# The $\kappa$-model under test of the SPARC database

Gianni Pascoli

► To cite this version:

Gianni Pascoli. The $\kappa$-model under test of the SPARC database. Universe, 2024, 10 (3), pp.151. 10.3390/universe10030151. hal-04533953

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**Title** The $\kappa$-model under test of the SPARC database

G. Pascoli

Email: pascoli@u-picardie.fr

Faculté des sciences

Département de physique

Université de Picardie Jules Verne (UPJV)

33 Rue Saint Leu, Amiens, France

**Abstract** Our main goal is here to make a comparative analysis between the well-known MOND theory and a more recent model called $\kappa$-model. An additional connection, between the $\kappa$-model and two other novel MOND-type theories: Newtonian Fractional-Dimension Gravity (NFDG) and Refracted Gravity (RG), is likewise presented. All these models are built to overtake the DM paradigm, or at least to strongly reduce the dark matter content. Whereas they rely on different formalisms, however, all four seem to suggest that the universal parameter, $a_0$, appearing in MOND theory could intrinsically be correlated to either the sole baryonic mean mass density (RG and $\kappa$-model) and/or to the dimension of the object under consideration (NFDG and $\kappa$-model). We could then confer to the parameter $a_0$ a more flexible status of multiscale parameter, as required to explain the dynamics together in galaxies and in galaxy clusters. Eventually, the conformal gravity theory (CFT) also seems to have some remote link with the $\kappa$-model, even though the first one is an extension of general relativity, and the second one is Newtonian in essence. The $\kappa$-model has been tested on a small sample of spiral galaxies and in galaxy clusters. Now we test this model on a large sample of galaxies issued from the SPARC database.

**Keywords** : SPARC database, galaxy, MOND, Newtonian Fractional-Dimension Gravity, Refracted Gravity, $\kappa$-model, dark matter
**Title** The $\kappa$-model under test of the SPARC database

G. Pascoli

Email: pascoli@u-picardie.fr

Faculté des sciences

Département de physique

Université de Picardie Jules Verne (UPJV)

33 Rue Saint Leu, Amiens, France

**Abstract** Our main goal is here to make a comparative analysis between the well-known MOND theory and a more recent model called $\kappa$-model. An additional connection, between the $\kappa$-model and two other novel MOND-type theories: Newtonian Fractional-Dimension Gravity (NFDG) and Refracted Gravity (RG), is likewise presented. All these models are built to overtake the DM paradigm, or at least to strongly reduce the dark matter content. Whereas they rely on different formalisms, however, all four seem to suggest that the universal parameter, $a_0$, appearing in MOND theory could intrinsically be correlated to either the sole baryonic mean mass density (RG and $\kappa$-model) and/or to the dimension of the object under consideration (NFDG and $\kappa$-model). We could then confer to the parameter $a_0$ a more flexible status of multiscale parameter, as required to explain the dynamics together in galaxies and in galaxy clusters. Eventually, the conformal gravity theory (CFT) also seems to have some remote link with the $\kappa$-model, even though the first one is an extension of general relativity, and the second one is Newtonian in essence. The $\kappa$-model has been tested on a small sample of spiral galaxies and in galaxy clusters. Now we test this model on a large sample of galaxies issued from the SPARC database.

**Keywords** : SPARC database, galaxy, MOND, Newtonian Fractional-Dimension Gravity, Refracted Gravity, $\kappa$-model, dark matter
# 1 Introduction

As it is well known, all the studies conducted on galaxies and galaxy clusters, lead to the seemingly firm conclusion that a significant portion of the mass in the Universe seems to be hidden from the view of the observers. This invisible (non-baryonic) matter is called dark matter (DM). It is true that DM is the more simple and economic hypothesis. However, the major problem with this paradigm is that the dark matter/baryonic (DM/B) mass ratio is incredibly huge, of the order of 6. It is not simply an addition of a small quantity of missing matter to a dominant form of visible (baryonic) matter. This is even the opposite, and the baryonic component eventually appears negligible in the Universe. This situation seems to be surprising and even rather unconfortable, the visible sector being explained by an undefined invisible sector about which we know nothing. De facto the explanation of the flatness of the rotation spiral galaxy curves with DM is fully indirect. A very pertinent parallel can be drawn with the phlogistic theory, a dominant theory in nascent chemistry during the 18th century. The phlogistic hypothesis was based on the existence of an illusive "substance" (the phlogiston) with indeterminate properties and thus without real physical support. The theory of phlogiston was finally disproved by the French chemist Antoine-Laurent Lavoisier through a series of experiments in the late eighteenth century. Is it the fate that awaits DM? At the present time the existence of DM is inferred only through gravitational effects. A direct proof is missing from both an observational and an experimental point of view.

The MACHOs (Massive Compact Halo Objects), possibly detected through gravitational microlensing in the Galactic halo, have been ruled out as a dark matter candidate [1]. Another interesting inter-
# 1 Introduction

As it is well known, all the studies conducted on galaxies and galaxy clusters, lead to the seemingly firm conclusion that a significant portion of the mass in the Universe seems to be hidden from the view of the observers. This invisible (non-baryonic) matter is called dark matter (DM). It is true that DM is the more simple and economic hypothesis. However, the major problem with this paradigm is that the dark matter/baryonic (DM/B) mass ratio is incredibly huge, of the order of 6. It is not simply an addition of a small quantity of missing matter to a dominant form of visible (baryonic) matter. This is even the opposite, and the baryonic component eventually appears negligible in the Universe. This situation seems to be surprising and even rather unconfortable, the visible sector being explained by an undefined invisible sector about which we know nothing. De facto the explanation of the flatness of the rotation spiral galaxy curves with DM is fully indirect. A very pertinent parallel can be drawn with the phlogistic theory, a dominant theory in nascent chemistry during the 18th century. The phlogistic hypothesis was based on the existence of an illusive "substance" (the phlogiston) with indeterminate properties and thus without real physical support. The theory of phlogiston was finally disproved by the French chemist Antoine-Laurent Lavoisier through a series of experiments in the late eighteenth century. Is it the fate that awaits DM? At the present time the existence of DM is inferred only through gravitational effects. A direct proof is missing from both an observational and an experimental point of view.

The MACHOs (Massive Compact Halo Objects), possibly detected through gravitational microlensing in the Galactic halo, have been ruled out as a dark matter candidate [1]. Another interesting inter-
rogation is that if DM particles are really existing, these particles
can very possibly decay. Strangely enough, X-ray space telescopes
like Chandra, XMM-Newton, and Fermi have not observed any ex-
cess of DM decay [2]. Eventually a major trouble for DM is the
tantamount difficulties observing the DM particles in the laboratory.
Large classes of candidates have been suggested following highly spec-
ulative theoretical models, such as Hidden-Sector Dark Matter parti-
cles, completely neutral under Standard Model forces, but interacting
through a new force; or still Ultra-Light Dark Matter particles with
predicted masses from 10-22 eV to about a keV, and that can be
produced during inflation or phase transitions in the very early Uni-
verse [3]. However the existing dark matter experimental programs
are now more reasonably focused on weakly-interacting massive par-
ticles (WIMPs) [4, 5, 6]. Unfortunately the conclusions of all these
very costly studies are always negative. All direct detections have
come up empty. The persisting non-detection in space and in the
laboratory of DM in spite of very intense efforts is rather discourag-
ing. A simple, but very frustating, conclusion would be that if DM
interacts uniquely gravitationally with baryonic matter and definitely
not through one of the other known three forces (the strong, weak
or electromagnetic forces), we might never detect it. Another possi-
bility is that DM interacts with itself and with baryonic matter but
via an unknown (fifth) force. In spite of all of that, DM remains the
leading explanation for the dynamics of galaxies, very likely for its
high flexibility adaptable to various situations (galaxies, galaxy clus-
ters and cosmology). This view can unfortunately persist for a very
long time because the DM paradigm seems to be unfalsifiable. Yet a
good question put by McGaugh [7] is however: Is it a missing mass
problem or rather an acceleration-velocity discrepancy when observ-
ing the galaxies? Indeed the mass is an indirect data contrarily to
rogation is that if DM particles are really existing, these particles
can very possibly decay. Strangely enough, X-ray space telescopes
like Chandra, XMM-Newton, and Fermi have not observed any ex-
cess of DM decay [2]. Eventually a major trouble for DM is the
tantamount difficulties observing the DM particles in the laboratory.
Large classes of candidates have been suggested following highly spec-
ulative theoretical models, such as Hidden-Sector Dark Matter parti-
cles, completely neutral under Standard Model forces, but interacting
through a new force; or still Ultra-Light Dark Matter particles with
predicted masses from 10-22 eV to about a keV, and that can be
produced during inflation or phase transitions in the very early Uni-
verse [3]. However the existing dark matter experimental programs
are now more reasonably focused on weakly-interacting massive par-
ticles (WIMPs) [4, 5, 6]. Unfortunately the conclusions of all these
very costly studies are always negative. All direct detections have
come up empty. The persisting non-detection in space and in the
laboratory of DM in spite of very intense efforts is rather discourag-
ing. A simple, but very frustating, conclusion would be that if DM
interacts uniquely gravitationally with baryonic matter and definitely
not through one of the other known three forces (the strong, weak
or electromagnetic forces), we might never detect it. Another possi-
bility is that DM interacts with itself and with baryonic matter but
via an unknown (fifth) force. In spite of all of that, DM remains the
leading explanation for the dynamics of galaxies, very likely for its
high flexibility adaptable to various situations (galaxies, galaxy clus-
ters and cosmology). This view can unfortunately persist for a very
long time because the DM paradigm seems to be unfalsifiable. Yet a
good question put by McGaugh [7] is however: Is it a missing mass
problem or rather an acceleration-velocity discrepancy when observ-
ing the galaxies? Indeed the mass is an indirect data contrarily to
the acceleration which can be directly measured. Di Paolo and coau-
thors [8] have remarked that there exists a mysterious link between
DM and the baryonic component. In fact this link is easily explained
if DM is a property of the baryonic mass itself.

Alternatively a lot of authors have tried to circumvent the trouble
by exploring other paths than DM. Without DM it is true that the
Newtonian theory of gravity, and even its basic relativistic version,
i.e. the general relativity, seem to fail on galactic scales. The first
model that has been developed in this sense is the Modified New-
tonian Dynamics or MOND [9, 10, 11]. Remarkably the basic idea
of this model is thus as simple and economic as DM concerning the
theoretical background. The initial aim was to explain the flatness
of the rotation velocity curves of the spiral galaxies uniquely with
help of the observed baryonic matter. In MOND the second law of
Newton (ma = F) is modified in the very low regime of acceleration
a ≤ a₀, a₀ ~ 1.2 10⁻¹⁰ m s⁻² being a universal constant. MOND
replaces the acceleration a by a²/a₀. Assuming then a test particle sur-
rounding an attractive mass M, with a circular orbit of radius r and
with F = GM/r², we have a²/a₀ = GM/r² or a = √GMa₀/r. For the velocity
we directly obtain v² = ar = √GMa₀ = Const. This leads to the
flatness of the observed rotation curves of spiral galaxies but, much
more, results in the Tully-Fisher law in a very natural manner [12].

Furthermore, MOND is sustained by the empirical Renzo's rule.
The empirical Renzo's rule highlights the correspondence between
detailed features observed in the observational rotational curves of
spiral galaxies and the same features seen in their Newtonian counter-
terparts [13]. This statement, that the observational rotation profiles
seem to be a magnification of the Newtonian counterparts, appears
quite natural when baryonic matter dominates the mass, but not if
DM is the dominant form of matter. Another strong support for
the acceleration which can be directly measured. Di Paolo and coau-
thors [8] have remarked that there exists a mysterious link between
DM and the baryonic component. In fact this link is easily explained
if DM is a property of the baryonic mass itself.

Alternatively a lot of authors have tried to circumvent the trouble
by exploring other paths than DM. Without DM it is true that the
Newtonian theory of gravity, and even its basic relativistic version,
i.e. the general relativity, seem to fail on galactic scales. The first
model that has been developed in this sense is the Modified New-
tonian Dynamics or MOND [9, 10, 11]. Remarkably the basic idea
of this model is thus as simple and economic as DM concerning the
theoretical background. The initial aim was to explain the flatness
of the rotation velocity curves of the spiral galaxies uniquely with
help of the observed baryonic matter. In MOND the second law of
Newton (ma = F) is modified in the very low regime of acceleration
a ≤ a₀, a₀ ~ 1.2 10⁻¹⁰ m s⁻² being a universal constant. MOND
replaces the acceleration a by $\frac{a^2}{a_0}$. Assuming then a test particle sur-
rounding an attractive mass *M*, with a circular orbit of radius *r* and
with *F* = $\frac{GM}{r^2}$, we have $\frac{a^2}{a_0} = \frac{GM}{r^2}$ or $a = \frac{\sqrt{GMa_0}}{r}$.
For the velocity
we directly obtain $v^2 = ar = \sqrt{GMa_0} = \text{Const.}$ This leads to the
flatness of the observed rotation curves of spiral galaxies but, much
more, results in the Tully-Fisher law in a very natural manner [12].

Furthermore, MOND is sustained by the empirical Renzo's rule.
The empirical Renzo's rule highlights the correspondence between
detailed features observed in the observational rotational curves of
spiral galaxies and the same features seen in their Newtonian counter-
terparts [13]. This statement, that the observational rotation profiles
seem to be a magnification of the Newtonian counterparts, appears
quite natural when baryonic matter dominates the mass, but not if
DM is the dominant form of matter. Another strong support for
MOND, as seen above, is the direct deduction, within a calculation
that takes just a few lines, of the Tully-Fisher relation. These two
facts are difficult to explain within the DM paradigm, except in an
ad hoc manner. Eventually MOND has predicted, well in advance,
the profile of the rotation curves in the case of low surface brightness
galaxies (LSB), once again a feat not possible for DM [13]. However,
the MOND phenomenology fails to explain the dynamics of galaxy
clusters. A natural remedy has been found by adopting a multi-
scale approach [14]<fnref n="1" />. In any way as for gravitational lensing and
cosmology, the classic modified-(gravity+inertia) MOND in its ini-
tial form [9] is not applicable. Various relativistic versions of MOND
(RMOND) have been proposed making clear predictions regarding
gravitational lensing and cosmology. The latest in date is that of
Skordis and Złośnik [15]. The latter version of RMOND reproduces
the galactic and lensing phenomenology and also the key cosmological
observables<fnref n="2" />.

Another well known modified-gravity theory is the covariant scalar-vector-tensor modified-gravity (MOG) built by J. Moffat [17]. MOG is based on a $D = 4$ pseudo-Riemannian metric, a spin 1 vector field, a corresponding second-rank skew field $B_{\mu\nu}$, and eventually three dynamical scalar fields $G$ (the gravitational constant), $\omega$ and $\mu$. The heavy price to be paid is the addition of extra vector and scalar fields to the gravity field. On the other hand in MOG the gravitational constant $G$ is assumed to vary with space and time. Moreover the introduction of new fields means that new particles are surreptitiously hypothesized. We are not far from DM with its elusive particles, even though MOG is much more subtle than DM because the particles in MOG are virtual, and may not be directly observable

<footnote n="1">¹ In the context of MOND, a multiscale approach adapts the parameter $a_0$ to the size of the system studied. However, in this case the parameter $a_0$ is no longer universal.</footnote>

<footnote n="2">² How the $\kappa$-model performs with lensing is presented in the reference [16].</footnote>
MOND, as seen above, is the direct deduction, within a calculation
that takes just a few lines, of the Tully-Fisher relation. These two
facts are difficult to explain within the DM paradigm, except in an
ad hoc manner. Eventually MOND has predicted, well in advance,
the profile of the rotation curves in the case of low surface brightness
galaxies (LSB), once again a feat not possible for DM [13]. However,
the MOND phenomenology fails to explain the dynamics of galaxy
clusters. A natural remedy has been found by adopting a multi-
scale approach [14]<fnref n="1" />. In any way as for gravitational lensing and
cosmology, the classic modified-(gravity+inertia) MOND in its ini-
tial form [9] is not applicable. Various relativistic versions of MOND
(RMOND) have been proposed making clear predictions regarding
gravitational lensing and cosmology. The latest in date is that of
Skordis and Złośnik [15]. The latter version of RMOND reproduces
the galactic and lensing phenomenology and also the key cosmological
observables<fnref n="2" />.

Another well known modified-gravity theory is the covariant scalar-vector-tensor modified-gravity (MOG) built by J. Moffat [17]. MOG is based on a $D = 4$ pseudo-Riemannian metric, a spin 1 vector field, a corresponding second-rank skew field $B_{\mu\nu}$, and eventually three dynamical scalar fields $G$ (the gravitational constant), $\omega$ and $\mu$. The heavy price to be paid is the addition of extra vector and scalar fields to the gravity field. On the other hand in MOG the gravitational constant $G$ is assumed to vary with space and time. Moreover the introduction of new fields means that new particles are surreptitiously hypothesized. We are not far from DM with its elusive particles, even though MOG is much more subtle than DM because the particles in MOG are virtual, and may not be directly observable

<footnote n="1">¹ In the context of MOND, a multiscale approach adapts the parameter $a_0$ to the size of the system studied. However, in this case the parameter $a_0$ is no longer universal.</footnote>

<footnote n="2">² How the $\kappa$-model performs with lensing is presented in the reference [16].</footnote>
in the laboratory. MOG has been largely applied with some success to spiral galaxy curves, to galaxy clusters, to gravitational lensing and eventually to cosmology [18]. RMOND and MOG are the two main models built to get rid of DM fairly efficiently. These two models are the only models that have been extensively studied and involved in concrete comparisons with the observational data. Unfortunately with the relativistic extension of MOND, or with its main concurrent MOG, one moves away from the beautiful simplicity of the Newtonian mechanics and even of general relativity. Let us note that RMOND and MOG appear very much alike. Thus the major pitfall of RMOND and MOG is the introduction of other subsidiary extra scalar and vector fields that have not been tested in the laboratory.

A broad number of other models also exist, but they have been more sporadically applied to real situations. Conformal gravity theories (CFT), which are compelling alternatives to general relativity theory, have been claimed to explain the observed flat rotation profiles of spiral galaxies, without invoking DM or other exotic modifications of gravity [19, 20]. Nevertheless the extension of this type of models to the field of cosmology appears to be questionable. Thus it seems that the Weyl CFT<fnref n="3" /> cannot accurately describe the stated lensing observations without again considering dark matter [21]. Eventually another very different way is to conceive gravity, not as a conventional interaction, but rather as an emergent property [22]. In this case, gravity is seen as an entropic force, i.e. closely related to thermodynamics. Testing this hypothesis in the galaxy world is underway.

Are there other options to get rid of DM? We can answer this question in the affirmative. Very recently and quasi-simultaneously, a lot of new models have been proposed by following original, even

<footnote n="3">³The Weyl CFT is built by replacing the Einstein-Hilbert Lagrangian density, proportional to the Ricci curvature scalar, by a quadratic contraction of the conformal Weyl tensor.</footnote>
in the laboratory. MOG has been largely applied with some success
to spiral galaxy curves, to galaxy clusters, to gravitational lensing and
eventually to cosmology [18]. RMOND and MOG are the two main
models built to get rid of DM fairly efficiently. These two models
are the only models that have been extensively studied and involved
in concrete comparisons with the observational data. Unfortunately
with the relativistic extension of MOND, or with its main concurrent
MOG, one moves away from the beautiful simplicity of the Newtonian
mechanics and even of general relativity. Let us note that RMOND
and MOG appear very much alike. Thus the major pitfall of RMOND
and MOG is the introduction of other subsidiary extra scalar and
vector fields that have not been tested in the laboratory.

A broad number of other models also exist, but they have been
more sporadically applied to real situations. Conformal gravity the-
ories (CFT), which are compelling alternatives to general relativity
theory, have been claimed to explain the observed flat rotation profiles
of spiral galaxies, without invoking DM or other exotic modifications
of gravity [19, 20]. Nevertheless the extension of this type of models to
the field of cosmology appears to be questionable. Thus it seems that
the Weyl CFT<fnref n="3" /> cannot accurately describe the stated lensing observa-
tions without again considering dark matter [21]. Eventually another
very different way is to conceive gravity, not as a conventional inter-
action, but rather as an emergent property [22]. In this case, gravity
is seen as an entropic force, i.e. closely related to thermodynamics.
Testing this hypothesis in the galaxy world is underway.

Are there other options to get rid of DM ? We can answer this
question in the affirmative. Very recently and quasi-simultaneously,
a lot of new models have been proposed by following original, even

<footnote n="3">³The Weyl CFT is built by replacing the Einstein-Hilbert Lagrangian density, proportional to the Ricci curvature scalar, by a quadratic contraction of the conformal Weyl tensor.</footnote>
though speculative, ways [23, 24, 25, 26, 27]. These models sound
similar, even though they use a different formalism. The aim is then
to satisfy a principle of parsimony in the introduced concepts. It is
indeed about three different strategies, but which share a number
of common features. All these ideas are new and still need deep
understanding.

One very aesthetic strategy is to assume that spacetime is multi-
fractal in nature. This property is revealed in the most prominent
quantum gravity theories in a natural manner [28]. This concept
of fractional-dimension space applied to Newtonian gravity has been
suggested as an alternative to DM [23, 29, 30, 31, 32]. In the lat-
ter work a connection has been established between the Newtonian
Fractional-Dimension Gravity (NFDG) with MOND. The MOND ac-
celeration constant $a_0$ can be related to a natural scale length $l_0$ in
NFDG, i.e., $a_0 \sim \frac{GM}{l_0^2}$, for any astrophysical structure of mass $M$,
and the deep-MOND regime appears in regions of space where the
dimension is reduced to $D \sim 2$.

A second strategy is Refracted Gravity [33, 34]. Refracted Grav-
ity mimics dark matter by introducing a gravitational equivalent to a
permittivity, seen as a monotonic function of the local mean volumet-
ric mass density. This function is parametrized by three coefficients
which are free as in the case of DM, but which are expected to be
universal, contrarily to DM where the parameters are free and, ad-
ditionally different for each galaxy. Once again even if this second
strategy apparently relies on a very different formalism than NFDG,
both share strong links with MOND.

We turn now to the third strategy, i.e. the $\kappa$-model. The aim of
the $\kappa$-model is to reflect on how the mean volumetric mass density
(estimated at a very large scale), surrounding a given observer, can
though speculative, ways [23, 24, 25, 26, 27]. These models sound
similar, even though they use a different formalism. The aim is then
to satisfy a principle of parsimony in the introduced concepts. It is
indeed about three different strategies, but which share a number
of common features. All these ideas are new and still need deep
understanding.

One very aesthetic strategy is to assume that spacetime is multi-
fractal in nature. This property is revealed in the most prominent
quantum gravity theories in a natural manner [28]. This concept
of fractional-dimension space applied to Newtonian gravity has been
suggested as an alternative to DM [23, 29, 30, 31, 32]. In the lat-
ter work a connection has been established between the Newtonian
Fractional-Dimension Gravity (NFDG) with MOND. The MOND ac-
celeration constant $a_0$ can be related to a natural scale length $l_0$ in
NFDG, i.e., $a_0 \sim \frac{GM}{l_0^2}$, for any astrophysical structure of mass $M$,
and the deep-MOND regime appears in regions of space where the
dimension is reduced to $D \sim 2$.

A second strategy is Refracted Gravity [33, 34]. Refracted Grav-
ity mimics dark matter by introducing a gravitational equivalent to a
permittivity, seen as a monotonic function of the local mean volumet-
ric mass density. This function is parametrized by three coefficients
which are free as in the case of DM, but which are expected to be
universal, contrarily to DM where the parameters are free and, ad-
ditionally different for each galaxy. Once again even if this second
strategy apparently relies on a very different formalism than NFDG,
both share strong links with MOND.

We turn now to the third strategy, i.e. the $\kappa$-model. The aim of
the $\kappa$-model is to reflect on how the mean volumetric mass density
(estimated at a very large scale), surrounding a given observer, can
modify his view of the Universe.

In the framework of this model [16, 25, 26, 27] it is hypothesized
that it is the perception of the observer, modified by his environment
(the local mean volumetric mass density, calculated at a very large
scale around him), that creates the observed anomalies and also his
proper experience of gravity (with today the need to call for a hypo-
thetical dark matter in order to explain these anomalies). This idea is
speculative, but it strongly resembles the models for which we have
given an overview above [23, 24]. However one point of difference
is that the effects described in the κ-model are only apparent, de-
pending on the observer (excepting the spectroscopic velocities whose
measurement is universal, see [16] par. 2 eq. 10). It is almost as if
we are looking at any object through a perfect, even though fictive,
optical device (such as an aberration-free flat superlens<fnref n="4" />, but without
being aware of the presence of this device (which obviously does not
exist). Clearly, the object has not changed but both its apparent size
and velocity can now appear magnified from the point of view of a
distant observer. Admittedly both the inertia and the gravity seem
to be modified in the κ-model, but it is a pseudo-modified-gravity, it
is not of the same nature that a real modified-gravity as introduced,
for instance in MOG or RMOND. Moreover, in the κ-model the grav-
itational constant (and the speed of light), locally measured by any
observer, are invariant. The gravitational constant, the speed of light
and all physical constants are universal in the κ-model. To make vari-
able a constant in physics, in our case *G* here, could require to make
variable other constants (for instance the speed of light) with, may
be, unpredictable consequences. Furthermore no new field or exotic
particles, undetected in the laboratory, are assumed in the κ-model.

<footnote n="4">⁴ A superlens is a flat, lightweight option that can replace bulky traditional lenses and other components in optical systems. It is a lens that goes beyond the diffraction limit.</footnote>
modify his view of the Universe.

In the framework of this model [16, 25, 26, 27] it is hypothesized
that it is the perception of the observer, modified by his environment
(the local mean volumetric mass density, calculated at a very large
scale around him), that creates the observed anomalies and also his
proper experience of gravity (with today the need to call for a hypo-
thetical dark matter in order to explain these anomalies). This idea is
speculative, but it strongly resembles the models for which we have
given an overview above [23, 24]. However one point of difference
is that the effects described in the κ-model are only apparent, de-
pending on the observer (excepting the spectroscopic velocities whose
measurement is universal, see [16] par. 2 eq. 10). It is almost as if
we are looking at any object through a perfect, even though fictive,
optical device (such as an aberration-free flat superlens<fnref n="4" />, but without
being aware of the presence of this device (which obviously does not
exist). Clearly, the object has not changed but both its apparent size
and velocity can now appear magnified from the point of view of a
distant observer. Admittedly both the inertia and the gravity seem
to be modified in the κ-model, but it is a pseudo-modified-gravity, it
is not of the same nature that a real modified-gravity as introduced,
for instance in MOG or RMOND. Moreover, in the κ-model the grav-
itational constant (and the speed of light), locally measured by any
observer, are invariant. The gravitational constant, the speed of light
and all physical constants are universal in the κ-model. To make vari-
able a constant in physics, in our case *G* here, could require to make
variable other constants (for instance the speed of light) with, may
be, unpredictable consequences. Furthermore no new field or exotic
particles, undetected in the laboratory, are assumed in the κ-model.

<footnote n="4">⁴ A superlens is a flat, lightweight option that can replace bulky traditional lenses and other components in optical systems. It is a lens that goes beyond the diffraction limit.</footnote>
We think that it is a very important point that obeys a principle of parsimony. Eventually even though the κ-model is Newtonian in essence, its great advantage is that it can be naturally made relativistic. A first draft of what might be a relativistic version of the κ-model is presented in the reference [25]. However in a galaxy the velocities $v$ of stars and gas are low compared to the speed of light $c$ (the ratio $v/c \sim 10^{-3}$), and the nonrelativistic approximation is sufficient, especially in the outskirts of galaxies where gravity is weak. The same arguments also apply to MOND. For MOND a notable relativistic version has however been proposed [15]. Nevertheless the elegant simplicity of the initial version of MOND has unfortunately disappeared in the operation. At the Newtonian level the κ-effect is mimicked by an apparent local scaling transformation applied in an Euclidean space [16, 26]. In a Riemannian structure of space a local scaling transformation could be applied exactly in the same manner. Eventually let us note that the multiscale approach already suggested in [14] is directly included in the κ-model, which assumes that the larger the characteristic dimension (the scale) of a system, the weaker the local mean volumetric mass density and the stronger the magnification [16].

In order to avoid any misunderstanding, three velocities are defined in the κ-model: the Newtonian velocities, $v_{New}$, which are directly calculable from the mean surface mass density profiles, but which are virtual and not measurable, the radial velocities, which are given by $v_{rad} \sim \kappa^{\frac{1}{2}} v_{New}$ (observationally the universal spectroscopic velocities, $v_{spec}$) and the tangential velocities which are given by $v_{tan} \sim \kappa v_{New}$ (observationally the proper motions). Following a more mathematical approach within the formalism of bundles, the Newtonian velocities are "located" in the base (not reachable) and both the measurable radial and tangent velocities are "located" in a
We think that it is a very important point that obeys a principle of parsimony. Eventually even though the κ-model is Newtonian in essence, its great advantage is that it can be naturally made relativistic. A first draft of what might be a relativistic version of the κ-model is presented in the reference [25]. However in a galaxy the velocities $v$ of stars and gas are low compared to the speed of light $c$ (the ratio $v/c \sim 10^{-3}$), and the nonrelativistic approximation is sufficient, especially in the outskirts of galaxies where gravity is weak. The same arguments also apply to MOND. For MOND a notable relativistic version has however been proposed [15]. Nevertheless the elegant simplicity of the initial version of MOND has unfortunately disappeared in the operation. At the Newtonian level the κ-effect is mimicked by an apparent local scaling transformation applied in an Euclidean space [16, 26]. In a Riemannian structure of space a local scaling transformation could be applied exactly in the same manner. Eventually let us note that the multiscale approach already suggested in [14] is directly included in the κ-model, which assumes that the larger the characteristic dimension (the scale) of a system, the weaker the local mean volumetric mass density and the stronger the magnification [16].

In order to avoid any misunderstanding, three velocities are defined in the κ-model: the Newtonian velocities, $v_{New}$, which are directly calculable from the mean surface mass density profiles, but which are virtual and not measurable, the radial velocities, which are given by $v_{rad} \sim \kappa^{\frac{1}{2}} v_{New}$ (observationally the universal spectroscopic velocities, $v_{spec}$) and the tangential velocities which are given by $v_{tan} \sim \kappa v_{New}$ (observationally the proper motions). Following a more mathematical approach within the formalism of bundles, the Newtonian velocities are "located" in the base (not reachable) and both the measurable radial and tangent velocities are "located" in a
sheet, attached to a given observer in the bundle situated "above" the base [16]. The latter mathematical considerations will be shortly developed in an up-coming paper. We are only concerned here with the observational aspect.

The synoptic table below summarizes the applicability domains of the different models discussed in this paper :

<table><thead><tr><th>Model</th><th>Main features</th></tr></thead><tbody><tr><td>MOND</td><td>Very low acceleration $a \lesssim a_0 \sim 10^{-10} m s^{-2}$</td></tr><tr><td>κ-model</td><td>Very low mean mass density $\lesssim 0.15 M_{\odot} pc^{-3}$<br>Geometry of the matter distribution (bulge, disk)<br>Compactness (stars, gas)</td></tr><tr><td>NFDG</td><td>Variable dimension of the matter distribution,<br>between $D = 3$ (sphere) and $D = 2$ (disk)</td></tr><tr><td>RG</td><td>Very low mean mass density $\ll 0.17 M_{\odot} pc^{-3}$<br>Geometry of the matter distribution (bulge, disk)</td></tr></tbody></table>

## 2 Calculation details

In the SPARC catalogue [35] each galaxy is usually identified by three independent main components for the densities : the bulge labeled *b* in the following, the stellar disk labeled *d.st* and the gaseous disk labeled *d.g*. This hierarchy is also preserved in the κ-model where both the geometry and the relative values taken by the mean densities (compact masses for the stellar component, or diffuse masses for the gaseous component) are now playing a new role by their implication in a magnification factor at a very large scale. A similar idea appears in the NFDG theory, but it is the dimension of the matter distribution that plays a major role. Let us note that the so-called κ-effect (a retranscription of the DM-like effect), said in a practical way, is a
sheet, attached to a given observer in the bundle situated "above" the base [16]. The latter mathematical considerations will be shortly developed in an up-coming paper. We are only concerned here with the observational aspect.

The synoptic table below summarizes the applicability domains of the different models discussed in this paper :

<table><thead><tr><th>Model</th><th>Main features</th></tr></thead><tbody><tr><td>MOND</td><td>Very low acceleration $a \lesssim a_0 \sim 10^{-10} m s^{-2}$</td></tr><tr><td>κ-model</td><td>Very low mean mass density $\lesssim 0.15 M_{\odot} pc^{-3}$<br>Geometry of the matter distribution (bulge, disk)<br>Compactness (stars, gas)</td></tr><tr><td>NFDG</td><td>Variable dimension of the matter distribution,<br>between $D = 3$ (sphere) and $D = 2$ (disk)</td></tr><tr><td>RG</td><td>Very low mean mass density $\ll 0.17 M_{\odot} pc^{-3}$<br>Geometry of the matter distribution (bulge, disk)</td></tr></tbody></table>

## 2 Calculation details

In the SPARC catalogue [35] each galaxy is usually identified by three independent main components for the densities : the bulge labeled *b* in the following, the stellar disk labeled *d.st* and the gaseous disk labeled *d.g*. This hierarchy is also preserved in the κ-model where both the geometry and the relative values taken by the mean densities (compact masses for the stellar component, or diffuse masses for the gaseous component) are now playing a new role by their implication in a magnification factor at a very large scale. A similar idea appears in the NFDG theory, but it is the dimension of the matter distribution that plays a major role. Let us note that the so-called κ-effect (a retranscription of the DM-like effect), said in a practical way, is a
"huge-volume-effect" and it only occurs at a very large scale; it is inexistent at the solar system level (a bit like the quantum effects are fully imperceptible at the macroscopic level). In the framework of the κ-model, the relationship associating the corresponding (fictive) Newtonian velocities to the measured spectroscopic velocity is [16, 26, 27]

$$v_{spec} = \left(\frac{\kappa M_t}{\kappa}\right)^{\frac{1}{2}} \left[ \frac{\kappa_{ref}}{\kappa_{Mst.b}} v_{b.st}^2 + \frac{\kappa_{ref}}{\kappa_{M_{d.st}}} v_{d.st}^2 + \frac{\kappa_{ref}}{\kappa_{M_{d.g}}} v_{d.g}^2 \right]^{\frac{1}{2}} \quad (1)$$

where each peculiar velocity is weighted by a κ-ratio. The origin of the κ-ratios results from the need to take into account explicitly both the matter distribution dimension (bulge or disk) and the compactness of this matter (stars or gas). In the κ-model all these coefficients are directly linked to the mean volumetric mass densities ρ by a simple and universal relationship (ln denotes the natural logarithm)

$$\frac{\kappa_1}{\kappa_2} = 1 + \ln \left[ \frac{\rho_1}{\rho_2} \right] \qquad (2)$$

with the necessary condition $\rho_1 / \rho_2 > 1$. The indexes "1,2" run on all the mentioned indexes. The relation (2) is called universal in the sense that this relation is valid whatever the type of galaxies, and also for galaxy clusters [16, 26]. In MOND the analog of κ is not a logarithmic function of the density, but a rational function of the distance [9, 10, 11] (but both are sensibly equivalent in the case of an exponential distribution of matter). In relation (1) the indexes ref, M_t, M_b.st, M_d.st and M_{d.g} respectively designate the reference value for the density, the maximum value M of the total density, t, (stellar bulge, b.st, + stellar disk, d.st, + gaseous disk, d.g) estimated at the center of the galaxy, and the maximum value M of each of the
"huge-volume-effect" and it only occurs at a very large scale; it is inexistent at the solar system level (a bit like the quantum effects are fully imperceptible at the macroscopic level). In the framework of the κ-model, the relationship associating the corresponding (fictive) Newtonian velocities to the measured spectroscopic velocity is [16, 26, 27]

$$v_{spec} = \left(\frac{\kappa M_t}{\kappa}\right)^{\frac{1}{2}} \left[ \frac{\kappa_{ref}}{\kappa_{Mst.b}} v_{b.st}^2 + \frac{\kappa_{ref}}{\kappa_{M_{d.st}}} v_{d.st}^2 + \frac{\kappa_{ref}}{\kappa_{M_{d.g}}} v_{d.g}^2 \right]^{\frac{1}{2}} \quad (1)$$

where each peculiar velocity is weighted by a κ-ratio. The origin of the κ-ratios results from the need to take into account explicitly both the matter distribution dimension (bulge or disk) and the compactness of this matter (stars or gas). In the κ-model all these coefficients are directly linked to the mean volumetric mass densities ρ by a simple and universal relationship (ln denotes the natural logarithm)

$$\frac{\kappa_1}{\kappa_2} = 1 + \ln \left[ \frac{\rho_1}{\rho_2} \right] \qquad (2)$$

with the necessary condition $\rho_1/\rho_2 > 1$. The indexes "1,2" run on all the mentioned indexes. The relation (2) is called universal in the sense that this relation is valid whatever the type of galaxies, and also for galaxy clusters [16, 26]. In MOND the analog of κ is not a logarithmic function of the density, but a rational function of the distance [9, 10, 11] (but both are sensibly equivalent in the case of an exponential distribution of matter). In relation (1) the indexes ref, M_t, M_b.st, M_d.st and M_{d.g} respectively designate the reference value for the density, the maximum value M of the total density, t, (stellar bulge, b.st, + stellar disk, d.st, + gaseous disk, d.g) estimated at the center of the galaxy, and the maximum value M of each of the
independent components, also estimated at the center of the galaxy.
The non-indexed coefficient $\kappa$ is the local one (there where resides
the observer who feels the gravitational field). For practical purposes
concerning the disk components, the density $\rho$ can be expressed as a
function of the observable surface mass density (indirectly obtained
from the brightness measurement), i.e. $\rho = \frac{\Sigma}{\delta}$ with the thickness $\delta$,
the latter quantity being here assumed to be constant throughout a
galaxy disk. Apparently, the thickness of the disks seems to play a
role in the $\kappa$-model, very similarly to what is assumed in the *NFDG*
model, even though in the *NFDG* model it is the dimension of the
mass distribution that intervenes instead of the thickness [23, 30]. A
variable thickness along a galactic radius in the $\kappa$-model could have a
close connection with the variable dimension $D$ in the *NFDG* model.
However it isn't as simple as it appears, and we return to this issue
in the following. The magnification coefficients of the active mass
composing both the stellar and gaseous components are expressed
separately, resp. $\frac{\kappa_{ref}}{\kappa_{M_{b.st}}} \frac{\kappa_{ref}}{\kappa_{M_{d.st}}}$ and $\frac{\kappa_{ref}}{\kappa_{M_{d.g}}}$ but are still calculated with
the same universal relationship (2). When the mean surface mass
density is larger than 500 $M_{\odot} pc^{-2}$, a saturation effect appears for
$\frac{\kappa_{ref}}{\kappa_{M}}$, and then in all the cases we put this factor invariably equal
to 0.45, as provided by the relation (2). However, in a few rare
situations, especially for galaxies with a big bulge, and in order to
adequately fit the observational profiles in the inner regions, we should
adjust the factor $\frac{\kappa_{ref}}{\kappa_{M_{b.st}}}$ to a value between 0.45 and 1. An explanation
to this statement is that, in fact, the relation (2) is valid for a thin
disk, but not for a 3D bulge. At this level a clear reference to the
NFDG model where the dimension plays a major role can be noticed.
Another explanation is that the bulge of a spiral galaxy is a very
complex system where the stellar orbits are randomly oriented. Then
independent components, also estimated at the center of the galaxy.
The non-indexed coefficient $\kappa$ is the local one (there where resides
the observer who feels the gravitational field). For practical purposes
concerning the disk components, the density $\rho$ can be expressed as a
function of the observable surface mass density (indirectly obtained
from the brightness measurement), i.e. $\rho = \frac{\Sigma}{\delta}$ with the thickness $\delta$,
the latter quantity being here assumed to be constant throughout a
galaxy disk. Apparently, the thickness of the disks seems to play a
role in the $\kappa$-model, very similarly to what is assumed in the *NFDG*
model, even though in the *NFDG* model it is the dimension of the
mass distribution that intervenes instead of the thickness [23, 30]. A
variable thickness along a galactic radius in the $\kappa$-model could have a
close connection with the variable dimension $D$ in the *NFDG* model.
However it isn't as simple as it appears, and we return to this issue
in the following. The magnification coefficients of the active mass
composing both the stellar and gaseous components are expressed
separately, resp. $\frac{\kappa_{ref}}{\kappa_{M_{b.st}}} \frac{\kappa_{ref}}{\kappa_{M_{d.st}}}$ and $\frac{\kappa_{ref}}{\kappa_{M_{d.g}}}$ but are still calculated with
the same universal relationship (2). When the mean surface mass
density is larger than 500 $M_{\odot} pc^{-2}$, a saturation effect appears for
$\frac{\kappa_{ref}}{\kappa_M}$, and then in all the cases we put this factor invariably equal
to 0.45, as provided by the relation (2). However, in a few rare
situations, especially for galaxies with a big bulge, and in order to
adequately fit the observational profiles in the inner regions, we should
adjust the factor $\frac{\kappa_{ref}}{\kappa_{M_{b.st}}}$ to a value between 0.45 and 1. An explanation
to this statement is that, in fact, the relation (2) is valid for a thin
disk, but not for a 3D bulge. At this level a clear reference to the
NFDG model where the dimension plays a major role can be noticed.
Another explanation is that the bulge of a spiral galaxy is a very
complex system where the stellar orbits are randomly oriented. Then
we know that a severe velocity dispersion, larger than ~50 km/s a few kpc from the center, can strongly affect the extraction of the pure rotation velocity (see for instance the reference [36] for the Milky Way). The part of the cylindrical rotational support in the inner regions of a spiral galaxy is generally difficult to estimate when the bulge is dominant.

A fundamental question is: how many free parameters are used in the κ-model? We know that in physics, the fewer parameters, the better the model. Yet by consulting relation (1), we see immediately that four parameters (the κ-ratios) appear. Following the parsimony principle it is not a "good" model. In fact, once the density in the bulge, in the stellar and gaseous disks is provided, the κ-ratios, which are directly issued from observational data, are automatically determined, there is no longer free parameters and the κ-model eventually becomes parameter-free (the only parameters being, as usually, the observables, i.e. the surface brightness, the inclination and the distance, even though unfortunately not very well known in some cases). This is in strong contrast with DM where two or three free external, and arbitrarily chosen, parameters are introduced to just obtain the expected results. However given that the κ-ratios are dependent on the densities, the parameters in the κ-model can now vary from one object to another, and this confers some flexibility to the model with no violation of the parsimony principle. For instance, the κ-model has been applied with success to the physics of galaxy clusters [16]. The mean mass density in a galaxy cluster is lower by three orders of magnitude compared to the mean mass density in a galaxy. The κ-model is then naturally a multiscale model (or density-dependent scale model), like the one proposed in [14] for the application of MOND to galaxy clusters. The difference is that in the κ-model the scaling is not imposed, but appears in essence, taking
we know that a severe velocity dispersion, larger than ~50 km/s a few kpc from the center, can strongly affect the extraction of the pure rotation velocity (see for instance the reference [36] for the Milky Way). The part of the cylindrical rotational support in the inner regions of a spiral galaxy is generally difficult to estimate when the bulge is dominant.

A fundamental question is: how many free parameters are used in the κ-model? We know that in physics, the fewer parameters, the better the model. Yet by consulting relation (1), we see immediately that four parameters (the κ-ratios) appear. Following the parsimony principle it is not a "good" model. In fact, once the density in the bulge, in the stellar and gaseous disks is provided, the κ-ratios, which are directly issued from observational data, are automatically determined, there is no longer free parameters and the κ-model eventually becomes parameter-free (the only parameters being, as usually, the observables, i.e. the surface brightness, the inclination and the distance, even though unfortunately not very well known in some cases). This is in strong contrast with DM where two or three free external, and arbitrarily chosen, parameters are introduced to just obtain the expected results. However given that the κ-ratios are dependent on the densities, the parameters in the κ-model can now vary from one object to another, and this confers some flexibility to the model with no violation of the parsimony principle. For instance, the κ-model has been applied with success to the physics of galaxy clusters [16]. The mean mass density in a galaxy cluster is lower by three orders of magnitude compared to the mean mass density in a galaxy. The κ-model is then naturally a multiscale model (or density-dependent scale model), like the one proposed in [14] for the application of MOND to galaxy clusters. The difference is that in the κ-model the scaling is not imposed, but appears in essence, taking
its origin in the hierarchy of the mean mass densities. By contrast, MOND [9, 10, 11] with just one universal parameter or even the Refracted Gravity [33, 34] with three universal parameters seems to be too rigid. On the other hand, the κ-model can naturally be made relativistic [25], making possible its extension to cosmology, especially to the analysis of the fluctuation density in the CMB. In this case it is the density anisotropies to mean density ratio which intervene in the relationship (2). The latter very important topic will be examined in a next paper.

Now, if we want to compare the κ-model and MOND, we must define a reference point for the mean mass density $\rho_{ref}$. Unfortunately, this quantity is only indirectly known by the ratio $\frac{\Sigma}{\delta}$ (surface mass density, $\Sigma$, over the disk thickness, $\delta$, in a spiral galaxy such as the Milky Way). The link between the acceleration parameter $a_0 \sim 1.2 \ 10^{-10} \ m \ s^{-2}$ of MOND and the reference surface density $\Sigma_{ref}$ is

$$ \Sigma_{ref} = \frac{a_0}{2\pi G} = 152 \ M_{\odot} \ pc^{-2} \qquad (3) $$

Let us note that this value is relatively close to the galactic surface mass density estimated in the solar region (~70 $M_{\odot} \ pc^{-2}$) (in comparison with the high range of surface densities seen in a disk galaxy, varying from ~1000 $M_{\odot} \ pc^{-2}$ in the inner regions, 1 $kpc$ from the center, to ~1 $M_{\odot} \ pc^{-2}$ in the outskirts, 20 $kpc$ from the centre). Taking into account the fact that the range of mean mass densities is very extended in the Universe, this appears indeed very odd if we see the parameter $a_0$ as a cosmological parameter; because in this case we must assume that our situation in the Universe is privileged. In reality, the κ-model easily explains this rather strange coincidence. We chose this reference taking into account our position in the galaxy,
its origin in the hierarchy of the mean mass densities. By contrast, MOND [9, 10, 11] with just one universal parameter or even the Refracted Gravity [33, 34] with three universal parameters seems to be too rigid. On the other hand, the $\kappa$-model can naturally be made relativistic [25], making possible its extension to cosmology, especially to the analysis of the fluctuation density in the CMB. In this case it is the density anisotropies to mean density ratio which intervene in the relationship (2). The latter very important topic will be examined in a next paper.

Now, if we want to compare the $\kappa$-model and MOND, we must define a reference point for the mean mass density $\rho_{ref}$. Unfortunately, this quantity is only indirectly known by the ratio $\frac{\Sigma}{\delta}$ (surface mass density, $\Sigma$, over the disk thickness, $\delta$, in a spiral galaxy such as the Milky Way). The link between the acceleration parameter $a_0 \sim 1.2 \ 10^{-10} \ m \ s^{-2}$ of MOND and the reference surface density $\Sigma_{ref}$ is

$$ \Sigma_{ref} = \frac{a_0}{2\pi G} = 152 \ M_{\odot} \ pc^{-2} \qquad (3) $$

Let us note that this value is relatively close to the galactic surface mass density estimated in the solar region ($\sim 70 \ M_{\odot} \ pc^{-2}$) (in comparison with the high range of surface densities seen in a disk galaxy, varying from $\sim 1000 \ M_{\odot} \ pc^{-2}$ in the inner regions, $1 \ kpc$ from the center, to $\sim 1 \ M_{\odot} \ pc^{-2}$ in the outskirts, $20 \ kpc$ from the centre). Taking into account the fact that the range of mean mass densities is very extended in the Universe, this appears indeed very odd if we see the parameter $a_0$ as a cosmological parameter; because in this case we must assume that our situation in the Universe is privileged. In reality, the $\kappa$-model easily explains this rather strange coincidence. We chose this reference taking into account our position in the galaxy,
but which has nothing special. Another observer, located elsewhere,
will take his own reference. The relation (1), giving a universal result
in the framework of the κ-model, his measurements would lead ex-
actly to the same results for the spectroscopic velocities as ours, even
though with his proper local reference for the mean mass density.

Figure 1 displays a panel of velocity profiles for MOND and the κ-model, in the schematic situation of a disk of matter where the mean surface mass density varies exponentially (the thickness is assumed to be constant following the radius r). The comparison between MOND and the κ-model shows that the logarithmic relation (2) is a very good choice. In MOND the function µ(r) plays a very similar role (see eqs. (7) and (8) of the reference [13]), even though in MOND µ(r) is not a logarithmic function, but a simple rational function. We can note that the κ-effect (or MOND-effect) plays a decreasing role, when going from low mass surface density (LSB galaxies) toward high mass surface density (HSB galaxies), as confirmed by the observations. This finding, naturally explained with MOND or the κ-model, remains unexplained in DM. A difference between MOND and the κ-model is however perceptible for the schematic representation of a so-called super spiral [37]. For a high surface density (Σ<sub>M</sub> ~ 10 000 M<sub>☉</sub> pc<sup>-2</sup>, see for instance the reference [38]) the κ-model curve is located more than 200 km/s above the MOND profile for the terminal velocity (fig. 1d).
but which has nothing special. Another observer, located elsewhere,
will take his own reference. The relation (1), giving a universal result
in the framework of the κ-model, his measurements would lead ex-
actly to the same results for the spectroscopic velocities as ours, even
though with his proper local reference for the mean mass density.

Figure 1 displays a panel of velocity profiles for MOND and the κ-model, in the schematic situation of a disk of matter where the mean surface mass density varies exponentially (the thickness is assumed to be constant following the radius r). The comparison between MOND and the κ-model shows that the logarithmic relation (2) is a very good choice. In MOND the function µ(r) plays a very similar role (see eqs. (7) and (8) of the reference [13]), even though in MOND µ(r) is not a logarithmic function, but a simple rational function. We can note that the κ-effect (or MOND-effect) plays a decreasing role, when going from low mass surface density (LSB galaxies) toward high mass surface density (HSB galaxies), as confirmed by the observations. This finding, naturally explained with MOND or the κ-model, remains unexplained in DM. A difference between MOND and the κ-model is however perceptible for the schematic representation of a so-called super spiral [37]. For a high surface density (Σ<sub>M</sub> ~ 10 000 M<sub>☉</sub> pc<sup>-2</sup>, see for instance the reference [38]) the κ-model curve is located more than 200 km/s above the MOND profile for the terminal velocity (fig. 1d).
<figure><img src="image_1.png" /><figcaption>Figure 1: Schematic galaxy velocity curves fitted with a simple exponential surface mass density profile of uniform thickness (in the approximation of the thin disk). MOND is the green line, and the $\kappa$-model is the amber line. The dashed red line is the Newtonian curve (baryons). The reference value for the surface density is $\Sigma_{ref} = \frac{a_0}{2\pi G} = 152 \ M_{\odot} \ pc^{-2}$. a. $\Sigma_M = 10$, b. $\Sigma_M = 100$, c. $\Sigma_M = 1000$, d. $\Sigma_M = 10\ 000 \ M_{\odot} \ pc^{-2}$.</figcaption></figure>

In the more concrete cases, the situation is obviously different from
<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic galaxy velocity curves fitted with a simple exponential surface mass density profile of uniform thickness (in the approximation of the thin disk). MOND is the green line, and the $\kappa$-model is the amber line. The dashed red line is the Newtonian curve (baryons). The reference value for the surface density is $\Sigma_{ref} = \frac{a_0}{2\pi G} = 152 \ M_{\odot} \ pc^{-2}$. a. $\Sigma_M = 10$, b. $\Sigma_M = 100$, c. $\Sigma_M = 1000$, d. $\Sigma_M = 10\ 000 \ M_{\odot} \ pc^{-2}$.</figcaption></figure>

In the more concrete cases, the situation is obviously different from
the previous trial examples with a simple exponential disk. In reality, we encounter in the SPARC catalog a number of situations where it is not possible to fit the mean surface mass density of both thin stellar, $\Sigma_{d.st}$, and gaseous, $\Sigma_{d.g}$, disks by just adopting a simple exponential fit. In these situations, we had to add to the exponential component one, or sometimes two, decentered gaussian components. The velocity curve $v_d(r)$ is then deduced from the self-evident, more general formula (still assuming an axisymmetric disk) valid for one component (stars or gas)

$$ \frac{v_d^2(r)}{r} = G \int_{\Omega_{\infty}} dx dy \frac{\Sigma(\sqrt{x^2 + y^2})(r-x)}{\kappa(r) [(r-x)^2 + y^2]^{\frac{3}{2}}} \quad (4) $$

As a first step the operational method consists to fit the Newtonian velocities available from the SPARC catalog, for the distributions of stars and gas, taken individually, and for each galaxy. In simple terms we fit the dashed-red (stars) and dashed-green (gas) curves of Figure A. In this case the relationship (4) is applied with $\kappa(r) \equiv 1$ (this is the usual Newtonian level). Secondly, the same relationship (4) is again used, but incorporating this time the coefficient $\kappa(r)$ that depends on the volumetric mass density (eq. 2). This second step automatically provides the corresponding $\kappa$-model curves. The great benefit of the method is that all parameters are internal to the theory, and supported by the sole observational data, essentially the baryonic mass density. There is no arbitrary parameter such as the ad hoc DM/B ratio in DM.

Eventually, when a bulge is present, a de Vaucouleurs formula [39] is used to fit the surface mass density of the bulge. Two other parameters, intervening in the $\kappa$-model, are still the thickness (scale height)
the previous trial examples with a simple exponential disk. In reality, we encounter in the SPARC catalog a number of situations where it is not possible to fit the mean surface mass density of both thin stellar, $\Sigma_{d.st}$, and gaseous, $\Sigma_{d.g}$, disks by just adopting a simple exponential fit. In these situations, we had to add to the exponential component one, or sometimes two, decentered gaussian components. The velocity curve $v_d(r)$ is then deduced from the self-evident, more general formula (still assuming an axisymmetric disk) valid for one component (stars or gas)

$$ \frac{v_d^2(r)}{r} = G \int_{\Omega_{\infty}} dx dy \frac{\Sigma(\sqrt{x^2 + y^2})(r-x)}{\kappa(r) [(r-x)^2 + y^2]^{\frac{3}{2}}} \quad (4) $$

As a first step the operational method consists to fit the Newtonian velocities available from the SPARC catalog, for the distributions of stars and gas, taken individually, and for each galaxy. In simple terms we fit the dashed-red (stars) and dashed-green (gas) curves of Figure A. In this case the relationship (4) is applied with $\kappa(r) \equiv 1$ (this is the usual Newtonian level). Secondly, the same relationship (4) is again used, but incorporating this time the coefficient $\kappa(r)$ that depends on the volumetric mass density (eq. 2). This second step automatically provides the corresponding $\kappa$-model curves. The great benefit of the method is that all parameters are internal to the theory, and supported by the sole observational data, essentially the baryonic mass density. There is no arbitrary parameter such as the ad hoc DM/B ratio in DM.

Eventually, when a bulge is present, a de Vaucouleurs formula [39] is used to fit the surface mass density of the bulge. Two other parameters, intervening in the $\kappa$-model, are still the thickness (scale height)
of the stellar (thick) disk, $\delta_{st}$, and that of the gaseous (thin) disk, $\delta_g$. For all the galaxies under study (SPARC catalog), these parameters have been taken to be equal to the reference values estimated for the Milky Way in the vicinity of the Sun, respectively $\delta_{\odot.st}$ and $\delta_{\odot.g} \sim 0.5 \delta_{\odot.st}$. Given that the galaxies are diversely oriented with any inclination angle, these parameters are difficult to estimate and certainly variable along a galactic radius. Our analysis of the SPARC galaxies seems to indicate a neat trend where the thickness decreases when going from the core regions to the outskirts in the flattened galaxies.

Globally, for a mean orientation of 45° the thickness along the line of view is increased by a factor of $\sqrt{2}$. In this case the rotation profiles provided by the model have to be magnified by a few percent. In fact, the logarithmic function flattens the density ratios in relation (2) and the influence of the variation of the thickness has a strongly reduced, even though not negligible, impact on the corresponding $\kappa$-ratios (of the order of 10% for a thickness variation of a factor 2. For orientations larger than 45° the magnification can obviously be much larger than 10%). Let us note that in other models where the density is assumed to play a role, for instance in [24, 30], the conclusions should be very similar when applied to a large sample of galaxies, such as the SPARC database. The thickness of various types of spiral galaxies has been estimated by different methods [40, 41, 42]. For irregular dwarf galaxies, the situation appears relatively confusing, but the latter category can exhibit quasi-round galaxies with a high mean thickness [40]. The measurement of the thickness seems to give values of the order of $\delta_{\odot.st}$ or $\delta_{\odot.g}$ to within a multiplicative in the range 0.25 (in the outer regions) to 4 (in the inner regions toward the bulge if existing), compared to the reference values, independent of the size of the galaxy (with a few exceptions for the very small
of the stellar (thick) disk, $\delta_{st}$, and that of the gaseous (thin) disk, $\delta_g$. For all the galaxies under study (SPARC catalog), these parameters have been taken to be equal to the reference values estimated for the Milky Way in the vicinity of the Sun, respectively $\delta_{\odot.st}$ and $\delta_{\odot.g} \sim 0.5 \delta_{\odot.st}$. Given that the galaxies are diversely oriented with any inclination angle, these parameters are difficult to estimate and certainly variable along a galactic radius. Our analysis of the SPARC galaxies seems to indicate a neat trend where the thickness decreases when going from the core regions to the outskirts in the flattened galaxies.

Globally, for a mean orientation of 45° the thickness along the line of view is increased by a factor of $\sqrt{2}$. In this case the rotation profiles provided by the model have to be magnified by a few percent. In fact, the logarithmic function flattens the density ratios in relation (2) and the influence of the variation of the thickness has a strongly reduced, even though not negligible, impact on the corresponding $\kappa$-ratios (of the order of 10% for a thickness variation of a factor 2. For orientations larger than 45° the magnification can obviously be much larger than 10%). Let us note that in other models where the density is assumed to play a role, for instance in [24, 30], the conclusions should be very similar when applied to a large sample of galaxies, such as the SPARC database. The thickness of various types of spiral galaxies has been estimated by different methods [40, 41, 42]. For irregular dwarf galaxies, the situation appears relatively confusing, but the latter category can exhibit quasi-round galaxies with a high mean thickness [40]. The measurement of the thickness seems to give values of the order of $\delta_{\odot.st}$ or $\delta_{\odot.g}$ to within a multiplicative in the range 0.25 (in the outer regions) to 4 (in the inner regions toward the bulge if existing), compared to the reference values, independent of the size of the galaxy (with a few exceptions for the very small
galaxies, where smaller values for the thickness are favored). Thus, a positive point is that an estimate of the thickness can be reached in the framework of the $\kappa$-model. However in figure A, for all the galaxies and for the sake of homogeneity, the thickness has been taken as invariable throughout the stellar and gaseous disks. The corresponding values are indicated in each individual figure. Taking into account a variable thickness would make it possible to obtain better profiles. A work that remains to be done.

Additionally, let us specify that the observations rather provide non-monotonous galactic rotation profiles. Nevertheless it is illusory to try to perfectly fit the rotational curves with their delicate patterns of bumps and wiggles. Very likely, these patterns are caused by the presence of spiral arms or portions of rings, a variable thickness or inclination, not taken into account by assuming smoothed axisymmetric and monotonous density profiles. Even DM with two or even three external parameters cannot make that<fnref n="5" />. One of the better DM methods, built on the Einasto profiles with three ad hoc parameters in the fits, is discussed in reference [45]. We can see that the fine details cannot be adequately fitted (see, for instance, NGC6015, NGC 7793, NGC3726, IC4202, NGC0289, UGC06787, etc). In any case a lot of physical parameters are very poorly known: the inclination of the galaxy (moreover, very likely variable along the galactic radius), the mass-to-light ratio, the thickness along the line of sight, the distance, etc. We must add that the observational profiles can substantially differ in some cases from one author to another, sometimes by more than 20 km/s. We can compare two different catalogs, for instance, that of Sofue [46] versus SPARC [35], when the rotation curve for the same galaxy is presented (see especially NGC 2903 where a discrepancy of 40 km/s can be notified). Even for the Milky Way, in

<footnote n="5">⁵ A list of DM methods with two or three ad hoc parameters is presented in the references [43, 44].</footnote>
galaxies, where smaller values for the thickness are favored). Thus, a positive point is that an estimate of the thickness can be reached in the framework of the $\kappa$-model. However in figure A, for all the galaxies and for the sake of homogeneity, the thickness has been taken as invariable throughout the stellar and gaseous disks. The corresponding values are indicated in each individual figure. Taking into account a variable thickness would make it possible to obtain better profiles. A work that remains to be done.

Additionally, let us specify that the observations rather provide non-monotonous galactic rotation profiles. Nevertheless it is illusory to try to perfectly fit the rotational curves with their delicate patterns of bumps and wiggles. Very likely, these patterns are caused by the presence of spiral arms or portions of rings, a variable thickness or inclination, not taken into account by assuming smoothed axisymmetric and monotonous density profiles. Even DM with two or even three external parameters cannot make that<fnref n="5" />. One of the better DM methods, built on the Einasto profiles with three ad hoc parameters in the fits, is discussed in reference [45]. We can see that the fine details cannot be adequately fitted (see, for instance, NGC6015, NGC 7793, NGC3726, IC4202, NGC0289, UGC06787, etc). In any case a lot of physical parameters are very poorly known: the inclination of the galaxy (moreover, very likely variable along the galactic radius), the mass-to-light ratio, the thickness along the line of sight, the distance, etc. We must add that the observational profiles can substantially differ in some cases from one author to another, sometimes by more than 20 km/s. We can compare two different catalogs, for instance, that of Sofue [46] versus SPARC [35], when the rotation curve for the same galaxy is presented (see especially NGC 2903 where a discrepancy of 40 km/s can be notified). Even for the Milky Way, in

<footnote n="5">⁵ A list of DM methods with two or three ad hoc parameters is presented in the references [43, 44].</footnote>
the vicinity of the Sun, divergences also exist [47]. Let us note that
the DM paradigm could, however, be made in agreement with any
inclination by adequately adjusting the DM/B rate! A contrario both
MOND and the κ-model apparently fail if the inclination is not ac-
curately estimated [26]. An example where the inclination factor can
sometimes play an important role in the determination of the rota-
tion velocity profiles is given in [48]. In the latter paper it is shown
that the inclination can vary by 20° following the authors, eventually
favoring a model rather than another one. Eventually, we can say
that, unfortunately, the determination of the inclination is not the
sole trouble. Additionally the gas and the stars in a galaxy, following
their types, do not rotate in the same manner, the velocities are not
circular, the galaxy disks are not symmetric, etc. The multiple con-
sequences on the observational profiles are difficult to estimate. This
is why various observational techniques can lead to different profiles
for the same galaxy.

In spite of these difficulties, and in order to make a valuable com-
parative analysis between different theoretical models, the idea is to
use a same set of extended data. For instance, the SPARC catalog
seems in this case necessary. This catalog gathers a large, and homo-
geneous, sample of rotation profiles. A very good point of the SPARC
database is that it represents a uniform estimate of the surface den-
sities of galaxies, starting from Spitzer near-infrared data [35]. Then
our procedure as to starting from mean fits of the Newtonian curves,
and then mean fits for the observed rotational curves can be deduced.
In some cases, the DM fits seem to be much more impressive [44, 45],
but a major drawback for a physical model is that the DM technique
of fitting is not at all predictive. Then, starting from any Newtonian
curve (even false), we can build any "good" predicted profiles, ob-
viously by adding the "good" rate of DM. Admittedly, MOND and
the vicinity of the Sun, divergences also exist [47]. Let us note that
the DM paradigm could, however, be made in agreement with any
inclination by adequately adjusting the DM/B rate! A contrario both
MOND and the κ-model apparently fail if the inclination is not ac-
curately estimated [26]. An example where the inclination factor can
sometimes play an important role in the determination of the rota-
tion velocity profiles is given in [48]. In the latter paper it is shown
that the inclination can vary by 20° following the authors, eventually
favoring a model rather than another one. Eventually, we can say
that, unfortunately, the determination of the inclination is not the
sole trouble. Additionally the gas and the stars in a galaxy, following
their types, do not rotate in the same manner, the velocities are not
circular, the galaxy disks are not symmetric, etc. The multiple con-
sequences on the observational profiles are difficult to estimate. This
is why various observational techniques can lead to different profiles
for the same galaxy.

In spite of these difficulties, and in order to make a valuable com-
parative analysis between different theoretical models, the idea is to
use a same set of extended data. For instance, the SPARC catalog
seems in this case necessary. This catalog gathers a large, and homo-
geneous, sample of rotation profiles. A very good point of the SPARC
database is that it represents a uniform estimate of the surface den-
sities of galaxies, starting from Spitzer near-infrared data [35]. Then
our procedure as to starting from mean fits of the Newtonian curves,
and then mean fits for the observed rotational curves can be deduced.
In some cases, the DM fits seem to be much more impressive [44, 45],
but a major drawback for a physical model is that the DM technique
of fitting is not at all predictive. Then, starting from any Newtonian
curve (even false), we can build any "good" predicted profiles, ob-
viously by adding the "good" rate of DM. Admittedly, MOND and
κ-model profiles are generally of lesser quality, but in most cases, both of them produce a good trend for the fits compared to the observational rotation curves. Let us specify again that the latter ones, impaired by various biases, are also not perfect either.

# 3 Results

## 3.1 MOND versus κ-model

The results of the calculations for the individual galaxies in the SPARC catalog are collected in Figure A in the Appendix. The galaxies have been classified in alphabetical order to facilitate the research. For the disks, it is assumed that the thickness is constant along a radius of the galaxy. In most cases, the thickness has been taken to be equal to the corresponding reference values taken at the Sun position in the Milky Way, for both the stellar, let $\delta_{\odot.st}$, and gaseous disk, let $\delta_{\odot.g}$. In view of the results, a first general remark is that the κ-model is clearly as predictive as MOND<fnref n="6" />. For both models, the results statistically deviate by less than 10% as for the prediction of the terminal velocities (Figure 2).

$$v_{MOND}^2 = v_{bew}^2 \left[ \frac{1}{2} + \frac{1}{2} \left[ 1 + 4 \left( \frac{a_0}{g_{new}} \right)^2 \right]^{\frac{1}{2}} \right]^{\frac{1}{2}} \quad (5)$$

where $v_{new}$ and $g_{new}$ are respectively the Newtonian velocity and acceleration.

<footnote n="6">⁶The MOND profiles have been obtained with the formula</footnote>
κ-model profiles are generally of lesser quality, but in most cases, both of them produce a good trend for the fits compared to the observational rotation curves. Let us specify again that the latter ones, impaired by various biases, are also not perfect either.

# 3 Results

## 3.1 MOND versus κ-model

The results of the calculations for the individual galaxies in the SPARC catalog are collected in Figure A in the Appendix. The galaxies have been classified in alphabetical order to facilitate the research. For the disks, it is assumed that the thickness is constant along a radius of the galaxy. In most cases, the thickness has been taken to be equal to the corresponding reference values taken at the Sun position in the Milky Way, for both the stellar, let $\delta_{\odot.st}$, and gaseous disk, let $\delta_{\odot.g}$. In view of the results, a first general remark is that the κ-model is clearly as predictive as MOND<fnref n="6" />. For both models, the results statistically deviate by less than 10% as for the prediction of the terminal velocities (Figure 2).

$$v_{MOND}^2 = v_{bew}^2 \left[ \frac{1}{2} + \frac{1}{2} \left[ 1 + 4 \left( \frac{a_0}{g_{new}} \right)^2 \right]^{\frac{1}{2}} \right]^{\frac{1}{2}} \quad (5)$$

where $v_{new}$ and $g_{new}$ are respectively the Newtonian velocity and acceleration.

<footnote n="6">⁶The MOND profiles have been obtained with the formula</footnote>
<figure><img src="image_2.png" /></figure>

Figure 2 The terminal velocities for the sample of galaxies (SPARC data base). MOND is in green, the κ-model is in amber, and the observed velocities, provided by the SPARC database, are represented in black. Linear regression lines are also represented.

Examining the individual cases, by browsing figure A, we can see
that the κ-model leads to predictions similar to the MOND phe-
nomenology, even though in some cases, the profiles are not quite
identical. Moreover, a comparison with the observational profiles
shows that the predicted curves for both MOND and the κ-model
don't perfectly match the observations<sup>7</sup>. Reporting to figure A we see
that the theoretical curves predicted by MOND or the κ-model can be
indifferently located slightly above or below the observational curves.
However, there are remedies for this. First, in the outer regions, the
predicted curve is quite often located above the observational one.
In MOND the bias can then be corrected by EFE (External Field
Effect) [52]. Likewise, this bias could be corrected by a diminishing

<footnote n="7"><sup>7</sup>We can remark that whereas some authors affirm that the MOND fits are fairly good [47, 49], on the other hand, other authors, who rather seem to defend the DM paradigm, conclude that the MOND fits are bad in a large percentage of analyzed individual cases [50, 51]. It is true that MOND gives fairly acceptable fits in a large number of cases and is less good in other cases. This situation can easily be explained if we admit that the mass-to-light ratios, the inclinations, and the distances are poorly known. We can then postulate that MOND systematically provides "perfect" fits and can then predict the inclinations and distances. A contrario DM fits are apparently better but, ironically enough, even if we choose a bad inclination or an erroneous distance.</footnote>
<figure><img src="image_3.png" /></figure>

Figure 2 The terminal velocities for the sample of galaxies (SPARC data base). MOND is in green, the κ-model is in amber, and the observed velocities, provided by the SPARC database, are represented in black. Linear regression lines are also represented.

Examining the individual cases, by browsing figure A, we can see
that the κ-model leads to predictions similar to the MOND phe-
nomenology, even though in some cases, the profiles are not quite
identical. Moreover, a comparison with the observational profiles
shows that the predicted curves for both MOND and the κ-model
don't perfectly match the observations<sup>7</sup>. Reporting to figure A we see
that the theoretical curves predicted by MOND or the κ-model can be
indifferently located slightly above or below the observational curves.
However, there are remedies for this. First, in the outer regions, the
predicted curve is quite often located above the observational one.
In MOND the bias can then be corrected by EFE (External Field
Effect) [52]. Likewise, this bias could be corrected by a diminishing

<footnote n="7"><sup>7</sup>We can remark that whereas some authors affirm that the MOND fits are fairly good [47, 49], on the other hand, other authors, who rather seem to defend the DM paradigm, conclude that the MOND fits are bad in a large percentage of analyzed individual cases [50, 51]. It is true that MOND gives fairly acceptable fits in a large number of cases and is less good in other cases. This situation can easily be explained if we admit that the mass-to-light ratios, the inclinations, and the distances are poorly known. We can then postulate that MOND systematically provides "perfect" fits and can then predict the inclinations and distances. A contrario DM fits are apparently better but, ironically enough, even if we choose a bad inclination or an erroneous distance.</footnote>
thickness of the disks (at constant surface density Σ) in the κ-model.
A contrario, there exist a number of cases where the predicted curve
is located below the observational one, especially in the innermost
regions of galaxies (for instance, in the more striking cases: F563-V2,
F568-1, F568-V1, F571-8, F579-V1, F583-1, NGC 2915, NGC 3992,
NGC 5907, NGC 5985, NGC 6674, UGC 00128, UGC 00731, UGC
02259, UGC 6446, UGC 06667, UGC 07399, UGC 7490, and UGC
8286). It is interesting to note that the magnitude of this bias is
very similar in both MOND and in the κ-model for a given galaxy.
Very possibly, a modification of the inclination in MOND (and also
in the κ-model) could partially remove the discrepancy in the inner
regions for these galaxies. As demonstrated in [48] the modification
of the inclination can substantially modify the profile of the rota-
tion curve. However this effect appears rather systematic throughout
figure A in the inner regions. In other words, the measurement of
the inclination would be systematically biased in the inner of spiral
galaxies and, strangely enough, always in the same direction. This
hypothesis is hardly acceptable. The fact that the MOND curve is
located below the observational one results from the fact that the
acceleration *a* is equal to or larger than the critical value *a*₀ in the
inner regions. In this case, we are in a domain where the Newto-
nian regime is still supposed to be valid. To save MOND, we can
then assume that the parameter *a*₀ is larger in the inner region, but
then this parameter is no longer universal. Another solution is that
the baryonic mass-to-light ratios are largely underestimated (by a
factor 2) in the inner regions of the quoted galaxies. Eventually, a
more credible explanation is to imagine that some non-exotic form
of DM exists in the innermost regions of galaxies. We could invoke,
for instance, a neutrino species with mass ~ eV (but in acceptable
quantity with DM/B ~ 2). A very similar idea has been assumed
thickness of the disks (at constant surface density Σ) in the κ-model.
A contrario, there exist a number of cases where the predicted curve
is located below the observational one, especially in the innermost
regions of galaxies (for instance, in the more striking cases: F563-V2,
F568-1, F568-V1, F571-8, F579-V1, F583-1, NGC 2915, NGC 3992,
NGC 5907, NGC 5985, NGC 6674, UGC 00128, UGC 00731, UGC
02259, UGC 6446, UGC 06667, UGC 07399, UGC 7490, and UGC
8286). It is interesting to note that the magnitude of this bias is
very similar in both MOND and in the κ-model for a given galaxy.
Very possibly, a modification of the inclination in MOND (and also
in the κ-model) could partially remove the discrepancy in the inner
regions for these galaxies. As demonstrated in [48] the modification
of the inclination can substantially modify the profile of the rota-
tion curve. However this effect appears rather systematic throughout
figure A in the inner regions. In other words, the measurement of
the inclination would be systematically biased in the inner of spiral
galaxies and, strangely enough, always in the same direction. This
hypothesis is hardly acceptable. The fact that the MOND curve is
located below the observational one results from the fact that the
acceleration *a* is equal to or larger than the critical value *a*₀ in the
inner regions. In this case, we are in a domain where the Newto-
nian regime is still supposed to be valid. To save MOND, we can
then assume that the parameter *a*₀ is larger in the inner region, but
then this parameter is no longer universal. Another solution is that
the baryonic mass-to-light ratios are largely underestimated (by a
factor 2) in the inner regions of the quoted galaxies. Eventually, a
more credible explanation is to imagine that some non-exotic form
of DM exists in the innermost regions of galaxies. We could invoke,
for instance, a neutrino species with mass ~ eV (but in acceptable
quantity with DM/B ~ 2). A very similar idea has been assumed
for the inner regions of galaxy clusters [53]. This hypothesis appears
admittedly reasonable; however, the κ-model can propose another
natural solution. In the κ-model, the leading role is not held by a
fixed parameter, i.e. the acceleration $a_0$, but by the mean volumetric
mass density. This hypothesis makes the κ-model more flexible than
MOND. Figure A displays the results under the reductive hypothesis
of a constant thickness throughout the galactic disks. However an
increase in disk thickness (at constant mean mass surface density) in
the inner regions of spiral galaxies could help to lessen the discrep-
ancy. An interesting conclusion is that the κ-model could hence help
to obtain an estimation of the mean thickness, a parameter difficult
to derive from the observations. In any case in the cases mentioned
above, even if the κ-model gives imperfect fits in the innermost re-
gions of these galaxies, we can see that the terminal velocities are
correctly predicted. A simple response to these statements is that
the empirical relationship (2) is very well adapted to a thin disk, but
far less applicable to a thick disk or a 3D bulge. For the galaxies
listed just above, where a discrepancy exists between the κ-model (or
equivalently MOND) and the observational data, a comparison with
DM profiles with two or three (ad hoc) parameters (as in reference
[44]) appears very interesting. Examining the cases displayed in figure
6.10 of [44], we can see that the underlined discrepancy also persists
in some of the cases, even though slightly lessened (see, for instance,
the rotation profiles for F568-1, F579-V1, NGC 2915, NGC 5907 and
NGC 6674). F571-8 is a pathological example where MOND, the
κ-model and DM yet provide very similar profiles, but paradoxically
enough far from the observational one in the outer regions. The three
theoretical profiles, even though very similar, are located 50 km/s
below the observational profile in the outer regions. In any way, we
know that trying to predict the rotation velocity curves with a better
for the inner regions of galaxy clusters [53]. This hypothesis appears
admittedly reasonable; however, the κ-model can propose another
natural solution. In the κ-model, the leading role is not held by a
fixed parameter, i.e. the acceleration $a_0$, but by the mean volumetric
mass density. This hypothesis makes the κ-model more flexible than
MOND. Figure A displays the results under the reductive hypothesis
of a constant thickness throughout the galactic disks. However an
increase in disk thickness (at constant mean mass surface density) in
the inner regions of spiral galaxies could help to lessen the discrep-
ancy. An interesting conclusion is that the κ-model could hence help
to obtain an estimation of the mean thickness, a parameter difficult
to derive from the observations. In any case in the cases mentioned
above, even if the κ-model gives imperfect fits in the innermost re-
gions of these galaxies, we can see that the terminal velocities are
correctly predicted. A simple response to these statements is that
the empirical relationship (2) is very well adapted to a thin disk, but
far less applicable to a thick disk or a 3D bulge. For the galaxies
listed just above, where a discrepancy exists between the κ-model (or
equivalently MOND) and the observational data, a comparison with
DM profiles with two or three (ad hoc) parameters (as in reference
[44]) appears very interesting. Examining the cases displayed in figure
6.10 of [44], we can see that the underlined discrepancy also persists
in some of the cases, even though slightly lessened (see, for instance,
the rotation profiles for F568-1, F579-V1, NGC 2915, NGC 5907 and
NGC 6674). F571-8 is a pathological example where MOND, the
κ-model and DM yet provide very similar profiles, but paradoxically
enough far from the observational one in the outer regions. The three
theoretical profiles, even though very similar, are located 50 km/s
below the observational profile in the outer regions. In any way, we
know that trying to predict the rotation velocity curves with a better
statistical precision than 10% (and even, in some pathological cases,
the incertitude can rise to 20%) appears unwarranted, considering
the dispersion in the observational data coming from various sources.
That matter aside, in the framework of the κ-model, at least we have
a fairly good estimation of the terminal velocities (Fig. 2 and see also
figure A for the individual cases), a conclusion that cannot be reached
by the ad hoc DM methodology. Obviously, the flexibility of the κ-
model by taking into account a variable thickness in the stellar and
gaseous disks would allow to fix the residual discrepancy between the
theoretical curves and the observational ones. In the same vein, this
statement is rather attractive because it implies that the κ-model
could predict the variation of the thickness in spiral galaxies along a
radius. This data is indeed difficult to obtain by sole observation.

## 3.2 Newtonian Fractional-Dimension Gravity

Newtonian Fractional-Dimension Gravity (NFDG) is an extension of
the laws of Newtonian gravitation to lower dimensional spaces, in-
cluding those with non-integer, "fractional" dimension (for a general
introduction see [23]). NFDG is based on a generalization of the
gravitational Gauss's law, replacing standard space integration over
$\mathbb{R}^3$ with an appropriate Hausdorff measure over the space, which was
related to Weyl's fractional integrals. As for MOND or $\kappa$-model, the
goal of NFDG is to describe galactic dynamics without using the con-
troversial DM component. A quick review of NFDG is presented in
the reference [30]. NFDG was introduced heuristically by extending
Gauss's law for gravitation to a lower dimensional space-time $D+1$,
where $D \le 3$ can be a non-integer space dimension. A scale length,
$l_0$, is needed to ensure the dimensional correctness of all expressions
for $D \ne 3$. Let us note that NFDG does not imply a change in the
statistical precision than 10% (and even, in some pathological cases,
the incertitude can rise to 20%) appears unwarranted, considering
the dispersion in the observational data coming from various sources.
That matter aside, in the framework of the κ-model, at least we have
a fairly good estimation of the terminal velocities (Fig. 2 and see also
figure A for the individual cases), a conclusion that cannot be reached
by the ad hoc DM methodology. Obviously, the flexibility of the κ-
model by taking into account a variable thickness in the stellar and
gaseous disks would allow to fix the residual discrepancy between the
theoretical curves and the observational ones. In the same vein, this
statement is rather attractive because it implies that the κ-model
could predict the variation of the thickness in spiral galaxies along a
radius. This data is indeed difficult to obtain by sole observation.

**3.2 Newtonian Fractional-Dimension Gravity**

Newtonian Fractional-Dimension Gravity (NFDG) is an extension of
the laws of Newtonian gravitation to lower dimensional spaces, in-
cluding those with non-integer, "fractional" dimension (for a general
introduction see [23]). NFDG is based on a generalization of the
gravitational Gauss's law, replacing standard space integration over
$\mathbb{R}^3$ with an appropriate Hausdorff measure over the space, which was
related to Weyl's fractional integrals. As for MOND or $\kappa$-model, the
goal of NFDG is to describe galactic dynamics without using the con-
troversial DM component. A quick review of NFDG is presented in
the reference [30]. NFDG was introduced heuristically by extending
Gauss's law for gravitation to a lower dimensional space-time $D+1$,
where $D \le 3$ can be a non-integer space dimension. A scale length,
$l_0$, is needed to ensure the dimensional correctness of all expressions
for $D \ne 3$. Let us note that NFDG does not imply a change in the
tri-dimensionality of space in galaxies, but rather the local Hausdorff
dimension *D* ≠ 3 is associated to the matter distribution (bulge or
disk). In this sense there is an analogy with the κ-model, where the
κ ratios (eq. 1) are assumed to be dependent on both the dimension
of the matter distribution (bulge or disk), and also the compactness
of matter (stars or gas).

In [29] Varieschi discusses in depth the case of NGC 6503. For
NFDG with a dimension *D* = 2, the theoretical curve is slightly
above the observational one and is remarkably flat (see Fig. 6 of [29]).
However, assuming that NGC 6503 behaves as a fractal medium, with
a variable fractional dimension, NFDG can produce a curve with a
perfect superimposition with the observational one. Reporting now
to Fig.A for this galaxy we can see that both the MOND and κ-model
curves are slightly below the observational curve in the inner regions
and slightly above in the outskirts. In the κ-model framework, the
statement of variable fractional dimension could be re-interpreted as
a variable thickness of the disk. In the case of NGC 6503 for in-
stance, an increase in the thickness in the inner regions (thick disk)
and, concomitantly, a decrease in the thickness in the outer regions
(thin disk), could also lead to an improved profile, such as in NFDG
theory. In [30] the same author applies his analysis to other rota-
tionally supported galaxies: NGC 7814 and NGC 3741, for which
very good NFDG fits are supplied. If we consider these galaxies,
MOND and the κ-model provide a fairly good value for the terminal
velocity. However, a same bias is perceptible for the inner veloci-
ties (the predicted curve is below the observational one). This bias
is not present on the NFDG profiles, which perfectly fit the corre-
sponding observational curves, even with their humps and wiggles.
However this perfect fit results from the fact that the NFDG the-
ory imposes on the fractional dimension to vary in "an appropriate
tri-dimensionality of space in galaxies, but rather the local Hausdorff
dimension *D* ≠ 3 is associated to the matter distribution (bulge or
disk). In this sense there is an analogy with the κ-model, where the
κ ratios (eq. 1) are assumed to be dependent on both the dimension
of the matter distribution (bulge or disk), and also the compactness
of matter (stars or gas).

In [29] Varieschi discusses in depth the case of NGC 6503. For
NFDG with a dimension *D* = 2, the theoretical curve is slightly
above the observational one and is remarkably flat (see Fig. 6 of [29]).
However, assuming that NGC 6503 behaves as a fractal medium, with
a variable fractional dimension, NFDG can produce a curve with a
perfect superimposition with the observational one. Reporting now
to Fig.A for this galaxy we can see that both the MOND and κ-model
curves are slightly below the observational curve in the inner regions
and slightly above in the outskirts. In the κ-model framework, the
statement of variable fractional dimension could be re-interpreted as
a variable thickness of the disk. In the case of NGC 6503 for in-
stance, an increase in the thickness in the inner regions (thick disk)
and, concomitantly, a decrease in the thickness in the outer regions
(thin disk), could also lead to an improved profile, such as in NFDG
theory. In [30] the same author applies his analysis to other rota-
tionally supported galaxies: NGC 7814 and NGC 3741, for which
very good NFDG fits are supplied. If we consider these galaxies,
MOND and the κ-model provide a fairly good value for the terminal
velocity. However, a same bias is perceptible for the inner veloci-
ties (the predicted curve is below the observational one). This bias
is not present on the NFDG profiles, which perfectly fit the corre-
sponding observational curves, even with their humps and wiggles.
However this perfect fit results from the fact that the NFDG the-
ory imposes on the fractional dimension to vary in "an appropriate
manner” along a galactic radius, in order to obtain a ”good” profile.
Nevertheless, the positive point of this procedure is that NFDG can
thus be predictive for variable dimension. Once again, the κ-model
can correct the mentioned bias by invoking a variable thickness. In
the framework of this model, a volume-effect, i.e. the influence of
the mean volumetric mass density surrounding a given observer, and
estimated at a very large scale, is playing a similar role to that of
the dimension in NFDG. Yet Varieschi underlines that the variable
dimension *D* should be interpreted as the dimension of the matter
distribution of the galactic structure and definitely not at all as the
local space dimension that an observer would measure at a specific
galactic location. In any event, the link between a variable dimension
in NFDG theory and a variable thickness in the κ-model could be
more subtle, and should be reconsidered in more depth. Furthemore,
examining relation (1), we can see that the coefficients κ for the bulge
and the stellar and gaseous disks are different. For the bulge and the
disk, the dimensions are admittedly different, but what about for the
stellar versus gas components ? All these questions deserve further
examination.

It will be very interesting for comparison with the κ-model that the
NFDG theory be applied to a larger sample of galaxies, for instance,
the totality of the galaxies of the SPARC database. A particular
attention must then be paid to the following cases : F563-V2, F568-
1, F568-V1, F571-8, F579-V1, F583-1, NGC 2915, NGC 3992, NGC
5907, NGC 5985, NGC 6674, UGC 00128, UGC 00731, UGC 02259,
UGC 6446, UGC 06667, UGC 07399, UGC 7490, and UGC 8286, for
which both MOND and κ-model substantially differ from the obser-
vational profiles in the innermost regions, while however providing
fairly good estimates in the outskirts of these galaxies (the terminal
velocities).
manner" along a galactic radius, in order to obtain a "good" profile. Nevertheless, the positive point of this procedure is that NFDG can thus be predictive for variable dimension. Once again, the κ-model can correct the mentioned bias by invoking a variable thickness. In the framework of this model, a volume-effect, i.e. the influence of the mean volumetric mass density surrounding a given observer, and estimated at a very large scale, is playing a similar role to that of the dimension in NFDG. Yet Varieschi underlines that the variable dimension *D* should be interpreted as the dimension of the matter distribution of the galactic structure and definitely not at all as the local space dimension that an observer would measure at a specific galactic location. In any event, the link between a variable dimension in NFDG theory and a variable thickness in the κ-model could be more subtle, and should be reconsidered in more depth. Furthermore, examining relation (1), we can see that the coefficients κ for the bulge and the stellar and gaseous disks are different. For the bulge and the disk, the dimensions are admittedly different, but what about for the stellar versus gas components ? All these questions deserve further examination.

It will be very interesting for comparison with the κ-model that the NFDG theory be applied to a larger sample of galaxies, for instance, the totality of the galaxies of the SPARC database. A particular attention must then be paid to the following cases : F563-V2, F568-1, F568-V1, F571-8, F579-V1, F583-1, NGC 2915, NGC 3992, NGC 5907, NGC 5985, NGC 6674, UGC 00128, UGC 00731, UGC 02259, UGC 6446, UGC 06667, UGC 07399, UGC 7490, and UGC 8286, for which both MOND and κ-model substantially differ from the observational profiles in the innermost regions, while however providing fairly good estimates in the outskirts of these galaxies (the terminal velocities).
## 3.3 Refracted Gravity

Along with the NFDG model, another new classical gravity modified theory is the so-called Refracted Gravity (RG) [24, 33, 34]. RG can be reformulated as a scalar-tensor theory [34]. RG mimics DM with a gravitational permittivity (a kind of variable gravitational "constant" G), and that boosts the gravitational field in low-density environments. In RG the link between the volumetric mass density ρ and the gravitational permittivity ϵ is expressed by using the relationship

$$ \epsilon(\rho) = \epsilon_0 + \frac{(1 - \epsilon_0)}{2} \left\{ \tanh\left[\ln\left(\frac{\rho}{\rho_c}\right)\right]^Q + 1 \right\} \quad (6) $$

where $ϵ_0$, $Q$ and $ρ_c$ are three free parameters. The formula (6) is an arbitrary monotonic function of the volumetric mass density with the asymptotic limits $ϵ(ρ) = 1$ for $ρ \gg ρ_c$ and $ϵ(ρ) = ϵ_0$ for $ρ \ll ρ_c$. This formula is the equivalent in RG of the relation (2) in the $κ$-model. This permittivity also shares a very strong analogy with the function $μ$ in MOND [13], or still the function $κ$ in the $κ$-model [16, 26]. However $ϵ$ is supported by three universal parameters, instead of just one, for instance as in MOND ($a_0$). Thus RG seems, at first sight, to be less economic than MOND, but its great interest is that it is now a multiscale version of MOND. In this sense, the objective of RG is very similar to that proposed by the $κ$-model; but with an essential difference : the $κ$-model model uses exclusively internal parameters (i.e. the mean volumetric mass densities of the bulge, stellar and gaseous disk components) and no free external parameters. Then, by contrast in RG the three arbitrary parameters still need to be obtained through a long statistical analysis of the observational data [24]. RG has been applied to both flattened galaxies [24] and a small number of elliptical galaxies [33].
### 3.3 Refracted Gravity

Along with the NFDG model, another new classical gravity modified theory is the so-called Refracted Gravity (RG) [24, 33, 34]. RG can be reformulated as a scalar-tensor theory [34]. RG mimics DM with a gravitational permittivity (a kind of variable gravitational "constant" G), and that boosts the gravitational field in low-density environments. In RG the link between the volumetric mass density ρ and the gravitational permittivity ϵ is expressed by using the relationship

$$ \epsilon(\rho) = \epsilon_0 + \frac{(1 - \epsilon_0)}{2} \left\{ \tanh\left[\ln\left(\frac{\rho}{\rho_c}\right)\right]^Q + 1 \right\} \quad (6) $$

where $ϵ_0$, $Q$ and $ρ_c$ are three free parameters. The formula (6) is an arbitrary monotonic function of the volumetric mass density with the asymptotic limits $ϵ(ρ) = 1$ for $ρ \gg ρ_c$ and $ϵ(ρ) = ϵ_0$ for $ρ \ll ρ_c$. This formula is the equivalent in RG of the relation (2) in the $κ$-model. This permittivity also shares a very strong analogy with the function $μ$ in MOND [13], or still the function $κ$ in the $κ$-model [16, 26]. However $ϵ$ is supported by three universal parameters, instead of just one, for instance as in MOND ($a_0$). Thus RG seems, at first sight, to be less economic than MOND, but its great interest is that it is now a multiscale version of MOND. In this sense, the objective of RG is very similar to that proposed by the $κ$-model; but with an essential difference : the $κ$-model model uses exclusively internal parameters (i.e. the mean volumetric mass densities of the bulge, stellar and gaseous disk components) and no free external parameters. Then, by contrast in RG the three arbitrary parameters still need to be obtained through a long statistical analysis of the observational data [24]. RG has been applied to both flattened galaxies [24] and a small number of elliptical galaxies [33].
The results presented in [24] rely on setting the three free parameters for each individual galaxy. However, the authors show that the variations of these parameters from galaxy to galaxy can, in principle, be ascribed to statistical fluctuations. Then the authors adopt an approximate procedure to estimate a single series of parameters that may properly describe the kinematics of the entire sample of galaxies, They eventually conclude that the gravitational permittivity is indeed a universal function. Unfortunately, a direct, and yet fruitful, comparison between RG and the $\kappa$-model is difficult because the galaxies under consideration are not issued from the same catalog. However, a close examination of the results displayed in [24] leads to the firm conclusion that the fits of the rotation profiles are of similar quality to those produced by MOND and the $\kappa$-model.

## 3.4 Conformal Gravity

Eventually, a comparison with conformal gravity can also be proven worthwhile. In the Conformal Gravity (CFT) [20] two universal parameters are introduced, setting apart the usual observational data, i.e. the luminosity and the $M/L$ ratio, the distance, and the inclination, common to any model. The first parameter ($\gamma*$) is related to the local geometry, while the second parameter ($\gamma_0$) describes the global geometry due to all the other galaxies in the Universe. These two parameters are statistically derived from the observational rotation curves of a chosen sample of 104 galaxies (this sample is limited to the galaxies whose mass density is fittable by a simple exponential thin disk). By comparison, we recall that in the $\kappa$-model the coefficients $\kappa$ are calculated from the mean mass density profiles attached to each galaxy. However, it is very difficult to decide which model is the best. Statistically, MOND, the $\kappa$-model and CFT pro-
The results presented in [24] rely on setting the three free parameters for each individual galaxy. However, the authors show that the variations of these parameters from galaxy to galaxy can, in principle, be ascribed to statistical fluctuations. Then the authors adopt an approximate procedure to estimate a single series of parameters that may properly describe the kinematics of the entire sample of galaxies, They eventually conclude that the gravitational permittivity is indeed a universal function. Unfortunately, a direct, and yet fruitful, comparison between RG and the $\kappa$-model is difficult because the galaxies under consideration are not issued from the same catalog. However, a close examination of the results displayed in [24] leads to the firm conclusion that the fits of the rotation profiles are of similar quality to those produced by MOND and the $\kappa$-model.

## 3.4 Conformal Gravity

Eventually, a comparison with conformal gravity can also be proven worthwhile. In the Conformal Gravity (CFT) [20] two universal parameters are introduced, setting apart the usual observational data, i.e. the luminosity and the $M/L$ ratio, the distance, and the inclination, common to any model. The first parameter ($\gamma*$) is related to the local geometry, while the second parameter ($\gamma_0$) describes the global geometry due to all the other galaxies in the Universe. These two parameters are statistically derived from the observational rotation curves of a chosen sample of 104 galaxies (this sample is limited to the galaxies whose mass density is fittable by a simple exponential thin disk). By comparison, we recall that in the $\kappa$-model the coefficients $\kappa$ are calculated from the mean mass density profiles attached to each galaxy. However, it is very difficult to decide which model is the best. Statistically, MOND, the $\kappa$-model and CFT pro-
vide equivalent results as for the proximity of the theoretical curves
to the observational ones. We can examine the fit through the indi-
vidual cases presented in [20]. For NGC 1003, NGC 3972, NGC 5585,
and UGC 7089, MOND and κ-model fit is better than the CFT fit.
For NGC 2903, UGC 5005,and UGC 5999 the fits are equivalent, For
NGC 4100 the CFT fit is better than the MOND and κ-model fits,
etc. Some cases are favorable to MOND or to the κ-model while in
other cases the CFT is better. At the present time, this situation is
very embarrassing because each author can validly support his own
model against that of others through a judicious choice of the data.
It is for that reason that the models have to be compared on a very
large sample of galaxies such as the SPARC database, and not on a
very small sample of a few galaxies.

4 Conclusion

This paper is a discussion on the capacity of a number of MOND-type
models [16, 23, 24] and a CFT-based model [20], which have been re-
cently proposed, to understand the dynamics of a large variety of
flattened galaxies. Admittedly, these models do not provide very
perfect fits (except may be NFDG that possesses a flexible dimension
associated to the mass distribution), but, nonetheless, they produce
fairly predictive mean rotational curves. It is nonetheless true that
DM can indeed lead to better fits with two or still three [44, 45]
parameters, but unfortunately, these parameters are freely adjusted
to each galaxy. This implies that by starting from any Newtonian
profile, even one strongly empaired by various biases, we can derive a
"very good" fit for any given observational rotation profile<fnref n="8" />. MOND

<footnote n="8">⁸Once again in the framework of the κ-model starting from the Newtonian curves, we generate a predictive profile for the observational one in an univocal manner. The baryonic mean mass density alone is the conductor of the situation.</footnote>
vide equivalent results as for the proximity of the theoretical curves to the observational ones. We can examine the fit through the individual cases presented in [20]. For NGC 1003, NGC 3972, NGC 5585, and UGC 7089, MOND and $\kappa$-model fit is better than the CFT fit. For NGC 2903, UGC 5005, and UGC 5999 the fits are equivalent, For NGC 4100 the CFT fit is better than the MOND and $\kappa$-model fits, etc. Some cases are favorable to MOND or to the $\kappa$-model while in other cases the CFT is better. At the present time, this situation is very embarrassing because each author can validly support his own model against that of others through a judicious choice of the data. It is for that reason that the models have to be compared on a very large sample of galaxies such as the SPARC database, and not on a very small sample of a few galaxies.

# 4 Conclusion

This paper is a discussion on the capacity of a number of MOND-type models [16, 23, 24] and a CFT-based model [20], which have been recently proposed, to understand the dynamics of a large variety of flattened galaxies. Admittedly, these models do not provide very perfect fits (except may be NFDG that possesses a flexible dimension associated to the mass distribution), but, nonetheless, they produce fairly predictive mean rotational curves. It is nonetheless true that DM can indeed lead to better fits with two or still three [44, 45] parameters, but unfortunately, these parameters are freely adjusted to each galaxy. This implies that by starting from any Newtonian profile, even one strongly empaired by various biases, we can derive a "very good" fit for any given observational rotation profile$^8$. MOND

<footnote>$^8$Once again in the framework of the $\kappa$-model starting from the Newtonian curves, we generate a predictive profile for the observational one in an univocal manner. The baryonic mean mass density alone is the conductor of the situation.</footnote>
and the κ-model are at least falsifiable and upgradable, while DM definitively not. For a physicist, the choice is quickly made. With no confirmation by experimental methods, DM unfortunately has very limited scientific significance. Obviously this conclusion would drastically change if, one day, we discover the signature of DM in the laboratory. We can always expect it over the next few years. Even though obviously the κ-model is not a definitive solution, at least it shows that the baryonic mean mass density could play an unexpected role in the determination of the galactic rotational velocities and that both are strongly correlated. If this model is on the right track, then the rotational velocities alone could allow us to directly determine the baryonic mean mass density (and not indirectly from the brightness measurements) and vice versa in a self-consistent manner. In this case, the delicate step, i.e. brightness → mean mass density would be short-circuited. After an analysis of spiral galaxies and galaxy clusters, the work is far from finished. The κ-model has to be still applied to the elliptical galaxies, to the globular clusters, to the formation and stability of primordial galaxies, and eventually to CMB/cosmology. Let us also note the very captivating open debate concerning the wide binary stars ([54] versus [55]). The κ-model obviously predicts a very weak κ-effect in the immediate vicinity of the Sun, i.e. the motion of the wide binaries is predicted to be quasi-keplerian in this region. Much work remains to be done. It would be interesting to concomitantly perform the same studies, on the same collection of galaxies, with other models, such as the Newtonian Fractional-Dimension Gravity [29, 30, 31, 32], the Refracted Gravity [33, 34] and also the CFT model [20].
and the κ-model are at least falsifiable and upgradable, while DM definitively not. For a physicist, the choice is quickly made. With no confirmation by experimental methods, DM unfortunately has very limited scientific significance. Obviously this conclusion would drastically change if, one day, we discover the signature of DM in the laboratory. We can always expect it over the next few years. Even though obviously the κ-model is not a definitive solution, at least it shows that the baryonic mean mass density could play an unexpected role in the determination of the galactic rotational velocities and that both are strongly correlated. If this model is on the right track, then the rotational velocities alone could allow us to directly determine the baryonic mean mass density (and not indirectly from the brightness measurements) and vice versa in a self-consistent manner. In this case, the delicate step, i.e. brightness → mean mass density would be short-circuited. After an analysis of spiral galaxies and galaxy clusters, the work is far from finished. The κ-model has to be still applied to the elliptical galaxies, to the globular clusters, to the formation and stability of primordial galaxies, and eventually to CMB/cosmology. Let us also note the very captivating open debate concerning the wide binary stars ([54] versus [55]). The κ-model obviously predicts a very weak κ-effect in the immediate vicinity of the Sun, i.e. the motion of the wide binaries is predicted to be quasi-keplerian in this region. Much work remains to be done. It would be interesting to concomitantly perform the same studies, on the same collection of galaxies, with other models, such as the Newtonian Fractional-Dimension Gravity [29, 30, 31, 32], the Refracted Gravity [33, 34] and also the CFT model [20].
# Appendices

## A Section Rotation curve fits results

<figure><img src="image_3.png" /></figure>

Figure A Rotation curves of the SPARC galaxies. The green line is predicted by MOND, the amber line is predicted by the $\kappa$-model, the red dashed line represents the stars, the green dotted line represents the gas, the blue line represents the sum of all baryonic components (stars + gas). The observed velocities are shown as a series of filled black circles.
# Appendices

## A Section Rotation curve fits results

<figure><img src="image_4.png" /></figure>

Figure A Rotation curves of the SPARC galaxies. The green line is predicted by MOND, the amber line is predicted by the $\kappa$-model, the red dashed line represents the stars, the green dotted line represents the gas, the blue line represents the sum of all baryonic components (stars + gas). The observed velocities are shown as a series of filled black circles.
<figure><img src="image_4.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_5.png" /></figure>

NGC0891 (Q = 1)

<figure><img src="image_6.png" /></figure>

NGC1003 (Q=1)

<figure><img src="image_7.png" /></figure>

NGC1090 (Q = 1)

<figure><img src="image_8.png" /></figure>

NGC1705 (Q = 3)

<figure><img src="image_9.png" /></figure>

NGC2366 (Q = 3)

<figure><img src="image_10.png" /></figure>

NGC2403 (Q = 1)

<figure><img src="image_11.png" /></figure>

NGC2683 (Q = 2)

<figure><img src="image_12.png" /></figure>

NGC2841 (Q = 1)

<figure><img src="image_13.png" /></figure>

NGC2903 (Q = 1)

<figure><img src="image_14.png" /></figure>

NGC2915 (Q = 2)

<figure><img src="image_15.png" /></figure>

NGC2955 (Q = 1)

<figure><img src="image_16.png" /></figure>

NGC2976 (Q = 2)

<figure><img src="image_17.png" /></figure>

NGC2998 (Q = 1)

<figure><img src="image_18.png" /></figure>

NGC3109 (Q = 1)

<figure><img src="image_19.png" /></figure>

NGC3198 (Q = 1)

<figure><img src="image_20.png" /></figure>

NGC3521 (Q = 1)

<figure><img src="image_21.png" /></figure>

NGC3726 (Q = 2)

<figure><img src="image_22.png" /></figure>

NGC3741 (Q = 1)

<caption>Figure A : Continued rotation profiles</caption>
<figure><img src="image_6.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure A : Continued rotation profiles</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Figure A: Continued rotation profiles</figcaption></figure>

**Acknowledgments:** The author sincerely thanks the three referees whose suggestions and corrections improved this paper.

**Data availability statement:** The author confirms that the data supporting the findings of this study are available within the article and the reference list.

**Conflicts of Interest:** The author declares no conflict of interest.

**References**

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<figure><img src="image_13.png" /><figcaption>Figure A: Continued rotation profiles</figcaption></figure>

**Acknowledgments:** The author sincerely thanks the three referees whose suggestions and corrections improved this paper.

**Data availability statement:** The author confirms that the data supporting the findings of this study are available within the article and the reference list.

**Conflicts of Interest:** The author declares no conflict of interest.

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