances in flat Friedmann-Lemaître-Robertson-Walker spacetime

Abstract

In the present paper we use the flat 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 Friedmann-Lemaitre-Robertson-Walker the radial physical distance is described by R(t) = a(t)r. In this equation the radial co-moving coordinate is named r and the time-depending scale parameter is named a(t). 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)r_a - a(t_e)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 a coordinate origin 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 we calculate the redshift distance and some relevant classical cosmological equations (effects) and compare these theoretical results with some measurements of astrophysics (quasars, SN Ia and black hole). We get the today’s Hubble parameter H_0a ≈ 65.66 km/(s Mpc) as a main result. This value is a little bit smaller than the Hubble parameter H_0,Planck ≈ 67.66 km/(s Mpc) resulting from Planck 2018 data which is discussed in the specialist literature. Furthermore, we find for the radius of the so-called Friedmann sphere R_0a ≈ 3,096.92 Mpc. This radius corresponds to the maximum possible distance of seeing within an expanding universe. Photons, which were emitted at this distance, are infinite red shifted. The today’s mass density of the Friedmann sphere results in ρ_0m ≈ 7.82 x 10-29 g/cm^3. For the mass of the Friedmann sphere we get M_Fs ≈ 2.86 x 10+56 g. The mass of black hole within the galaxy M87 has the value M_BH, M87 ≈ 2.36 x 10+45 g. The redshift distance of this object is D ≈ 19.45 Mpc but its today’s distance is only D_0 ≈ 6.27 Mpc.