# Observational tests in scale invariance I: galaxy clusters and rotation of
galaxies

Andre Maeder

###### Abstract

Galaxy velocities in clusters, rotation curves of galaxies, and ”vertical”
oscillations in the Milky Way itself currently show too high velocities with
respect to the masses thought to be involved. While these velocity excesses
are currently interpreted as the consequence of dark matter, it can also be
naturally explained as a consequence of scale invariant theory, which rests on
a very simple first principle: the addition of a new fundamental symmetry. In
the present work, the case of scale invariance, present in General Relativity
and Maxwell equations for the empty space without charge and current, is
considered. Cosmological models predict a rapid decrease of these effects with
increasing mean density up to the critical density, where they totally
disappear. Starting from the scale invariant geodesic equation by Dirac
(1973), for which a demonstration by an action principle is presented, a
modified Newton equation is derived. The solutions of this equation are
applied to clusters of galaxies, galactic rotation and ”vertical” motions in
the Milky Way. In this new framework the following results emerge: \- 1. the
mass estimates of galaxy clusters are very much reduced with respect to the
standard case, \- 2. the flat rotation curves of galaxies are beatifully
accounted for as a result of the dynamical evolution, \- 3. steeper rotation
curves are predicted in significantly redshifted galaxies, progressively
evolving to Keplerian at higher $z$, all in a consistent way with recent
observations showing less dark matter at higher $z$. \- 4. In clusters of
galaxies and in galaxies, both at low and at high $z$, the amount of dark
matter necessary to account for the velocities seems to be always determined
by the amount of baryonic matter, which points in favour of a gravity effect.
\- 5. The growth of the ”vertical” velocity dispersion, i.e., perpendicular to
the plane, with stellar ages in the Milky Way is also positively tested. The
convergence of these results, in very different gravitational systems, epochs,
mass range and spatial scales, opens very interesting perspectives that
deserve to be explored further.

##  1 Introduction

It is a general trend in dynamical systems that the observed velocities are
too high with respect to current mass estimates. This was first observed in
clusters of galaxies by Zwicky [65], and later by Karachantsev [22] who
obtained mass excesses reaching a factor 80. Further studies have well
confirmed the above trend, for example mass-luminosity ratios up to a factor
700 were found by Bahcall [1], while Newtonian Virial masses reaching factors
30-50 were estimated by Proctor [51] for large clusters. Noticeably, the total
dynamical masses of clusters (and galaxies) appears, as shown by Chan [7],
always related to the baryonic masses of the central regions of the clusters.

The rotation curves of galaxies raised several problems since the early
discovery of flat rotation curves in spiral galaxies up to galactocentric
distances of a few tens of kpcs [53]. A complete review of the observations
and their analysis has been given by Sofue and Rubin [56], while the
challenges to the dark matter interpretation have also been reviewed [14]. The
flat rotation curves played a major role in the raise of the hypothesis of
dark matter (DM), which, with the accumulation of observations, left the
characteristics of an astronomical hypothesis of the late 70’s to
progressively get in the 21st century the status of a material reality.

The radial acceleration relation (RAR) studied by Lelli et al. [29] compares
the centrifugal accelerations $\frac{\upsilon^{2}}{R}$ and the gravity
$\frac{GM}{R^{2}}$ (masses being estimated from the luminosities) in different
locations of a large number of galaxies of different types. The corresponding
points are deviating from a 1:1 relation for gravities below a gravity
$a_{0}=1.2\times 10^{-8}$ cm s-2. Such properties, in particular the flat
rotation curves were at the origin of the MOND theory by Milgrom, suggesting a
modification of the gravity law [44, 45]. Several observations in favour of
MOND have also been identified [14]. The deviations from the 1:1 relation are
taken as an indication of the amount of dark matter. A most puzzling fact is
that the amount of dark matter appears to be always directly determined by the
amount of baryonic matter [29], although a total of 2700 measurements have
been made in galaxies of different types and at different distances from their
centers. Such a strong connection between DM and baryons appears as a general
property, it was also recently found in clusters [7] and in a large sample of
massive star forming galaxies at $z=0.6-2.5$ by Nestor Shachar et al. [47].

The recent observations of rotation curves in galaxies at the peak of their
star formation is bringing new questions. Surveys by Genzel et al. [17] and
Lang et al. [27] showed that the rotation curves in galaxies with redshifts
$z$ between 0.6 and 2.6 were not flat, but steeply decreasing with radius, a
point which raised much discussions before being well confirmed by a further
analysis [18]. Recently new observations with ESO-VLT, LUCI, NOEMA at IRAM and
ALMA of hundreds of galaxies at redshifts $z=0.6-2.5$ have confirmed the steep
decrease of the velocity curves with galactocentric distance in the past, thus
implying according to the authors a decrease of the amount of dark matter down
to 17% at $z=2.44$ [47]. These results, based on new observations of a larger
sample of galaxies, give a remarkable support to the earlier findings by the
Genzel group. They raise new questions, since this picture does not appear to
correspond to the $\Lambda$CDM models, where the baryonic matter sets into the
potential well early formed by DM.

Within galaxies, several critical dynamical problems were already found long
ago. Spitzer and Schwarzschild [58] showed that the ”vertical” velocity
dispersion (perpendicular to the galactic plane) was much increasing with the
age of the stellar group considered and this was interpreted as a due to
collisions with large interstellar clouds. This dispersion increase was
confirmed by further studies [54]. Since 75 years, many other astrophysical
effects were proposed to account for the growth of velocities with age.

The aim of this work is to examine the above dynamical effects within the
scale invariant vacuum (SIV) theory and its recent developments [32, 35]. This
theory is resting on a most simple and fundamental question: is there still
another fundamental symmetry in the sequence of Galilean invariance, Lorentz
invariance and Einstein general covariance ? The case of scale invariance
which is present in General Relativity and Maxwell equations for the empty
space without charge and currents. Cosmological models predicts a rapid
decrease of these effects with increasing mean density up to the critical
density where they vanish. The deep MOND limit, where a gravity
$g=\sqrt{a_{0}\,g_{\mathrm{N}}}$ (for Newtonian gravities
$g_{\mathrm{N}}<a_{0}$), appears as an approximation of the SIV theory when
the scale factor $\lambda$ can be considered as a constant. Such a condition
is satisfied with deviations smaller than 1% for systems with dynamical
timescales $\leq$ 200 Myr. The scale invariant effects are generally
negligible in current life, they may however play a role in some astrophysical
cases. The SIV theory already shows several positive results: on basic
cosmological tests [32, 38, 40] with the occurrence of an accelerated
expansion, on the growth of density fluctuations [37], as well on first
dynamical tests [33].

Section 2 gives a presentation of the fundamental properties and equations of
the scale invariant theory. Section 3 is devoted to clusters of galaxies,
Section 4 to rotation of galaxies at the present time, as well as for
redshifted galaxies. In Section 5, the case of the velocity dispersion
perpendicular to the galactic plane is examined. Section 6 gives the
conclusion.

##  2 Fundamental properties of the scale invariant theory

Scale invariance is another symmetry in addition to the covariance of General
Relativity. In this new symmetry, the equation of the gravitational field
allow possible changes of the length of the line element $ds$.

The scale invariant vacuum (SIV) theory tries to follow the statement by Dirac
[10]: _“It appears as one of the fundamental principles in Nature that the
equations expressing basic laws should be invariant under the widest possible
group of transformations”_. Thus, are there further invariances to be
considered in the theory of gravitation ? This simple question is very
meaningful since scale invariance is present in General Relativity (GR) in
empty space without a cosmological constant and is also present in Maxwell
equations without charges and currents. Thus, we may wonder whether some
limited effects of scale invariance are still present in our very low density
Universe. The point is of interest regarding the long standing uncertainties
regarding the assumed dark components.

The problem is not new. Weyl [62] and Eddington [12] considered scale
invariance in an attempt to account for electromagnetism by a geometrical
property of the space-time. The proposition was then abandonned because
Einstein [13] had shown that the properties of a particle would have depended
on its past world line, so that atoms in an electromagnetic field would not
show sharp lines. Dirac [10] and Canuto et al. [5] considered an interesting
new possibility: the Weyl Integrable Geometry (WIG) where Einstein’s criticism
does not apply as shown below.

###  2.1 Basic assumptions

We refer to the above works by Dirac and Canuto et al. for a more detailed
study of the WIG space. In this Section we just give a short summary of the
main poperties of the SIV theory, see also [35]. The WIG space is the
appropriate framework to study scale invariance. Alike GR, it is endowed with
a metrical connection $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ and a line element
$ds^{\prime}$ of GR transforms into $ds$ in the WIG space,

$ds^{\prime}=\lambda(x^{\mu})\;ds,\quad\mathrm{with}\;\lambda\mathrm{\;the\;scale\;factor.}$
(2.1)

Thus, there is a conformal relation
$g^{\prime}_{\mu\nu}=\lambda^{2}g_{\mu\nu}$ between the two spaces. Moreover,
in the transport from a point $P_{0}(x^{\mu})$ to a nearby point
$P_{1}(x^{\mu}+dx^{\mu})$, the length $\ell$ of a vector is assumed to change
by $\;d\ell\,=\,\ell\,\kappa_{\nu}\,dx^{\nu}.$ There, $\kappa_{\nu}$ is called
the coefficient of metrical connection, it is a fundamental characteristics of
the geometry alike $g_{\mu\nu}$. We note that in GR $\kappa^{\prime}_{\nu}=0$.

The key difference between Weyl’s Geometry [62] and the WIG is that in the
latter $\kappa_{\nu}$ is the gradient of a scalar field $\Psi$ [10, 5],

$\kappa_{\nu}=-\Psi_{,\nu}=-\frac{\partial\ln\lambda}{dx^{\nu}},\quad\quad\mathrm{with}\;\;\Psi=\ln\lambda.$
(2.2)

This implies that $\kappa_{\nu}$ is a perfect differential with
$\partial_{\nu}\kappa_{\mu}=\partial_{\mu}\kappa_{\nu}$ and thus the parallel
displacement of a vector along a closed loop in WIG does not change its
length. Thus, the length change of a vector does not depend on the path
followed. This is the specificity of WIG [5] with respect to the classical
Weyl’s Geometry [62] and the criticism by Einstein does not apply in WIG.

The inclusion of scale invariance demands further developments of the tensor
calculus, with the introduction of _cotensors_. These mathematical expressions
enjoy both the covariance of GR and the scale covariance. A quantity $Y,$
scalar, vector or tensor, which by a scale transformation changes like
$Y\rightarrow\,Y^{\prime}\,=\,\lambda^{n}(x)\,Y$, is said to be a _coscalar,
covector, or cotensor_ of power $\Pi(Y)=n$; one then speaks of _scale
covariance_. For $n=0$ (_scale invariance_), one has an _inscalar, invector,
or intensor_. The ordinary derivatives $Y_{,\mu}$ as well as the covariant
derivative $Y_{;\mu}$ are not necessarily _co-covariant_ (a definition by
Dirac) or invariant, but derivatives with such properties can be defined and
are noted by $Y_{*\mu}$.

A brief summary of co-tensor calculus has been given by Canuto et al. [5].
Many useful expressions can also be found in [10], as well as in [4]. These
developments are essential when studying the scale invariant field equation in
order to derive the cosmological equations [32]. However at the level of the
geodesics we are considering here, we can manage without calling for
cotensors.

###  2.2 The geodesics from an action principle

The equation of geodesics is so fundamental that we provide herewith a
detailed demonstration. This equation leads to: - the weak field or Newtonian
approximation, - the derivation of post-Newtonian effects, - and the lensing
effect. In WIG can be obtained in different ways, as is the case in GR:

\- A geodesic is a curve such that a co-vector tangent to the curve is always
transported by parallel displacement along the curve. In GR, one has
$u^{\prime\mu}_{;\nu}u^{\prime\nu}=0$, and in WIG $u^{\mu}_{*\nu}u^{\nu}=0$
[5].

\- It can be derived from the Equivalence Principle, which states [61] that at
every point of the space-time there is a local inertial coordinate system
$x^{\prime\alpha}$ such that $\frac{d^{2}x^{\prime\alpha}}{ds^{\prime 2}}=0$,
a condition applied by [36].

\- The geodesic can also be derived from an action principle [10]. Below, we
give the derivation of this equation which is basic for the studies of motions
in planetary systems and galaxies, as well as for the gravitational lensing
[4].

We could think to define the geodesics as the path along which the change of
length of a vector is minimum. This could apply to the classical Weyl’s
geometry, but not to WIG, since as said above, the change of length during a
displacement is independent of the path followed. Thus, we simply express that
the path $\mathcal{I}$ followed by a particle between two fixed points $P_{0}$
and $P_{1}$ should be an extremum and develop its implication in the WIG space
in terms of the two geometrical factors $g_{\mu\nu}$ and $\kappa_{\nu}$, see
Eq. 8.1 [10].

$\mathcal{I}=\int^{P_{1}}_{P_{0}}ds^{\prime}=\int^{P_{1}}_{P_{0}}\lambda ds.$
(2.3)

We express the following action principle,

$\delta\mathcal{I}=\delta\int^{P_{1}}_{P_{0}}\lambda
ds=\int^{P_{1}}_{P_{0}}\delta\left(\lambda{ds}\right)=0\,,$ (2.4)

with

$\delta\left(\lambda\;{ds}\right)^{2}=2\lambda\,ds\,\delta\left(\lambda\,ds\right).$
(2.5)

Thus, $\delta\mathcal{I}$ becomes,

$\delta\mathcal{I}=\int\frac{1}{2\lambda}\delta\left(\lambda^{2}\;g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}\right)ds=0,$
(2.6)

the fixed limits being omitted since $\delta x^{\nu}(P_{0})=0$ and the same
for $P_{1}$. Expressing the integrand, we get,

$g_{\mu\nu}\,u^{\mu}u^{\nu}\delta\lambda+\frac{\lambda}{2}\;\left(\frac{\partial}{\partial
x^{\,\rho}}g_{\mu\nu}\right)\;u^{\mu}u^{\nu}\,\delta
x^{\,\rho}+\lambda\,g_{\mu\nu}\,u^{\mu}\delta u^{\nu},$ (2.7)

where $u^{\mu}=dx^{\mu}/ds$ and $u^{\nu}=dx^{\nu}/ds$ are tangent vectors. The
last term can be written
$\delta\,u^{\nu}=\delta\left(\frac{dx^{\nu}}{ds}\right)=\frac{d}{ds}(\delta
x^{\nu})$ and this last term can be integrated by parts so that $\delta I$
becomes,

$\delta\mathcal{I}=\int\left[g_{\mu\nu}\frac{\partial\lambda}{\partial
x^{\,\rho}}\,\ u^{\mu}u^{\nu}+\frac{1}{2}\lambda\left(\frac{\partial
g_{\mu\nu}}{\partial
x^{\,\rho}}\right)\,u^{\mu}u^{\nu}-\frac{d}{ds}\left(g_{\,\rho\mu}\,u^{\mu}\lambda\right)\right]\delta
x^{\rho}=0\,.$ (2.8)

There, the integrated part which vanishes at the limits has disappeared. Now
we can develop the last term in the integrand,

$-\frac{d}{ds}\left(g_{\,\rho\mu}\,u^{\mu}\lambda\right)=-\left(\frac{\partial}{\partial
x^{\nu}}g_{\rho\mu}\right)\lambda\frac{\partial x^{\nu}}{ds}u^{\mu}-\lambda
g_{\rho\mu}\frac{du^{\mu}}{ds}-\left(\frac{\partial\lambda}{\partial
x^{\nu}}\right)g_{\rho\mu}\,u^{\mu}\frac{\partial x^{\nu}}{ds}\,.$ (2.9)

The action principle now becomes,

$\displaystyle\quad\quad\quad\delta\mathcal{I}$ $\displaystyle=$
$\displaystyle\left[-g_{\mu\nu}\frac{\partial\lambda}{\partial x^{\,\rho}}\,\
u^{\mu}u^{\nu}-\frac{1}{2}\lambda\left(\frac{\partial g_{\mu\nu}}{\partial
x^{\,\rho}}\right)\,u^{\mu}u^{\nu}\right]\delta x^{\,\rho}$ (2.10)
$\displaystyle+$ $\displaystyle\left[\lambda\frac{\partial
g_{\,\rho\mu}}{\partial x^{\nu}}u^{\mu}u^{\nu}+\lambda
g_{\,\rho\mu}\frac{du^{\mu}}{ds}+g_{\,\rho\mu}\frac{\partial\lambda}{\partial
x^{\nu}}u^{\mu}u^{\nu}\right]\delta x^{\,\rho}=0\,.$

This applies at all points of the curve for any displacements $\delta
x^{\,\rho}$, thus the integrand vanishes at all points. Expressing the
metrical connection
$\kappa_{\nu}=-\frac{1}{\lambda}\frac{\partial\lambda}{\partial x^{\nu}}$, we
get

$\quad g_{\,\rho\mu}\frac{du^{\mu}}{ds}-\left(\frac{1}{2}\frac{\partial
g_{\mu\nu}}{\partial x^{\,\rho}}+\frac{\partial g_{\,\rho\mu}}{\partial
x^{\nu}}+g_{\mu\nu}\kappa_{\,\rho}-g_{\,\rho\mu}\kappa_{\nu}\right)u^{\mu}u^{\nu}=0.$
(2.11)

We can write $\frac{\partial g_{\,\rho\mu}}{\partial
x^{\nu}}u^{\mu}u^{\nu}=\frac{1}{2}\left(\frac{\partial g_{\,\rho\mu}}{\partial
x^{\nu}}+\frac{\partial g_{\,\nu\rho}}{\partial x^{\mu}}\right)u^{\mu}u^{\nu}$
and obtain,

$\quad
g_{\,\rho\mu}\frac{du^{\mu}}{ds}+\Gamma_{\rho,\mu\nu}\,u^{\mu}u^{\nu}+\kappa_{\,\rho}\,g_{\mu\nu}u^{\mu}u^{\nu}-\kappa_{\nu}g_{\,\rho\mu}u^{\mu}u^{\nu}=0,$
(2.12)

Multiplying this equation by $g^{\,\rho\alpha}$, one obtains

$\quad\frac{du^{\alpha}}{ds}+\Gamma^{\alpha}_{\mu\nu}\,u^{\mu}u^{\nu}+\kappa^{\alpha}\,g_{\mu\nu}u^{\mu}u^{\nu}-\kappa_{\nu}u^{\nu}u^{\alpha}=0.$
(2.13)

Using the above expressions of $u^{\mu}$ and $u^{\nu}$, we get simply

$\frac{du^{\alpha}}{ds}+\Gamma^{\alpha}_{\mu\nu}\,u^{\mu}u^{\nu}-\kappa_{\nu}u^{\nu}u^{\alpha}+\kappa^{\alpha}=0,$
(2.14)
$\mathrm{with}\quad\Gamma^{\alpha}_{\mu\nu}=g^{\rho\alpha}\Gamma_{\rho,\mu\nu}\quad\mathrm{with}\;\;\Gamma_{\rho,\mu\nu}=\frac{1}{2}\left(\frac{\partial
g_{\mu\,\rho}}{\partial x^{\nu}}+\frac{\partial g_{\nu\,\rho}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\,\rho}}\right),$ (2.15)

and

$\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\sigma\alpha}\left(\frac{\partial
g_{\nu\,\sigma}}{\partial x^{\mu}}+\frac{\partial g_{\mu\,\sigma}}{\partial
x^{\nu}}-\frac{\partial g_{\nu\mu}}{\partial x^{\,\sigma}}\right)\,.$ (2.16)

Eq. (2.14) only depends on the WIG geometrical factors $g_{\nu\mu}$ and
$\kappa_{\nu}$. It is the same expression as that obtained by the other
methods. If $\lambda$ is a constant, one has $\kappa_{\nu}=0$ and the geodetic
equation is the same as the classical one, cf. [61].

###  2.3 The Newtonian approximation in weak fields

The Newtonian approximation was obtained by [36] for a weak stationary
potential $\Phi$, with a line element only slightly differing from
Minkowski’s, with

$g_{ii}\cong-1,\quad i=1,2,3\quad\mathrm{and}\;\;g_{00}=1+2\,{\Phi}\,,$ (2.17)

setting $c=1$ and the same below. With the above $g_{\mu\nu}$, the
$\Gamma^{\alpha}_{\mu\nu}$ vanish, except
$\Gamma^{i}_{00}=g^{i\alpha}\frac{\partial\Phi}{\partial x^{\alpha}}$ and for
a potential of the classical form $\Phi=GM/r$, this just gives
$\Gamma^{i}_{00}=-\frac{GM}{r^{2}}$ For slow motions vs. $c$,
$u^{i}\rightarrow\upsilon^{i}=\frac{dx^{i}}{dt}$ and $u^{0}\rightarrow 1$,
with $ds\rightarrow dt$. All these are just the approximations currently made
in the weak field low velocity approximation [61]. At this stage, the
expressions of $\lambda$ and thus of $\kappa_{\nu}$ are not specified, this
will be done in Sect. 2.4. For now, let us consider the case of the empty
space with $\kappa_{\nu}=(0,0,0,\kappa_{0})$. Eq. (2.14) is leading to,

$\frac{d\upsilon^{i}}{dt}+g^{i\alpha}\frac{\partial\Phi}{\partial
x^{\alpha}}-\kappa_{0}\upsilon^{i}=0,$ (2.18)

and in vectorial form,

$\frac{d^{2}\vec{r}}{dt^{2}}=-\frac{GM}{r^{2}}\frac{\vec{r}}{r}+\kappa_{0}\frac{d\vec{r}}{dt}\,.$
(2.19)

An acceleration term proportional to the velocity and in the same direction
appears in the 2nd member. It is such as to accelerate an expansion or a
contraction depending on circumstances. Such effects appear _e.g._ in the
cosmological models [32] and in the rapid growth of density fluctuations in
the early Universe [37].

Some properties of the theory, such as $\lambda$, $\kappa_{\nu}$ and the way
the differents physical quantities depend on the scale $\lambda$, are
determined by the conservation laws and the considered cosmological models.
Therefore, it is now necessary to specify a few physical properties in
presence of a varying gauge.

###  2.4 The scale invariant field equation and the gauge

For detailed developments of the scale invariant theory and cosmology, the
reader may refer to [32, 38, 35]. Here we limit the presentation to a few
relevant points. A general scale invariant field equation was obtained by [5],

$\displaystyle R^{\prime}_{\mu\nu}-\frac{1}{2}\
g_{\mu\nu}R^{\prime}-\kappa_{\mu,\nu}-\kappa_{\nu,\mu}-2\kappa_{\mu}\kappa_{\nu}+2g_{\mu\nu}\kappa^{\alpha}_{,\alpha}-g_{\mu\nu}\kappa^{\alpha}\kappa_{\alpha}=-8\pi
GT_{\mu\nu}-\lambda^{2}\Lambda_{\mathrm{E}}\,g_{\mu\nu},\,$ (2.20)

where $R^{\prime}_{\mu\nu}$ is the Ricci tensor in GR and $R^{\prime}$ its
contracted form, $\Lambda_{\mathrm{E}}$ the cosmological constant in GR, $G$
is a true constant. A fully detailed demonstration of this field equation by
an action principle has also been recently established [41] in the line of
earlier works by [10], who established the expression of the first member
which is scale invariant, and by [5] who also further developed the WIG
concepts.

The second member of the field equation must also be scale invariant, and the
energy-momentum tensor $G\,T_{\mu\nu}$ has the same property. Thus,
$T_{\mu\nu}\,=\,T^{\prime}_{\mu\nu}$, which has implications on the relevant
densities, _e.g._ for a perfect fluid,

$(p+\varrho)u_{\mu}u_{\nu}-g_{\mu\nu}p=(p^{\prime}+\varrho^{\prime})u^{\prime}_{\mu}u^{\prime}_{\nu}-g^{\prime}_{\mu\nu}p^{\prime}\,.$
(2.21)

The velocities $u^{\prime\mu}$ and $u^{\prime}_{\mu}$ transform like,

$\displaystyle u^{\prime\mu}$ $\displaystyle=$
$\displaystyle\frac{dx^{\mu}}{ds^{\prime}}=\lambda^{-1}\frac{dx^{\mu}}{ds}=\lambda^{-1}u^{\mu}\,,$
$\displaystyle\mathrm{and}\;\;u^{\prime}_{\mu}$ $\displaystyle=$
$\displaystyle
g^{\prime}_{\mu\nu}u^{\prime\nu}=\lambda^{2}g_{\mu\nu}\lambda^{-1}u^{\nu}=\lambda\,u_{\mu}\,.$
(2.22)

The contravariant and covariant components of a vector have different powers
and $T_{\mu\nu}$ is scaling like,

$(p+\varrho)u_{\mu}u_{\nu}-g_{\mu\nu}p=(p^{\prime}+\varrho^{\prime})\lambda^{2}u_{\mu}u_{\nu}-\lambda^{2}g_{\mu\nu}p^{\prime}\,,$
(2.23)

implying for $p$ and $\varrho$ [5],

$p=p^{\prime}\,\lambda^{2}\,\quad\mathrm{and}\quad\varrho=\varrho^{\prime}\,\lambda^{2}\,.$
(2.24)

The consistency of the field equation implies that $p$ and $\varrho$ are not
scale invariant, but coscalars of power $\Pi(\rho)=-2$. These properties are
consistent with the equation of conservation (2.34) discussed below.

Chosing a metric is necessary to fix the $g_{\mu\nu}$, similarly a gauging
condition is necessary to fix the scale or gauge $\lambda$. Dirac [10] and
Canuto et al. [5] had chosen the so-called “Large Number Hypothesis“, see also
[11]. The author’s choice is to adopt the following statement [32]: _The
macroscopic empty space is scale invariant, homogeneous and isotropic_. The
equation of state of the vacuum
$p_{\mathrm{vac}}=-\varrho_{\mathrm{vac}}c^{2}$ is precisely the relationship
permitting the vacuum density to remain constant for an adiabatic expansion or
contraction [6]. The above gauging condition also implies
$\lambda=\lambda(t)$, since the empty space is homogeneous and isotropic.
Applied to the field equation in empty space, the gauging condition is leading
to two differential equation for $\lambda$ which are then only a function of
time [32],

$\displaystyle\
3\,\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,\quad\mathrm{and}\quad\,2\frac{\ddot{\lambda}}{\lambda}-\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,,$
(2.25)

Impressively enough, the trace of Eq. (2.20) in empty space is also leading to
these two equations [41]. Their solution is,

$\lambda(t)=\sqrt{\frac{3}{\Lambda_{\mathrm{E}}}}\frac{1}{ct}\,\quad\mathrm{or}\;\;\lambda(t)=\frac{t}{t_{0}},$
(2.26)

with $\lambda=1$ for $t_{0}=1$. This implies that the only component of
$\kappa_{\nu}$ is $\kappa_{0}=-\dot{\lambda}/\lambda=1/t.$ Thus, we see that
the above Eq. (2.19) has consistent unities. The form of Eqs. (2.25) and
(2.26) is independent of the matter content according to the gauging
condition, but the range of variations of $t$, and thus of $\lambda$ and
$\kappa_{\nu}$, are set by the considered comological models [32].

###  2.5 Cosmological constraints

The application of the FLWR metric to Eq. (2.20) led to rather ”heavy”
cosmological equations [5], however the equations (2.25) bring interesting
simplifications, letting only one term with a $\lambda$-function [32],

$\displaystyle\frac{8\,\pi G\varrho}{3}\,$ $\displaystyle=$
$\displaystyle\,\frac{k}{a^{2}}+\frac{\dot{a}^{2}}{a^{2}}+2\,\frac{\dot{a}\dot{\lambda}}{a\lambda},$
(2.27) $\displaystyle-8\,\pi Gp\,$ $\displaystyle=$
$\displaystyle\,\frac{k}{a^{2}}+2\frac{\ddot{a}}{a}+\frac{\dot{a^{2}}}{a^{2}}+4\frac{\dot{a}\dot{\lambda}}{a\lambda},$
(2.28) $\displaystyle-\frac{4\pi G}{3}\left(3p+\varrho\right)\,$
$\displaystyle=$
$\displaystyle\,\frac{\ddot{a}}{a}+\frac{\dot{a}\dot{\lambda}}{a\lambda}.$
(2.29)

These equations differ from Friedmann’s by only one term. For a constant
$\lambda$, Friedmann’s equations are recovered. The 3rd equation is derived
from the first two. Since $\dot{\lambda}/{\lambda}$ is negative, the extra
term represents an additional acceleration _in the direction of motion_ and is
thus fundamentally different from that of the cosmological constant.

Analytical solutions for the flat SIV models with $k=0$ were found for the
matter era by [21],

$a(t)=\left[\frac{t^{3}-\Omega_{\mathrm{m}}}{1-\Omega_{\mathrm{m}}}\right]^{2/3},\quad\quad\mathrm{the\;initial\;time\;is}\;\;t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}},$
(2.30)

and for radiation dominated models by [34]. These equations allow flatness for
different values of $\Omega_{\mathrm{m}}$, unlike Friedmann’s models. At
present $t_{0}=1$ and $a(t_{0})=1$. The graphical solutions are relatively
close to the corresponding $\Lambda$CDM models [32], with larger differences
for very low $\Omega_{\mathrm{m}}$. The initial time $t_{\mathrm{in}}$ of the
origin is defined by $a(t_{\mathrm{in}})=0$. For
$\Omega_{\mathrm{m}}=0,0.01,0.1,0.3,0.5$, the values of $t_{\mathrm{in}}$ are
0, 0.215, 0.464, 0.669, 0.794 respectively. This strongly reduces the possible
range of $\lambda(t)=(t_{0}/t)$, between $1/t_{\mathrm{in}}$ and $1/t_{0}=1$.
Scale invariant effects are drastically reduced in Universe models with
increasing $\Omega_{\mathrm{m}}$. They are killed for $\Omega_{\mathrm{m}}\geq
1$.

While $t$ varies between $t_{\mathrm{in}}$ and $t_{0}=1$ at present; the usual
timescale $\tau$ in years or seconds varies from $\tau_{\mathrm{in}}=0$ and
$\tau_{0}=13.8$ Gyr at present [15]. The relation between these age units is
$\frac{\tau-\tau_{\mathrm{in}}}{\tau_{0}-\tau_{\mathrm{in}}}=\frac{t-t_{\mathrm{in}}}{t_{0}-t_{\mathrm{in}}}$,
expressing that the age fractions of an event are the same. This gives

$\displaystyle\tau\,=\,\tau_{0}\,\frac{t-t_{\mathrm{in}}}{1-t_{\mathrm{in}}}\,\quad\mathrm{and}\;\;t\,=\,t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})\,,$
(2.31)
$\displaystyle\mathrm{with}\quad\frac{d\tau}{dt}\,=\,\frac{\tau_{0}}{1-t_{\mathrm{in}}}\,,\quad\mathrm{and}\;\;\frac{dt}{d\tau}\,=\,\frac{1-t_{\mathrm{in}}}{\tau_{0}}\,.$
(2.32)

Both timescales are evidently linearly related
$\frac{dX}{d\tau}=\frac{dX}{dt}\frac{dt}{d\tau}$, with a constant
$\frac{dt}{d\tau}$ for a given $\Omega_{\mathrm{m}}$.

Eqs. (2.27) and (2.28) lead to the scale invariant mass-energy conservation
law [5, 32],

$\displaystyle\frac{d(\varrho
a^{3})}{da}+3\,pa^{2}+(\varrho+3p)\frac{a^{3}}{\lambda}\frac{d\lambda}{da}=0,$
(2.33)
$\displaystyle\mathrm{and}\quad\varrho\,a^{3(1+w)}\lambda^{1+3w}=\mathrm{const.}\quad\mathrm{with\;p=w\varrho},$
(2.34)

(there $c=1$). For ordinary matter $w=0$ and
$\varrho_{\mathrm{m}}a^{3}\lambda=const.$ A length element scales like
$\lambda^{-1}$, thus from (2.34) the mass also varies like $\lambda^{-1}\sim
t$. Such a mass variation might at first be surprising, however let us recall
that the invariance to Lorentz transformations, assumed in Special Relativity,
is also leading to the concept of non-constant masses, which may vary up to
infinity as a function of the velocities. Here, as in Special Relativity the
number of particles remains the same, but their inertial and gravitational
properties may vary. This property is quite consistent with the above
mentioned scale invariance of the energy momentum tensor $T_{\mu\nu}$ in the
second member of the general field equation (2.20). It implies that
$\varrho=\varrho^{\prime}\lambda^{2}$ (cf. Eq. 2.24) or
$\frac{M}{R^{3}}=\lambda^{2}\frac{M^{\prime}}{R^{\prime 3}}$, since
$R^{\prime}=\lambda R$ one has $M=M^{\prime}/\lambda$, meaning that $M\sim t$.
A noticeable consequence of the mass variation is that the gravitational
potential $\Phi=-GM/R$ of a given mass appears as a more fundamental quantity
than the mass, being scale invariant and thus independent of time through the
evolution of the Universe. These results about an additional physical
invariance are consistently derived from first principles, however this does
not prove that Nature is proceeding this way. The purpose of the present work
is precisely to proceed to some observational tests.

The mass changes are very limited, from (2.31) they behave like
$M(\tau)=M(\tau_{0})[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]$.
As examples for $\Omega_{\mathrm{m}}=0.30$, the initial mass of the Sun was
0.891 of the present one and 400 Myr ago, masses were differing by less than
1% from today’s. Thus, galactic masses are about constant over such a time
interval and in this case (only) the deep MOND limit appears as an
approximation of the scale invariant theory [35].

Equation (2.19) was expressed in variable $t$, where the age of the Universe
is $(1-t_{\mathrm{in}})$. One has
$\frac{d^{2}r}{dt^{2}}=\frac{d^{2}r}{d\tau^{2}}(\frac{d\tau}{dt})^{2}$, and
the constant $G$ in $t$-units becomes $G(\frac{dt}{d\tau})^{2}\;(\equiv G$ in
current units). Thus at present $\tau_{0}$, one has

$\frac{d^{2}\bf{r}}{d\tau^{2}}=-\frac{GM}{r^{2}}\frac{\bf{r}}{r}+\frac{\psi_{0}}{\tau_{0}}\frac{d\bf{r}}{d\tau},\quad\mathrm{with}\;\;\psi_{0}=1-t_{\mathrm{in}}\,.$
(2.35)

For $\Omega_{\mathrm{m}}=0,0.05,0.10,0.20,0.30,1$, one has
$\psi_{0}=1,0.632,0.536,0.415,0.331,0.$ At other epochs $\tau$, instead of
$\psi_{0}$ one has the numerical factor

$\psi(\tau)=\frac{t_{0}-t_{\mathrm{in}}}{\left(t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})\right)},\quad\quad\left(=\frac{t_{0}-t_{\mathrm{in}}}{t}\;\right).$
(2.36)

The additional acceleration, the dynamical gravity, is generally small since
$\tau_{0}$ is large. In an empty universe, $\psi_{0}=1$ and the dynamical
acceleration is the largest, while for $\Omega_{\mathrm{m}}\geq 1$,
$\psi_{0}=0$ and the additional term consistently disappears.

##  3 Dynamics of the clusters of galaxies

The above prescriptions are now submitted to major observational tests,
starting with the velocities in clusters of galaxies.

###  3.1 Observational constraints

Since the historical discovery by [65], later followed by [22], of large mass
excesses in clusters of galaxies when comparing galaxy velocities and
luminosities, evidences have further accumulated (e.g. [3], [51], [57]).
Virial masses in excess with respect to stellar luminosities have been
confirmed, reaching a factor of about 30-50 [51] for large clusters with
masses between $10^{14}$ and $10^{15}$ M⊙. This implies that the stellar mass
fraction is only of the order of 0.02-0.03. The mass estimates can be affected
by uncertainties in the mass-luminosity ratio $(M/L)$ and also by different
dynamical substructures within clusters [49]. However, the above quantitative
estimates are unchanged. As mentioned above, there is a relation between the
total dynamical mass of clusters and the baryonic masses of central regions
[7]. Such an unexpected relation, also found by [29] in galaxies, is
challenging for the current $\Lambda$CDM models.

X-ray observations of the emitting gas allow estimates of the gas mass
fraction (0.10-0.15) in clusters [30, 28, 19, 55, 16, 9]. This largely
dominates over the stellar mass fraction mentioned above. These two fractions
contribute to the baryon mass fraction,
$f_{\mathrm{bar}}=\frac{M_{\mathrm{star}}+M_{\mathrm{gas}}}{M_{\mathrm{tot}}}$.
The total baryon fraction appeared nearly constant with cluster mass between
0.12 and 0.15 [19]. From a sample of 120 clusters, a gas mass fraction of
0.156($\pm 0.003$) was estimated [50]. From a sample of 40 clusters with deep
Chandra, X-ray observations determined [43] a gas mass fraction on the average
similar, but with a trend to increase with the overdensity $\Delta$ (the ratio
to the critical density at the considered redshift). In the Perseus cluster,
the gas mass fractions are in the range of 0.09 to 0.20 for $\Delta$ going
from $10^{4}$ to 500. Gas mass fractions between 0.12 and 0.16 were found
[63], with increasing values with redshift $z$, also increasing as a function
of their masses, an effect possibly due to a bias due the hypothesis of
hydrostatic equilibrium in cluster models.

Summing the stellar and gas mass fractions one gets the baryon mass fraction
which turns around 0.15 - 0.18, a factor of about 6 smaller than unity, a
result in agreement with [50] for matter, baryon and dark matter density
parameters: $\Omega_{\mathrm{m}}=0.315\pm 0.007$,
$\Omega_{\mathrm{b}}=0.050\pm 0.0002$ and $\Omega_{\mathrm{DM}}=0.265\pm
0.002$.

###  3.2 Cluster dynamics in the scale invariant context

This matter was already considered [33], however an update to include the
limited $\lambda$-variations and new observations is necessary. This reduction
of the scale invariant effects is expressed by the factor $\psi_{0}$ (or
$\psi(\tau)$) in the equation of motion (2.35), as well as in its various
applications. The question is now whether in the scale invariant context the
baryonic matter alone is sufficient or not to account for the high observed
velocities in galaxy clusters.

As the current Virial theorem does not apply in an expanding system, we start
from the equation of motion (2.35) and consider a galaxy $i$ with velocity
$\upsilon_{i}$ attracted by another $j$-one at time $\tau$ and distance
$r_{ij}$. Its acceleration is

$\displaystyle\frac{d\upsilon_{i}}{d\tau}=\frac{Gm_{j}(\tau)}{r^{2}_{ij}}+\frac{\psi(\tau)}{\tau_{0}}\upsilon_{i},\quad\mathrm{with}\;\;\psi(\tau)=\frac{1}{\frac{\Omega^{1/3}_{\mathrm{m}}}{1-\Omega^{1/3}_{\mathrm{m}}}+\frac{\tau}{\tau_{0}}}.$
(3.1)
$\displaystyle\mathrm{thus}\quad\frac{\psi(\tau)}{\tau_{0}}=\frac{1}{\tau_{\Omega}+\tau}\,,\quad\mathrm{with}\;\;\tau_{\Omega}=\tau_{0}\,\frac{\Omega^{1/3}_{\mathrm{m}}}{1-\Omega^{1/3}_{\mathrm{m}}},$
(3.2)

where $\Omega^{1/3}_{\mathrm{m}}=t_{\mathrm{in}}$, the initial time (Big
Bang). We multiply the first equation by $\upsilon_{i}=dr_{ij}/d\tau$,
integrate and sum on all the other $(N-1)$ $m_{j}$-masses. The variations of
the mass $m_{j}(\tau)$ are very limited, thus we consider a mean value
$p\,m_{j}(\tau_{0})$ of masses $m_{j}$ at the middle age of the Universe, thus
$p$ is given by $p\approx(1+\Omega^{1/3}_{\mathrm{m}})/2$. Values of $p$ for
different $\Omega_{\mathrm{m}}$ are given in Table 1 (the range of the
$p$-values is small). One is getting,

$\frac{1}{2}\upsilon^{2}_{i}\,=\,p\sum_{j\neq
i}\frac{G\,m_{j}(\tau_{0})}{r_{ij}}+\int\frac{1}{\left[\tau_{\Omega}+\tau\right]}\,\upsilon^{2}_{i}d\tau\,.$
(3.3)

Then, we are also summing on all the N $i$-masses and get,

$\frac{1}{2}\sum_{i}\upsilon^{2}_{i}=\frac{1}{2}p\sum_{i}\sum_{j\neq
i}\frac{G\,m_{j}(\tau_{0})}{r_{ij}}+\sum_{i}\int\frac{1}{\left[\tau_{\Omega}+\tau\right]}\,\upsilon^{2}_{i}d\tau\,.$
(3.4)

The factor $1/2$ in front of the double summation is necessary in order not to
account twice the same masses.

Table 1:  Parameters for cluster dynamics and ratios of the mass estimates in
the standard case and in the scale invariant theory, calculated with
$\tau_{0}=13.8$ Gyr and $\tau_{1}=400$ Myr.

$\Omega_{\mathrm{m}}$ | $p$ | $\tau_{\Omega}$ | $x(\Omega_{\mathrm{m}})$ | $M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ | $M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ | $M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ | $M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ | $M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ | $M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ |
---|---|---|---|---|---|---|---|---|---|---
|  | [Gyr] |  | $f=0.5$ | $f=0.6$ | $f=0.7$ | $f=0.8$ | $f=0.9$ | $f=1.0$ |
0.05 | 0.684 | 8.05 | 0.950 | 13.68 | $\infty$ | $\infty$ | $\infty$ | $\infty$ | $\infty$ |
0.10 | 0.732 | 11.95 | 0.735 | 2.76 | 6.20 | $\infty$ | $\infty$ | $\infty$ | $\infty$ |
0.20 | 0.792 | 19.44 | 0.516 | 1.64 | 2.07 | 2.85 | $4.54$ | $11.12$ | $\infty$ |
0.30 | 0.835 | 27.95 | 0.387 | 1.36 | 1.56 | 1.82 | $2.19$ | $2.75$ | $3.69$ |

Note: the signs $\infty$ indicate a divergence of the ratio
$M_{\mathrm{Std}}/M_{\mathrm{SIV}}$ in Eq. (3.9) .

Now, we divide the above equation by $N$. The term on the left becomes
$\frac{1}{2}\overline{\upsilon^{2}}$. The first term on the right gives the
potential energy $pqGM/R$ of the cluster with a total present mass $M$ and a
radius $R$, with $q$ a structural factor depending on the density profile
($q=3/5$ for an homogeneous distribution). The last term is an adiabatic
invariant, for which we take an average over the lifetime of the cluster
between $\tau_{1}$ and $\tau_{0}$, where $\tau_{1}$ is the formation epoch. It
is then easy to integrate it,

$\displaystyle
f\overline{\upsilon^{2}}\,\int^{\tau_{0}}_{\tau_{1}}\frac{d\tau}{\left[\tau_{\Omega}+\tau\right]}\,=f\overline{\upsilon^{2}}\,\ln\frac{\tau_{\Omega}+\tau_{0}}{\tau_{\Omega}+\tau_{1}}=f\overline{\upsilon^{2}}\times
x(\Omega_{\mathrm{m}}),$ (3.5)
$\displaystyle\mathrm{with}\;\;x(\Omega_{\mathrm{m}})=\ln\frac{\tau_{\Omega}+\tau_{0}}{\tau_{\Omega}+\tau_{1}}.\quad\quad$
(3.6)

The factor $f$ accounts for the integration of $\upsilon^{2}$ over time
$\tau$; different values of $f$ between 0.5 and 1.0 are considered in Table 1.
The values of $x(\Omega_{\mathrm{m}})$ as a function of different
$\Omega_{\mathrm{m}}$ are also given in the table. The integration constant on
the first term in (3.4) may be considered as zero, since a zero velocity is an
appropriate value for the end of the collapse; the same for the potential
which is vanishing at large distances. Equation (3.4) can be written as,

$\displaystyle\overline{\upsilon^{2}}\,=\,2\,\frac{pq\,GM_{\mathrm{SIV}}}{R}+2\,x(\Omega_{\mathrm{m}})\times
f\overline{\upsilon^{2}}\,,$ (3.7)

where $M_{\mathrm{SIV}}$ is the present mass in the scale invariant theory. In
the standard case, a value $M_{\mathrm{std}}$ would be derived from the same
value of the velocity dispersion $\overline{\upsilon^{2}}$ by the relation,

$\overline{\upsilon^{2}}\,=\,2\,q\,GM_{\mathrm{std}}/{R}.$ (3.8)

The ratio of the two mass estimates is given by,

$\frac{M_{\mathrm{std}}}{M_{SIV}}\,\approx\,\frac{p}{\left[1-2\,f\,x(\Omega_{\mathrm{m}})\right]}.$
(3.9)

The same structural factor $q$ is taken in the standard and scale invariant
cases. Also, we may consider that the statistical factors, which account for
the fact that the observed velocities are the radial ones, are the same in the
two theories. The factors $x(\Omega_{\mathrm{m}})$ are estimated by Eq. (3.6),
taking a ratio $\tau_{1}/\tau_{0}=3\%$, which corresponds to galaxy formation
at a cosmic time of 400 Myr, the choice of the initial time being not
critical. We note that the factor $p$ has no effect in the term responsible of
the divergence.

Table 1 gives the ratios of the mass estimates
$\frac{M_{\mathrm{std}}}{M_{SIV}}$. We see that for the whole range of
$\Omega_{\mathrm{m}}$ values, the standard mass estimates are much larger than
the SIV ones. The infinite values in the table result from the vanishing
denominators of Eq. (3.9) for the mass ratios
$\frac{M_{\mathrm{std}}}{M_{SIV}}$. Particularly, for low values of
$\Omega_{\mathrm{m}}$ such as $\Omega_{\mathrm{m}}=0.05$, the overestimate of
the masses becomes very important. Thus, if scale invariant effects are
present in Nature, we may expect, by applying standard estimates, large
overestimates of the real masses of the galaxy clusters, with then the
necessity of advocating large amounts of dark matter. Of course, in the
process of standard determinations from an expression like Eq. (3.8) no
mathematical divergence is occurring. Nevertheless, in the past ratios
$\frac{M_{\mathrm{std}}}{M_{SIV}}$ as high as 80 [22] or mass-luminosity
ratios $M/L$ of about 700 [1] have been obtained.

The conclusion of this analytical study is that scale invariant dynamics in
our low density Universe could account for the high observed velocities of
galaxies in clusters, despite the reduction factor $\psi(\tau)$ in the
equation of motion. Of course, numerical simulations would be needed to
further quantify the scale invariant effects; at least the present analytical
developments are an encouragement to the undertaking of such calculations.
Globally, it appears in this new context that the hypothesis of large amounts
of dark matter may no longer be necessary, the dynamical acceleration
$\left(\frac{\psi(\tau)}{\tau_{0}}\frac{d\bf{r}}{d\tau}\right)$ producing a
sufficient increase of the velocity dispersion during the ages. Now, we will
examine whether such a conclusion is supported by other dynamical problems.

##  4 Galactic rotation

###  4.1 The possible evolution of the rotation curves: theory and
observations

In the outer regions of spiral galaxies, the orbital motions should in
principle be close to Keplerian behaving like $1/\sqrt{r}$ being dominated by
the central mass concentration. The large differences from what is generally
observed with the flat or relatively flat rotation curves over a few tens of
kpc (see review by [56]) have greatly promoted, since the beginning, the
hypothesis of dark matter [60]. Such dark components have been searched over
the last forty years. We have already considered the topic previously [33],
but new observational and theoretical results have considerably enriched and
strengthened the subject.

###  4.2 Properties of orbital motions

The two-body problem in SIV theory has been studied by [36] and [33]. The
equation of motion (2.19) becomes in plane polar coordinates $r$ and
$\varphi$,

$\displaystyle\ddot{r}-r\dot{\varphi}^{2}=-\frac{G\,M}{r^{2}}+\kappa(t)\dot{r},\quad\quad
r\ \ddot{\varphi}+2\ \dot{r}\dot{\varphi}=\kappa(t)r\dot{\varphi}.$ (4.1)

The time variable $t$ is provisionally considered, since the equations are
much simpler; however we turn to the current time units $\tau$ for the main
conclusions below. Integration gives the equivalent law of angular momentum
conservation,

$\kappa\,r^{2}\,\dot{\varphi}=L=const.\quad\mathrm{with}\;\;\kappa=1/t.$ (4.2)

Some manipulations give the corresponding Binet equation [36],

$\frac{d^{2}u}{d\varphi^{2}}+u\,=\,\frac{GM\kappa^{2}}{L^{2}}\,,\quad\mathrm{with}\;\;u=1/r.$
(4.3)

The second member is scaling like $\lambda\sim 1/t$. This equation as in the
classical case is leading to a family of conics,

$r(\varphi)=\frac{p}{1+e\;cos\varphi},\;\;\mathrm{with}\;p=\frac{L^{2}}{GM\kappa^{2}}=\frac{b^{2}}{a},\;b=a\sqrt{1-e^{2}}.$
(4.4)

$a$ and $b$ are the semi-major and semi-minor axes. Eccentricity $e$ is scale
invariant; for $e=0$, one has $r=p$. Both $p$ and $r$ are of power -1 and thus
slightly increase with time. Differentiating $r$ gives,

$\frac{\dot{r}}{r}\,=\,1/t\,\quad\mathrm{implying}\;r\,\sim\,t\,,\;\mathrm{and}\;\frac{\Delta
r}{r}\,=\,\frac{\Delta t}{t}\,.$ (4.5)

The orbital motion of a bound system is a circle or an ellipse with a slight
outwards spiraling motion. With the above $t-\tau$ relations (2.31) and (2.32)
we get in current units,

$\displaystyle\frac{dr}{r}=\frac{dt}{t}=\frac{dt}{d\tau}\frac{d\tau}{t}=\
\frac{(t_{0}-t_{\mathrm{in}})}{(t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}}))}\frac{d\tau}{\tau_{0}}=\psi(\tau)\frac{d\tau\,}{\tau_{0}}.$
(4.6)

where $\psi(\tau)$ was already defined above (2.36). At present time
$\tau_{0}$, the relative change of an orbital radius is

$\frac{\Delta
r}{\Delta\tau}=\psi_{0}\,\frac{r_{0}}{\tau_{0}},\quad\mathrm{for\;a\;circle\;or\;an\;ellipse.}$
(4.7)

At a time $\tau$, one has $\psi(\tau)$ instead of $\psi_{0}$. The relation
between $r$ and $\tau$ is not linear, due to the term $\psi(\tau)$. However,
up to time intervals of 400 Myr the deviations from $\psi_{0}$ are of the
order of 1%, see also Eq. (4.10).

Remarkably, the effect of the additional acceleration term just compensates
the usual slowing down due to orbital expansion, thus keeping the orbital
$\upsilon$ constant,

$\upsilon=r\,\dot{\varphi}=\frac{L}{r\,\kappa}=\frac{GM}{r}=const.\quad\mathrm{since}\;\;r\sim
t\;\mathrm{and}\;\kappa=\frac{1}{t},$ (4.8)

an effect already found by [36].

###  4.3 Application to galactic rotation

Fig. 1 shows in red the present observed rotation curve of the Milky Way as a
function of the galactocentric distance $R(kpc)$, as given by [20]. Up to
about 26 kpcs there are some dips and bumps in the rotation curve, likely
corresponding to spiral arms. Over that part, the rotation curve may be
considered as globally flat. Then, beyond 26 kpc a progressive decrease is
observed up to distances as large as 100 kpc. This extended rotation curve is
much flatter than Keplerian. It is based on the observations of about 16’000
red clump giants in the outer disk and 5700 K giants in the halo.

Some basic properties of the scale invariant two-body mechanics have been
recalled in Sect. 4.2. With these laws we can easily reconstruct the evolution
of the galactic rotation over the ages due to scale invariant effects:

  * •

The orbital radius (here the galactocentric radius) increases with time (Eq.
4.6).

  * •

The orbital velocity $\upsilon$ of a given star around the galactic center is
keeping constant during the additional expansion (Eq. 4.8).

If the time intervals considered in the evolution of galaxies are not
negligible with respect to the age of the Universe, Eq. (4.7) cannot be used;
for such cases we must integrate Eq. (4.6) over time $\tau$ between $\tau_{1}$
(in Gyr) the epoch considered in the past with size $r_{1}$, and the present
time $\tau_{0}$ with size $r_{0}$,

$\displaystyle\int^{r_{0}}_{r_{1}}\frac{dr}{r}=\frac{(1-\Omega_{\mathrm{m}})^{1/3}}{\tau_{0}}\int^{\tau_{0}}_{\tau_{1}}\frac{d\tau}{\Omega_{\mathrm{m}}^{1/3}+\frac{\tau}{\tau_{0}}(1-\Omega_{\mathrm{m}}^{1/3})}=\int^{\tau_{0}}_{\tau_{1}}\frac{d\tau}{\tau_{0}\frac{\Omega_{\mathrm{m}}^{1/3}}{(1-\Omega_{\mathrm{m}})^{1/3}}+\tau}=\int^{\tau_{0}}_{\tau_{1}}\frac{d\tau}{\tau_{\Omega}+\tau},$
(4.9)

where we use the constant $\tau_{\Omega}$ defined in Eq. (3.2). The
integration gives

$\quad\quad\quad\quad\quad\quad\quad\frac{r_{0}}{r_{1}}\,=\,\frac{\tau_{\Omega}+\tau_{0}}{\tau_{\Omega}+\tau_{1}}.$
(4.10)

For example, for $\Omega_{\mathrm{m}}=0.045$, $\tau_{0}=13.8$ Gyr and an age
$\tau_{1}$ of 6 Gyr, one has $\tau_{\Omega}=7.62$ Gyr and a scaling of the
distances $\frac{r_{0}}{r_{1}}=1.57$. This means that at an age of 6 Gyr, the
same velocity as one on the red curve was reached at a galactocentric distance
$r_{1}$ a factor 1.57 smaller than the present distance $r_{0}$. This enables
the construction of the past rotation curves of galaxies at different past
ages 0.2, 3, 6, 9 Gyr, as shown by the blue curves in Fig. 1, on the basis of
the present red curve based on data by [20]. Among the blue curves, the left
one corresponds to the early age of galaxy formation (0.2 Gyr). This early
curve, although not identical, is nevertheless much closer to a Keplerian
curve than the present one. The differences to a Keplerian distribution are
currently interpreted as differences in the amount of dark matter present. For
consistency, we have considered here a cosmological model without dark matter,
thus where $\Omega_{\mathrm{m}}=\Omega_{\mathrm{b}}=0.045$ [50].

We can draw the following predictions about the past effects of scale
invariance in galaxies from the above Fig. 1:

  1. 1.

In the past, the extensions of galaxies were smaller than today. Galaxies were
more compact according to the ratio (4.10).

  2. 2.

Some part of the flat rotation curves is resulting from the cosmic expansion
associated to scale invariance.

  3. 3.

In the past, the rotation curves started to decrease at much smaller distance
from the galactic center than today, with a much steeper profile,
progressively tending to Keplerian near early times. Within the classical
interpretation in term of dark matter, the rotation curves would require
smaller fractions of dark matter in the past.

This describes only the effects resulting from scale invariance and does not
preclude all the other current effects of stellar dynamics to operate over the
ages.

Figure 1: Evolution of the rotation curve of the Milky Way. The grey points
are the observed velocity averages by [20] with their error bars. The thick
red line represents a mean rotation curve. The thin broken red line describes
the undulations in the globally flat part of the velocity distribution; up to
a radius of 26 kpc, we have drawn a flat thick red line. The blue lines show
the rotation curves predicted by the scale invariant theory for different ages
in the past history of the Universe. Calculations have been performed with no
dark matter in a model with $\Omega_{\mathrm{m}}=\Omega_{\mathrm{b}}$, where
$\Omega_{\mathrm{b}}=0.045$. The thick broken green line shows a Keplerian
curve in $1/\sqrt{r}$ near the age of galaxy formation.

It is interesting to compare these theoretical results with some recent
observations. Six star forming galaxies with redshifts between $z=0.8-2.4$
were analyzed by [17], and a large sample of 101 other galaxies from the KMOS
and SINS/zC-SINF surveys with redshifts between $z=0.6$ and 2.6 were studied
by [27]. Both surveys give trends corresponding to the above point 3: the
observed past rotation curves were not flat, but were exhibiting steeply
decreasing velocities with their galactocentric radius. Thus, even at
distances of several effective radii, the authors found that the dark matter
fractions were modest or negligible in the past. A variety of discussions and
interpretations of the above results have taken place (see next ref.).
However, a further analysis by [18] from the same survey confirmed that at
$z\sim 2$ galaxies have very low dark matter fractions (¡ 20%).

The observations are also improving rapidly, and very recently the above
results were confirmed by new data from different instruments and wavelengths.
In particular, a hundred galaxies at the peak of their star formation
($z=0.6-2.5$) were observed with ESO-VLT, LUCI, NOEMA at IRAM and ALMA, and
further analyzed so doubling the previous samples [47]. These new observations
show rather steep rotation curves, which according to the authors demonstrate
that the fractions of dark matter strongly decrease with increasing redshifts:
for example at $z=0.85$ the fraction of dark matter is estimated to be 38% of
the typical value, and this fraction goes down to 17% at $z=2.44$.

In standard models with dark matter, the above decrease of DM with redshifts
is difficult to interpret. It is not clear where at high $z$ the usual dark
matter fraction is located, in order to let only such small dynamical
signatures. The possibility that the missing dark matter is lying in more
external regions would be a rather troublesome one. Indeed, in the CDM theory
the baryons set in the potential wells already created by dark matter during
the radiative era. Thus, it is not clear why galaxies would contain less dark
matter in the past, since the accumulation of DM is considered preceding that
of baryons.

At the opposite, we point out that the above mentioned results are correctly
corresponding to the predictions of the SIV theory concerning the evolution of
rotation curves (Fig. 1). In itself it is not at all a proof of the theory,
but an interesting indication, which is perfectly consistent with previous
results on the origin of the high velocities of galaxies in clusters (Sect.
3), and with other results quoted in the introduction, in particular the
growth of density fluctuations [37].

###  4.4 The limited connection with MOND

The modified Newton equation, in the approximation of a constant scale factor
$\lambda$, is leading to the deep MOND limit [45] and [46] with a gravity
given by [35],

$\displaystyle
g=\sqrt{a_{0}\;g_{\mathrm{N}}},\quad\mathrm{and}\;\;\;a_{0}=\frac{nc\,H_{0}\,(1-\Omega_{\mathrm{m}})^{2}}{4},\quad$
(4.11)
$\displaystyle\mathrm{with}\;H_{0}=\frac{2}{\tau_{0}}\frac{(1-\Omega^{1/3}_{\mathrm{m}})}{(1-\Omega_{\mathrm{m}})}\quad
a_{0}=\frac{nc\,(1-\Omega_{\mathrm{m}})(1-\Omega^{1/3}_{\mathrm{m}})}{2\,\tau_{0}},$
(4.12)

where $g_{\mathrm{N}}$ is the Newtonian gravity. There $n$ is of the order of
unity and $c$ is the velocity of light. As the range of $\lambda$-variations
is generally limited (Sect. 2), over timescales less than about $2\times
10^{8}$ yr, the scale factor $\lambda$ undergoes changes smaller than 1%. For
example, in Universe models with a value $\Omega_{\mathrm{m}}=0.3$, for
galaxies with dynamical timescales (_e.g._ orbital period) of 200 Myr, the
approximation of constant $\lambda$ is in error of only 0.5 %. For the extreme
case with $\Omega_{\mathrm{m}}=0.045$, the corresponding errors amount to 1%.
Thus, MOND may be considered as an approximation tangent to the SIV theory
over one to two galactic rotations. However, within SIV the approximation is
not valid over longer timescales covering the global evolution of a galaxy.

In SIV theory, the deep limit regime is entered when the dynamical
acceleration (the additional term in the modified Newton equation (2.35))
becomes larger than the usual Newtonian gravity term [35], (this limit may
also be influenced by the opposite gravity of nearby objects). This condition
leads to the above value of $a_{0}$ (Eq. 4.11). The product $c\,H_{0}$ is
equal to 6.80 $10^{-8}$ cm s-2 for $H_{0}=70$ km/(s Mpc). For
$\Omega_{\mathrm{m}}$=0, 0.10, 0.20, 0.30 and 0.50, with $n\approx 1$ one has
$a_{0}\approx$ (1.70, 1.38, 1.09, 0.83, 0.43) $\;\times\;10^{-8}$ cm s-2
respectively. These well encompass the observed value $a_{0}$ of about 1.2
$\times\,10^{-8}$ cm s-2 [46].

There are some similarities, but also differences, between the two theories.
In MOND, equation $g=\sqrt{a_{0}\;g_{\mathrm{N}}}$ directly leads to a
constant circular velocity $\upsilon^{2}=\sqrt{a_{0}GM}$ independent from the
orbital radius in the low gravity regime. In SIV theory, the two-body system
predicts a constant orbital velocity during the increase of radius resulting
from the expansion due to the scale invariant effects (4.8). This is not the
same, in SIV the velocities may depend on their initial values and on the
duration of the expansion, i.e. the ages. Moreover, in SIV it is not necessary
that the gravity is smaller than $a_{0}$ to have significant deviations from
Newtonian. The dynamics is always governed by the full equation (2.35) with
both Newtonian and dynamical accelerations.

Turning to observations, we see that the galactic rotation curve (Fig. 1)
indicates that the flat part is relatively limited. Over large distances, the
orbital velocity is not a constant, but a slowly decreasing function. Also
MOND is not in agreement with the new results by the Genzel group presented
above. The deep limit, which accounts for the flat rotation curves, applies in
principle the same way at all epochs, while the observations of galaxies at
$z=2$ do not show flat curves thus predicting much less dark matter. The
correspondence between MOND and SIV appears limited to objects of low
redshifts and to a certain domain of radii in galaxies (a few tens of kpcs).

In the case of the clusters of galaxies, the MOND approximation should barely
apply, since their dynamical timescale is much longer than for galaxies due to
their lower densities. For Universe models with a value
$\Omega_{\mathrm{m}}=0.3$, clusters with dynamical timescales of 1, 3 or 5
Gyr, the approximation of constant $\lambda$ would be in error of 2.4, 7.1 and
11.9 %. For the case with $\Omega_{\mathrm{m}}=0.045$, the corresponding
errors amount to 4.6, 14.0 and 23.3 % respectively. Thus, MOND might be a
possible approximation of scale invariant dynamics for small clusters, but not
for large clusters, with dynamical timescales of the order of the age of the
Universe.

###  4.5 The connection between the assumed dark matter and the baryons: the
signature of a gravity effect

An interesting observation concerns the radial acceleration (RAR) studied by
Lelli et al. [29], who showed that the relation between average accelerations
$\frac{\upsilon^{2}}{R}$ and $\frac{GM}{R^{2}}$ for galaxies deviates from a
1:1 relation for gravities below $a_{0}$. The RAR is well accounted for by
both MOND (cf. Lelli et al.) and the SIV theory [39], with however a possible
difference concerning the extreme case of spheroidal galaxies. Remarkably as
found by Lelli et al., see also [7], the amount of dark matter (DM, measured
by the deviations from the 1:1 line) seems to be always directly determined by
the amount of baryonic matter, although in total 2700 measurements have been
made in very different galaxies and _at different distances from their
centers_. Such a result is pointing in favour of a gravity effect rather than
for DM.

As mentioned above, the same kind of relation between the dynamical and
baryonic masses of central regions is also present in clusters of galaxies
[7]. Moreover, such a relation is also observed in redshifted galaxies, where
Nestor Shachar et al. [47] are finding a strong correlation between the
fraction of DM and the baryon surface density within the effective radius (the
part corresponding to the flat curve). The convergence of such findings in
three so different environments confirms the intimate relation between the
baryons and the assumed dark matter. It is clearly supporting the views that
this general connection is the signature of gravity effect.

##  5 Dispersion of the vertical velocities

The velocities are too high not only in the galactic plane as seen above, but
also in the ”vertical” direction of the W-motions perpendicular to the
galactic plane. The old problem of the dispersion of vertical velocities
cannot be considered as a test, since its 70 year history has shown that a lot
of astrophysical effects may significantly intervene, as recalled in the
introduction. Scale invariant effects, if there would be no interaction with
other effects, only account for a small fraction of the total vertical
velocity dispersion, thus confirming the richness of the involved processes.

Spitzer and Schwarzschild [58], who showed that the ”vertical” velocity
dispersion (perpendicular to the galactic plane) was increasing with the age
of the stellar group interpreted this effect as due to collisions with large
interstellar clouds. This effect was confirmed by further studies, see
Seabroke & Gilmore [54]. Among the other astrophysical effects proposed to
account for the growth of velocities with age, we may mention the effects of
spiral waves by Barbanis & Woltjer [2], of the driving be an unknown diffusive
process by Wielen [64], of massive halo black holes by Lacey & Ostriker [25],
of mergers of dwarf galaxies by Toth & Ostriker [59], of evaporating star
clusters by Kroupa [23] and of the evolution of the interstellar medium by
Kumamoto et al. [24].

Recent valuable observations of the age effect on the dispersion of vertical
velocities have been providedby Cheng et al. [8] and by Raddi et al. [52] on
the basis of the kinematical data for white dwarfs from Gaia DR2 in the solar
neighborhood. Their data are in general agreement with the results of [54]
based on the large survey of nearby F and G dwarf stars by [48]. In Fig. 2,
the agreement is excellent for the thin disk up to ages of about 7 Gyr. For
larger ages, the scatter of the data is larger, but both samples well
illustrate the large increase of the vertical velocity dispersion for the old
star population of the thick disk. As mentioned in the introduction, a number
of authors have identified many astrophysical effects which may possibly
contribute to the increase with time of the observed vertical velocity
dispersion.

Some characteristics of the acting astrophysical processes have been
enlightened by [54]. They pointed out that the effect [58] of collisions with
giant clouds saturates after some time with no further increase of the
dispersion. They also demonstrated that the effect producing the growth of the
dispersion is acting continuously throughout the motions around the galactic
plane and not only when the stars are crossing it.

Figure 2: The vertical velocity dispersion as a function of the age of the
star population considered. The thin and the thick disks are represented by
black broken lines according to the WD data by [8], see also [52]. Data from
[54] are shown as blue triangles (with a broken blue line for their average
trend), these are in good agreement with the WD data for the thin disk. The
results for scale invariant effects are represented by red lines for
$\Omega_{\mathrm{m}}=0.30$ and 0.045 in absence of dark matter; as usual, the
effects are smaller for denser Universe models.

A study of the scale invariant effects in the oscillations of stars around the
galactic plane has been performed by [42]. From the modified Newton equation
(2.19), they showed that the vertical motions in the $z$-directions obey the
following equation of motion (in variable $t$),

$\ddot{z}-\frac{1}{t}\dot{z}+\omega^{2}(t)z=0,\quad\mathrm{with}\;\omega^{2}(t)=\left(\frac{\partial
K_{z}}{\partial z}\right)=4\pi\,G\varrho\,.$ (5.1)

There, $K_{z}$ is the vertical component of the acceleration, $\varrho$ the
mean spatial density at the level of the galactic plane and $\omega$ the
oscillation frequency. The oscillatory solutions around the galactic plane are
of the form,

$z=\left(\frac{z_{\mathrm{form}}}{t_{\mathrm{form}}}\right)\,t\,\sin(s\,\ln
t+\varphi),\quad\mathrm{with}\quad s=\sqrt{\omega^{2}_{0}\,t^{2}_{0}-1}\,,$
(5.2)

where the index “form” indicates the time of formation and “0” the present
time. This shows that the oscillation amplitudes are increasing with time t.
The amplitudes of the velocity when crossing the galactic plane behave like,

$W\equiv\dot{z}\sim\frac{z_{\mathrm{form}}}{t_{\mathrm{form}}}\,\sim\frac{1}{t_{\mathrm{form}}},\quad\mathrm{thus}\;W(t_{\mathrm{form}})=W(t_{0})\frac{t_{0}}{t_{\mathrm{form}}},$
(5.3)

under the assumption that stars preferentially form in the galactic plane. If
the stars are formed outside the galactic plane, i.e. at a larger
$z_{\mathrm{form}}$, the velocity is increased as discussed below for the
thick disk. There, $W(t_{\mathrm{form}})$ is the vertical velocity (in the
galactic plane) nowadays of a star born at time $t_{\mathrm{form}}$, while
$W(t_{0})$ is the velocity of stars recently born. Turning now to the current
time units with (2.31) , we get the velocity $W(\tau_{\mathrm{form}})$,

$W(\tau_{\mathrm{form}})=W(\tau_{0})\,\frac{1}{t_{\mathrm{in}}+\frac{\tau_{\mathrm{form}}}{\tau_{0}}(1-t_{\mathrm{in}})}.$
(5.4)

This possible cosmological effects of scale invariance are illustrated by the
red lines in Fig. 2, for the cases $\Omega_{\mathrm{m}}=0.045$ and 0.30. The
account in variable $\tau$ of the limited variations of $\lambda$ reduces the
effect, compared to the results by [33]. As the observations show a steep
initial increase from about 10 km/s during the first Gyr, we start with a
value of 10 km/s at an age of 1 Gyr. Fig. 2 shows a small regular increase of
the dispersion: for $\Omega_{\mathrm{m}}=0.045$ it is of 4.9 km/s at an age of
7 Gyr, at the end of thick disk phase. For $\Omega_{\mathrm{m}}=0.30$, the
increase amounts to 2.0 km/s, which only accounts for a fraction of the
observed effect.

Clearly, this lets room for other effects. The thickness of the disk at the
time of star formation is a key point. This is shown by the value
$z_{\mathrm{form}}$ in the first of Eqs. (5.3): if the stars are “launched” at
their birth from a more distant place, the amplitudes will be proportionally
larger. This accounts for the thick disk effect in Fig. 2. Also, the collision
with massive interstellar clouds, partially favoring an equipartition of the
energy, would speed up the velocity by about 7 km/s at an age of 7 Gyr
according to [58]. The other effects mentioned above would also contribute to
several km/s at 7 Gyr, in particular the formation of star clusters [23]. The
gravitational field associated to the spiral structure would also contribute
[2], but probably more significantly for the horizontal dispersion of the $U$
and $V$ components.

The conclusion we get from Fig. (2) is that scale invariant may play some
role. However, this role is for now probably largely underestimated since the
red lines show these effects without any interaction with the other dynamical
mechanisms and this is a major point. Indeed, the additional acceleration term
in the modified Newton equation (2.35) depends on the effective total
velocity, thus the influence of scale invariance would be magnified all the
way through the ages by its interaction with the various dynamical processes.
The solution of these interactions is beyond the scope of the present study,
where only the effects acting on an initial velocity of 10 km/s, typical of
the situation after 1 Gyr, were considered. Nevertheless, it shows that, if
scale invariant effects do exist, they are significant actors of the problem.

##  6 Conclusion

Overall, there is an impressive convergence of results when interpreting well-
known astrophysical facts: the observed excesses of velocity in clusters of
galaxies, the observed flat rotation curves of low redshift galaxies, the
observed steeper rotation curves of high-redshift galaxies, the unicity of the
relation between dark and baryonic matters repeatedly observed in galaxies
[29], in galaxies and galaxy clusters [7] and in significantly redshifted
galaxies [47]. All these results come from a broad range of sources and
authors, and happen in very different astronomical objects spanning themselves
a vast range in stages of evolution, masses and spatial scales. It would be
estonishing that Nature ”conspires” to make all these results pointing to the
same direction for no reason. This is in fact questioning the very existence
of dark matter.

These results correspond to the predictions of the scale invariant theory,
which is resting on a simple fundamental principle: enlarging the group of
invariances subtending the theory of gravitation as suggested by Dirac [10].
Moreover, the theory contains no ”adjustment parameter”. The resulting effects
depend on the inverse of the age of the Universe and are thus negligible in
current life, while their cumulative effects may play some role in the cosmic
evolution. They are also consistent with the results of some other tests
recalled in the introduction. The global conclusion is that it is worth
pursuing further observational tests of scale invariance and to perform
numerical simulations. It would be a pity to miss the possibility offered by
this additional symmetry over standard $\Lambda$CDM.

Acknowledgements:

The author expresses his gratitude to Prof. James Lequeux for his support and
encouragements, to Dr. Vesselin Gueorguiev for fruitful collaboration over the
years and to Prof. Frederic Courbin for very helpful comments.

## References

  * [1] Bahcall, N., _The Perseus Cluster: Galaxy Distribution, Anisotropy, and the Mass/Luminosity Ratio_ , ApJ, 187, 439 (1974)
  * [2] Barbanis, B., Woltjer, L. , _Unresolved stellar companions with Gaia DR2 astrometry_ , MNRAS, 496, 1922 (2020)
  * [3] Blindert, K., Yee, H.K.C., Gladders, M.D. et al., _Dynamical masses of RCS galaxy clusters_ , IAU Coll. 195, p. 215 (2004)
  * [4] Bouvier, P., Maeder, A., _Consistency of Weyl’s Geometry as a Framework for Gravitation_ , Astrophys. Space Science, 54, 497 (1978)
  * [5] Canuto, V., Adams, P. J., Hsieh, S.-H., & Tsiang, E., _Scale-covariant theory of gravitation and astrophysical applications_ , Physical Rev. D, 16, 1643 (1977)
  * [6] Carroll, S. M., Press, W. H., & Turner, E. L., _The cosmological constant_ , Annual Rev. Astron. Astrophs., 30, 499 (1992)
  * [7] Chan, M. H., _Two mysterious universal dark matter-baryon relations in galaxies and galaxy clusters_ , Physics of the Dark Universe, 38, 101142 (2022)
  * [8] Cheng, S., Cummings, J.D., Menard, B., _A Cooling Anomaly of High-mass White Dwarfs_ , Astrophys. J. 886, 100 (2019)
  * [9] Chiu, I., Mohr, C.L., McDonald, M. et al., _Baryon content of massive galaxy clusters at 0.57 ¡ z ¡ 1.33_ , MNRAS, 455, 258 (2016)
  * [10] Dirac, P. A. M., _Long range forces and broken symmetries_ , Proceedings of the Royal Society of London Series A, 333, 403 (1973)
  * [11] Dirac, P. A. M., _Cosmological Models and the Large Numbers Hypothesis_ , Proceedings of the Royal Society of London Series A, 338, 439 (1974)
  * [12] Eddington, A.S., _The Mathematical Theory of Relativity_ , Chelsea Publ. Co. New York, 266 p. (1923)
  * [13] Einstein, A., _Kommentar zu ”Hermann Weyl: Gravitation und Elektrizität”_ , Sitzung Berichte der Königlich Preussischen Akademie des Wissenschaften (Berlin), p. 478. (1918)
  * [14] Famaey, B., McGaugh, S.S., _Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions_ , Living Rev. Relativity, 15, 10 (2012)
  * [15] Frieman, J. A., Turner, M. S., & Huterer, D., _Dark energy and the accelerating universe_ , Annual Rev. Astron. Astrophs., 46, 385 (2008)
  * [16] Ge, C., Wang, Q.D., Tripp, T.M. et al., _Baryon content and dynamic state of galaxy clusters: XMM-Newton observations of A1095 and A1926_ , MNRAS, 459, 366 (2016)
  * [17] Genzel, R., Forster Schreiber, N.M., Ubler, H., _Strongly baryon-dominated disk galaxies at the peak of galaxy formation ten billion years ago_ , Nature, 543, 397 (2017)
  * [18] Genzel, R., Price, S. H., Ubler, H., et al. _Rotation Curves in z 1-2 Star-forming Disks: Evidence for Cored Dark Matter Distributions_ , ApJ, 902, 98 (2020)
  * [19] Gonzalez, A.H., Sivanandam, S., Zabludoff, A.I. et al., _Tracing Cosmic Evolution with Clusters of Galaxies_ http://www.sexten-cfa.eu/en/conferences/2013/details/34-SestoClusters2013.html, id.123 (2013)
  * [20] Huang, Y., Lin, X.-W., Yuan, H.-B. et al., _The Milky Way’s rotation curve out to 100 kpc and its constraint on the Galactic mass distribution_ , MNRAS, 463. 2623 (2016)
  * [21] Jesus, J.F., _Exact solution for flat scale-invariant cosmology_ Rev. Mex. Astron. Astrophys. 55, 17, (2018)
  * [22] Karachantsev, I. D. 1966, Astrofizika, 2, 81, (In English: 1967, Astrophysics, 2, 39).
  * [23] Kroupa, P.,_Thickening of galactic discs through clustered star formation_ , MNRAS, 330, 707 (2002)
  * [24] Kumamoto, J., Baba, J., Saitoh, T.R., _Imprints of zero-age velocity dispersions and dynamical heating on the age-velocity dispersion relation_ , Publ. Astron. Soc. Japan, 69, 31 (2017)
  * [25] Lacey, C.G., Ostriker, J.P., _Massive black holes in galactic halos ?_ , ApJ, 299, 633 (1985)
  * [26] Landau, L. & Lifchitz, E. Mecanique, Ed. Mir Moscou, 236 p. (1960)
  * [27] Lang, P., Forster Schreiber, N.M., Genzel, R., _Falling Outer Rotation Curves of Star-forming Galaxies at 0 ¡z ¡ 2.6 Probed with KMOS3D and SINS/zC-SINF_ , ApJ, 840, 92L (2017)
  * [28] Leauthaud, A., George, M.R., Behroozi, P.S. et al., _The Integrated Stellar Content of Dark Matter Halos_ , ApJ, 746, 95 (2012)
  * [29] Lelli, F., McGaugh, S.S., Schombert, J.M., Pawlowski, M.S., _One Law to Rule Them All: The Radial Acceleration Relation of Galaxies_ , ApJ, 836, 152 (2017)
  * [30] Lin, Y.-T., Stanford, S.A., Eisenhardt,P.R.M. et al., _Baryon Content of Massive Galaxy Clusters at z = 0-0.6_ , ApJ, 745, L3 (2012)
  * [31] Maeder, A., _Metrical connection in space-time, Newton’s and Hubble’s laws_ , Astron. Astrophys. 65, 337 (1978)
  * [32] Maeder, A., _An Alternative to the LambdaCDM Model: The Case of Scale Invariance_ Astrophys. J., 834, 194 (2017a)
  * [33] Maeder, A., _Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?_ Astrophys. J., 849, 158 (2017c)
  * [34] Maeder, A.._Evolution of the early Universe in the scale invariant theory_ , arXiv:1902.10115 (2019)
  * [35] Maeder, A., _MOND as a peculiar case of the SIV theory_ , MNRAS, 520, 1447 (2023)
  * [36] Maeder, A., Bouvier, P., _Scale invariance, metrical connection and the motions of astronomical bodies_ , Astron. Astrophys., 73, 82 (1979)
  * [37] Maeder, A.; Gueorguiev, V.G., _The growth of the density fluctuations in the scale-invariant vacuum theory_ , Physics of the Dark Universe, 25, 100315 (2019)
  * [38] Maeder, A.; Gueorguiev, V.G., _The Scale-Invariant Vacuum (SIV) Theory: A Possible Origin of Dark Matter and Dark Energy_ , Universe, vol. 6, 46 (2020a)
  * [39] Maeder, A.; Gueorguiev, V.G., _Scale-invariant dynamics of galaxies, MOND, dark matter, and the dwarf spheroidal_ , MNRAS, 492, 2698 (2020b)
  * [40] Maeder, A.; Gueorguiev, V.G., _Scale invariance, horizons, and inflation_ , MNRAS, 504, 4005 (2021a)
  * [41] Maeder, A.; Gueorguiev, V.G., _Action Principle for Scale Invariance and Applications (Part I)_ , Symmetry 2023, 15, 1966; arXiv 2310.16913 (2023)
  * [42] Magnenat, P., Martinet, L., Maeder, A., _The age dependence of stellar velocity dispersion in a scale-covariant (theory of) gravitation_ , Astron. Astrophys., 67, 51 (1978)
  * [43] Mantz, A.B., Morris, R.G., Allen, S.V., _Cosmological constraints from gas mass fractions of massive, relaxed galaxy clusters_ , MNRAS, 510, 131 (2022)
  * [44] Milgrom, M., _A modification of the newtonian dynamics : implications for galaxy systems_ , Astrophys. J., 270, 365 (1983)
  * [45] Milgrom, M., _The Mond Limit from Spacetime Scale Invariance_ , Astrophys. J., 698, 1630 (2009)
  * [46] Milgrom, M., _The MOND paradigm of modified dynamics_ , Scholarpedia, 9, 31410 (2014a)
  * [47] Nestor Shachar, A., Price, S.H., Forster Schreiber, N. M. et al., _RC100: Rotation Curves of 100 Massive Star-forming Galaxies at z = 0.6-2.5 Reveal Little Dark Matter on Galactic Scales_ , ApJ 944, 78 (2023)
  * [48] Nordstrom, B., Mayor, M., Andersen, J. et al., _The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of 14 000 F and G dwarfs_ , Astron. Astrophys. 418, 989 (2004)
  * [49] Old, L., Wojtak, R., Pearce, F.R. et al., _Galaxy Cluster Mass Reconstruction Project - III. The impact of dynamical substructure on cluster mass estimates_ , MNRAS, 475, 852 (2018)
  * [50] Planck Collaboration 2020, Aghanim, N., Akrami, Y. Ashdown, M., _Planck 2018 results. VI. Cosmological parameters_ , Astron. Astrophys. 641, A6 (2020)
  * [51] Proctor, R.N., Mendes de Oliveira, C., Azanha, L., _A derivation of masses and total luminosities of galaxy groups and clusters in the maxBCG catalogue_ , MNRAS, 449, 2345 (2015)
  * [52] Raddi, R., Torres, S., Rebassa-Mansergas, A. et al., _Kinematic properties of white dwarfs. Galactic orbital parameters and age-velocity dispersion relation_ , Astron. Astrophys. 658, A22 R (2022)
  * [53] Rubin, V.C., Ford, W.K. Jr., Thonnard, N., _Radial velocities of spiral galaxies determined from 21-cm neutral hydrogen observations_ , ApJ, 225, L107 (1978)
  * [54] Seabroke, G.M., Gilmore, G., _Revisiting the relations: Galactic thin disc age-velocity dispersion relation_ , MNRAS, 380, 1348 (2007)
  * [55] Shan, Y., McDonald, M., Courteau, S. , _Revised Mass-to-light Ratios for Nearby Galaxy Groups and Clusters_ , ApJ, 800, 122 (2015)
  * [56] Sofue, Y., Rubin, V., _Rotation Curves of Spiral Galaxies_ , Annual Rev. Astron. Astrophys., 39, 137 (2001)
  * [57] Sohn, J., Geller , M.J., Zahid, H.J. et al., _The Velocity Dispersion Function of Very Massive Galaxy Clusters: Abell 2029 and Coma_ , ApJ Suppl., 229, 20 (2016)
  * [58] Spitzer, L., Schwarzschild, M., _The Possible Influence of Interstellar Clouds on Stellar Velocities_ , ApJ, 114, 385 (1951)
  * [59] Toth, G., Ostriker, J.P., _Galactic Disks, Infall, and the Global Value of Omega_ , ApJ, 389, 5 (1992)
  * [60] Trimble, V., _Existence and nature of dark matter in the universe_ , Annual Rev. Astron. Astrophs., 25, 425 (1987)
  * [61] Weinberg, S. _Gravitation and Cosmology: Principles and applications of the General Theory of Relativity_ , John Wiley & Sons, Inc, New York, London, Sydney, Toronto (1972)
  * [62] Weyl, H., _Raum, Zeit, Materie. Vorlesungen über allgemeine Relativitätstheorie_ , Re-edited 1970 by Springer Verlag, Berlin, (1923)
  * [63] Wicker, R., Douspis, M., Salvati, L. et al., _Constraining the mass and redshift evolution of the hydrostatic mass bias using the gas mass fraction in galaxy clusters_ , Astron. Astrophys. 674, A48 (2023)
  * [64] Wielen, R. 1977, _The Diffusion of Stellar Orbits Derived from the Observed Age-Dependence of the Velocity Dispersion_ , Astron. Astrophys., 60, 263
  * [65] Zwicky, F., _Die Rotverschiebung von extragalaktischen Nebeln_ , Helevetica Physica Acta, 6, 10 (1933)