Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?

To express the scale invariance of the empty space intervening through ${{\rm{\Lambda }}}_{{\rm{E}}}$ at large scales, we must consistently do it in a theoretical framework that permits scale invariance (but does not necessarily demand it!). GR does not offer this possibility; however, a framework—the cotensor analysis—that allows it has been worked out in detail by Weyl (1923), Eddington (1923), Dirac (1973), and Canuto et al. (1977) (this was often in the context of the studies on variable G, but this is not what we do here). Short summaries of cotensor analysis are given by Dirac (1973) and in the Appendix of Canuto et al. (1977); see also Bouvier & Maeder (1978). In addition to the general covariance of tensor analysis used in GR, cotensor analysis also admits the possibility of scale invariance of the form

$$\begin{eqnarray}&&{ds}^{\prime} =\lambda ({x}^{\mu })\,{ds}.\end{eqnarray} \tag{ 1 }$$

There, ${ds}{{\prime} }^{2}=g{{\prime} }_{\mu \nu }{dx}^{\prime} {}^{\mu }\,{dx}^{\prime} {}^{\nu }$ is the line element in the GR framework with coordinates $x^{\prime} {}^{\mu }$ and ${{ds}}^{2}={g}_{\mu \nu }{{dx}}^{\mu }\,{{dx}}^{\nu }$ is the line element in a new more general framework, where we examine scale invariance. Parameter $\lambda ({x}^{\mu })$ is the scale factor connecting the two line elements. If $\lambda ({x}^{\mu })=1$, the two frameworks are the same. In addition, we also make here a transformation of coordinates from $x^{\prime} {}^{\mu }$ to $x{}^{\mu }$, because we want to study simultaneously the effects of transformation of coordinates as in GR, together with the effects of a change of scale.

When the various steps of the development of cotensorial analysis are followed, a general scale-invariant field equation can be written (see Paper I). With respect to the usual field equation, it contains additional terms depending only on ${g}_{\mu \nu }$ and ${\kappa }_{\nu }$, where

$$\begin{eqnarray}&&{\kappa }_{\nu }=-\displaystyle \frac{\partial }{\partial {x}^{\nu }}\,\mathrm{ln}\lambda .\end{eqnarray} \tag{ 2 }$$

The term ${\kappa }_{\nu }$ is called the coefficient of metrical connection. It is as fundamental a quantity as are the ${g}_{\mu \nu }$ in GR. The field equation is written as (Canuto et al. 1977)

$$\begin{eqnarray}&&R{{\prime} }_{\mu \nu }-\displaystyle \frac{1}{2}\ {g}_{\mu \nu }R^{\prime} -{\kappa }_{\mu ;\nu }-{\kappa }_{\nu ;\mu }\\ &&\quad -\,2{\kappa }_{\mu }{\kappa }_{\nu }+2{g}_{\mu \nu }{\kappa }_{;\alpha }^{\alpha }-{g}_{\mu \nu }{\kappa }^{\alpha }{\kappa }_{\alpha }\\ &&\qquad =\,-\,8\pi {{GT}}_{\mu \nu }-{\lambda }^{2}{{\rm{\Lambda }}}_{{\rm{E}}}\,{g}_{\mu \nu }.\end{eqnarray} \tag{ 3 }$$

The terms with a prime are those of GR. The gravitational constant G is a true constant, and ${{\rm{\Lambda }}}_{{\rm{E}}}$ is the Einstein cosmological constant. A subscript ";" means a derivative. The application of the general field equation to the empty space has led in Paper I to some relations between the cosmological constant and the scale factor λ. The assumption is also made that the empty space is homogeneous and isotropic, which implies that scale factor λ is only a function of the cosmic time t. The 1, 2, 3 components (the three give the same result) and the 0 component of the above field equation become, respectively, for the empty space (Maeder & Bouvier 1979; Maeder 2017a)

$$\begin{eqnarray}&&2\,\displaystyle \frac{\dot{{\kappa }_{0}}}{c}-{\kappa }_{0}^{2}=-{\lambda }^{2}\,{{\rm{\Lambda }}}_{{\rm{E}}},\quad \mathrm{and}\quad 3\,{\kappa }_{0}^{2}={\lambda }^{2}\,{{\rm{\Lambda }}}_{{\rm{E}}}.\end{eqnarray} \tag{ 4 }$$

The addition of these two equations gives $\dot{{\kappa }_{0}}/c=-{\kappa }_{0}^{2}$, the solution of which is

$$\begin{eqnarray}&&{\kappa }_{0}=\displaystyle \frac{1}{c\,t}.\end{eqnarray} \tag{ 5 }$$

Here we keep the velocity of light c in the equations in order to write the weak-field equations with the appropriate units. From Equation (2), one also has ${\kappa }_{0}=-\tfrac{\dot{\lambda }}{c\,\lambda }$. This expression, together with Equation (4), leads to

$$\begin{eqnarray}&&\ 3\,\displaystyle \frac{{\dot{\lambda }}^{2}}{{c}^{2}\,{\lambda }^{2}}={\lambda }^{2}\,{{\rm{\Lambda }}}_{{\rm{E}}}\\ &&\quad \mathrm{and}\quad 2\,\displaystyle \frac{\ddot{\lambda }}{{c}^{2}\,\lambda }-\displaystyle \frac{{\dot{\lambda }}^{2}}{{c}^{2}\,{\lambda }^{2}}={\lambda }^{2}\,{{\rm{\Lambda }}}_{{\rm{E}}},\end{eqnarray} \tag{ 6 }$$

which give the fundamental relations between ${{\rm{\Lambda }}}_{{\rm{E}}}$ and the scale factor λ. We see that if ${{\rm{\Lambda }}}_{{\rm{E}}}=0$, the scale factor would be a constant, that is to say, the scale-invariant framework would be strictly identical to GR. The first of the above equations leads to $\lambda =A/t$, where A is a constant. Taking $\lambda =1$ at the present time t₀, one has

$$\begin{eqnarray}&&\lambda =\displaystyle \frac{{t}_{0}}{t}.\end{eqnarray} \tag{ 7 }$$

The origin of time t depends on the cosmological models. For example, the numerical models in Paper I show that for ${t}_{0}=1$ the origin lies at ${t}_{\mathrm{in}}=0.6694$ for a value of ${{\rm{\Omega }}}_{{\rm{m}}}=0.30$. This means that the variations of the scale factor λ are small, being limited to a change from 1.4938 at the big bang to 1.0 at the present time. For ${{\rm{\Omega }}}_{{\rm{m}}}=0$, one has ${t}_{\mathrm{in}}=0$ and the variations of λ would go from infinity to zero. These examples show that the presence of matter rapidly reduces the cosmological effects of scale invariance (see Feynman 1963).

To study the dynamics of systems, we need an equation of motion. For that, we may derive the geodesic equation in the scale-invariant framework in various ways (Maeder & Bouvier 1979). Let us do it in a straightforward way, starting from the equivalence principle as expressed by Weinberg (1972). At every point of the spacetime, there is a local inertial system $x{{\prime} }^{\alpha }$ such that the motion in GR may be described by

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}x{{\prime} }^{\alpha }}{{ds}{{\prime} }^{2}}=0.\end{eqnarray} \tag{ 8 }$$

Let us develop this expression in the new framework (defined by ds²),

$$\begin{eqnarray}&&\displaystyle \frac{d}{{ds}^{\prime} }\left(\displaystyle \frac{\partial x{{\prime} }^{\alpha }}{\partial {x}^{\mu }}\displaystyle \frac{{{dx}}^{\mu }}{{ds}^{\prime} }\right)=\displaystyle \frac{d}{\lambda \,{ds}}\left(\displaystyle \frac{\partial x{{\prime} }^{\alpha }}{\partial {x}^{\mu }}\displaystyle \frac{{{dx}}^{\mu }}{\lambda {ds}}\right)=0,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{x}^{\rho }}{{{ds}}^{2}}+\displaystyle \frac{{\partial }^{2}x{{\prime} }^{\alpha }}{\partial {x}^{\mu }\partial {x}^{\nu }}\displaystyle \frac{\partial {x}^{\rho }}{\partial x{{\prime} }^{\alpha }}\displaystyle \frac{{{dx}}^{\mu }}{{ds}}\displaystyle \frac{{{dx}}^{\nu }}{{ds}}+{\kappa }_{\nu }\displaystyle \frac{{{dx}}^{\rho }}{{ds}}\displaystyle \frac{{{dx}}^{\nu }}{{ds}}=0,\end{eqnarray} \tag{ 10 }$$

In cotensor analysis, scale-invariant derivatives of the first and second order have been developed preserving scale invariance. Other scale-invariant quantities are also defined; they are noted by an asterisk (Dirac 1973; Canuto et al. 1977). The modified form of the Christoffel symbol ${}^{* }{{\rm{\Gamma }}}_{\mu \,\nu }^{\rho }$ corresponds to the first two derivatives in the second term on the left-hand side of Equation (10),

$$\begin{eqnarray}&&{}^{* }{{\rm{\Gamma }}}_{\mu \,\nu }^{\rho }=\displaystyle \frac{{\partial }^{2}x{{\prime} }^{\alpha }}{\partial {x}^{\mu }\partial {x}^{\nu }}\displaystyle \frac{\partial {x}^{\rho }}{\partial x{{\prime} }^{\alpha }}.\end{eqnarray} \tag{ 11 }$$

With Equations (2) and (11), we may write the equation of motion in the scale-invariant framework,

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{x}^{\rho }}{{{ds}}^{2}}{+}^{* }{{\rm{\Gamma }}}_{\mu \,\nu }^{\rho }\displaystyle \frac{{{dx}}^{\mu }}{{ds}}\displaystyle \frac{{{dx}}^{\nu }}{{ds}}+{\kappa }_{\nu }\displaystyle \frac{{{dx}}^{\rho }}{{ds}}\displaystyle \frac{{{dx}}^{\nu }}{{ds}}=0,\end{eqnarray} \tag{ 12 }$$

The modified Christoffel symbol is also written as (see relation (A5) by Canuto et al. (1977), relation (3.2) by Dirac (1973), or relation (86.3) by Eddington (1923))

$$\begin{eqnarray}&&{}^{* }{{\rm{\Gamma }}}_{\mu \,\nu }^{\rho }={{\rm{\Gamma }}}_{\mu \,\nu }^{\rho }\,-{g}_{\mu }^{\rho }{\kappa }_{\nu }-{g}_{\nu }^{\rho }{\kappa }_{\mu }+{g}_{\mu \nu }\,{\kappa }^{\rho }.\end{eqnarray} \tag{ 13 }$$

Here, ${{\rm{\Gamma }}}_{\mu \nu }^{\rho }$ is the usual Christoffel symbol, and the term ${\kappa }_{\nu }$ is defined by Equation (2). Quite generally, as shown by the field equation, the scale-invariant terms are given by the corresponding usual terms in GR, plus or minus some functions of the ${g}_{\mu \nu }$ and ${\kappa }_{\nu }$. From relations (12) and (13), one has

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{x}^{\rho }}{{{ds}}^{2}}+({{\rm{\Gamma }}}_{\mu \,\nu }^{\rho }-{g}_{\mu }^{\rho }{\kappa }_{\nu }-{g}_{\nu }^{\rho }{\kappa }_{\mu }+{g}_{\mu \nu }\,{\kappa }^{\rho })\\ &&\quad \times \,\displaystyle \frac{{{dx}}^{\mu }}{{ds}}\displaystyle \frac{{{dx}}^{\nu }}{{ds}}+{\kappa }_{\nu }\displaystyle \frac{{{dx}}^{\rho }}{{ds}}\displaystyle \frac{{{dx}}^{\nu }}{{ds}}=0.\end{eqnarray} \tag{ 14 }$$

The third and the last terms simplify, and one is left with the following geodesic equation:

$$\begin{eqnarray}&&\displaystyle \frac{{{du}}^{\rho }}{{ds}}+{{\rm{\Gamma }}}_{\mu \nu }^{\rho }{u}^{\mu }{u}^{\nu }-{\kappa }_{\mu }{u}^{\mu }{u}^{\rho }+{\kappa }^{\rho }=0,\end{eqnarray} \tag{ 15 }$$

with the velocity ${u}^{\mu }={{dx}}^{\mu }/{ds}$. This equation allows one to study the motion of astronomical bodies for various conditions.

The Robertson–Walker metric was used to derive the cosmological equations from the general field equations in Paper I. These equations were greatly simplified thanks to relations (6), which allow us to express ${{\rm{\Lambda }}}_{{\rm{E}}}$. Compared to the usual standard equations of cosmology, they only contain one additional term representing an acceleration opposed to gravity; see Equation (32) below. In view of Equation (7), the effects due to the evolution over a long period of time are expected to be the largest ones. The effects not depending on a time evolution are in principle the same as in GR.

Now, let us consider a test particle in the weak field of a potential Φ created by a central mass point. We now develop this nonrelativistic approximation, with $v/c\ll 1$, of the geodesic Equation (15), which in the classical framework would lead to Newton's equation. The adopted metric only slightly deviates from the Minkowski metric,

$$\begin{eqnarray}&&{g}_{ii}=-1,\ \mathrm{for}\ i=1,2,3\quad \mathrm{and}\quad {g}_{00}=1+(2{\rm{\Phi }}/{c}^{2}).\end{eqnarray} \tag{ 16 }$$

This implies that the only nonzero component of the Christoffel symbols is (Tolman 1934)

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{00}^{i}=\displaystyle \frac{1}{2}\,\displaystyle \frac{\partial {g}_{00}}{\partial {x}^{i}}=\displaystyle \frac{1}{2}\,\displaystyle \frac{\partial (1+(2{\rm{\Phi }}/{c}^{2}))}{\partial {x}^{i}}=\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial {x}^{i}}.\end{eqnarray} \tag{ 17 }$$

We also have ${ds}\approx {cdt}$, and the velocities are ${u}^{i}\approx \tfrac{{v}^{i}}{c}=\tfrac{{{dx}}^{i}}{{cdt}}$ and ${u}^{0}\approx 1$. The only nonzero component of the coefficient of metrical connection ${\kappa }_{\nu }$ is ${\kappa }_{0}$. Thus, the last term in Equation (15) vanishes. In the Newtonian-like approximation of the equation of motion, we have

$$\begin{eqnarray}&&\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{{dv}}^{i}}{{dt}}+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial {x}^{i}}-{\kappa }_{0}\displaystyle \frac{{v}^{i}}{c}=0.\end{eqnarray} \tag{ 18 }$$

In the cosmological models of Paper I, we have put c = 1, while this is not the case here. Also, since ${\kappa }_{0}$ is a function of time (see Equation (5)), we define, in order to avoid any ambiguity hereafter,

$$\begin{eqnarray}&&\kappa (t)\,\equiv \,c\,{\kappa }_{0}=1/t.\end{eqnarray} \tag{ 19 }$$

Thus, one has

$$\begin{eqnarray}&&\displaystyle \frac{{{dv}}^{i}}{{dt}}+\displaystyle \frac{\partial {\rm{\Phi }}}{\partial {x}^{i}}-\kappa (t){v}^{i}=0.\end{eqnarray} \tag{ 20 }$$

We need to express the appropriate potential Φ. In the framework of GR, we would consider a central mass point $M^{\prime}$ and examine the situation at a distance $r^{\prime}$ in a spherically symmetric system with a potential ${\rm{\Phi }}^{\prime} =-{GM}^{\prime} /r^{\prime}$. In the scale-invariant system, from Equation (1) we have the correspondence $r^{\prime} =\lambda r$. For the density, it is $\rho =\rho ^{\prime} \,{\lambda }^{2}$ according to Equation (11) in Paper I. Thus, $\tfrac{M}{{r}^{3}}=\tfrac{M^{\prime} }{r{{\prime} }^{3}}\,{\lambda }^{2}$ and the relation between the Einsteinian mass $M^{\prime}$ and the scale-invariant one is

$$\begin{eqnarray}&&M^{\prime} =\lambda \,M.\end{eqnarray} \tag{ 21 }$$

The number of particles forming an object evidently does not change with time. Expression (21) is quite interesting: since the mass is changing like the length is doing, this means that the curvature of spacetime (or the gravitational potential) associated with a massive object is a scale-invariant quantity,

$$\begin{eqnarray}&&{\rm{\Phi }}^{\prime} =-\displaystyle \frac{{GM}^{\prime} }{r^{\prime} }=-\displaystyle \frac{{GM}}{r}={\rm{\Phi }},\end{eqnarray} \tag{ 22 }$$

being the same in the GR and scale-invariant frameworks. Equation (18) applies to each of the i-components. In Cartesian coordinates we may write

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{x}^{i}}{{{dt}}^{2}}=-\displaystyle \frac{G\,M}{{r}^{2}}\,\displaystyle \frac{{x}^{i}}{r}+\,\kappa (t)\displaystyle \frac{{{dx}}^{i}}{{dt}}.\end{eqnarray} \tag{ 23 }$$

In spherical coordinates, we can write the vectorial form of the equation of motion:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{\boldsymbol{r}}}{{{dt}}^{2}}=-\displaystyle \frac{G\,M}{{r}^{2}}\,\displaystyle \frac{{\boldsymbol{r}}}{r}+\,\kappa (t)\displaystyle \frac{d{\boldsymbol{r}}}{{dt}}.\end{eqnarray} \tag{ 24 }$$

We recognize Newton's law plus an additional term opposed to gravity. This expression means that in addition to the curvature of space associated with a mass element, a particle may experience some outward acceleration associated with the nonconstancy of the scale factor λ. This additional term is generally very small, as discussed in Section 2.3.

We have to take the proper units of time in Equation (24). In current units, the present age t₀ of the universe is 13.8 Gyr (Frieman et al. 2008) or 4.355 × 10¹⁷s. (This is an observed age value independent of cosmological models, resting essentially on the rather uncertain ages of globular clusters; see Catelan (2017) for a review. It is clear, however, that the relation between the age and a parameter like H₀ depends on the cosmological models; see below.) The inverse of the above age is $2.295\times {10}^{-18}$ s⁻¹ or 70.85 km s⁻¹ Mpc⁻¹. Thus, the empirical value of $\kappa ({t}_{0})=1/{t}_{0}$ is a quantity very close to the current value of the Hubble constant H₀, which lies between 73 and 67 km s⁻¹ Mpc⁻¹ (Chen & Ratra 2011; Aubourg et al. 2015; Chen et al. 2016; Riess et al. 2016). We may write the relation between the Hubble constant and the age t₀ of the universe in some chosen cosmological models as follows:

$$\begin{eqnarray}&&{H}_{0}=\xi \,\displaystyle \frac{1}{{t}_{0}},\end{eqnarray} \tag{ 25 }$$

which may be written for other times with appropriate ξ. The numerical factor ξ depends on the cosmological model. For scale-invariant models with ${{\rm{\Omega }}}_{k}=0$ and values of ${{\rm{\Omega }}}_{{\rm{m}}}=0.10,0.20,0.25,0.30,0.40$ we have $\xi =1.191,1.038,0.987,0.945,0.878$, respectively (see column (7) in Table 1 of Paper I).

Some developments of the weak-field approximation were already performed (Maeder & Bouvier 1979). However, at that time $\kappa (t)$ was identified with the Hubble constant. Although the numerical values are very close to each other, there is an important physical difference between the two. The Hubble constant depends on the cosmological models, with ${H}_{0}={(\dot{R}/R)}_{0}$ being the result of the evolution of the universe for the appropriate parameters ${{\rm{\Omega }}}_{{\rm{m}}}$ and ${{\rm{\Omega }}}_{k}$. This is physically different from the properties of the scale factor λ, which results from Equation (6).

Below, we shall carefully explore some first consequences of the above law of mechanics (24). Observations, rather than dogmas, will tell us whether the above modified Newton's law should be supported or rejected.

Let us estimate numerically the relative importance of the additional acceleration term with respect to the Newtonian attraction at the present time t₀. We consider a test particle orbiting with a circular velocity v at a distance r of a point mass M. The ratio x of the outward acceleration term resulting from the scale invariance of the empty space with respect to the Newtonian inward attraction term in Equation (24) is given by

$$\begin{eqnarray}&&x=\displaystyle \frac{v\,{r}^{2}}{G\,M\,{t}_{0}}.\end{eqnarray} \tag{ 26 }$$

We may now relate the present time t₀ to H₀ by expression (25) with the appropriate value of ξ, recalling that for the most realistic values of the density parameters ${{\rm{\Omega }}}_{{\rm{m}}}$ the value of ξ is of the order of unity. In turn, H₀ may be related to the critical density of the universe at the present time. We have seen in Paper I that the true critical density ${\varrho }_{{\rm{c}}}^{* }$ corresponding to k = 0 in scale-invariant models is given by (see Equation (39) of Paper I)

$$\begin{eqnarray}&&{\varrho }_{{\rm{c}}}^{* }=\displaystyle \frac{3}{8\pi \,G}\left({H}_{0}^{2}-2\displaystyle \frac{{H}_{0}}{t}\right)={\varrho }_{{\rm{c}}}\left(1-\displaystyle \frac{2}{3}\ \displaystyle \frac{8\pi \,G}{{H}_{0}\,t}\right).\end{eqnarray} \tag{ 27 }$$

Here,

$$\begin{eqnarray}&&{\varrho }_{{\rm{c}}}=\displaystyle \frac{3\,{H}_{0}^{2}}{8\,\pi \,G}\end{eqnarray} \tag{ 28 }$$

is the standard critical density in Friedman's models. These densities are usually considered at the present time t₀, but the above forms could also be used for other epochs with the appropriate H- and t-values. Now, the above x-ratio can be written in terms of the standard density ${\varrho }_{{\rm{c}}}$ (the use of ${\varrho }_{{\rm{c}}}^{* }$ brings other expressions with no particular interest for the numerical estimates below),

$$\begin{eqnarray}&&x=\displaystyle \frac{{H}_{0}\,v\,{r}^{2}}{\xi \,G\,M}=\displaystyle \frac{\sqrt{2}}{\xi }{\left(\displaystyle \frac{{\varrho }_{{\rm{c}}}}{\varrho }\displaystyle \frac{{v}^{2}}{({GM}/r)}\right)}^{1/2}.\end{eqnarray} \tag{ 29 }$$

Here, ρ is the mean density associated with the mass M within the radius r considered. (At a time t different from the present one, the corresponding values of the parameters need to be taken.) We will see, when studying the energy properties in Section 3.1, that the ratio $\tfrac{{v}^{2}}{{GM}/r}$ is not necessarily always equal to unity. As the dynamical evolution of a system proceeds, the additional acceleration term in Equation (24) may introduce progressive deviations from the classical relation ${v}^{2}\simeq G\,M/r$. According to Section 3, the above ratio $\tfrac{{v}^{2}}{{GM}/r}$ significantly differs from unity only for systems with a density within less than about 3 orders of magnitude from the critical density ${\varrho }_{{\rm{c}}}$. Thus, we write

$$\begin{eqnarray}&&x\,\geqslant \,\displaystyle \frac{\sqrt{2}}{\xi }{\left(\displaystyle \frac{{\varrho }_{{\rm{c}}}}{\varrho }\right)}^{1/2}.\end{eqnarray} \tag{ 30 }$$

For systems with $\varrho \gt {10}^{3}\,{\varrho }_{{\rm{c}}}$, we may consider the equality in the above expression. The ratio x is thus mainly given by the ratio of the critical density to the average density of the dynamical system considered. We see that the dynamical effects of the scale invariance of the empty space are particularly significant in systems of very low density, such as clusters of galaxies and possibly galaxies.

The acceleration term would dominate over gravitation ($x\gt 1$) only for systems with ϱ smaller than about $2\,{\varrho }_{{\rm{c}}}$. The only such system known is the universe, which currently shows some cosmic acceleration. The matter density and the critical density have different time dependences. The matter density evolves according to the conservation law given by Equation (61) in Paper I, while the critical density varies like H² (the variations of H(z) are given in Table 2 of Paper I for two useful models). The result is that the acceleration term dominates over braking only after a transition phase, which is located near z = 0.75 for models with ${{\rm{\Omega }}}_{{\rm{k}}}=0$ and ${{\rm{\Omega }}}_{{\rm{m}}}=0.30$ (Maeder 2017a).

The scale-invariant cosmological models depend on the usual density parameters ${{\rm{\Omega }}}_{{\rm{m}}}$ and ${{\rm{\Omega }}}_{{\rm{k}}}$, which now satisfy a relation of the form (see Equation (45) in Paper I)

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{m}}}+{{\rm{\Omega }}}_{{\rm{k}}}\,+{{\rm{\Omega }}}_{\lambda }=1,\quad \mathrm{with}\quad \quad {{\rm{\Omega }}}_{\lambda }\equiv \displaystyle \frac{2}{H\,t}.\end{eqnarray} \tag{ 31 }$$

For models with k = 0 supported by the observations of the CMB radiation (de Bernardis et al. 2000; Bennett et al. 2003), expansion implies $H\gt 0$ and thus ${{\rm{\Omega }}}_{\lambda }\gt 0$ with ${{\rm{\Omega }}}_{{\rm{m}}}\lt 1$. As in standard cosmology, from the two fundamental cosmological equations, a third one may be derived by elimination of the terms depending on the space curvature (Paper I); it is

$$\begin{eqnarray}&&-\displaystyle \frac{4\,\pi G}{3}\,(3p+\varrho )=\displaystyle \frac{\ddot{R}}{R}+\displaystyle \frac{\dot{R}\dot{\lambda }}{R\lambda }.\end{eqnarray} \tag{ 32 }$$

Terms p and ϱ are the pressure and density in the scale-invariant system, respectively. R(t) is the expansion function. Taking p = 0 and considering that the density ϱ is the average density in a sphere of radius R and central mass M, we get

$$\begin{eqnarray}&&\ddot{R}=-\displaystyle \frac{G\,M}{{R}^{2}}-\displaystyle \frac{\dot{\lambda }}{\lambda }\,\dot{R},\end{eqnarray} \tag{ 33 }$$

which compares with Equation (24). This shows the consistency of the above modified Newton equation with the scale-invariant cosmological equations in their limit.

Let us now consider the case of the empty space. In the Newtonian framework, a test particle would have a constant velocity with ${dv}/{dt}=0$. In the scale-invariant case, it would experience a slow acceleration. From the additional term in Equation (24) we have $\tfrac{{dv}}{{dt}}=\tfrac{v}{t}$ and thus $v=a\,t\mathrm{and}\ \ r-{r}_{0}=a\,({t}^{2}-{t}_{0}^{2})$. This is quite consistent with the results of Paper I, which show that the expansion function R(t) of an empty universe would behave like $R(t)\,\sim \,{t}^{2}$ in the scale-invariant cosmology, while the empty Friedman model would expand like $R(t)\sim t$.

We may not directly apply the virial theorem in the context of the scale-invariant theory, because of the additional term in Equation (24), which produces the expansion of a gravitational system (see Section 4.1). The additional radial outward acceleration term in Equation (24) may influence the relation between the motions and the present mass in a cluster of galaxies (Maeder 1978). We consider in a simplified way a spherical cluster containing N mass points of mass m_i and velocity v_i governed by the above modified Newton Equation (24). According to this equation, the acceleration of an object i interacting with another one of mass m_j is

$$\begin{eqnarray}&&\displaystyle \frac{{{dv}}_{i}}{{dt}}=-\displaystyle \frac{G\,{m}_{j}}{{r}_{{ij}}^{2}}+\kappa (t)\,{v}_{i},\end{eqnarray} \tag{ 35 }$$

where r_ij is the distance between objects i and j and $\kappa (t)=1/t$ according to Equation (19). Multiplying the above equation by ${v}_{i}=\tfrac{{{dr}}_{{ij}}}{{dt}}$, we get

$$\begin{eqnarray}&&{v}_{i}\,{{dv}}_{i}=-\displaystyle \frac{G\,{m}_{j}}{{r}_{{ij}}^{2}}\,{{dr}}_{{ij}}+\kappa (t)\,{v}_{i}^{2}\,{dt}.\end{eqnarray} \tag{ 36 }$$

This equation accounts for only one interaction i − j, and we have to sum up to account for all the gravitational interactions of the object i with the other masses m_j in the cluster. Thus, we have

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\,d({v}_{i}^{2})=-\displaystyle \sum _{j\ne i}\displaystyle \frac{G\,{m}_{j}{{dr}}_{{ij}}}{{r}_{{ij}}^{2}}+\kappa (t)\,{v}_{i}^{2}\,{dt},\end{eqnarray} \tag{ 37 }$$

We now integrate the above differential equation. The system is nonconservative, because the additional outward acceleration term cannot be derived as the gradient of a potential. The non-Newtonian term is an "adiabatic invariant," since the rate of its effects is generally very slow. The usual treatment is to consider a limited but significant interval of time and to obtain relations between time averages. The integration of the above equation between time t₁ and time t_z, where z is the cluster redshift, gives

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}[{v}_{i}^{2}({t}_{z})-{v}_{i}^{2}({t}_{1})]=\displaystyle \sum _{j\ne i}\left[\displaystyle \frac{G\,{m}_{j}({t}_{z})}{{r}_{{ij}}({t}_{z})}-\displaystyle \frac{G\,{m}_{j}({t}_{1})}{{r}_{{ij}}({t}_{1})}\right]\\ &&\quad +\,{\displaystyle \int }_{{t}_{1}}^{{t}_{z}}\kappa (t)\,{v}_{i}^{2}(t)\,{dt}.\end{eqnarray} \tag{ 38 }$$

Let us take t₁ as the time of the formation of the system. The effects of the nonconservative term in the initial collapse of the system are limited, and we have at equilibrium $\tfrac{1}{2}\,{v}_{i}^{2}({t}_{1})\,={\sum }_{j\ne i}\,\tfrac{G\,{m}_{j}({t}_{1})}{{r}_{{ij}}({t}_{1})}$. The above expression simplifies, and summing over all objects i, we get

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\,\displaystyle \sum _{i}\,{v}_{i}^{2}({t}_{z})=\displaystyle \frac{1}{2}\,\displaystyle \sum _{i}\displaystyle \sum _{j\ne i}\displaystyle \frac{G\,{m}_{j}({t}_{z})}{{r}_{{ij}}({t}_{z})}\\ &&\quad +\,\displaystyle \sum _{i}{\displaystyle \int }_{{t}_{1}}^{{t}_{z}}\kappa (t)\,{v}_{i}^{2}(t)\,{dt}.\end{eqnarray} \tag{ 39 }$$

In the above expression, there is a factor 1/2 in front of the double summation in order not to count twice the same interaction between a mass m_i and the surrounding masses m_j. We now take the mean over the N masses of the cluster considered to be spherical. The term on the left-hand side gives $\tfrac{1}{2\,N}\,{\sum }_{i}\,{v}_{i}^{2}({t}_{z})=(1/2)\,\overline{{v}^{2}({t}_{z})}$, while the first term on the right-hand side of Equation (39) leads to $0.5\,q^{\prime} {{GM}}_{\mathrm{sc}.\mathrm{inv}}/R$, where R is the cluster radius and ${M}_{\mathrm{sc}.\mathrm{inv}}$ the mass determined in the scale-invariant theory. Here $q^{\prime}$ is an appropriate structural factor, which has no influence on the final result; see Equations (45) and (48). For the non-Newtonian term, we need to know how the velocities are varying with t. In an empty space, $v=a\,t$ (Section 2.4), while in a bound two-body system the velocity is a constant (Section 4.1). Thus, we write $v{(t)=v({t}_{z})(t/{t}_{z})}^{\beta }$, with β between 0 and 1. To express the last term in Equation (39), we define

$$\begin{eqnarray}&&F\,\overline{{v}^{2}({t}_{z})}\,\equiv \,\displaystyle \frac{1}{N}\displaystyle \sum _{i}{\displaystyle \int }_{{t}_{1}}^{{t}_{z}}\kappa (t)\,{v}_{i}^{2}(t)\,{dt}\\ &&\quad =\,\displaystyle \frac{\overline{{v}^{2}({t}_{z})}}{2\,\beta }\left[1-{\left(\displaystyle \frac{{t}_{1}}{{t}_{z}}\right)}^{2\beta }\right],\\ &&\qquad \mathrm{with}\ F=\mathrm{ln}\displaystyle \frac{{t}_{z}}{{t}_{1}},\ \mathrm{for}\ \beta =0.\end{eqnarray} \tag{ 40 }$$

The above-mentioned replacements lead to

$$\begin{eqnarray}&&\overline{{v}^{2}({t}_{z})}(1-2\,F)\,\simeq q^{\prime} \displaystyle \frac{G\,{M}_{\mathrm{sc}.\mathrm{inv}}}{R}.\end{eqnarray} \tag{ 41 }$$

The observed velocities are radial velocities, and we may write their relations to the total velocities

$$\begin{eqnarray}&&\overline{{v}_{\mathrm{rad}}^{2}}=p\,\overline{{v}^{2}},\quad \mathrm{and}\quad \overline{| {v}_{\mathrm{rad}}| }=p^{\prime} \,\overline{| v| }.\end{eqnarray} \tag{ 42 }$$

For isotropic motions of the galaxies within the cluster, we would have $p=1/3$ and $p^{\prime} =1/2$, values that we adopt below. We finally write the expression corresponding to the virial theorem in the scale-invariant framework

$$\begin{eqnarray}&&\overline{{v}_{\mathrm{rad}}^{2}}(1\,-2\,F)\simeq \,p\,q^{\prime} \displaystyle \frac{G\,{M}_{\mathrm{sc}.\mathrm{inv}}}{R}.\end{eqnarray} \tag{ 43 }$$

This expression differs from the classical one by the parentheses on the left-hand side. The dynamical masses of clusters of galaxies published in the literature are based on the standard virial theorem. Some improvements in order to take into account the differences of the concentration of galaxies in clusters and other differences have been proposed (e.g., Rood 1974). The standard cluster masses ${M}_{\mathrm{std}}$ are based on a relation of the form

$$\begin{eqnarray}&&\overline{{v}_{\mathrm{rad}}^{2}}\simeq \,p\,q^{\prime} \,\displaystyle \frac{G\,{M}_{\mathrm{std}}}{R}.\end{eqnarray} \tag{ 44 }$$

The ratio of the standard masses ${M}_{\mathrm{std}}$ from Equation (44) to the masses ${M}_{\mathrm{sc}.\mathrm{inv}}$ given by the scale-invariant theory in Equation (43) is equal to

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}\,\simeq \,\displaystyle \frac{1}{1\,-\,2\,F}.\end{eqnarray} \tag{ 45 }$$

Two protoclusters of galaxies in a forming stage have been observed at z = 5.7 (Ouchi et al. 2005). This corresponds to ages between 1.2 and 1.8 Gyr, for models with ${{\rm{\Omega }}}_{{\rm{m}}}$ between 0.3 and 0.1 for k = 0, giving an upper limit of ${t}_{0}/{t}_{1}\simeq 10$. With $\beta =1$, for ${t}_{z}/{t}_{1}=1.5,2,4$, and 10, we have $\tfrac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}\,\simeq 2.2,4,16$, and 100, respectively. With $\beta =0$, $\tfrac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}$ rapidly diverges for ${t}_{z}/{t}_{1}\gt 1.6$. Thus, we may conclude that, except for clusters still in formation, the masses obtained by the standard virial theorem are often much larger than given by the scale-invariant theory.

Another estimate of F can be made by considering an average over an interval of time ${\rm{\Delta }}t$ equal to the radius crossing time $R/\overline{v}$, which often represents a large fraction of the cluster lifetime, especially when massive clusters are considered. This offers the advantage to provide an estimate based on observed parameters. We write

$$\begin{eqnarray}F\,\overline{{v}^{2}({t}_{z})} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{i}{\displaystyle \int }_{{t}_{1}}^{{t}_{z}}\kappa (t)\,{v}_{i}^{2}(t)\,{dt}\,\simeq \,\overline{{v}^{2}(t^{\prime} )}\displaystyle \frac{R}{t^{\prime} \overline{| v(t^{\prime} )| }}\\ & \simeq & f\,\overline{{v}^{2}({t}_{z})}\displaystyle \frac{R}{t^{\prime} {f}^{1/2}\,\overline{| v({t}_{z})| }}.\end{eqnarray} \tag{ 46 }$$

The intermediate time $t^{\prime}$ is about $(1/2){t}_{z}$. For f, we take 1 as for equilibrium ($\beta =0$). From Equation (39), we get

$$\begin{eqnarray}&&\overline{{v}^{2}({t}_{z})}\left(1\,-\displaystyle \frac{4\,R}{{t}_{z}\,\overline{| v({t}_{z})| }}\right)\simeq \,q^{\prime} \displaystyle \frac{G\,{M}_{\mathrm{sc}.\mathrm{inv}}}{R},\end{eqnarray} \tag{ 47 }$$

which leads to the following mass ratio for radial velocities with $p^{\prime} =1/2$:

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}\,\simeq \,\displaystyle \frac{1}{1\,-\,\tfrac{2\,R}{{t}_{z}\,\overline{| {v}_{\mathrm{rad}}({t}_{z})| }}}.\end{eqnarray} \tag{ 48 }$$

This confirms that the masses derived in the present theory may be much smaller than the standard masses. A prediction of the theory is that forming clusters have little or no dark matter.

The ratio ${M}_{\mathrm{tot}}/L$ of the mass to the luminosity of the observed clusters is considered in general. As the luminosities are essentially independent of the dynamical state of the clusters, we also have

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{M}_{\mathrm{tot}}}{L}\right)}_{\mathrm{std}}\,\simeq {\left(\displaystyle \frac{{M}_{\mathrm{tot}}}{L}\right)}_{\mathrm{sc}.\mathrm{inv}}\,\displaystyle \frac{1}{1\,-\,\tfrac{2\,R}{{t}_{z}\,\overline{| {v}_{\mathrm{rad}}({t}_{z})| }}}.\end{eqnarray} \tag{ 49 }$$

The standard mass–luminosity ratios are also larger than those from the scale-invariant framework. There are uncertainties; nevertheless, these estimates confirm that some substantial part of the dark matter could be due to scale-invariant effects. As the dynamical masses have contributed to ascertaining the concept of dark matter, we now examine in two quantitative examples what fraction of the dark matter could possibly be due to the above effects.

The Coma Cluster and Abell 2029 are the most studied massive clusters of galaxies, and they both have about 1000 member galaxies. A recent study by Sohn et al. (2016) provides a very complete and detailed study of their luminosity, stellar mass, and velocity dispersion functions. For the Coma Cluster, they found a mass ${M}_{200}={1.29}_{-.15}^{+.15}\times {10}^{15}$ ${M}_{\odot }$ and a radius ${R}_{200}={2.23}_{-0.09}^{+0.08}$ Mpc (R₂₀₀ and M₂₀₀ indicate the values up to which the enclosed density is equal to 200 times the critical density). The velocity dispersion σ is 947 (±31) km s⁻¹. For Abell 2029, the corresponding data are ${M}_{200}={0.94}_{-.27}^{+.30}\times {10}^{15}$ ${M}_{\odot }$, ${R}_{200}={1.97}_{-0.21}^{+0.20}$ Mpc, and $\sigma =973$ (±31) km s⁻¹. We point out that the radii R₂₀₀ only encompass $\sim 5 \%$ of the total volume of these clusters, which have observed total radii of about 6 Mpc, according to the data by Sohn et al. (2016). These authors have published the plot of the clustercentric velocities versus clustercentric distances for the Coma and Abell 2020 clusters. These plots support that the radial extension of these clusters reaches 6 Mpc.

For Coma the redshift is z = 0.0235, and for Abell 2029 it is z = 0.0784. According to the scale-invariant cosmological models with k = 0 and ${{\rm{\Omega }}}_{{\rm{m}}}=0.3$–0.1 of Paper I, this corresponds to ages of about 13.5 and 12.8 Gyr, respectively (for these small z, different models make small differences). As for the radius crossing times, for more realistic radii of 5 or 6 Mpc, we get 5.16 or 6.20 Gyr (Coma) and 5.03 or 6.03 Gyr (Abell 2029), respectively. For the factor $2\,F=\tfrac{2\,R}{{t}_{z}\,\overline{| {v}_{\mathrm{rad}}({t}_{z})| }}$, we get $2\,F=0.764\,\mathrm{or}\,0.919\,(\mathrm{Coma})\,\mathrm{and}\,0.786\,\mathrm{or}\,0.942\,(\mathrm{Abell}\,2029)$. The corresponding estimates of the ratios $\tfrac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}$ in Equation (48) are

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}\,\simeq \,4.2\,\,\mathrm{or}\,12.3\,(\mathrm{Coma})\\ &&\quad \mathrm{and}\simeq \,4.7\,\mathrm{or}\,17.2\,\,(\mathrm{Abell}\,\,\,2029).\end{eqnarray} \tag{ 50 }$$

As a matter of fact, the above numerical values likely are not overestimated for two reasons. First, the above radii of 6 Mpc may still be too low. For example, in the case of Abell 2029, the concentration of points in Figure 5 of Sohn et al. (2016) may extend up to 8 Mpc. Second, in both clusters at large clustercentric distances the velocities are much smaller than in the cluster core. In Figures 5 and 6 of Sohn et al. (2016), the caustics defining the velocity limit decrease by about a factor of 2 from R₂₀₀ to a distance of 6 Mpc. Even if the average velocity is reduced by 5% or 10%, this would significantly increase the ratios of the standard to the scale-invariant mass in both clusters. In fact, both the first and second effects intervene.

Thus, we see that the dynamical masses estimated in the scale-invariant system are smaller by a large factor (of about 4–12) with respect to the standard case. In this context, we recall that the baryon fraction from WMAP 7 yr and Planck 2013 turns around 0.16–0.17. Thus, with the above numerical figures, we see that there would be not much room, and maybe no room at all, left for dark matter in the context of the scale-invariant theory.

We conclude that a large fraction of the dark matter, and possibly the whole of it, is no longer demanded in the framework of the scale-invariant dynamics. More detailed analyses with extensive numerical simulations of the dynamical evolution of clusters of galaxies in the framework of the scale-invariant theory need to be performed in the future.

We now examine the relation between the excesses of the standard masses (with respect to the scale-invariant results) and the average cluster densities. Expressing the term $2F\,=\tfrac{4\,R}{{t}_{z}\,\overline{| v({t}_{z})| }}$ with $H=\xi /t$ (see Equation (25)) and the usual critical density ${\varrho }_{{\rm{c}}}=\tfrac{3\,{H}^{2}}{8\pi \,G}$, we get

$$\begin{eqnarray}&&2F=\displaystyle \frac{4}{\xi }{\left(\displaystyle \frac{{R}^{2}{H}^{2}}{{\overline{| v({t}_{z})| }}^{2}}\right)}^{1/2}\simeq \,\displaystyle \frac{4}{\xi }{\left(\displaystyle \frac{2{\varrho }_{{\rm{c}}}}{q^{\prime} \varrho }\right)}^{1/2},\end{eqnarray} \tag{ 51 }$$

where we have used Equation (44), also identifying the quadratic and arithmetic means of the velocities. According to Section 3.1, the critical density ${\varrho }_{{\rm{c}}}$ must be taken at the redshift corresponding to t_z. For the mass we take the standard mass ${M}_{\mathrm{std}}$; thus, the density is the corresponding density ${\varrho }_{\mathrm{std}}$ of the cluster. The ratio $\tfrac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}$ may be written as

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}\,\simeq \,\displaystyle \frac{1}{1-\tfrac{4}{\xi }{\left(\tfrac{2{\varrho }_{{\rm{c}}}({t}_{z})}{q^{\prime} {\varrho }_{\mathrm{std}}({t}_{z})}\right)}^{\tfrac{1}{2}}},\end{eqnarray} \tag{ 52 }$$

where, as seen above, we adopt $\xi \sim 1$, $q^{\prime} \sim 1$. We see that the ratio of the standard to the scale-invariant masses increases for object of lower densities, consistently with the remarks in Section 2.3. For astronomical systems with densities much above the critical density of the universe, the two mass estimates are similar. Let us recall that Table 2 of Paper I allows one to estimate H and thus the critical densities at different redshifts.

Figure 1 shows the ratio $\tfrac{{M}_{\mathrm{std}}}{{M}_{\mathrm{sc}.\mathrm{inv}}}$ as a function of ratio of the cluster density ${\varrho }_{\mathrm{std}}$ with respect to the critical density at time t_z. We see that the excess of the standard cluster masses with respect to the values in the scale-invariant theory rapidly diverges for values of the density ratio ${\varrho }_{\mathrm{std}}/{\varrho }_{\mathrm{sc}.\mathrm{inv}}$ below 10², consistently with the results for the Coma Cluster and Abell 2029. Finally, we recall that, even within a given cluster, the standard estimates of the (M/L) ratio steeply increase for larger radii (Lewis et al. 2003), i.e., for decreasing average internal densities. This remarkable fact is quite in agreement with the above results and does not demand any peculiar distribution of dark matter according to clustercentric distances.

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We start by studying the classical case of the two-body problem in the scale-invariant framework, following some early developments by Maeder & Bouvier (1979). The specific angular momentum in the classical case is in Cartesian coordinates ${\boldsymbol{x}}^{\prime} \,\times \,\tfrac{d{\boldsymbol{x}}^{\prime} }{{dt}}$, which is constant in time. Let us examine the product ${\boldsymbol{x}}\times \kappa (t)\,\tfrac{d{\boldsymbol{x}}}{{dt}}$ and its derivative,

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left({\boldsymbol{x}}\times \kappa (t)\,\displaystyle \frac{d{\boldsymbol{x}}}{{dt}}\right)=\displaystyle \frac{d\kappa }{{dt}}\left({\boldsymbol{x}}\times \displaystyle \frac{d{\boldsymbol{x}}}{{dt}}\right)+\kappa (t)\,\displaystyle \frac{d}{{dt}}\left({\boldsymbol{x}}\times \displaystyle \frac{d{\boldsymbol{x}}}{{dt}}\right).\end{eqnarray} \tag{ 53 }$$

Now, according to the expression of $\kappa (t)$ in Equation (19), one has

$$\begin{eqnarray}&&\displaystyle \frac{d\kappa }{{dt}}=-{\kappa }^{2}(t).\end{eqnarray} \tag{ 54 }$$

Let us develop the two terms on the right-hand side of Equation (53),

$$\begin{eqnarray}&&-{\kappa }^{2}(t)\left({\boldsymbol{x}}\times \displaystyle \frac{d{\boldsymbol{x}}}{{dt}}\right)=-{\kappa }^{2}(t)\left({x}^{1}\,\displaystyle \frac{{{dx}}^{2}}{{dt}}-{x}^{2}\displaystyle \frac{{{dx}}^{1}}{{dt}}\right)\\ &&\quad +\,\mathrm{the}\,\mathrm{same}\,\mathrm{for}\,\,(2,3)\,\mathrm{and}\,(3,1),\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&\kappa (t)\displaystyle \frac{d}{{dt}}\left({\boldsymbol{x}}\times \displaystyle \frac{d{\boldsymbol{x}}}{{dt}}\right)\\ &&\quad =\,\kappa (t)\left(\displaystyle \frac{{{dx}}^{1}}{{dt}}\displaystyle \frac{{{dx}}^{2}}{{dt}}+{x}^{1}\displaystyle \frac{{d}^{2}{x}^{2}}{{{dt}}^{2}}-\displaystyle \frac{{{dx}}^{2}}{{dt}}\displaystyle \frac{{{dx}}^{1}}{{dt}}-{x}^{2}\displaystyle \frac{{d}^{2}{x}^{1}}{{{dt}}^{2}}\right)\\ &&\qquad +\,\mathrm{the}\,\mathrm{same}\,\mathrm{for}\,\,(2,3)\,\mathrm{and}\,(3,1).\end{eqnarray} \tag{ 56 }$$

In this last expression, the first and third terms on the right-hand side cancel each other, and the second and fourth terms are, according to Equation (23),

$$\begin{eqnarray}&&{x}^{1}\,\displaystyle \frac{{d}^{2}{x}^{2}}{{{dt}}^{2}}=-\displaystyle \frac{G\,M}{{r}^{2}}\,\displaystyle \frac{{x}^{1}\,{x}^{2}}{r}+\,\kappa (t)\,{x}^{1}\displaystyle \frac{{{dx}}^{2}}{{dt}}.\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{x}^{2}\,\displaystyle \frac{{d}^{2}{x}^{1}}{{{dt}}^{2}}=-\displaystyle \frac{G\,M}{{r}^{2}}\,\displaystyle \frac{{x}^{1}\,{x}^{2}}{r}+\,\kappa (t)\,{x}^{2}\displaystyle \frac{{{dx}}^{1}}{{dt}}.\end{eqnarray} \tag{ 58 }$$

Now, we can write the complete expression

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left(\,{\boldsymbol{x}}\,\times \,\kappa (t)\,\displaystyle \frac{d{\boldsymbol{x}}}{{dt}}\right)=-{\kappa }^{2}(t)\left({x}^{1}\,\displaystyle \frac{{{dx}}^{2}}{{dt}}-{x}^{2}\displaystyle \frac{{{dx}}^{1}}{{dt}}\right)\\ &&\quad +\,\kappa (t)\left(-\displaystyle \frac{G\,M}{{r}^{2}}\,\displaystyle \frac{{x}^{1}\,{x}^{2}}{r}+\kappa (t){x}^{1}\displaystyle \frac{{{dx}}^{2}}{{dt}}\right)\\ &&\quad -\,\kappa (t)\left(-\displaystyle \frac{G\,M}{{r}^{2}}\,\displaystyle \frac{{x}^{1}\,{x}^{2}}{r}+\kappa (t)\,{x}^{2}\displaystyle \frac{{{dx}}^{1}}{{dt}}\right)\\ &&\qquad +\,\mathrm{the}\,\,\,\mathrm{same}\,\,\,\mathrm{for}\,\,(2,3)\,\mathrm{and}\,(3,1)={\bf{0}}.\end{eqnarray} \tag{ 59 }$$

The two Newtonian terms cancel each other, and the same goes for the other terms. Thus, the above equation expresses the angular momentum conservation in the scale-invariant framework. The vector $\left(\kappa (t)\,{\boldsymbol{x}}\times \tfrac{d{\boldsymbol{x}}}{{dt}}\right)$ is always orthogonal to the orbital motion, which indicates that the problem is two-dimensional. The angular momentum conservation is written in polar coordinates $(r,\vartheta )$ as

$$\begin{eqnarray}&&\kappa (t)\,{r}^{2}\,\dot{\vartheta }=L=\mathrm{const}.\end{eqnarray} \tag{ 60 }$$

It is a scale-invariant term. At a fixed time, the above expression is similar to the usual conservation law. Now, the equation of motion (24) is written in the two polar coordinates as

$$\begin{eqnarray}&&\ddot{r}-r\,{\dot{\vartheta }}^{2}=-\displaystyle \frac{G\,M}{{r}^{2}}+\kappa (t)\,\dot{r},\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&r\,\ddot{\vartheta }+2\,\dot{r}\,\dot{\vartheta }=\kappa (t)\,r\,\dot{\vartheta }.\end{eqnarray} \tag{ 62 }$$

The mass M is the mass in the scale-invariant framework; see expression (21). We easily verify the compatibility of expression (60) with the above Equation (62). The radial Equation (61) can be expressed with L,

$$\begin{eqnarray}&&\ddot{r}-{\left(\displaystyle \frac{L}{\kappa (t)}\right)}^{2}\,\displaystyle \frac{1}{{r}^{3}}+\displaystyle \frac{G\,M}{{r}^{2}}-\kappa (t)\,\dot{r}=0.\end{eqnarray} \tag{ 63 }$$

We may transform the time derivatives into derivatives with respect to ϑ; with $\tfrac{{dr}}{d\vartheta }=\tfrac{\dot{r}}{\dot{\vartheta }}=\tfrac{\dot{r}}{L}\kappa (t)\,{r}^{2}$ we have

$$\begin{eqnarray}\dot{r} & = & \displaystyle \frac{L({dr}/d\vartheta )}{\kappa (t)\,{r}^{2}}\,\quad \mathrm{and}\quad \ddot{r}=\displaystyle \frac{{L}^{2}}{{H}^{2}\,{r}^{4}}\left(\displaystyle \frac{{d}^{2}r}{d{\vartheta }^{2}}-2\displaystyle \frac{{({dr}/d\vartheta )}^{2}}{r}\right)\\ & & -\displaystyle \frac{\dot{\kappa (t)}\,L\,({dr}/d\vartheta )}{{\kappa }^{2}(t)\,{r}^{2}}\,.\end{eqnarray} \tag{ 64 }$$

These replacements lead to an equation in $(r,\vartheta )$ giving the curve described by a test particle in the central field of the scale-invariant mass. Most remarkably, the last term in the modified Newton Equation (63) simplifies with the last one in expression (64) for $\ddot{r}$, and we have

$$\begin{eqnarray}&&\displaystyle \frac{{L}^{2}}{{\kappa }^{2}(t)\,{r}^{4}}\left(\displaystyle \frac{{d}^{2}r}{d{\vartheta }^{2}}-\displaystyle \frac{2}{r}{\left(\displaystyle \frac{{dr}}{d\vartheta }\right)}^{2}\right)-{\left(\displaystyle \frac{L}{\kappa (t)}\right)}^{2}\displaystyle \frac{1}{{r}^{3}}+\displaystyle \frac{{GM}}{{r}^{2}}=0.\end{eqnarray} \tag{ 65 }$$

This allows us with the transformation $\rho =1/r$ to write

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\rho }{d{\vartheta }^{2}}+\rho =\displaystyle \frac{G\,M\,{\kappa }^{2}(t)}{{L}^{2}}.\end{eqnarray} \tag{ 66 }$$

This expression is identical to the classical Binet equation except for the κ-term on the right-hand side. Thus, we may immediately write the solution $\rho =(1/{r}_{0})+C\,\cos (\vartheta )$ or

$$\begin{eqnarray}&&r=\displaystyle \frac{{r}_{0}}{1+e\cos (\vartheta )},\quad \mathrm{with}\quad {r}_{0}=\displaystyle \frac{{L}^{2}}{G\,M\,{\kappa }^{2}(t)}.\end{eqnarray} \tag{ 67 }$$

Here, r₀ is the radius of a circular orbit (for e = 0). It is not a scale-invariant quantity. Recalling once more that the Einsteinian mass is $M^{\prime} =\lambda \,M$, we see that r₀ grows like t, consistently with the basic relation (1). The eccentricity e is given by

$$\begin{eqnarray}&&e=C\,\displaystyle \frac{{L}^{2}}{G\,M\,{\kappa }^{2}(t)},\quad {\rm{i}}.{\rm{e}}.,\quad C=e/{r}_{0}.\end{eqnarray} \tag{ 68 }$$

We verify that the eccentricity e is scale invariant, which is satisfactory. The above Equation (67) is that of a conic, ellipse, parabola, or hyperbola depending on the eccentricity, however with a secular variation of the orbital radius r₀, or semimajor axis as shown below.

The solutions of the two-body problem are similar to those of the standard case, with, in addition, slow secular variations of the orbital radius. More generally, if we consider the semimajor axis a of an orbital motion,

$$\begin{eqnarray}&&a=\displaystyle \frac{{r}_{0}}{1-{e}^{2}},\end{eqnarray} \tag{ 69 }$$

we have from Equations (67) and (69), together with Equations (7) and (19),

$$\begin{eqnarray}&&\displaystyle \frac{\dot{a}}{a}=\displaystyle \frac{\dot{\lambda }}{\lambda }-2\displaystyle \frac{\dot{\kappa }}{\kappa }=-\displaystyle \frac{\dot{\kappa }}{\kappa }=\displaystyle \frac{1}{t}.\end{eqnarray} \tag{ 70 }$$

Thus, we see that the semimajor axis increases linearly with time t. The behavior of the circular velocity ${v}_{\mathrm{circ}}$ is also interesting. From Equation (60) of the conservation of the angular momentum, we get

$$\begin{eqnarray}&&{v}_{\mathrm{circ}}={r}_{0}\,\dot{\vartheta }=\displaystyle \frac{L}{\kappa (t)\,{r}_{0}}=\mathrm{const}.\end{eqnarray} \tag{ 71 }$$

The constancy results from the fact that $\kappa (t)$ behaves like $1/t$ and r₀ like t, L being a constant. This is consistent with the fact that the gravitational potential is an invariant as shown by Equation (22). The constancy of the circular velocity over time is of great importance for the study of the rotation curves of galaxies below. From the conservation law (60), we also see that the orbital period P similarly varies like $\dot{P}/P=1/t$. This is also evident since the radius increases linearly and both the eccentricity and the circular velocity are constant.

Thus, the scale-invariant two-body problem leads essentially to the same solutions as the Newtonian case, with a slight supplementary outward expansion at a rate that is not far from the Hubble expansion. These conclusions consistently come from the hypothesis we have made (see Section 1). Now, whether this corresponds to nature or not can only be decided on the basis of careful comparisons with observations.

The rotation curves of nearby spiral galaxies, i.e., the circular velocities as a function of the galactocentric distances r, generally remain flat in the outer regions, instead of having a Keplerian decrease like $\sim 1/\sqrt{r}$, as expected if most of the mass lies in inner regions. The velocity determinations are mainly based on optical observations of Hα, N ii, and S ii lines and on radio observations of H i and CO lines. There is a long history of the problem of the flat rotation curves, as reviewed by Sofue & Rubin (2001), who report that already in 1940 Oort noticed that "the distribution of mass [in NGC 3115] appears to bear no relation to that of the light." Such facts were further confirmed by other precursors. From a sample of 10 high-luminosity spiral galaxies, Rubin et al. (1978) stated that "all rotation curves are approximately flat, to a distance as great as r = 50 kpc." The sample was extended to 21 galaxies (Rubin et al. 1980), further supporting the previous conclusions. Nowadays, the observations of thousands of galaxies confirm the difference of the matter and luminosity distributions and support the existence of a halo of dark matter around the Milky Way and other galaxies (e.g., Persic et al. 1996; Sofue & Rubin 2001; Sofue 2012; Huang et al. 2016).

We concentrate on the case of the Milky Way where the rotation curve is known to the largest distances from the center. On the basis of the velocities of about 16,000 red clump giants in the outer disk, as well as ∼5700 halo K giants in the halo, Huang et al. (2016) have constructed the rotation curve of the Milky Way up to about 100 kpc. The average data as a function of the galactocentric distance are given in their Table 3, which indicates the various segments of the curve and the source of their measurements.

The curve by Huang et al. (2016) is illustrated in Figure 2; it shows a flat rotation curve with a circular velocity of 240 km s⁻¹ up to galactocentric distances R of about 25–30 kpc and then slowly decreases down to 150 km s⁻¹ at 100 kpc. There are also some prominent dips at R = 11 and 19 kpc (represented by the thin dashed red line in Figure 2). The error bars on the velocities are rather small ($\sigma \approx 7$ km s⁻¹) for R between 4.6 and about 13 kpc, so that the dip at 11 kpc appears as very significant. From R = 15 to 20 kpc, the error bars are much larger, so that the dip at 19 kpc may be less significant. Nevertheless, in view of the small amplitude of the dips with respect to the velocity, the rotation curve may be considered as globally flat up to at least 25 kpc (Huang et al. 2016). We note that the 11 kpc dip is often interpreted as due to a ring of dark matter at that location. The dip at 19 kpc may have the same origin; however, it could also be artificial owing to the use of different data sets. We note that Rubin et al. (1978) already pointed out secondary velocity undulations in various rotation curves, with rotational velocities lower by about 20 km s⁻¹ on the inner edge than on the outer edges of spiral features.

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The decrease in the external regions reaches about 100 km s⁻¹, which is about five times larger than the error bars. Moreover, it is supported by all measurements beyond a galactocentric distance $R\approx 25\,\mathrm{kpc}$. The observed points then form a rather smoothly decreasing curve.

The red curve in Figure 2 is the velocity distribution at the present cosmic time t₀. (In the cosmological models of Paper I, the present age is fixed to ${t}_{0}=1$, which corresponds to 13.8 Gyr. The correspondence between t₀ and H₀ is expressed by Equation (23) with the appropriate ξ-values.) We can find the corresponding velocity distributions at past epochs, 0.8 t₀, 0.6 t₀, 0.4 t₀, etc., by applying the properties of Equations (70) and (71) derived from the equivalent Binet equation (Equation (66)) in the scale-invariant framework. At past epochs, the radii were smaller, while the circular velocities remained constant. Thus, we apply these simple evolution laws to the present rotation curve to deduce the curves at past epochs. Of course, this does not preclude the various dynamical effects that currently are at work in galaxies to be simultaneously operating: interactions due to spiral waves, effects of bars, nonaxisymmetric perturbations, radial motions, cloud collisions, mergers, etc. For now, we ignore these various effects in order to just examine the consequences of scale invariance.

Figure 2 shows that at earlier epochs the outer velocity distributions derived from the scale-invariant predictions were increasingly steeper with decreasing time. At the same time, the Galaxy was more compact. The galaxy formation occurred on a relatively short timescale compared to the age of the universe. At a given location, the infalling matter stops its collapse when the centrifugal force equilibrates gravity, thus establishing a Keplerian law. Later, during the aging of the Galaxy, the dynamical effects of scale invariance intervene, leading to a flatter distribution.

Let us consider that the initial Keplerian velocity distribution was of the form $v(r,{t}_{\mathrm{in}})={v}_{\mathrm{in}}\,\sqrt{{r}_{\mathrm{core}}({t}_{\mathrm{in}})/r({t}_{\mathrm{in}})}$, with the assumption of a relatively constant circular velocity ${v}_{\mathrm{in}}$ up to a distance ${r}_{\mathrm{core}}({t}_{\mathrm{in}})$, followed beyond ${r}_{\mathrm{core}}$ by a Keplerian decrease. As time progresses, the orbital radii increase by a factor of $\lambda (t)$; thus, at time t the velocity distribution becomes

$$\begin{eqnarray}&&v(r,t)={v}_{\mathrm{in}}\,\sqrt{\displaystyle \frac{\lambda (t)\,{r}_{\mathrm{core}}({t}_{\mathrm{in}})}{\lambda (t)\,r({t}_{\mathrm{in}})}}={v}_{\mathrm{in}}\,\sqrt{\displaystyle \frac{{r}_{\mathrm{core}}(t)}{\,r(t)}},\\ &&\quad \mathrm{for}\,r(t)\,\gt \,{r}_{\mathrm{core}}(t).\end{eqnarray} \tag{ 72 }$$

We see that the scale transformation conserves the Keplerian law in $1/\sqrt{r}$. As a matter of fact, the velocity distribution found by Huang et al. (2016) in the external regions of the Galaxy is close to a Keplerian law starting from $R\approx 30\,\mathrm{kpc}$. Consequently, the curve at past epochs, like $0.1\,{t}_{0}$, derived by a backward scaling from the present curve by Huang et al. (2016), is also close to a steep Keplerian distribution as shown in Figure 2.

Two important remarks need to be made. (1) There are a variety of rotation curves of galaxies as shown by Sofue & Rubin (2001). The available data generally concern radial extensions smaller than 20 or 30 kpc. Two very massive galaxies, UGC 2953 and UGC 2487, have been observed up to radial distances of 60 and 80 kpc, respectively (Sanders & Noordermeer 2007; Famaey & McGaugh 2012). Over these ranges, they only show a decline of 40–50 km s⁻¹, smaller than the decrease of about 100 km s⁻¹ for the Milky Way. However, these two galaxies are among the most massive and fastest-rotating galaxies, with maximum velocities of about 300 and 380 km s⁻¹, respectively, much higher than in the Milky Way or in the galaxies studied by Sofue & Rubin (2001). Thus, it would be extremely interesting to know the rotation curves in the further outer layers of these extreme galaxies to see whether the decrease goes on, and whether their data can be interpreted in the context of the scale-invariant dynamics. (2) We also note that a remarkable correlation between the radial acceleration derived from the rotation curves and the distribution of baryons has been found (McGaugh et al. 2016; Lelli et al. 2017), implying that the dark matter is fully specified by the baryons. The obtained relation indicates the absence of dark matter at high acceleration and a systematic deviation for acceleration lower than about 10⁻¹⁰ m s⁻². These findings that imply deviations from standard dynamics at the lower densities might provide further tests of the scale-invariant dynamics and will be studied in a future work (I am very indebted to the referee for these remarks).

Thus, we tend to conclude that the relatively flat rotation curves of spiral galaxies are an age effect from the mechanical laws, which account for the scale-invariant properties of the empty space at large scales. These laws predict that the circular velocities remain the same, while a very low expansion at a rate not far from the Hubble rate progressively extends the outer layers, increasing the radius of the Galaxy and decreasing its surface density like $1/t$ when account is given to Equation (21). It is interesting that both the mass excesses derived from the virial in clusters of galaxies and from the flat rotation curves tend to find an explanation within the scale-invariant theory. In both cases, there is apparently no need of dark matter and unknown particles.

The age effect in the rotation curves of galaxies is nicely confirmed by recent works by Genzel et al. (2017) and Lang et al. (2017). Six star-forming galaxies in the range $z=0.8\mbox{--}2.4$ were studied in detail by Genzel et al. (2017), and a sample of 101 other galaxies between z = 0.6 and 2.6 were studied by Lang et al. (2017). The rotation curves they derive for these early objects show, with a high statistical significance, that the rotation velocities are not constant, but decrease in a compelling way with radius. They show that no dark matter is required to interpret the data, and the rotation curve is consistent with a pure baryonic disk. Even at a distance of several effective radii, the authors find that the dark matter fractions are modest or negligible, the results being essentially insensitive to the M/L ratios.

Figure 3 shows the stacked rotation curves with their error bars as derived by Lang et al. (2017). The points beyond the radius with maximum velocity show a decrease, which is not far from a $1/\sqrt{r}$ Keplerian curve (in green). Most of the galaxies of the sample are observed at epochs before the peak of star formation. This shows that the usual flatness of the rotation curve is a characteristic of the present epoch, but it is a property absent in the early stages. We emphasize that it is a bit worrying that the concentrations of dark matter, in the potential wells of which galaxies are supposed to form, are not present in epochs close to the formation time. Moreover, there is a progression in the presence of dark matter in spiral galaxies with time, since the observations by Wuyts et al. (2016) indicate that galaxies at z = 1 contain more dark matter than galaxies at z = 2, and in turn the local present galaxies show more dark matter than those at z = 1.

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In Section 4.2, we have described a possible sequence in the dynamical evolution: cloud collapse–equilibrium–steep Keplerian velocity distribution–secular evolution according to Equations (70) and (71)–flatter rotation curve of galaxies. This scenario appears to account simultaneously for the observations of Genzel et al. (2017) and Lang et al. (2017), concerning the steep Keplerian rotation curve and the absence of dark matter at significant redshifts $z\geqslant 2$, for the intermediate situation at medium redshifts $z\approx 1$ (Wuyts et al. 2016), as well for the present flat rotation curves of most galaxies. These results appear to give some support to the above scale-invariant dynamics based on the modified Newton Equation (24).

Now, we may wonder whether the progressive flattening of the galaxy rotation curve is the only consequence of the scale-invariant stellar dynamics. As a matter of fact, the velocity dispersion, in particular in the so-called "vertical direction," shows a strong increase with age, known as the age–velocity dispersion relation (AVR; Seabroke & Gilmore 2007). This relation has received a variety of explanations over the past few decades without any clear consensus (see, e.g., Kroupa 2002; Kumamoto et al. 2017). The velocity dispersion in the Galactic plane is dominated by the effects of spiral waves, as well as by the collisions with giant molecular clouds that are strongly concentrated in the Galactic plane. However, in the directions orthogonal to the plane, there is little interaction since the stars spend most of their lifetimes out of the Galactic plane (Seabroke & Gilmore 2007). Thus, we may wonder whether the secular effects of scale invariance may play some role. The answer is positive; this problem is examined in the Appendix.

We also emphasize that the two problems of velocity dispersion and rotation curves are related. The vertical dispersion is an expression of the support in the vertical direction, while the rotation curves express the mechanical support in the horizontal direction. The results by Genzel et al. (2017) and Lang et al. (2017) show that the horizontal support is increasing with age, and the AVR shows a similar result for the vertical support. Thus, both mechanical supports of the Galaxy, vertical and horizontal, show an increasing trend with age.