extra-dimensional space-time. Thus, for the TeV energy scale new research
windows will be opened by the current generation of cosmic-ray observatories
(e.g. H.E.S.S.[339, 340, 341], IceCube [342]) and the next generation of
experiments, e.g. Cherenkov Telescope Array (CTA) [343, 344] and the Square
Kilometre Array (SKA) [345]. These experiments, beside the detection, will
able to constraint the WIMPS mass to much higher value than the current one.

A careful analysis of high redshift RCs will be crucial for validating any new
particle proposed to address the cusp-core issue emerging from all similar
analysis of RCs of local galaxies.

One of these is fuzzy DM (FDM). FDM consists of ultra-light bosons that have
mass in the range $10^{-23}$–$10^{-20}$ eV, and arise from symmetry breaking
due to the misalignment mechanism in the string theory landscape [303, 346].
Since the de Broglie wavelength of a FDM particle is a few kpc, the density
fluctuations on a scale below the de Broglie wavelength are unstable and lead
to the formation of a halo whose minimum mass is $\sim 10^{7}\,{\rm
M}_{\odot}$ for a boson mass of $\sim 10^{-22}$ eV. Differently, density
fluctuations on a scale above the de Broglie wavelength lead to a large scale
structure of a DM halo indistinguishable from CDM [347]. As recently shown in
numerical simulations, an imprint of FDM particles is the formation of a core
of DM in the innermost part of each virialized halo that exhibits a flat
density profiles [347]. In recent analyses of the kinematic of dwarf
spheroidals, ultra-faint and ultra diffuse galaxies favour FDM with boson mass
$\sim 10^{-22}$ eV over cuspy NFW profile [348, 349, 350]. Nevertheless, the
debate on the fuzzy nature of DM is far to be closed. The existence of the
core in each FDM halo enhances the circular velocity in the inner part of the
rotation curve [351], questioning the model. Additionally, the boson mass
constrained with the analysis of the Lyman-$\alpha$ forest data ($>7\times
10^{-21}$ eV) [352] is almost two order of magnitudes larger than the one
required to account for the kinematic of stars in dwarf galaxies ($\sim
10^{-22}$ eV). Noticeably, for this particle, high redshift rotation curves
will provide us with a decisive test, since in this scenario the DM halos are
born with the central density core that we observe today, differently from
other scenarios in which the DM cores develop with time from originally cuspy
distributions ([319]). At the same time, SKA and Pulsar Timing Array
experiments may play an extremely important role providing us with the smoking
gun for FDM by measuring the Compton and de-Broglie scale modulation of Pulsar
Timing [353].

In general, it is very likely once we obtain $\rho_{DM}(r,z)$, where $\rho$ is
the DM density , $r$ is the radial coordinate and $z$ the redshift at which
this quantity has been measured, we will be able to single out the DM
particle.

##  18 Conclusions

The association of Einstein, Planck and Vera Rubin is made because these great
scientists have at least two aspects in common. First, they, as seen together,
represent the conjunction point of very different research lines of Physics.
Always more people think that such point must be reached in order if we have
to succeed in working out that New Physics which is required to frame a
currently very large number of unexplained observations and experiment
outcomes. The second is that they all provide examples of the time delay,
often existing in Physics, between the formulation of a theory or the
publication of an intriguing discovery and that of their general understanding
and validation.

The idea of the great usefulness, for the growth of Physics and our
understanding of the Universe, of a contamination between the fields of
research of Astrophysics, Cosmology, GR and Elementary Particles is not a
recent one but one has to date it 50 years back and recognize it as one of the
extraordinary intuitions of Dennis Sciama [354]. Later on, other scientists
have much contributed to it: [355, 356, 357, 358, 359]

Here, we have shown that this idea emerges naturally and roots many sub-fields
of Astrophysic, Astroparticle and Cosmology. These include: the cosmological
validation of the beyond GR scalar tensor theories, the (possibly not
$\Lambda$CDM) cosmological model for the expansion of the Universe, tensions
between different measurements of the same (cosmological model) parameters,
the nature of dark matter, the evolution of galaxy halos and their stellar
counterparts, the cosmological importance of the stellar BH in galaxies, the
evolution of the dark matter perturbations at large scale, the properties of
the biggest virialized objects of the universe, the Fundamental Physics
related to the scattering angle of two body systems and in curved space-time
backgrounds, the bridge between classical and quantum world, the complex
intrinsic neutrino nature and its cosmological importance, the tomographic
description of the Universe, the origin of dark Energy, and the Holy Grail of
the coupling between Gravity and Quantum Mechanics.

Let us stress that this list of fields of investigation covers only a part of
all the active fields that also entangle the GR, the quantum world and the
physics of the Universe and that can be successfully explored by means of the
multi-lateral approach we have advocated. These are, therefore, subjects for
the works of the present Special Topic. A short list of them includes: the
baryogenesis, the inflation, the formation of primordial and Supermassive BH,
the dark Ages of the Universe, the Astrophysical and cosmological impact of
Gravitational Waves.

Finally, it could come as a surprise but, in the above discussion, an unknown
entity, that we call dark matter, takes very often the center of the scene. In
detail, a large number of issues related to this mystery emerge in many sub-
fields of Cosmology, Physics and Elementary Particle Physics. However, one
point must be stressed. The investigations of the past decades have not been
able to tear the Maja veil off the Dark Matter Phenomenon. Our current
knowledge presents mysterious evidences, lacks of congruence and shortcomings
in both theoretical and experimental/observational sides. Theories like
General Relativity and Standard Model of Elementary particles seem unable to
account for the ”darkness” of the Universe. As result the answers of our
questions about dark Matter do not appear to be straightforward and shining
but seem to follow the Nietzsche’s directive: Beauty is False, Truth is Ugly.
New, multi-lateral approaches to the mysteries of the Universe are likely to
be the way to unveil the thick veil above.

##  19 APPENDIX A: COSMOGRAPHY

Cosmography is a technique to describe the expansion of the universe without
assuming any cosmological model. Such a tool is based only on the cosmological
principle, and consists in expanding the scale factor, $a(t)$, in Taylor
series around the present time ($t_{0}$):

$a(t)=1+\sum_{k=1}^{\infty}\dfrac{1}{k!}\dfrac{d^{k}a}{dt^{k}}\bigg{|}_{t=t_{0}}(t-t_{0})^{k}\
.$ (24)

The different orders of this expansion define the cosmographic series, i.e.
the Hubble, deceleration, jerk and snap parameters:

$\displaystyle H(t)\equiv\dfrac{1}{a}\dfrac{da}{dt}\ ,\hskip
28.45274ptq(t)\equiv-\dfrac{1}{aH^{2}}\dfrac{d^{2}a}{dt^{2}}\ ,$ (25a)
$\displaystyle j(t)\equiv\dfrac{1}{aH^{3}}\dfrac{d^{3}a}{dt^{3}}\ ,\hskip
14.22636pts(t)\equiv\dfrac{1}{aH^{4}}\dfrac{d^{4}a}{dt^{4}}\ .$ (25b)

Noteworthy, the above quantities take on important physical meanings such as
the acceleration or deceleration of the universe (by the sign of the parameter
$q$), the change of the Universe’s dynamics (by the sign of $j$) and the
discrimination between an evolving dark energy or a cosmological constant (by
the value of $s$).

The Taylor expansion of the luminosity distance can be obtained by the
definition $z=a^{-1}-1$ and Eq. (24):

$\displaystyle d_{L}(z)=$ $\displaystyle\
\dfrac{z}{H_{0}}\bigg{[}1+\dfrac{z}{2}(1-q_{0})-\dfrac{z^{2}}{6}\left(1-q_{0}-3q_{0}^{2}+j_{0}\right)+$
$\displaystyle\hskip
22.76228pt+\dfrac{z^{3}}{24}\left(2-2q_{0}-15q_{0}^{2}-15q_{0}^{3}+5j_{0}+10q_{0}j_{0}+s_{0}\right)+\mathcal{O}(z^{4})\bigg{]},$
(26)

which can be fitted by the cosmographic series to trace the expansion of the
universe without assuming any _a priori_ cosmological model. Indeed, the
background evolution can be derived as

$H(z)=\left[\dfrac{d}{dz}\left(\dfrac{d_{L}(z)}{1+z}\right)\right]^{-1}\,,$
(27)

and find

$H(z)\simeq
H_{0}\left[1+H^{(1)}z+H^{(2)}\dfrac{z^{2}}{2}+H^{(3)}\dfrac{z^{3}}{6}\right],$
(28)

where

$H^{(1)}=1+q_{0}\ ,\quad H^{(2)}=j_{0}-q_{0}^{2}\ ,\quad
H^{(3)}=3q_{0}^{2}+3q_{0}^{3}-j_{0}(3+4q_{0})-s_{0}\ .$ (29)

Therefore, it is possible, for a given cosmological model, to calculate the
cosmografic coefficients. Furthermore, it is possible to relate the
cosmographic series to the cosmological parameters characteristic of a
specific model. In the case of the $\Lambda$CDM model:

$q_{0}=-1+\dfrac{3}{2}\Omega_{m}\ ,\quad j_{0}=1\ ,\quad
s_{0}=1-\dfrac{9}{2}\Omega_{m}\ .$ (30)

Similarly, for the $w$CDM model, one finds

$\displaystyle q_{0}=\dfrac{1}{2}\left[1+3w(1-\Omega_{m})\right],\quad
j_{0}=1+\dfrac{9}{2}w(w+1)(1-\Omega_{m})\ ,$ (31) $\displaystyle
s_{0}=\dfrac{1}{4}\left[-14-81w(1-\Omega_{m})-27w^{3}(3-4\Omega_{m}+\Omega_{m}^{2})-9w^{2}(16-19\Omega_{m}+3\Omega_{m}^{2})\right],$
(32)

and the equation of state of dark energy reads

$w=-\dfrac{1+2j_{0}-6q_{0}}{3(1-2q_{0})}\ .$ (33)

##  20 Appendix B

We list below the level-5 post-Newtonian-accurate value for the level-4 post-
Minkowskian local part of the scattering angle ${\chi_{4}}_{\rm loc}^{5\rm
PN}(p_{\infty})$

$\displaystyle{\chi_{4}}_{\rm loc}^{5\rm
PN}(p_{\infty})=\pi\left(-\frac{94899}{32768}\pi^{2}\nu^{2}+\frac{93031}{32768}\pi^{2}\nu-\frac{1945583}{33600}\nu+\frac{1937}{16}\nu^{2}-\frac{2895}{32}\nu^{3}\right.$
$\displaystyle\left.\quad+\frac{525}{64}\nu^{4}+\frac{1845}{2048}\pi^{2}\nu^{3}\right)p_{\infty}^{6}\,.$
(34)

##  21 Appendix C: Constraining dark energy models at high redshifts

In this Appendix we show the analytical form of the basic cosmological
quantities in the different cosmological models investigated in our analisys.
For the CPL model we have:

$\displaystyle H(z,{\mathrm{\theta}})_{CPL}$ $\displaystyle=$ $\displaystyle
H_{0}\sqrt{(1-\Omega_{m})g(z,{\mathrm{\theta}})+\Omega_{m}(z+1)^{3}+(z+1)^{4}\Omega_{\gamma}{\tilde{N}}}\,$

where

$\displaystyle g(z,{\mathrm{\theta}})$
$\displaystyle=\exp^{3\int_{0}^{z}\frac{w(x,{\mathrm{\theta}})+1}{x+1}dx}\,,$
(35) $w(z)=w_{0}+w_{1}z(1+z)^{-1}\,,$ (36)
${\tilde{N}}=\left(\frac{7}{8}\left(\frac{4}{11}\right)^{\frac{4}{3}}N_{eff}+1\right)\,,$
(37)

and we set $N_{eff}=3$.

| CPL Parametrization
---|---
$Id$ | $\langle x\rangle$ | $\tilde{x}$ | $68\%\ {\rm CL}$ | $\langle x\rangle$ | $\tilde{x}$ | $68\%\ {\rm CL}$ |
| Full dataset | No SNIa
$\Omega_{m}$ | 0.23 | 0.24 | (0.19, 0.27) | 0.19 | 0.2 | (0.16, 0.22) |
$\Omega_{b}$ | 0.046 | 0.046 | (0.04, 0.047) | 0.055 | 0.054 | (0.045, 0.068) |
$w_{0}$ | -0.91 | -0.92 | (-1.1, -0.74) | -0.7 | -0.7 | (-0.8, -0.62) |
$w_{1}$ | -0.8 | -0.7 | (-1.05,-0.5) | -0.75 | -0.72 | (-0.93,-0.48) |
$h$ | 0.69 | 0.69 | (0.68, 0.71) | 0.67 | 0.67 | (0.64, 0.69) |
| Scalar field
$\Omega_{b}$ | 0.051 | 0.051 | (0.049, 0.052) | 0.051 | 0.051 | (0.050, 0.0514) |
$\mathcal{H}_{0}$ | 0.98 | 0.98 | (0.95,0.99) | 0.96 | 0.96 | (0.94, 0.98) |
$h$ | 0.69 | 0.68 | (0.67,0.695) | 0.67 | 0.67 | (0.65, 0.68) |
| Early Dark Energy
$\Omega_{m}$ | 0.29 | 0.29 | (0.27, 0.31) | 0.285 | 0.285 | (0.271, 0.298 ) |
$\Omega_{b}$ | 0.047 | 0.048 | (0.037, 0.052) | 0.045 | 0.048 | (0.035, 0.047) |
$w_{0}$ | -0.66 | -0.67 | (-0.85, -0.56) | -0.65 | -0.63 | (-0.75, -0.53) |
$\Omega_{e}$ | 0.04 | 0.035 | (0.032, 0.043) | 0.025 | 0.023 | (0.009, 0.039) |
$h$ | 0.71 | 0.71 | (0.69, 0.71) | 0.71 | 0.71 | (0.67,0.73) |

Table 1: Constraints on the parameters characterizing the different dark
energy models described above. Columns show the mean $\langle x\rangle$ and
median $\tilde{x}$ values and the $68\%$ confidence limits.

For the exponential potential we have:

$\displaystyle a^{3}(t)={t^{2}\over
2}[(3\mathcal{H}_{0}-2)t^{2}+4-3\mathcal{H}_{0}],$ (38) $\displaystyle
H(t)={{2\left(2(3\mathcal{H}_{0}-2)t^{2}+4-3\mathcal{H}_{0}\right)}\over{3t\left((3\mathcal{H}_{0}-2)t^{2}+4-3\mathcal{H}_{0}\right)}},$
(39)
$\displaystyle\Omega_{m}={{(4-3\mathcal{H}_{0})\left((3\mathcal{H}_{0}-2)t^{2}+4-3\mathcal{H}_{0}\right)}\over[2(3\mathcal{H}_{0}-2)t^{2}+4-3\mathcal{H}_{0}]^{2}},$
(40)
$\displaystyle\Omega_{\varphi}={{(3\mathcal{H}_{0}-2)t^{2}\left(4(3\mathcal{H}_{0}-2)t^{2}+3(4-3\mathcal{H}_{0})\right)}\over[2(3\mathcal{H}_{0}-2)t^{2}+4-3\mathcal{H}_{0}]^{2}}\,$
(41)
$\displaystyle\varphi(t)=-\sqrt{\frac{2}{3}}\log\left(\frac{6.48}{\left(3\mathcal{H}_{0}-2\right)t^{2}-3\mathcal{H}_{0}+4}\right)\,,$
(42)

where $\mathcal{H}_{0}$ is a constant. $\Omega_{\varphi}$ is the scalar field
density parameter. In order to determine the integration constants we set the
age of the universe, $t_{0}$, as a unit of time, $t_{0}=1$, so that
$a_{0}=a(1)=1$, and $\mathcal{H}_{0}=H(1)$.

For the EDE model we have that:

$\Omega_{DE}(z,\Omega_{m},\Omega_{e},w_{0})_{EDE}=\frac{\Omega_{e}\left(-\left(1-(z+1)^{3w_{0}}\right)\right)-\Omega_{m}+1}{\Omega_{m}(z+1)^{-3w_{0}}-\Omega_{m}+1}+\Omega_{e}\left(1-(z+1)^{3w_{0}}\right)\,.$

Therefore, the Hubble function is:

$H^{2}(z,\Omega_{m},\Omega_{e},w_{0},\Omega_{\gamma},N_{eff})_{EDE}=\Omega_{DE}(z,\Omega_{m},\Omega_{e},w_{0})+(z+1)^{3}\Omega_{m}+(z+1)^{4}\Omega_{\gamma}{\tilde{N}}\,.$
(43)

Here ${\tilde{N}}$ is the same defined in Eq.(37).

##  22 Appendix E: On the inverse $\beta$-decay in the comoving frame

This Appendix contains the calculation of the inverse $\beta$-decay rate in
the comoving frame. As remarked in Sec. 7, in this case the process occurs due
to the interaction of the proton with the thermal bath of electrons and anti-
neutrinos (Unruh effect). At tree-level, the following three processes occur
(see Fig. 6) [139]:

$\mathrm{(ii)}\,\,\,p^{+}\,+\,e^{-}\,\rightarrow\,n\,+\,\nu_{e}\,,\quad\mathrm{(iii)}\,\,\,p^{+}\,+\,\overline{\nu}_{e}\,\rightarrow\,n\,+\,e^{+}\,,\quad\mathrm{(iv)}\,\,\,p^{+}\,+\,e^{-}\,+\,\overline{\nu}_{e}\,\rightarrow\,n$
(44)

Upon quantizing the lepton fields as in [137], the evaluation of the decay
rate leads to

$\Gamma^{(\mathrm{com})}\,\equiv\,\Gamma^{(\mathrm{ii})}\,+\,\Gamma^{(\mathrm{iii})}\,+\,\Gamma^{(\mathrm{iv})}\,=\,\cos^{4}\theta\,\widetilde{\Gamma}_{1}\,+\,\sin^{4}\theta\,\widetilde{\Gamma}_{2}\,+\,\cos^{2}\theta\sin^{2}\theta\,\widetilde{\Gamma}_{12}\,.$
(45)

The explicit expression of these three terms is given in [137], where it is
shown that

$\widetilde{\Gamma}_{1}\,=\,\Gamma_{1}\,,\quad\widetilde{\Gamma}_{2}\,=\,\Gamma_{2}\,,\quad\widetilde{\Gamma}_{12}\,=\,\Gamma_{12}\,,$
(46)

with $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{12}$ given in Eq. (5). Then, from
comparison of Eqs. (45) and (5), it follows that the rates in the laboratory
and comoving frames are equal when the flavour representation for asymptotic
states is adopted.

##  23 Appendix F: Geometric phases for neutrinos propagating in matter

The notion of geometric phase can be defined for any curve in a Hilbert space
$\mathcal{H}$. In the kinematical approach of Mukunda-Simon the geometric
phase of a curve $\Gamma:[s_{1},s_{2}]\rightarrow\mathcal{H}$ is given by

$\Phi_{G}(\Gamma)=\arg\langle\psi(s_{1})|\psi(s_{2})\rangle-\Im\int_{s_{1}}^{s_{2}}\langle\psi(s)|\dot{\psi}(s)\rangle
ds$ (47)

where $|\psi(s)\rangle\equiv\Gamma(s)$. Recalling that the effect of matter on
neutrino oscillations can be included by introducing modified mixing angle
$\theta_{m}$ and squared mass difference $\Delta m^{2}_{m}$, we apply the
definition of Eq. (47) to the propagation of neutrinos in a medium. For the
phases associated to a single flavour, we obtain

$\Phi^{g}_{\nu_{e}}(t)=\arg\left[\cos\left(\frac{\Delta
m_{m}^{2}t}{4E}\right)+i\cos 2\theta_{m}\sin\left(\frac{\Delta
m_{m}^{2}t}{4E}\right)\right]-\frac{\Delta m_{m}^{2}t}{4E}\,\cos 2\theta_{m}\
,$ (48)

with $E$ neutrino energy, and $t$ time. On the other hand, for the phases
associated to flavour mixing we find:

$\Phi_{\nu_{e}\rightarrow\nu_{\mu}}(t)=\arg\left[\langle\nu_{e}(0)|\nu_{\mu}(t)\rangle\right]-\Im\int_{0}^{t}\langle\nu_{e}(t^{\prime})|\dot{\nu}_{\mu}(t^{\prime})\rangle
dt^{\prime}\\\ =\frac{3\pi}{2}+\phi+\left(\frac{\Delta m_{m}^{2}}{4E}\,\sin
2\theta_{m}\;\cos\phi\;\right)t$ (49)

$\Phi_{\nu_{\mu}\rightarrow\nu_{e}}(t)=\arg\left[\langle\nu_{\mu}(0)|\nu_{e}(t)\rangle\right]-\Im\int_{0}^{t}\langle\nu_{\mu}(t^{\prime})|\dot{\nu}_{e}(t^{\prime})\rangle
dt^{\prime}\\\ =\frac{3\pi}{2}-\phi+\left(\frac{\Delta m_{m}^{2}}{4E}\sin
2\theta_{m}\;\cos\phi\ \right)t\ .$ (50)

##  24 Appendix G: Rotation Curve Data and Instrument Description

KROSS is an Integrated Filed Spectroscopic (IFS) survey using the KMOS
instrument on ESO/VLT. KMOS consists of 24 Integrated Field Units (IFUs);
those can place within 7.2 arcminutes diameter field. Each IFU covers the
$2.8\times 2.8$ arcsec in size with $0.2$ arcsec pixels. The targets for the
survey are selected from extragalactic deep fields covered by multi-wavelength
photometric and spectroscopic data: 1) Extended Chandra Deep Field Survey
(E-CDFS: [360, 361]), 2) Cosmic Evolution Survey (COSMOS: [362]), 3) Ultra-
Deep Survey (UKIDSS: [363]), 4) SA22 field (see: [364]). Some of the targets
were selected from CF-HiZELS survey. With these targets the $H\alpha$ emission
is shifted into J-band. The median redshift of parent sample (KROSS full data)
is $z=0.85^{+0.11}_{-0.04}$. The median J-band seeing of the objects was $0.7$
arcsec, with 92% of the targets observed within seeing $<1$ arcsec. The data
were reduced using ESOREX/SPARK pipeline ([365]). The end product of the
process is a 3D datacube consisting of 2048 channels. These provide us with
spectra, the line and continuum images and the moment maps (see [323] that are
available at KROSS-website.

##  25 APPENDIX H: INFN - IS QGSKY

It is not a coincidence that the authors of this review belong(ed) to the
INFN-Iniziativa Specifica QGSKY. The National Institute of Nuclear Physics,
INFN, is a main Italian research agency with a portfolio of studies ranging
from the infinitely small to the infinitely large. Theoretical, experimental
and observational research in subnuclear, nuclear and astroparticle physics
are topics of interest for the INFN, which has created and supports special
projects, called ”Iniziative Specifiche”. These connect scientists with
different skills and competences to reach the excellence in the scientific
results in Fields that are at the forefront of Physics. This IS has a
particular attention to Quantum Cosmology and towards the Gravitation and the
physics of the Universe both within the framework of standard quantum field
theory and General Relativity, and in that of extended theories of Gravity.
Most of the researches are focused or end up into the elusive nature of the
Dark Matter phenomenon or concern the origin of the accelerated expansion of
the Universe. QGSKY is therefore a network devoted at theoretical and
observational studies lying at the crossroad among Cosmology, Relativity and
Quantum World with a flexibility that allows to tackle this frontier region of
knowledge. The recent results obtained inside this IS show very naturally the
complex entanglement between Dark Matter and the rest of the Physics, which is
also the aim of the current ST.

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