\label{h_function} In standard cosmology is assumed that the isotropic away of galaxies is influenced by the gravitational force and if \(v_{\text{rec}} = v_{\text{exp}}\), than the Mass-Energy can decelerate the expansion of the Universe.
See instead the universe expansion as increasing space it is equivalent to reaffirm that the expansion is the expression of a particular property of the space (indicated with \(P_{exp}\)) distinguished by gravity. We remind that gravity is understood as curvature of spacetime, property denoted by us with \(P_{grv}\).
This distinction denies that gravity (property \(P_{grv}\)) can influence the expansion (property \(P_{exp}\)) and vice versa.
Thus it can be argued that the velocity of a galaxy \(v_{gal}\), revealed experimentally, consists of a \(v_{fall}\), speed of gravitational fall (property \(P_{grv}\)) and a recession velocity \(v_{exp}\) for expansion (property \(P_{exp}\)): \[\begin{aligned}
\vec v_{gal} = \vec v_{exp} + \vec v_{fall} \label{galaxyspeed}\\
v_{gal} = v_{exp} + v_{grv} \nonumber
\end{aligned}\] The Hubble law will be annotated with: \[v_{gal} = H_U d_{gal}\] where \(H_U\) is the Hubble’s function in case of gravitation.
This is different from: \[v_{exp} = H_H d_{gal}\] Where \(H_H\) is the Hubble’s constant (in space) without gravity influence, being a direct expression of the property \(P_{exp}\). A similar situation is found in the Milne cosmological model \cite{milne1935} corresponding to a universe without matter and with open metric, \(k < 0\). Nevertheless in our model a universe without mass could never exist.
The Einstein-de Sitter universe \cite{britannica_einstein2014}, with a flat metric, is different from the Milne model because the recession velocity coincides with the escape velocity, \(v_{rec} = v_{esc} = v_{exp}\): this relation corresponds to the condition of a space whose curvature is zero, \(k = 0\), and the expansion is described by the function \(H_0 \not= H_H\).
Distinguishing between different forms of the Hubble law, \(H_U, H_0, H_H\), we expect (using a correct distance indicator, you see the (Ia) supernovae) the different observational results (for the two forms) concerning the galactic red shift.
Two galaxies \(S_U\) and \(S_H\) initially placed at an equal distance from us \(S_0\) after a certain time would be on different distance \(d_U \not= d_H\).
If it is present gravity (case with curvature \(k > 0\)) the recession will be described by a not linear Hubble function \(H_{U^-}\), (see the figure \ref{fig_hubble_law}) finding \(d_{U^-} < d_H\) (see the red curve in figure \ref{fig_hubble_law}).
If, always in presence of gravity, \(v_{rec} > v_{esc}\) with curvature \(k < 0\), thus we have \(H_{U^+}\) and we find \(d_{U^+} > d_H\) (see the blue curve in figure \ref{fig_hubble_law}).
Let us assume that \[v_{gal} = H_U (t) d_{gal} \Rightarrow \left \lbrace
\begin{array}{cc}
v_{gal} = & H_{U^-} (t) d_{gal}\\
v_{gal} = & H_{U^0} (t) d_{gal}\\
v_{gal} = & H_{U^+} (t) d_{gal}
\end{array} \right.\] Then we can suppose that the correct form of the Hubble’s law, with \(H_{U^-}, H_{U^\circ}, H_{U^+}\), is a combination between the gravity (\(P_{grv}\)) and the standard Hubble’s law (\(P_{esp}\)), with \(H = H_H\).
Note that the three cases \(H_{U^-}, H_{U^\circ}, H_{U^+}\), correspond to the three Friedmann solutions in general relativity, that always include the gravity.
Writing then \[\begin{aligned}
(\vec v_{all})_k = (\vec v_{exp} + \vec v_{grv})_k\\
(v_{all})_k = (v_{exp} - v_{grv})_k \nonumber
\end{aligned}\] where \((v_{all})_{k=1} < (v_{all})_{k=0} < (v_{all})_{k=-1}\).
Following the Friedmann’s equation \cite{friedmann1922,friedmann1924}, we can obtain the function \(H_U (t)\): \[H_U^2 (t) - \frac{8 \pi}{3} G \rho_U (t) = - \frac{k c^2}{R^2 (t)} \Rightarrow H_U^2 (t) = \frac{8 \pi}{3} G \rho_U (t) - \frac{k c^2}{R^2 (t)}\] Remembering that, \(\forall k\): \[\label{eqgravity}
g = - \frac{4 \pi G R}{3} \left ( \rho + \frac{3P}{c^2} \right )\] in the case of inconsiderable pressure (\(P_{gas} \approx 0\)), then \[\label{hubblelaweq}
H_U^2 (t) = \frac{2 g_{grav}}{R(t)} - \frac{k c^2}{R^2 (t)} \Rightarrow H_U(t) = \frac{\sqrt{2 g_{grav} R(t) - kc^2}}{R(t)}\] If \(k = 0\), \(H_U = H_0\), it follows from (\ref{eqgravity}) and (\ref{hubblelaweq}): \[g (U_c) = \frac{4 \pi G R \rho_c}{3} = \frac{4 \pi G R}{3} \frac{3 H_0^2}{8 \pi G} = \frac{R (t) H_0^2}{2}\] Thus obtaining an identity. We get here that even in a flat space the gravity has its influence. It follows then that: \[ H_U (t) = \sqrt{H_0^2 - \frac{kc^2}{R^2 (t)}}\] where \(H_{U^+} (t) > H_0\) for \(k<0\), \(H_U (t) = H_0\) for \(k=0\), \(H_{U^-} (t) < H_0\) for \(k>0\). The three curves in figure \ref{fig_hubble_law} correspond to the three different values.
However, the experimental data seem to admit a \(v^*_{rec}\) greater than that of an open universe \(v^*_{rec} > (v_{rec})_{k<1}\). In particular, it is as if there was an extra speed \(v_{ext}\) obtained by an acceleration opposite to that of gravity \(g\) (see figure \ref{fig_hubble_law_scp}).
In this specific case we can rewrite the (\ref{galaxyspeed}) like this: \[\begin{aligned}
\vec v_{gal} = \vec v_{exp} + \vec v_{ext} + \vec v_{fall}\\
v_{gal} = v_{exp} + v_{ext} + v_{grv} \nonumber
\end{aligned}\] In standard cosmology, it is believed that the \(v_{ext}\) is consequence of an action of repulsion generated by a form of dark energy (or by the energy of quantum vacuum), that accelerate the galaxies.
Instead, if the expansion is an effect of increasing mass-space then we could think that \(v_{exp} + v_{ext}\) is a consequence of a metric variable which in the past has been open (curvature negative) but with a accelerated growth rate caused by the presence of an additional pressure that can only result from increasing mass-space (as will be shown later).