Structure of the equivalent Newtonian systems in MOND N-body simulations. Density profiles and the core-cusp problem

Abstract

Aims. We investigate the core-cusp problem of the Λ cold dark matter (ΛCDM) scenario in the context of the modified Newtonian dynamics (MOND) paradigm while exploiting the concept of an equivalent Newtonian system (ENS).
Methods. By means of particle-mesh N -body simulations in MOND, we explored the processes of galaxy formation via cold dissipationless collapse and the merging of smaller substructures. From the end states of our simulations, we recovered the associated ENS and studied the properties of their dark matter halos. We compared the simulation results with simple analytical estimates with a family of γ -models.
Results. We find that the dark matter density of ENSs of most spherical cold collapses have a markedly cored structure, particularly for the lowest values of the initial virial ratios. End states of some simulations with initially clumpy conditions have more complex profiles, and some of their ENSs exhibit a moderate cusp, with the logarithmic density slope always shallower than one.
Conclusions. In contrast to what one would expect from theoretical and numerical arguments in ΛCDM, these results seem to point towards the fact that the absence of a central DM cusp in most observed galaxies would be totally consistent in a MONDian description.

Full text

Astronomy &Astrophysics manuscript no. output ©ESO 2023
October 18, 2023
Structure of the equivalent Newtonian systems in MOND N-body
simulations
Density profiles and the core-cusp problem
Federico Re1,2and Pierfrancesco Di Cintio3,4,5
1Dipartimento di Fisica ”Giuseppe Occhialini", Universitá di Milano Bicocca, Piazza della Scienza 3 20126, Milano, Italy
2INFN-Sezione di Milano Via Celoria 15 20133, Milano, Italy
e-mail: federico.re@unimib.it
3CNR-ISC, via Madonna del Piano 17 50022 Sesto Fiorentino, Italy
4INAF-Osservatorio Astronomico di Arcetri, Largo Enrico Fermi 5 50125 Firenze Italy
5INFN-Sezione di Firenze, via Sansone 1 50022 Sesto Fiorentino, Italy
e-mail: pierfrancesco.dicintio@cnr.it
Received ??; accepted ??
ABSTRACT
Aims. We investigate the core-cusp problem of the Λcold dark matter (ΛCDM) scenario in the context of Modified Newtonian
Dynamics (MOND) paradigm exploiting the concept of equivalent Newtonian system (ENS)
Methods. By means of particle-mesh N−body simulations in MOND we explore processes of galaxy formation via cold dissipation-
less collapse or merging of smaller substructures. From the end states of our simulations, we recover the associated ENS and study the
properties of their dark matter halos. We compare the simulation results with simple analytical estimates with a family of γ−models.
Results. We find that the dark matter density of ENSs of most spherical cold collapses have a markedly cored structure, in particular
for the lowest values of the initial virial ratios. End states of some simulations with clumpy initial conditions have more complex
profiles and some of their ENSs exhibit a moderate cusp, with logarithmic density slope always shallower than 1.
Conclusions. These results seem to point towards the fact that the absence in most observed galaxies of a central DM cusp, at
variance with what one would expect from theoretical and numerical arguments in ΛCDM, would be totally consistent in a MONDian
description.
Key words. Galaxies: kinematics and dynamics - Galaxies: formation - Gravitation - Methods: numerical - Methods: analytical
1. Introduction
In the Λcold dark matter scenario (hereafter ΛCDM), theoret-
ical arguments and collisionless N−body simulations (Navarro
et al. 1997) predict that galaxies are embedded in dark matter
(DM) halos characterized by a ρ(r)∝r−1central cusp. Obser-
vational results seem to suggest, from the analysis of the central
velocity dispersion profiles of dwarf galaxies, that the DM dis-
tribution has a cored density distribution1(see Moore 1994; Di
Cintio et al. 2014).
Several solutions to this apparent contradiction -often re-
ferred to as “the core-cusp problem"- such as self-interacting
DM (e.g. see Lovell et al. 2012; Nguyen et al. 2021; Eckert et al.
2022), DM annihilation (e.g. see Vasiliev 2007), baryon feed-
back (e.g. see Governato et al. 2010; Cole et al. 2011; Pontzen
& Governato 2012; Del Popolo & Pace 2016) or simply a mis-
interpretation of the observational data (McGaugh et al. 2003),
have been proposed so far. However, notwithstanding the great
1Technically speaking, in multi component equilibrium self gravitat-
ing systems there exist analytical constraints on the magnitude of a
component’s density given the logarithmic slope of the other, e.g. see
Dubinski & Carlberg 1991; Ciotti & Pellegrini 1992; Ciotti 1996, 1999.
Moreover for a broad range of spherical density profiles, the central
density slope constraints the value of the central anisotropy profile (An
& Evans 2006)
amount of theoretical and observational work, a clear answer is
still far from being obtained. Moreover, given the large inter-
est in alternative theories of gravity, such as among the others
f(R) gravities (Buchdahl 1970; Sotiriou & Faraoni 2010); Mod-
ified Gravity (MoG, Moffat 2006; Moffat & Rahvar 2013); Re-
tarded Gravity (Raju 2012; Yahalom 2022); Emergent Gravity
(Verlinde 2011, 2017); Refracted Gravity (Cesare et al. 2020;
Sanna et al. 2023); Fractional Gravity (Giusti 2020; Benetti et al.
2023), proposed to avoid introducing the DM as a collisionless
fluid of exotic particles, it is natural to ask what becomes of the
core-cusp problem in those proposals.
In this work we investigate this matter in the modified New-
tonian dynamics (hereafter MOND, Milgrom 1983) paradigm.
We recall that in the Bekenstein & Milgrom (1984) Lagrangian
formulation of MOND (sometimes referred to as AQUAL) the
classical Poisson equation for a density-potential pair (ρ;Φ)
∆Φ = 4πGρ(1)
is substituted by the non-linear field equation
∇ · "µ ||∇Φ||
a0!∇Φ#=4πGρ. (2)
In the equation above where a0≈10−8cm s−2is a scale acceler-
ation and µ(x) is the MOND interpolating (monotonic) function
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10-8
10-6
10-4
10-2
100
102
104
106
10-2 10-1 100101102
ρDM
r/rc
10-2
100
102
104
106
108
1010
ρ*/ρDM
10-2 10-1 100101102
10-8
10-6
10-4
10-2
100
102
104
106
ρ*
r/rc
γ=0.0
γ=0.5
γ=1.0
γ=1.5
γ=2.0
γ=2.5
Fig. 1. Ratio of the stellar to dark density in the ENS (top) and DM
and stellar density profiles (bottom left and bottom right) in units of
3M/4πr3
cfor κ=100 and γ=0, 0.5, 1, 1.5, 2 and 2.5.
known only by its asymptotic limits
µ(x)∼(1,x≫1,
x,x≪1; (3)
so that for ||∇Φ|| ≫ a0Eq. (2) one recovers the Newtonian
regime, while for ||∇Φ|| ≪ a0one obtains the so-called deep-
MOND (hereafter dMOND) regime and Eq. (2) simplifies to
∇ · [||∇Φ||∇Φ]=4πGρa0.(4)
Note that, the non-linear operator in Equation (4) is the special
case of the p−Laplace operator (see e.g. Stein 1970) for p=
3, while Eq. (1) would correspond to the p=2 case. In this
respect, Equation (2) somewhat ”interpolates" between the two
regimes via the µfunction. Note also that, in both cases, any
given baryonic mass density ρcan be taken out from Equation
(1) obtaining the relation
µ ||gM||
a0!gM=gN+S(5)
between the MOND and Newtonian force fields gMand gN, and
where S≡ ∇ × h(ρ) is a density-dependent solenoidal field. It
can be proved that the latter is identically null for systems in
spherical, cylindrical or planar symmetry, while it is generally
non-zero for arbitrary configurations of mass. On which extent
the stellar system at hand with mass Mis dominated by MOND
effects, is usually quantified by the dimensionless parameter
κ≡GM
r2
ca0
,(6)
where rcis the scale of the baryon distribution. That is, for κ≫1
the system is mainly in Newtonian regime, vice versa for κ≤1
MOND effects become strong at all scales.
For any given stationary model in MOND one can always
build the equivalent Newtonian system (hereafter ENS), defined
as a system with the same baryonic mass density ρ∗plus a
DM halo with density ρDM such that their total potential Φ
satisfying Eq. (1) is the same as the MOND potential enter-
ing Eq. (2) for the sole density ρ∗. We note that, in principle,
the positivity of the DM density of the ENS is not always as-
sured (see Milgrom 1986), in particular for flattened systems
(see Ciotti et al. 2006, 2012; Ko 2016). We recall that Mil-
grom (2010) introduced a Quasi-linear formulation of MOND
(hereafter QuMOND) where the modified field equation has the
same form as Eq. (2), with ν(||gN/a0||) in lieu of ν(||gM/a0||).
The QuMOND interpolating function ν(y) can be recovered from
µ(x) appearing in Eq. (2) as
ν=1
µ.(7)
It is easy to show that, from a given baryonic density distribu-
tion, one obtains the MONDian potential Φ=ΦN+ΦpDM by first
solving a classical Poisson equation for the Newtonian potential
ΦN, that trough an algebraic passage involving νbecomes the
source for the potential ΦpDM of the so-called "phantom Dark
Matter" via a second application of the Poisson equation. No-
tably, in this alternative bi-potential MOND formulation, the DM
is de facto interpreted as the effect of the second potential. As in
AQUAL, in QuMOND one can retrieve a dMOND regime, that
for spherical systems easily reads gM=pa0/gNgN(see Mil-
grom 2021).
If on one hand several workers have investigated the dif-
ferences between static equilibrium models in Newtonian and
MOND gravities, or the interpretation of observations in both
theories, on the other much less is known about the formation
and evolution of stellar systems. For obvious reasons, in ob-
served systems one has access only to de-projected properties
for both stellar and dark components in the Newtonian frame-
work. Numerical experiments, though with their intrinsic limita-
tions, yield information on the full phase-space of the simulated
models; in particular the 3D density profiles. In this paper, we ex-
plore the structure of the ENS of MOND N−body simulations of
galaxy formation, in order to shed some light on the possibility
that the core-cusp problem is a MOND artifact in this paradigm
of gravity. We stress the fact that the MOND core-cusp prob-
lem discussed here, is different from that introduced recently by
Eriksen et al. (2021) that deals with the modified gravity versus
modified inertia hypothesis (see Milgrom 2022).
The rest of this paper is structured as follows. In Sect. 2 we
revise the definition of ENS and discuss their properties. In Sect.
3 we introduce the numerical models and the analysis of the sim-
ulations. In Sect. 4 we discuss the properties of the simulations’
end states. Finally, in Sect. 5 we summarize and discusses the
implications and the relations to previous work.
2. Equivalent Newtonian systems
As anticipated above, the ENS of a MOND model is the New-
tonian system with the same stellar (baryonic) mass distribution
ρ∗with an additional dark component ρDM such that the total po-
tential (and thus the force field) is the same of the parent MOND
system (see Sanders & Begeman 1994; Angus et al. 2006). For
the case of an isolated spherical system one has
ρDM =(4πG)−1∇ · (gM−gN),(8)
since the solenoidal term Svanishes. We stress the fact that,
Equation (5) in QuMOND can be rewritten exactly as
gM=ν ||gN||
a0!gN.(9)
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Federico Re and Pierfrancesco Di Cintio : Core-cusp problem in MOND
10-8
10-6
10-4
10-2
100
102
104
106
10-2 10-1 100101102
ρDM
r/rc
10-2
100
102
104
106
108
1010
ρ*/ρDM
10-2 10-1 100101102
10-8
10-6
10-4
10-2
100
102
104
106
ρ*
r/rc
γ=0.0
γ=0.5
γ=1.0
γ=1.5
γ=2.0
γ=2.5
Fig. 2. Same as in Fig. 1 but for κ=1
If Equation (9) is applied to a spherically symmetric system one
has ν(y)=x/y, and the total density of its ENS (baryonic plus
phantom DM, see e.g. Hodson et al. 2020; Oria et al. 2021) be-
comes
ρ∗(r)+ρDM (r)=d(yν)
dy ρ∗(r)−ydν
dy
2
r3Zr
0
ρ∗(r)r2dr.(10)
Let us consider the family of spherical γ−models (Dehnen 1993;
Tremaine et al. 1994), with density profile given by
ρ∗(r)=3−γ
4π
Mrc
rγ(r+rc)4−γ,(11)
where Mis the total baryonic mass, 0 ≤γ < 3 is the logarithmic
density slope and rcthe scale radius.
If the density profile (11) is substituted in Eq. (10) one ob-
tains
ρ∗+ρDM =ρ∗"rc
r+rc
d(yν)
dy −2
3−γ
ydν
dy #,(12)
where
y=||gN||
a0
=κ r
rc!1−γ 1+r
rc!γ−3
,(13)
with κdefined in Equation (6). We note that, for small radii r,
Equation (13) tends to zero if γ < 1, while it diverges for γ > 1.
In practice, at least for the γ < 1 case, even in central regions the
model falls in the MOND regime. Strong MOND corrections in
the centre are therefore associated with a dominant DM compo-
nent ρDM in the ENS.
2.1. Massive galaxies
Let us consider a typical 1012 M⊙massive elliptical galaxy with
a scale radius of 3 kpc, modelled with a γ-model. In this case
κ≈102. Due to discreteness effects of the underlying stellar
system Equation (11) can be considered reliable until the radius
that contains a fraction of roughly 10−3of the total mass M(in
this case 109M⊙, i.e. the typical mass of its central supermassive
black hole). The Lagrangian radius enclosing such mass fraction
is
r10−3=rc
10 3
3−γ−1
.(14)
The region in MOND regime has a far smaller radius, that for
γ≤1 is obtained by y(r10−3)105r2
10−3/r2
c, varying between
2×102and 2×103. And thus, even in the framework of
(Qu)MOND the phantom DM halo does not really dominate in
the central region for a cored stellar density profile. In Figure
1 we plot for γ=0, 0.5, 1, 1.5, 2 and 2.5 the ratio of stellar
to phantom DM and their respective radial density profiles for
κ=102. We note that, remarkably, models with a strong cusp
(i.e. γ > 1) have phantom DM halos in their ENS characterized
by a decreasing density inside the scale radius. Vice versa, cored
models are associated with ENS having halos with a weak cusp
and several slope changes.
2.2. Diffuse galaxies
Let us now consider a diffuse galaxy such that κ∼1; so that
its central region can fall in the MOND regime, even for radii
bigger than r10−3. Typically, this occurs again if γ < 1. We find
that limr→0y(r)=0 in the central region, hence ν(y)∼y−1/2. If
substituted in (12), this yields
ρ∗+ρDM
ρ∗∼5−γ
6−2γra0r2
c
GM rc
r1−γ
2; (15)
that is, the phantom DM component dominates also at small
radii. In particular, the latter has a central profile given by
ρDM ∼(5 −γ)a0
8πGrcra0r2
c
GM rc
r1+γ
2.(16)
The equation above is characterized by a weak cusp with a log-
arithmic density slope α=−1+γ
2>−1. For example, for γ=0,
the DM component in the ENS would have a cusp ∝r−1/2. We
note that this trend is valid for any spherically symmetric stel-
lar distribution with a central core, and not only for the γ=0
Dehnen model. We note also it always implies a central weak
cusp with logarithmic density slope α=−1/2 for the phantom
dark matter. This has the interesting astrophysical implication
that a galaxy with a cored stellar density profile could be indeed
interpreted in the DM scenario as having a cored halo, due to the
fact that week cusps can be often mistaken for cores.
For the cases with γ > 1 and κ=1, for which the gravita-
tional field diverges in the centre, even though the stellar density
is diffuse, the ENS is DM-dominated only in the external region.
This can be easily checked by substituting the asymptotic be-
haviour ν(y)∼1+1
y−1
y2+o(y−2), and finding from (12) that
ρDM
ρ∗∼2
3−γ
a0r2
c
GM r
rc!γ−1
−r
rc
+O(rγ),(17)
implying a vanishingly small central DM density. This is sum-
marized in Figure 2 where we plot the same quantities as in the
previous Fig. but κ=1. As expected, in the upper plot showing
ρ∗/ρDM , the γ=0 and 0.5 (red and orange lines) are every-
where below 1, i.e. the system is dominated by the phantom DM
distribution at all radii. Vice versa, for the γ≥1 cases, the phan-
tom DM of the ENS dominates only in the external regions. We
recall that Sánchez Almeida (2022) showed that galaxies with
central regions in MOND regime imply ENS characterized by a
decreasing baryon density and a cored DM.
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3. Numerical code and models
3.1. Numerical code and initial conditions
The N−body simulations discussed here have been performed
with a modified version of the publicly available nmody particle-
mesh MOND code (Nipoti et al. 2007a, see also Londrillo &
Nipoti 2011 for additional technical details). The latter uses a
non-linear Poisson solver to compute Φfrom Eq. (2) on a Nr×
Nϑ×Nφspherical grid in polar coordinates, using an iterative
relaxation procedure starting from a guess solution (here given
by Eq. 5 neglecting S), as for the linear Poisson methods (see
Londrillo & Messina 1990; Londrillo et al. 1991). As a rule, in
the simulations discussed here we used a 128 ×32 ×64 grid.
In this work we adopt the following form for the interpolation
function
µ(x)=x
√1+x2.(18)
Alternative choices can also be implement always leading to
qualitatively similar end states.
The equation of motion are integrated using a standard 4th
order leapfrog scheme (see e.g. Dehnen & Read 2011) with
an adaptive timestep ∆tconditioned by the stability threshold
∆t=C/pmax|∇ · g|, where the Courant-Friedrichs-Lewy con-
dition Cwas taken in the range 0.01 ≤C≤0.1.
We performed two sets of numerical simulations with initial
conditions defined as follows.
In the first, the particles positions were sampled from Equa-
tion (11) while in the second, following Hansen et al. (2006),
we first distribute according to a Poissonian distribution inside a
larger γmodel the centres of NCclumps2also described by Eq.
(11) with different choices of rc, and γand later populate them
with particles.
In both cases, the initial particle velocities are extracted from
a position-independent isotropic Maxwell-Boltzmann distribu-
tion and normalized to obtain the wanted value of the initial
virial ratio 2K/|W|, where Kis the total kinetic energy and W
the virial function, defined for a (finite mass) continuum system
of density ρand potential Φas
W=−Zρ(r)⟨r,∇Φ⟩d3r.(19)
We recall that in isolated dMOND systems of finite mass W=
−2pGM3a0/3 is constant (see Nipoti et al. 2007a). Curiously,
even in systems of particles interacting with additive 1/rforces
with logarithmic potential the virial function is constant (see Di
Cintio et al. 2013, 2017).
The simulations of this work span a range of Nbetween 104
and 106. All simulations were extended up to t=300tDyn, where
tDyn ≡q2r3
h/GMtot and rhis the radius containing half of the to-
tal mass of the system Mtot , so that virial oscillations and phase-
mixing are likely to be complete.
Following Ciotti et al. (2007) in some cases we enforce the
spherical symmetry during the collapse by propagating parti-
cles only using the radial part of the evaluated force field, so
that the system behaves effectively as a spherical shell model
introduced in Newtonian gravity by Hénon (1964) and used in
MOND among the others by Sanders (2008); Malekjani et al.
2Clumpy initial conditions were also explored in the context of New-
tonian simulations by Nipoti (2015) and Ludlow & Angulo (2017) when
investigating the relation of the initial density fluctuation power spec-
trum with the Sérsic index m(see below) conjectured by Cen (2014).
(2009) and by Di Cintio & Ciotti (2011) for systems interacting
with 1/rαforces.
3.2. Analysis of the end products
For all simulations presented here we first extract the intrinsic
properties of the end products from their phase-space positions.
We first evaluate the triaxiality of the final particle distribution
(see e.g. Nipoti et al. 2006a; Di Cintio et al. 2013 and references
therein) by defining the tensor
Ii j ≡m
N
X
k=1
r(k)
ir(k)
j(20)
for the particles with positions riwithin the Lagrangian radius
r70 containing the 70% of the stellar mass of the system and
evaluating with a standard iterative procedure its three eigenval-
ues I1≥I2≥I3. By applying a rotation Rto all particles of
the system so that the three associated eigenvectors are now ori-
ented along the coordinate axes we then get that the three semi-
axes a≥b≥cfrom I1=Aa2,I2=Ab2and I3=Ac2, where
Ais a numerical constant depending on the density profile. Fi-
nally, we define the axial ratios b/a=√I2/I1and c/a=√I3/I1,
and the ellipticities in the principal planes ϵ1=1−√I2/I1and
ϵ2=1−√I3/I1.
Following Nipoti et al. (2007a) and Di Cintio et al. (2013)
we compare the surface density profiles of the end products with
the Sersic (1968) law
Σ(R)= Σee−bR
Re1/m−1,(21)
where Σeis the projected mass density at effective radius Re, the
radius of the circle containing half of the projected mass, and
the dimensionless parameters b,mare related by b≃2m−1/3+
4/405mas found by Ciotti & Bertin (1999).
Once the projected density in the 3 principal planes is cir-
cularized over elliptical shells, we determine the corresponding
pair (Re,Σe) by particle counts (i.e. we are assuming a constant
mass to light ratio for each particle), and fit Eq. (21) for the three
projections. We find that, in general, all 3 sets of (Σe,Re,m) are
rather similar (differing only for less than the 5%), we therefore
chose randomly only one.
In addition, for all simulations we also evaluate the so-called
anisotropy index (see Binney & Tremaine 2008) defined by
ξ=2Kr
Kt
,(22)
where Krand Kt=Kθ+Kϕare the radial and tangential compo-
nents of the kinetic energy tensor, respectively and read
Kr=2πZρ(r)σ2
r(r)r2dr,Kt=2πZρ(r)σ2
t(r)r2dr.(23)
In the expressions above, σ2
rand σ2
tare the radial and tangential
phase-space averaged square velocity components and are ob-
tained for the end products of the simulations by particle counts
over radial shells.
For each simulation we recover the (spherical) DM density
of the ENS from Equation (8) where the Newtonian force field
gNhas been evaluated and averaged on the radial coordinate. In
practice, we are assuming a ”sphericized" system. Finally, for the
density distribution ρDM so obtained we evaluate the logarithmic
Article number, page 4 of 10

Federico Re and Pierfrancesco Di Cintio : Core-cusp problem in MOND
Table 1. Summary of the simulation properties: After the name of each simulation (Col. 1), we report the number of particles (Col. 2), the gravity
law (MOND or dMOND, Col. 3), the initial density profile (Col. 4), the initial virial ratio (Col. 5) the axial ratios (Cols. 6 and 7) , the Sérsic (Col.
8), the final anisotropy index (Col. 9) and the virial velocity dispersion (Col. 10).
Name Gravity Initial profile N2K0/|W0|c/a b/a m ξ σvir α
gamma0v0 MOND γ=0 3 ×10410−40.36 0.56 2.45 3.04 0.83 0.25
gamma05v0 MOND γ=0.5 3 ×10410−40.42 0.62 2.33 2.84 0.84 −0.15
gamma1v0 MOND γ=1 3 ×10410−40.54 0.98 2.07 2.73 0.89 −0.11
gamma15v0 MOND γ=1.5 3 ×10410−40.53 0.94 0.89 2.91 1.03 −0.23
gamma2v0 MOND γ=2 3 ×10410−40.57 0.92 0.72 2.26 1.05 −0.35
gamma0ve1m3 MOND γ=0 3 ×10410−30.36 0.56 3.34 3.13 0.83 0.21
gamma0ve3m3 MOND γ=0 3 ×1043×10−30.33 0.50 4.28 3.15 0.85 0.50
gamma0ve1m2 MOND γ=0 3 ×10410−20.34 0.57 2.53 3.10 0.84 0.40
gamma0ve3m2 MOND γ=0 3 ×1043×10−20.32 0.52 3.06 3.33 0.86 0.52
gamma0ve1m1 MOND γ=0 3 ×1040.1 0.32 0.46 2.23 2.98 0.84 0.70
gamma0ve2m1 MOND γ=0 3 ×1040.2 0.33 0.38 3.07 3.03 0.83 0.75
gamma0ve3m1 MOND γ=0 3 ×1040.3 0.36 0.37 3.29 3.02 0.83 0.85
gamma0ve4m1 MOND γ=0 3 ×1040.4 0.37 0.38 2.91 3.01 0.82 0.89
gamma0ve5m1 MOND γ=0 3 ×1040.5 0.40 0.42 2.50 2.92 0.83 1.10
gamma1ve1m3 MOND γ=1 3 ×10410−30.49 0.87 1.90 2.86 0.90 −0.21
gamma1ve3m3 MOND γ=1 3 ×1043×10−30.50 0.91 3.75 2.96 0.91 −0.14
gamma1ve1m2 MOND γ=1 3 ×10410−20.48 0.67 2.48 3.18 0.91 −0.12
gamma1ve3m2 MOND γ=1 3 ×1043×10−20.47 0.82 4.10 3.20 0.93 −0.10
gamma1ve1m1 MOND γ=1 3 ×1040.1 0.42 0.62 2.66 3.32 0.90 0.00
gamma1ve2m1 MOND γ=1 3 ×1040.2 0.45 0.46 3.10 2.99 0.87 0.15
gamma1ve3m1 MOND γ=1 3 ×1040.3 0.51 0.52 2.25 3.33 0.88 0.21
gamma1ve4m1 MOND γ=1 3 ×1040.4 0.54 0.54 2.87 3.46 0.88 0.25
gamma1ve5m1 MOND γ=1 3 ×1040.5 0.95 0.96 3.04 3.71 0.87 0.45
gamma1v1em1 MOND γ=1 5 ×1040.1 0.51 0.95 1.24 2.43 0.88 −0.15
gamma1v0b MOND γ=1 2.1×1050 0.37 0.69 2.58 2.90 0.89 −0.15
gamma1v0dmd dMOND γ=1 2.1×1050 0.24 0.41 2.45 3.49 0.82 0.50
clumpy1 MOND clumpy 8.7×1040.1 0.46 0.85 1.91 2.72 0.87 0.70
clumpy2 MOND clumpy 8.7×1040.1 0.46 0.67 2.86 2.48 0.84 0.50
clumpy3 MOND clumpy 8.7×1040.1 0.60 0.97 3.32 1.44 1.11 −0.40
clumpy4 MOND clumpy 8.7×1040.1 0.57 0.67 1.98 2.07 0.88 0.05
clumpy5 MOND clumpy 8.7×1040.1 0.66 0.94 3.47 1.62 1.18 −0.75
clumpy6 MOND clumpy 8.7×1040.1 0.41 0.70 1.75 1.98 0.85 −0.80
clumpy7 MOND clumpy 8.7×1040.1 0.30 0.54 1.67 2.22 0.83 1.10
clumpy8 MOND clumpy 8.7×1040.1 0.33 0.59 3.39 2.02 0.84 0.95
clumpy9 MOND clumpy 8.7×1040.1 0.80 0.98 1.64 1.41 1.08 −0.40
clumpy10 MOND clumpy 8.7×1040.1 0.56 0.71 0.98 1.89 0.85 −0.99
clumpy1dmd dMOND clumpy 8.7×1040.1 0.13 0.28 3.46 7.45 0.77 1.01
clumpy2dmd dMOND clumpy 8.7×1040.1 0.36 0.61 3.16 2.69 0.83 1.00
clumpy3dmd dMOND clumpy 8.7×1040.1 0.35 0.38 1.40 1.94 0.82 −1.99
clumpy4dmd dMOND clumpy 8.7×1040.1 0.30 0.37 1.74 2.34 0.81 0.70
clumpy5dmd dMOND clumpy 8.7×1040.1 0.55 0.56 0.82 2.08 0.82 −0.50
clumpy6dmd dMOND clumpy 8.7×1040.1 0.33 0.62 1.55 2.00 0.83 0.90
clumpy7dmd dMOND clumpy 8.7×1040.1 0.26 0.57 1.44 2.24 0.82 0.91
clumpy8dmd dMOND clumpy 8.7×1040.1 0.29 0.59 1.89 2.00 0.81 0.99
clumpy9dmd dMOND clumpy 8.7×1040.1 0.35 0.39 1.56 2.10 0.81 0.80
clumpy10dmd dMOND clumpy 8.7×1040.1 0.42 0.60 1.76 1.70 0.75 0.61
gamma0v5em51D MOND (1D) γ=0 3 ×10410−40.96 0.98 2.26 20.5 0.71 −0.01
gamma05v5em51D MOND (1D) γ=0.5 3 ×10410−40.98 0.99 2.27 33.1 0.98 −0.45
gamma1v5em51D MOND (1D) γ=1 3 ×10410−40.97 0.97 1.62 88.0 1.63 −0.40
gamma15v5em51D MOND (1D) γ=1.5 3 ×10410−40.97 0.99 0.71 44.8 1.48 −0.21
gamma2v5em51D MOND (1D) γ=2 3 ×10410−40.96 0.97 0.51 451 4.72 −0.35
gamma1v3em31D MOND (1D) γ=1 3 ×1043×10−30.97 1.00 2.10 4.47 0.66 0.01
gamma1v1em21D MOND (1D) γ=1 3 ×10410−20.97 0.99 3.78 15.8 0.60 −0.10
gamma1v3em21D MOND (1D) γ=1 3 ×1043×10−20.96 0.98 3.11 5.19 0.63 −0.51
gamma1v1em11D MOND (1D) γ=1 3 ×1040.1 0.97 0.99 2.42 4.02 0.64 −0.62
gamma1v2em11D MOND (1D) γ=1 3 ×1040.2 0.98 0.99 3.61 3.19 0.62 −0.45
gamma1v3em11D MOND (1D) γ=1 3 ×1040.3 0.99 0.99 4.09 2.57 0.63 −0.35
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10-1
100
101
102
10-1 100101
ρDM/ρ*,50
r/r50
10-4
10-3
10-2
10-1
100
101
102
103
104
γ0=0
ρ*/ρ*,50
γ0=1 clumpy
10-1 100101
r/r50
10-1 100101
r/r50
Fig. 3. Final (t=300tDyn) density profiles from MOND simulations (top panels) and the DM halo of the ENS (bottom panels) for cored γ0=0
(left), moderately cuspy γ=1 (centre) and clumpy (right) initial conditions. The increasing initial values of the virial ratio in the models with
spherical initial conditions with γ=0 and 1 is mapped with increasingly lighter tones of blue and green respectively. All clumpy initial conditions
start with 2K0/|W|0=0.1
density slope α. We find that the profiles of ρDM are generally
well fitted by the empirical law
ρ(r)=ραr2
α
rα(r2+r2
α)2−α
2
,(24)
where rαis a scale radius and ραis the associated scale density.
Equation (24) above recovers the 1/r2trend of the density of the
ENS as predicted by the logarithmic behaviour of the far field
MOND potential. The properties of the simulations and their ini-
tial conditions are summarized in Tab. 1 below.
4. N−body simulations
4.1. Spherical collapses
One of the main motivations of the present work is to establish
whether the end products of MOND dissipationless collapses
could, in principle, reproduce the structural properties of ellipti-
cal galaxies together with their inferred dark halos. Single com-
ponent Newtonian collapses with spherical initial conditions, are
known to produce flatter end states for increasing values of their
initial virial ratio (see Nipoti et al. 2006a,b; Di Cintio et al. 2013
and references therein) at fixed initial density profile.
We find that, this (partially) holds true for MOND spherical
collapses, as shown in Fig. 3 (top left and top mid panels), where
we plot the baryon density distribution at 300tDyn for γ=0 and 1
and increasing values of the virial ratio with increasingly lighter
tones of blue and green in the range 10−3≤2K0/|W0| ≤ 0.5.
Using Equation (8) for the angle-averaged final density profile
on a spherical grid, we evaluated the density distribution of the
DM component of the parent Newtonian model (see bottom pan-
els, same figure). We find that, in qualitative agreement with
the structural properties of the ENS (see Figs. 1 and 2 in Sect.
2), cuspy end systems can be associated with cored or weakly
cuspy phantom halos. In general, the end products of spherical
collapses have always inner regions that are baryon dominated
when building their ENS, even if the initial conditions are such
that κ=1 (in particular for the γ=0 cases).
Consistently with Nipoti et al. (2007a), we observe that, in-
dependently on the specific value of the initial virial ratio, ini-
tial conditions characterized by a moderate density cusp (i.e.
0.5≤γ≤2) tend to yield end products that are in general
oblate (i.e. 0.5≲c/a≲b/a), as for Newtonian single component
collapses. We typically observe major ellipticities up to ∼0.63
(corresponding to the gamma1v0b case, see Tab. 1). Remarkably,
MOND collapses with cored initial conditions (i.e. γ=0) evolve
into rather prolate end states for 2K0/|W0|≳0.1, and markedly
triaxial end states for lower values of the initial virial ratio. For
both cored and moderately cuspy initial conditions, the inner
slope αof the DM halo of the ENS, obtained by fitting with Eq.
(24) increases for increasing values of the baryon initial virial ra-
tio in the MOND simulation, as shown in Fig. 4 (top panel). The
Article number, page 6 of 10

Federico Re and Pierfrancesco Di Cintio : Core-cusp problem in MOND
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
10-3 10-2 10-1 100
ε
2K0/|W0|
1
1.5
2
2.5
3
3.5
4
4.5
5
m
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
α
γ=0
γ=1
Fig. 4. Inner density slope of the ENS halo (top panel), best fit Sérsic
index (middle panel) and minor ellipticity ϵ=1−c/aas function of the
initial virial ratio for initial conditions with Dehnen profiles with γ=0
(circles) and 1 (triangles)
best fit Sérsic index m, measuring the concentration of the pro-
jected stellar density profile is always in the range 2 ≤m≤4.5
for both choices of the initial density profile (mid panel, same
figure), while the major ellipticity ϵ=1−c/ais typically larger
when the initial condition has a lower virial ratio, being smaller
for larger values of the initial γat fixed 2K0/|W0|(bottom panel,
same figure). Remarkably no system is found being more flat-
tened than an E7 galaxy. However, as also found by Nipoti et al.
(2007a), dMOND collapses may produce even flatter end states
as in the case of the gamma1v0dmd run, for which c/a∼0.24 so
that ϵ=0.76. For fixed initial virial ratio, the end states attain
larger values of the central virial velocity dispersion σvir for in-
creasing values of the initial density slope, while the anisotropy
index ξdecreases (cfr. 1). At fixed initial density profile, the fi-
nal values of σvir have little variation with 2K0/|W0|, while ξis
usually lower for the relaxed states of hotter initial conditions.
In order to clarify whether the properties of the halo in the
ENS are or not an artifact of the angle averaging procedure, we
also performed a set of simulation in enforced spherical sym-
metry by propagating particles only using the radial component
of the force field. By doing so, the system remains spherically
symmetric (as no radial orbit instability is possible) and S=0
de facto hold true at all times, so that one could apply Eq. (8)
exactly. In Figure 5 we show the same quantities as in Fig. 4
as function of the initial values of γfor systems starting with
a virial ratio of 10−4with (empty symbols) and without (filled
symbols) enforced spherical symmetry. The trend as well as the
values of α3D and effective 1D simulations are comparable, and
the same could be noted also for the Sérsic index mthat attains
considerably lower values (associated with a more concentrated
density profile) for larger values of the initial logarithmic den-
sity slope. In all cases (cfr. 1), as expected, 1D collapses relax to
final stated with rather large values of the orbital anisotropy ξ.
Figure 6 shows the final angle averaged density profiles for
γ0=0, 1 and 1.5 in 3D and 1D simulations (solid lines) as well
as the density profiles of the ENS halos (dashed lines). Notably,
if on one hand the large rbehaviour of the baryon density pro-
files ρ∗(where the systems are mostly dominated by radial or-
bits) does not change significantly, on the other, the inner slope
of ρ∗is always higher for the end products of the 1D simulations
and typically settles around 2.5. With the sole exception of the
cored initial conditions (γ0=0), the DM halo of the ENS of the
end products is denser (in units of the baryon component density
ρ∗,50 evaluated at the half mass radius r50) for the 1D simula-
tions, being in both cases considerably shallower than the parent
baryon density.
4.2. Clumpy collapses
The numerical studies in MOND carried out so far, have typ-
ically explored spherical initial conditions (see Nipoti et al.
2007a; Ciotti et al. 2007; Sanders 2008; Malekjani et al. 2009;
Nipoti et al. 2011), disks (Brada & Milgrom 1999; Tiret &
Combes 2007, 2008a; Nipoti et al. 2007c; Ghafourian & Roshan
2017; Wittenburg et al. 2020) or galaxy merging (Nipoti et al.
2007b; Tiret & Combes 2008b) and references therein. Here,
in addition to the usual spherical collapses we also explored
clumpy initial conditions. When starting with such initial states,
MOND simulations tend (as expected) to yield markedly triaxial
end states with broader ranges of both c/aand b/a. In general,
for fixed values of the initial virial ratio, the systems tend to relax
at later times with respect to their initially spherical counterparts
(the oscillations of 2K/Wdamp out at about 50tDyn in spherical
collapses, see Nipoti et al. 2007a, while in clumpy systems this
happens on average at around 140tDyn) for analogous choices of
the virial ratio. We report here only the runs corresponding to
2K0/|W0|=0.1, see Tab. 1.
The final three dimensional (angle averaged) density profiles
(see top right panel in Fig. 3) are strikingly more complex than
those obtained from spherical initial conditions and bare indi-
vidually more slope changes. The projected 2D density profiles
are fitted by the Sérsic law with roughly the same (percentage)
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2
ε
γ
0.5
1
1.5
2
2.5
m
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
α
3D
1D
Fig. 5. Inner density slope of the ENS halo (top panel), best fit Sérsic
index (middle panel) and minor ellipticity ϵ=1−c/aas function of
the logarithmic density slope γof the initial condition for full 3D (filled
symbols) and 1D simulations (empty symbols).
asymptotic standard error of about, on average, 3% as for the
spherical collapses, while the scatter in the mSérsic parameter
is slightly smaller for MOND clumpy systems (see top panel in
Fig. 7). For comparison, we also run the same clumpy initial con-
ditions in dMOND finding a larger scatter in m.
As a general trend, the DM halo of the circularized ENS of
clumpy collapses are significantly more cored3than what is typ-
ically obtained in spherical collapses. In several cases the inner
density slopes are negative, down to ∼ −0.99, corresponding to
a DM density profile that decreases in the central regions (mid-
dle panels in Fig. 7). Interestingly, no initially clumpy system is
found to evolve into a state flatter than an E7 galaxy (thin dashed
line in bottom panels of Fig. 7) in MOND simulations. However,
some dMOND collapses result in considerably flatter end states
(and often prolate) with major ellipticity reaching 0.87 for the
clumpy1dmd.
We observe that, final states with larger values of the
anisotropy index ξ(i.e. more and more dominated by low-
angular momentum orbits) are always associated to larger ellip-
ticities ϵand Sérsic indexes. A similar, though somewhat weaker,
correlation is also found between αand ϵ, that could be read in
the DM scenario as steeper inner DM profiles producing flatter
stellar distributions.
4.3. The MOND mass-to-light ratio - ellipticity relation
Deur (2014, 2020) and more recently Winters et al. (2023), using
a broad sample of elliptical galaxies from independent surveys,
and different methods to evaluate the mass to light ratio M/L(i.e.
Jeans anisotropic modelling, gravitational lensing, X-ray spectra
and the dynamics of satellite star clusters) and the ellipticity ϵ,
recovered the linear relation
M/L=(14.1±5.4)ϵ, (25)
where the M/Lis normalized such that M/L(ϵapp =0.3) ≡
8M⊙/L⊙≡4M/M∗(ϵapp =0.3), and the intrinsic ellipticity ϵ
is extrapolated from its observed 2D projected value ϵapp assum-
ing that all systems are oblate with a Gaussian distribution of
projection angles θso that
ϵapp =1−q(1 −ϵ)2sin2θ+cos2θ. (26)
The Equation above in the context of ΛCDM implies that a larger
contribution of the DM mass MDM to the total mass Mcorre-
sponds to a larger departure from the spherical symmetry (here
quantified by larger major ellipticity) for the stellar component.
Winters et al. (2023) argue that, if true, such a correlation would
be contrasting the standard ΛCDM scenario of galaxy formation,
where more massive (and rather spherical) DM halos are embed
less flattened stellar systems. We note that some peculiar ellipti-
cal galaxies (though excluded by the original sample of Winters
et al. 2023) such as the ultrafaint dwarfs (Simon 2019) appear to
go against the trend given by Eq. (25), having usually ϵ≲0.1
with M/Lin some cases up to 103.
Using the simulations discussed in the previous sections, we
have investigated the relation (25) in the context of MOND, eval-
uating the effective DM mass MDM in the ENSs of both clumpy
and spherical collapses. To do so, after recovering the ρDM from
the angle averaged ENS, we integrate it radially up to the radius
containing all simulation particles.
In Figure 8 we show the total to stellar mass (here we have
assumed units such that M∗/L=1) ratio M/M∗versus major
ellipticity ϵfor collapses with both spherical and clumpy initial
states, here indicated by circles and diamonds respectively, as
well as the observational relation given in Eq. (25). We found
that the end products of initially clumpy systems fall in (almost)
3Notably, in Newtonian simulations of clumps in fall in DM halo Cole
et al. (2011) found that the central DM cusp is considerably weakened
by the collapsing clumpy satellites.
Article number, page 8 of 10

Federico Re and Pierfrancesco Di Cintio : Core-cusp problem in MOND
10-4
10-3
10-2
10-1
100
101
102
103
104
105
10-2 10-1 100101
γ0=0
ρ/ρ*,50
r/r50
Baryons (1D)
Halo ENS (1D)
Baryons (3D)
Halo ENS (3D)
10-2 10-1 100101
γ0=1
r/r50
10-2 10-1 100101
γ0=1.5
r/r50
Fig. 6. Final baryon density profiles (coloured solid lines) and ENS halos (coloured dashed lines) for γ=0,1 and 1.5. The black lines refer to the
1D the cases with the same initial conditions.
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3 3.5 4 4.5 5
ε
ξ
Clumpy
Clumpy (dMOND)
Spherical
-1.5
-1
-0.5
0
0.5
1
1.5
α
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
m
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
m
-1.5 -1 -0.5 0 0.5 1 1.5
α
Fig. 7. Matrix plot of the Sérsic index, slope of the DM profile in the
ENS, major ellipticity and anisotropy index for simulations with clumpy
(diamonds) and spherical (circles) initial conditions. Empty symbol
mark the dMOND runs.
all cases within Winters et al. (2023)’s relation and its error range
(indicated in figure by the shaded area), while for the spherical
collapses fall on a rather steeper relation. We performed a linear
fit (marked in figure by the orange dotted line) obtaining
M/M∗=(23.24 ±0.59)ϵ. (27)
We stress the fact, that none of the simulations discussed above
produces final states that could be interpreted as ultrafaint dwarfs
(except, possibly, some dMOND collapses), that in the standard
cosmological scenario are supposed to be DM dominated at all
radii (i.e. even in the central region where our simulations, when
interpreted in the context of DM have baryon dominated cores).
5. Discussion and conclusions
In this work we have investigated the structure of the dark matter
density profiles of the (angular averaged) equivalent Newtonian
0
2
4
6
8
10
12
14
16
18
20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Mtot/M*
ε
Deur (2014)
this work
Spherical IC
Clumpy IC
Fig. 8. Mass ratio against ellipticity relation for the final states of spher-
ical (red circles) and clumpy (green diamonds) initial conditions. The
purple dashed line marks the Deur (2014) relation with its uncertainty
(blue shaded area), while the orange dotted line marks the linear fit for
the models with spherical initial conditions.
systems of the end states of MOND dissipationless collapse sim-
ulations. We studied a broader range of initial conditions than
those discussed by Nipoti et al. (2007a), including non spherical
ones.
The main results of this work can be summarized as follows:
Simple analytical estimates in spherical symmetry suggest that
the presence of a core or even centrally decreasing DM distri-
bution in ENS of MOND models with cuspy stellar profiles.
Vice versa, cored stellar profiles are associated with ENS DM
central density profiles ρDM ∝1/rαwith α≲1. Our MOND
N−body simulations and the angle averaged ENS of their end
states nicely confirm this. This established, we can conclude that
he flat-cored halos invoked by some observational studies, can
be reasonably considered in agreement with our numerical find-
ing, as the dynamical effect of a weak cusp, independently on
the specific value of the central logarithmic density slope of the
baryons, can be easily mistaken for that of a cored dark mass
distribution in the DM paradigm.
Article number, page 9 of 10

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In general, we observe that as for the simulations in New-
tonian gravity, in MOND the stronger is the collapse (i.e. lower
initial virial ratio and/or larger initial density slope), the steeper
is the final density profile, and thus the dark halo of the ENS
has a markedly cored, or sometimes even depleted, inner den-
sity. Obviously, the end product of simplified MOND N−body
simulations with enforced spherical symmetry have ENS with
markedly flat cores, for a broad spectrum of initial values of den-
sity slope and virial ratio, with baryon density always dominated
by a rather strong cusp at inner radii. Moreover, we also find that,
if interpreted in the context of DM, the relaxed end states with
smaller values of the ellipticity (i.e. less flattened) should have
cuspier DM halos. In general, independently on the specific form
of the initial density profile, colder initial conditions are always
associated to flatter end states.
As a by-product of this simulation study on ENSs, we have
also recovered a numerical confirmation of the claimed Deur
(2014) observational linear correlation between M/L(or M/M∗)
and ϵ, though with seemingly different slope, when evaluating
the dark matter content of ENSs in units of the baryon mass (the
latter being a pre-defined simulation parameter).
Our findings lead us to speculate that in the context of
MOND the core cusp problem could be a “MOND artifact" in
the same sense as rings and DM shells discussed by Milgrom &
Sanders (2008). Moreover, we stress the fact that in the DM ha-
los reconstructed from observational data using the line-of-sight
velocity dispersion of a given tracer stellar population, the ef-
fect of the velocity anisotropy profiles β(r)=1−σ2
t(r)/2σ2
r(r)
(and the intrinsic departure from the spherical symmetry) is ne-
glected, as noted by Evans et al. (2009) for the case of dwarf
spheroids. In fact, since the central stellar βprofile imposes a
constraint on the slope of the DM component in the form of the
inequality β≤α/2 (see An & Evans 2006; Ciotti & Morganti
2009, 2010) the entity of the central density cusp or core inferred
for observed galaxies is likely to bare a rather large uncertainty.
In the context of (single component) MOND models, the rela-
tion between anisotropy and central density cusps has not been
explored in detail, neither analytically nor in simulations. Sim-
ple numerical experiments (Di Cintio et al. 2013) with inverse
power-law radial forces seem to suggest that the density slope
anisotropy inequality is a rather general property of the relaxed
states of collapses with long-range interactions.
A natural follow up of this work will be a systematic study
of the interplay of the βprofiles in MOND systems and the DM
density profiles of the parent ENSs.
Acknowledgements. We would like to express gratitude to Carlo Nipoti for the
assistance with the simulation in nmody and Michal Bílek for the discussions at
an early stage of this work. One of us (PFDC) wishes to acknowledge funding
by “Fondazione Cassa di Risparmio di Firenze" under the project HIPERCRHEL
for the use of high performance computing resources at the university of Firenze.
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