Redshift distances in modified flat Friedmann-Lemaître-Robertson-Walker spacetime containing g00(t)
Authors: Haase Steffen 
Published: December 2023

In the present paper we use a modified flat Friedmann-Lemaitre-Robertson-Walker metric containing g_00(t) 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 flat Friedmann-Lemaitre-Robertson-Walker metric 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 an origin of coordinates being on r = 0.
Nobody can doubt this real travel way of light: The photons are emitted on a 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 some different redshift distances 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 ≈ 65.2 km/(s Mpc) as a main result. This value is a little smaller than the Hubble parameter H_0,Planck ≈ 67.66 km/(s Mpc) resulting from Planck 2018 data.
Furthermore, we find for the radius of the so-called Friedmann sphere R_0a ≈ 2,586.94 Mpc. This radius is not the maximum possible distance of seeing within an expanding universe. Photons, which were emitted at this distance, are not infinite red shifted.
The today’s mass density of the Friedmann sphere results in ρ_0m ≈ 9.09 x E-30 g/cm3. For the mass of the Friedmann sphere we get M_Fs ≈ 1.94 x E+55 g.
The mass of black hole within the galaxy M87 has the value M_BH,M87 ≈ 1.56 x E+43 g. The redshift distance of this object is D ≈ 19.60 Mpc but its today’s distance is only D_0 ≈ 12.27 Mpc. The radius of this black hole is R_S ≈ 1.498 x E-3 pc.

Key words:	relativistic astrophysics, theoretical and observational cosmology, redshift, Hubble parameter, quasar, galaxy, M87, SN Ia, black hole, flat universe, Friedmann-Lemaitre-Robertson-Walker metric, general relativity

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Abstract

In the present paper we use a modified flat Friedmann-Lemaitre-Robertson-Walker metric containing g00(t) 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 flat Friedmann-Lemaitre-Robertson-Walker metric 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 re (the subscript e indicates emission) describing the place of a galaxy which is emitting photons and ra (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(t0)ra - a(te)re ≡ R0a - Ree. Here means a(t0) the today’s (t0) scale parameter and a(te) the scale parameter at the time te 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 a co-moving coordinate place re and are than traveling to the co-moving coordinate place ra. During this traveling the time is moving from te to t0 (te ≤ t0) and therefore the scale parameter is changing in the meantime from a(te) to a(t0).
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 H0a ≈ 65.2 km/(s Mpc) as a main result. This value is a little smaller than the Hubble parameter H0,Planck ≈ 67.66 km/(s Mpc) resulting from Planck 2018 data.
Furthermore, we find for the radius of the so-called Friedmann sphere R0a ≈ 2,586.94 Mpc. This radius is not the maximum possible distance of seeing within an expanding universe. Photons, which were emitted at this distance, are not infinite red shifted.
The today’s mass density of the Friedmann sphere results in ρ0m ≈ 9.09 x 10-30 g/cm3. For the mass of the Friedmann sphere we get MFs ≈ 1.94 x 1055 g.
The mass of black hole within the galaxy M87 has the value MBH,M87 ≈ 1.56 x 1043 g. The redshift distance of this object is D ≈ 19.60 Mpc but its today’s distance is only D0 ≈ 12.27 Mpc. The radius of this black hole is RS ≈ 1.498 x 10-3 pc.