# Jet hadronization at LHCb

University of Michigan
E-mail

###### Abstract:

In high energy proton-proton collisions, collimated sprays of particles,
called jets, result from hard scattered quarks or gluons. Jets are copiously
produced in these collisions; however, the dynamic process through which
quarks and gluons, collectively referred to as partons, become bound state
hadrons is still not well understood. Jets provide an excellent tool to study
this process as they are proxies for the scattered parton; therefore, final-
state hadrons can be measured with respect to an observable that is correlated
to the scattered parton. The LHCb experiment is in an excellent position to
measure hadrons within jets due to its excellent tracking and particle-
identification capabilities. In this talk, new measurements of charged hadrons
within jets measured opposite a $Z$ boson will be presented from the LHCb
collaboration.

##  1 Introduction

Quantum chromodynamics (QCD) is unique amongst the four fundamental forces due
to the nonperturbative dynamics that bind quarks and gluons within hadrons.
Although quarks and gluons, referred to as partons, are the fundamental
particles in QCD, they can never be observed freely due to the principle of
confinement. The partonic structure of hadrons, with a focus on the proton,
has been a deeply studied field of particle physics for decades. However, the
process by which hadrons are formed from scattered partons is less understood
than their initial-state partonic structure counterparts. Generally,
fragmentation functions (FFs) are used to describe the probability of a
particular parton fragmenting into a particular hadron [1, 2]. Global fits to
experimental data have improved FF phenomenology; for a recent review see Ref.
[3]. However, FFs are ultimately parameterizations of the underlying QCD
interactions and represent a first approximation to understanding the
mechanisms through which hadrons are formed from partons. Since hadronization
is an inherently dynamic nonperturbative process and thus cannot be calculated
perturbatively, additional data are required to further understand how hadrons
form from scattered partons.

When a high-energy parton is scattered from a proton, the resulting collimated
spray of particles is referred to as a jet. Jets are an excellent tool to
study the hadronization process since they are a proxy for the scattered
parton. Thus, unlike inclusive observables where only one hadron is measured
in the final state, hadrons can be measured with respect to an object that is
correlated with the fragmenting parton. The measurement of the jet allows for
the transverse distribution of hadrons to be measured in addition to the more
typical longitudinal momentum distribution of hadrons with respect to the
scattered parton. Additionally, the jet provides an object which is calculable
within perturbative QCD; thus, measurements of hadrons within jets utilize a
final-state that is already perturbatively understood, the jet, to study
something that is less understood and not perturbatively calculable, the
result of the hadronization process.

In this talk, new measurements of charged hadron production within jets
opposite a $Z$ boson are presented from the LHCb experiment [4]. Previous
measurements of hadrons within jets at the Large Hadron Collider (LHC) have
been made almost exclusively in the inclusive jet channel [5, 6, 7]. At LHC
energies, the inclusive jet channel is dominated by gluon jets, whereas the
$Z$-jet channel is predominantly sensitive to light quark jets [8]. Thus,
these new measurements provide an opportunity to study phenomenological
differences between light quark and gluon hadronization. A very recent
measurement, studying the isolated photon-jet channel, is also dominated by
quark jets and provides an opportunity to study hadronization differences
between jets that are measured opposite a massive vs. massless vector boson
[9]. Additionally, previous measurements generally focus on the longitudinal
momentum fraction of hadrons with respect to the jet, while the measurements
presented here also study the charged hadron distributions transverse to the
jet axis.

##  2 Methods

The methods used to construct charged hadron distributions within jets are
similar to those in Ref. [5]. Charged hadrons are measured as a function of
three observables, and are normalized by the total number of $Z$-jet pairs
measured in a given jet transverse momentum ($p_{\rm T}$) bin. This
normalization ensures that the integral of the distributions gives the average
multiplicity of the jets in a given jet $p_{\rm T}$ bin. Three charged hadron-
in-jet observables are studied, defined as

$z\equiv\frac{{\bf{p}}_{\rm{jet}}\cdot{\bf{p}}_{\rm{hadron}}}{|{\bf{p}}_{\rm{jet}}|^{2}},$
(1)
$\mbox{$j_{\rm{T}}$}\equiv\frac{|{\bf{p}}_{\rm{jet}}\times{\bf{p}}_{\rm{hadron}}|}{|{\bf{p}}_{\rm{jet}}|},$
(2)

and

$r\equiv\sqrt{(\phi_{\rm{jet}}-\phi_{\rm{hadron}})^{2}+(y_{\rm{jet}}-y_{\rm{hadron}})^{2}}.$
(3)

Here $\bf{p}$ is the 3-momentum vector, $\phi$ is the azimuthal angle, and $y$
is the rapidity. The observables give the longitudinal momentum fraction
($z$), the transverse momentum ($j_{\rm{T}}$), and the radial distribution
($r$) of charged hadrons with respect to the jet axis.

The measurements are made by the LHCb experiment in $\sqrt{s}=8$ TeV proton-
proton collisions. $Z$ bosons are measured via their dimuon decay. The dimuon
pair is required to satisfy $60<M_{\mu^{+}\mu^{-}}<120$ GeV, where
$M_{\mu^{+}\mu^{-}}$ is the invariant mass of the dimuon pair. Additionally,
the muons must have $2<\eta<4.5$, where $\eta$ is the pseudorapidity. Jets are
clustered with the $R=0.5$ anti-$k_{T}$ algorithm [10] within $2.5<\eta<4$ and
must have $\mbox{$p_{\rm T}$}>20$ GeV. Requirements are placed on the $Z$-jet
pair to better identify events that correspond to a two-to-two partonic hard
scattering process. Only events with one reconstructed primary vertex are
analyzed, and the $Z$-jet pair is required to be nearly back-to-back in
azimuth such that $|\Delta\phi_{Z-{\rm{jet}}}|>7\pi/8$. Charged hadrons are
required to have $|{\bf{p}}|>4$ GeV, $p_{T}>0.25$ GeV, and be within the jet
such that $\Delta
R\equiv\sqrt{(\phi_{\rm{jet}}-\phi_{\rm{hadron}})^{2}+(\eta_{\rm{jet}}-\eta_{\rm{hadron}})^{2}}<0.5$.
Results are efficiency corrected and unfolded with a Bayesian unfolding
procedure as implemented in Ref. [11] to facilitate comparisons with
theoretical calculations and predictions from event generators. The
measurements are performed integrated over the $Z$ kinematics, so that the
charged hadron-in-jet distributions can be studied in three bins of jet
$p_{\rm T}$. Reference [4] describes the analysis procedures and methods in
greater detail.

##  3 Results

Figure 1: Measured longitudinal momentum fractions of charged hadrons in jets
opposite a $Z$ boson.

Figure 2: Comparisons of the longitudinal momentum fractions of charged
hadrons in $Z$-jet with similar measurements in midrapidity inclusive jets
(left) and midrapidity isolated photon-jets (right).

Figure 1 shows the charged hadron longitudinal momentum fractions in three jet
$p_{\rm T}$ bins. The distributions of $z$ are relatively constant as a
function of jet $p_{\rm T}$ at large $z$. At low $z$ the distributions
diverge, which is a kinematic effect due to the requirement that the track
momentum be greater than 4 GeV; therefore, higher $p_{\rm T}$ jets can probe
lower $z$. Figure 2 shows comparisons of the longitudinal momentum fractions
in the $Z$-jet process to similar measurements in the midrapidity inclusive
jet channel [5] (left) and midrapidity isolated photon-jet channel [9]
(right). The comparison with inclusive jets shows that the fragmentation
distributions are more steeply falling at high $z$ in the gluon dominated
process when compared to the light-quark dominated $Z$-jet process.
Interestingly, the $Z$-jet and isolated photon-jet distributions are extremely
similar, indicating that, within the current uncertainties, the longitudinal
hadronization is similar when the jet is measured opposite a massive $Z$ boson
or a massless photon.

Figure 3: Measured transverse momentum with respect to the jet axis
distributions as a function of jet $p_{\rm T}$ (left), and comparisons of
these distributions to similar results from the midrapidity inclusive jet
channel (right).

The charged hadron transverse momentum with respect to the jet axis
distributions are shown in Fig. 3 (left), while the comparison of these
distributions with similar measurements in the midrapidity inclusive jet
channel are also shown in Fig. 3 (right). The distributions show a sensitivity
to nonperturbative transverse momentum, indicated by the rounder shape at
small $j_{\rm{T}}$ which transitions to a perturbative tail at large
$j_{\rm{T}}$. Performing a similar comparison to the midrapidity inclusive jet
results shows that the distributions in the $Z$-jet channel tend to lie at
smaller $j_{\rm{T}}$, which can be seen at $\mbox{$j_{\rm{T}}$}<0.5$ GeV. At
large $j_{\rm{T}}$, the distributions are relatively similar; this may
indicate that hadrons generated from perturbative radiation at large
$j_{\rm{T}}$ are independent of the initial fragmenting parton.

The charged hadron radial distributions as a function of jet $p_{\rm T}$
(left) and the comparison of these distributions to similar measurements in
the midrapidity inclusive jet channel (right) are shown in Fig. 4. The radial
distributions are largely independent of jet $p_{\rm T}$ at large $r$, away
from the jet axis. At small $r$, the multiplicity of hadrons rises sharply
with the jet $p_{\rm T}$ which is a feature of the anti-kT algorithm. The
independence of the distributions away from the jet axis may indicate that
nonperturbative contributions to the jet are largely independent of jet
$p_{\rm T}$. Comparisons with the midrapidity inclusive jet data show that the
radial distributions in the $Z$-jet process are more collimated than those in
the inclusive jet process. This same conclusion can also be drawn from the $z$
and $j_{\rm{T}}$ distributions, which display a more collimated behavior in
the respective observable when compared to the midrapidity inclusive jet
distributions.

Figure 4: Measured radial distributions as a function of jet $p_{\rm T}$
(left), and comparisons of these distributions to similar results from the
midrapidity inclusive jet channel (right).

In summary, new measurements of charged hadrons within jets measured opposite
a $Z$ boson from the LHCb experiment are presented. These measurements are
dominated by light quark jets, which is contrast to the majority of previous
measurements from the LHC which are sensitive to predominantly gluon jets.
Thus, the results offer the opportunity to study hadronization differences
between light quarks and gluons. These measurements provide new information
into the dynamic process of hadronization in both the longitudinal and
transverse directions with respect to the jet axis. Future hadronization
measurements from LHCb will utilize the excellent particle identification and
heavy flavor jet identification capabilities already demonstrated by the
experiment.

## References

  * [1] J. C. Collins and D. E. Soper, “Back-to-back jets in QCD,” Nucl. Phys., vol. B193, p. 381, 1981.
  * [2] J. C. Collins and D. E. Soper, “Parton distribution and decay functions,” Nucl. Phys., vol. B194, pp. 445–492, 1982.
  * [3] A. Metz and A. Vossen, “Parton fragmentation functions,” Prog. Part. Nucl. Phys., vol. 91, pp. 136–202, 2016.
  * [4] R. Aaij et al., “Measurement of charged hadron production in $Z$-tagged jets in proton-proton collisions at $\sqrt{s}=8$ TeV,” arXiv:1904.08878, 2019.
  * [5] G. Aad et al., “Measurement of the jet fragmentation function and transverse profile in proton-proton collisions at a center-of-mass energy of 7 TeV with the ATLAS detector,” Eur. Phys. J., vol. C71, p. 1795, 2011\.
  * [6] S. Chatrchyan et al., “Measurement of jet fragmentation in PbPb and pp collisions at $\mbox{$\sqrt{s_{{}_{NN}}}$}=2.76$ TeV,” Phys. Rev., vol. C90, no. 2, p. 024908, 2014.
  * [7] S. Acharya et al., “Charged jet cross section and fragmentation in proton-proton collisions at $\sqrt{s}$ = 7 TeV,” Phys. Rev., vol. D99, no. 1, p. 012016, 2019.
  * [8] T. Sjöstrand, S. Mrenna, and P. Skands, “A brief introduction to PYTHIA 8.1,” Comput. Phys. Commun., vol. 178, pp. 852–867, 2008.
  * [9] M. Aaboud et al., “Comparison of fragmentation functions for light-quark- and gluon-dominated jets from $pp$ and Pb+Pb collisions in ATLAS,” arXiv:1902.10007, 2019.
  * [10] M. Cacciari, G. P. Salam, and G. Soyez, “The anti-$k_{t}$ jet clustering algorithm,” JHEP, vol. 04, p. 063, 2008.
  * [11] T. Adye, “Unfolding algorithms and tests using RooUnfold,” in Proceedings, PHYSTAT 2011 Workshop on Statistical Issues Related to Discovery Claims in Search Experiments and Unfolding, CERN,Geneva, Switzerland 17-20 January 2011, (Geneva), pp. 313–318, CERN, CERN, 2011.


# MOND as a peculiar case of the SIV theory

Andre Maeder
Geneva Observatory - chemin Pegasi 51, CH-1290 Sauverny, Switzerland E-mail:
Andre.Maeder at UniGe.ch

(Accepted 2023 January 3. Received 2022 December 13; in original form 2022
October 20)

###### Abstract

The scale invariant theory is preserving the fundamental physical properties
of General Relativity, while enlarging the group of invariances subtending
gravitation theory (Dirac1973; Canuto et al.1977). The Scale Invariant Vacuum
(SIV) theory assumes, as gauging condition, that “The macroscopic empty space
is scale invariant, homogeneous and isotropic”. Some basic properties in
Weyl’s Integrable Geometry and cotensor calculus are examined in relation with
scalar-tensor theories. Possible scale invariant effects are strongly reduced
by matter density, both at the cosmological and local levels. The weak field
limit of SIV tends to MOND, when the scale factor is taken as constant, an
approximation valid (<1%) over the last 400 Myr. A better understanding of the
$a_{0}$-parameter is obtained: it corresponds to the equilibrium point of the
Newtonian and SIV dynamical acceleration. Parameter $a_{0}$ is not a universal
constant, it depends on the density and age of the Universe. As MOND is doing,
SIV theory avoids the call to dark matter, moreover the cosmological models
predict accelerated expansion.

###### keywords:

Cosmology: theory – dark energy–dark matter

††pubyear: 2021††pagerange: MOND as a peculiar case of the SIV theory–A

##  1 Introduction

In the context of the dark matter problem, the Modified Newtonian Dynamics
(MOND) was proposed by (Milgrom, 1983); this dynamics was accounting for the
flat rotation curves of galaxies. Over the following decades, this theory
received a number of further extensions and applications, e.g.Milgrom (2009,
2014a, 2014b). The application of MOND to observations is meeting a number of
positive results, see review by Famaey & McGaugh (2012). Agreement has been
obtained for the flat rotation curves of spiral galaxies (Lelli et al., 2017),
also for clusters of galaxies (Sanders, 2003; Milgrom, 2018), as well as in
the Local Group (Pawlowski et al., 2012; Pawlowski & Kroupa, 2022) and in the
Fornax Cluster (Asencio et al. , 2022).

The dynamical acceleration ($V^{2}/R$, where $V$ is the velocity and $R$ the
galactocentric distance) is related to the baryonic gravitiy ($GM/R^{2}$) for
spiral, irregular, elliptical, lenticular and spheroidals by a unique thin
relation that significantly deviates from the Newtonian Law at very low
gravities (McGaugh et al., 2016; Lelli et al., 2017; Li et al., 2018). The
unicity of this relation is striking, because it concerns observations made in
different types of galaxies and at different distances from their centers.
Such a result is uneasy to account for by dark matter, and it has been
recognized to better correspond to a gravity effect (Lelli et al., 2017).

In MOND, the usual Newtonian gravity law remains unmodified for gravities
above a constant value $a_{0}\approx 1.2\cdot 10^{-8}$ cm s-2. In the so-
called deep-MOND limit, for gravities much lower than $a_{0}$, a different
gravitation law applies with a gravity $g$ given by,

$g\,=\,\sqrt{a_{0}\;g_{\mathrm{N}}}\,,$ (1)

where $g_{\mathrm{N}}$ is the usual Newtonian gravity.

In attempts to generalize the theory, developments of the basic MOND have been
made in a variety of theoretical directions (Milgrom, 2015): in the non-
relativistic regime the Modified Poisson Gravity and  Quasilinear MOND, both
space dilation invariant; in the relativistc regime the Tensor-Vector-Scalar
(TeVeS), MOND adaptations of Einstein–Aether theories, Bimetric MOND gravity
(BIMOND) (Milgrom, 2022). Properties of General Relativity (GR), such as the
weak Principle of Equivalence, the Lorentz invariance and the general
covariance, become invalid in some of the MOND extensions (Milgrom, 2019).

Scale invariance was first considered by Weyl (1923) and Eddington (1923) in
order to account by the geometry of space-time for both gravitation and
electromagnetism, before being abandonned because the properties of a particle
would have depended on its past worldline (Einstein, 1918). A revival of the
theory was first brought about by Dirac (1973) and then by Canuto et al.
(1977), who considered the so-called Weyl Integrable Geometry (WIG), where
Einstein’s criticism does not apply (see Sect. 2.2). Dirac (1973) emphasized
that: “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”. Scale invariance means that the basic equations do
not change upon a transformation of the line element of the form,

$ds^{\prime}\,=\,\lambda(x^{\mu})\,ds\,,$ (2)

where $\lambda(x^{\mu})$ is the scale factor, $ds^{\prime}$ refers to GR while
$ds$ refers to the WIG space, for example. Scale invariance is present in
Maxwell’s equations in absence of charge and currents, as well as in General
Relativity (GR) in the case of the empty space (Bondi, 1990).

Geodesics and geometrical properties of the WIG were studied by Bouvier &
Maeder (1978) and the weak field limit, which shows an additional acceleration
in the direction of motion, by Maeder & Bouvier  (1979). Instead of the Large
Number Hypothesis (Dirac, 1974), a new gauging condition is now applied to
derive the cosmological equations, which naturally show an accelerated
expansion; several observational tests were performed (Maeder, 2017a, b). A
number of studies followed: on the rotation of galaxies and the dynamics of
clusters of galaxies (Maeder, 2017c; Maeder and Gueorguiev, 2020b); on the
growth of density fluctuations in the early Universe, where the formation of
galaxies is favoured by the additional acceleration without the need of dark
matter (Maeder and Gueorguiev, 2019); a general review (Maeder and Gueorguiev,
2020a); on horizons and inflation (Maeder and Gueorguiev, 2021a) and on the
lunar recession (Maeder & Gueorguiev, 2022). The tests were positive and cast
doubts on the need of dark components.

The aim of this work is to study the relations between MOND and SIV theories.
In Section 2, we recall the basics of the SIV theory and examine the relation
with scalar-tensor theories. In Section 3, some cosmological properties of the
SIV theory are emphasized in view of Section 4, where the Newtonian and MOND
approximations in the SIV theory are derived. The meaning and numerical value
of the $a_{0}$-parameter are examined. Section 5 contains the conclusions.

##  2 The basic theoretical context

###  2.1 Scalar-tensor versus cotensor theories

Among alternative theories of gravity, the scalar-tensor theories represent
one of the most elaborated developments111See a 312 pages review by Clifton,
Ferreira, Padilla and Skordis on _“Modified Gravity and Cosmology”_ (Clifton
et al., 2012). In particular, the scalar-tensor theories also offer an
appropriate framework for the implementation of scale invariance. A
fundamental property of these theories is the presence of an extra scalar
field $\varphi$, acting in parameter domains where General Relativity is not
currently tested. The occurrence of dark components has stimulated such
researches. The starting point is generally the expression of the action,

$S\,=\,\int
d^{4}x\sqrt{-g}\left[\frac{1}{12}\alpha\varphi^{2}R+\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi+\lambda\varphi^{4}\right]\,,$
(3)

here in a simple version given by Ferreira & Tattersall (2020), with $\varphi$
the scalar field. As pointed out by these authors, the theory is invariant
under $g_{\mu\nu}\rightarrow\lambda^{2}g_{\mu\nu}$ and
$\varphi\rightarrow\lambda^{-1}\varphi$ where $\lambda$ is a constant. With
$\alpha=1$, the theory is conformally invariant. Decades ago, applications to
varying $G$ were often considered (see also Fujii (2003)), while present
perspectives more concern early phases and inflation as well as advanced
phases of accelerated expansion. Observational signatures of scale invariant
gravity predicted in the scalar-tensor theories are quite rare, however a
pertinent signature to distinguish such effects from those of General
Relativity may be provided by perturbations in the ring down phase of black
holes, with signatures in their resulting gravitational wave emission
(Ferreira & Tattersall, 2020).

In the scale invariant vacuum (SIV) theory, an action with great similarities
to the above Eq.(3) can also be expressed, see below expression (37). However,
rather than a scalar-tensor theory, SIV should better be called _“a cotensor
theory”_. Now, some preliminaries are required. SIV applies properties of the
Weyl’s Geometry (Weyl, 1923; Eddington, 1923; Dirac, 1973), where the metrical
determination is given by the usual quadratic form
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$. Moreover, the length $\ell$ of any vector
with contravariant components $a^{\mu}$ is determined by a scale factor
$\lambda(x^{\mu})$ and the same for the line element $ds$,

$\ell^{2}\,=\,\lambda^{2}(x^{\mu})\,g_{\mu\nu}\,a^{\mu}a^{\nu}\,,\quad\mathrm{and}\;\;ds^{\prime}=\lambda(x^{\mu})\,ds\,.$
(4)

This last relation with the quadratic form implies that Weyl’s space is
conformally equivalent to an other space222In the original Weyl’s Geometry,
this other space was not necessarily General Relativity, while in WIG it is
always the case. (defined by $g^{\prime}_{\mu\nu}$) through the gauge
transformation,

$g^{\prime}_{\mu\nu}\,=\,\lambda^{2}\,g_{\mu\nu}\,.$ (5)

In the transport from a point $P_{1}(x^{\mu})$ to a nearby point
$P_{2}(x^{\mu}+dx^{\mu})$, the length $\ell$ of a vector is assumed to change
by,

$d\ell\,=\,\ell\,\kappa_{\nu}\,dx^{\nu}\,.$ (6)

There, $\kappa_{\nu}$ is called the coefficient of metrical connection, as a
fundamental characteristics of the geometry alike $g_{\mu\nu}$. If we change
the standards of length $\ell$ of any vector to
$\ell^{\prime}=\lambda(x)\,\ell$, one has

$\displaystyle\ell^{\prime}+d\ell^{\prime}=(\ell+d\ell)\,\lambda(x+dx)=(\ell+d\ell)\lambda(x)+\ell\,\lambda,_{\mu}dx^{\mu}\,,$
(7)
$\displaystyle\mathrm{thus}\;\;d\ell^{\prime}=\lambda(x)\,d\ell+\ell\,\lambda,_{\mu}dx^{\mu}=\lambda(x)\ell\,(\kappa_{\mu}+\Phi,_{\mu})dx^{\mu}\,,$
$\displaystyle\mathrm{with}\;\;\Phi,_{\mu}\,=\,\frac{\lambda,_{\mu}}{\lambda}\,,\;\;\mathrm{and}\;\,\Phi\,=\,\ln\lambda.$

Thus, we get for
$d\ell^{\prime}=\ell^{\prime}\,\kappa^{\prime}_{\nu}\,dx^{\mu}\,,$ (note that
different derivatives, e.g. with respect to $x^{\mu}$ are indicated by “,μ “,
or a by “;μ “ or even simply by “μ”, see definitions by Dirac (1973)),

$\kappa^{\prime}_{\nu}=\kappa_{\nu}\,+\Phi,_{\mu}\,.$ (8)

A quantity $Y$, scalar, vector or tensor, which in a scale transformation
changes like $Y^{\prime}\,=\,\lambda^{n}(x)\,Y$ is said to be _coscalar,
covector or cotensor_ of power $\Pi(Y)=n$, this is called _scale covariance_.
For $n=0$, we have an _inscalar, invector or intensor_ , this is the
particular case of scale invariance. The derivatives $Y,_{\mu}$ do not
necessarily enjoy the _co-covariant_ (a defintion by Dirac) or invariant
properties, but derivatives with such properties can be defined. Let us take
for example a scalar $S$ of power $n$. Its ordinary (covariant) derivative is
$S,_{\mu}$ (also noted $S_{\mu}$ when no ambiguity). Following Dirac (1973),
let us perform a change of scale, we have

$S^{\prime}_{\mu}\,=\,(\lambda^{n}S)_{\mu}=\lambda^{n}S_{\mu}+n\lambda^{n-1}\lambda_{\mu}S=\lambda^{n}(S_{\mu}+n\,\Phi_{\mu})S\,.$
(9)

With Eqs. (8) and (9), we get

$(S_{\mu}-n\kappa_{\mu}S)^{\prime}\,=\,\lambda^{n}(S_{\mu}-n\kappa_{\mu}S)\,.$
(10)

Now, we see that the covector $S_{*\mu}=(S_{\mu}-n\kappa_{\mu}S)$ of power $n$
is the co-covariant derivative of scalar $S$ (such derivatives are always
indicated with “∗”). This ensures that the co-covariant derivatives preserve
the power of the object they are applied on, thus the co-covariant derivative
is also preserving scale covariance.

For a covector of power $n$, similar developments (Dirac, 1973) lead to the
derivatives of covectors,

$\displaystyle A_{\mu*\nu}\,$ $\displaystyle=$
$\displaystyle\,\partial_{\nu}A_{\mu}-^{*}\Gamma_{\mu\nu}^{\alpha}A_{\alpha}-n\kappa_{\nu}A_{\mu},\,$
(11) $\displaystyle{}A_{*\nu}^{\mu}\,$ $\displaystyle=$
$\displaystyle\,\partial_{\nu}A^{\mu}+^{*}\Gamma_{\nu\alpha}^{\mu}A^{\alpha}-n\kappa_{\nu}A^{\mu},\,$
(12) $\displaystyle\mathrm{with}\quad^{*}\Gamma_{\mu\nu}^{\alpha}$
$\displaystyle=$
$\displaystyle\Gamma_{\mu\nu}^{\alpha}+g_{\mu\nu}\kappa^{\alpha}\,-g_{\mu}^{\alpha}\kappa_{\nu}-g_{\nu}^{\alpha}\kappa_{\mu},$
(13)
$\mathrm{with}\quad^{*}\Gamma_{\sigma,\mu\nu}=\,g_{\sigma\alpha}^{*}\Gamma_{\mu\nu}^{\alpha}\,.$
(14)

There, ${}^{*}\Gamma_{\mu\nu}^{\alpha}$ is a modified Christoffel symbol,
while $\Gamma_{\mu\nu}^{\alpha}$ is the usual form. As a further example, the
first derivatives of a co-tensor have the following expressions,

$\displaystyle
T^{\mu\nu}_{*\rho}=T^{\mu\nu}_{,\rho}+^{*}\Gamma_{\rho\sigma}^{\mu}T^{\sigma\nu}+^{*}\Gamma_{\rho\sigma}^{\nu}T^{\mu\sigma}-n\kappa_{\rho}T^{\mu\nu}\;,$
(15) $\displaystyle
T_{\mu\nu*\rho}=T_{\mu\nu,\rho}-^{*}\Gamma_{\mu\rho}^{\sigma}T_{\sigma\nu}-^{*}\Gamma_{\nu\rho}^{\sigma}T_{\mu\sigma}-n\kappa_{\rho}T_{\mu\nu}\,,$
(16)

which also ensures their scale covariance.

Second co-covariant derivatives can also be expressed. As an example, the
second derivative of coscalar $S$ becomes,

$S_{*\mu*\nu}\,=\,S_{*\mu,\nu}-(n-1)\kappa_{\nu}S_{*\mu}+\kappa_{\mu}S_{*\nu}-g_{\mu\nu}\kappa^{\sigma}S_{*\sigma}\,.$
(17)

An appropriate expression for the second derivative of a covector can also be
derived. See Canuto et al. (1977) for a summary of cotensor calculus.
Operations on covectors and cotensors can also be performed. Briefly, a
contravariant covector $a^{\mu}$ with power $m$ multiplied by a contravariant
vector $b^{\nu}$ with power $n$ will form a cotensor $a^{\mu}b^{\nu}$ of power
(m+n). The corresponding covariant components $a_{\mu}$ and $b_{\nu}$ have
power (m+2) and (n+2) respectively. Also, the scalar product
$\vec{a}\cdot\vec{b}=g_{\mu\nu}a^{\mu}b^{\nu}$ gives a coscalar of power
(m+n+2), the angle $\vartheta$ between the two vectors is conserved in
displacement on a geodesics as shown by (Bouvier & Maeder, 1978), who also
studied geodesics, isometries and the Killing vectors.

###  2.2 Weyl’s Integrable Geometry (WIG) and relation with scalar-tensor
theories

If a vector is parallelly transported along a closed loop, the total change of
the length of the vector can be expressed as,

$\Delta\ell\,=\,\ell\left(\partial_{\nu}\kappa_{\mu}-\partial_{\mu}\kappa_{\nu}\right)\,d\sigma^{\mu\nu}\,,$
(18)

where $d\sigma^{\mu\nu}=dx^{\mu}\wedge dx^{\nu}$ is an infinitesimal surface
element. Weyl identified the tensor
$F_{\mu\nu}\,=\,\left(\kappa_{\mu,\nu}-\kappa_{\nu,\mu}\right)$ with the
electromagnetic field, as its original aim was to also provide a geometrical
interpretation of electromagnetism. The problem is that this formulation (if
the parenthesis does not vanish) implies non-integrable lengths so that the
properties of an atom, such as its emission frequencies, would be influenced
by its past world line. Thus, one could not observe sharp atomic lines in the
presence of an electromagnetic field. This was the key point of Einstein
(1918), who criticized the use of Weyl’s geometry to describe electromagnetism
and gravitation.

In the line of the developments by Dirac (1973), a modified version of Weyl’s
Geometry called Weyl’s Integrable Geometry (WIG) was proposed by Canuto et al.
(1977). In WIG, the above Einstein’s remark no longer applies and it may thus
form a consistent framework for the study of gravitation as emphasized by
Canuto et al. (1977), see also Bouvier & Maeder (1978). Let us consider a line
element $ds^{\prime}=g^{\prime}_{\mu\nu}dx^{\mu}dx^{\nu}$, where the prime
symbols apply to Riemann space, while $ds$ refers to the WIG, which in
addition to the quadratic form $g_{\mu\nu}$ also has a scalar gauge field
$\lambda(x)$. In this case, in Riemann space we have,

$\kappa^{\prime}_{\nu}=0\,,$ (19)

since there is no change length in Riemann space. According to Eq. (8), we
thus have,

$\kappa_{\nu}\,=\,-\Phi_{,\nu}\,=\,-\frac{\partial\ln\lambda}{\partial
x^{\nu}}\,.$ (20)

This means that the metrical connexion $\kappa_{\nu}$ in WIG space is the
gradient of a scalar field,

$\Phi\,=\,\ln\lambda\,,$ (21)

and $\kappa_{\nu}dx^{\nu}$ is an exact differential,

$\partial_{\nu}\kappa_{\mu}\,=\,\partial_{\mu}\kappa_{\nu}\,.$ (22)

This implies that the parallel displacement of a vector along a closed loop
does not change its length, see Eq. (18). This means that the change of the
length does not depend on the path followed and thus Einstein’s objection does
not hold in WIG. Nevertheless, the nice mathematical tools of Weyl’s geometry
designed for preserving scale covariance (and invariance) also work in the
integrable form of this geometry and this is what we are using in this work.

At this point, it is appropriate to comment on the relation between the SIV
theory and the scalar-tensor theories. Alike these theories, but in the
context of WIG, SIV enjoys the property of conformal scale invariance. In
addition, there is also a coupled scalar field $\Phi$, expressed in Eqs. (20)
and (21). However, there is a major difference with scalar tensor theories. In
these, the scalar field is generally defined in an independent way, it has
some specific coupling to standard gravitation, but is not directly determined
by the scale factor. On the contrary, in SIV theory the scalar field is
entierely defined by the scale factor. Thus, there is no degree of freedom to
choose the scalar field and the theory is very constrained.

Now, the question is what is then constraining the scale factor $\lambda$ in
SIV. Alike the field equation of GR needs the specification of the metric
(Minkowski, Schwarschild, FLWR, etc) characterizing the physical system under
investigation, SIV needs one more specification in the form of a gauging
condition, necessary for fixing the gauge $\lambda$. Canuto et al. (1977) have
chosen “The Large Number Hypothesis” (Dirac, 1974), while a different choice
is made in Sect. (2.4), see also Maeder (2017a).

Many expressions were originally developed by Weyl (1923), Eddington (1923),
then extended by Dirac (1973) and applied by Canuto et al. (1977) in the
integrable form of Weyl’s Geometry. On the whole, WIG together with cotensor
calculus forms a complete scale covariant framework for gravitation.

###  2.3 The geodesic and the field equations

The objective of this work is to show that the Newtonian approximation in SIV
is just leading to the MOND rules for a valid approximation. For this purpose
we need the geodesic equation in our WIG framework. It has been obtained in
different consistent ways. \- 1. It has first been established by Dirac (1973)
from an action principle. \- 2. It was obtained by the covariant
generalisation of the usual relation
$u^{\prime\mu}_{,\nu}\,u^{\prime\nu}\,=\,0$ (Canuto et al., 1977),

$u^{\mu}_{*\nu}\,u^{\nu}\,=\,0.$ (23)

\- 3. The geodesic in WIG is the curve between two points so that the change
of the length of a vector is minimum: $\int^{P_{1}}_{P_{0}}\delta(dl)=0$
(Bouvier & Maeder, 1978). \- 4. As in GR, the geodesics in WIG are also direct
consequences of the Equivalence Principle (Maeder & Bouvier , 1979) in the
scale invariant framework,

$\frac{d^{2}x^{\alpha}}{ds^{2}}+^{*}\Gamma^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}+\kappa_{\nu}\frac{dx^{\alpha}}{ds}\frac{dx^{\nu}}{ds}\,=\,0\,,$
(24)

with ${}^{*}\Gamma^{\alpha}_{\mu\nu}$ given by Eq.(13). Thus, we get

$\frac{du^{\alpha}}{ds}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}-\kappa_{\mu}u^{\mu}u^{\alpha}+\kappa^{\alpha}=0\,,$
(25)

with the velocity $u^{\mu}\,=\,dx^{\mu}/ds$. This expression satisfying the
requirement of scale invariance will be used to derive the weak field
approximation (Sect. 4.1). At this stage, the expression of the metrical
connexion $\kappa_{\mu}$ is still unknown, since the gauge is not yet fixed.

The developments of cotensor calculus of Sect. (2.1) can be pursued, and they
are leading to a corresponding Riemann–Christoffel tensor
${}^{*}R^{\mu}_{\nu\lambda\rho}$ (see Sect. 87 by Eddington (1923); Dirac
(1973)),

${}^{*}R^{\mu}_{\nu\lambda\rho}\,=\,\frac{\partial^{*}\Gamma^{\mu}_{\nu\lambda}}{\partial
x^{\rho}}+\frac{\partial^{*}\Gamma^{\mu}_{\nu\rho}}{\partial
x^{\lambda}}+{{}^{*}\Gamma^{\eta}_{\nu\lambda}}{{}^{*}\Gamma^{\mu}_{\eta\rho}}+{{}^{*}\Gamma^{\eta}_{\nu\rho}}{{}^{*}\Gamma^{\mu}_{\lambda\eta}}\,.$
(26)

The contracted Riemann–Christoffel tensor or Ricci tensor appears to be an
intensor, it writes in the cotensor form (Eddington, 1923; Dirac, 1973)

${}^{*}R_{\mu\nu}=R^{\prime}_{\mu\nu}-\kappa_{\mu,\nu}-\kappa_{\nu,\mu}-g_{\mu\nu}\kappa^{\alpha}_{,\alpha}-2\kappa_{\mu}\kappa_{\nu}+2g_{\mu\nu}\kappa^{\alpha}\kappa_{\alpha}\,,$
(27)

where $R^{\prime}_{\mu\nu}$ is the usual expression in General Relativity. The
total curvature $R$ in the scale-invariant context is,

${}^{*}R\,=R^{\alpha}_{\alpha}\,=\,R^{\prime}-6\kappa^{\alpha}_{;\alpha}+6\kappa^{\alpha}\kappa_{\alpha},\
$ (28)

where $R^{\prime}$ is the total curvature in a standard Riemann geometry. All
these expressions, first and second derivatives, modified Christoffel symbols
${}^{*}\Gamma_{\mu\nu}^{\alpha}$, Riemann–Christoffel tensor
${}^{*}R^{\mu}_{\nu\lambda\rho}$, Ricci tensor and total curvature are scale
invariant by construction. The summation and products are keeping the
cotensoral character and scale covariance. They all contain additional terms
depending on $\kappa_{\nu}$. The major difference with these former references
rests on the fact that the metrical connexion $\kappa_{\nu}$ is the gradient
of the scale factor $\lambda$ by Eq. (20).

The scale invariant field equation has also been derived in different ways
(Canuto et al., 1977): parallelly to GR but with cotensor expressions taking
care of co-covariant expressions, and from an action principle (see below). In
the first method, the first member
${}^{*}R_{\mu\nu}-\frac{1}{2}{g_{\mu\nu}}{{}^{*}R}$ in the scale invariant
context is obtained from Eqs. (27) and (28). Its scale invariant properties
are ensured by the cotensoral nature of these expressions.

The second member of the field equation must also be scale invariant, implying
that the product of the gravitational constant $G$ and the energy-momentum
tensor $G\,T_{\mu\nu}$ should have the same property. Unlike Dirac (1974) and
Canuto et al. (1977), $G$ is considered as a constant as in Maeder (2017a).
Thus, one has,

$T_{\mu\nu}\,=\,T^{\prime}_{\mu\nu}\,.$ (29)

This expression has implications on the behavior of the relevant densities.
The scale invariance of tensor $T_{\mu\nu}$ implies,

$(p+\varrho)u_{\mu}u_{\nu}-g_{\mu\nu}p=(p^{\prime}+\varrho^{\prime})u^{\prime}_{\mu}u^{\prime}_{\nu}-g^{\prime}_{\mu\nu}p^{\prime}\,.$
(30)

There, velocities $u^{\prime\mu}$ and $u^{\prime}_{\mu}$ transform as follows,

$\displaystyle u^{\prime\mu}$ $\displaystyle=$
$\displaystyle\frac{dx^{\mu}}{ds^{\prime}}=\lambda^{-1}\frac{dx^{\mu}}{ds}=\lambda^{-1}u^{\mu}\,,$
$\displaystyle\mathrm{and}\;\;u^{\prime}_{\mu}$ $\displaystyle=$
$\displaystyle
g^{\prime}_{\mu\nu}u^{\prime\nu}=\lambda^{2}g_{\mu\nu}\lambda^{-1}u^{\nu}=\lambda\,u_{\mu}\,.$
(31)

The contravariant and covariant components of a vector have different power,
we have seen above that their covariant derivatives are different. Thus, the
energy-momentum tensor is scaling like,

$(p+\varrho)u_{\mu}u_{\nu}-g_{\mu\nu}p=(p^{\prime}+\varrho^{\prime})\lambda^{2}u_{\mu}u_{\nu}-\lambda^{2}g_{\mu\nu}p^{\prime}\,,$
(32)

implying the following behaviour for $p$ and $\varrho$ (Canuto et al., 1977),

$p=p^{\prime}\,\lambda^{2}\,\quad\mathrm{and}\quad\varrho=\varrho^{\prime}\,\lambda^{2}\,.$
(33)

The consistency of the field equation means that pressure and density are not
scale invariant, but coscalars of power $\Pi(\rho)=-2$.

The term with the cosmological constant in Einstein’s equation of GR is
$\Lambda_{\mathrm{E}}\,g^{\prime}_{\mu\nu}$. According to Eq. (5), we have

$\Lambda_{\mathrm{E}}\,g^{\prime}_{\mu\nu}\,=\,\Lambda_{\mathrm{E}}\lambda^{2}\,g_{\mu\nu}\,\equiv\,\Lambda\,g_{\mu\nu}\,.$
(34)

Notation $\Lambda_{E}$ is adopted to avoid any confusion with $\Lambda$ in WIG
($\Lambda_{\mathrm{E}}$ is not necessarily the value of Einstein static
model). In the SIV context, $\Lambda=\lambda^{2}\Lambda_{\mathrm{E}}$, it is a
coscalar of power $\Pi(\Lambda)=-2$, quite consistently with the previous
results for pressure and density. Also, this correspondance will insure the
scale iinvariance of the second member of the general field equation.
Assembling the above expressions developed in the WIG cotensoral context, we
may express the corresponding scale invariant field equation (Canuto et al.,
1977),

$\displaystyle R^{\prime}_{\mu\nu}-\frac{1}{2}\
g_{\mu\nu}R^{\prime}-\kappa_{\mu,\nu}-\kappa_{\nu,\mu}-2\kappa_{\mu}\kappa_{\nu}+2g_{\mu\nu}\kappa^{\alpha}_{,\alpha}-g_{\mu\nu}\kappa^{\alpha}\kappa_{\alpha}=$
$\displaystyle-8\pi
GT_{\mu\nu}-\lambda^{2}\Lambda_{\mathrm{E}}\,g_{\mu\nu},\,$ (35)

where $G$ is a true constant, as seen above. This is a generalization of
Einstein equation in WIG. In addition to the general covariance of Einstein
equation, it is also scale invariant, since each term in the equation
satisfies this requirement. The terms with a prime are the same as in GR, the
equation contains additional terms depending on $\kappa_{\nu}$. This
coefficient will be determined by the gauging condition we may adopt for the
physical system considered, in the same way as the $g_{\mu\nu}$ and their
sequence of derivatives (up to the Ricci tensor) are determined by the metric,
_i.e._ some sort of topography of the space-time system.

The above coordinate and scale invariant field equation (35) has also been
derived from an action principle. Such a so-called co-covariant equation has
been developed by Dirac (1973) for the vacuum and thus it concerns the first
member of the general field equation. As this was originally performed in the
classical Weyl’s geometry, the additional field Dirac introduced in the action
is independent of the scale factor $\lambda$, in this sense Dirac’s results
belong to scalar-tensor theories, as he was pointing it himself.

The study by Canuto et al. (1977) was made in the line of that by Dirac
(1973), with the difference that it is formally related to the WIG framework.
The action is a generalization of Einstein action dealing with both coordinate
and scale invariant equations,

$I\,=\,\int\lambda^{2}\left(*R\right)\sqrt{g}\,d^{4}x\,.$ (36)

The scale factor $\lambda$ is a coscalar of power $\Pi(\lambda)=-1$, while
$\left(*R\right)$ (so noted as a coscalar) has a power $\Pi(*R)=-2$, since it
is a contraction of the intensor $R_{\mu\nu}$. We also have
$\Pi(g_{\mu\nu})=2$, and thus $g=\det(g_{\mu\nu})$ is a co-scalar of power
$\Pi(g)=8$. The multiplication by $\lambda^{2}$ garantees the invariant
property. Terms being functions of $\lambda$ and of its cotensor derivatives
are added to the above equation, no other new field is introduced. The action
principle writes,

$\delta
I=\delta\int\left(-\lambda^{2}\,*R+c_{1}\lambda^{*\mu}\lambda_{*\mu}+c_{2}\lambda^{4}\right)\sqrt{g}\,d^{4}x=0\,,$
(37)

where $c_{1}$ and $c_{2}$ are constants. The quartic term then is related to
the cosmological constant. A matter Lagrangian $\mathcal{L}$ may be included,
its relation with the energy momentum tensor is,

$G\,T^{\mu\nu}\,=\,\lambda^{-2}\,\frac{2}{\sqrt{g}}\frac{\delta\sqrt{g}\;\mathcal{L}}{\delta
g_{\mu\nu}}\,,$ (38)

with the term $\lambda^{-2}$ for power consistency, while the Lagrangian
density $\mathcal{L}$ must be an inscalar expression. As a result, the
development of the above expressions confirms the above scale invariant field
equation (35) and, consistently enough, produces no new field equations.

###  2.4 Fixing the gauge

Since scale covariance is considered in addition to the coordinate covariance
of GR, an additional condition is necessary to specify the gauge $\lambda$.
Dirac (1973) and Canuto et al. (1977) had chosen the so-called “Large Number
Hypothesis“, see also Dirac (1974). The author’s choice is to adopt the
following statement (Maeder, 2017a): _The macroscopic empty space is scale
invariant, homogeneous and isotropic_. This assumption is consistent with the
scale invariance of GR in empty space and with Maxwell’s equations in absence
of charges and currents. Moreover, 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 (Carroll et al., 1992).

Under the above key hypothesis, one is left from the general field Eq.(35)
with the following condition for the empty spacetime,

$\kappa_{\mu;\nu}+\kappa_{\nu;\mu}+2\kappa_{\mu}\kappa_{\nu}-2g_{\mu\nu}\kappa_{;\alpha}^{\alpha}+g_{\mu\nu}\kappa^{\alpha}\kappa_{\alpha}=\lambda^{2}\Lambda_{\mathrm{E}}\,g_{\mu\nu}.$
(39)

The geometrical terms $R^{\prime}_{\mu\nu}$ and $R^{\prime}$ of the field
equation have disappeared, since the de Sitter metric for an empty space
endowed with a cosmological constant is conformal to the Minkowski metric
where $R^{\prime}_{\mu\nu}=0$ and $R^{\prime}=0$. The conformal relation
becomes an identity, if $3\lambda^{-2}/(\Lambda_{\mathrm{E}}t^{2})=1$ (Maeder,
2017a), a condition which is noticeably fully consistent with the solution of
(39), as shown below.

We assume that the macroscopic empty space characterized by the above equation
is homogeneous and isoptropic. This is consistent with the hypothesis that the
scale factor $\lambda$ is a function of time only. Thus, only the zero
component of $\kappa_{\mu}$ is non-vanishing and the coefficient of metrical
connection becomes,

$\displaystyle\kappa_{\mu;\nu}=\kappa_{0;0}=\partial_{0}\kappa_{0}=\frac{d\kappa_{0}}{dt}\equiv{\dot{\kappa}_{0}}\,=-\frac{\dot{\lambda}}{\lambda}\,,$
(40)

The 0 and the 1, 2, 3 components of what remains from Eq. (39) become
respectively:

$3\kappa^{2}_{0}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,,\quad\mathrm{and}\;\;2\dot{\kappa}_{0}-\kappa_{0}^{2}=-\lambda^{2}\Lambda_{\mathrm{E}}\,.$
(41)

and we get the two most important equations,

$\displaystyle\
3\,\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,\quad\mathrm{and}\quad\,2\frac{\ddot{\lambda}}{\lambda}-\frac{\dot{\lambda}^{2}}{\lambda^{2}}\,=\,\lambda^{2}\,\Lambda_{\mathrm{E}}\,,$
(42)

or some combinations of them. These two expressions have important
consequences:

\- In GR, $\Lambda_{\mathrm{E}}$ and the properties of the empty space are
considered not to depend on the matter content of the Universe. The same
applies to the above two equations and to the scale factor $\lambda$. This is
also consistent with the fact that matter density does not appear in these two
equations.

\- These differential equations establish a relation of the cosmological
constant, or the energy density of the vacuum, with the scale factor $\lambda$
and its variations.

\- From the relation $\Lambda=8\pi G\varrho_{\mathrm{vac}}$ and the first of
the Eqs. (42), the energy density of the vacuum can be expressed in term of a
scalar field $\psi$,

$\varrho\,=\,\frac{1}{2}\,C\,\dot{\psi}^{2}\,\quad\mathrm{with}\;\,\dot{\psi}\,=\,\kappa_{0}=\,-\frac{\dot{\lambda}}{\lambda}\,.$
(43)

with constant $C=3/(4\pi G)$. The field $\psi$ obeys a modified Klein-Gordon
equation (Maeder and Gueorguiev, 2021a).

\- For a solution of Eqs. (42) of the form $\lambda=a(t-b)^{n}+d$, we get
$d=0$, $n=-1$ with $a=\sqrt{\frac{3}{\Lambda_{\mathrm{E}}}}$. There is no
condition on $b$ from Eqs.(42), any value would fit. However, the solutions of
the cosmological equations may put some conditions on the origin of time,
depending on model parameters, see Sect. 3. Thus, with this remark the general
solution is,

$\lambda(t)=\sqrt{\frac{3}{\Lambda_{\mathrm{E}}}}\frac{1}{ct}\,.$ (44)

Thus, if we adopt the scale factor $\lambda_{0}=1$ at the present time
$t_{0}=1$, we just have $\lambda(t)=(t_{0}/t)$ in a system of units where
$\sqrt{\frac{3}{c^{2}\,\Lambda_{\mathrm{E}}}}=1$.

##  3 Cosmological solutions and their implications

The application of the FLWR metric to Eq.(35) leads to cosmological equations
(Canuto et al., 1977),

$\displaystyle\frac{8\,\pi
G\varrho}{3}=\frac{k}{a^{2}}+\frac{\dot{a}^{2}}{a^{2}}+2\,\frac{\dot{\lambda}\,\dot{a}}{\lambda\,a}+\frac{\dot{\lambda}^{2}}{\lambda^{2}}-\frac{\Lambda_{\mathrm{E}}\lambda^{2}}{3}\,,$
(45) $\displaystyle-8\,\pi
Gp=\frac{k}{a^{2}}+2\frac{\ddot{a}}{a}+2\frac{\ddot{\lambda}}{\lambda}+\frac{\dot{a}}{a}^{2}+4\frac{\dot{a}\,\dot{\lambda}}{a\,\lambda}-\frac{\dot{\lambda^{2}}}{\lambda^{2}}-\Lambda_{\mathrm{E}}\,\lambda^{2}\,,$
(46)
$\displaystyle\mathrm{with}\;\;\varrho\,a^{3(1+c^{2}_{s})}\lambda^{1+3c^{2}_{s}}=\mathrm{const.}$
(47)

The last expression is the conservation law, with a sound velocity
$c^{2}_{s}=0$ for a dust model and $c^{2}_{s}=1/3$ for the radiative era. If
$\lambda$ is a constant, the derivatives of $\lambda$ vanish and one is
brought back to the equations of GR. Solutions of these equations have been
searched by Canuto et al. (1977) for two variant cases of the Large Number
Hypothesis. The major point is that the above equations are leading to
solutions characterized by expansion factors $a(t)\,\sim\,t$ at large
cosmological times, a situation no longer supported nowadays.

Remarkably, the gauging condition, which implies Eqs. (42), leads to major
simplifications of Eqs. (45) and (46), which become (Maeder, 2017a),

$\displaystyle\frac{8\,\pi
G\varrho}{3}=\frac{k}{a^{2}}+\frac{\dot{a}^{2}}{a^{2}}+2\,\frac{\dot{a}\dot{\lambda}}{a\lambda}\,,$
(48) $\displaystyle-8\,\pi
Gp=\frac{k}{a^{2}}+2\frac{\ddot{a}}{a}+\frac{\dot{a^{2}}}{a^{2}}+4\frac{\dot{a}\dot{\lambda}}{a\lambda}\,.$
(49)

For a constant $\lambda$, Friedmann’s equations are recovered. A third
equation may be derived from the above two,

$-\frac{4\pi
G}{3}\left(3p+\varrho\right)=\frac{\ddot{a}}{a}+\frac{\dot{a}\dot{\lambda}}{a\lambda}\,.$
(50)

Since $\dot{\lambda}/{\lambda}$ is negative, the extra term represents an
additional acceleration _in the direction of motion_. This effect of the scale
invariance is fundamentally different from that of the cosmological constant.
Now, for an expanding Universe, this extra force produces an accelerated
expansion, without requiring dark energy particles. For a contraction, the
additional term favours collapse, as shown in the study of the growth of
density fluctuations (Maeder and Gueorguiev, 2019), where an early formation
of galaxies is resulting without the need of dark matter.

Figure 1: Top: The expansion factor $a(t)$ for the $\Lambda$CDM and SIV models
as a function of $\Omega_{\mathrm{m}}$. Bottom: The scale factor
$\lambda_{\mathrm{in}}=1/t_{\mathrm{in}}$ at the initial time
$t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{in}}$ as a function of
$\Omega_{\mathrm{m}}$. The yellow zone shows, vs. $\Omega_{\mathrm{m}}$, the
range of $\lambda(t)$ from the Big-Bang (broken red line) to the present time
(continuous red line). For $\Omega_{\mathrm{m}}=0.3$, $\lambda(t)$ only varies
between 1.494 at the origin and 1.0 at present.

The solutions of these equations have been discussed in details in Maeder
(2017a), together with various cosmological properties concerning the Hubble-
Lemaître and deceleration parameters, the cosmological distances and different
cosmological tests. The redshift drifts appear as one of the most promising
cosmological tests (Maeder and Gueorguiev, 2020a). Here, we limit the
discussion to a few points pertinent to the subject of the paper. Analytical
solutions for the flat SIV models with $k=0$, considered here, have been found
for the matter (Jesus, 2018) and radiation (Maeder, 2019) dominated models. In
the former case, we have

$a(t)\,=\,\left[\frac{t^{3}-\Omega_{\mathrm{m}}}{1-\Omega_{\mathrm{m}}}\right]^{2/3}\,.$
(51)

It is expressed in the timescale $t$ where at present $t_{0}=1$ and
$a(t_{0})=1$. Such solutions are illustrated in Fig. 1, top. They are lying
relatively close to the $\Lambda$CDM ones, the differences being larger for
lower $\Omega_{\mathrm{m}}$. This is a general property: the effects of scale
invariance are always larger for the lower matter densities, being the largest
ones for the empty space.  There
$\Omega_{\mathrm{m}}=\varrho/\varrho_{\mathrm{c}}$ with
$\varrho_{\mathrm{c}}=3H^{2}_{0}/(8\pi G)$. Remarkably, Eqs. (48) and (49)
allow flatness for different values of $\Omega_{\mathrm{m}}$, unlike the
classical Friedmann models.

The initial time when $a(t_{\mathrm{in}})=0$ is,

$t_{\mathrm{in}}\,=\,\Omega^{1/3}_{\mathrm{m}}\,.$ (52)

This dependence in 1/3 produces a rapid increase of $t_{\mathrm{in}}$ for
increasing $\Omega_{\mathrm{m}}$. 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. The key point is that this leads
to a strong reduction of the range of $\lambda(t)$ for increasing
$\Omega_{\mathrm{m}}$ (Fig. 1, bottom): while the range of $\lambda$ is
infinite for an empty model, it is very limited for significant
$\Omega_{\mathrm{m}}$-values. Thus, the presence of matter through
$\Omega_{\mathrm{m}}$ drastically reduces the range of variation of the
universal scale factor $\lambda$. For $\Omega_{\mathrm{m}}>1$ scale invariance
is killed, which makes sense in view of the remarks by Feynman (1963). This is
a global effect associated to the range of $\lambda$ in Universe models. This
does not prevent other effects due to local variations of density to also
intervene, as shown in Sect. (4.3), see Eq. (74).

The Hubble parameter is , in the timescale $t$ (which goes from
$t_{\mathrm{in}}$ at Big-Bang to $t_{0}=1$ at present),

$H(t)=\frac{2\,t^{2}}{t^{3}-\Omega_{\mathrm{m}}}\,.$ (53)

From Eqs. (51) and (53), we see that there is no meaningful scale invariant
solution for an expanding Universe model with $\Omega_{\mathrm{m}}$ equal or
larger than 1. Thus, the model solutions are quite consistent with the
causality relations discussed by Maeder and Gueorguiev (2021a).

One can also define

$\Omega_{\mathrm{k}}=-\frac{k}{a^{2}H_{0}^{2}}\,\quad\mathrm{and}\quad\Omega_{\lambda}\,=\,-\frac{2}{H_{0}}\left(\frac{\dot{\lambda}}{\lambda}\right)_{0}\,=\,\frac{2}{H_{0}\,t_{0}}\,.$
(54)

These are the normalized contributions vs. $\varrho_{\mathrm{c}}$ respectively
of the matter, space curvature, and scale factor $\lambda$. With these
definitions, the first cosmological equation (48) leads to,

$\Omega_{\mathrm{m}}\,+\,\Omega_{\mathrm{k}}\,+\Omega_{\lambda}=\,1\,.$ (55)

These quantities are usually considered at the present time.

##  4 The Newtonian and MOND approximations

###  4.1 The weak field approximation

The scale invariant expression of the geodesic equation was derived by Dirac
(1973) and also from an action principle by Bouvier & Maeder (1978), see Sect.
2.3. The weak field low velocity approximation has been obtained by Maeder &
Bouvier  (1979), see also Maeder (2017c),

$\frac{d^{2}\bf{r}}{dt^{2}}\,=\,-\frac{G_{t}\,M(t)}{r^{2}}\,\frac{\bf{r}}{r}+\kappa(t)\,\frac{d\bf{r}}{dt}\,,$
(56)

in spherical coordinates. It contains an additional acceleration term in the
direction of motion, the dynamical gravity. This term, proportional to the
velocity, favours collapse during a contraction, and outwards motion if
expansion.

The conservation law (Eq. 47) imposes for a dust Universe a relation $\varrho
a^{3}\lambda=const.$, meaning that the inertial mass of a particle is not a
constant and that it depends on the scale factor $\lambda$. We note that the
non-constancy of mass is also a common situation in Special Relativity. Here,
masses vary like $M(t)=M(t_{0})(t/t_{0})$. Interestingly enough, rather than
the inertial and gravitational mass, the gravitational potential $\Phi=G\,M/r$
of an object, thus the field, appears as a more fundamental quantity, being
scale-invariant through the evolution of Universe. As an example, for
$\Omega_{\mathrm{m}}=0.3$, the mass at the Big-Bang was,
$M(t_{\mathrm{in}})=\Omega_{\mathrm{m}}^{1/3}\,M(t_{0})=0.6694\;M(t_{0})$, the
variations are smaller than 1% over the last 400 million years (Sect.4.2).

In the above cosmological models, the age $t$ is $t_{0}=1$ at present and
$t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}}$ at the origin. The usual timescale
$\tau$ in years or seconds is $\tau_{0}=13.8$ Gyr at present (Frieman et al.,
2008) and $\tau_{\mathrm{in}}=0$ at the Big-Bang. Thus, the relation between
these ages is,

$\frac{\tau-\tau_{\mathrm{in}}}{\tau_{0}-\tau_{\mathrm{in}}}=\frac{t-t_{\mathrm{in}}}{t_{0}-t_{\mathrm{in}}}\,,$
(57)

expressing that the age fraction with respect to the present age is the same
in both timescales. This gives

$\tau\,=\,\tau_{0}\,\frac{t-\Omega^{1/3}_{\mathrm{m}}}{1-\Omega^{1/3}_{\mathrm{m}}}\,\quad\mathrm{and}\;\;t\,=\,\Omega^{1/3}_{\mathrm{m}}+\frac{\tau}{\tau_{0}}(1-\Omega^{1/3}_{\mathrm{m}})\,,$
(58)

and for the derivatives,

$\frac{d\tau}{dt}\,=\,\frac{\tau_{0}}{1-\Omega^{1/3}_{\mathrm{m}}}\,,\quad\mathrm{and}\;\;\frac{dt}{d\tau}\,=\,\frac{1-\Omega^{1/3}_{\mathrm{m}}}{\tau_{0}}\,.$
(59)

For larger $\Omega_{\mathrm{m}}$, timescale $t$ is squeezed over a smaller
fraction of the interval 0 to 1.0, (which reduces the range of $\lambda$ over
the ages).

We need to convert the equation of motion (56) expressed with variable $t$
into the usual time $\tau$. Equation (56) becomes,

$\frac{d^{2}\bf{r}}{d\tau^{2}}\left(\frac{d\tau}{dt}\right)^{2}\,=\,-\frac{G_{t}\,M(t)}{r^{2}}\,\frac{\bf{r}}{r}+\,\frac{1}{t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})}\,\frac{d\tau}{dt}\frac{d\bf{r}}{d\tau}\,\,.$
(60)

Here $G_{t}$ is used to specify that the gravitational constant is expressed
with time units $t$. In the $\tau$-scale, the units of $G$ are $[cm^{3}\cdot
g^{-1}s^{-2}]$, thus, the correspondence is
$G_{t}\,\left(\frac{dt}{d\tau}\right)^{2}=G$. At present, the masses
$M(t_{0})$ and $M(\tau_{0})$ are evidently equal. At other epochs, the
relation is,

$M(t)=\frac{t}{t_{0}}M(t_{0}),\;\mathrm{thus}\;M(\tau)=\left[\Omega^{1/3}_{\mathrm{m}}+\frac{\tau}{\tau_{0}}(1-\Omega^{1/3}_{\mathrm{m}})\right]M(\tau_{0}).$
(61)

Now, multiplying both members of Eq.(60) by
$\left(\frac{dt}{d\tau}\right)^{2}$, we get at time $\tau/\tau_{0}$,

$\frac{d^{2}\bf{r}}{d\tau^{2}}\,=\,-\frac{G\,M(\tau)}{r^{2}}\,\frac{\bf{r}}{r}+\,\frac{1}{t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})}\,\frac{t_{0}-t_{\mathrm{in}}}{\tau_{0}}\,\frac{d\bf{r}}{d\tau}\,\,.$
(62)

We define the numerical factor $\psi$,

$\psi(\tau)\,=\,\frac{t_{0}-t_{\mathrm{in}}}{t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})}\,;\;\mathrm{thus}\;\psi_{0}=\psi(\tau_{0})\,=1-\Omega^{1/3}_{\mathrm{m}}\,.$
(63)

The modified Newton’s equation at present time $\tau_{0}$ is then,

$\frac{d^{2}\bf{r}}{d\tau^{2}}\,=\,-\frac{G\,M(\tau_{0})}{r^{2}}\,\frac{\bf{r}}{r}+\frac{\psi_{0}}{\tau_{0}}\frac{d\bf{r}}{d\tau}\,\,.$
(64)

The additional term, the dynamical gravity, which is generally extremely small
(cf. Eq. 74), also depends on $\Omega_{\mathrm{m}}$: in an empty Universe,
$\psi_{0}=1$, the effect being maximum, while for $\Omega_{\mathrm{m}}=1$, one
consistently has $\psi_{0}=0$, scale invariance has no effect. For
$\Omega_{\mathrm{m}}=0.30$, 0.20, 0.10 and 0.05 one has
$\psi_{0}=0.331,0.415,0.536$ and 0.632, which reduces the dynamical gravity.

###  4.2 The MOND approximation: first approach

Let us first examine how the scale factor $\lambda(\tau)$ and consequently the
masses $M(\tau)$ are varying over the past. For the case
$\Omega_{\mathrm{m}}=0.30$ as an example, over the last 100 Myr, 200 Myr and
0.5 Gyr the mass increase predicted by Eq. (61) amounts to a factor 1.0024,
1.0048 and 1.012 respectively. For $\Omega_{\mathrm{m}}=0.10$, these values
would be 1.0039, 1.0078 and 1.020.

Thus, for galaxies where the rotation periods are a few hundered millions
years, we can consider that both $\lambda$ and masses are constant with an
accuracy equal or better than 1%. Indeed, this is quite consistent with MOND,
which is also known to be scale invariant with a constant scale factor
$\lambda$ (Milgrom, 2015). Such an approximation is much less satisfactory for
clusters of galaxies where the time scales, e.g. the crossing times, are of
the order of a few Gyr.

With a constant $\lambda$, the coefficient
$\kappa=-\frac{\dot{\lambda}}{\lambda}$ is equal to zero and the dynamical
gravity in Eq. (64) disappears. We are left with transformations
$r=\lambda\;r^{\prime}$ and $t=\lambda\;t^{\prime}$ with a constant $\lambda$
(and thus $M$) applied to the Newton equation expressed in the prime
coordinates,

$\frac{d^{2}r^{\prime}}{dt^{\prime
2}}\,=\,-\frac{GM}{r^{\prime}{{}^{2}}}\,\equiv\,g^{\prime}_{N}.$ (65)

The total acceleration $g$ in system $(r,t)$ becomes,

$g\,=\,\frac{d^{2}r}{dt^{2}}\,=\,\frac{1}{\lambda}\,\frac{d^{2}r^{\prime}}{dt^{\prime
2}}\,.$ (66)

The Newtonian gravitational accelerations $g_{N}$ and $g^{\prime}_{N}$ are
related by,

$g_{N}\,\equiv\,-\frac{G\,M}{r^{2}}\,=\,-\frac{1}{\lambda^{2}}\frac{G\,M}{r^{\prime
2}}\,\equiv\frac{1}{\lambda^{2}}\,g^{\prime}_{N}\,.$ (67)

Eq. (66) can thus be developed as follows,

$g\,=\,\frac{d^{2}r}{dt^{2}}\,=\,\frac{1}{\lambda}\,\frac{d^{2}r^{\prime}}{dt^{\prime
2}}\,=\,\frac{1}{\lambda}\,g^{\prime}_{N}\,=\,{\lambda}\,g_{N}\,,$ (68)

according to (67). This last relation also implies,

$g\,=\,\frac{d^{2}r}{dt^{2}}\,=\,{\lambda}\,g_{N}\,=\,\left(\frac{g^{\prime}_{N}}{g_{N}}\right)^{1/2}g_{N}\,=\left(g^{\prime}_{N}\,g_{N}\right)^{1/2}\,.$
(69)

This is to be compared to the deep-MOND limit given by Eq. (1). We note a
correspondence between constant $a_{0}$ and $g^{\prime}_{N}$. At this stage,
we have no information on what kind of value should be used for
$g^{\prime}_{N}$, and for what range of gravities it may apply. In a second
approach below, we will get more information on these points. For now, we note
that the approximation of constant $\lambda$ and masses over a few hundred
millions years in the scale invariant theory just leads to a form analog to
the deep-MOND limit.

The constant $a_{0}$ is related to $c\,H_{0}$ (Milgrom, 1983, 2015). We may
wonder why a constant, considered as a universal constant, thus being the same
at any time, should just be related to the present value $H_{0}$, see also
(Milgrom, 2020). This is highly suggestive that $a_{0}$, alike $\lambda$, is
in fact time dependent, contrarily to MOND assumption, but in agreement with
the scale invariant theory.

###  4.3 The MOND approximation: second approach

We may also derive the MOND behaviour at very low gravities from the equation
of motion Eq. (64). The ratio $x$ of the radial components of the dynamical
gravity to the Newtonian one is given by,

$x\,=\,\frac{\psi_{0}\,\upsilon r^{2}}{\tau_{0}\,GM}\,.$ (70)

where $\upsilon$ is the radial component of the velocity. The ratio $x$ may
become larger than 1 in two particular cases:

\- 1. In very early stages of the Universe, $\tau_{0}$ is very small and
favours a large dynamical acceleration $(\psi\,\upsilon)/\tau_{0}$ in the
sense of motion. This effect is likely to have favoured the early galaxy
formation without the need of dark matter (Maeder and Gueorguiev, 2019).

\- 2. At large distances from a central body, the Newtonian gravity $g_{N}$
may become smaller than $(\psi\,\upsilon)/\tau_{0}$. This may typically occur
in very wide binaries and in the outer layers of galaxies and clusters. This
situation is favoured by the fact that both the deep-MOND Limit and SIV theory
predict that the orbital velocities in two-body systems are independant of the
orbital radius (Milgrom, 2014b; Maeder & Bouvier , 1979). Thus, in both cases
larger orbital velocities may favour the dynamical acceleration
$(\psi\,\upsilon)/\tau_{0}$ in the outer layers of gravitational systems.

We use the fact that $H_{0}=\frac{\xi}{\tau_{0}}$ and
$\varrho_{\mathrm{c}}=\frac{3\,H_{0}^{2}}{8\,\pi\,G}$ to express $\tau_{0}$ in
term of $\varrho_{\mathrm{c}}$ in Eq.(70). To obtain $\xi$, we use the
following expression of the Hubble-Lemaître parameter (see Appendix),

$\displaystyle
H(\tau_{0})\,=\,\frac{2}{1-\Omega_{\mathrm{m}}}\,\frac{(1-\Omega^{1/3}_{\mathrm{m}})}{\tau_{0}}\,,$
(71)
$\displaystyle\quad\mathrm{thus}\;\;\xi\,=\,\frac{2\,(1-\Omega^{1/3}_{\mathrm{m}})}{(1-\Omega_{\mathrm{m}})}.$
(72)

The ratio $\xi$ is about unity in the SIV theory for
$\Omega_{\mathrm{m}}=0.10,0.20$ or 0.30, there one has $\xi=1.191,1.038$ or
0.945 respectively. We consider a mass $M$ spherically distributed in a radius
$r$ with a mean density $\varrho$ and get,

$x\,=\,\frac{\sqrt{2}\,\psi_{0}}{\xi}\left(\frac{\upsilon^{2}}{(GM/r)}\frac{\varrho_{\mathrm{c}}}{\varrho}\right)^{1/2}\,.$
(73)

Let us consider a two-body system formed by a massive central mass $M$ and a
test object of negligible mass, there the instantaneous equilibrium of forces
determined by Eq. (64) along the radial direction is just
$\frac{\upsilon^{2}}{r}=\frac{GM}{r^{2}}$, since the additional dynamical
acceleration is in the direction of the motion333When one considers the
average velocity dispersion in a long-time evolution, this type of relation
does not necessarily hold any more (Maeder, 2017c)). Thus, in external regions
of galaxies where normally the motion should be mainly determined by the
central mass concentration, the ratio $x$ becomes,

$x\,=\,\frac{\sqrt{2}\,\psi_{0}}{\xi}\left(\frac{\varrho_{\mathrm{c}}}{\varrho}\right)^{1/2}\,.$
(74)

For high values of the density $\varrho$ with respect to the critical density,
the $x$-parameter becomes negligible and thus the gravity is just determined
by the usual Newton Law. At the edge of the sphere of radius $r$ and mean
density $\varrho$, the Newtonian gravity $g_{N}=(4/3)\pi G\,\varrho\,r$, so
that we also have,

$x\,=\,\frac{\sqrt{2}\,\psi_{0}}{\xi}\left(\frac{g_{\mathrm{c}}}{g_{\mathrm{N}}}\right)^{1/2}\,,$
(75)

where $g_{\mathrm{c}}$ is the mean gravity at the edge of a similar sphere
having the critical density. Thus, the total gravity $g$ given by the the
first member of Eq.(64) is

$g\,=\,g_{\mathrm{N}}+x\,g_{\mathrm{N}}\,=g_{\mathrm{N}}\left[1+\frac{\sqrt{2}\,\psi_{0}}{\xi}\,\left(\frac{g_{\mathrm{c}}}{g_{\mathrm{N}}}\right)^{1/2}\right]\,.$
(76)

Let us consider regions at large distances from the galactic center, or in a
binary system at large distances from the other body. There, the Newtonian
gravity $g_{\mathrm{N}}$, resulting from the attraction of the central or
other body, can be counterbalanced by the gravity of external galaxies or
other star systems so that the resulting $g_{\mathrm{N}}$ essentially vanishes
and $x$ becomes bigger than 1 in large regions. When this happens, we get a
total resulting gravity $g$ behaving like,

$g\,\rightarrow\,\frac{\sqrt{2}\,\psi_{0}}{\xi}\,\left(g_{\mathrm{c}}\,g_{\mathrm{N}}\right)^{1/2}\,.$
(77)

The numerical factor $\frac{\sqrt{2}\,\psi_{0}}{\xi}$ becomes,

$\frac{\sqrt{2}\,\psi_{0}}{\xi}=\frac{\sqrt{2}\,(1-\Omega^{1/3}_{\mathrm{m}})}{H(\tau_{0})\,\tau_{0}}=\frac{(1-\Omega_{\mathrm{m}})}{\sqrt{2}}\,,$
(78)

where we have used Eq.(72) for $\xi$. For $\Omega_{\mathrm{m}}=$ 0, 0.1, 0.2,
0.3, 0.5, one has $\frac{\sqrt{2}\,\psi_{0}}{\xi}=$ 0.707, 0.636, 0.566,
0.495, 0.354. Thus, the weak field equation (64) leads to relation (77)
equivalent to the deep-MOND Limit $g\,=\,\sqrt{a_{0}\;g_{\mathrm{N}}}$. In
this second approach, we may learn much more on the significance of $a_{0}$
and its numerical value.

###  4.4 The significance of the $a_{0}$-parameter

We have the correspondence

$a_{0}\,\Longleftrightarrow\,\frac{(1-\Omega_{\mathrm{m}})^{2}}{2}g_{\mathrm{c}}\,.$
(79)

We may explicit the limiting value $g_{\mathrm{c}}$ in term of the critical
density over the radius $r_{\mathrm{H_{0}}}$ of the Hubble sphere, defined by
$n\,c\,=r_{\mathrm{H_{0}}}H_{0}$. There, $n$ depends on the cosmological
model. As an example, for the EdS model, $n=2$, for SIV or $\Lambda$CDM models
with $\Omega_{\mathrm{m}}=0.2-0.3$, the initial braking and recent
acceleration almost compensate each other, so that $n\simeq 1$. We get,

$\displaystyle
a_{0}\,=\,\frac{(1-\Omega_{\mathrm{m}})^{2}}{2}\,\frac{4\pi}{3}G\varrho_{\mathrm{c}}r_{\mathrm{H_{0}}}=\frac{(1-\Omega_{\mathrm{m}})^{2}}{4}\,n\,c\,H_{0}\,$
(80) $\displaystyle\mathrm{or}\quad
a_{0}\,=\,\frac{n\,c\,(1-\Omega_{\mathrm{m}})(1-\Omega^{1/3}_{\mathrm{m}})}{2\,\tau_{0}}$
(81)

The product $c\,H_{0}$ is equal to 6.80 $10^{-8}$ cm s-2. For
$\Omega_{\mathrm{m}}$=0, 0.10, 0.20, 0.30 and 0.50, we get $a_{0}\approx$
(1.70, 1.36, 1.09, 0.83, 0.43) $\;\cdot\;10^{-8}$ cm s-2 respectively. These
values obtained from the SIV theory are remarkably close to the value $a_{0}$
about 1.2 $\cdot\,10^{-8}$ cm s-2 derived from observations by Milgrom (2015).

Let us make several remarks on the $a_{0}$-parameter and its meaning:

\- 1. The equation of the deep-MOND limit is reproduced by the SIV theory both
analytically and numerically if $\lambda$ and $M$ can be considered as
constant. This may apply to systems with a typical dynamical timescale up to a
few hundred million years.

-2. Parameter $a_{0}$ is not a universal constant. It depends on the Hubble-Lemaître $H_{0}$ parameter (or the age of the Universe) and on $\Omega_{\mathrm{m}}$ in the model Universe, cf. Eq. (80). The value of $a_{0}$ applies to the present epoch.

-3 . Parameter $a_{0}$ is defined by the condition that $x>1$, _i.e._ when the dynamical gravity $(\psi_{0}\upsilon)/\tau_{0})$ in the equation of motion (Eq. 64) becomes larger than the Newtonian gravity. This situation may occur over large regions at the edge of gravitational systems.

##  5 Conclusions

The basic properties of SIV and its relations with the scalar-tensor theories
of gravity have been reviewed. Similarities and differences have been
enlightened. The deep-MOND limit is found to be an approximation of the SIV
theory for low enough densities and for systems with timescales smaller than a
few Myr.

SIV theory preserves the physical properties of General Relativity and
enlarges the group of symmetries by the inclusion of scale covariance (Dirac,
1973). In fact, this also avoid the call to dark matter, and at the same time
as shown by Fig.1 the SIV theory predicts an accelerated expansion. Results of
a number of cosmological and astrophysical applications have been quoted in
the introduction.

Finally, while the present approach may for the moment look “non standard” or
out of the main stream, we point out that the present work is not in
contradiction with the following statement by Einstein (1949):  “…the
existence of rigid standard rulers is an assumption suggested by our
approximate experience, assumption which is arbitrary in its principle”.

## Acknowledgements

I express my gratitude to Dr. Vesselin Gueorguiev for many years of support
and excellent collaboration.

## Data availability

No new data were generated or analyzed in this research.

## References

  * Asencio et al. (2022) Asencio, E, Banik, I., Mieske, S. et al. 2022, MNRAS, 515, 2981
  * Bondi (1990) Bondi, H. 1990, in Modern Cosmology in Retrospect, Eds. Bertotti, B., Balbinot, R., & Bergia, S.,Cambridge Univ. Press., 426 pp.
  * Bouvier & Maeder (1978) Bouvier, P., Maeder, A. 1978, Astrophys. Space Science, 54, 497
  * Canuto et al. (1977) Canuto, V., Adams, P. J., Hsieh, S.-H., & Tsiang, E. 1977, Phys. Rev. D, 16, 1643
  * Carroll et al. (1992) Carroll, S. M., Press, W. H., & Turner, E. L. 1992, ARA&A, 30, 499
  * Clifton et al. (2012) Clifton, T., Ferreira, P.G., Padilla, A., Skordis, C. 2012, Physics Reports, 513, 1
  * Dirac (1973) Dirac, P. A. M. 1973, Proceedings of the Royal Society of London Series A, 333, 403
  * Dirac (1974) Dirac, P. A. M. 1974, Proceedings of the Royal Society of London Series A, 338, 439
  * Eddington (1923) Eddington, A.S., 1923, The Mathematical Theory of Relativity, Chelsea Publ. Co. New York, 266 p.
  * Einstein (1918) Einstein, A. 1918, Sitzung Berichte der Königlich Preussischen Akademie des Wissenschaften (Berlin), p. 478.
  * Einstein (1949) Einstein, A. 1949, Autobiographical Notes, Open Court Publ. Co., La Salle, Illinois and The Library of Living Philosophers Inc. and Estate of Albert Einstein.
  * Famaey & McGaugh (2012) Famaey, B. and McGaugh, S. 2012 Living Rev. Relativity 15, 10
  * Ferreira & Tattersall (2020) Ferreira, P.G., Tattersall, O.J. 2020, Phys. Rev. D 101.024011
  * Feynman (1963) Feynman, R. P. 1963, Feynman Lectures on Physics, Vol. 1: Mainly Mechanics, Radiation and Heat, 560 p.
  * Frieman et al. (2008) Frieman, J. A., Turner, M. S., & Huterer, D. 2008, ARA&A, 46, 385
  * Fujii (2003) Fujii, Y., Maeda, K.-I. 2003, The Scalar-Tensor Theory of Gravity, Cambridge Monographs Math. Physics, 258 p.
  * Jesus (2018) Jesus, J.F. 2018, Rev. Mex. Astron. Astrophys. 55, 17.
  * Lelli et al. (2017) Lelli, F., McGaugh, S.S., Schombert, J.M., Pawlowski, M.S., 2017, ApJ, 836, 152
  * Li et al. (2018) Li, P., Lelli, F., McGaugh, S. et al. 2018, A&A, 615, A3, 70
  * Maeder (2017a) Maeder, A. 2017a, Astrophys. J., 834, 194.
  * Maeder (2017b) Maeder, A. 2017b, Astrophys. J., 847, 65
  * Maeder (2017c) Maeder, A. 2017c, Astrophys. J., 849,158.
  * Maeder (2019) Maeder, A. 2019, Evolution of the early Universe in the scale invariant theory. arXiv:1902.10115.
  * Maeder & Bouvier (1979) Maeder, A., Bouvier, P. 1979, Astron. Astrophys., 73, 82
  * Maeder and Gueorguiev (2019) Maeder, A.; Gueorguiev, V.G. 2019, Physics of the Dark Universe, 25, 100315
  * Maeder and Gueorguiev (2020a) Maeder, A.; Gueorguiev, V.G. 2020a, Universe, vol. 6, 46
  * Maeder and Gueorguiev (2020b) Maeder, A.; Gueorguiev, V.G. 2020b, MNRAS, 492, 2698
  * Maeder and Gueorguiev (2021a) Maeder, A.; Gueorguiev, V.G. 2021a, MNRAS, 504, 4005
  * Maeder & Gueorguiev (2022) Maeder, A., Gueorguiev, V. 2022, arXiv:2204.13560
  * McGaugh et al. (2016) McGaugh, S. S., Lelli, F., & Schombert, J. M. 2016, Phys. Rev. Lett., 117, 201101
  * Milgrom (1983) Milgrom, M. 1983, Astrophys. J., 270, 365
  * Milgrom (2009) Milgrom, M. 2009, Astrophys. J., 698, 1630
  * Milgrom (2014a) Milgrom, M. 2014a, Scholarpedia, 9, 31410
  * Milgrom (2014b) Milgrom, M. 2014b, MNRAS, 437, 2531
  * Milgrom (2015) Milgrom, M. 2015, Phys. Rev. D. 91, 044009
  * Milgrom (2018) Milgrom, M. 2018, Physicl Review D, 99, 044041
  * Milgrom (2019) Milgrom, M. 2019, Physicl Review D, 100, 084039
  * Milgrom (2020) Milgrom, M. 2020, arXiv:2001.09729
  * Milgrom (2022) Milgrom, M. 2022, arXiv:2208.10882
  * Pawlowski & Kroupa (2022) Pawlowski, , M.S., Kroupa, P. 2022, MNRAS, 491, 3042
  * Pawlowski et al. (2012) Pawlowski M. S., Pflamm-Altenburg J., Kroupa P., 2012a, MNRAS, 423, 1109
  * Sanders (2003) Sanders, R.H. 2003, MNRAS, 342, 901
  * Weyl (1923) Weyl, H. 1923, Raum, Zeit, Materie. Vorlesungen über allgemei ne Relativitätstheorie. Re-edited by Springer Verlag, Berlin, 1970

##  Appendix A The Hubble constant in current units

Equation (53) gives the Hubble-Lemaître parameter $H(t)$ in a timescale $t$
varying from the initial time $\Omega^{1/3}_{\mathrm{m}}$ to 1. We want to
express it in timescale $\tau$ varying from 0 to 13.8 Gyr. We first have
according to Eq.(59),

$H(\tau)\,=\,\frac{\dot{a}(\tau)}{a(\tau)}=\frac{da}{d\tau}\frac{1}{a}=\frac{da}{dt}\frac{dt}{d\tau}\frac{1}{a}=H(t)\frac{1-\Omega^{1/3}_{\mathrm{m}}}{\tau_{0}}\,\left[\frac{km}{s\cdot
Mpc}\right],$ (82)

For $\tau_{0}=$ 13.8 Gyr, the inverse $1/\tau_{0}$ expressed in the current
units for $H_{0}$ is 70.85 km/(s Mpc), thus we get,

$H(\tau)=70.85\,\cdot H(t)(1-\Omega^{1/3}_{\mathrm{m}})\,,$ (83)
$\mathrm{with}\;H(t)=\frac{2\,t^{2}}{t^{3}-\Omega_{\mathrm{m}}}\,$ (84)

and relations (58). For the Hubble-Lemaître parameter $H(\tau_{0})$ at the
present time $\tau_{0}$, this becomes

$H(\tau_{0})=2\,\cdot\,70.85\;\frac{1-\Omega^{1/3}_{\mathrm{m}}}{1-\Omega_{\mathrm{m}}}\;\left[\frac{km}{s\;Mpc}\right]\,.$
(85)

For $\Omega_{\mathrm{m}}=0,$ 0.10, 0.20, 0.30, 0.50, the value of
$\xi=(H(\tau_{0})\cdot\tau_{0})$ are 2, 1.191, 1.038, 0.945, 0.814
respectively (see Sect. 4.3). Interestingly enough, for
$\Omega_{\mathrm{m}}=1,$ we have $\xi=2/3$, scale invariance disappears and as
expected the flat model converges towards the EdS model. The above
corresponding values of $H(\tau_{0})$ are 141.70, 84.38, 73.54, 66.95, 57.70
in km/(s Mpc).


# Two Flavor Baryon Number Susceptibilities of Non-equilibrium State

A-Meng Zhao1<EMAIL_ADDRESS>1 Department of Foundation,
Southeast University Chengxian College, Nanjing 210088, China

###### Abstract

In this paper, to be closer to the realities of the relativistic heavy ion
collisions than the thermal equilibrium assumption often used in the
calculations of baryon number susceptibilities, the creatures of the
collisions are treated as isotropic systems with temperature gradient. In
addition, the quarks are thought as the quasi-particles with thermal masses.
By doing these and according to Boltzmann transport equation, we get our
results of the experimental observables $S\sigma$ and $\kappa\sigma^{2}$. We
found that the results show a gap in value and a centrality dependence, which
might indicate the relation between the baryon number susceptibilities and the
temperature gradient in QGP.

Keywords: baryon number susceptibilities, temperature gradient, Boltzmann
transport equation, non-equilibrium state.

##  I Introduction

RHIC has undertaken its first phase BES(Beam Energy Scan) Program to search
for the critical point and phase boundary in the QCD phase diagramb1 . Since
the moments of the distributions of conserved quantities, for example net-
baryon number, in the relativistic heavy ion collisions are related to the
correlation length $\xi$ of the systemb2 , they are believed to be good
signatures of the QCD phase transition and its critical point. By relating the
moments of the baryon number to the various order baryon-number
susceptibilitiesb3 , the experimental observables $S\sigma$ and
$\kappa\sigma^{2}$ could be calculated.

In the former works of our study on the baryon-number susceptibilitiesb4 ; b5
; b6 , the quark-gluon plasma is often treated as a system of grand canonical
ensemble. At the same time, the quark-gluon plasma created in the relativistic
heavy ion collisions is considered to be undertaken the hydrodynamic
evolutionb7 . That is to say, the distributions of the quarks in the
experiments deviate from those of thermal equilibrium state. Since the
complexity of the quantum chromodynamics and the hydrodynamic evolution, we
try to make a simplification to show the influence of the evolution on the the
baryon-number susceptibilities qualitatively. To do this, we describe the
quarks as quasi-particles with thermal massesb8 ; b9  and treat the creatures
of the heavy ion collisions as isotropic systems with temperature gradient
only. Though it can not describe the all realities in the heavy ion
collisions, our attempt is closer to the realities comparing with the thermal
equilibrium assumption. In addition, the relation between the susceptibilities
and the factor of temperature gradient, among a set of factors, could be
demonstrated more clearly.

##  II QGP with Temperature Gradient

According to Boltzmann transport equation:

$\frac{\partial{f}}{\partial{t}}+\frac{\partial{f}}{\partial{x}}\frac{dx}{dt}+\frac{\partial{f}}{\partial{y}}\frac{dy}{dt}+\frac{\partial{f}}{\partial{z}}\frac{dz}{dt}+\frac{\partial{f}}{\partial{v_{x}}}\frac{dv_{x}}{dt}+\frac{\partial{f}}{\partial{v_{y}}}\frac{dv_{y}}{dt}+\frac{\partial{f}}{\partial{v_{z}}}\frac{dv_{z}}{dt}=-\frac{f-f^{0}}{\tau_{0}},$
(1)

as we have mentioned above, if we take the temperature gradient only into
account, then the equation is written as:

$\frac{\partial{f}}{\partial{T}}\frac{\partial{T}}{\partial{x}}\frac{dx}{dt}+\frac{\partial{f}}{\partial{T}}\frac{\partial{T}}{\partial{y}}\frac{dy}{dt}+\frac{\partial{f}}{\partial{T}}\frac{\partial{T}}{\partial{z}}\frac{dz}{dt}=-\frac{f-f^{0}}{\tau_{0}},$
(2)

furthermore, if the distribution is isotropic, that is to say
$\frac{\partial{T}}{\partial{x}}=\frac{\partial{T}}{\partial{y}}=\frac{\partial{T}}{\partial{z}}$
, we can get it as:

$f=f^{0}-\alpha v_{x}\frac{\partial{f^{0}}}{\partial{T}}-\alpha
v_{y}\frac{\partial{f^{0}}}{\partial{T}}-\alpha
v_{z}\frac{\partial{f^{0}}}{\partial{T}},$ (3)

where $\alpha=\tau_{0}\frac{\partial{T}}{\partial{x}}$. Commonly, $\alpha$, as
a function of time and position, is determined by the nature of QCD and the
hydrodynamic evolution. Here we choose a constant $\bar{\alpha}$ instead of
$\alpha$. And $\bar{\alpha}$ is the average of $\alpha$ over time and
position.

In this paper, we adopt the quasi-particle model of QGP in which the
interaction of quarks and gluons is treated as an effective mass termb9 . The
effective mass of quark is made up of the rest mass and the thermal mass. The
relation is expressed as follow:

$m_{q}^{2}=m_{q0}^{2}+\sqrt{2}m_{q0}m_{th}+m_{th}^{2},$ (4)

where $m_{q0}$ is the rest mass of up or down quark, and the temperature
dependent quark mass $m_{th}$ is

$m_{th}^{2}(T)=\frac{g^{2}T^{2}}{18}N_{f},$ (5)

where $N_{f}$ is the number of quark flavors. Since only up and down quarks
are considered here, $N_{f}=2$. And $g^{2}$ is related to the two-loop order
running coupling constant,

$g^{2}=\frac{24\pi}{(33-2N_{f})\ln\frac{T}{\Lambda_{T}}}(1-\frac{3(153-19N_{f})}{(33-2N_{f})^{2}}\frac{\ln(2\ln{\frac{T}{\Lambda_{T}}})}{\ln{\frac{T}{\Lambda_{T}}}}).$
(6)

As in b10 , we choose $m_{q0}=6.5$MeV and $\Lambda_{T}=122.5MeV$ in this
paper.

Now the number density of the quarks with a single flavor and color is:

$n=\frac{1}{4\pi^{3}}\int d^{3}pf=\frac{1}{4\pi^{3}}\int
d^{3}p(f^{0}-3\bar{\alpha}v_{x}\frac{\partial{f^{0}}}{\partial{T}}).$ (7)

where
$f^{0}=\frac{1}{1+e^{\beta(\epsilon-\mu)}}-\frac{1}{1+e^{\beta(\epsilon+\mu)}}$b4
; b11  and $\epsilon=(p^{2}+m_{q}^{2})^{1/2}$.

According to the simulations of the hydrodynamic evolution in heavy ion
collisionsb12 , we can estimate the value of $\bar{\alpha}$ approximately. For
example, in the Au-Au collisions at $\sqrt{S_{NN}}=200GeV$, the temperature
decreases from $0.35GeV$ to $0.05GeV$ in $4-6fm$ at the beginning and the
temperature gradient reduces to an insignificant value at about
$\tau_{0}=10fm/c$ . That is to say, the average $\bar{\alpha}$ of the
parameter $\alpha=\tau_{0}\frac{\partial{T}}{\partial{x}}$ is about $0.7GeV/c$
in that case. In this paper, $\bar{\alpha}$ is varied from $0$ to $1GeV/c$ and
from now on we simply use $\alpha$ to represent $\bar{\alpha}$.

It is mentioned in b6  that,while considering the statistic independence of
the flavors and the confinement of the colors, the variance of baryon number
density is

$\displaystyle\begin{split}\sigma^{2}&=<(n_{B}-<n_{B}>)^{2}>=\frac{N_{f}N_{c}^{2}}{3^{2}}<(n-<n>)^{2}>\\\
&=N_{f}<(n-<n>)^{2}>=N_{f}\cdot\chi^{(2)}.\end{split}$ (8)

Similarly, we could get

$\displaystyle\begin{split}S\sigma&=\frac{<(n_{B}-<n_{B}>)^{3}>}{<(n_{B}-<n_{B}>)^{2}>}=\frac{N_{f}<(n-<n>)^{3}>}{N_{f}<(n-<n>)^{2}>}\\\
&=\frac{T^{2}\chi^{(3)}}{T\chi^{(2)}}=T\frac{\chi^{(3)}}{\chi^{(2)}},\end{split}$
(9)

and

$\displaystyle\begin{split}\kappa\sigma^{2}&=\frac{<(n_{B}-<n_{B}>)^{4}>-3<(n_{B}-<n_{B}>)^{3}>}{<(n_{B}-<n_{B}>)^{2}>}\\\
&=\frac{<(n-<n>)^{4}>-3<(n-<n>)^{3}>}{<(n-<n>)^{2}>}\\\
&=T^{2}\frac{\chi^{(4)}}{\chi^{(2)}}.\end{split}$ (10)

where $\frac{\partial n}{\partial\mu}=\cdot\chi^{(2)}$,
$\frac{\partial^{2}n}{\partial\mu^{2}}=\chi^{(3)}$ and
$\frac{\partial^{3}n}{\partial\mu^{3}}=\chi^{(4)}$. .

##  III Results and Comparings with Experiments

Firstly, we shown the results of $S\sigma$ and $\kappa\sigma^{2}$ as a
function of the temperature $T$ and the chemical potential of quark $\mu$. By
Eq.(9) and Eq.(10), $S\sigma$ and $\kappa\sigma^{2}$ are calculated at the
parameter $\alpha=0,0.2,0.53$ and $1GeV/c$ respectively and the results are
shown in Fig.1 and Fig.2. When $\alpha=0$, both $S\sigma$ and
$\kappa\sigma^{2}$ show a continuous change with $T$ and $\mu$. Meanwhile, the
gap appears at $\alpha=0.2$ and increases at $\alpha=0.53$ and $1GeV/c$ . Now
it might be concluded that the transport effect caused by the temperature
gradient in QGP is not neglectable in the study of $S\sigma$ and
$\kappa\sigma^{2}$.

Figure 1: $S\sigma$ is calculated at $\alpha=0,0.2,0.53$ and $1GeV/c$
respectively. Figure 2: $\kappa\sigma^{2}$ is calculated at
$\alpha=0,0.2,0.53$ and $1GeV/c$ respectively.

To test the reasonableness of our calculations, we compare the results of
$S\sigma$ and $\kappa\sigma^{2}$ with the experimental observables in RHIC.
The $S\sigma$ and $\kappa\sigma^{2}$ for net-proton distributions as a
function of $\sqrt{S_{NN}}$ at different centralities of Au+Au collisions
measured by STARb13  are shown in Fig.3 and Fig.4. The correspondence of the
freeze-out $T$ and $\mu$ in the calculation of quasi-particle model to the
collision energies $\sqrt{S_{NN}}$ comes from the experimental resultsb14 . At
both $\alpha=0$ and $\alpha=1GeV/c$, the $S\sigma$ and $\kappa\sigma^{2}$
results are comparable to the experimental data, which shows the
reasonableness of the quasi-particle model we adopted.

Figure 3: $S\sigma$ of the quasi-particle model and the experiments. And
experimental data comes from Ref.b13 .

At the same time, it should be noticed that the results in RHIC of these
observables show not only an energy but also a centrality dependenceb15 . As
for our results at $\alpha=0$ , the dependence on the centrality could hardly
be seen. Meanwhile at $\alpha=1GeV/c$, the centrality dependence is obvious.
It possibly indicates that the transport effects in the relativistic heavy ion
collisions might be related to the centrality dependence of $S\sigma$ and
$\kappa\sigma^{2}$.

Figure 4: $\kappa\sigma^{2}$ of the quasi-particle model and the experiments.
And experimental data comes from Ref.b13 .

##  IV Summary

Baryon number fluctuations are believed to be good signatures of the QCD phase
transition and its critical point. And the fluctuations can be directly
connected to the susceptibilities of the system computed in theoretical
calculations. In this paper, be different from a system of the grand canonical
ensemble, we treat the creatures of the heavy ion collisions as an isotropic
system with temperature gradient only. By doing this, our calculations are
closer to the realities comparing with the thermal equilibrium assumption. In
addition, the relation between the susceptibilities and the factor of
temperature gradient, among a set of factors, could be demonstrated more
clearly. And for simplicity, we treat the QGP as the quasi-particles with a
temperature related mass term.

According to our results of $S\sigma$ and $\kappa\sigma^{2}$, there are two
point should be noticed. Firstly, with the increase of $\alpha$, the results
of $S\sigma$ and $\kappa\sigma^{2}$ as a function of $T$ and $\mu$ get the
gaps. Secondly, comparing with those at $\alpha=0$, the results at
$\alpha=1GeV/c$ show obvious centrality dependence of the collisions. Now it
might be concluded that the thermal transport effect is related to the results
of baryon number susceptibilities in the relativistic heavy ion collisions.

For the further study of baryon number fluctuations, the hydrodynamic
evolution and more complicated transport effects in QGP should be taken into
account. And the exploration of the relation between the local thermalization
and the phase transition of QGP is of the same importance.

## Acknowledgements

This work is supported by the University Natural Science Foundation of JiangSu
Province China (under Grants No.17KJB140003 ).

## References

  * (1) M. M. Aggarwal et al., Phys. Rev. Lett. 105, 022302(2010); B. Abelev et al. (STAR Collaboration), Phys. Rev. C 81, 024911 (2010);B. Mohanty, Nucl. Phys. A 830, 899c (2009); L. Kumar (STAR Collaboration), Nucl. Phys. A 904, 256c (2013); L. Kumar, Mod. Phys. Lett. A 28, 1330033 (2013).
  * (2) M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009); C. Athanasiou et al., Phys. Rev. D 82, 074008 (2010).
  * (3) S. Gupta et al., Science 332, 1525 (2011).
  * (4) M. He, J.F. Li, W.M. Sun and H.S. Zong, Phys.Rev.D79, 036001 (2009).
  * (5) A.M. Zhao, Z.F. Cui, Y. Jiang, and H. S. Zong, Phys.Rev.D90, 114031 (2014).
  * (6) A.M. Zhao, X.F. Luo, H.S. Zong, Eur. Phys. J. C 77:207 (2017).
  * (7) W. Busza, K. Rajagopal,and W. Schee, Annu. Rev. Nucl. Part. Sci. 68:339-76 (2018).
  * (8) A. Peshier, B. Kampfer, G. Soff, Phys.Rev. D 66, 094003 (2002); H. Abuki, K. Fukushima, Phys. Lett. B 676, 57 (2006).
  * (9) V. M. Bannur, JHEP 0709046 (2007).
  * (10) J. Cao et al., Chin. Phys. Lett. 27(3), 031201 (2010).
  * (11) H.S. Zong and W.M. Sun, Phys.Rev.D78,054001(2008).
  * (12) R. Baier, P. Romatschke, and U. A. Wiedemann, Phys. Rev. C 73 064903 (2006); P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99 172301 (2007).
  * (13) X. Luo, PoS(CPOD2014)019. arXiv: 1503.02558; X. Luo, Nucl. Phys. A, 1-9 (2016). arXiv:1512.09215.
  * (14) L. Adamczyk et al.(STAR Collaboration),arxiv:1701.07065 (2017).
  * (15) L. Adamczyk et al., Phys. Rev. Lett. 112, 032302 (2014).