###### Abstract

General relativity and its Newtonian weak field limit are not sufficient to
explain the observed phenomenology in the Universe, from the formation of
large-scale structures to the dynamics of galaxies, with the only presence of
baryonic matter. The most investigated cosmological model, the $\Lambda$CDM,
accounts for the majority of observations by introducing two dark components,
dark energy and dark matter, which represent $\sim$95% of the mass-energy
budget of the Universe. Nevertheless, the $\Lambda$CDM model faces important
challenges on the scale of galaxies. For example, some very tight relations
between the properties of dark and baryonic matters in disk galaxies, such as
the baryonic Tully–Fisher relation (BTFR), the mass discrepancy–acceleration
relation (MDAR), and the radial acceleration relation (RAR), which see the
emergence of the acceleration scale $a_{0}\simeq 1.2\times 10^{-10}$ m s-2,
cannot be intuitively explained by the CDM paradigm, where cosmic structures
form through a stochastic merging process. An even more outstanding
coincidence is due to the fact that the acceleration scale $a_{0}$, emerging
from galaxy dynamics, also seems to be related to the cosmological constant
$\Lambda$. Another challenge is provided by dwarf galaxies, which are darker
than what is expected in their innermost regions. These pieces of evidence can
be more naturally explained, or sometimes even predicted, by modified theories
of gravity, that do not introduce any dark fluid. I illustrate possible
solutions to these problems with the modified theory of gravity MOND, which
departs from Newtonian gravity for accelerations smaller than $a_{0}$, and
with Refracted Gravity, a novel classical theory of gravity introduced in
2016, where the modification of the law of gravity is instead regulated by a
density scale.

###### keywords:

modified gravity; dark matter; galaxy dynamics; acceleration scale; scaling
relations; LSB galaxies; globular clusters

9 1 56 Academic Editors: Ivan De Martino and Mahmood Roshan 30 November 2022
11 January 2023 12 January 2023 16 January 2023
https://doi.org/10.3390/universe9010056 Dark Coincidences: Small-Scale
Solutions with Refracted Gravity and MOND Dark Coincidences: Small-Scale
Solutions with Refracted Gravity and MOND Valentina Cesare Valentina Cesare
Cesare, V.

##  1 Introduction

If we assume the validity of standard gravity, that is, General Relativity
(GR), and the presence of ordinary (baryonic) matter alone, we cannot explain
the observations from the largest to the smallest scales of the Universe. The
cosmic microwave background (CMB) radiation Planck Collaboration et al.
(2020), the dynamics and the large-scale distribution of the cosmic structure
Davis et al. (1985); Springel et al. (2006), the gravitational lensing, with
particular reference to the Bullet Cluster Markevitch (2006); Clowe et al.
(2006); Paraficz et al. (2016), the dynamics of galaxy clusters Zwicky (1933),
and the flat trend of the rotation curves of disk galaxies Rubin and Ford Jr.
(1970); Bosma (1978); Sanders (1990) all show a mass discrepancy of
$\sim$80–90%. Moreover, the Hubble diagram of Ia Supernovae Kirshner (1996);
Peebles and Ratra (2003) proved the expansion of the Universe to be
accelerated, which is not what we expect from the attractive nature of the
gravity force. This probably represents the most important open question in
modern cosmology and it launches several lines of research.

With the only presence of baryonic matter, GR cannot justify $\sim$95% of the
components of the Universe. The most investigated solution is provided by the
$\Lambda$ cold dark matter ($\Lambda$CDM) cosmological model, which assumes
the validity of GR and introduces two dark components besides baryonic matter
to explain this missing $\sim$95%.

The first dark component is an exotic fluid with negative pressure called dark
energy (DE), which causes the accelerated expansion of the Universe and
explains $\sim$70% of its mass-energy budget. The DE can be identified with
the cosmological constant $\Lambda$, present in the Einstein equations. The
second dark component is an invisible form of matter with a nonbaryonic
nature, the dark matter (DM), which interacts with baryonic matter only
gravitationally and explains $\sim$25% of the mass-energy budget of the
Universe. To account for the observed bottom-up formation scenario of cosmic
structures, DM has also to be “cold”, i.e., nonrelativistic at the epoch of
decoupling from radiation.

The $\Lambda$CDM model can account for almost all the mentioned pieces of
evidence. Yet, it presents some issues, both on cosmological and small scales.
Three of the most puzzling challenges on large scales are the cosmological
constant problem Weinberg (1989); Luo (2014), the coincidence problem Velten
et al. (2014), and the tensions between the cosmological parameters inferred
from the late and the early Universe Planck Collaboration et al. (2020);
Fleury et al. (2013); Douspis et al. (2019). However, the most important
problems of the $\Lambda$CDM paradigm are observed on the scale of galaxies.
The cusp/core, missing satellites, too-big-to-fail, and planes of satellite
galaxies problems are among the most investigated (see Del Popolo and Le
Delliou (2017); de Martino et al. (2020) for a review).

Among the small-scale problems, particularly relevant is also the presence of
very tight scaling relations between the properties of dark and baryonic
matters in galaxies. These are the baryonic Tully–Fisher relation (BTFR)
McGaugh et al. (2000), the mass discrepancy–acceleration relation (MDAR)
McGaugh (2004), and the radial acceleration relation (RAR) McGaugh et al.
(2016), whose scatter is consistent with the observational errors. In a
$\Lambda$CDM context, it is not trivial to explain why quantities related to
DM, which represents $\sim$90% of the galaxy mass, are so tightly regulated by
baryonic matter, which only represents $\sim$10% of the galaxy mass. Moreover,
these pieces of evidence are not intuitively explained in a Universe where the
formation of structures occurs hierarchically through a stochastic merging
process dominated by DM.

An even more noteworthy coincidence, which might be difficult to interpret in
$\Lambda$CDM, is the emergence of an acceleration scale, $a_{0}\simeq
1.2\times 10^{-10}$ m s-2, from the three relations. This acceleration scale,
expressed in natural units, can also be written as $a_{0}\sim
H_{0}\sim{\Lambda}^{1/2}$, which is even more puzzling since it seems to
suggest that DM and DE phenomenologies are regulated by the same acceleration
scale Famaey and McGaugh (2012). This is even less intuitive in $\Lambda$CDM.

At last, another issue on the galaxy scale is provided by the complex dynamics
of some categories of systems. The relative contributions of the stellar disk,
made of baryonic matter, and of the DM halo to the overall rotation curves of
disk galaxies are degenerate to each other, a problem known as the “disk-halo
conspiracy” van Albada et al. (1985); van Albada and Sancisi (1986). Many
pieces of evidence suggest the “maximum-disk hypothesis” van Albada and
Sancisi (1986); Sackett (1997); Courteau and Rix (1999); Bissantz and Gerhard
(2002); Sellwood and Debattista (2014); McGaugh and Schombert (2015), where
the disk maximally contributes to the inner part of the rotation curve.
However, this assumption is only valid for high surface brightness (HSB)
galaxies, whose rotation curves are steeply rising in their central regions
and are described with a cuspy DM density profile. Instead, dwarf, low surface
brightness (LSB), and dwarf spheroidal (dSph) galaxies are DM-dominated even
in their innermost regions and, thus, their rotation curves, slowly rising in
their central part and modelled with a cored DM density profile, cannot be
described by a maximum-disk model Strigari et al. (2008); Di Paolo et al.
(2019). These categories of galaxies represent one of the best pieces of
evidence of the mass-discrepancy problem on the scale of galaxies since they
are among the darkest galaxies in the Universe, e.g., Mateo (1998), and they
give rise to the cusp/core problem. Intriguingly, there are systems with
similar baryonic masses but completely different dynamics. They are globular
clusters (GCs), which seem to be nearly DM-free Baumgardt et al. (2005); Jordi
et al. (2009); Baumgardt et al. (2009); Sollima and Nipoti (2010); Ibata et
al. (2011a, b); Frank et al. (2012).

Some of these problems are more intuitively explained, rather than with the
$\Lambda$CDM paradigm, with a modification of the law of gravity with respect
to standard gravity, without introducing any dark component. According to the
theory modified Newtonian dynamics (MOND) Milgrom (1983a, b, c), the law of
gravity departs from the Newtonian one when the acceleration goes below
$a_{0}$. This theory not only reproduces but actually predicted some pieces of
evidence on the galaxy scale, such as the three mentioned scaling relations
and the difference between the dynamics of HSB and LSB galaxies.

Another theory of modified gravity, more recently introduced, is refracted
gravity (RG) Matsakos and Diaferio (2016), a classical theory where the
modification of the law of gravity is regulated by the value of the local mass
density, rather than of the acceleration. RG has shown some encouraging
results in describing the dynamics of galaxies of different shapes, such as
disk Cesare et al. (2020) and elliptical E0 galaxies with nearly spherical
morphology Cesare et al. (2022). In RG, the shape of the gravitational field
lines depends on the morphology of the system: they are refracted towards the
mid-plane of flattened systems and they remain radial for spherical systems,
which might intuitively explain the different dynamics of dwarf galaxies (more
flattened) and globular clusters (nearly spherical). Moreover, the
relativistic formulation of RG Sanna et al. (2021) belongs to the class of
scalar-tensor theories and introduces a single scalar field that explains the
phenomenology of both DM and DE. The covariant RG provides a natural
explanation for the $a_{0}\sim\Lambda^{1/2}$ relation, suggesting a
unification of the two dark sectors.

In this review, we present more intuitive explanations, with the theories MOND
and RG, for the three scaling relations, for the emergence of the same
acceleration scale from these relations and from the DE sector, and for the
different dynamics of LSB, dwarf, and dSph galaxies and of GCs. The outline of
the paper develops as follows. In Section 2, we detail the BTFR, the MDAR, and
the RAR (Section 2.1) and we explain the different interpretations of these
relations in a $\Lambda$CDM framework (Section 2.2). Section 3 describes the
formulation of MOND and RG theories (Section 3.1) and the interpretation of
the three scaling relations with these theories (Section 3.2). Section 4
illustrates how the $a_{0}$ acceleration scale and the $a_{0}$–$\Lambda$
relation emerge in Newtonian, MOND, and RG theories. Section 5 describes the
different dynamical properties of LSB, dwarf, and dSph galaxies and of GCs and
how they can be modelled or interpreted in Newtonian, MOND, and RG gravities.
Section 6 concludes the paper.

##  2 The Baryonic Scaling Relations

The mass discrepancy on the galaxy scale can be neatly quantified by three
relations that tightly correlate the properties of dark and baryonic matters
in galaxies: the BTFR, the MDAR, and the RAR.

###  2.1 Description of the Three Relations

The BTFR McGaugh et al. (2000) (Figure 1) correlates the total baryonic mass,
$M_{\rm bar}$, and the asymptotic value of the flat part of the rotation curve
of galaxies, $V_{\rm f}$, according to the relation, e.g., McGaugh (2012):

$M_{\rm bar}=AV_{\rm f}^{b},$ (1)

where the normalisation $A$ and the slope $b$ are free parameters. In the
BTFR, each point corresponds to one galaxy. McGaugh McGaugh (2012) fits the
($V_{\rm f}$,$M_{\rm bar}$) data points from 47 gas-rich galaxies with
Equation (1), adopting different techniques. The normalisations resulting from
these fits are in agreement with each other and the slopes are consistent with
4. In the BTFR, the acceleration scale $a_{0}$ emerges from its normalisation.
Setting the slope to 4, McGaugh McGaugh (2012) found the normalisation
$A=(47\pm 6)$ M⊙ km-4 s4, comparable to the expression $(Ga_{0})^{-1}$. For
mass-to-light ratios in the 3.6 $\upmu$m band $M/L_{[3.6]}\gtrsim 0.5$ M⊙/L⊙,
the BTFR intrinsic scatter is minimised to $\sim$0.10 dex Lelli et al. (2016).
The residuals of the measured BTFR from the model (Equation (1)) do not
correlate with galaxy properties, such as the radius or the surface brightness
Desmond and Wechsler (2015).

Figure 1: The baryonic Tully–Fisher relation (BTFR) built for a subsample of
123 galaxies from the SPARC catalogue Lelli et al. (2016). $M_{\rm b}$ is the
total galaxy baryonic mass, accounting for the stars and the gas, and $V_{\rm
f}$ is the mean circular velocity along the flat region of the rotation curve.
The black solid line represents the linear fit to the data with the relation
$\log(M_{\rm b}/{\rm M}_{\odot})=s\log(V_{\rm f}/\text{km~{}s}^{-1})+I$.
$\sigma_{\perp}$ denotes the intrinsic scatter of the BTFR, expressed in dex.
The galaxies are colour-coded according to the gas fraction $F_{\rm g}$ to the
total baryonic mass. The figure is re-adapted from Figure 2 in Lelli et al.
(2019).

Two local, rather than global, scaling relations related to the BTFR are the
MDAR and the RAR Lelli et al. (2017). The MDAR McGaugh (2004) anti-correlates,
at each distance $R$ from the galaxy centre, the Newtonian acceleration
generated by the baryons distribution, $g_{\rm bar}$, and the squared ratio
$(V/V_{\rm bar})^{2}$, where $V$ and $V_{\rm bar}$ are the total and the
baryons-only velocities (Figure 2, top panel). If we assume spherical
symmetry, the ratio $(V/V_{\rm bar})^{2}$ coincides with the mass discrepancy,
$M/M_{\rm bar}$, where $M$ and $M_{\rm bar}$ are the masses of the entire
galaxy and of its baryonic component. The MDAR can be modelled by the
following relation, e.g., Mayer et al. (2023):

$\frac{M}{M_{\rm bar}}(R)=1+\frac{a_{0}}{g_{\rm bar}(R)}.$ (2)

As long as the acceleration $g_{\rm bar}$ is $\gtrsim a_{0}$, the mass
discrepancy maintains around 1. Instead, when $g_{\rm bar}$ goes below
$a_{0}$, the mass discrepancy starts to increase. The intrinsic scatter in
both the MDAR and the BTFR is minimised by the same mass-to-light ratio,
consistent with the estimates from stellar population synthesis (SPS) models
McGaugh (2004). In fact, the MDAR can also be expressed as a relation between
the mass discrepancy and $g_{\rm obs}$ (Figure 2, bottom panel), that is, the
total observed acceleration, which presents a slightly larger scatter than the
relation with $g_{\rm bar}$ Famaey and McGaugh (2012). This relation can be
reproduced by replacing $g_{\rm bar}$ with $g_{\rm obs}$ in Equation (2).

Figure 2: The mass discrepancy–acceleration relation (MDAR) represented as the
squared ratio between the total ($V$) and the baryonic ($V_{\rm b}$)
velocities as a function of the Newtonian acceleration due to baryonic matter,
$g_{N}$ (top panel) and the total acceleration $a$ (bottom panel). We can
clearly see that the mass discrepancy is observed from accelerations smaller
than $a_{0}\simeq 10^{-10}$ m s-2 (indicated in the figure as a vertical blue
dashed line, for reference). The $(V/V_{\rm b})^{2}-a$ relation shows a
slightly larger scatter than the $(V/V_{\rm b})^{2}-g_{N}$ relation. The black
dots represent hundreds of individual resolved data points belonging to the
rotation curves of about one hundred spiral galaxies. The black solid line
$(V/V_{\rm b})^{2}=1$, showing the no mass discrepancy case, is illustrated as
a reference in both panels. The figure is re-adapted from Figure 10 in Famaey
and McGaugh (2012).

A slightly different perspective with respect to the MDAR is provided by the
RAR Milgrom (2016) (Figure 3), which links the centripetal acceleration
inferred from the observed rotation curve $V(R)$, $g_{\rm obs}=V^{2}/R$, with
the Newtonian acceleration due to the baryons alone, $g_{\rm bar}$ McGaugh et
al. (2016). The data in the ($g_{\rm bar}$,$g_{\rm obs}$) plane of the RAR
present an even tighter relation with respect to the data in the ($g_{\rm
bar}$,$(V/V_{\rm bar})^{2}$) plane of the MDAR. Moreover, the ($g_{\rm
bar}$,$g_{\rm obs}$) plane presents an advantage compared to the ($g_{\rm
bar}$,$(V/V_{\rm bar})^{2}$) plane, since the $g_{\rm bar}$ and $g_{\rm obs}$
quantities and their corresponding uncertainties are completely independent
from each other Li et al. (2018).

McGaugh and coauthors McGaugh et al. (2016) fitted the ($g_{\rm bar}$,$g_{\rm
obs}$) data from 153 edge-on disk galaxies belonging to Spitzer photometry and
accurate rotation curves (SPARC) catalogue Lelli et al. (2016) with the
relation:

$g_{\rm obs}(R)=\frac{g_{\rm bar}(R)}{1-\exp\left(-\sqrt{\frac{g_{\rm
bar}(R)}{g_{\dagger}}}\right)},$ (3)

where the only free parameter $g_{\dagger}=(1.20\pm 0.02\pm 0.24)\times
10^{-10}$ m s-2 is 1$\sigma$ consistent with $a_{0}$. The errors of
$0.02\times 10^{-10}$ m s-2 and of $0.24\times 10^{-10}$ m s-2 represent the
random and the systematic contributions to the uncertainty, respectively. In
particular, the random error represents the 1$\sigma$ confidence interval and
the systematic error represents the 20% normalisation uncertainty due to the
fact that the mass-to-light ratios of the disks and the bulges of the galaxies
are kept fixed across the SPARC sample.

Assuming a disk and a bulge mass-to-light ratio $M/L_{[3.6]}$ of 0.5 and 0.7
M⊙/L⊙, respectively, which are reasonable values in the 3.6 $\upmu$m band, the
RAR of SPARC galaxies is retrieved with an observed scatter of 0.13 dex
McGaugh et al. (2016). This value closely coincides with the scatter of 0.12
dex due to the observational errors of the measured rotation curves,
distances, and galaxy inclinations, and to the possible variation of the mass-
to-light ratio among galaxies, which leaves little room for intrinsic scatter
McGaugh et al. (2016); Lelli et al. (2017).

Li and collaborators Li et al. (2018) wanted to test whether the RAR was
followed by the individual galaxies in the SPARC sample. They fitted Equation
(3) to the observed RAR of the individual SPARC galaxies, finding a RAR with a
smaller scatter of [0.054–0.057] dex, both by fixing $g_{\dagger}$ to $a_{0}$
and by leaving it free to vary. Since, differently from McGaugh et al. (2016),
they estimated the mass-to-light ratios and marginalised over the errors on
the galaxy distances and inclinations from the RAR of the single SPARC
galaxies, the obtained scatter might be assimilated to the intrinsic one,
rather than to the observed one.

Figure 3: The radial acceleration relation (RAR) built from Spitzer photometry
and accurate rotation curves (SPARC) galaxies. For each galaxy, a mass-to-
light ratio in the 3.6 $\upmu$m band for the disk and the bulge equal to 0.5
and 0.7 M⊙/L⊙, respectively, is adopted. The black solid line represents the
fit to the data with Equation (3) and the black dashed line is the $g_{\rm
obs}=g_{\rm bar}$ relation, for reference. The only best-fit parameter,
$g_{\dagger}$, is highlighted in the top-left corner of the panel and it is
1$\sigma$ consistent with the acceleration scale $a_{0}$. The histogram and
the bottom panel show the distribution of the residuals of the observed RAR
from Equation (3), and its standard deviation $\sigma$, which quantifies the
RAR observed scatter, is shown in the top-left corner of the panel. The figure
is re-adapted from Figure 2 in Ghari et al. (2019).

The three relations cover a baryonic mass range of six orders of magnitude,
from $M_{\rm bar}\sim 10^{12}$ $M_{\odot}$, correspondent to the most massive
HSB spiral galaxies, to $M_{\rm bar}\sim 10^{6}$ $M_{\odot}$, correspondent to
the dwarf and the LSB galaxies. However, the observed scatter of the RAR might
increase from 0.13 dex to 0.24 dex for small $g_{\rm bar}$ due to dwarf and
LSB galaxies with slowly-rising rotation curves Santos-Santos et al. (2020),
which requires further investigation.

The works of McGaugh et al. and of Kroupa et al. McGaugh et al. (2018); Kroupa
et al. (2018) might demonstrate a fundamental origin for the acceleration
scale $a_{0}$, and, thus, for the RAR, since they show a consistency of
$g_{\dagger}$ among different galaxies. The work of Li et al. Li et al. (2018)
might suggest the same result, since fitting Equation (3) to the RAR of the
individual SPARC galaxies by leaving $g_{\dagger}$ free to vary does neither
improve the $\chi^{2}$ nor reduce the obtained scatter with respect to the
fits in which $g_{\dagger}$ is fixed to $a_{0}=1.2\times 10^{-10}$ m s-2.
However, several works question this result. For example, Rodrigues et al.
Rodrigues et al. (2018) estimated the $g_{\dagger}$ parameter with Bayesian
inference from 193 individual galaxies from SPARC Lelli et al. (2016) and
THINGS Walter et al. (2008); de Blok et al. (2008) samples, rejecting the
consistency of $g_{\dagger}$ among the galaxies at the 10$\sigma$ level. The
discrepancy reduces but does not disappear by removing some approximations,
becoming equal to 5$\sigma$ Marra et al. (2020). The work of Zhou et al. Zhou
et al. (2020) leads to a similar result. If this is the case, it would
represent an important challenge for MOND (see Section 3.2). A debate about
the existence or the absence of a universal $a_{0}$ is presented in Rodrigues
et al. (2018). Another study sees the emergence of an acceleration scale
consistent with $a_{0}$, besides from the BTFR, built from rotationally-
supported galaxies, from the baryonic Faber–Jackson relation built from
pressure-supported galaxies, such as elliptical galaxies and GCs, and from the
baryonic Faber–Jackson relation built from galaxy clusters (Edmonds et al.
(2020) and references). The fact that the same acceleration scale also emerges
from the dynamics of galaxy clusters, which are different from galaxies both
in terms of scale and evolutionary histories, might pose an even more severe
issue for $\Lambda$CDM.

###  2.2 Interpretation of the Three Relations in Newtonian Gravity

The $\Lambda$CDM paradigm does not provide a natural explanation for these
three relations. Indeed, these relations correlate quantities related to the
dynamics of a galaxy to its baryonic content, which is counterintuitive in a
universe where the dynamics of structures is dominated by DM. The pieces of
evidence that are most difficult to interpret are the small scatter, in
agreement with observational uncertainties, and the lack of correlations of
the residuals between the observed and the modelled relations with the galaxy
properties. This phenomenology indicates a quite precise fine-tuning between
dark and baryonic matter in galaxies. Moreover, the MDAR and the RAR pose a
more serious issue for $\Lambda$CDM compared to the BTFR due to their local,
rather than global, nature.

The three relations resulting from semi-analytical models of galaxy formation
and DM-only cosmological simulations in $\Lambda$CDM do not completely agree
with the observational data. The predicted minimum scatter of the BTFR is
larger than the observed one (0.17 dex vs 0.10 dex) Dutton (2012); Di Cintio
and Lelli (2016) and the simulated BTFR slope $b$ is equal to 3, 8$\sigma$
discrepant from the value $b=3.98\pm 0.12$, fitted by McGaugh and coauthors
McGaugh et al. (2000) from the observed BTFR. However, some successful
explanations for the three relations remaining within the $\Lambda$CDM
paradigm exist, even if with some issues.

Ludlow et al. Ludlow et al. (2017) demonstrated that a set of galaxies
resulting from the EAGLE suite of $\Lambda$CDM hydrodynamical simulations, run
with the same initial conditions but with different stellar and active
galactic nuclei (AGN) feedback processes for the baryonic component, follow a
RAR-like relation, independently from the considered feedback. Different
feedback models make simulated galaxies moving along the RAR and not
perpendicular to it, which produces a small RAR scatter $\gtrsim 0.08$ dex.
However, the $g_{\dagger}$ parameter fitted with Equation (3) from the
galaxies simulated in Ludlow et al. (2017) results in a 70$\sigma$
inconsistency with $a_{0}=1.2\times 10^{-10}$ m s-2. Moreover, measurement
errors are not included in the simulated galaxies and the obtained result
about the scatter should be further investigated Li et al. (2018). The same
consideration can be drawn for the scatter of 0.06 dex of the RAR obtained
from the 32 galaxies resulting from the MUGS2 “zoom-in” hydrodynamic
simulations in $\Lambda$CDM Keller and Wadsley (2017). On the other hand,
Stone and Courteau Stone and Courteau (2019) found that the intrinsic scatter
of the stellar RAR, where only stellar mass is considered to compute the
baryonic acceleration, is of $(0.11\pm 0.02)$ dex, in agreement, despite being
slightly larger, with $\Lambda$CDM predictions. They obtained this result from
PROBES, a catalogue made of more than 2500 spiral galaxies taken from six deep
imaging and spectroscopic surveys. Yet, neglecting gas masses might affect the
obtained result. In fact, the question of the RAR raises further complications
since for small $g_{\rm bar}$, namely for dwarf and LSB galaxies, its scatter
increases Santos-Santos et al. (2020) (see Section 2.1), and the RAR built
from some galaxy samples different from SPARC shows some correlations between
its residuals and certain galaxy properties Di Paolo et al. (2019); Cesare et
al. (2020).

Some hydrodynamical simulations, which are simulations that include the
presence of baryons, can reproduce the slope of the BTFR Ferrero et al.
(2017); Ferrero and Abadi (2017). Yet, the small scatter of the BTFR can only
be explained with a quite precise balance between star formation efficiency
and stellar feedback processes McGaugh (2012); Lelli et al. (2016).

With the semi-empirical model proposed by Di Cintio and Lelli Di Cintio and
Lelli (2016), the shape and the scatter of the MDAR are reproduced but this
does not simultaneously account for the small scatter of $\sim$0.10 dex of the
BTFR, which instead results in 0.17 dex.

Mayer et al. Mayer et al. (2023) used the Magneticum hydrodynamical
simulation, which provides a large and representative sample of galaxies
covering a large range of masses and a variety of morphologies, from $z\sim 0$
to $z\sim 3$, to see whether the baryonic scaling relations (BTFR, MDAR, and
RAR) were reproduced in a $\Lambda$CDM context. The resulting BTFR, built for
galaxies at different redshifts in the range of $0.1<z<2.3$, has a slope more
consistent with 3, rather than with 4, as in the observed BTFR. The MDAR and
the RAR built from simulated Magneticum galaxies at redshift $z\sim 0.1$
reproduce the observed relations (Equations (2) and (3)) with a fitted
acceleration scale consistent with $a_{0}=1.2\times 10^{-10}$ m s-2 and a
scatter in agreement with the observations from the SPARC sample. However, the
simulated MDAR and RAR show a positive correlation between the total baryonic
mass and the mass discrepancy which is not observed in SPARC data. Other
$\Lambda$CDM hydrodynamical simulations of single galaxies might indicate that
the baryonic scaling relations naturally emerge in this framework as well
Navarro et al. (2017); Dutton et al. (2019).

It is important to note that many small-scale problems of $\Lambda$CDM, such
as too large bulges in disk galaxies, the cusp/core problem, and the too-big-
to-fail problem, have been, at least partially, solved by introducing baryonic
feedback mechanisms, such as outflows due to AGN and Supernovae, and,
therefore, it is not so unlikely to suppose that galaxy dynamics is regulated
by baryonic physics Mayer et al. (2023).

##  3 MOND and RG

A totally different scenario for the three scaling relations is provided by
theories of modified gravity that do not imply the presence of DM. Without DM,
the fine-tuning issue between the properties of dark and baryonic matters
disappears. A modified gravity theory that not only describes but even
predicted these scaling relations is MOND. Another theory of modified gravity
that seems to account for these relations is RG. In the following subsections,
I present the formulation of these two theories of gravity and how they
reproduce the mentioned scaling relations.

###  3.1 Summary of Theories Formulation

####  3.1.1 MOND

In 1983, Milgrom Milgrom (1983a, b, c) formulated MOND, a theory of gravity
that mimics the effect of DM with a boost of the gravitational field compared
to the Newtonian one in low-acceleration environments. MOND is a general
paradigm that assumes spacetime scale-invariance when the acceleration is
$a\ll a_{0}$. Specifically, the acceleration $a$ presents the following
asymptotic values:

$a\simeq\begin{cases}g_{\rm N},&a\gg a_{0}\\\ \sqrt{g_{\rm N}a_{0}},&a\ll
a_{0},\end{cases}$ (4)

where $g_{\rm N}$ is the Newtonian acceleration.

The MOND paradigm is obtained by modifying either gravity, e.g., Bekenstein
and Milgrom (1984) or inertia Milgrom (1994), where the modified-inertia
version Milgrom (1994) was less developed. In the first nonrelativistic
modified gravity version of MOND Bekenstein and Milgrom (1984), the following
Poisson equation was defined:

$\nabla\cdot\left[\mu\left(\frac{\left|\nabla\phi\right|}{a_{0}}\right)\nabla\phi\right]=4\pi
G\rho,$ (5)

where the interpolating function $\mu$, monotonic in its argument, has these
two asymptotic behaviours:

$\mu\left(\frac{\left|\nabla\phi\right|}{a_{0}}\right)=\begin{cases}1,&a\gg
a_{0}\\\ \frac{\left|\nabla\phi\right|}{a_{0}},&a\ll a_{0}.\end{cases}$ (6)

For $a\gg a_{0}$, the Newtonian Poisson equation is retrieved:

$\nabla^{2}\phi_{\rm N}=4\pi G\rho,$ (7)

and for $a\ll a_{0}$, we observe a boost of the gravitational field over the
Newtonian one. In this regime, whereas the Newtonian field is $\propto
R^{-2}$, the MOND field is $\propto R^{-1}$, deviating from the Newtonian
inverse square law and reproducing the flat trend of the rotation curves
without the presence of DM.

The MOND paradigm was also defined with other modified gravity formulations,
such as QUMOND Milgrom (2010), where the MONDian behaviour of the
gravitational field is obtained with the Poisson equation:

$\nabla^{2}\phi=\nabla\cdot\left[\nu\left(\frac{\left|\nabla\phi_{\rm
N}\right|}{a_{0}}\right)\nabla\phi_{\rm N}\right],$ (8)

where $\phi_{\rm N}$ is Newtonian gravitational potential. A possible form of
the interpolating function $\nu$ is given by the “simple $\nu$-function”
(Equation (50) with $n=1$ in Famaey and McGaugh (2012)):

$\nu(y)=\frac{1}{2}\left(1+\sqrt{1+\frac{4}{y}}\right),$ (9)

with $y$ being equal to $\frac{\left|\nabla\phi_{\rm N}\right|}{a_{0}}$.

####  3.1.2 Refracted Gravity

RG is a novel theory of modified gravity inspired by the behaviour of
electrodynamics in matter that does not resort to DM Matsakos and Diaferio
(2016). RG was formulated in a nonrelativistic way by Matsakos and Diaferio
(2016) and its gravitational potential $\phi$ obeys the modified Poisson
equation:

$\nabla\cdot\left[\epsilon(\rho)\nabla\phi\right]=4\pi G\rho,$ (10)

where the gravitational permittivity $\epsilon(\rho)$ mimics the DM
phenomenology.

Whereas in MOND the modification of the law of gravity is regulated by an
acceleration scale, in RG it is regulated by a mass density scale. The
gravitational permittivity $\epsilon(\rho)$ is a monotonic increasing function
of the local mass density $\rho$, it depends on three universal free
parameters, and it presents the following asymptotic limits in the high and
low-density regimes:

$\epsilon(\rho)\simeq\begin{cases}1,&\rho\gg\rho_{\rm c}\\\
\epsilon_{0},&\rho\ll\rho_{\rm c},\end{cases}$ (11)

where the permittivity of vacuum $\epsilon_{0}$ and the critical density
$\rho_{\rm c}$ are two of the three free parameters of the theory.

As in MOND for $a\gg a_{0}$, for $\rho\gg\rho_{\rm c}$, Equation (10) reduces
to the Newtonian Poisson equation (7). When $\rho\ll\rho_{\rm c}$, the RG
field is boosted compared to the Newtonian one thanks to the value of
$\epsilon_{0}\in(0,1)$.

RG predicts a different behaviour for the gravitational field in the low-
density environments of spherical and flattened systems. In the external
regions of spherical systems, where $\rho\ll\rho_{\rm c}$, the gravitational
field does not deviate from the inverse square law, i.e.,
$\partial\phi/\partial r\propto r^{-2}$, as in the Newtonian case, but it is
boosted with respect to the Newtonian field by the inverse of the
gravitational permittivity, as obtained by integrating Equation (10):

$\frac{\partial\phi}{\partial
r}=\frac{1}{\epsilon(\rho)}\frac{GM(<r)}{r^{2}}=\frac{1}{\epsilon(\rho)}\frac{\partial\phi_{\rm
N}}{\partial r},$ (12)

where $M(<r)$ is the system mass within the spherical radius $r$.

Whereas for spherical systems we still observe a Newtonian trend, in the
outskirts of flattened systems the field lines are refracted toward the mid-
plane of the object. This can be seen by expanding the left-hand side of
Equation (10):

$\frac{\partial\epsilon}{\partial\rho}\nabla\rho\cdot\nabla\phi+\epsilon(\rho)\nabla^{2}\phi=4\pi
G\rho,$ (13)

where the term
“$\frac{\partial\epsilon}{\partial\rho}\nabla\rho\cdot\nabla\phi$”, different
from zero in nonspherical systems, is responsible for the focussing of the
field lines. In nonspherical configurations, we thus observe an analogy with
electrodynamics in matter: the gravitational field lines behave like electric
field lines when they cross a dielectric medium with a nonuniform
permittivity, changing both in direction and in magnitude.

This redirection effect in the low-density regions of flattened systems yields
the trend $\partial\phi/\partial R\sim\sqrt{a_{0}|g_{\rm N}|}\propto R^{-1}$
for the RG gravitational field Matsakos and Diaferio (2016), where $|g_{\rm
N}|=|\partial\phi_{\rm N}/\partial R|$ is the Newtonian field. In this regime,
the RG field deviates from the Newtonian inverse square law and is subject to
a boost that in Newtonian gravity is obtained with the presence of DM. This
limit also coincides with the MOND asymptotic behaviour for $a\ll a_{0}$ (see
Equation (4)) and it suggests that the ability of MOND in describing galaxy
dynamics is shared by RG, as demonstrated in Cesare et al. (2020). According
to Equation (13), the flatter the system, the larger the boost of the
gravitational field, and, thus, the larger the mass discrepancy, as
interpreted in Newtonian theory. Figure 4 summarises the analogies and the
differences between Newtonian (top panels) and RG (bottom panels)
gravitational fields for flat (left panels) and spherical (right panels)
systems.

Figure 4: Comparison between the behaviour of Newtonian (top panels) and
refracted gravity (RG) (bottom panels) gravitational fields for flat (left
panels) and spherical (right panels) systems. The figure is taken from Figure
16 in Matsakos and Diaferio (2016).

The behaviour of the RG gravitational field in the low-density regime is
comparable to the behaviour of the MOND gravitational field in the low-
acceleration regime only for nonspherical configurations. In the low-
acceleration regions of spherical systems, MOND field is $\propto R^{-1}$, as
in the low-acceleration regions of flattened systems, whereas, in the low-
density regions of spherical systems, the $R$-dependence of RG field remains
Newtonian. Moreover, RG and MOND fields also present a difference for
flattened systems. As we can see in the left panels of Figure 4, the
refraction effect of the RG field lines in a flattened object already begins
where the local density $\rho$ is still above $\rho_{\rm c}$ and, in that
region, the RG field already begins to be boosted compared to Newtonian one.
Instead, MOND modification with respect to the Newtonian field only appears in
regions where $a<a_{0}$.

In all the analyses of galaxy dynamics performed with RG Matsakos and Diaferio
(2016); Cesare et al. (2020, 2022), the following smooth step function of the
gravitational permittivity was adopted:

$\epsilon(\rho)=\epsilon_{0}+(1-\epsilon_{0})\frac{1}{2}\left\\{\tanh\left[\text{ln}\left(\frac{\rho}{\rho_{\mathrm{c}}}\right)^{Q}\right]+1\right\\},$
(14)

where the power index $Q$ is the third free parameter of the theory and it
regulates the transition speed between Newtonian and RG regimes (the larger
its value the steeper the transition).

Being that the modification of the law of gravity is dependent on a scalar
quantity, the mass density of baryonic matterIn fact, the gravitational
sources depend on other scalar quantities besides the mass density, such as
their total mechanical and thermodynamical energy or their entropy. Yet, these
quantities depend in turn on the mass density and, thus, adopting a
gravitational permittivity also dependent on these quantities would be likely
to produce a phenomenology comparable to the one obtained with the simple
dependence on the mass density alone Matsakos and Diaferio (2016); Cesare et
al. (2020)., it was possible to build a covariant formulation of RG Sanna et
al. (2021) without the challenges encountered in defining a relativistic
extension of MOND Bekenstein and Milgrom (1984); Bekenstein (1988); Sanders
(1988); Skordis (2009); Bekenstein (2011). On the other hand, a modification
of the law of gravity that is dependent on a density scale might appear not so
intuitive, given that the majority of the pieces of evidence on galaxy scale
rather see the emergence of an acceleration scale, $a_{0}$, below which a
departure from Newtonian gravity is observed. However, the acceleration scale
$a_{0}$ also seems to appear in RG, from the weak field limit (WFL) of
covariant refracted gravity (CRG). This point will be better addressed in
Sections 4 and 6.

###  3.2 Interpretation of the Three Scaling Relations in MOND and RG

The fact that the acceleration scale $a_{0}$ emerges from the three scaling
relations seems to identify MOND as the most natural solution to explain them.
MOND not only reproduces but actually predicted the three relations with a
zero intrinsic scatter many years before they were observed, besides other
pieces of evidence on the galaxy scale. It is the only theory of gravity that
has this peculiar feature. Already in its first formulation of 1983 Milgrom
(1983b), Milgrom concludes that “The $V^{4}_{\infty}=a_{0}GM$ relation should
hold exactly”, where $V_{\infty}$ is the asymptotic flat velocity of the
rotation curve, $M$ is the total baryonic mass of the galaxy (the total mass
in MOND), and the mentioned equation coincides with the BTFR (Equation (1)).
The acceleration scale is set by its normalisation. Later observations
confirmed this prediction (see Section 2.1). In his first works of 1983,
Milgrom calculated in several independent ways the value of $a_{0}$ Milgrom
(1983b), which turned out to be consistent with the value observed years after
from the scaling relations. Moreover, Milgrom predicted that the BTFR does not
depend on the galaxy type or on any other galaxy property Milgrom (1983b), in
agreement with the future observations.

The condition on the acceleration $a<a_{0}$, where the departure from
Newtonian dynamics begins to be observed, can be translated in a condition on
the surface mass density, $\Sigma<\Sigma_{0}$, where $\Sigma_{0}=a_{0}/G$
Famaey and McGaugh (2012). This implies a relation between the mass
discrepancy in galaxies and the acceleration due to baryons, which is observed
to hold from HSB to LSB galaxies with a very narrow intrinsic scatter and no
dependency on galaxy properties Famaey and McGaugh (2012).

Concerning the RAR, MOND predicted this relation with a null intrinsic scatter
only for the modified inertia version of MOND and for circular orbits Milgrom
(1994). For the modified gravity versions of MOND, the RAR is recovered with
zero intrinsic scatter only for spherical systems, whereas, for other systems
morphologies, it is retrieved with a very small intrinsic scatter, in
agreement with the observations Bekenstein and Milgrom (1984); Lelli et al.
(2017); Brada and Milgrom (1995).

A work of Eriksen et al. Eriksen et al. (2019) seems to challenge MOND in
modelling the RAR. Specifically, MOND might present a “cusp/core like” issue,
different from the classical cusp/core problem of $\Lambda$CDM. This issue is
particularly relevant for the modified inertia version of MOND. Moreover, by
fitting the RAR relation (Equation (3)), in both modified gravity and modified
inertia versions of MOND, from the observational data of SPARC galaxies, the
best fit acceleration scale $g_{\dagger}$ might result inconsistent among
different galaxies.

Matsakos and Diaferio Matsakos and Diaferio (2016) demonstrated that RG
reproduces both the BTFR and the MDAR. In Section 3.1.2, I showed that the
asymptotic limit for the gravitational field in low-density environments of
nonspherical systems is $\partial\phi/\partial R\sim\sqrt{a_{0}|g_{\rm N}|}$,
as in MOND, in the low-acceleration regime. In fact, Matsakos and Diaferio
Matsakos and Diaferio (2016) write this asymptotic limit as:

$\frac{\partial\phi}{\partial R}\sim\sqrt{b|g_{\rm N}|},$ (15)

where $b$ is an acceleration scale that can be expressed as:

$b=\frac{Gm}{h^{2}},$ (16)

and $\pm h$ is the height from the disk plane where the condition
$\rho=\rho_{\rm c}$ is reached. In a simplified formulation for RG (SRG),
whose conclusions can be extended to a more generic RG framework, the volume
within the disk planes $z=-h$ and $z=+h$ is where the redirection effect of
the field lines occurs.

In the low-density regime, the condition on the gravitational field given by
Equation (15) can be translated in a condition on the rotation velocity:

$V_{\rm f}=\sqrt{R\frac{\partial\phi}{\partial R}}\sim\sqrt[4]{bGM_{\rm
bar}},$ (17)

being $|g_{\rm N}|=GM_{\rm bar}/R^{2}$. Inverting the above equation,

$M_{\rm bar}=(Gb)^{-1}V_{\rm f}^{4},$ (18)

we obtain the BTFR (Equation (1)) with the correct slope. To make the
normalisation of Equation (18) in agreement with the normalisation of the
observed BTFR, the acceleration $b$ has to coincide with $a_{0}$, which sets a
condition on the $z=\pm h$ planes that depends on the galaxy baryonic mass.

Matsakos and Diaferio Matsakos and Diaferio (2016) plotted the $(V_{\rm
f},M_{\rm bar})$ points from real data McGaugh (2005) (black dots in Figure 5)
and from some disk galaxy models of star- and gas-dominated disk galaxies,
with typical values of the central surface mass density, $\sigma_{0}$, and of
the disk scale-length, $h_{R}$ (open circles and squares in Figure 5). Both
sets of points follow relation (18) (black solid line in Figure 5).

Figure 5: The BTFR built in RG using Equation (18) (black solid line) and from
the observational data of McGaugh (2005) (black dots) and some models of star-
and gas-dominated disk galaxies (open circles and squares). The RG BTFR
properly describes both the data and the model points. The figure is re-
adapted from Figure 9 in Matsakos and Diaferio (2016).

RG also recovers the MDAR of the galaxies (Figure 6). Matsakos and Diaferio
Matsakos and Diaferio (2016) plotted the squared ratio between the rotation
velocity predicted by the SRG framework and the Newtonian theory ($v_{F}$ and
$v_{N}$ in Figure 6) against the Newtonian and the SRG accelerations ($-N$ and
$-F$ in Figure 6) for the same models used to build the BTFR and for a set of
point masses having $m=10^{x}$ M⊙, with $x=\\{7,8,9,10,11,12\\}$, settling
between two parallel planes with $z=\pm h=\pm\sqrt{Gm/b}$. The models are
plotted together with a set of data points, taken from McGaugh (2004); Famaey
and McGaugh (2012), and they properly reproduce the data, both in the $(g_{\rm
bar},(V/V_{\rm bar})^{2})$ and in the $(g_{\rm SRG},(V/V_{\rm bar})^{2})$
planes. The gas-rich galaxies models (where the gas represents more than 20%
of the total baryonic mass) have a smaller $\sigma_{0}$ and they distribute in
the leftmost part of the MDAR, in agreement with MOND prediction according to
which a larger mass discrepancy is observed when $\Sigma<\Sigma_{0}$. On the
contrary, the star-rich galaxies curves distribute on the rightmost part of
the MDAR.

Figure 6: The MDAR built in RG as the squared ratio between the SRG and the
Newtonian speeds, $v_{F}$ and $v_{N}$, respectively, as a function of the
Newtonian (top panel) and the SRG (bottom panel) accelerations, $-N$ and $-F$,
respectively, for the same set of models represented as open circles and
squares in Figure 5 (black solid lines) and for a set of point masses having
$m=10^{x}$ M⊙, with $x=\\{7,8,9,10,11,12\\}$ (black dotted lines). The solid
and dotted lines appear almost indistinguishable in the
($-N$,$(v_{F}/v_{N})^{2}$) and ($-F$,$(v_{F}/v_{N})^{2}$) planes. The gas-rich
galaxies models appear in the leftmost part of the plots, indicating that they
are always in the “mass discrepancy regime”. The curves are plotted together
with observational measurements from Famaey and McGaugh (2012); McGaugh (2004)
(black dots), showing that they properly reproduce the data. The black dashed
line $(v_{F}/v_{N})^{2}=1$, showing the no mass discrepancy case, and the blue
dashed lines $-N=10^{-10}$ m s-2 and $-F=10^{-10}$ m s-2 are illustrated as a
reference in both panels. The figure is re-adapted from Figure 10 in Matsakos
and Diaferio (2016).

Cesare et al. Cesare et al. (2020) built the RAR of 30 galaxies from the
DiskMass Survey (DMS) Bershady et al. (2010), adopting the general RG
framework and the QUMOND formulation of MOND (blue and green solid lines in
Figure 7). They modelled with RG, at the same time, the rotation curve and the
vertical velocity dispersion profile of each DMS galaxy, obtaining mass-to-
light ratios consistent with SPS models, disk scale heights in agreement with
the observations of edge-on galaxies, and RG parameters consistent among the
different galaxies, suggesting their universality. To build the RG
acceleration, they numerically solved the RG Poisson equation (Equation (10)),
adopting the mass-to-light ratios, the disk-scale heights, and the three RG
parameters estimated from the kinematic profiles of each DMS galaxy. To
compute the QUMOND acceleration, they numerically solved the QUMOND Poisson
equation (Equation (8)) with the same mass-to-light ratios and disk scale
heights found in RG, since they are fully in agreement with the parameters
found by Angus et al. Angus et al. (2015) by performing the same analysis of
the dynamics of DMS galaxies in QUMOND. The Newtonian acceleration, $g_{\rm
bar}$, was calculated by solving the Newtonian Poisson equation (Equation (7))
with the same mass-to-light ratios used to compute the RG and QUMOND
accelerations, being in agreement with SPS models, and with disk-scale heights
$h_{z,{\rm SR}}$ derived from the scale relation between the disk-scale
lengths and heights:

$\log_{10}\left(\frac{h_{R}}{h_{z,{\rm
SR}}}\right)=0.367\log_{10}\left(\frac{h_{R}}{\rm kpc}\right)+0.708\pm 0.095,$
(19)

estimated from 60 edge-on late-type galaxies Bershady et al. (2010); Cesare et
al. (2020).

RG properly reproduces the asymptotic limits of the observed RAR (Equation
(3), black solid line in Figure 7) but it tends to underestimate relation (3)
at low $g_{\rm bar}$, even if it generally interpolates the observational data
(red dots with error bars in Figure 7). Instead, QUMOND reproduces the RAR
with the correct shape. This can be interpreted by the fact that RG might
attribute more luminous mass than QUMOND.

A more serious problem for RG is that the RAR presents a too large intrinsic
scatter (0.11 dex, whereas the possible intrinsic scatter of the RAR found by
Li et al. Li et al. (2018) for SPARC galaxies is 0.057 dex) and some
correlations between the residuals from Equation (3) and certain galaxy
properties, in disagreement with the observations McGaugh et al. (2016); Lelli
et al. (2017). However, this question has to be further deepened by building
the RAR in RG for a larger sample of disk galaxies with more accurate rotation
curves, such as SPARC, before concluding that this is due to an issue of RG
theory. Indeed, the RAR of DMS data also shows some correlations between the
residuals and some galaxy properties, which might not be observed in the SPARC
sample and suggests that the DMS could not be the most suitable sample where
to investigate the RAR.

In contrast, the RAR curves computed in QUMOND, very neatly distribute around
Equation (3) with an intrinsic scatter of 0.017 dex. This is consistent with
expectations, since QUMOND is a modified gravity version of MOND, and, thus,
it does not provide a RAR with zero intrinsic scatter for nonspherical systems
as disk galaxies Bekenstein and Milgrom (1984). Moreover, QUMOND RAR also
presents correlations between its residuals and some galaxy properties, again
in agreement with the fact that, in this case, the scatter of the RAR is not
equal to zero and which might further suggest that the DMS sample was not the
most suitable to investigate the RAR.

Figure 7: The RAR built in RG (left panel, blue solid lines) and in QUMOND
(right panel, green solid lines) for DMS galaxies. The RG and QUMOND models
are superimposed onto the observed RAR of DMS galaxies (red dots with error
bars). Equation (3) (black solid line) and the $g_{\rm obs}=g_{\rm bar}$ line
(black dashed line) are shown as a reference. The figure is taken from Figure
11 in Cesare et al. (2020). Credit: Cesare V., Diaferio A., Matsakos T., and
Angus G., A&A, 637, A70, 2020, reproduced with permission $\copyright$ ESO.

##  4 Possible Interpretations for an Intriguing Acceleration Scale

If the observed DM-baryons scaling relations might appear not so intuitive in
the $\Lambda$CDM context, a piece of evidence even more difficult to interpret
in this framework is the emergence of an acceleration scale, $a_{0}$, from the
three relations. However, this acceleration scale seems to be retrieved in the
$\Lambda$CDM model as well, for example, from the MDAR and the RAR built from
the simulated galaxies in the Magneticum simulation at $z\sim 0.1$ Mayer et
al. (2023). These relations were fitted with Equations (2) and (3), obtaining
an $a_{0}$ consistent with $1.2\times 10^{-10}$ m s-2 Mayer et al. (2023) (see
Section 2.2). Mayer et al. Mayer et al. (2023) also found that Magneticum
galaxies followed a MDAR (Equation (2)) and a RAR (Equation (3)) relation at
higher redshifts but with a fitted $a_{0}$ substantially different.
Specifically, $a_{0}$ decreases for decreasing redshift, that is, in the more
recent Universe. This trend means that the mass discrepancy decreases, and,
thus, galaxies become more baryons-dominated, as cosmic time advances, and
this can be explained by the progressive cooling of the gas during time.

Mayer et al. Mayer et al. (2023) tried to assess whether the $a_{0}(z)$
relation observed in simulated Magneticum galaxies is consistent or not with
MOND predictions, which is not trivial since MOND is not formulated as a
relativistic theory. Milgrom Milgrom (1983a) pointed out that
$a_{0}\thickapprox cH_{0}$, which might suggest that the redshift dependence
of $a_{0}$ is similar to the one of the Hubble parameter $H(z)$. This would
imply the following relation:

$a_{0}(z)\thickapprox a_{0}(0)\times\sqrt{\Omega_{\rm
m}(1+z)^{3}+\Omega_{\Lambda}},$ (20)

where the fraction of baryonic matter $\Omega_{\rm m}$ and of the cosmological
constant $\Omega_{\Lambda}$ with respect to the total mass-energy budget of
the Universe are put equal to $0.25$ and $0.75$, following Bekenstein and Sagi
(2008), and $a_{0}(0)$ is the value of the acceleration scale $a_{0}=1.2\times
10^{-10}$ m s-2 observed today. However, this relation is too steep compared
to the $a_{0}(z)$ relation observed in Magneticum simulation and it also seems
to disagree with observational data. More realistic predictions of the
$a_{0}(z)$ relation in MOND might be provided by relativistic theories that
have MOND as a limiting case in a nonrelativistic regime, such as tensor-
vector-scalar gravity (TeVeS) Bekenstein (2004) and covariant emergent gravity
(CEG) Hossenfelder (2017). TeVeS might suggest that $a_{0}$ varies on
timescales larger than the Hubble time, even if, according to Bekenstein and
Sagi (2008), this change is also possible on cosmological timescales. Anyway,
the $a_{0}(z)$ relation predicted by TeVeS is not consistent with the one
observed in Mayer et al. (2023) from Magneticum galaxies. CEG predicts an
$a_{0}(z)$ relation dependent on the variation of the size of the cosmological
horizon Hossenfelder and Mistele (2018). However, the redshift dependence of
this $a_{0}(z)$ relation is smaller than the one observed from the galaxies in
the MUGS2 simulated sample in $\Lambda$CDM Keller and Wadsley (2017).

An even more striking coincidence is given by the near equality between
$a_{0}$ emerging from the local universe, that is, from galactic dynamics, and
from cosmology. Given the two acceleration cosmological parameters Milgrom
(2020):

$a_{H}\equiv cH_{0}$ (21)

and

$a_{\Lambda}\equiv c^{2}\left(\frac{\Lambda}{3}\right)^{1/2},$ (22)

where $H_{0}$ and $\Lambda$ are the present values of the Hubble parameter and
of the cosmological constant, the following equivalence subsists:

$\bar{a_{0}}\equiv 2\pi a_{0}\thickapprox a_{H}\thickapprox a_{\Lambda}.$ (23)

This arises an outstanding coincidence that needs careful investigation since
$\Lambda$ is assumed to be constant across cosmic time and $H_{0}$ is the
value at the present epoch of the Hubble parameter, $H(t)$, that varies across
cosmic time.

Some interpretations of the relation $a_{0}\sim\Lambda^{1/2}$ in the MOND
context were provided by Famaey and McGaugh (2012); Milgrom (1989, 1999,
2020). In the most recent work among these, Milgrom Milgrom (2020) re-wrote
Equation (23) in terms of length or mass, introducing the “MOND length”,
$l_{\rm M}$, and the “MOND mass”, $M_{\rm M}$:

$l_{\rm M}\equiv\frac{c^{2}}{a_{0}}\thickapprox 2\pi l_{H}\thickapprox 2\pi
l_{\Lambda}$ (24)

and

$M_{\rm M}\equiv\frac{c^{4}}{Ga_{0}}\thickapprox M_{H}\thickapprox
M_{\Lambda},$ (25)

where $l_{H}\equiv cH_{0}^{-1}$ is the Hubble radius,
$l_{\Lambda}\equiv(\Lambda/3)^{-1/2}$ is the de Sitter radius related to
$\Lambda$, and $M_{H}$ and $M_{\Lambda}$ are the total mass of the Universe
enclosed within $l_{H}$ and $l_{\Lambda}$, respectively.

Equation (23) emerges in some particular effective-field, relativistic
extensions of MOND, where MOND is retrieved in their WFL. In these MOND
relativistic formulations, an additional Lagrangian term,

$\mathcal{L}_{\rm
M}=\frac{c^{4}}{G}\sqrt{g}\frac{1}{l^{2}}\mathcal{F}(l^{2}Q),$ (26)

is added to the GR Einstein–Hilbert (EH) Lagrangian density:

$\mathcal{L}_{\rm EH}\propto\frac{c^{4}}{G}\sqrt{g}R,$ (27)

where $R$ is the Riemann curvature scalar. In Equation (26), $l$ is a length
constant, needed to provide the Lagrangian with the correct dimension,
$\mathcal{F}$ is a dimensionless function, and $Q$ has the dimension of a
length-2 and it is built from the first space-time derivatives of the
gravitational degrees of freedom Milgrom (2020). Some examples of MOND
relativistic extensions where a Lagrangian-like term as Equation (26) is
included are the MOND adaptations of the Einstein–Aether theories Zlosnik et
al. (2007), bimetric MOND (BIMOND) Milgrom (2009), bimetric massive gravity
Blanchet and Heisenberg (2017), and the noncovariant theory presented in
Milgrom (2019).

If we include the cosmological constant term in the EH Lagrangian, Equation
(27) becomes:

$\mathcal{L}_{\rm EH}\propto\frac{c^{4}}{G}\sqrt{g}(R-2\Lambda).$ (28)

Therefore, we can see from Equation (26) that any constant term added to
$\mathcal{F}(x)$, which has to be of the order of unity due to naturalness,
can be identified with a cosmological constant term of this kind:

$\Lambda\sim l^{-2}=l_{\rm M}^{-2},$ (29)

that, combined with Equation (24), gives the $a_{0}$-cosmology coincidence
given byEquations (22) and (23) Milgrom (2020). It is important to point out
that this result is not obtained by adding a cosmological constant term,
$\Lambda$, “ad-hoc” to the Lagrangian, verifying a posteriori that
$\Lambda\sim(a_{0}/c^{2})^{2}$, but by the presence of the two $l$-terms in
the Lagrangian (26), appearing inside and outside $\mathcal{F}(x)$ for
dimensional reasons and playing different roles. This suggests that the two
$l$-terms derive from the same underlying physics, whereas a $\Lambda$
constant added “ad-hoc” might have suggested a different physical origin.

Milgrom Milgrom (2020) also suggested that the $a_{0}$-cosmology connection
can also emerge from a scenario in which the Universe is seen as a sphere-like
submanifold, that is a brane, of radius $l_{H}$ or $l_{\Lambda}$, embedded in
a space-time with higher dimension Milgrom (2020). The dynamics of the
submanifold is the one we observe, that is the MOND one, and it emerges from
the dynamics of the higher dimension space-time. In fact, the acceleration
scale $a_{0}$ is seen as an emergent acceleration constant in a brane
scenario. Further details can be found in Milgrom (2020).

In RG, where the transition between Newtonian and modified gravity regimes is
regulated by a density scale $\rho_{\rm c}$, rather than by an acceleration
scale $a_{0}$, the $a_{0}$-cosmology relation might appear harder to
interpret. However, Sanna et al. Sanna et al. (2021) recently formulated a
covariant extension of RG, where $a_{0}$ seems to emerge from the WFL of the
theory and the $a_{0}$-$\Lambda$ coincidence might be explained.

CRG is a scalar-tensor theory that introduces a single scalar field $\varphi$,
nonminimally coupled to the metric, that mimics both the DM effect on galaxy
scale and the DE effect on cosmic scale, namely, the accelerated expansion of
the Universe. Therefore, CRG belongs to the restricted class of modified
gravity models that invoke a unified dark sector, that is, that attribute the
phenomenologies of DM and DE to a single cause.

CRG is derived from a general scalar-tensor action:

$\mathcal{S}=\frac{1}{16\pi G}\int d^{4}x\sqrt{g}\left[\varphi
R+\frac{\mathcal{W}(\varphi)}{\varphi}\nabla^{\alpha}\varphi\nabla_{\alpha}\varphi+2\mathcal{V}(\varphi)\right]+\int
d^{4}x\sqrt{g}\mathcal{L}_{\rm m}(g_{\mu\nu},\psi_{\rm m})$ (30)

(see Sanna et al. (2021) for the explanation of the symbols). In CRG, the
general differentiable function of the scalar field $\mathcal{W}(\varphi)$ is:

$\mathcal{W}(\varphi)=-1,$ (31)

and the potential $\mathcal{V}(\varphi)$ has a self-interaction form:

$\mathcal{V}(\varphi)=-\Xi\varphi,$ (32)

where $\Xi$ is a constant parameter. With the definitions adopted for
$\mathcal{W}(\varphi)$ and $\mathcal{V}(\varphi)$, the CRG equations below are
obtained:

$\varphi
R_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}\varphi-\frac{1}{\varphi}\nabla_{\mu}\varphi\nabla_{\nu}\varphi=-8\pi
GT_{\mu\nu},$ (33)

and

$\square\varphi-2\Xi\varphi=8\pi GT.$ (34)

CRG is based on a chameleon screening mechanism, e.g., Khoury and Weltman
(2004), that is, in regions where the Newtonian WFL holds, the extra degree of
freedom of the scalar field $\varphi$ mediates a fifth force which can be
detected, whereas, in high-density regions, this degree of freedom is
screened. This behaviour is also what we expect from the RG gravitational
permittivity $\epsilon(\rho)$, which, therefore, might be related to
$\varphi$.

The WFL of CRG holds the following equation:

$\nabla\cdot(\varphi\nabla\phi)\simeq 8\pi G\rho,$ (35)

which reduces to RG Poisson equation (10) if the scalar field is twice the
permittivity, $\varphi=2\epsilon$. This is an important result since it
confirms that the scalar field is associated with the phenomenology mimicked
by the gravitational permittivity, that is of DM on galaxy scale. Instead, the
Newtonian Poisson equation is recovered for a constant scalar field
$\varphi=2$.

Calculating the CRG gravitational field in the WFL from Equation (35) for a
spherical source with density $\rho_{\rm s}(r)$, monotonically decreasing with
$r$, immersed in a homogeneous background with constant density $\rho_{\rm
bg}$, Sanna et al. Sanna et al. (2021) found that an acceleration scale can be
set. At large distances from the source, the scalar and the gravitational
fields, $\varphi$ and $d\phi/dr$, are linked by the relation:

$\frac{d\ln{\varphi}}{dr}=\frac{d\phi}{dr}\left\\{-1-\left[1+\left(\frac{d\phi}{dr}\right)^{-2}\left(2\Xi-\frac{8\pi
G\rho}{\varphi}\right)\right]^{1/2}\right\\},$ (36)

where $\rho(r)$ is $\rho_{\rm s}(r)+\rho_{\rm bg}$. The acceleration scale:

$a_{\Xi}=\left(2\Xi-\frac{8\pi G\rho}{\varphi}\right)^{1/2}$ (37)

is set from Equation (36). In regions where $d\phi/dr\gg a_{\Xi}$, the
gravitational field has a similar $r$-dependence as the gravitational field
calculated close to the source, that is, the Newtonian field. Instead, for
$d\phi/dr\ll a_{\Xi}$, it departs from the Newtonian one. From this result,
the acceleration $a_{\Xi}$ recalls $a_{0}$ since it demarcates Newtonian from
modified gravity regimes. Solving the CRG field equations for a homogeneous
and isotropic universe with flat curvature, Sanna et al. Sanna et al. (2021)
found that $\Xi\sim\Lambda$, and, thus, $\Xi$ plays the role of the
cosmological constant in $\Lambda$CDM. In the limit obtained at large
distances from the source,

$2\Xi\gg 8\pi G\rho/\varphi,$ (38)

Equation (37) becomes:

$a_{\Xi}\sim(2\Xi)^{1/2}.$ (39)

Using the observed value of $\Lambda$ at the present epoch, Equation (39)
holds $a_{\Xi}\sim 10^{-10}$ m s-2, fully in agreement with the value of
$a_{0}$, and it also provides the relation $a_{0}\sim\Lambda^{1/2}$, which is
the observed $a_{0}$-cosmology connection.

The difference between $a_{\Xi}$ and $a_{0}$ consists in the fact that,
whereas $a_{0}$ is a constant independent of the gravitational field source,
$\rho$, $a_{\Xi}$ depends on the source, given Equation (37), even if for
large distances from the source this dependence drops since $\rho_{\rm
s}\ll\rho_{\rm bg}$. Future investigations have to verify whether by repeating
the calculations for a generic case and not for a specific source, the
connection between $a_{\Xi}$ and $a_{0}$ continues to hold in a real Universe.

Given the connection between $\Xi$ and the cosmological constant $\Lambda$, it
can be concluded that the scalar field $\varphi$ is also responsible for the
accelerated expansion of the Universe, that is for the DE phenomenology,
besides the DM phenomenology on galaxy scale, given its connection to the
gravitational permittivity $\epsilon$. The fact that the DM and the DE effects
are described by a single scalar field might provide an advantage for CRG
since the idea of a unified dark sector is theoretically justified. The
$a_{0}\sim\Lambda^{1/2}$ relation itself represents one possible piece of
evidence for the unification of the dark sector. Moreover, Martin Kunz claims
that gravity can only probe the total energy-momentum tensor of the Universe,
implying a degeneracy between the dark components, which further goes in this
direction Kunz et al. (2009); Kunz (2009).

CRG is not the only theory that provides a single explanation for the DM and
DE phenomenologies. Among the other theories that invoke a unified dark sector
we can mention: (I) Conformal gravity Mannheim and Kazanas (1989); Nesbet
(2013); Campigotto et al. (2019); (II) the models of Martin Kunz Kunz et al.
(2009); Kunz (2009); (III) the quartessence model Makler et al. (2003);
Brandenberger et al. (2019); Ferreira et al. (2019) (known also as Unified
Dark Matter or Unified Dark Energy), which includes the generalised Chaplygin
gas Bento et al. (2002); Carturan and Finelli (2003); Sandvik et al. (2004),
the $K$-essence Scherrer (2004); Giannakis and Hu (2005), the fast transition
models Bruni et al. (2013); Leanizbarrutia et al. (2017), and other models
that assume the presence of condensates Cadoni et al. (2018a, b); Alexander et
al. (2010); (IV) $f(R)$ theories, e.g., Sotiriou and Faraoni (2010); (V)
mimetic gravity Sebastiani et al. (2016); (VI) a class of models where the
cosmic acceleration emerges from the interactions between DM and baryons
Berezhiani et al. (2017); (VII) the unified superfluid dark sector Ferreira et
al. (2019); (VIII) the fuzzy dark fluid Arbey and Coupechoux (2021).

Among these models, Conformal gravity is one of the first to invoke the
unified dark sector, even if it might present some problems in modelling the
rotation curves of galaxies and the gravitational lensing effects Campigotto
et al. (2019). Two recent and novel models invoking the unified dark sector
are the unified superfluid dark sector Ferreira et al. (2019) and the fuzzy
dark fluid Arbey and Coupechoux (2021). In the unified superfluid dark sector,
the Universe is dominated by a unique DM superfluid made of axion-like
particles, with two energy states having an energy gap smaller than $H_{0}$
that can interact with each other. These interactions at the microscopic level
change the macroscopic behaviour of the fluid, producing an accelerated
expansion of the Universe that mimics DE. Besides the effect of DE, this fluid
can also mimic the galaxy phenomenology due to DM without facing some problems
encountered by others of these models, such as superluminal sound speeds or
the need for a UV completion.

In the fuzzy dark fluid model, besides the behaviours of DM and DE, a single
scalar field also mimics the behaviour of inflation, by assuming a nonminimal
coupling to the gravitational field, a Mexican hat-shape potential, and a
spontaneous symmetry breaking before the inflationary period. This peculiar
feature of a unique description of the DM, DE, and inflation phenomenologies
is also shared by mimetic gravity Sebastiani et al. (2016).

##  5 Dwarf Galaxies and Globular Clusters

The flatness of the rotation curves of disk galaxies and the three scaling
relations, BTFR, MDAR, and RAR, observed on the galaxy scale, are only some of
the predictions of MOND. MOND predicted additional pieces of evidence about
galactic dynamics before they were observed. Among them, the dynamics of LSB
galaxies is worth mentioning.

As already mentioned in the Introduction (Section 1), HSB disk galaxies are
dominated by stellar mass in their central regions, where their rotation
curves are steeply rising toward their asymptotic values. Instead, the
dynamics of LSB galaxies, generally dwarf and dSph galaxies, is observed to be
different. Their rotation curves are slowly rising toward their flat region
and they appear to be DM-dominated even in their innermost regions. Therefore,
the maximum-disk hypothesis cannot be applied as for HSB galaxies Strigari et
al. (2008); Di Paolo et al. (2019). Specifically, dwarf galaxies are among the
known darkest galaxies observed: they have an inner velocity dispersion
$\sigma\sim 10$ km s-1, an order of magnitude larger than the velocity
dispersion $\sigma\sim 1$ km s-1 expected for systems having the same
luminosity and scale radius ($\sim$100 pc) at equilibrium Mateo (1998). Their
luminosity varies in the range $\sim$[$10^{2}$, $10^{10}$] L⊙ McConnachie
(2012); Javanmardi et al. (2016) but their velocity dispersions are similar,
which might indicate that they are dominated by a similar DM distribution
Mateo et al. (1993); de Martino et al. (2020).

This different shape of the rotation curves of HSB and LSB galaxies reflects
in one of the small-scale issues of the $\Lambda$CDM model, the cusp/core
problem. Indeed, steeply-rising rotation curves can be modelled by a Navarro
Frenk White (NFW) DM density profile Navarro et al. (1996); Dubinski and
Carlberg (1991), cuspy in its innermost part and predicted by collisionless
N-body simulations, whereas slowly-rising rotation curves can only be
reproduced by a cored DM density profile, which might be accounted for in
$\Lambda$CDM only introducing baryonic feedback and tidal effects Torrealba et
al. (2019).

The different dynamic properties of these two categories of galaxies were
instead predicted by MOND some years before they were observed Milgrom
(1983b). Milgrom Milgrom (1983b) predicted that dwarf galaxies would have
shown strong deviations from standard gravity and in particular that when the
velocity dispersion data of dwarf galaxies had been available, these galaxies
would have presented a mass discrepancy equal to 10 or larger, depending on
their distance from the Milky Way, when modelled in standard gravity.

As already anticipated in Section 3.2, the MOND acceleration scale $a_{0}$ can
be translated on a surface mass density scale $\Sigma_{0}=a_{0}/G$ Milgrom
(1983b); Famaey and McGaugh (2012). Small surface mass densities also indicate
small surface brightnesses and Milgrom predicted that LSB galaxies would have
shown stronger effects of the modification of the law of gravity with respect
to the Newtonian one. The effect of gravity modification already appears in
the innermost radii of these galaxies. Milgrom Milgrom (1983b) predicted that
a transition radius $r_{0}$ between the standard and the modified gravity
regimes, dependent on the local value of the rotation velocity $V$, would have
been set when the equality $V^{2}/r_{0}\thickapprox a_{0}$ occurs. In
particular, where $V^{2}/r\gg a_{0}$, the local mass-to-light ratio of the
galaxy should not indicate the presence of a hidden mass, and where $V^{2}/r$
starts to go below $a_{0}$, the local $M/L$ should begin to rapidly increase.
The smaller the average surface brightness of the galaxy, the smaller the
$r_{0}$ in units of the galaxy scale length $h_{R}$. A similar result was also
found by Matsakos and Diaferio (2016) for RG (see the top panel of Figure 10
of Matsakos and Diaferio (2016)). Moreover, Milgrom Milgrom (1983b) also
predicted a correlation between the average surface mass density or surface
brightness and the steepness of the rotation curve in reaching its asymptotic
limit, which states that galaxies with small surface densities show a slowly-
rising rotation curve and vice-versa, as observed by successive measurements.

A possible challenge for MOND on galaxy scale is instead provided by the
internal dynamics of GCs, which have baryonic masses similar to the ones of
LSB galaxies, settling in the outermost regions of the Milky Way. In those
regions, the background acceleration is much smaller than $a_{0}$ and the
external field effect is negligible. Therefore, we expect that their stellar
velocity dispersion profiles present a MONDian behaviour but this is not the
case. Newtonian theory without the presence of a DM halo can better fit these
kinematic profiles than MOND, whose predicted velocity dispersions can exceed
the Newtonian ones also by a factor of $\sim$3 Baumgardt et al. (2005); Jordi
et al. (2009); Baumgardt et al. (2009); Sollima and Nipoti (2010); Ibata et
al. (2011a, b); Frank et al. (2012). Relevant examples for this result are
provided by the GCs NGC 2419, Palomar 14, and Palomar 4 Jordi et al. (2009);
Baumgardt et al. (2009); Ibata et al. (2011a, b); Frank et al. (2012). Yet,
this tension between MOND predictions and measurements might not indicate an
issue of the theory but it might be due to inaccurate data or approximate
modelling.

Concerning the former case, inaccurate data can derive from low-resolution
spectroscopy and from errors on GCs distances larger than 10% of their values
(e.g., Sollima and Nipoti (2010)). The Gaia mission, which provides the
parallaxes of the stars from which to derive their distances, accurate at the
$\mu$arcsec level, might represent a turning point in this sense Gaia
Collaboration et al. (2018). Concerning the latter case, most of the adopted
models assume spherical symmetry, absence of rotation, and orbital isotropy.
Whereas the first two assumptions can be justified by observations, orbital
anisotropies in GCs are predicted by $N$-body simulations Giersz (2006) and
including strong radial anisotropies in the modelling can reconcile MOND
expectations with the observed velocity dispersions. Further studies with
high-precision measurements, which allow us to neatly disentangle the effect
of a strong radial anisotropy and the adopted theory of gravity, need to be
performed Sollima and Nipoti (2010). Moreover, the question might be even
harder to understand, since some theories of formation and evolution of GCs
predict the presence of DM in these objects and its observational evidence is
debated Mashchenko and Sills (2005); Moore (1996); Forbes et al. (2008).

Instead, RG theory provides a more natural solution for the different dynamic
properties of LSB, dwarf, and dSph galaxies and GCs. As anticipated in Section
3.1.2, RG predicts a different shape for the gravitational field lines in flat
and spherical systems. Specifically, whereas in spherical systems the
gravitational field lines always maintain a radial direction following the
Newtonian trend and become enhanced compared to the Newtonian field in the
external regions of the systems, where the density goes below the critical
value $\rho_{\rm c}$, in nonspherical configurations we observe a refraction
of the field lines toward the equatorial plane of the object and the
deviations from Newtonian gravity already begin to appear in regions where
$\rho>\rho_{\rm c}$. In particular, the flatter the system the stronger the
redirection effect of the field lines and the larger the mass discrepancy if
interpreted in a Newtonian framework. This can intuitively explain the diverse
dynamic behaviour of LSB, dwarf, and dSph galaxies and GCs, the former
generally having a flatter shape and the latter a more spherical one Matsakos
and Diaferio (2016).

This RG prediction is also in agreement with a claimed positive correlation
between the elliptical galaxies’ ellipticities and their DM content Deur
(2014, 2020). As mentioned inSections 1 and 3.2, Cesare et al. Cesare et al.
(2020, 2022) demonstrated that RG can model the dynamics of flat (30 disk
galaxies from the DMS) and spherical (three E0 galaxies from the SLUGGS
survey) systems. The modelling of the two classes of systems is obtained with
statistically consistent $\\{\epsilon_{0},Q,\rho_{\rm c}\\}$ parameters,
showing that the gravitational permittivity is independent of the shape of the
considered system. To perform a more robust test of the theory, Cesare et al.
Cesare et al. (2020) estimated the three RG parameters both from each
individual DMS galaxy, simultaneously modelling its rotation curve and
vertical velocity dispersion profile, and from the kinematic profiles of the
entire DMS sample considered at the same time. The two sets of values are
consistent within 2$\sigma$. The average $Q$ and $\rho_{\rm c}$ derived from
the simultaneous modelling of the velocity dispersions of the stars and the
blue and red GCs in each E0 galaxy are consistent within 1$\sigma$ with the
$Q$ and $\rho_{\rm c}$ averaged from the values obtained from each DMS galaxy,
whereas the $\epsilon_{0}$ parameters are still in agreement within 3$\sigma$.
The average $Q$ and $\rho_{\rm c}$ from the E0 galaxies are also consistent,
within 3$\sigma$, with the unique combination of $Q$ and $\rho_{\rm c}$
derived from the entire DMS sample. Instead, the tension increases to
14.8$\sigma$ for the $\epsilon_{0}$ parameter. However, this does not
necessarily indicate an issue for RG. This might be due to the approximate
procedure with which the single combination of RG parameters is estimated from
the entire DMS sample, which results in error bars of the $\epsilon_{0}$
parameter that are much smaller than the error bars of the average
$\epsilon_{0}$ of DMS galaxies. Moreover, it might be due to incorrect
modelling of the dynamics of elliptical galaxies, which are treated as
isolated systems without net rotation, or to a wrong assumption for the
gravitational permittivity functional form Cesare et al. (2022). A review of
both the works about disk and elliptical galaxies in RG is presented in Cesare
(2021).

##  6 Discussion and Conclusions

The most investigated cosmological model is $\Lambda$CDM, which assumes the
validity of GR and the inclusion of two dark components, DE and DM, besides
baryonic matter, which only represents the $\sim$5% of the mass-energy budget
of the Universe. The $\Lambda$CDM paradigm reconciles with the majority of the
observations, from the largest to the smallest scales. However, the results of
the detection, through direct, indirect, or collider experiments, of the most
investigated DM candidate, the weakly interacting massive particles (WIMPs),
are still under debate Tanabashi et al. (2018). Moreover, the nature of DE is
still unknown. Future experiments, such as Euclid, might shed light on its
nature.

Furthermore, the $\Lambda$CDM model presents some issues, both on large and on
small scales. Particularly relevant are the problems observed on the galaxy
scale, such as some remarkable coincidences that can hardly be explained by
the stochastic merging process of structure formation predicted by
$\Lambda$CDM, unless precise fine-tuning between DM and baryonic processes is
invoked. Among these coincidences, three scaling relations between dark and
baryonic matters in galaxies, the BTFR, the MDAR, and the RAR, that neatly
quantify the mass discrepancy on galaxy scale, are observed and they see the
appearance of the acceleration scale $a_{0}=1.2\times 10^{-10}$ m s-2. The
results of Mayer et al. Mayer et al. (2023) from the galaxies in the
Magneticum simulation can reproduce in $\Lambda$CDM the emergence of this
acceleration scale at redshift $z\sim 0.1$. They also predicted an evolution
of $a_{0}$ with redshift, where $a_{0}$ decreases with increasing redshift,
which still has to be confirmed. Intriguingly, the acceleration scale $a_{0}$
presents another coincidence, being its value consistent with the combination
of some cosmological parameters: $a_{0}\sim H_{0}\sim\Lambda^{1/2}$. In
particular, the relation $a_{0}\sim\Lambda^{1/2}$ links a quantity observed on
galaxy scale and the parameter that regulates the Universe accelerated
expansion, which suggests a unification of the two dark sectors and connects
the physics on small and large scales. This is even less intuitive to
interpret in a $\Lambda$CDM framework.

On the other hand, MOND theory not only reproduces but even predicted, with
either a small or a null intrinsic scatter, these three relations, assuming a
modification of the law of gravity for accelerations smaller than $a_{0}$. The
value of $a_{0}$ was estimated by Milgrom Milgrom (1983b) in several
independent ways before it emerged from observations and it turned out to be
consistent with the value of $a_{0}$ observed some years later from the DM-
baryons scaling relations. MOND predicted other pieces of evidence on the
galaxy scale, such as the fact that LSB galaxies appear “darker” and with more
slowly-rising rotation curves than HSB galaxies. However, it presents some
issues in describing the dynamics of GCs residing in the Milky Way outskirts,
which present a Newtonian behaviour even if the background acceleration is
below $a_{0}$. MOND also reproduces the $a_{0}\sim\Lambda^{1/2}$ relation, as
presented in several studies Milgrom (1989, 1999); Famaey and McGaugh (2012);
Milgrom (2020).

A more recent theory of modified gravity, RG, formulated in a nonrelativistic
way by Matsakos and Diaferio in 2016 Matsakos and Diaferio (2016), might shed
further light on galaxy dynamics. RG has already presented some encouraging
results on galaxy scale, reproducing the dynamics of both flat and spherical
systems with a consistent set of RG parameters Cesare et al. (2020, 2022);
Cesare (2021), the BTFR, and the MDAR of real and simulated galaxies. It also
models the RAR of DMS galaxies, even if with some issues, which requires
further investigations to assess whether these problems are due to the chosen
galaxy sample or to RG theory. RG predicts a different shape for the
gravitational field lines in spherical and nonspherical systems. Specifically,
the field lines remain radial in spherical systems and become increasingly
refracted toward the equatorial plane of increasingly flat systems. The
refraction of the field lines produces a boost of the gravitational field that
mimics the presence of a DM halo in Newtonian gravity. This means that an
increasingly flat system is increasingly DM-dominated, if interpreted in the
Newtonian context. With this feature, RG naturally due to the different
dynamic properties of LSB galaxies and GCs to their diverse shape, the former
generally being flatter and the latter nearly spherical. This system
morphology–mass discrepancy relation predicted by RG is also consistent with
the possible ellipticity–total M/L correlation of elliptical galaxies
estimated by Deur Deur (2014, 2020).

Despite the promising results shown by different theories, we are still far
from the answer for a final scenario of the cosmological model. Given the
emergence of $a_{0}$ from several pieces of evidence on galaxy scale, MOND
might seem the most intuitive solution. However, it presents several issues on
a larger scale. For example, it can reduce but not eliminate all the mass
discrepancy in galaxy clusters Sanders (1999, 2003); Hodson and Zhao (2017).
Moreover, the building of a covariant version of MOND seems to appear
challenging. Some attempts at formulating this relativistic extension failed
to describe the features of gravitational lenses, provided superluminal
speeds, or were not in agreement with the post-Newtonian tests of General
Relativity Bekenstein and Milgrom (1984); Bekenstein (1988); Sanders (1988). A
relativistic extension of MOND is TeVeS Bekenstein (2004), which solved some
of these problems but was unable to reproduce cosmological pieces of evidence,
such as the CMB or the matter power spectra Skordis (2009); Bekenstein (2011).
However, further studies about covariant MOND are still ongoing and some
recent results might look promising Hernandez et al. (2019); Skordis and
Złośnik (2021).

RG might provide an alternative solution. Despite being based on a density
scale-dependent modification of the law of gravity, which is not what
observations might suggest, a covariant formulation of RG, CRG, seems to be
promising, given the results of Sanna et al. (2021) that also show that the
acceleration scale $a_{0}$ might emerge from the WFL of the theory. In
particular, building a relativistic extension of RG was possible since the
modification of the law of gravity depends on a scalar quantity, in this case
the density, whereas for MOND it depends on a vector quantity, namely, the
acceleration. CRG describes the DM and DE phenomenologies with a single scalar
field, suggesting a unified dark sector, and retrieves the
$a_{0}\sim\Lambda^{1/2}$ relation, which is a remarkable result.

However, further studies have to be performed to validate RG. A more accurate
study of elliptical galaxies, removing the assumptions adopted in Cesare et
al. (2022) and considering a larger sample with different ellipticities and
extended kinematic profiles (e.g., SLUGGS Forbes et al. (2017) and ePN.S
Pulsoni et al. (2018) surveys), have to be made to better assess if RG can
reproduce the dynamics of these systems. The fact that RG can account for the
different dynamics of dwarf galaxies and GCs is only a hypothesis that should
be tested on real samples, such as the dwarf galaxies surrounding the Milky
Way, e.g., Salucci et al. (2012) and belonging to the LITTLE THINGS survey Oh
et al. (2015), and the GCs settling in the Milky Way outskirts, in particular
NGC 2419, Palomar 14, and Palomar 4, e.g., Baumgardt et al. (2009); Jordi et
al. (2009); Ibata et al. (2011b); Frank et al. (2012). Moreover, RG should be
tested on larger scales, to verify whether it can describe the dynamics of
galaxy clusters. Some preliminary encouraging results in this sense were
obtained by Matsakos and Diaferio Matsakos and Diaferio (2016) but these
studies have to be extended to larger samples (e.g., CIRS and HeCS Rines and
Diaferio (2006); Rines et al. (2013)) and with less approximate modelling. At
last, the studies on cosmological scales have to be completed with CRG.

MOND and RG are only two possible theories of modified gravity but many other
theories that might provide a solution for small- and large-scale problems
have been built. Another theory that is worth mentioning is scalar-vector-
tensor gravity (SVTG), better known as modified gravity (MOG) Moffat (2006).
Whereas the modification of the law of gravity in MOND and RG depends on an
acceleration and on a density scale, respectively, in MOG it depends on a
length scale. MOG is a theory of gravity built in a covariant way that
introduces a scalar, a tensor, and a massive vector field, whose contributions
are added to the classical EH action. MOG assumes that the gravitational
constant $G$, the coupling constant $\omega$ and the mass $\mu$ of the vector
field, and the cosmological constant $\Lambda$ are dynamical scalar fields
which vary with space and time Moffat (2006).

MOG has two progenitor theories, nonsymmetric gravity theory (NGT) Moffat
(1995) and metric-skew-tensor gravity (MSTG) theory Moffat (2005), which
produce the same modified acceleration law as MOG for weak gravitational
fields. NGT, MSTG, and MOG presented several encouraging results on different
scales. They can reproduce solar system and terrestrial gravitational tests,
the observations of the binary pulsar PSR 1913+16 Brownstein and Moffat
(2006), the rotation curves of both HSB and LSB galaxies and the BTFR Moffat
(2005); Brownstein and Moffat (2006); Moffat and Toth (2009), the dynamics of
an elliptical galaxy Brownstein and Moffat (2006), the velocity dispersion of
Milky Way (MW) satellite galaxies Moffat and Toth (2007), the internal
velocity dispersion profiles of GCs Moffat and Toth (2008), the cluster
lensing Moffat (2005), the mass profiles of galaxy clusters derived from
$X$-ray emitting gas Brownstein and Moffat (2006) and their thermal profiles
Moffat and Toth (2009), the Bullet cluster Brownstein and Moffat (2007), and
several cosmological observations, such as the CMB temperature anisotropy, the
galaxy power spectrum, and the supernova luminosity-distance measurements
Moffat (2007); Moffat and Toth (2007, 2009, 2009). A more recent work seems to
demonstrate that MOG is able to reproduce also the RAR of 153 spiral galaxies
from SPARC sample Moffat (2016).

The dynamics of GCs, of galaxy clusters, and of systems at sufficiently large
distances from their centres provide an important test to distinguish among
MOG, MOND, and RG. MOG reproduces the internal velocity dispersions of GCs
around the MW, independently from their distance from the Galaxy centre,
consistently with Newtonian expectations. This is not the case for MOND, which
predicts a MONDian behaviour for the dynamics of GCs sufficiently distant from
the MW centre, such that the background acceleration goes below $a_{0}$.
However, observations suggest a Newtonian behaviour also for these GCs, where
MOND predictions may exceed the measured velocity dispersions also by a factor
of $\sim$3, e.g., Jordi et al. (2009). Moreover, as already specified, whereas
MOG can reproduce the masses of galaxy clusters, and also RG is likely to do
this, MOND can only retrieve them with an additional DM component.

To distinguish among MOG, MOND, and CRG, it would be ideal to have the
kinematic data of different systems at large distances from their centres.
Indeed, at these distances, MOG predicts a Newtonian keplerian trend of the
rotation velocity, i.e., $V(R)\propto{R^{-1/2}}$, consistent with the results
of Prada et al. (2003) and with the gravitational lensing results of Sheldon
et al. (2004). However, this trend is not predicted by MOND and RG (see
Section 3).

Both MOG and RG, also with its covariant extension CRG, reproduce pieces of
evidence on different scales. However, some fundamental features distinguish
MOG from CRG. Whereas MOG introduces a scalar, a tensor, and a vector fields,
CRG only includes a single scalar field. Moreover, whereas the CRG scalar
field is responsible for both the DM and the DE phenomenologies, MOG
attributes the two dark sectors to two different causes and it introduces a
cosmological constant, dependent on space and time, to mimic DE effect.
Concerning the $a_{0}\sim\Lambda^{1/2}$ relation, whereas CRG retrieves it as
a consequence, MOG imposes it Brownstein and Moffat (2006) to constrain some
MOG parameters ($G_{0}$, $M_{0}$, and $r_{0}$). Several works, e.g.,
Brownstein and Moffat (2006); Moffat and Toth (2009), demonstrate that MOG can
model different pieces of evidence with a minimal number or with no free
parameters. The results of Cesare et al. Cesare et al. (2020, 2022) show that
a universal combination of RG free parameters might exist. Additional studies
have to be performed to verify if the phenomenology at several scales in the
Universe can be modelled with this unique combination of RG free parameters.
Generally, further tests have to assess which of the three theories of
modified gravity, MOND, RG, or MOG, better describe the pieces of evidence on
different scales.

To conclude, the correct cosmological model might, at last, be $\Lambda$CDM
and further baryonic processes might still be discovered to properly reconcile
the theory with all the observed coincidences on the galaxy scale.

This research received no external funding.

Not applicable.

###### Acknowledgements.

I sincerely thank the four referees whose suggestions and corrections improved
my review. Part of this work is the result of my Ph. D. activity at the
Physics Department of the University of Turin (2017–2021). I sincerely thank
all the people that helped me during my Ph. D., in particular Antonaldo
Diaferio, who supervised my Ph. D. activity. The authors declare no conflict
of interest. Abbreviations The following abbreviations are used in this
manuscript:
AGN | Active Galactic Nuclei
---|---
BIMOND | bimetric MOND
BTFR | baryonic Tully–Fisher relation
CEG | Covariant Emergent Gravity
CMB | cosmic microwave background
CRG | covariant refracted gravity
DE | dark energy
DM | dark matter
DMS | DiskMass Survey
dSph | dwarf spheroidal
EH | Einstein-Hilbert
GCs | globular clusters
GR | General Relativity
HSB | high surface brightness
$\Lambda$CDM |  $\Lambda$ cold dark matter
LSB | low surface brightness
MDAR | mass discrepancy–acceleration relation
MOG | MOdified Gravity
MOND | MOdified Newtonian Dynamics
MW | Milky Way
NFW | Navarro Frenk White
RAR | radial acceleration relation
RG | Refracted Gravity
SPARC | Spitzer Photometry and Accurate Rotation Curves
SPS | stellar population synthesis
SRG | simplified refracted gravity
SVTG | Scalar-Vector-Tensor gravity
TeVeS | Tensor Vector Scalar gravity
WFL | weak field limit
WIMPs | weakly interacting massive particles
list-name=Note [custom] References

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