###### Abstract

This review aims at proposing to the field an overview of the Cusp-core
problem, including a discussion of its advocated solutions, assessing how each
can satisfactorily provide a description of central densities. Whether the
Cusp-core problem reflects our insufficient grasp on the nature of dark
matter, of gravity, on the impact of baryonic interactions with dark matter at
those scales, as included in semi-analytical models or fully numerical codes,
the solutions to it can point either to the need for a paradigm change in
cosmology, or to to our lack of success in ironing out the finer details of
the $\Lambda$CDM paradigm.

###### keywords:

cosmology ; dark matter; cusp-core problem; $\Lambda$CDM model

?? 10.3390/galaxies??? ?? Academic Editors: Jose Gaite and Antonaldo Diaferio?
Received: ? June 2021; Accepted: ???; Published: ??? Review of solutions to
the Cusp-core problem of the $\Lambda$CDM Model Antonino Del Popolo 1,2,3, and
Morgan Le Delliou 4,5,6* Antonino Del Popolo and Morgan Le Delliou
Correspondence: (delliou<EMAIL_ADDRESS><EMAIL_ADDRESS>(delliou<EMAIL_ADDRESS>
##  1 Introduction

Notwithstanding the many successes of the concordance cosmology model, also
known as the $\Lambda$CDM model, fundamental issues remain open. Most of the
cosmology community considers the $\Lambda$CDM paradigm to accurately describe
the Universe’s large scale structure formation and evolution, its early state
and the proportions of its content in matter and energy Spergel et al. (2003);
Komatsu et al. (2011); Del Popolo (2007, 2013, 2014). This view has been
reinforced with each new discovery, such as the kinetic Sunyaev-Zel’Dovich
(SZ) effect, the Cosmic Microwave Background (CMB) B-mode polarisation Hanson
et al. (2013), or its lensing Smith et al. (2007); Das et al. (2011),… Among
its most famous resisting issues, the questions of the nature, “fine tuning”
and “cosmic coincidence” problems of the cosmological constant $\Lambda$
remain open Weinberg (1989); Astashenok and del Popolo (2012).

The “cosmological constant fine tuning problem” emerges from the large
discrepancy between the observed value of $\Lambda$ and the huge values
predicted by quantum field theories for the present quantum vacuum energy, the
latter exceeding the former by more than 100 orders of magnitude (Weinberg,
1989; Martin, 2012; Astashenok and del Popolo, 2012): the cosmological
observation upper bounds are limited to $\rho_{\Lambda}\simeq 10^{-47}~{}\rm
GeV^{4}$, while the naive theoretical expectations obtain
$\rho_{\Lambda}\simeq 10^{71}~{}\rm GeV^{4}$. This is the most extreme fine
tuning problem known to physics. As $\Lambda$ is constant, the concordance
model also presents a strong fine tuning of the dark energy (DE) initial
conditions. The “cosmic coincidence problem” questions the relative
coincidence of the orders of magnitude at present of dark matter (DM) and DE
energy densities (Sivanandam, 2013).

Further issues of the $\Lambda$CDM paradigm revolve around observations of the
CMB or of smaller scales structures. Unexplained statistical anomalies in the
CMB large-angle fluctuations contradict the $\Lambda$CDM CMB multipoles
statistical independence, which reflects the assumption that the content of
our Universe should realise a Gaussian and statistically isotropic random
field. These anomalies comprise detection in the CMB of a power hemispherical
asymmetry Eriksen et al. (2004); Hansen et al. (2004); Jaffe et al. (2005);
Hoftuft et al. (2009); Planck Collaboration et al. (2014); Akrami et al.
(2014), of a cold spot, marking an underdense region, Cruz et al. (2005, 2006,
2007), and of a quadrupole-octupole alignment Schwarz et al. (2004); Copi et
al. (2006, 2007, 2010, 2015). Such anomalies could either mark new physics, be
caused by statistical effects Bennett et al. (2011), or reflect unknown
systematics.

Furthermore, puzzling tensions have been found between the cosmological
fluctuations $\sigma_{8}$ parameters measured by cluster number counts and
weak lensing, compared with the 2013 Planck results Planck Collaboration et
al. (2014), as well as between the 2013 Planck $H_{0}$ Hubble parameter,
compared with SNIa data. Those tensions have persisted between the Planck 2015
data and the $\sigma_{8}$ growth rate Macaulay et al. (2013), as well as with
the CFHTLenS weak lensing Raveri (2015).

In addition to those CMB issues, the description of small scale structures in
the $\Lambda$CDM model also raise problems (e.g. Moore, 1994; Moore et al.,
1999; Ostriker and Steinhardt, 2003; Boylan-Kolchin et al., 2011, 2012; Oh et
al., 2011). They count the “missing satellite problem” (MSP Klypin et al.,
1999; Moore et al., 1999), the “Too Big To Fail” problem (TBTF Boylan-Kolchin
et al., 2011, 2012), the angular momentum catastrophe (AMC van den Bosch et
al., 2001; Cardone et al., 2009), the satellites planes problem Pawlowski et
al. (2014), and the baryonic Tully-Fisher relation reproduction problem
McGaugh (2011). Among those small-scale problems, the so called Cusp-core
problem (CC Moore, 1994; Flores and Primack, 1994) retains a central role. The
CC coins the gap between the flat central slopes of density profiles in dwarf
galaxies (i.e. dwarfs), Low Surface Brightness (LSBs), and Irregulars galaxies
on one hand, while they all are DM dominated, and on the other hand, the cuspy
central slopes of dissipationless N-body simulations density profile
predictions Navarro et al. (1996, 2010); Saburova and Del Popolo (2014).

Besides the aforementioned issues, further problems of the $\Lambda$CDM
paradigm Kroupa et al. (2005, 2010); Kroupa (2012); Kroupa et al. (2012);
Kroupa (2015) are left out of this review, as we focus here on the CC problem.
Note however that the formation of cusps and issues of satellites in
$\Lambda$CDM are tightly connected, as shown through the mechanism in some
unified models Zolotov et al. (2012); Brooks and Zolotov (2014); Del Popolo et
al. (2014) that can turn cusps into cores as well as alleviate the MSP, since
the shape of the parent halo determines strongly its tidal effect on
satellites (e.g. Mashchenko et al., 2006, 2008; Peñarrubia et al., 2010), with
cored profiles even capable of stripping the satellite out of existence
Peñarrubia et al. (2010).

We organise the discussion as follows: starting with a short exposition of the
initial discussions on the CC problem, Sec. 2 will summarise its issues, while
early solutions involving baryons, DM matter type changes or gravity
modifications will be presented in Sec. 3. A review of the baryonic solutions
to SPH simulations collapse models through gas heating feedback effects from
dynamical friction of baryonic clumps will be discussed in Sec. 4, including a
discussion in Sec. 4.1, where the supernovae explosions Sommer-Larsen et al.
(1999) solutions of core heating will be presented, a Sec. 4.2 that will
discuss both baryonic approaches and a unified baryonic solution (Del Popolo
et al., 2014) and a presentation of the resulting mass dependent density
profile in Sec. 4.3. Finally, Sec. 5 will review and discuss solutions
modifying the DM nature, before to conclude in Sec. 6.

##  2 Exposition of the Cusp/Core Problem

The CC problem emerged after Flores & Primack Flores and Primack (1994) and
Moore Moore (1994) showed DDO galaxies’ rotation curves could be better fitted
with cored (or pseudo-)isothermal density profiles, ruling out cuspy profiles.
As dissipationless CDM model simulations persisted to produce cuspy profiles
(see Fig. 1), this was coined as the CC problem.

Figure 1: Illustration of the CC problem. The cuspy NFW profile is displayed
on the left panel, with the dotted vertical line representing the profile’s
scale radius. The right panel exemplifies the cored profile, with the dotted
vertical line marking the profile’s core radius.

Indeed, such cuspy DM profile was obtained in the dissipationless N-body
simulations of Navarro, Frenk, & White Navarro et al. (1996, 1997), yielding
inner density $\rho\propto r^{\alpha}$, for the power index $\alpha=-1$.
Moreover, they found such profile (thus coined NFW profile) to emerge
universally, thus independently of cosmology and scale, from dissipationless
simulations. This universality was challenged by the obtaining of a steeper
profile at $\alpha=-1.5$ Moore et al. (1998); Fukushige and Makino (2001), or
even by studies observing dependence of the inner slope on the nature of the
object and/or on its mass (Jing and Suto, 2000; Ricotti, 2003; Ricotti and
Wilkinson, 2004; Ricotti et al., 2007; Del Popolo, 2010; Cardone et al., 2011;
Del Popolo, 2011; Del Popolo et al., 2013; Di Cintio et al., 2014). Finally
(see Fig. 2), the so-called Einasto profile, yielding a central flattening
down to $\simeq\\!\\!-0.8$ (Stadel et al., 2009), tend to better fit more
recent N-body dissipationless simulations (Gao et al., 2008). Nevertheless,
the CC problem persists, and has been strongly debated for two decades, as
even this smallest predicted inner slope from dissipationless N-body
simulations exceeds slopes outputted from SPH simulations (Governato et al.,
2010, 2012), semi-analytical models (Del Popolo, 2009; Cardone and Del Popolo,
2012; Del Popolo, 2012a, b; Del Popolo and Cardone, 2012; Del Popolo and
Hiotelis, 2014; Popolo, 2001) or even observations (Burkert, 1995; de Blok et
al., 2003; Swaters et al., 2003; Kuzio de Naray and Kaufmann, 2011; Oh et al.,
2011, 2011).

Historically, the first contradicting results originated from dwarf and LSB
observation in HI: their rotation curves (RCs) did not match the NFW halo
Hernandez and Gilmore (1998); McGaugh and de Blok (1998); Côté et al. (2000).
Fueling the debate, steep profiles could not be excluded for several objects
studied by van den Bosch & Swaters van den Bosch and Swaters (2001). However
evidence against steep profiles further mounted through H$\alpha$ data, in de
Blok & Bosma de Blok and Bosma (2002), in the inner slopes $\alpha=-0.2\pm
0.2$ measured by de Blok et al. de Blok et al. (2003), and $\alpha=-0.22\pm
0.08$ obtained by Spekkens et al. Spekkens et al. (2005), although
contradicted by RCs agreeing with cusply profiles produced by Hayashi et al.
Hayashi et al. (2004). Nevertheless, flat profiles were usually in agreement
with high resolution observations (Blais-Ouellette et al., 2001; Kuzio de
Naray et al., 2008, 2009; Weldrake et al., 2003; Trachternach et al., 2008,
etc.).

Understanding of the construction of RCs was refined with their decomposition
for some spiral galaxies into their components in terms of stars, gas and DM,
fitted with various models in Gentile et al. Gentile et al. (2004, 2007);
Salucci et al. (2007). The results favoured constant density core profiles
over cuspy models, in the same way as the THINGS (The HI Nearby Galaxy Survey)
7 dwarf galaxies processed by Oh et al. Oh et al. (2011). Fig. 3 compares the
RCs of the latter, among others, with those of NFW and pseudo-isothermal (ISO)
profiles, clearly showing the ISO fits them best. The LITTLE THINGS survey
also produced similar results Oh et al. (2015).

However, as the previous results emanate mainly from low brightness objects
such as dwarfs or LSBs, while large galaxies or high-surface brightness
objects inner structure determination is more complex, general statements on
inner density profiles of all galaxies, in particular on their cored or cuspy
nature, are far from settled. Indeed, while the high-surface brightness
galaxies studied by Spano et al. (2008) present cored centres, their analysis
by (e.g., Simon et al., 2005; de Blok et al., 2008; Del Popolo and Cardone,
2012; Del Popolo et al., 2013; Martinsson et al., 2013) disagree. Other
surveys contain a mix of cored and cuspy objects. In particular, in the THING
sample, galaxies with low luminosity, $M_{B}>-19$, tend to follow a cored ISO
profile, while luminous galaxies, $M_{B}<-19$, are equally well fitted by
cored or cuspy profiles.

Such confused picture is reinforced by the lack of systematically flat slopes
in dwarf galaxies: in Simon et al. Simon et al. (2005), a large scatter in the
inner slopes of the low mass spirals NGC2976, NGC4605, NGC5949, NGC5963, and
NGC6689, was obtained, with a cored profile
$\alpha\simeq{\color[rgb]{0,0,0}{-}}0.01$ for NGC2976, a cuspy slope
$\alpha\simeq{\color[rgb]{0,0,0}{-}}1.28$ for NGC5963, while
$\alpha\simeq{\color[rgb]{0,0,0}{-}}0.88$ for NGC4605,
$\alpha\simeq{\color[rgb]{0,0,0}{-}}0.88$ for NGC5949 and
$\alpha\simeq{\color[rgb]{0,0,0}{-}}0.80$ for NGC6689. We illustrate this in
Fig. 4: its top left panel confronts NGC5963’s DM halo RC (black dots with
error-bars) to RCs computed from  a) the (Del Popolo, 2009) model that
accounts for baryonic physics (yellow dashed line),  b) a fitted NFW profile
(cyan line),  c) a fitted pseudo-isothermal profile (ISO, short-dashed magenta
line),  while its top right panel confronts the same kinds of fits to the RC
from NGC5949, and the bottom panel only displays NGC2976’s RC with a flat
power law (black line) and model a) (dashed line). The NGC5963 RC is well
fitted by the cuspy NFW profile’s. The NGC5949 RC is equally well

Figure 2: Comparison of density profiles with simulations. The NFW profile
(solid lines) is compared in the top left panel with simulations from Ref.
Navarro et al. (2004). Note that the Einasto profile fits also well those
simulations Navarro et al. (2010). The same comparison for the Moore profile
is displayed in the top right panel, and each panel represents simulations of
dwarfs galaxies (red line), medium size galaxies (green line), and galaxy
clusters (blue line). The bottom left and right panels focus on the residuals
between the corresponding theoretical profiles and the compared simulations
(this figure is reproduced from Navarro et al., 2004, Fig. 1).

approximated by RCs from a cuspy NFW or ISO cored profiles. Finally, the
NGC2976 RC reflects a very flat inner profile ($\alpha\simeq 0.01$). The three
RCs displayed in Fig. 4 all agree with the Del Popolo (2012a) model.

It appears then that if a cored profile agrees with RCs of many dwarfs, some
remain clearly cuspy.

In addition, opposite conclusions on the core/cusp question have been drawn
from the same objects with similar techniques by different studies: Simon et
al. (2003) analysed NGC2976’s DM inner profile to lie within
$-0.17<\alpha<-0.01$, when the same object yielded sometimes incompatible
profiles in (Adams et al., 2014), traced either by gas, $\alpha=-0.30\pm
0.18$, or stars, $\alpha=-0.53\pm 0.14$.

In agreement with the preceding considerations, galaxies’ observed RCs were
recognised by Oman et al. (2015) to exhibit much more diversity than what
simulations produced.

The conclusion this brings emphasizes the difficulty of galaxies’ inner slope
assessment, including for dwarfs. This moreover calls for a redefinition of
the CC problem in terms of inner mass, rather than inner slope of galaxies
density or RCs. Indeed, as shown by several studies, including those mentioned
here, and despite kinematic maps recent improvements, there exists no
agreement on DM slope distributions correlated with morphologies (Simon et
al., 2005; Oh et al., 2011; Adams et al., 2014), but rather a range of
profiles.

This lack of agreement appears even more obvious for the extreme ends of the
mass spectrum of galaxies: for larger masses, dominated by stars, as in spiral
galaxies, and for lighter galaxies, such as dwarf spheroidals (dSphs), which
models require the account of biases, that result in opposite consequences.

e

Figure 3:  Confrontations between RCs (left panel) or density profiles (right
panel) from the pseudo-isothermal (ISO, red dashed line) and the NFW (solid
lines) profiles, to thoses from (a) the Governato et al. (2010) simulated
galaxies marked DG1 and DG2, and (b) 7 dwarf galaxies selected from THINGS.
The small red dots emphasize the NFW RCs for $V_{200}\in$ [10–90] km/s. RCs
are normalised to the rotation velocity $V$ at the distance where $\frac{d\log
V}{d\log R}=0.3$, noted $R_{0.3}$ (this figure is reproduced from Oh et al.,
2011, Fig. 6).

To interpret and assess these results on dSphs, a range of approaches have
been tried. As stellar orbit anisotropy and mass are degenerate in the
spherical Jeans equation (Evans et al., 2009), its results highly vary with
assumptions. Similar degeneracies also hinder the approach applying parameter
space maximum likelihood to Jeans modelling (Wolf and Bullock, 2012; Hayashi
and Chiba, 2012; Richardson and Fairbairn, 2013). With Schwarzschild
modelling, cored profiles are found (Jardel and Gebhardt, 2012; Breddels et
al., 2013; Jardel and Gebhardt, 2013; Jardel et al., 2013), for, e.g.,
Sculptor or Fornax dSphs. More recently a study by Hayashi et al. (2020) has
found that the majority of dSPhs favor cuspy profiles. Multiple stellar
populations methods also obtained cored profile (Battaglia et al., 2008;
Walker and Peñarrubia, 2011; Agnello and Evans, 2012; Amorisco and Evans,
2012) for Fornax (measuring its slope at $\simeq$ 1 kpc) and Sculptor (for its
slope at $\simeq$ 500 pc). Nevertheless, the lack of agreement on a single
inner distribution for dSphs still remains, with contradictory results such as
the cusp obtained from Schwarzschild modelling for Draco (Jardel et al.,
2013).

Such scatter preventing a unified understanding occurs as well for galaxy
clusters scales. In clusters MACS1206, MS2137-23, RXJ1133, A383, A1201, A963
and RXJ1133, studies with a combination of BCG’s (Brightest Central Galaxy)
stars velocity dispersion, weak and strong lensing, Sand et al. Sand et al.
(2004) found cores, and only the RXJ1133 profile could be compatible with the
NFW model. Such flatter profiles were also obtained, in similar studies (see
also (Del Popolo, 2014)), for A611 by Newman et al. (Newman et al., 2009,
2009), for A383 by Newman et al. (Newman et al., 2011, 2011) and for MS2137,
A963, A383, A611, A2537, A2667, A2390 by Newman et al. (Newman et al., 2013,
2013). This contradicts other studies such as the cuspy profile in the strong
lensing and X-ray observations of A611, among seven cuspy or flat profiles of
relaxed massive clusters with average slope $\alpha=0.50\pm 0.1$, by Donnaruma
et al. Donnarumma et al. (2011).

More generally, conflicting estimates with simulations are common with
gravitational lensing Dahle et al. (2003); Gavazzi et al. (2003); Donnarumma
et al. (2011), although much shallower slopes ($-$0.5) can also result from
that technique Sand et al. (2002, 2004); Newman et al. (2009, 2011, 2013); Del
Popolo (2002); Del Popolo et al. (2005). A similar wide slopes spanning can
stem from X-ray analyses, ranging from cores at $-$0.6 Ettori et al. (2002),

Figure 4: Presentation of the DM halo RCs (error-barred, black dots) for the
galaxies NGC5963, NGC5949 and NGC2976, provided in Ref. Simon et al. (2005).
In the two top panels showing resp. NGC5963, on top left, and NGC5949, on top
right, are represented NFW profile fitted RCs (dotted cyan lines), pseudo-
isothermal profile fitted RCs (ISO, short-dashed magenta lines) and the RCs
from Ref. Del Popolo (2012a)’s model (yellow dashed lines). The bottom panel,
displaying NGC2976, compares its observations with a density power-law fit
model (slope $\simeq 0.01$, solid line) and with Ref. Del Popolo (2012a)’s
model (dashed line; this figure is adapted from Ref. (Del Popolo, 2012a, Figs.
4 and 5)).

to NFW-like $-$1.2 Lewis et al. (2003), up to very cuspy profiles $-$1.9
Arabadjis et al. (2002). However, NFW also agrees with some X-ray profiles
Schmidt and Allen (2007).

Although the situation improved with more recent, higher resolution,
observations, as they agree on flatter-than-NFW profiles on average, compared
with the early data’s conflicting density profiles inner structures, such high
resolution pictures still display galaxy to galaxy inner structure diversity,
reflected in Oman et al. (2015) simulations.

Despite the agreement of the IC2574 galaxy’s RC with the simulations from Ref.
Oman et al. (2015) (presented in Fig. 5, shaded green band) for radii $>6$
kpc, the galaxy’s inner region behaviour differs completely from their
simulations. Such divergence emphasizes that correct RC slope predictions for
some inner regime doesn’t entail that the successful simulation remains
accurate in all regimes of the RC. As the main characteristic of the CC
problem resides in the inner galaxy’s mass deficit Oman et al. (2015), the
focus should lie in the DM excess in the central distribution produced by CDM
rather than on the density profile shape. The observation vs prediction
discrepancy already appears where the asymptotic value of the circular
velocity is reached Kuzio de Naray and Spekkens (2011).

It is therefore unfortunate that practically no observational papers in the
field depart from the inner slope estimation method using $\alpha$. A notable
exception can be found in the inner slope estimation via the integrated
indicator $\Gamma\equiv\Delta\log{M}/\Delta\log{r}<3-\alpha$ performed by
Walker and Peñarrubia (2011) in the cases of the Fornax and Sculptor galaxy
clusters. As $\Gamma$ is integrated, contrary to the local $\alpha$, its
evaluation is easier, but provides a lesser constraint on the inner slopes.

Finally, an alternate probe of the DM inner profile flattening to rotation
curves consists in the dynamical friction timing (DFT) technique, that is
assessing the time an object like a globular clusters (GC) requires to move
inside the distribution of mass. It clearly represents an indirect method to
obtain the DM inner slope, however it presents the advantage of directly
probing the density, rather that the integrated mass furnished by rotation
curves. Thanks to the presence of GCs in dwarf galaxies, like Sagittarius or
Fornax, (Hernandez and Gilmore, 1998) was able to show analytically that the
inner density profile must be cored. This behaviour focused on GCs was
confirmed by several simulations (Sánchez-Salcedo et al., 2006; Read et al.,
2006; Inoue, 2009; Dutta Chowdhury et al., 2020). The key interests of the DFT
technique are twofold:  1) it concentrates on smallest scale galactic systems
like dwarf galaxies, on which the proposed solutions to the CC problem show
their main drawbacks, and  2) its method is independent from that of rotation
curves, giving information on the structure of the mass distribution in the
studied systems.  Recently, DFT studies have shifted their focus from GCs to
DF on stars. Using analytical approaches, as in Hernandez Hernandez (2016), or
numerical techniques, in Inoue (Inoue, 2017), on ultra-faint satellite
galaxies of our galaxy, was shown that DM fractions imply DFT for the stars
themselves and are incompatible with cusped dark matter haloes, at these tiny
scales, out of reach of SNF models, and even maybe of DFBC solutions.

Since smaller mass objects are more likely to harbour inner profiles in
agreement with predictions of dissipationless, N-body, codes (cuspy), a
definite verdict on the nature of the inner slope, cored or cuspy, of the
smaller end of the mass spectrum, i.e. dSphs, is critical.

Figure 5: Observed RC from galaxy IC2574 (filled blue circles with error bars)
compared with the DG1 (open red circles) and DG2 (open red triangles)
simulated galaxies from (Governato et al., 2010). The green shaded region and
line, as well as the black line, respectively stands for the scatter of one
and both median RCs for simulated galaxies from Ref. (Oman et al., 2015, which
this figure, Fig.1, is extracted from).

##  3 Initially proposed CC problem solutions

In the wealth of propositions to solve the CC problem, the initial attempts
were determined by the limited state of knowledge and observations. Systematic
effect and poor resolution were first invoked as the origin for the CC
inconsistencies (e.g. van den Bosch et al., 2000; van den Bosch and Swaters,
2001): they blamed beam smearing, slit-misplacement, non-circular motions or
beam off-centering, as they are sources of systematically lowering slopes.

For instance, finite HI emission beam size tends to smear the HI observations,
inducing a larger measured disk, depending on the HI distribution, beam size,
intrinsic velocity gradients and galaxy inclination angle. With modern high
spatial resolution ($<$1 kpc, see below), the smear can be resolved.

The slit misplacement for an H$\alpha$ emitting galaxy can entail missing the
galaxy’s dynamic centre, inducing a flatter measured profile. One of the
solutions to avoid this slope underestimate have used 3d spectroscopy (Kuzio
de Naray et al., 2008; Spano et al., 2008). As H$\alpha$ observations assume
circular motions for the gas, any non-circular orbits underestimate the slope.
Some have found Trachternach et al. (2008) that those departures from circular
velocity remain within the order of a few km/s. Current high resolution
rotation curve data are able to discriminate cuspy from cored haloes with
asymptotic inner slopes Kuzio de Naray and Kaufmann (2011).

Projection effects were put forward by several papers Power et al. (2003);
Navarro et al. (2004); Hayashi et al. (2004) to explain the discrepancy of
their simulations with observations, despite their claims of good agreement of
their profiles, becoming shallower in the range the virial radius towards the
centre. Their argument lies on the practice of representing the 3D DM haloes
motion with a spherically averaged circular velocity, traced by the gas disk
rotation speeds. Their claim is to blame the disagreement with data on the
correspondence fitting formulae rather than their simulations Hayashi et al.
(2004). Current DM-only high resolution simulations clearly reject this
explanation, as their inner profiles cannot reach shallower slope than
$\simeq-0.8$ Stadel et al. (2009), compared with high resolution galaxy
observations revealing much smaller values.

More generally, simulations limitations, such as over-merging or lack of
relaxation or resolution, have been blamed to source the CDM model’s failure
de Blok et al. (2003, 2001); Borriello and Salucci (2001). This was already
questioned then by Ref. Diemand et al. (2004)’s convergence tests proving the
dissipationless N-body simulations of the time were adequate to properly
obtain the CDM density profiles, despite their neglecting the galaxies inner
kpc regions effects of baryons. Such effects were shown to dominate the DM
generated profiles in clusters’ central 10 kpc regions Sand et al. (2004);
Newman et al. (2009, 2011, 2013). Modern simulations no longer suffer from the
blamed limitations and hydrodynamic cosmological high resolution models now
account for baryons, as will be discussed in the next section on baryonic
resolutions of the CC Problem.

Questioning of the very paradigm of CDM opened the door to speculations on the
nature of DM, such as warm protect(WDM Sommer-Larsen and Dolgov, 2001), fluid
Peebles (2000), repulsive Goodman (2000), fuzzy Hu et al. (2000), decaying Cen
(2001), self-annihilating Kaplinghat et al. (2000), or self-interacting
Spergel and Steinhardt (2000) DM. More fundamental questioning led to altered
small scale DM power spectrum (e.g., Zentner and Bullock, 2003) or
modifications of gravity itself, such as the so-called $f(R)$ (Buchdahl, 1970;
Starobinsky, 1980) or $f(T)$ (Bengochea and Ferraro, 2009; Linder, 2010; Dent
et al., 2011; Zheng and Huang, 2011) theories, or the so-called Modified
Newton Dynamics (MOND, Milgrom, 1983a, b)111 Generalisations of Einstein’s
General Relativity, $f(R)$, and $f(T)$ theories modify gravity in distinct
ways. Replacing, in their Lagrangian, the Ricci scalar by a function $f(R)$,
the first type dates from the Buchdahl Buchdahl (1970) introduction in 1970
but picked up interest with Starobinsky Starobinsky (1980) a decade later.
Developed from the Teleparallel Equivalent of GR, $f(T)$ theories also modify
their Lagrangian from the torsion scalar to a function. On one hand, both
types were introduced to absorb the origin of cosmic acceleration into gravity
and dispense from the need for DE (Bengochea and Ferraro, 2009). On the other
hand, MOND, introduced to fit galaxies rotation curves by Milgrom in 1983
Milgrom (1983a, b), purports to replace DM. The DM effects of the MOND
Newtonian modification also lead to other GR generalisations with the same aim
Mendoza et al. (2013); Campigotto et al. (2017); Barrientos et al. (2018);
Barrientos and Mendoza (2018). Such paradigmatic modifications, whether
touching DM nature, initial conditions or the nature of gravity, will
hereafter be denoted, in their impact the CC problem, as "cosmological
solutions", and developed in Sec. 5.

##  4 Baryonic Resolutions of the CC Problem

Before resorting to the above mentioned “cosmological solutions” to deal with
the CC problem, a thorough check, whether neglected or poorly understood local
physics could explain it, would be wise. Indeed, discarding the concordance,
i.e. $\Lambda$CDM, model would risk to significantly alter the otherwise very
successful predictions $\Lambda$CDM provided, thus loosing the explanations
for many observations of our Universe. This is the aim of the current section.

Keeping the framework of the $\Lambda$CDM model, “astrophysical solutions”
alter the results of pure CDM structure formation with some “heating”
mechanism applied to inner DM distribution, originated in the effects of
baryons. The inner density profile gets then flatter than for pure CDM
evolution. Such “heating” mechanism could be provided by

  1. 1.

accounting for the action, in structure formation, of angular momentum;

  2. 2.

the induced dynamics from a central black hole, claimed to shallow the DM cusp
Le Delliou et al. (2011, 2010, 2011);

  3. 3.

the modification of equilibrium from the presence of a rotating bar;

  4. 4.

the mechanism of dynamical friction that induces baryons-DM transfer of
angular momentum El-Zant et al. (2001, 2004); Del Popolo (2009);

  5. 5.

the supernova (SN) explosions-generated bulk motions in gas, dubbed AGN
feedback Mashchenko et al. (2006, 2008); Governato et al. (2010).

Larger angular momentum acquired by a proto-structure has been correlated to
flatter inner density profile by many contributors within the framework of
spherical Nusser (2001); Hiotelis (2002); Le Delliou and Henriksen (2003);
Ascasibar et al. (2004); Williams et al. (2004); Del Popolo et al. (2013a, b,
c, 2001); Del Popolo and Gambera (1997); Cardone et al. (2011); Del Popolo
(2006), or even elliptic Popolo (2002), infall models.222A General
Relativistic version of secondary infall models is at the root of works by a
group including Mimoso and Le Delliou Mimoso et al. (2010, 2013); Le Delliou
et al. (2011, 2013). Such models were able to produce RCs agreeing with those
of dwarf galaxies Williams et al. (2004), which profile builds flatter than
giant galaxies’ because structure’s density peak height333A proto-structure’s
peak height measures the ratio of its central peak overdensity $\delta(0)$ to
its mass variance $\sigma$ (see Del Popolo and Gambera, 1996), as
$\nu=\delta(0)/\sigma$. More massive objects are characterised by larger
$\nu$. anticorrelate with the angular momentum they get during their
formation.

Dynamical friction was shown to flatten, and even erase, the DM formed cusps
in both galaxy clusters and dwarfs by transferring energy from baryon clumps
to DM El-Zant et al. (2001, 2004). Conversely, DM adiabatic contraction (AC)
in haloes steepens their density profiles Blumenthal et al. (1986); Gnedin et
al. (2004).444As computed with iterative methods (e.g., Spedicato et al.,
2002).

The above effects (see also Del Popolo and Kroupa, 2009) were all combined in
the model from Del Popolo Del Popolo (2009), including:

  * -

tidal torques caused ordered angular momentum acquired during the structure
formation;

  * -

its obtained random angular momentum;

  * -

its dynamical friction induced baryons-DM energy and angular momentum
exchanges;

  * -

and the effects of its DM adiabatic contraction (hereafter, AC).

Figure 6: Halo density profile evolutions with redshift (following the DFBC
scenario of Del Popolo, 2009). The evolution for a $10^{9}M_{\odot}$ halo
yields profiles at $z=10,3,2,1$, and 0, characterised resp. by the solid,
long-dashed, short-dashed, dot-dashed and dotted lines, and presented in the
left panel. The profile evolution, ending at $z=0$ with the total (DM+Baryons)
mass density profile of a $10^{14}M_{\odot}$ halo and marked by the dot-dashed
line, is reproduced in the right panel. That DM profile is characterised at
$z=3,1.5,$ $1$ and 0 resp. by the solid, dotted, short-dashed, and long-dashed
lines (this figure is extracted from Del Popolo, 2009, Figs. 3 and 5).

In addition to inducing flatter density profiles in the center of galaxies and
clusters, dynamical friction and angular momentum also tend to modify their
overall internal structure Del Popolo (2002); Del Popolo and Gambera (2000);
Colafrancesco et al. (1995); Del Popolo (2003). The simple confrontation of
real structures, including baryons, with the outputs of dissipationless
simulations is demonstrably incorrect Del Popolo (2009): the non-negligible
action of baryons in the inner regions of structures explains the failures of
N-body simulations to reproduce observed density profiles. This is illustrated
in Fig. 6, showing halo evolution for two masses, $10^{9}$ and
$10^{14}M_{\odot}$, modeled using the “Dynamical Friction from Baryonic
Clumps” scenario (hereafter DFBC scenario) from Ref. Del Popolo (2009) and
discussed below.

The impact of baryons content, environment and formation history on dwarf
galaxies density profiles was demonstrated with the Del Popolo DFBC model Del
Popolo (2012a).

The “heating” mechanism 5 was first put forward by Navarro et al. Navarro et
al. (1996), flattening the inner profile from the feedback of SN explosion
into the medium, as discussed below.

In the present state of the field, the discussion on astrophysical solutions
to the CC problem have narrowed down to the following mechanisms

  1. 1.

the cusp reduction through “supernovae feedback flattening” (SNFF) (Navarro et
al., 1996; Gelato and Sommer-Larsen, 1999; Read and Gilmore, 2005; Mashchenko
et al., 2006, 2008; Governato et al., 2010, 2012), and

  2. 2.

the cusp flattening via the “Dynamical Friction from Baryonic Clumps” (DFBC)
(El-Zant et al., 2001, 2004; Ma and Boylan-Kolchin, 2004; Nipoti et al., 2004;
Romano-Díaz et al., 2008, 2009; Del Popolo, 2009; Cole et al., 2011; Inoue and
Saitoh, 2011; Nipoti and Binney, 2015).

They will be discussed in more details in Secs. 4.1 and 4.2, and are put in
context below.

In general, baryons and DM usually “interact” in smooth distributions via AC,
that can be counterbalanced by transferring energy back from baryons to DM. AC
leads DM to collapse towards the centre of the halo, steepens its profile and
increase its central density. The counteracting transfer has the opposite
effect. It can occur via

  1. 1.

Dynamical friction transfer of incoming clumps orbital energy to DM.
This leads DM particles to drift towards the halo edges and thus to flatten
the DM density profile (El-Zant et al., 2001, 2004; Ma and Boylan-Kolchin,
2004; Nipoti et al., 2004; Romano-Díaz et al., 2008, 2009; Del Popolo, 2009;
Cole et al., 2011; Inoue and Saitoh, 2011; Nipoti and Binney, 2015, see also
the review of Pontzen and Governato (2014)).

  2. 2.

DM particles “heating” from galactic internal energy sources (Pontzen and
Governato, 2014).
Such source expelling from, or relocating baryons in, the halo (for instance,
bulk motions induced from SN explosions, (Mashchenko et al., 2006, 2008)) can
temporarily flatten the gravitational potential, shift DM particles outwards,
and flatten the cusp.

We will now discuss the two favoured “heating” mechanisms in the CC problem
solution, and how those solutions are synthesised into the proposal of a mass
dependent density profile for DM haloes with baryons.

###  4.1 Discussion on Supernovae Feedback Flattening

Historically, the first solution proposed to the CC problem while accounting
for the effects of baryons, as initially suggested in Ref. Flores and Primack
(1994) and confirmed by many efforts, employed SNF (Navarro et al., 1996;
Gelato and Sommer-Larsen, 1999; Read and Gilmore, 2005; Mashchenko et al.,
2006, 2008; Governato et al., 2010, 2012; Teyssier et al., 2013; Sawala et
al., 2016).

Profile flattening by a single sudden baryons expulsion into the halo was
first shown by Ref. Navarro et al. (1996), and found to be most effective in
dwarfs and other shallow potential structures. However, energy from a single
explosion was shown Gnedin and Zhao (2002) to remain insufficient to drive the
profile to a core, while repeated moderately violent explosive events would be
required (see Garrison-Kimmel et al., 2013, for a different point of view).

A more detailed study of the Navarro et al. (1996) approach, focused on
evolving from NFW profiles to DDO154’s RC via a gas outflow event modeled from
an abrupt change in the disk potential, by Gelato and Sommer-Larsen Gelato and
Sommer-Larsen (1999), established that at least 75% of the disk mass should be
expelled that way to generate this RC.

In the case of larger galaxies and deeper potential structures, a core was
shown by Read and Gilmore Read and Gilmore (2005) to result from repeated and
alternate outflows and re-accretions of gas.

In primordial galaxies, a core was determined to follow from SN-explosions-
driven gas random bulk motions Mashchenko et al. (2006, 2008), as confirmed in
Governato et al. (2010) simulations. The average slope of the latter
simulations were compared with observations of THINGS dwarfs in Refs. Oh et
al. (2011, 2011). Producing simulations for larger galaxies than Governato et
al. (2010), and comparing them with observations Governato et al. (2012),
correlation between the stellar mass $M_{*}$ and the inner slope were obtained
for $M_{*}>10^{7}M_{\odot}$ galaxies were obtained. These results were
obtained from simulated galaxies thanks to the N-Body+SPH code GASOLINE
(Wadsley et al., 2004). With an 86 kpc softening, gas particles and DM
particles resolution were reduced to $M_{\rm p,gas}=3\times 10^{3}M_{\odot}$
and $M_{\rm p,DM}=1.6\times 10^{4}M_{\odot}$, reps., thanks to the “zoom”
technique of Ref. (Katz and White, 1993). Two realisations were produced,
distinguished by their star formation thresholds: a High Threshold run (HT),
with star forming hydrogen cloud densities $>$$100/$cm3, and a Low Threshold
run (LT), with hydrogen densities $>$$0.1/$cm3. Both SNF blast wave mechanism
(Stinson et al., 2006) and/or early stellar feedback (Stinson et al., 2013)
were implemented in those simulations, as later by Di Cintio et al. (2014) in
the same way. The Governato simulations resulting energy feedback to the
interstellar medium (ISM) amounted to $10^{51}$ ergs from $>$$8$$M_{\odot}$
stars, while the ISM gas ejecting energy was coupled to the SN expelled energy
via the coefficient $\epsilon_{\rm esf}$. The fiducial $\epsilon_{\rm
esf}=0.1$ was selected for the MaGICC simulations (Stinson et al., 2013).

Comparably, a combination of bursty star formation and SNF, inducing inner (1
kpc) halo potential fast oscillations and expanding gas bubbles when the
central cold gas density reaches $>$100/cm3 and stars start to form555This
process agrees with the assumptions from Ref. Governato et al. (2010); the
bulk gas flows start beyond $>$10/cm3 (Ceverino and Klypin, 2009)., was shown
to flatten the cusp Pontzen and Governato (2012). However in this model, no
noticeable modifications of the DM inner density profiles are induced for
smaller densities (e.g., 0.1/cm3). The case of larger mass galaxies was
treated by Governato et al. Governato et al. (2012) with the same method (also
see Fig. 7, top panels). Agreement with the results of (Pontzen and Governato,
2012) (Fig. 7, bottom left panel) was obtained from introducing a new SNF
scheme in the RAMSES adaptive mesh refinement code by Teyssier et al. Teyssier
et al. (2013), showing flat galaxies inner profile for
$M_{*}>10^{7}M_{\odot}$. Similar outcomes with a lower threshold at
$M_{*}>10^{6}M_{\odot}$ were produced by Onorbe et al. Oñorbe et al. (2015)
(Fig. 7 bottom right panel)666 Essentially, $M_{*}<10^{6}M_{\odot}$ galaxies
are not able to turn cusps into cores under the SNF mechanism..

Since their simulations found little effect of baryons on rotation curves of
galaxies with $V_{\rm max}<60$ km/s, even in their inner regions, this
perspective was denounced by Oman et al. (2015). They argued that the cores
obtained by Pontzen and Governato (2012), since no evidence of core formation
was detected in their Oman et al. (2015) work, were inherently reflecting an
ad hoc choice of parameters.

Figure 7: Density profiles including baryonic effect. The Ref. (Governato et
al., 2012, Fig. 1)’s hydrodynamic simulations present density profile
evolution (top left panel) reflected on their inner logarithmic slopes (at 0.5
kpc). The Ref. Governato et al. (2012)’s results (red crosses) are compared,
in the top right panel, for different stellar mass galaxies, with THINGS
galaxies’ results (open squares), and with DM-only N-body simulation
(reproduced from Governato et al., 2012, Fig. 3), shown as a solid line.
Results from hydrodynamic simulations are presented in the bottom panels:
density profile evolution from Ref. (Teyssier et al., 2013, Fig. 5) are shown
on the bottom left panel, while the bottom right presents hydrodynamic
evolution for three different types of dwarfs, where stars all form either in
early times, in medium or in late times (Oñorbe et al., 2015, Fig. 6).

Such understanding agrees with the findings of Ref. González-Samaniego et al.
(2014), which 7 high-resolution dwarfs simulations, included in halos with
$1-2\times 10^{10}M_{\odot}$ and following different assembly history, could
not reveal any inner core flattening. A very extended assembly history dwarf,
thus also having an extended star formation rate (SFR) history, exhibits their
lowest detected inner slope: they measured it to reach, at 0.01–0.02 $R_{\rm
vir}$, up to $-$0.8. It also remains consistent with the core-less simulated
realistic galaxies of Ref. Schaller et al. (2015).

In addition to those contradictions in core formation between various high
resolution hydrodynamical simulations, several criticisms have arisen on the
methods and assumptions used in simulation models of the types of (Governato
et al., 2010; Pontzen and Governato, 2011), as they obtain faulty

  1. 1.

core formation energetics: the number of stars, obtained in their
$M_{*}<10^{7}M_{\odot}$ galaxies, generates an insufficient energy to flatten
the galaxies cusp (Peñarrubia et al., 2012). Moreover, their required core-
forming baryonic mass marginally exceeds the dSphs observed baryon content
(Garrison-Kimmel et al., 2013). That problem in Penarrubia Peñarrubia et al.
(2012) is illustrated in Fig. 8’s left panel. However, the right panel
displays the opposite results from Maxwell’s study.
In addition, the SNFF solution they propose for the CC problem, requiring the
formation of numerous SNs, entails a star formation efficiency (SFE)
constraint towards larger values, opposite to the SFE constraint induced by
the complementary solution to the Too-Big-To-Fail (TBTF) problem Boylan-
Kolchin et al. (2011, 2012)777“Too Big to Fail” is used in the context of
Milky Way (MW) satellite simulations producing bigger objects than observed MW
satellites without any mechanism explaining why such object would fail to be
detected.

Figure 8:  Minimal SN energy/mass output, $\Delta E$, needed to convert a DM
cusp into a core as a function of halo mass, $M_{h}$, measured at virial mass,
$M_{\rm vir}$, in the left panel. The constraints on the minimal amount of
energy as a function of mass can also be expressed in terms of stellar mass,
standing for luminosity, shown in the right axes of each panel. Conversion
from stellar mass proceeds, in the left panel, from a luminous MW satellites
constrained star formation efficiency $F_{*}=F_{*}(Mvir)$, the resulting
luminosities converting into SNeII energy output via Ref. Peñarrubia et al.
(2012)’s Eq. (6) with strong energy coupling $\epsilon_{DM}=0.4$, while it
assumes, in the right panel, 100% efficiency ($\epsilon_{DM}=1$).
The left panel, reproduced from (Peñarrubia et al., 2012, Fig. 2), presents:
a) the range of minimum SN explosions energy required to obtain a core of size
$r_{c}$ given within $0.1<r_{c}/r_{s}<c$, with $c$ and $r_{s}$, respectively
being the concentration parameter and the scale radius of the NFW profile at
given virial mass $M_{\rm vir}$, as the red shaded area; b) for a fixed core
size $r_{c}=1$ kpc, the dotted black line is obtained; c) the SN explosion
energy outputs, indicated from two different studies Koposov et al. (2009);
Kravtsov (2010) as dot-dashed green and dashed blue lines, denote those
compatible with the “missing satellite” Klypin et al. (1999); Moore et al.
(1999) problem and reveal, for haloes with $M_{\rm vir}<10M_{\odot}$, the
tension with the “core/cusp” problem.
The right panel, extracted from (Maxwell et al., 2015, Fig. 3), shows: a) the
left panel red shaded area of Penarrubia et al. Peñarrubia et al. (2012)’s
energy estimates, as shaded grey area; b) solid lines with symbols for the
conversion energy of cuspy to pseudo-isothermal density profiles at fixed core
sizes, indicated in the legend by the symbol types; c) the solid black line
for the $\Delta E$ scaling with halo mass $M_{h}$, while the ratio of the cusp
mass redistribution limit radius, $r_{m}$, over the halo radius, $r_{h}$, is
fixed; d) the dotted, dashed, and dot-dashed lines, respectively, for the
Behroozi et al. Behroozi et al. (2010), Kravtsov Kravtsov (2010), and Moster
et al. Moster et al. (2010) $M_{\star}-M_{h}$ relations.

  2. 2.

energy coupling: their $\epsilon_{\rm SN}\simeq 0.4$ exceeds the 0.05 coupling
that studies such as Revaz and Jablonka (2012) deduced;

  3. 3.

star formation threshold: the Governato et al. (2010) results have been found
in (Sawala et al., ; Oman et al., 2015) to require a very high star formation
threshold;

  4. 4.

tensions with TBTF solution: those models simply encounter adverse conditions
(Ferrero et al., 2012; Garrison-Kimmel et al., 2013; Papastergis et al., 2015)
when trying to solve the TBTF problem;

  5. 5.

feedback resolution: accurately describing the cusp transformation into a core
via their feedback processes would require better resolution than they present
(Choi et al., 2015; Laporte and Peñarrubia, 2015; Laporte and White, 2015;
Marinacci et al., 2014).

Despite the SNF limitations,888Recall footnote 6.for
$M_{\ast}>10^{6}M_{\odot}$, correlations between $M_{\ast}$ and galaxies inner
slope have been found, comparing THINGS dwarves average slopes with Governato
et al. (2010)’s simulations in Refs. Oh et al. (2011, 2011), and similarly in
Governato et al. (2012) for larger objects.

Inversely, hydrodynamic simulations for $M_{*}<10^{6}M_{\odot}$ produce cuspy
profiles, in contradiction with the Fornax and Sculptor cored inner structure
found by Ref. Walker and Peñarrubia (2011) thanks to their more reliable DM
haloes’ slopes from mass profiles $\Gamma\equiv\frac{d\log M}{d\log
r}<3-\alpha$. Only with 50% more SN energy injections could Ref. Madau et al.
(2014) claim agreement with Walker and Peñarrubia (2011)’s cores.

Assuming SN feedback and other specific assumptions, a Semi-Analytic model
from Gnedin and Zhao Gnedin and Zhao (2002) claimed impossibility to obtain
cores. As, among details of stellar feedback, the core formation impact of
baryonic clumps, such as in Ref. Del Popolo (2009), is ignored, such objection
is weakened.

However, there exists one core-forming SNF dwarf galaxies simulation for
masses $<10^{6}M_{\odot}$: with the P-SPH mode in the GIZMO code, Oñorbe et
al. (2015) obtained cores in a much less natural way than in the DFBC
scenario, discussed in the next section.

###  4.2 Discussion of Gas Clumps Dynamical Friction

As discussed in the introduction of the present Sec. 4, Refs. El-Zant et al.
(2001, 2004) proposed an alternate path from SNF for the turning of cusps into
cores: baryon clumps with mass ratios to the system above $\simeq\\!0.01\%$
can transfer, through dynamical friction (DF), clumps orbital energy to DM.
Such heating induces similar effects to SNF without the need to wait for a
full massive stellar cycle, moving to outer orbits the central DM particles
and thus flattening the inner DM density profile. It is more efficient on
earliest, smallest haloes of the hierarchical structure formation.

DFBC was first shown to function in galaxies El-Zant et al. (2001), then in
clusters El-Zant et al. (2004). Cluster C0337-2522 evolution with such model
Nipoti et al. (2004) showed the BCG generation precedes DFBC inner slope
formation, resulting in flatter than NFW inner profiles with
$0.49<\alpha<0.90$.

Implemented in a hybrid N-body/SPH simulation, DFBC galaxy evolution showed
heated up cusps by baryons subhalos into a core of $\simeq$1 kpc, and was
compared with DM-only systems and mixed DM/baryons systems (Romano-Díaz et
al., 2008).

Refs. (Cole et al., 2011; Inoue and Saitoh, 2011; Nipoti and Binney, 2015)’s
simulations confirmed such outcomes.

The main steps of the DFBC can be outlined in the following:

  * •

the linear phase develops from initial DM and diffuse gas proto-structures.

  * •

DM gravitational collapse first form non-linear potential wells, attracting
then the baryons.

  * •

Unstable accreting gas then fragment into clumpy structures (e.g., Noguchi,
1998, 1999; Immeli et al., 2004a, b; Bournaud et al., 2007; Agertz et al.,
2009; Aumer et al., 2010; Ceverino et al., 2010, 2012), leading to highly gas-
rich discs.

  * •

Further instability in the rotating discs, triggered by their surface density,
$\Sigma$, exceeding a threshold at $Q\simeq\sigma\Omega/(\pi G\Sigma)<1$
(Toomre, 1964, with $\sigma$, the disc’s 1-D vertical velocity dispersion and
$\Omega$, its angular velocity, while $Q$ is related with the instability of
the vertical kinetic to gravitational balance to the excess of centrifugal
angular velocity), fragment it into clumps, which largest can reach mass
ratios with the disc of a few percent and 1 kpc radii (e.g., Krumholz and
Dekel, 2010). Typical clumps, in galaxies with baryon mass
$10^{10}$–$10^{11}M_{\odot}$, range within $\simeq 10^{8}$–$10^{9}M_{\odot}$
(see Agertz et al., 2009; Ceverino et al., 2010, 2012), are rotationally
supported, in Jeans equilibrium Ceverino et al. (2010) and stable for long
periods ($\simeq 2\times 10^{8}$ Myr).

  * •

As in local star-generating systems and following the Kennicutt-Schmidt law, a
few percents of those clumps’ gas converts to stars Krumholz and Dekel (2010),
while DF migrates them to the DM halo (galactic) center.

  * •

The first collapse phase’s AC compressed baryons (Blumenthal et al., 1986;
Gnedin et al., 2004, and e.g., $10^{9}M_{\odot}$ galaxy at $z\simeq 5$ in our
Fig. 6), reenforcing the DM cusp.

  * •

The previous effect is countered by DF-induced migration of clumps to galaxy
center, as DF transfers energy and angular momentum from baryons to DM,
heating up the cusp into a core.

  * •

SNF only intervenes later (e.g., around $z=2$), when SN explosions expel gas,
decreasing surrounding stellar density. Moreover, such feedback destroys the
smallest gas clumps as soon as a small fraction of their mass turns to
stars.999Star formation is not an efficient process.

Although sharing some common features (e.g., clumps to DM energy transfer via
gravitational interaction), leading some to consider SNFF and DFBC as the same
idea implemented differently, they fundamentally differ on two crucial points:

  1. 1.

their respective profile flattening epochs differ markedly: while pure SNFF DM
profile does not depart noticeably from NFW at $z\simeq 3$ (Pontzen and
Governato, 2011; Macciò et al., 2012; Oñorbe et al., 2015), DFBC flattening
initiate at higher redshifts ($z<5$, see Fig. 6);

  2. 2.

their clumps moving energy sources diverge in nature: in Refs. (Mashchenko et
al., 2006, 2008)’s definition, while clumps motion is driven, in the SNFF
model, by SN explosions energy, the DFBC views them to just “passively” infall
to the halo centre.

The two competing scenarios predictions for the central density slope vs.
stellar mass and vs. circular velocity relationships were confronted in Refs.
(Del Popolo and Pace, 2016; Del Popolo, 2016) with a series of high resolution
data predictions from the Milky Way, Sculptor, Fornax, LITTLE THINGS (Oh et
al., 2015), THINGS dwarfs (Oh et al., 2011, 2011) and Refs. (Adams et al.,
2014; Simon et al., 2005). Such confrontation showed slightly better
performance from the DFBC model compared with the SNFF scenario, as the former
predicts core emergence at smaller stellar masses than $10^{6}M_{\odot}$,
limiting the latter. Although very small dwarfs ($M_{*}\leq 10^{4}M_{\odot}$)
cannot produce cores in DFBC, this nevertheless agrees with results from Ref.
Weidner et al. (2013).

Both competing model face the same caveats in their resolution of the CC
problem: galaxies containing a bulge have been demonstrably able to
reconstitute their cusp (Del Popolo and Hiotelis, 2014). This effect can even
been found in dwarf galaxies (Laporte and Peñarrubia, 2015).

###  4.3 Mass dependent DM density profiles

Assuming that baryons are indeed responsible for some cored DM haloes, while
allowing some others to be cuspy, and using the results of SPH simulations,
the SNFF or DFBC models, some authors synthesised the diversity of DM density
profiles and generalized the usual density profiles, giving the mass dependent
empirical law that would solve the CC problem. For example, Lazar et al.
(2020) generalized the Einasto profile into a mass dependent parameterised
profile, encompassing both cuspy and cored cases, while (Freundlich et al.,
2020) obtained such mass dependent profile by modifying the Zhao (1996)
profile. In the following we will give the details concerning the works of Di
Cintio et al. (2014), and (Freundlich et al., 2020).

In previous discussions, we saw that the DM density profile is well
approximated by the NFW law, and even better fitted by the Einasto profile.
Those two profiles are obtained by DM-only simulations. There, the
hydrodynamical processes, fundamental in the determination of the structure of
the density profile, are neglected. For example AC Blumenthal et al. (1986);
Gnedin et al. (2004) produced by gas cooling, which strengthens cusps, is not
taken into account by N-body, DM-only simulations. At the same time, the
presence of baryons can expand haloes. As discussed in Secs. 4.1 and 4.2,
stellar feedback and DFBC can produce a halo expansion and a flattening of the
inner density profile. As discussed there, Di Cintio et al. (2014) showed how
the inner slope is modified by SNF. The results of this paper were used to
obtain a DM halo whose slope depends from the ratio between stellar and DM
masses, $\frac{M_{*}}{M_{\rm halo}}$.

The generalised NFW profile (gNFW) is a specific form of a law depending on
three parameters $(\alpha,\beta,\gamma)$:

$\rho(r)=\frac{\rho_{s}}{\left(\frac{r}{r_{s}}\right)^{\gamma}\left[1+\left(\frac{r}{r_{s}}\right)^{\alpha}\right]^{(\beta-\gamma)/\alpha}}$
(1)

being $r_{s}$ the scale radius and $\rho_{s}$ the scale density. $-\gamma$ and
$-\beta$, are the logarithmic slopes of the inner and outer regions
respectively, while the sharpness of the transition from inner to outer region
is specified by $\alpha$. The peculiar case $(\alpha,\beta,\gamma)=(1,3,1)$
corresponds to the NFW profile. The goal of Di Cintio et al. (2014) was that
of making the profile given in Eq. (1) mass dependent, in the sense that its
three parameters would depend on the ratio $\frac{M_{*}}{M_{\rm halo}}$, using
haloes obtained, in Di Cintio et al. (2014) from SPH simulations. The density
profile of the haloes were fitted, using Eq. (1). The behavior of the three
parameters was captured using fitting laws as a function of
$\frac{M_{*}}{M_{\rm halo}}$. The outer slope $\beta$ was modeled with a
parabola. In the cases of the inner slope, $\gamma$, and of the transition
parameter, $\alpha$, double power law model were used.

Figure 9: Best fits to the generalised NFW parameters $\gamma$ (green),
$\beta$ (red), and $\alpha$ (black) plotted in terms of $\frac{M_{*}}{M_{\rm
halo}}$. The symbols mark various SPH simulated galaxies (Figure reproduced
from Di Cintio et al., 2014, Figs. 1).

The best fits for these parameters to the values for SPH galaxies are shown as
dotted lines in Fig. 9. The functional forms of the fits are:

$\displaystyle\alpha=2.94-\log_{10}[(10^{X+2.33})^{-1.08}+(10^{X+2.33})^{2.29}]$
(2) $\displaystyle\beta=4.23+1.34X+0.26X^{2}$
$\displaystyle\gamma=-0.06+\log_{10}[(10^{X+2.56})^{-0.68}+(10^{X+2.56})]$

where $X=\log_{10}(\frac{M_{*}}{M_{\rm halo}})$.

Given the stellar-to-halo mass ratio of a galaxy, by means of Eqs. (2) and
(1), it is possible to compute the entire DM profiles.

Di Cintio et al. (2014) also showed that the concentration parameter is
dependent on $\frac{M_{*}}{M_{\rm halo}}$. The dependence of the concentration
parameter on $\frac{M_{*}}{M_{\rm halo}}$ is shown in Fig. 10. Each simulation
is represented by its symbol and size as described in (Di Cintio et al., 2014,
Table 1). The dependence of the ratio between concentration parameters
obtained from SPH and pure DM N-body simulations, $C_{SPH}/C_{DM}$, on
$\frac{M_{*}}{M_{\rm halo}}$ is nearly exponential. Its best fit is:

$\frac{C_{SPH}}{C_{DM}}=1.0+0.00003e^{3.4X_{+}}$ (3)

where $X_{+}=\log_{10}(\frac{M_{*}}{M_{\rm halo}})+4.5$.

Figure 10: Ratio, for a given halo, between concentration parameters of its
models, in the SPH simulation and in the DM-only simulation, as a function of
$\frac{M_{*}}{M_{\rm halo}}$. The best fit to $C_{SPH}/C_{DM}$ is represented
by the dashed red line (Figure reproduced from Di Cintio et al., 2014, Figs.
4).

$C_{DM}$ has been derived fitting a NFW profile to the DM-only version of each
galaxy, while $C_{SPH}$ has been obtained from hydrodynamical runs where their
galaxies DM halo model profile had their scale radius correspondence converted
between $r_{s}$ into $r_{-2}$.

A comparison of the model with simulated data of the density profiles, and
rotation velocities gives a good agreement.

The mass dependent density profile idea from Ref. (Di Cintio et al., 2014) was
applied by several other authors. As mentioned above, it was applied by Lazar
et al. (2020) to the Einasto profile and by Freundlich et al. (2020) to the
so-called Dekel-Zhao density profile, a peculiar case of Eq. (1). We now focus
on another popular profile, similar in performance to the Einasto profile, the
modified NFW model, adopted by Freundlich et al. (2020) in the form from Zhao
(1996)

$\rho(r)=\frac{\rho_{c}}{x^{a}(1+x^{1/b})^{b(g-a)}}$ (4)

where the radial dependence is encoded in $x=r/r_{c}$, with respect to the
characteristic radius $r_{c}$, and the density scale is set by the typical
density $\rho_{c}$. It is equivalent to the generalised NFW profile, noting
the parameters $\alpha=1/b$, $\beta=g$ and $\gamma=a$. Note that, for any
natural numbers $n$ and $k$, setting $g=3+k/n$ and $b=n$, it is possible to
integrate the mass and obtain the corresponding equilibrium velocity
dispersion and gravitational potential analytically. Some choices of integers
actually yield very good DM simulations fits, and are even able to accommodate
baryons’ presence, as well as cusps or core.

The profile referred to as the Dekel-Zhao (DZ) profile, found for $k=1$ and
$n=2$, which corresponds in Eq. (4) to $b=2$ and $g=3.5$, and leaves two free
parameters, $a$ and a concentration parameter $c$, have been found, using the
NIHAO suite of simulations (Wang et al., 2015), particularly efficient at
representing cored profiles (Dekel et al., 2017, but also cuspy profiles).

Following Eq. (4), it takes the form

$\rho(r)=\frac{\rho_{c}}{x^{a}(1+x^{1/2})^{2(3.5-a)}}$ (5)

using $x=r/r_{c}$ and defining the concentration $c=R_{\rm vir}/r_{c}$ from
the virial equilibirum radius of the spherical mass distribution $R_{\rm
vir}$, where the average density at that radius computes as
$\overline{\rho_{\rm vir}}=3M_{\rm vir}/4\pi R_{\rm vir}^{3}$, while the shape
factor $\mu=c^{a-3}(1+c^{1/2})^{2(3-a)}$ is involved in the profile mass
integration, yielding the average critical density
$\overline{\rho_{c}}=c^{3}\mu\overline{\rho_{\rm vir}}$, and thus relating the
density scale to the shape parameters, $c$ and the central slope $a$, as
$\rho_{c}=(1-a/3)\overline{\rho_{c}}$.

As measurements are made at fixed given resolution, the central slope is
actually an integrated slope with respect to the actual profile. Freundlich et
al. (2020) explicits, for a fixed resolution $r_{1}$, an inner logarithmic
slope

$s_{1}=\frac{a+3.5c^{1/2}(r_{1}/R_{\rm vir})^{1/2}}{1+c^{1/2}(r_{1}/R_{\rm
vir})^{1/2}},$ (6)

and the corresponding averaged concentration parameter

$c_{2}=c\left(\frac{1.5}{2-a}\right)^{2}.$ (7)

The physically motivated restrictions to positive density and inner slope
result in the conditions $a\leq 3$ and $a+3.5c^{1/2}(r_{1}/R_{vir})^{1/2}\geq
0$, respectively. One can reobtain the intrinsic parameters $(a,c)$, employed
in analytic expressions, from the observed $(s_{\rm 1},c_{\rm 2})$, usually
bestowed to numerical tests, by inverting Eqs. (6) and (7), as

$a=\frac{1.5s_{1}-2\left(3.5-s_{1}\right)\left(r_{1}/R_{vir}\right)^{1/2}c_{2}^{1/2}}{1.5-\left(3.5-s_{1}\right)\left(r_{1}/R_{vir}\right)^{1/2}c_{2}^{1/2}}$
(8)

and

$c=\left(\frac{s_{1}-2}{\left(3.5-s_{1}\right)\left(r_{1}/R_{vir}\right)^{1/2}-1.5c_{2}^{-1/2}}\right)^{2}.$
(9)

Freundlich et al. (2020, Eq. (18), Eq. (21), and Section 2.3, respectively)
provided analytic expressions for the DZ profile’s gravitational potential,
velocity dispersion and lensing properties, as well as, using some NiHAO SPH
simulations suite’s profiles (see their Fig. 3), compared the DZ, gNFW and
Einasto profiles through fits, assuming their parameters free, and finally
finding the DZ profile performs significantly better than Einasto’s and
marginally better than the gNFW on NiHAO.

In this context, the mass dependent, DZ-based profile from Freundlich et al.
(2020), was constructed, as in Di Cintio et al. (2014), from the NIHAO
simulations suite (Wang et al., 2015), to set the DZ parameters functional
forms. SPH simulations were fitted by Di Cintio et al. (2014) with a similar
profile to Eq. (4), to set the free parameters forms as a function of $M_{\rm
star}/M_{\rm vir}$, as did Lazar et al. (2020) with the Einasto profile and
Freundlich et al. (2020) with the DZ profile. However, the mass dependence
introduced, similarly to Eqs. (2) by Di Cintio et al. (2014), to improve
galaxies and clusters density profiles fits, such as in the modified Einasto
profile from Lazar et al. (2020) or in the modified NFW from Read et al.
(2016), does not allow to keep any analytic form for the mass, as detected
through lensing, velocity dispersion or gravitational potential. On the
contrary, the DZ profile keeps that possibility intact.

This is the advantage of the DZ profile fit by Freundlich et al. (2020),
applied to simulated NIHAO haloes’ density profile logarithms, optimising the
parameters, spacing the $N\simeq 100$ grid profile radii $r$ in the range 0.01
$R_{\rm vir}$-$R_{\rm vir}$, through a least-square minimization. Setting the
virial mass $M_{\rm vir}$, the profile can set the scale by extracting the
corresponding $R_{\rm vir}$, leaving the remaining free parameters $a$ and
$c$. The star distribution can be samely fitted to get the star mass at virial
radius $M_{\rm star}$ and obtain the corresponding $M_{\rm star}/M_{\rm vir}$.
The theoretical slope and concentration are related to the integrated
(observed) ones through Eqs. (6) and (7), and Eqs. (8) and (9), respectively.

Figure 11:  Ratios, for integrated slopes and concentrations $s_{1}$ (lower
panel) and $c_{2}$ (upper panel), between the DZ model parameters and their
DM-only NFW profiles counterparts, as a function of mass ratios $M_{\rm
star}/M_{\rm vir}$. The NFW integrated parameters, $s_{DMO}$ and $c_{DMO}$,
were obtained from the Dutton and Macciò (2014) relation (Eqs. 47 and 48 of
Freundlich et al., 2020, and their best fits, Eqs. (45) and (49) on their Fig.
9, here marked as plain black lines). The best fit residual rms $\sigma$ is
shown in gray. The dashed black line in the $c_{2}/c_{DMO}$ panel represents
the best-fit function from Di Cintio et al. (2014). Color coding, indicated on
the right side in each panel, for each halo compounds the information of the
other panel.

This fitting procedure allowed to obtain, for each NIHAO haloes, their $s_{1}$
and $c_{2}$ as a function of $M_{\rm star}/M_{\rm vir}$, as seen Freundlich et
al. (2020, Fig. 8, black dots).

The obtained $s_{1}$ and $c_{2}$ behaviour were synthesised in two
parameterised functions of $x=\frac{M_{*}}{M_{\rm halo}}$, where $x_{0}$,
$s^{\prime}$, $s^{\prime\prime}$, and $\nu$ are ajustable and which used
values were given in Freundlich et al. (2020, Table 1)

$s_{1}(x)=\frac{s^{\prime}}{1+\left(\frac{x}{x_{0}}\right)^{\nu}}+s^{\prime\prime}\log\left(1+\left(\frac{x}{x_{0}}\right)^{\nu}\right),$
(10)

while the parameter $c^{\prime}$ is introduced for

$c_{2}(x)=c^{\prime}\left(1+\left(\frac{x}{x_{0}}\right)^{\nu}\right).$ (11)

The resulting mass-dependent profile presents the advantages, over the Di
Cintio et al. (2014) approach, to require less free parameters as well as to
allow integration into analytic expressions for lensing properties, dispersion
velocity and gravitational potential.

##  5 Cosmological Solutions to the CC Problem

The CC problem, together with the other small scale problems encountered by
the $\Lambda$CDM model, open the question of a more general failure of the
CDM’s paradigm. This justifies models in which the nature of DM or gravity
itself are modified. Such models encountered various degrees of success in
their already widely checked instances. We will here focus on alternate models
of DM. Several approaches have been used and we will detail the two main
avenues, before discussing other possibilities.

###  5.1 Warm DM

Warm DM (WDM) presents the simplest departure from CDM, endowing DM with a
small velocity dispersion ($\sigma\simeq 100$ m/s nowadays), also linked, for
thermally produced DM, to a small DM relic mass (Colín et al., 2000; Sommer-
Larsen and Dolgov, 2001). Since velocity dispersion is expected to decrease
with time, it should be more important in the past and thus smear small scale
structure formation. Such smear induces similar DM “heating” as the baryonic
solutions of Sec. 4: in this case the WDM particles higher velocity than for
CDM smears the haloes to flatter profiles as well as produce fewer low mass
haloes. These effects of WDM on structure formation have been copiously
simulated (e.g., Polisensky and Ricotti, 2011; Lovell et al., 2012; Macciò et
al., 2012; Angulo et al., 2013).

Despite solving the CC problem for haloes with their corresponding scales on a
case by case basis by tuning its particle mass, WDM does not manage to solve
it with one mass for all galaxies, nor in the entire CC problem mass range
(Kuzio de Naray et al., 2010). As shown in Ref. Wu and Kroupa (2015), neither
WDM or pure collisionless CDM models can match observed disk galaxies rotation
curves, while they can agree with hydrodynamical simulations including baryons
(see Sec. 2’s discussion).

In addition, Ref. (Polisensky and Ricotti, 2011, for a $m=2$ keV thermal
relic), and several other authors (e.g., Dalal and Kochanek, 2002; Fadely and
Keeton, 2011, 2012), shown that, for instance, the strong-lensing subhalo
fraction is too high compared with WDM produced subhaloes. Furthermore, WDM
thermal mass displays tensions between expected core size (1 kpc core
corresponding to 0.1 keV) and large scale structure ($m\simeq$ 1–2 keV leads
to 10–20 pc cores, see Fig. 12 and Ref. Macciò et al., 2012). In conclusion,
because its power spectrum falls off too sharply, WDM does not improve on CDM
(Schneider et al., 2014). Finally, since WDM structure formation is also shown
to modify the Lyman-$\alpha$ forest (Narayanan et al., 2000), the situation is
even worse, as WDM solution to the CC problem in our vicinity cannot, at the
same time, be consistent with observed high redshift Lyman-$\alpha$ forest.

Figure 12:  WDM effect on the CC problem, reproduced from (Macciò et al.,
2012, Figs. 2 and 8). The WDM core formation is illustrated against the CDM
density profile (solid black line) by showing profiles from a range of five
WDM particle masses, from $m=2keV$ (WDM1, as per the legend) to 0.05 keV
(WDM5), in the left panel. The right panel synthesises the WDM core radius
obtained for its given particle mass with other constraints: while the solid
black line marks the core size dependence at fixed halo mass, the shaded area
limits the plane to the cosmological constraints for $0.15<\Omega_{m}<0.6$ and
the vertical dashed line sets the upper limit from large scale observations.

###  5.2 Self-Interacting DM

The second simplest variation away from CDM, coined Self-interacting DM
(hereafter SIDM, Spergel and Steinhardt, 2000), consists in granting to DM a
self-interaction cross-section, within the same magnitude of nucleon-nucleon
interactions ($\simeq$(m/g)-1cm2).101010Note that 1 cm2/g $\simeq$ 1 barn/GeV,
so multiplying by the DM particle mass yields the cross-section. The resulting
elastic scattering in regions of highest density (inner galactic regions)
redistributes energy and angular momentum so that tri-axiality is reduces and
a Burkert profile core forms (Burkert, 2000).

The CC problem has been claimed to be solved in dwarfs, MW-sized galaxies, and
galaxy clusters by SIDM from some cosmological simulations (Newman et al.,
2013, 2013; Rocha et al., 2013; Peter et al., 2013) consistent with merging
galaxy clusters observations (Clowe et al., 2006; Randall et al., 2008; Dawson
et al., 2012, 0.1–0.5 cm2/g cross sections). Fig. 13 presents simulations of
haloes from galaxies to clusters masses, with $\sigma/m$ set within two
values, in the case of Ref. (Rocha et al., 2013). Although SIDM appears to
solve the CC problem, since its cored subhaloes are more sensitive to
disruption and tidal stripping than CDM’s, and to improve on WDM (Rocha et
al., 2013; Peter et al., 2013), as enough subhaloes survive, a different point
of view still remains (Kuzio de Naray et al., 2010).

The SIDM’s attraction does not entirely resides in its structure formation
effects. As “hidden sector” particle models naturally produce them (e.g.
Spergel and Steinhardt, 2000; Yoshida et al., 2000; Davé et al., 2001), it
also retains appeal from the particle theory point of view.

Figure 13: SIDM simulations compared with CDM, indicated by black dots, in
density profiles, reproducing a figure from Ref. (Rocha et al., 2013, Fig. 4).
Two SIDM models, designated in the legend as $SIDM_{1}$ and $SIDM_{0.1}$, are
shown, using DM particle masses $\sigma/m=1$, marked with blue stars, and
$0.1$, characterised with green triangles, respectively. They are ovelayed
with the NFW, plotted as a black line, and the Burkert, traced in blue,
density profiles. The core radii of Burkert’s profiles are located with
arrows.

###  5.3 Other DM models

Other proposals to change from the CDM model are more or less loose
declinations altering around the SIDM concept. Here is a list of the main
variations:

RDM

switching interaction to negative scattering results in repulsive DM (RDM,
Goodman, 2000).

The next two variations from SIDM are very commonly related to indirect DM
detection:

SADM

if self-interaction produces a DM particles annihilation, the result is coined
Self-Annihilating DM (SADM, Kaplinghat et al., 2000, proposing cross section-
velocity $\sigma v\simeq 10^{-29}$ (m/GeV) cm2). Such annihilation decreases
the dense regions’ particle numbers, in particular in the halo’s centre,
reducing central gravity, therefore allowing to expand central particles’
orbits and thus to flatten the central profile. At the same time, annihilation
results in radiation emissions that are possibly detectable.

DDM

Alternately, self-interaction can result in DM decay into relativistic
particles, designated as Decaying DM (DDM, Cen, 2001). Their gravitational
effect on structures is similar to SADM, as they equally deplete galaxies’
central density, since the relativistic particles escape away, while larger
scales structures behaviour remain similar to CDM. Similarly, the relativistic
decay products produce radiations that can also be detected.

The last main variations are more loosely related to SIDM:

BCDM

Superfluid behaviour of non-relativistic, massive boson condensates in haloes
centre can also result in smoothing down their profiles from cusp to cored
(Harko, 2011). Recent structure formation simulations (Schive et al., 2014)
demonstrated that the Bose condensate DM (BCDM) small scale gravity vs.
uncertainty principle opposition reduced substructures and flattened the
density profile at those scales, while producing larger scale structures
indistinguishable from CDM outputs.

SFDM

As scalar field condensation also forms a Bose condensate, similar flat
galaxies inner profiles were obtained from such implementation of the BCDM
(Robles and Matos, 2012), that was called Scalar Field Dark Matter (SFDM)

FDM

Further implementation of the gravity vs. uncertainty principle exploited the
wave-particle duality to obtain Fuzzy DM (FDM, Hu et al., 2000), using
galactic core sized Compton wavelength, ultra-light ($m\simeq 10^{-22}$ eV)
scalar particles. As they cannot be “squeezed” below their Compton wavelength,
they also develop flatter profiles and form less substructures.

##  6 Conclusions

Since the first assessment by Öpik of the amount of matter surrounding our
solar system through vertical star motion with respect to the plane of the
ecliptic more than a century ago Öpik (1915), current data indicates a
dominant amount of matter should be in the form of DM. However, such dynamics
and geometric data has still not been confirmed, to date, by direct or
indirect DM particles detection Del Popolo (2013, 2014).

Grasping the essence of DM from a gravitational point of view should privilege
the study of galaxy satellites, dwarf galaxies and other small scale
structures, dominated by DM, all the more as they present significant
departures from the $\Lambda$CDM model predictions. In this review, we focused
on the departure constituted by the CC problem, formulated more than twenty
years ago. Although we now have some competing explanations, there is no
concensus on the CC problem’s cause.

In this review, we have discussed the focus on baryonic effects on small scale
structures. We recalled that the DFBC model can explain cores in
$M>10^{5}M_{\odot}$ dwarfs from baryon clumps interactions with DM, while
cusps form in MW-type galaxies Del Popolo and Pace (2016). Although the SNFF
model gets similar results, its prediction cover structures in a more
restricted range $M_{*}>10^{6}M_{\odot}$ Governato et al. (2012). Note that
both models, by depleting the DM density in central halo regions, also explain
other small scale problems such as the TBTF or the subhaloes abundance
problems. However, baryonic solutions to the CC and other small scale problems
are still debated, as some issues remain, such as with the SNFF in classical
MW dwarfs Peñarrubia et al. (2012); Garrison-Kimmel et al. (2013), or the
possible future discovery of $M_{*}<10^{6}M_{\odot}$ dwarfs with flat inner
profiles which would drive the SNFF model to conclude that DM is actually not
cold111111Note that such conclusion is not true for the DFBC model., or even
that no baryonic physics solutions clearly appear for the TBTF problem in
isolated galaxies. Escaping this debate would entail understanding how the
SIDM model or one its variants, as we have discussed, could solve the CC
problem. Future surveys, including measurements of star velocities, would
anyhow be required to further our understanding of the CC problem and other
small scale issues.

From an observational point of view, as discriminating cores from cusps in
dSphs remains delicate, the discussion on their structure persists (see the
introduction and, e.g., Strigari et al., 2014). Although the Subaru Hyper-
Supreme-Camera Takada (2010) and GAIA de Bruijne (2012) were foreseen as
providing possible resolution for the question, their abilities would not
exceed larger dwarf galaxies, of the size of Sagittarius Richardson et al.
(2014). Based on Jeans equations, their determination of density profile using
stars line of sight velocity and 2D projection radius are plagued with
degeneracy with the anisotropy parameter. Notwithstanding the challenges GAIA
faces from it Richardson et al. (2014), improvements to measure dwarfs’s stars
proper motions of the Strigari et al. (2007), that yield density slope at
half-light radius, were recommended, that focused on only one component out of
the three velocities and on just two components out of three positions (see
Battaglia et al., 2013). However successful these efforts may be,
understanding dwarfs inner structure will be crucial. In contrast, the
difficulties of the SNFF tensions do not appear fundamental: baryons, through
the more efficient DFBC model, can still turn cusps into cores sooner than gas
turns into stars. Consequently, should cored dwarfs with
$M_{*}<10^{6}M_{\odot}$ be discovered, despite their inconsistency with the
SNFF, provided $M_{*}>10^{5}M_{\odot}$, the $\Lambda$CDM model correct with
baryonic effects of the DFBC model would remain consistent with observations.

Beyond the CC problem and the other documented issues in small scales of the
$\Lambda$CDM model, further testing could involve

  * •

checking the predicted large subhaloes number within galaxies’ virial radius,
barely agreeing with anomalies in the gravitational lensing fluxes Metcalf and
Amara (2012);

  * •

measuring subhaloes perturbations of the MW’s cold tidal streams (see Carlberg
and Grillmair, 2013).

Finally, the nature and existence of DM would find a definitive solution in
detection of DM particles, whether in colliders, direct or indirect means.
Despite the hopes raised by a di-photon excess decay at 750 GeV at 3.9
$\sigma$ significance in the LHC, as it was later ruled out as a statistical
fluctuation, no evidence of super-symmetry (SUSY), that would allow WIMPS
particles DM, have so far emerged and collider detection of DM is yet elusive.
Although controversial, the annual modulation claimed in DAMA/LIBRA/CoGeNT
remains the only possible direct DM particle detection. To date, no
incontrovertible direct, indirect or collider evidence for the nature or
existence of DM has been announced.

###### Acknowledgements.

MLeD acknowledges the financial support by the Lanzhou University starting
fund, the Fundamental Research Funds for the Central Universities (Grant No.
lzujbky-2019-25), National Science Foundation of China (grant No. 12047501)
and the 111 Project under Grant No. B20063. All authors have contributed
equally in this work. The authors declare no conflict of interest.

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