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\section*{Conclusions}   Please state what you think are \subsubsection{The 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 main conclusions space (indicated with $P_{exp}$) distinguished by gravity. We remind  that gravity is understood as curvature of Space-Time, 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 realistically drawn argued that the velocity of a galaxy $v_{gal}$, revealed experimentally, consists of a $v_{fall}$, \emph{speed of gravitational fall} (property $P_{grv}$) and a recession velocity $v_{exp}$ for expansion (property $P_{exp}$):  \begin{align}  \vec v_{gal} = \vec v_{exp} + \vec v_{fall} \label{galaxyspeed}\\  v_{gal} = v_{exp} + v_{grv} \nonumber  \end{align}  The Hubble law will be annotated with:  \begin{equation}  v_{gal} = H_U d_{gal}  \end{equation}  This is different from:  \begin{equation}  v_{exp} = H_H d_{gal}  \end{equation}  Where $H_H$ is the Hubble’s constant (in space) without \emph{gravity} influence, being a direct expression of the property $P_{exp}$. A similar situation is found in the Milne cosmological model\footnote{ref} 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\footnote{ref}, with a flat metric, is different  from the findings 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 galaxy 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 $d_U \not= d_H$.\\  In an expanding universe (with law of increasing mass-space $P_{exp}$) the relative velocity between two points  in space (eg. galaxies $S_0$ and $S$) increases to spend proper time of each galaxy.\\  If it is present gravity (case with curvature $k > 0$)  the paper, taking care recession will be described by a  not linear Hubble function $H_{U^-}$, you see the Fig. 4, finding $d_{U^-} < d_H$, you see in fig. 4 the red curve.\\  If, always in presence of gravity, $v_{rec} > v_{esc}$ with curvature $k < 0$, thus we have $H_{U^+}$ and we find $d_{U^+} > d_H$, you see in fig. 4 the blue curve.\\  (fig. 4)\\  Let us assume that  \begin{equation}  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.  \end{equation}  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 $H = H_H$.\\  In a graph\footnote{You see the Supernovae cosmology projcet} showing the magnitude variation as a function of the recession velocity or the parameter $z$ of red shift, we have (as from the literature) that\\  (fig. 5)\\  Or in logarithmic scale\\  (fig. 6)\\  Note that the three cases $H_{U^-}, H_{U^\circ}, H_{U^+}$, correspond  to make claims the three Friedmann solutions in GR,  that cannot always include the gravity.\\  Writing then  \begin{align}  (\vec v_{all})_k = (\vec v_{exp} + \vec v_{grv})_k\\  (v_{all})_k = (v_{exp} - v_{grv})_k \nonumber  \end{align}  where $(v_{all})_{k=1} < (v_{all})_{k=0} < (v_{all})_{k=-1}$.\\  Following the Friedmann's equation, we can obtain the function $H_U (t)$:  \begin{equation}  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)}   \end{equation}  Remembering that, $\forall k$:  \begin{equation}\label{eqgravity}  g = - \frac{4 \pi G R}{3} \left ( \rho + \frac{3P}{c^2} \right )  \end{equation}  in the case of inconsiderable pressure ($P_{gas} \tilde = 0$), then  \begin{equation}\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)}  \end{equation}  If, in (\ref{hubblelaweq}), $k = 0$:  \begin{equation}  H_U^2 (t) = \frac{\sqrt{2 g_{grav} R(t)}}{R(t)} = \sqrt{\frac{2 g_{grav}}{R(t)}} \Leftrightarrow g_{grav} (t) = \frac{H_U^2 (t) R(t)}{2}  \end{equation}  and, from (\ref{eqgravity}), with $H_U = H_0$, it follows:  \begin{equation}  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}  \end{equation}  Thus obtaining an identity. We get here that even in a flat space the gravity has its influence. It follows then that:  \begin{equation}  H_U^2 = \frac{\sqrt{2 g_{grav} R(t) - kc^2}}{R(t)} = \frac{\sqrt{2 \frac{R (t) H_0^2}{2} R(t) - kc^2}}{R(t)} \Rightarrow H_U (t) = \sqrt{H_0^2 - \frac{kc^2}{R^2 (t)}}  \end{equation}  where $H_{U^+} (t) > H_0$ for $k<0$, $H_U (t) = H_0$ for $k=0$, $H_{U^-} (t) < H_0$ for $k>0$.\\  Three curves correspond to the three different values, see Fig. 6\\  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 \emph{extra speed} $v_{ext}$ obtained by an acceleration opposite to that of gravity $g$ (see fig. 7, as from the Supernovae cosmology projcet).\\  (fig 7)\\  In this specific case we can rewrite the (\ref{galaxyspeed}) like this:  \begin{align}  \vec v_{gal} = \vec v_{exp} + \vec v_{ext} + \vec v_{fall}\\  v_{gal} = v_{exp} + v_{ext} + v_{grv} \nonumber  \end{align}  In standard cosmology, it is believed that the $v_{ext}$ is consequence of an action of repulsion generated by a form of \emph{dark} energy (or by the energy of empty), that accelerate the galaxies (allontamanento).\\  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 \emph{open} (curvature negative) but with a \emph{accelerated} growth rate caused by the presence of an additional pressure that can only result from increasing mass-space (as will  be supported. shown later).