# Einstein, Planck and Vera Rubin: relevant encounters between the
Cosmological and the Quantum Worlds

Paolo Salucci 1,∗, Giampiero Esposito 2, Gaetano Lambiase 3, Emmanuele
Battista 2,4,5, Micol Benetti 2,6, Donato Bini 2,7, Lumen Boco 1, Gauri Sharma
1, Valerio Bozza 3, Luca Buoninfante 3, Antonio Capolupo 3, Salvatore
Capozziello 2,6, Giovanni Covone 2,6, Rocco D’Agostino 2,6, Mariafelicia
DeLaurentis 2,6, Ivan De Martino 2,8, Giulia De Somma 2,6, Elisabetta Di
Grezia 2, Chiara Di Paolo 1, Lorenzo Fatibene 9, Viviana Gammaldi 10, Andrea
Geralico 2,7, Lorenzo Ingoglia 2, Andrea Lapi 1, Giuseppe G. Luciano 3,
Leonardo Mastrototaro 3, Adele Naddeo 2, Lara Pantoni 1, Luciano Petruzziello
2,11, Ester Piedipalumbo 2,6, Silvia Pietroni 3, Aniello Quaranta 3, Paolo
Rota 3, Giuseppe Sarracino 2, Francesco Sorge 2, Antonio Stabile 3, Cosimo
Stornaiolo 2, Antonio Tedesco 3, Riccardo Valdarnini 1, Stefano Viaggiu 2,12,
Andy A. V. Yunge 2

###### Abstract

In Cosmology and in Fundamental Physics there is a crucial question like:
where the elusive substance that we call Dark Matter is hidden in the Universe
and what is it made of?, that, even after 40 years from the Vera Rubin seminal
discovery does not have a proper answer. Actually, the more we have
investigated, the more this issue has become strongly entangled with aspects
that go beyond the established Quantum Physics, the Standard Model of
Elementary particles and the General Relativity and related to processes like
the Inflation, the accelerated expansion of the Universe and High Energy
Phenomena around compact objects. Even Quantum Gravity and very exotic DM
particle candidates may play a role in framing the Dark Matter mystery that
seems to be accomplice of new unknown Physics.

Observations and experiments have clearly indicated that the above phenomenon
cannot be considered as already theoretically framed, as hoped for decades.
The Special Topic to which this review belongs wants to penetrate this newly
realized mystery from different angles, including that of a contamination of
different fields of Physics apparently unrelated. We show with the works of
this ST that this contamination is able to guide us into the required new
Physics.

This review wants to provide a good number of these ”paths or contamination”
beyond/among the three worlds above; in most of the cases, the results
presented here open a direct link with the multi-scale dark matter phenomenon,
enlightening some of its important aspects. Also in the remaining cases,
possible interesting contacts emerges. Finally, a very complete and accurate
bibliography is provided to help the reader in navigating all these issues.

##  1 Introduction

The phenomenal roles in investigating the Universe and its content and
governing rules, of these three great scientists is without discussion.
However, despite that we have today a well organized system of laws of Nature
and a formidable set of experimental and observational discoveries, the status
of the Universe seems to require that researchers in Observational Cosmology,
General Relativity and Quantum Physics concur together and join forces in
building new effective paths of knowledge. In fact, it is well known that
about $95\%$ of the energy density of the Universe is of dubious invisible
nature and cannot be framed within the ”presently verified” physics. Moreover,
a number of phenomena like e.g. ”inflation”, ”matter-antimatter asymmetry”,
”cosmological and astrophysical quantum relativistic objects”, ” dark matter”
just to name a few, are far from being understood and good ideas like ”the
Unification of Gravity with the other forces” and ”the intrinsic Symmetry of
the Universe” are also stalling.

In these circumstances, conceptually new directions of investigation have been
proposed. The Universe is considered not only as the arena in which theories
(e.g. on the Dark Matter) get falsified/verified, but, through the properties
of its content, a strong motivator for creation of totally new ones. A
contamination among Cosmology, Astrophysics, Relativity, Theoretical Physics,
Physics of the Elementary particles theory is taking place in many recent
investigations aimed at discovering new Physics. This is the leitmotiv of the
Special Topic collection to which this work belongs; we want to contribute to
it by providing a review of 17 examples of such new and in some case very
different paths of contaminated knowledge.

This review confirms that just the complementary of physical, astrophysical
and cosmological probes leads to a successful approach to the (always more
required) New Physics and Cosmology. This is done by describing a substantial
number of topics of the above kind. Of course this work is not fully complete
in reporting such investigations, so as also in exhaustively discussing all
the most pressing arguments of Cosmology and Fundamental Physics. About this,
let us stress, however, that this review is conceived in relation to the
collection of articles of the Special Issue for which it works as the
Introductory Paper. In addition, in order to widen the scope of the present
work, we indicate to the reader the following reviews on crucial issues of
Cosmology and Fundamental Physics which are related to those accurately dealt
in the present work: ”Cosmology Intertwined: Perspectives for the Next
Decade”[2, 3], ”Lectures on Black Holes and linear waves”[4], “The
distribution of dark matter in galaxies”[5], ”Status of dark matter in the
Universe” [6], ”Testing dark energy models in the light of $\sigma_{8}$
tension” [7]. Finally, a large and accurate list of references will contribute
to frame this review within the Cosmological and Fundamental Physics
Literature.

##  2 Cosmography in General Relativity and in Extended Theories of Gravity

The standard cosmological model is also known as $\Lambda$CDM due to the dark
components it takes on, i.e $\Lambda$ stands for the cosmological constant
while CDM points to the Cold Dark Matter fluid. Such a model provides a
successful description of the evolution and the structure of the universe,
requiring just six parameters, but leaves several questions open including the
nature of the dark matter and the cosmological constant. Recently, with the
improvement of the accuracy of the cosmological observations, tensions between
different probes have emerged. Among the most debated at the moment, there is
certainly the tension in the Hubble constant measurements, $H_{0}$ when it is
derived at early Universe, $z=1100$, from the CMB measurements and when it is
measured at low redshift by exploiting the SNIa distance indicator calibrated
by Cepheids. Let us stress that, in the first case, the $H_{0}$ estimation
strictly depends on the model used to describe the CMB data. In particular,
the position of the main CMB peak reflects the size of the horizon at last
scattering of the CMB photons and is determined almost entirely by the
geometry of the Universe. The height of the peak is directly proportional to
the fractional mass in baryons $\Omega_{b}$ and also varies according to the
expansion rate of the Universe as specified by the Hubble constant $H_{0}$
[8]. In the latter case it is instead an accurate estimate of the magnitude
$\mu$ of the SNIa distance indicator calibrated by Cepheids, which implies a
series of accurate techniques which also consider the Cepheid period,
metallicity and luminosity (see Ref.[9] for details). The magnitude is then
related to the luminosity distance, $D_{L}$, via $\mu=5log\,D_{L}+25$, and it
is possible to obtain the direct measurement (therefore independent of the
model) of $H_{0}$ from the $D_{L}$ relationship, e.g
$D_{L}=(1+z)\frac{zc}{H_{0}}$, that is valid for low redshift.

The tension on H0 mentioned above is quantified in about $4.4\sigma$ between
the Planck collaboration estimation [10] $H_{0}=\left(67.27\pm 0.60\right)$
km/s/Mpc and the SH0ES collaboration measurement [11]) $H_{0}=(74.03\pm 1.42)$
km/s/Mpc using the Hubble Space Telescope observations of 70 long-period
Cepheids in the Large Magellanic Cloud.

Furthermore, the existence of other mutually discrepant probes with no common
observational systematics reinforces the idea that the above tension is
systematics-free and due to unknown properties of the actual cosmological
scenario [12, 13, 14]. As an example, we can cite a joint data analysis of
Baryon Acoustic Oscillation (BAO) from BOSS and eBOSS and Big Bang
Nucleosyntesis (BBN) [15] that finds $H_{0}=(67.35\pm 0.97)$, while SPTpol
[13] finds $H_{0}=(71.3\pm 2.1)$ km/s/M

Moreover, standard distance ladder and time delay distances agree with the
SH0ES estimate. Finally, the H0LiCOW [16] value $H_{0}=(73.3^{+1.7}_{-1.8})$
km/s/Mpc, based on strong gravitational lensing effects on quasars favour high
values for the Hubble constant as the result of the Maser Cosmology Project
[17]: $H_{0}=(73.9\pm 3.0)$ km/s/Mpc.

These tensions may indicate an inadequacy in the Standard Cosmological model
in providing a correct description of the evolution of the universe since the
last scattering of the Cosmic Background Radiation till recent times.

Extensions of the standard model, i.e. including new physics in the
$\Lambda$CDM theory to obtain changes in observational predictions, were
explored to solve the $H_{0}$ tension. Among the proposals, we can mention the
inclusion of extra relativistic species at recombination [18, 20, 19, 21, 22]
and the dynamical interaction between dark matter and dark energy [23, 24,
25], as well as the investigation of the cosmological inflationary context
[26]. Indeed, all these sectors can modify the value of the current expansion
rate of the universe. For example, in the case of further relativistic species
at CMB epoch, it is worth mentioning that the standard cosmological model
considers three families of neutrinos as relativistic species at decoupling
epoch. Nevertheless, also extra contributions can be considered as, for
example, non-standard particle physics like eV-scale sterile neutrinos,
Goldstone Bosons, thermal axions and even relativistic dark matter, or also
the stochastic background of primordial gravitational waves [27, 28]. These
additional free-streaming relativistic particles are often called dark
radiation [29]. Intuitively, increasing the number of species increases the
contribution of these species in the calculation of background evolution, thus
increasing the value of the expansion rate. Despite the efforts spent to cover
the shortcomings of the standard model by including new factors, the current
conclusion is that such tension cannot be dealt within reasonable physics
beyond the standard cosmological model since any resolution of this
disagreement produces even greater tensions with other parameters of the model
such as $\Omega_{m}$ and $\sigma_{8}$ [160, 161, 32, 33].

The perspectives therefore open up, to explore drastically new theories, as
the Extended Theories of Gravitation, or to discover the existence of subtle
parameters’ degeneracies plaguing the standard model. Cosmographic studies can
give important hints also about the existing cosmological tensions (as those
on $H_{0}$ or $\sigma_{8}$) since they are able to fully describe the
expansion of the universe independently of any a-priori model, and therefore
best highlight all the anomalies in the cosmological parameters that have not
a priori a known theoretical explanation [34, 35].

The only assumptions cosmography relies on are the homogeneity and isotropy of
the Universe. This model-independent technique permits the reconstruction of
the dark energy dynamical evolution, with no need of assuming any particular
cosmological model at late-time epochs [36, 37] (see Appendix 19). Cosmography
involves Taylor expansions of the observable quantities that get directly
compared with data. The results of this procedure ensure the independence from
a postulated equation of state for the Universe’s evolution and, thus, help us
to break the above degeneracy. However, this approach is plagued [38] by the
divergence of the Taylor polynomials at high redshifts that leads to non-
accurate numerical results when data at $z>1$ are used. To alleviate this
problem one can use other polynomial expansions like the rational Padé
polynomials, which extend the convergence radius of the standard cosmographic
series [39], or the Chebyshev polynomials which reduce the systematics on the
fitting coefficients and remain stable at high-redshifts. [40] (cf. Fig. 1).

Figure 1: Dimensionless luminosity distance curves for the $\Lambda$CDM model,
the Taylor expansion at fourth-order, the Padé(2,1) and Chebyshev(2,1)
rational approximations.

At late times, assuming an arbitrary fiducial value for the current (dark +
baryonic) matter density fraction, $\Omega_{m}$, the cosmography can be used
to reconstruct the Hubble parameter in function of redshift using low redshift
probes, as the Ia Supernovae. An extension of this strategy is to link
together the parametric cosmographic behaviour of the late universe expansion
with the small scale universe, i.e joining such a parametric late time
evolution with a cosmological model as the $\Lambda$CDM (replacing the intake
of dark energy with a cosmographic description of the current expansion of the
universe). In principle, one can obtain a snapshot of the universe at any
epoch. Following this strategy, it is possible to develop a $f(z)$CDM model,
where $z$ is the redshift and the function $f(z)$ is a suitable combination of
polynomials which track the cosmic luminosity distance and replace the
cosmological constant $\Lambda$ at any $z$. This allows one to avoid the
assumption of a fiducial value for $\Omega_{m}$, since the latter can be
constrained by early universe data [41].

Figure 2: $H(z)$ evolution expansion of $f(z)$CDM model, where $j_{0}=1.97$
and $s_{0}=l_{0}=0$ with different values of deceleration parameter:
$q_{0}=-0.5$ (magenta curve), $q_{0}=-1$ (green curve), $q_{0}=-1.5$ (red
curve). For comparison, the minimal $\Lambda$CDM model is also drawn (black
solid line) and the Cosmic Clock data are reported (see [41]). Moreover,
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=1-\Omega_{m}$ and $H_{0}=70$ Km/s/Mpc.

The background evolution reads as:

$E(z)^{2}=\left(\frac{H(z)}{H_{0}}\right)^{2}=\Omega_{r}(1+z)^{4}+\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{f}f(z)\,,$
(1)

with $\Omega_{k}$ the spatial curvature density, $\Omega_{r}$ the radiation
density and $\Omega_{f}$ the density that drives the current accelerated
universe expansion. In the Standard Model the last term reduces to
$\Omega_{f}f(z)=\Lambda$. Clearly the class of cosmological models to test (or
the given equations of state) can be reconstructed, in principle, starting
from $f(z)$( for details see Sec. 3 of [41]. It is then possible to calculate
the predictions of the various cosmological models and to compare them with
the data by a Markov chain Monte Carlo process. This approach was applied to a
second order Padé polynomials (see Fig. 2) in order to analyse data at both
small and large scales [41]i.e. by using recent $H(z)$ measurements from the
1048 measurements, in the redshift range $0.01-2.3$, of the Supernovae
Pantheon sample [42], the galaxy clustering (DES sample) that combine galaxy
clustering and weak gravitational lensing measurements, using 1321 square
degrees of imaging data [12] and the early universe data i.e. the Cosmic
Background Radiation anisotropies by Planck Collaboration and the Baryonic
Acoustic Oscillation measurements [43, 44, 45, 46].

A dependence of the $\Omega_{m}$ value with the cosmographic parameters
emerges in the analysis of Ref. [41], indicating the importance of not fixing
this parameter in cosmographic analyses. Furthermore, the authors show an
improvement on the parameter constraint using the Cosmic Background Radiation
temperature anisotropy power spectrum. Moreover, it has been found that the
cosmographic series truncated at third order (i.e., one uses just two
cosmographic parameters: the deceleration and the jerk), better describe the
current observations with respect to the minimal $\Lambda$-CDM cosmological
model, showing a better $\Delta\chi^{2}$ with respect to the minimal standard
cosmological model (see [41]). This result acquires great significance when
one recalls that General Relativity is a second order theory. Indeed, the
evidence that the third order is required in cosmography to fully describe the
data could be interpreted as that a higher order theory is needed to correctly
describe the recent expansion of the Universe. In other words, $\Lambda$CDM
may be a coarse-grained model that needs higher-order corrections in order to
reproduce in a self-consistent way the existent phenomenology.

Furthermore, it is possible to go beyond General Relativity [47] and the
related problems of the (homogeneous and isotropic) $\Lambda$CDM model that
includes a cold dark matter (CDM) component and a no-evolving dark energy term
with $w=-1$. This model provides the best fit to most of the available
cosmological data [10] at present epoch but it is plagued with conceptual and
theoretical shortcomings [48]. This state of art may indicate that General
Relativity has to be, in some sense, ”improved” at cosmological scales [35].
To this purpose one can look to the Extended Theories of Gravity representing
a semi-classical approach, in which General Relativity is recovered in the
weak energy limit [49]. Within this approach, beside the common metric
formalism, one can also apply the Palatini formalism in which the field
equations are obtained through the variation of the gravity action with
respect to both metric and connection, treated as independent variables. This
provides different field equations with respect to those of General Relativity
when non-linear terms in the Ricci scalar $R$, or scalar fields non-minimally
coupled to gravity are present [50]. One can adopt _a priori_ the function of
the Ricci scalar $f(R)$ but, in cosmographic context, it is possible to
reconstruct the $f(R)$ function without assuming any specific functional form
[51, 52]. The free parameters of the models are therefore obtained by
observational data. The results indicate slight departures from the
$\Lambda$CDM model, with a time-evolving dark energy term [40].

Therefore, better performing cosmographic series and more precise data at low
redshift by future experiments [53, 54, 55], will help toward the final goal
to reconstruct the actual cosmic history, to test the effectiveness of General
Relativity at any redshift and to investigate the DM and DE mysteries. Indeed,
cosmography allows us not only to avoid to choose a-priori a particular
cosmological model, but also to reconstruct it by an inverse scattering
procedure by which it is possible to obtain reliable equations of state.

From the current state of the art of this topic it can be concluded that the
standard cosmological model shows significant gaps in describing low redshift
data, and this is also confirmed using higher-redshift probes such as quasars,
as shown in the following session.

##  3 Constraining dark energy models at high redshifts

Over the last decades, observations of high-redshift supernovae of type Ia
revealed the current accelerated expansion of the Universe ([56, 57, 58, 59]).
This result has been confirmed by temperature anisotropies of the cosmic
microwave background radiation ([60],[61]), and by many different data.
According to the most recent estimates, dark energy provides about 75% of
matter-energy content in the universe.

The nature of dark energy is unknown. Models of dark energy include, the
standard one with the cosmological constant having a constant equation of
state, $w$, (in this case: $w=-1$), but also one or more of the following: 1)
a potential energy of some scalar field [62, 63, 64], 2) effects due to a
possible inhomogeneous distribution of matter or to errors in averaging
procedures 3) the standard GR being substituted by extended theories of
gravity where dark energy can be described in terms of scalar non minimally
coupled scalar fields ([65],[66] and references therein). Moreover, these
scalar fields might be not fundamental and consist of fermion condensates
[67]. The effects of these condensates on the expansion of the Universe can be
seen in [68]. There are also other possibilities for dark energy see [69, 70,
71], and [72]. Interacting dark matter/dark energy models have been recently
introduced in different contexts [73, 74, 75, 76]. In all these cases, in
general, $w\not=-1$ and it is not constant with redshift.

Extracting the information on the effective equation of state $p_{\Lambda}=w\
\rho_{\Lambda}$ of dark energy from observational data is, therefore, at the
same time a fundamental problem and a challenging task in Cosmology and
Physics: new, independent measurements of the rate of expansion of the
Universe, especially in different ranges of redshifts, can provide us with a
non-trivial test of the standard cosmological model (see [77, 78]).

Other cosmological probes of distances have entered the game. Ghirlanda et al.
(2004), Fermiano et al.( 2005) used long GRBs (gamma ray burst) to build up
the Hubble diagram up to high redshifts ($z\simeq 9$) ([79], and [80]). GRBs
are sudden bursts of hard X–ray / soft gamma–ray emissions, and can be as
short as tens of millisecond and as long as several minutes. They are
classified as long GRBs if the pulse lasts more than $2$ secs, and show a non-
thermal spectrum that generally peaks at few hundreds keV. These objects
release enormous amounts of energy (the isotropic equivalent radiated energy
is of order of $10^{52}-10^{54}$ erg) and therefore are visible up to very
high redshifts being so good candidates as cosmological probes. Their non-
thermal spectra can be modeled with the Band function $N(E)$, a broken power
law with the following parameters: the low–energy spectral index $\alpha$, the
high energy spectral index $\beta$ and the roll–over energy $E_{0}$:

$N(E)=\left\\{\begin{array}[]{ll}A\left(\frac{E}{100keV}\right)^{\alpha}\exp{\left(\frac{-E}{E_{0}}\right)}\,,&\left(\alpha-\beta\right)\geq
0\,,\\\
A\left(\frac{\left(\alpha-\beta\right)E}{100keV}\right)^{\alpha-\beta}\exp{\left(\alpha-\beta\right)\left(\frac{E}{100keV}\right)^{\beta}}\,,&\left(\alpha-\beta\right)E_{0}\leq
E\,,\\\ \end{array}\right.$ (2)

The spectra have a peak in correspondence of the photon energy $E_{\rm
p}=E_{0}(2+\alpha)$. For GRBs with measured spectrum and redshift it is
possible to evaluate the intrinsic peak energy, $E_{\rm p,i}=E_{\rm p}(1+z)$
and the isotropic equivalent radiated energy

$E_{\rm iso}=4\pi
d_{L}(z,{\mathrm{\theta}})\left(1+z\right)^{-1}\int^{10^{4}/(1+z)}_{1/(1+z)}EN(E)dE\,.$
(3)

It turns out that both $E_{\rm p,i}$ and $E_{\rm iso}$ span orders of
magnitude and their distributions can be approximated by Gaussians plus a tail
at low values of energy. In 2002, for long GRBs the Amati relation was
discovered: $E_{\rm p,i}$ was found strongly correlated with $E_{\rm iso}$
([81]):

$\displaystyle\log\left[\frac{E_{\rm
iso}}{1\;\mathrm{erg}}\right]=b+a\log\left[\frac{E_{\mathrm{p,i}}}{300\;\mathrm{keV}}\right]\,.$

This correlation can be properly calibrated and allows us to standardize the
GRBs as a distance indicators in a way similar to that used for the Phillips
relation concerning SNeIas ([82] and reference therein). Then, once the
correlation between $E_{\rm p,i}$ and $E_{\rm iso}$ is calibrated, it is
possible to estimate the luminosity distance $d_{L}(z,{\mathrm{\theta}})$ by
using Eq. (3), and therefore to build up the Gamma Ray Burst Hubble diagram
[83, 84, 85, 86, 87]. Furthermore, Eisenstein et al. (2005) introduced BAOs as
standard rulers ([88]), Chavez (2016) used HII galaxies ([89]), and Negrete et
al. (2017) used extreme quasars [90]. Recently Risaliti and Lusso [91, 92]
found a physical relation between the ultraviolet (at $2500\AA$, $L_{UV}$) and
the X-ray (at $2\ keV$, $L_{X}$) emissions in quasars. This correlation has an
intrinsic dispersion between 0.35 to 0.4 dex. By eliminating quasars with host
galaxy contamination, reddening, X-ray obscured objects and radio loudness
([91]) the dispersion decreases to 0.21-0.24 dex. This correlation provides a
way to standardize quasars as distance indicators similar to SNeIa’s.

One investigates the dynamical evolution of dark energy by parametrizing the
equation of state,

$\displaystyle w(z)=\frac{p(z)}{\rho(z)}\,,$

assuming some analytical form of $w$ as function of $z$ [83]. Otherwise, we
can follow a cosmographic approach relying upon quantities that are not model
dependent. In [91],[92] a tension appears between the $\Lambda$CDM model and a
Hubble diagram extended behind the supernovae of type Ia, built up by using a
quasar sample with high-quality ultraviolet and X-ray flux measurements. This
sample spans a huge range in red-shift, $0.04<z<5.1$. In [93]. By adding other
162 Gamma Ray Bursts data, with an even larger range in redshift:
$0.03<z<9.3$, the existence of such tension has been reinforced (see Fig. (3)
in [93]). The fitting cosmographic models deviate from the standard
$\Lambda$CDM model at $>4\sigma$ by using the high-redshift Hubble diagrams of
SNe Ia, quasars and GRBs. Since the diagrams of quasars and GRBs are
completely independent, the found tension with the standard model is unlikely
to arise from unknown systematic effects and seems to imply new physics.
Moreover, such tension is confirmed also in [85, 86] where the Gamma Ray Burst
Hubble diagram was analyzed.

These results suggest investigating the $\Lambda$CDM tension by assuming
several well known cosmological scenarios with different dark energy equation
of state and comparing them by means of the Akaike Information Criterion [94].
These are:

  * (i)

a model with an evolving equation of state empirically parametrized, the CPL
model [95], [96] see Eq.(21) in Appendix D.

  * (ii)

a model where dark energy is due to a self-interacting scalar field,
$\varphi$, which drives the acceleration. We consider the potential
$V(\varphi)\propto\exp\left\\{-\sqrt{\frac{3}{2}}\varphi\right\\}$ [87, 84].
For this model exact solutions of the cosmological equations are known, and
are shown in the Appendix C: it turns out that all basic cosmological
functions surprisingly depend on a single parameter, $\mathcal{H}_{0}$, which
is the Hubble constant when we use the age of the universe as a unit of time.

  * (iii)

Early dark energy models, where a non-negligible fraction of dark energy
exists at early stages of the Universe.

In these models the dark energy density parameter, $\Omega_{DE}$, depends on
the present matter fraction $\Omega_{m}$ dominated by the dark component, the
early dark energy density parameter $\Omega_{e}$, and the present dark energy
equation of state parameter $w_{0}$ [97]. The form of $\Omega_{DE}$ and the
Hubble function of these models are provided in Appendix C by Eqs. (21) and
(43). The redshift investigation uses the Supernovae Cosmology Project Union
2.1 compilation, containing 580 SNIa, spanning the redshift range $0.015\leq
z\leq 1.4.$, a Gamma Ray Bursts (GRBs) Hubble diagram, consisting of 193
objects, a set of 28 independent measurements of the Hubble parameter,
compiled in [98], and several baryon acoustic oscillations (BAO) measurements
compiled in [99]. The authors obtain the probability distributions of the
cosmological parameters for each of the competing models. Their region of
confidence is obtained by maximizing the appropriate likelihood functions by
using the Monte Carlo Markov chains method.

Figure 3: 2D confidence regions in the $w_{0}-w_{1}$ plane for the CPL model,
obtained from our full data set. The red points correspond to the best fit
value and the mean, respectively.

In Tab. (1) of Appendix A are shown the results of our analysis. In Fig. (3)
we plot the 2-dim confidence regions in the $w_{0}-w_{1}$ plane for the CPL
model,obtained from the full data set. The $\Lambda$CDM model corresponds to
the case $w_{0}=-1\,,w_{1}=0$, and is out at $>3\sigma$. It is worth noting
that this result adds value to that obtained in Sec. 2 showing that third
order cosmographic series describe the currently available cosmological
observations better than the standard $\Lambda$CDM cosmological model. To
bring out more clearly this result one plots, in Fig. (4), the counterpart of
the confidence regions in Fig. (3) in terms of cosmographic parameters. We
actually implemented a standard cosmographic series in the auxiliary variable
$y=\displaystyle\frac{z}{1+z}$, which maps the z-range [ $0,\infty$] into the
y-range [$0,1$]. It turns out that the value $j_{0}=1.0$ is ruled out at 3
$\sigma$.

We then apply to the above data the Bayesian Akaike criterion and found that
among the three models the evolving dark energy, described by the exponential
scalar field potential (EXP), is favoured. Let us notice that such evolution
of the Dark Energy reverberates also in the Dark Matter component, whose
evolution of perturbations with redshift, will now differ from the
$\Lambda$CDM case.

Figure 4: 2D confidence regions in the $q_{0}-j_{0}$ plane obtained from our
full data set. For $\Lambda$ CDM we have: $j_{0}=1$, and
$q_{0}=-1+\frac{3}{2}\Omega_{m}$. The pink point corresponds to
$\Omega_{m}=0.3$.

Future missions such as the THESEUS observatory for Gamma Ray bursts will
substantially increase the number of Gamma Ray bursts in their Hubble diagram,
therefore, providing us with the opportunity to confirm these results.
Reliable high redshift cosmological probes alongside with the cosmography
approach might indicate the actual dark energy model and the correct gravity
theory.

##  4 The scattering angle of a two-body system

This research issue covers one of the most important topics in classical
general relativity today. In fact, the last few years have witnessed the
beginning of the era of gravitational-wave astronomy, with the detection of
the first signals by the LIGO and Virgo interferometers (see e.g., the web
sites https://www.ligo.caltech.edu/ and http://www.virgo-gw.eu/). These
detectors have the potential to accurately measure the two-body dynamics up to
the region where the gravitational interaction is very strong, so that there
is now a pressing need to go beyond the post-Newtonian approximation by
combining results from different approaches. The Hamiltonian of a two-body
system takes information (encoded in several building-block potentials) from
any theoretical computation of gauge-invariant quantities. After having
studied several orbital invariants like the redshift and the periastron
advance, all recent literature is mainly focusing on the scattering angle in
hyperbolic-like orbits. This is particularly promising in view of the bridge
recently discovered between the classical scattering angle in post-Minkowskian
context [100, 101] and the quantum scattering amplitudes for the same problem,
computed by a completely different method of high-energy physics. The reached
accuracy is now limited to the third post-Minkowskian order [102, 103] and to
the local part of the associated Hamiltonian, but the aim is to go beyond this
level, including gravitational self-force information and to study the effects
related to the non-local (non-trivial) part of the Hamiltonian.

The topics included in this research involve, besides classical GR, also
effective field theories and high-energy physics, with a direct superposition
with other research interests. More in detail: in the gravitational
interaction of a two-body system, i.e., the capture of one body by the other
as well as the hyperbolic scattering, two different scenarios must be
considered, in view of the detection of the associated gravitational wave
signals. While several analytical and semi-analytical methods have been
developed so far to study the first situation (“capture”), this is not the
case for the second one (“scattering”). A renewed interest in the scattering
problem has recently emerged with studies involving post-Newtonian (PN) and
post-Minkowskian (PM) expansions of the scattering angle. The latter has been
computed within the post-Newtonian theory at the fourth post-Newtonian level
of accuracy in [104], including both local and non-local (tail) contributions,
whereas the general framework of the post-Minkowskian treatment of the
scattering problem has been established in Refs. [100, 101]. The level-3 post-
Minkowskian Hamiltonian, describing the conservative dynamics of a two-body
system, has been obtained in [102, 103], where the two-loop scattering
amplitude of the process and the associated scattering angle at third post-
Minkowskian level were computed. This result in the high-energy limit can be
properly compared with previous results [105]. In an effort to clarify this
situation, Ref. [106] suggested a novel approach, aiming at improving the
current knowledge of the post-Newtonian Hamiltonian of a two-body system. This
approach combines various different theoretical formalisms: post-Newtonian,
post-Minkowskian, multipolar-post-Minkowskian, gravitational self-force, and
effective-one-body. [106] has given an independent check of the level-3 post-
Minkowskian result, limited at the fifth post-Newtonian level of accuracy, in
terms of the scattering angle $\chi_{3}$, i.e.,

$\displaystyle\chi_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{3p_{\infty}^{3}}+\frac{4}{p_{\infty}}+(-8\nu+24)p_{\infty}+\left(-36\nu+\frac{64}{3}+8\nu^{2}\right)p_{\infty}^{3}$
(4) $\displaystyle+$
$\displaystyle\left(-\frac{91}{5}\nu+34\nu^{2}-8\nu^{3}\right)p_{\infty}^{5}+\left(\frac{69}{70}\nu+\frac{51}{5}\nu^{2}-32\nu^{3}+8\nu^{4}\right)p_{\infty}^{7}+o(p_{\infty}^{7})\,,$

where the label 3 refers to the third post-Minkowskian level,
$\nu=m_{1}m_{2}/(m_{1}+m_{2})^{2}$ denotes the symmetric mass ratio of the two
bodies and the relation between the momentum at infinity and the (effective)
energy of the process is given by
$p_{\infty}\equiv\sqrt{{\widehat{\mathcal{E}}_{\rm eff}}^{2}-1}$. In the above
expression, the last term proportional to $p_{\infty}^{7}$ is exactly the
level-5 post-Newtonian contribution to $\chi_{3}$. In addition, Bini et al.
have anticipated the level-5 post-Newtonian-accurate value for the level-4
post-Minkowskian scattering angle $\chi_{4}=\chi_{4}^{\rm loc}+\chi_{4}^{\rm
nonloc}$, with the local contribution $\chi_{4}^{\rm loc}(p_{\infty})$ given
in Appendix 20, Eq. (20).

Finally, the analysis developed in [106] has allowed to identify the local
part of the level-5 post-Newtonian Hamiltonian modulo two unknown functions,
$a_{6}^{\nu^{2}}$ and $\bar{d}_{5}^{\nu^{2}}$ that, however, are of the second
order in the symmetric mass-ratio of the two bodies, see Appendix 20.

Further studies and possible generalizations are currently being considered
[107, 108, 109, 110, 111], including both the 6PN local and non-local
contribution to the scattering angle.

We point out that measuring the scattering angle in a two-body system implies
the determination of (gauge-invariant) information about the radiated
gravitational wave energy along the full scattering process. Therefore, the
analytical modeling for this quantity is important in the analysis of future
detections of gravitational wave signals by gravitational wave
interferometers. Noticeably, an accurate investigation of the merging of two
BHs may yield to a precious test about them being the DM in galaxies.

##  5 Classical physics in curved spacetime backgrounds

(i) Scalar wave equation and Ermakov-Pinney.

The field equations of classical field theory in curved spacetime, i.e.,
scalar wave equation, Maxwell equations and Einstein equations, are wave
equations with variable coefficients, whose solution is known only in a few
cases. Thus, a systematic technique for dealing with such hyperbolic equations
may lead to a valuable physical insight.

In the work in [112], the first part of the paper proves that a subset of the
Ermakov-Pinney equations can be obtained by differentiation of a first-order
non-linear differential equation. The second part of the paper proves that the
equation for the amplitude function for the parametrix of the scalar wave
equation can be obtained by differentiating covariantly a first-order non-
linear equation. The construction of such a first-order non-linear equation
relies upon a pair of auxiliary $1$-forms $(\psi,\rho)$. The $1$-form $\psi$
satisfies the vanishing divergence condition ${\rm div}(\psi)=0$, whereas the
$1$-form $\rho$ fulfills the non-linear equation

${\rm div}(\rho)+\langle\rho,\rho\rangle=0.$

The auxiliary $1$-forms $(\psi,\rho)$ are evaluated explicitly in Kasner
spacetime. Therefore, amplitude and phase function in the parametrix are
obtained. The novel method developed in this paper can be used in studying the
electromagnetic field of binary systems in relativistic astrophysics.

The key property to be used is the fact that the technical difficulty of
dealing with a coupled set of partial differential equations for the
electromagnetic potential can be overcome by exploiting the Hertz potentials,
that lead eventually to a linear wave equation for a complex scalar field
[113]. The real and imaginary part of such a scalar field are then subject to
a linear wave equation, and the original method relying upon two auxiliary
$1$-forms may come into play.

In [114], the authors have solved the Ermakov-Pinney equation associated to
the scalar wave equation in Schwarzschild, de Sitter and gravitational-wave
spacetimes. For example, in the Schwarzschild case it is found that a positive
coupling constant leads to spatially damped oscillations of the field (see
Fig. 5), whereas a negative or vanishing one is associated with a blow-up of
the solutions.

Figure 5: Amplitude function for a timelike current in Schwarzschild space-
time. The real part of the Heun function $H_{1}(x)$ is plotted, which
contributes to the amplitude for different values of the coupling constant
$f_{c}=\frac{1}{2},1,2$. All curves show a spatially damped oscillating
behaviour with a wavelength qualitatively proportional to the inverse of the
coupling constant. The imaginary parts of the amplitude have a similar
behaviour.

In light of this, and bearing in mind the interest of studying in greater
detail the quantization of fields, it seems more important to focus on the
case of a positive coupling constant, a topic to be developed in future work.
It would also be valuable to understand the relation (if it exists) with the
asymptotic behaviour of solutions of the scalar wave equation discovered by
Schmidt and Stewart [115].

(ii) Trapped surfaces in general relativity. After the proof by Christodoulou
and Klainerman of the global non-linear stability of Minkowski spacetime
[116], Christodoulou decided to apply his optical formalism in order to prove
that, in the non-linear equation that describes the formation of trapped
surfaces, some terms can be omitted while keeping under control the resulting
error, that remains suitably small [117]. More recently, in order to deal with
exact equations, without any approximation, in [118] trapped surfaces have
been studied instead from the point of view of local isometric embedding into
three-dimensional Riemannian manifolds. When a two-surface is embedded into
three-dimensional Euclidean space, the problem of finding all surfaces
applicable upon it gives rise to a non-linear partial differential equation of
the Monge-Ampère type, first discovered by Darboux [119], and later
reformulated by Weingarten [120]. Even today, this problem remains very
difficult, despite some remarkable results. In [118], the authors have found
an original way of generalizing the Darboux technique, which leads to a
coupled set of six non-linear partial differential equations. For the
$3$-manifolds occurring in Friedmann-(Lemaitre)-Robertson-Walker cosmologies,
they have shown that the local isometric embedding of trapped surfaces into
such cosmologies can be proved by solving just one non-linear equation. Such
an equation has been solved for the Friedmann models associated with positive,
zero, negative curvature of the spatial sections, respectively. However, the
task of solving such a non-linear equation in the most general case remains
extremely difficult, and deserves a substantial effort, because there is a
proliferation of terms in the non-linear equation, without any simplifying
ansatz.

##  6 GR as the bridge between classical and quantum worlds

Motivated by recent efforts to apply the asymptotic symmetry group of
asymptotically flat space-times to the investigation of black holes’ soft hair
[121], in [122] the authors remark that half of the Bondi-Metzner-Sachs (BMS)
transformations are made of orientation-preserving homeomorphisms of the
extended complex plane known as fractional linear (or Möbius) transformations.
These can be of four kinds, i.e., they are parabolic, or hyperbolic, or
elliptic, or loxodromic, depending on the number of fixed points and on the
value of the trace of the associated $2\times 2$ matrix in the projective
version of the $SL(2,{\bf C})$ group. The resulting particular forms of
$SL(2,{\bf C})$ matrices affect also the other half of BMS transformations.
They have been used in [122] to propose four realizations of the asymptotic
symmetry group that are called, again, parabolic, or hyperbolic, or elliptic,
or loxodromic.

Moreover, it has been proved that a particular subset of hyperbolic and
loxodromic transformations, i.e., those that have a trace that approaches
$\infty$, correspond to the fulfillment of limit-point condition for singular
Sturm-Liouville problems. Thus, a very deep link might exist between the
language for describing asymptotically flat spacetimes and the world of
complex analysis and self-adjoint problems in ordinary quantum mechanics. In
other words, we are suggesting that General Relativity plays the role of
gateway that establishes a correspondence between classical and quantum world.
Such a viewpoint is of particular interest for the theoretical analysis of
fundamental interactions.

So far, relativists have never considered the possibility of defining the
concept of boundary in a measure-theoretic way. On the other hand, in a branch
of mathematics known as geometric measure theory, the usefulness has been
discovered long ago of yet another concept, i.e. the reduced boundary of a
finite-perimeter set. In [123] the authors have proposed therefore a
definition of finite-perimeter sets and their reduced boundary in general
relativity. Moreover, an integral formula of geometric measure theory has been
evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry,
for the first time in the literature. This prepares the ground for a measure-
theoretic approach to several concepts in gravitational physics, supplemented
by geometric insight.

Interestingly, some observational tests of quantum gravity in the weak-gravity
regime has become, in principle, conceivable by applying the effective-gravity
picture of Donoghue [124]. In particular, within the framework of the solar
system, approximate realizations of the three-body problem occur when a comet
approaches a planet; such a configuration was investigated by Tisserand [125]
in Newtonian gravity. The exact relativistic treatment of the problem is not
easy, but the work in [126] develops an approximate computational scheme which
evaluates for the first time the very tiny effective-gravity correction to the
equation of the surface for all points of which it is equally legitimate to
regard the heliocentric motion as being perturbed by the attraction of
Jupiter, or the jovicentric motion as being perturbed by the gravitational
attraction of the Sun. In the second part of [126], comet trajectories for
effective and Newtonian gravity are obtained and compared in detail. In the
years to come, the relativistic analysis of cometary motions will require the
inclusion of high orders of the post-Newtonian formalism, a task extremely
difficult for three-body systems. This investigation will lead to a better
understanding of the occurrence of chaotic behaviour in general relativity.

Remarkably, such behaviour is very distinct from the eventual effects, in the
Solar System, of alternatives to the DM that, on the other hand, can affect
the cometary motions at the level of the above high orders.

##  7 Quantum Field Theory in curved space-time and neutrino physics

The search for a merging between General Relativity and Quantum Field Theory
is the most demanding task of theoretical physics nowadays. Notwithstanding
the zoo of models developed in recent years, the problem of finding a
consistent theory of quantum gravity is still open, due to the lack of
experimental guidance. Groping in the dark, significant progress is then
expected to be achieved via indirect investigation. For instance, in [127] it
was argued that quantum gravity may involve foam models endowed with
stochastic fluctuations of the space-time background. One of the most
sensitive probes of these fluctuations could be _neutrinos_ through
decoherence effects on _flavour oscillations_. Similarly, in [128] neutrinos
were selected as best candidates to witness the gravity-induced entanglement
between massive systems, being only affected by the weak and gravitational
interactions.

On the other hand, in the cosmological framework it has been widely discussed
[129, 130, 131] the active rôle played by primordial neutrinos in the
formation of large-scale structures (LSS) of the Universe and, in particular,
in giving a power at a large scale that otherwise could not be explained in
the ordinary cold dark matter scenario at the critical density. Due to the
considerable impact of neutrinos on the process of structure formation,
observations of matter clustering in different epochs of the Universe have
then contributed to build a solid bridge between cosmology and neutrino
physics. This has also enabled us to put very competitive constraints on
characteristic neutrino parameters, such as the neutrino mass sum
$\sum_{i}m_{\nu_{i}}<0.72\,eV$ at 95% CL (through measurements of large scale
polarization and CMB temperature anisotropies) or
$\sum_{i}m_{\nu_{i}}<0.14\,eV$ at 95% CL (including measurements of
Lyman-$\alpha$ forests), and the number of relativistic species at
recombination $N_{eff}=3.03\pm 0.18$ at 68% CL (exploiting LSS and CMB data).
More details can be found in Ref. [132] and therein.

In light of the above discussion, it is believed that a deeper study of the
fundamental nature of neutrinos could not only improve our knowledge of
particle physics beyond the Standard Model, but also provide novel insights
into a series of related problems, the most prominent ones being the
development of a quantum theory of gravity and the understanding of the
formation of the current Universe from small early density fluctuations.

Neutrinos are the most puzzling particles currently known. Although a series
of findings have confirmed Pontecorvo’s idea of flavour mixing and
oscillations, several questions remain unsolved. Among these, the problem of
the very nature of asymptotic states – flavour or mass – is still a vibrant
subject of analysis [133, 134, 135]. Recently, a challenging test bench has
been provided in [136, 137] within the framework of the weak decay of
uniformly accelerated protons (inverse $\beta$-decay). The idea that non-
inertial protons are allowed to decay if exposed to sufficiently large
accelerations traces back to [138]. In [139] the inverse $\beta$-decay was
exhibited as a theoretical proof of Unruh effect [140] for the consistency of
QFT in curved space-time. However, in these works neutrinos were
simplistically considered as massless. The first successful attempt to embed
mixing was made in [137], concluding that the use of the flavour basis is
mandatory both to preserve the general covariance of QFT and to account for
neutrino oscillations.

Starting from the outlined picture, let us review the rôle of the
superposition of neutrino mass states in the inverse $\beta$-decay. By
employing the $S$-matrix formalism, the scalar decay rate of accelerated
protons is calculated in both the inertial and comoving frames in a two-
flavour model (the three-flavour description including CP violation effects is
contained in [141]). Perspectives and possible experimental implications are
then discussed.

Figure 6: Proton decay in the laboratory (i) and comoving (ii, iii, iv) frames
(time is on the vertical axis).

_Laboratory frame_. Concerning an inertial observer, the accelerating source
supplies the proton with the missing energy to decay as $(i)\,\,p\rightarrow
n\hskip 0.56905pt+\hskip 0.56905pte^{+}\hskip 0.56905pt+\hskip
0.56905pt\nu_{e}$ (see the diagram $(i)$ in Fig. 6). If one assumes the
acceleration to be much smaller than the masses of the ${Z^{0}}$ and
$W^{\pm}$, the weak coupling can be described by a Fermi-like action $S_{I}$
with a classical (quantum) hadronic (lepton) current [137, 139]. Accordingly,
the tree-level transition amplitude reads
$\mathcal{A}^{\mathrm{lab}}\equiv\langle n|\otimes\langle
e^{+},\nu_{e}|{S}_{I}|0\rangle\otimes|p\rangle$. This leads to the following
decay rate

$\Gamma^{(\mathrm{i})}\,\equiv\,\frac{1}{T}\int
d^{3}k_{\nu}\,d^{3}k_{e}\sum_{\sigma_{e},\sigma_{\nu}}\big{|}\mathcal{A}^{{\mathrm{lab}}}\big{|}^{2}\,=\,\cos^{4}\theta\,\Gamma_{1}\,+\,\sin^{4}\theta\,\Gamma_{2}\,+\,\cos^{2}\theta\sin^{2}\theta\,\Gamma_{12}\,,$
(5)

where the Pontecorvo state
$|\nu_{e}\rangle=\cos\theta|\nu_{1}\rangle+\sin\theta|\nu_{2}\rangle$ has been
used for the asymptotic neutrino, $\theta$ is the mixing angle and
$|\nu_{i}\rangle$ ($i=1,2$) denote the mass states (the expressions of
$\Gamma_{i}$ and $\Gamma_{12}$ are given in [137]).

Remarkably, in [142] it has been proved that the flavour-violating process
$p\rightarrow n\hskip 0.56905pt+\hskip 0.56905pte^{+}\hskip 0.56905pt+\hskip
0.56905pt\nu_{\mu}$ has a non-vanishing probability as well, due to the
asymptotic occurrence of neutrino oscillations. Accounting for such a behavior
further strengthens the use of flavour (rather than mass) states for neutrinos
[142, 143].

_Comoving frame_. The question now arises as how to describe the above process
for an observer comoving with the proton. Although one may envisage that the
decay is kinematically forbidden in this frame, this is not the case, since
the proton interacts with electrons and anti-neutrinos popping out from the
thermal vacuum due to Unruh effect (see the last three diagrams in Fig. 6).
Details on the calculation of the decay rate are carried out in Appendix 22,
showing the result to be consistent with Eq. (5). Therefore, unlike previous
claims in literature [136], flavour mixing is perfectly consistent with the
covariance of QFT, provided that asymptotic neutrinos are assumed to be
_flavour states_.

The rôle of the non-thermality of Unruh spectrum for mixed fields [144, 145,
146] and the effects induced by gravity [49, 147, 148, 149, 150] and
acceleration [151, 152, 153] on the standard oscillation formula have still to
be fully addressed. Worthy of attention are also the entanglement properties
that emerge among neutrinos and other decay products in interaction processes.
Beyond its inherent theoretical relevance, the tension between flavour and
mass states may be of interest at the phenomenological level, as well.
Recently, indeed, it has been shown that the predicted spectra of the neutrino
capture on tritium and of the tritium $\beta$-decay are sensitive to whether
neutrinos interact as massive or flavour eigenstates. Thus, it is expected
that the output of experiments such as KATRIN [154] (that aims at measuring
the mass of the electron antineutrino by investigating the spectrum of
electrons emitted by the $\beta$-decay of tritium) or PTOLEMY [155, 156]
(which is projected to reveal the Cosmic Neutrino Background via capture on
tritium), may provide important pieces of information in the considered
problem, deepening our understanding of neutrino physics. In addition, let us
notice that neutrinos might add up to the total dark matter amount of the
Universe. In this context, KATRIN and PTOLEMY setups could detect these
particles in the keV mass range, that is the scale at which they should form
the DM of the Universe. Potential connections between neutrinos (and, more
general, particle) physics and dark matter are also under the active
investigation of upcoming experiments such as AEDGE [157].

##  8 Dirac and Majorana neutrinos

The fundamental nature of neutrinos has eluded physicists so far. Its
determination is important due to the prominent role that neutrinos play in
Cosmology. In particular, both the known active neutrinos and the hypothetical
sterile neutrinos have been proposed as dark matter candidates, respectively
as hot dark matter and warm dark matter [158]. The issue suffers from a
substantial lack of experimental tests, the only attempts are based on the
phenomenon of neutrinoless double beta decay [159]. While Dirac neutrinos have
a distinct anti-particle (anti-neutrinos), Majorana neutrinos coincide with
their own anti-particle.

The most promising way to reveal the possible Majorana nature of neutrinos is
the experimental observation of the double beta decay without neutrinos [160]
(see Refs. [161] for alternative possibilities). The main difficulty in
determining the Majorana character is that in theories with $V-A$
interactions, this is the case of the Standard Model, any observable
difference between Dirac and Majorana neutrinos is suppressed by a factor
$(m_{\mu}/E)^{2}$, being $E$ the energy scale of the process [164]. However,
an interesting possibility arises in the case in which neutrinos have new
interactions beyond the SM [165]. In is worth of mentioning, for example, the
proposal of Ref. [166] in which the author considers the most general Lorentz-
invariant form of neutrino-fermion interactions. This includes scalar, pseudo-
scalar, vector, axial-vector and tensor couplings. In such a circumstance, it
follows that in elastic neutrino-electron scattering, the ratio of forward to
backward scattering cross sections $R_{p}$ could be used for distinguishing
Dirac and Majorana neutrinos. More specifically, $R_{p}\leqslant 2$ for
Majorana neutrinos, and $R_{p}>4$ for Dirac neutrinos. Other alternative
methods to determine the Majorana and Dirac nature stem from the differences
that arise in presence of decoherence [167, 168] and in the propagation
through a medium [169]. Let us recall the main distinctions between the two
kinds of neutrinos. Let $\nu$ denote the neutrino field and let
$\nu_{R,L}=\frac{1\pm\gamma_{5}}{2}\nu$ be its left–handed and right–handed
components. The Dirac mass term in the neutrino Lagrangian has the form

$L_{D}=\frac{m_{LR}}{2}(\bar{\nu}_{L}\nu_{R}+\bar{\nu}_{R}\nu_{L})\,$ (6)

with $\bar{\nu}=\nu^{\dagger}\gamma^{0}$. On the other hand the Majorana mass
term has the structure

$L_{M,L}=\frac{m_{LL}}{2}(\overline{(\nu_{L})^{c}}\nu_{L}+\bar{\nu}_{L}(\nu_{L})^{c})\,$
(7)

where the superscript $c$ denotes the charge conjugated field. If right–handed
neutrinos are allowed for, one has an analogous Majorana mass in terms of the
right–handed components, namely

$L_{M,R}=\frac{m_{RR}}{2}(\overline{(\nu_{R})^{c}}\nu_{R}+\bar{\nu}_{R}(\nu_{R})^{c})\
.$ (8)

A heavy right–handed Majorana mass $m_{RR}$ might explain the smallness of the
mass of the observed neutrinos via the see–saw mechanism (see ref. [170] for a
review on several neutrino mass models). The different structure in terms of
left–handed and right–handed components implies different transformation
properties for the two kinds of neutrinos. Indeed, while the Dirac Lagrangian
is invariant under global $U(1)$ transformations, the Majorana Lagrangian is
not. As a consequence, Dirac neutrinos preserve the total lepton number,
whereas Majorana neutrinos allow for lepton number violating processes. In
addition, a different number of phases appears in the mixing matrix. For $n$
flavours, there are $N_{D}=\frac{(n-1)(n-2)}{2}$ Dirac phases and
$N_{M}=\frac{n(n-1)}{2}$ Majorana phases. This work is focused on $2$-flavour
mixing, which features a single Majorana phase $\phi$. The Majorana mixing
matrix admits various parametrizations

$U_{1}=\left({\begin{array}[]{cc}\cos\theta&\sin\theta\,e^{i\phi}\\\
-\sin\theta&\cos\theta\,e^{i\phi}\\\ \end{array}}\right)\ \ \ \ \ \
U_{2}=\left({\begin{array}[]{cc}\cos\theta&\sin\theta\,e^{-i\phi}\\\
-\sin\theta\,e^{i\phi}&\cos\theta\\\ \end{array}}\right)$ (9)

which lead to the same oscillation formulae in absence of decoherence [171,
172, 173].

The Dirac mixing matrix is in any case obtained for $\phi=0$. To study
propagation with decoherence, it is necessary to model neutrinos as open
quantum systems, described by means of density matrices $\rho(t)$. The time
evolution of $\rho(t)$ is driven by a Lindblad-Kossakowski master equation
[174, 175]: $\frac{\partial\rho(t)}{\partial
t}=-\frac{i}{\hbar}[H_{eff},\rho(t)]+D[\rho(t)]$. The effect of decoherence is
contained in the dissipator $D[\rho(t)]$, which for the case at hand is a
$3\times 3$ matrix on the space of bounded operators $B(\mathcal{H})$ on the
two-level Hilbert space $\mathcal{H}$.

The elements of the dissipation matrix $D$ are phenomenological parameters, to
be extracted from the experimental setup. If $D$ is diagonal, Dirac and
Majorana neutrinos happen to obey the same oscillation formulae. For a non-
diagonal dissipator, however, the two are distinct. One considers a dissipator
with two off-diagonal components and plugs it in the master equation.
Projecting on the $SU(2)$ basis, one obtains a linear system for the
components $\rho_{\mu}(t)=\mathrm{Tr}(\rho(t)\sigma_{\mu})$ which is easily
solved. Then the transition probability from flavour $\sigma$ to flavour
$\varrho$ can be computed as
$P_{\nu_{\sigma}\rightarrow\nu_{\varrho}}(t)=\mathrm{Tr}\left[\rho_{\varrho}(t)\rho_{\sigma}(0)\right]$.
Notice that, at this stage, the choice of a parametrization for the mixing
matrix becomes relevant, as $U_{1}$ and $U_{2}$ produce different formulae. In
particular, choosing $U_{2}$ results in probabilities that have an explicit
dependence on the Majorana phase $\phi$, which prompts an asymmetry between
the particle and anti-particle transitions. One can introduce the
$CP$-violating quantity
$\Delta_{CP}^{\sigma\rightarrow\varrho}(t)=P_{\nu_{\sigma}\rightarrow\nu_{\varrho}}(t)-P_{\overline{\nu}_{\sigma}\rightarrow\overline{\nu}_{\varrho}}(t)$
as a measure of the $CP$ asymmetry. For Majorana neutrinos, even the survival
probability $P_{\nu_{e}\rightarrow\nu_{e}}(t)$ shows a $CP$ asymmetry. It
should be remarked that this $CP$ violation, depending exclusively on the
Majorana phase $\phi$, is absent in the standard 3-flavour mixing of Dirac
neutrinos, and is therefore distinct from the $CP$ violation arising from the
Dirac phase $\delta_{CP}$. On the other hand, $\Delta_{CP}=0$ for any
transition involving Dirac neutrinos. This can be generalized to three
flavours in a straightforward way.

Figure 7: Upper panel: plot of the oscillation formula
$P_{\nu_{\mu}\rightarrow\nu_{\tau}}$ for Majorana neutrinos and for Dirac
neutrinos ($\phi=0$, the black lines), as a function of the energy $E$. The
experimental bounds from long baseline experiments on the decoherence
parameters are used; Lower panel: plot of the geometric phases
$\Phi^{g}_{\nu_{e}\rightarrow\nu_{\mu}}$ (the blue dashed line) and
$\Phi^{g}_{\nu_{\mu}\rightarrow\nu_{e}}$ (the red dot–dashed line) for
Majorana neutrinos as a function of the neutrino energy $E$, for a distance
length $z=300km$. The geometric phase
$\Phi^{g}_{\nu_{e}\rightarrow\nu_{\mu}}=\Phi^{g}_{\nu_{\mu}\rightarrow\nu_{e}}$
for Dirac neutrinos is represented by the black solid line.

Another distinction between Dirac and Majorana neutrinos comes up from the
analysis of geometric phases for neutrinos propagating in matter (see
appendix). Using the Mukunda-Simon definition, one concludes that the
geometric phase associated to a single flavour Eq. (48) is not affected by the
Dirac/Majorana distinction [169]. On the contrary, the phases associated with
the mixing, Eqs. (49) and (50) show an explicit dependence on the Majorana
phase $\phi$. As for decoherence, we find an asymmetry in the oscillations of
Majorana neutrinos, which is absent in the Dirac case. The asymmetries in the
transition probabilities and geometric phases are exemplified in the figure
(7). In the upper panel we show the transition probability
$P_{\nu_{\mu}\rightarrow\nu_{\tau}}(z)$ for Dirac and Majorana neutrinos when
decoherence is taken into account, for a distance $z=300km$ as a function of
energy. The dissipator parameters are chosen in correspondence with the bounds
from long baseline experiments (see [167] and the references therein). In the
lower panel we plot the geometric phases associated to neutrino oscillations
(see Appendix F) as a function of energy.

Future studies [176] shall be aimed at finding additional tools and phenomena
that can highlight the fundamental nature of neutrinos. The determination of
the latter is important to shed light on one of the most elusive particles in
the universe, which perhaps constitutes the strongest evidence for physics
beyond the standard model of particles and may direct us towards the searched
new ”dark sector”.

##  9 Quantum fields in curved space-times

Several unresolved questions concerning the origin of our Universe, its very
evolution and fate, are still waiting for quantum gravity - a theory allowing
to manage and reconcile general relativity with quantum mechanics. In that
respect, also the apparent presence in the Universe of ”dark substances” such
as Dark Energy and Dark Matter could play a crucial role. One of the
difficulties of developing a quantum gravity theory is that quantum
gravitational effects are expected near the Planck scale, around $10^{-35}$
meters, so by far out of reach, given the present technology. On the other
hand, the approach to quantum cosmology through a suitable quantum-mechanical
translation of some known solutions in classical general relativity (and its
possible extensions) could perhaps offer a back-door entry towards quantum
gravity. Quantum effects involved in the early stages of a cosmological
evolution may be of some relevance also at a low-energy microscopical level,
influencing the microphysics in a classical background. In that respect, the
study of the behaviour of quantum fields in a curved space-time background
represents an inescapable first step towards the quest for a comprehensive,
self-consistent quantum theory of gravity [177]. In the semi-classical
approach, the background - although allowed to evolve in time - is assumed not
to be affected by the quantum field itself (i.e., back-reaction is not taken
into account), the former being only sourced by some given classical
distribution of mass-energy and completely described by the Einstein field
equations. Nevertheless, even in such semi-classical scenario, several non-
trivial manifestations of the background influence upon a quantum field
appear, often related to the lack of a privileged reference frame in presence
of a curved background. Hence, special attention has to be payed to the role
of the physical observer’s frame, in which the relevant measurements are
performed.

Vacuum fluctuations are a distinctive feature pertaining to the quantum nature
of a matter field. One of their most famous manifestations is the well-known
Casimir effect [178, 179, 180, 181, 182]. Modification in the vacuum energy
can be also induced by external fields, including gravity [183, 184, 185,
186]. In such latter case, the space-time background and the quantum field
confinement may conspire together, giving rise to several interesting effects
[187, 188].

Looking at those effects, the Casimir energy has been extensively
investigated, taking into account a small Casimir apparatus both in the weak
and in the strong gravitational field limit. Analysis of a Casimir cavity at
rest in a weak, static gravitational field has shown a small change in the
value of the Casimir energy. Such a result has been first obtained by using a
field mode decomposition technique [189]. Recently, the same result has been
confirmed by means of a quite different approach based upon the Schwinger
effective-action method [190]. Furthermore, a similar behaviour has been
obtained by considering a cavity in free fall into a black hole [191, 192,
193]. The Casimir effect has been investigated also in presence of (weak)
gravitomagnetism. An interesting result is that - up to the lowest order of
approximation - a gravitomagnetic field causes no distortion in the vacuum
energy [194] (the extension to modified theories of gravity has been studied
in [195, 196, 197]). However, a deeper and more exhaustive analysis in the
field of a Kerr black hole, has revealed that the Casimir energy density is
indeed sensitive to the Kerr gravitational dragging [198]. Actually, a small
Casimir apparatus orbiting in the Kerr equatorial plane suffers a change in
the Casimir energy density. Such a change vanishes in the case of a zero-
angular-momentum-observer orbit, suggesting that modifications in the Casimir
energy do appear when the azimuthal symmetry is broken by the equatorial
circular motion of the cavity with respect to the local zero-angular-momentum-
observer frame. A similar analysis has been carried on taking into account a
Casimir apparatus orbiting an Ellis wormhole [199]. While obtaining basically
the same results, e.g., a decrease in the absolute value of the vacuum energy
density inside the cavity, the above work has shown that the Casimir energy is
affected by the orbital motion of the cavity nearby the wormhole, also when
other local classical measurements \- performed by a co-moving observer - as
the acceleration or the Fermi rotation coefficients \- yield a vanishing
result. In that respect, it is believed that the Casimir effect might
represent an interesting quantum probe for testing the (non-)inertial
character of a given reference frame.

Figure 8: A Casimir cavity freely falling from infinity onto a Schwarzschild
black hole. The cavity size is assumed to be small with respect to the
Schwarzschild radius, ($L/r_{g}\ll 1$). The cavity is rigid, i.e., the plate
separation $L$ is constant according to a comoving observer.

.

Concerning time-dependent backgrounds, particle creation out of quantum vacuum
has been obtained by analyzing a “kicked” Casimir cavity [200] and, more
recently, a Casimir cavity in free fall into a black hole [192, 193]. Such
effect can be considered as a time-dependent tidal one, due to the non-
uniformity of the gravitational field experienced by the quantum field
confined to the cavity, as the black hole horizon is approached. In figure (8)
a schematic picture of a small Casimir cavity radially falling onto a
Schwarzschild black hole is shown. The cavity is rigid, namely the plate
separation $L$ is constant with respect to a comoving observer.

Similar particle creation effects can be found when the confined quantum field
experiences a time-dependent background describing a (weak) gravitational
wave. In an early related study on a small three-dimensional cavity [201]
hosting a scalar quantum field, the relevance of confinement from the point of
view of a particle-creation effect has been highlighted. Last, within that
framework the excitation of acoustic phonons out of the vacuum in a toy-model
one-dimensional elastic medium has been considered, as a consequence of the
interaction with a gravitational wave [191].

Future investigations will be devoted to a deeper understanding of the
interplay between gravity and quantum fields both in the static and in the
dynamical regime, the latter being typically related to particle creation.
Moreover, mesoscopic quantum systems interacting with a gravitational wave
[202] will be considered in detail. Quantum effects induced by gravitational
waves at nanoscales might represent a new stimulating research arena, both
from a theoretical and the experimental point of view.

Finally, quantum vacuum will be considered at cosmological level, being
strongly related to the cosmological constant problem in non-trivial spacetime
topologies as well as in models with compact extra dimensions. Let us stress
that, along these lines, Gamma-Ray-Bursts [203] \- will be investigated, as
they could be deeply rooted in the intriguing interplay between GR and QFT and
their capability to be indicators of the Universe expansion will be put in
question.

##  10 Tomographic description of quantum and classical states of the universe

Modern cosmology was born after the formulation of general relativity by
Albert Einstein. The evolution of the universe is described by solutions of
the Einstein equations. However, Hawking-Penrose theorems show that these
solutions inescapably present singularities if the energy-momentum tensor has
properties which are regular when ordinary matter and radiation are
considered. The existence of these singularities, which appear to be a
pathological aspect of the theory, can be interpreted as a signal that the
theory is no longer reliable at particular conditions, for this reason many
researchers postulated that general relativity, which is a classical theory,
must be replaced by a quantum theory of gravitation. In particular, cosmology
seems a topic that lends itself well to the study of the quantum properties of
a gravitational system such as the universe, which is homogeneous to a very
high degree; it is therefore sufficient to study the problem of quantization
in the minisuperspace, which is the space of all homogeneous three-metrics.

The study of the evolution of the universe is the story of the evolution of
the states of the universe. The state of a particle in classical physics is
described by its position and momentum in phase space. More generally, for a
particle in a thermal bath, its state can be described by a probability
function. For a constrained system where not all the dynamical variables are
independent (for example the Friedmann equations show that expansion factor
$a(t)$ and its time derivative $\dot{a}(t)$ and are not independent), it is
sufficient to use a smaller number of variables, and the states are confined
in a subset of phase space. Given one state and the equation of motion, one
can predict the evolution of the physical system.

On the other side, in quantum theory the state of a particle, or more
generally a quantum system, is defined by a wave function (or functional in
field theory). It can be used to evaluate the probability that a physical
system is in a particular state and to obtain the probability amplitudes of
its transition from a state to another. Alternatively to the wave function,
other descriptions of the physical state have been introduced like the density
matrix or the Wigner function. The last one describes the physical state in a
phase space, even if, differently from the classical particle, it cannot be
considered as a probability function since it can take negative values.

In general relativity the existence of singularities does not make it possible
to predict the evolution of the universe from an initial undetermined state.
On the other side the quantum description of the initial states of the
universe needs to be extended to the final states crossed by the classical
universe. Unfortunately, there is no a simple way to perform this extension by
using the kind of representations used so far. For this reason, we introduce a
new description in terms of tomograms which are marginal probability
functions, and as such they are positive-norm functions and are observables
(see for example[204] and [205]) They describe equally well quantum and
classical states. For this reason, tomograms are suitable for a good
description of the quantum-to-classical transition of the universe, and
eventually can be defined phenomenologically by cosmological observations,
leading to a reconstruction of the early universe [206] [207] [208] [209]
[210] [211] [212] with obvious relations with its dark components in matter
and in energy.

For this purpose, recently a de Sitter universe was considered and the wave
functions, solutions of the cosmological quantum equation (the Wheeler-DeWitt
equation) and were translated in the tomographic formalism [213][214].

Figure 9: The tomogram of a classical de Sitter universe

If we describe the de Sitter space-time with the closed metric

$ds^{2}=-\frac{N^{2}}{q}d\tau^{2}+\frac{q}{1-R^{2}}dr^{2}+qr^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (10)

where $q=a^{2}$, $N$ is the lapse function (related to the freedom of choice
of the time variable). The classical distribution is represented by the delta
function of the constraint equation

$f(q,p)=\delta\left(-4p^{2}+\lambda q-1\right)$ (11)

where $p=-\dot{q}/{4N}$ is the conjugate momentum, and the corresponding
classical tomogram is (see fig.(9)

$\mathcal{W}\left(X,\mu,\nu\right)=\int\delta\left(-4p^{2}+\lambda
q-1\right)\delta(X-\mu q-\nu
p)dqdp=\frac{1}{2|\mu|}\frac{1}{\left|\sqrt{\frac{\lambda^{2}\nu^{2}}{16\mu^{2}}+\frac{\lambda
X}{\mu}-1}\right|}\,.$ (12)

Figure 10: The tomogram of a quantum de Sitter universe according to the
Hartle and Hawking initial conditions

The tomogram corresponding to the Hartle-Hawking wave function is (see also
fig.10)

$\mathcal{W}(X,\mu,\nu)=\frac{A^{2}}{|\mu|}\left|Ai\left(\frac{1}{(2\hbar\lambda)^{2/3}}\left(1-\frac{\lambda
X}{\mu}-\frac{\lambda^{2}}{16}\frac{\nu^{2}}{\mu^{2}}\right)\right)\right|^{2}$
(13)

where the constant $A=2^{5/6}\pi/\hbar^{1/6}\lambda^{2/3}$. These two
tomograms can be compared. Then, first we see that the tomogram resulting from
taking the limit for $\hbar\to 0$ or the limit for $\lambda\to 0$ is the same.
This circumstance suggests that a decay of the cosmological constant from the
Planck scale to its present value causes a transition from the quantum to the
classical regime, as the quantum tomogram becomes

$\mathcal{W}(X,\mu,\nu)\approx\frac{1}{2|\mu|}\frac{1}{\left|1-\frac{\lambda
X}{\mu}-\frac{\lambda^{2}}{16}\frac{\nu^{2}}{\mu^{2}}\right|^{1/2}}\left|\cos\left(\frac{2}{3}S^{3/2}-\frac{\pi}{4}\right)\right|^{2}\,,$
(14)

The Hartle-Hawking model does not converge to the classical tomogram and does
not converge at all. But if nevertheless instead of taking $\lambda\to 0$, we
take $\lambda\sim 10^{-122}$ we have the possibility of examining a new
“classical” model of universe with peculiar features characterized by the
presence of the cosine factor in (14). Therefore, the Hartle-Hawking model is
not ruled out, but its asymptotic expression is a possible ”classical”
description of the present universe not derived from the classical equations.
In other words quantum cosmology offers the possibility of formulating new
classical models that are not necessarily derived from the classical general
theory of relativity.

On the other side the tomogram

$\displaystyle\mathcal{W}(X,\mu,\nu)$
$\displaystyle=\frac{A^{2}}{|\mu|}\left|\frac{1}{2}\left(Ai\left(\frac{1}{(2\hbar\lambda)^{2/3}}\left(1-\frac{\lambda
X}{\mu}-\frac{\lambda^{2}}{16}\frac{\nu^{2}}{\mu^{2}}\right)\right)\right.\right.$
$\displaystyle\left.\left.+iBi\left(\frac{1}{(2\hbar\lambda)^{2/3}}\left(1-\frac{\lambda
X}{\mu}-\frac{\lambda^{2}}{16}\frac{\nu^{2}}{\mu^{2}}\right)\right)\right)\right|^{2},$
(15)

which is the tomogram obtained from Vilenkin’s initial conditions (see fig.11)
[215], in the limit $(2\hbar\lambda)^{2/3}\to 0$ converges asymptotically to

$\mathcal{W}(X,\mu,\nu)\approx\frac{1}{2|\mu|}\frac{1}{\left|1-\frac{\lambda
X}{\mu}-\frac{\lambda^{2}}{16}\frac{\nu^{2}}{\mu^{2}}\right|^{1/2}}\,.$ (16)

Remarkably, the quantum to classical transition found here can be extended to
more general cosmological models involving either scalar fields or
cosmological fluids [215]. Many of these models predict that a classical
inflationary universe emerges after the decay of the cosmological constant
problem during the quantum epoch, and a further decay of the cosmological
constant guarantees the exit from the inflationary epoch. A result of this
analysis is that the so called ”cosmological constant problem” can be
addressed in quantum epoch rather than in the classical one.

Figure 11: The tomogram of a quantum de Sitter universe according to
Vilenkin’s ”tunneling from nothing” initial conditions

Last, tomograms are in principle observables. The form of the classical
tomogram (11) suggests that the phenomenological tomogram of the universe
should be a function of the cosmographic parameters extending the analysis in
sect. 2 to the variables of the tomogram.

##  11 On the origin of the cosmological constant: Planckian fluctuations,
decoherence scale

One of the biggest problems in modern physics and cosmology is the nature and
origin of dark energy. There exist many attempts to explain the current
acceleration phase of our universe, by adopting ‘modified gravity theories’ or
introducing ad hoc running (in time) scalar fields. In the standard
cosmological model dark energy is interpreted as vacuum energy and it is
simply depicted in terms of a cosmological constant $\Lambda$, both in time
and space. The observed value for $\Lambda$ is very small and in complete
disagreement with theoretical predictions (about $122$ orders of magnitude
smaller than the value predicted by Quantum Field Theory). Scarsely convincing
explanations concerning the issues related to $\Lambda$ have been put forward
(see for example [216], [217], [218], [219] and [220] and references therein).
It is important to point out also that all the above bounces on the other
great darkness of the Universe: the dark matter component, whose interaction
with the dark energy is as important as presently unknown.

Remarkably, a completely new point of view, regarding the origin of $\Lambda$
has been recently put forward [221], [222] and [223] The main idea relies on
the nature of $\Lambda$ and its very small observed value and states that
quantum fluctuations can explain the equation of state of the cosmological
constant. In fact, a radiation field sufficiently close to the Planck scale
can allow quantum fluctuations so strong to permit the transformation of this
radiation field into one with the equation of state that $\Lambda$ shows. The
effective value of the cosmological constant depends on the physical scale
under consideration. Thus, in the new view, Planckian fluctuations average on
bigger and bigger scales: the cosmological constant is watered down on scales
bigger than the Planck one. The important concept of decoherence scale $L_{D}$
is also introduced: this is the scale determining the observed value of the
cosmological constant and determines the crossover to classicality. The
calculations concerning the formula of the observed cosmological constant
$\overline{\Lambda}$ in terms of the decoherence scale $L_{D}$ are performed
in the context of a semi-classical solution where the effective metric at the
Planck scale is averaged. In formulas we have:

$\overline{\Lambda}=\frac{\pi^{3}}{90}\frac{L_{P}^{2}}{L_{D}^{4}},$ (17)

where $L_{P}$ is the Planck length and the constant $\frac{\pi^{3}}{90}$ has
been fixed in [222] by analogy with Casimir effect. From a mathematical point
of view the Buchert formalism in [224], known since 2001 and used to model
inhomogeneities in our universe by means of a template homogeneous metric, is
properly modified and applied at Planckian scales. Also this research field
represents a novelty and, in absence of a sound quantum gravity proposal,
provides a practical way to perform physically sound calculations.

A comparison with the Casimir effect is outlined in [222]. In this regard, it
should be noticed that an expression for a quantum modification of the Misner-
Sharp mass, introduced in [225], can be written in the following way:

$E(L)=\frac{c^{4}}{2G}\frac{L^{3}}{L_{A}^{2}}+\xi\frac{c^{4}}{2G}\frac{L_{P}^{2}}{L}.$
(18)

In this context, as usual, the Casimir energy $E_{C}$ can be expressed as the
difference between vacuum energy in presence of boundary conditions $E_{B}$
and the one of a space $E_{0}$ without boundary conditions:

$E_{C}=E_{B}-E_{0}.$ (19)

and as a result one obtains the expression

$E_{C}=E(L,\overline{\Lambda})-E(L,\Lambda)=\frac{\xi
c^{4}}{2G}\frac{L_{P}^{2}}{L}$

, in perfect agreement with the Casimir energy $E_{C}$ obtained in the
standard Casimir effect for a spherical conductor with
$\xi=\frac{\pi^{3}}{360}$. This comparison allows to fix the decoherence scale
$L_{D}$ to be of the order of about $10^{-2}$ millimeters. Note that this is
the scale where thermodynamic fluctuations are expected to be of the same
order of the quantum ones in the standard Casimir effect. Future studies must
be focused in order to depict the decoherence scale in terms of Quantum Field
Theory approach. By virtue of the well known lack of a well posed quantum
gravity theory, we need an effective treatment of the Einstein’s equations at
Planckian scales incorporating sound arguments of general relativity and
quantum mechanics. The semi-classical models so obtained can guide us towards
a complete and accepted quantum gravity theory. Moreover, extended theories of
gravity naturally generalize the Misner-Sharp mass: it is thus intriguing to
explore such modifications according to the Casimir effect for spherical
configurations.

Finally, in the same line of research, in [226] and [227] the logarithmic
corrections to Black Hole entropy have been outlined in a very simple and
physically sound way. A similar formula can be applied to the whole universe
represented by its apparent horizon, allowing the study of thermodynamics’
laws at a cosmological scales. Then, in [228] and in [229], a generalization
of the well known Bekenstein-Hawking entropy formula at a cosmological level
has been obtained: in an expanding universe a further term arises depicting
the degree of freedom due to the non-static nature of our universe. Recently,
in [230] Saha showed that with this new proposal, according to recent
astrophysical data, the phantom era is ruled out. This intriguing fact places
in an even more important position the role of $\overline{\Lambda}$.

One can study a relation between the model above for the cosmological constant
and the renormalization group (RG) approach. In particular, one chooses a
suitable renormalization scale $\mu$ or a subtraction point, where the physics
is calculated and fixed, as suggested by [221],[222] and in [223] in a
cosmological context. A physically sound well motivated expression for $\mu$
will represent the starting point to formulate a RG equation for the
cosmological constant. In this regard, the main goal is to obtain a non-
trivial (non-vanishing) infrared (IR) fixed point at $\mu=\mu_{D}$, thus
identifying the fixed point $\mu_{D}$ with the decoherence scale $L_{D}$. This
result will allow to firmly establish the decoherence scale $L_{D}$ as an
invariant one where the value of the cosmological is fixed, thus representing
the crossover to classicality.

The physical scales where quantum fluctuations act in a non-trivial way can be
of interest, as an example, also in the fate of a collapsing star. In
particular, are possible modifications of the Chandrasekhar-mass limit (i.e.,
the maximum mass of a stable white dwarf star) leading the the formation of
more massive stable stars without forming a black hole?. For this purpose, it
should be noticed that a possible presence of gravitational-echoes in the
Ligo-Virgo detection [231] of post-merger signals of binary coalescent stars
could model some deviations from General Relativity in the near-horizon
structures or more generally for ultra-compact stars made of possible exotic
matter, where quantum effects are expected to come into action. For example,
gravitational-echoes are supposed to be generated by the collision of two
black holes and may indicate possible new physics beyond General Relativity.
Detections of gravitational-echoes will be the smoking gun of quantum effects
in a very strong gravitational regime. Moreover, these objects may be part of
the elusive Dark Matter component of the Universe.

##  12 Gravity meets quantum mechanics: a simple quantum low-energy gravity
model at work

The idea that quantum mechanics (QM) should be characterized by a fundamental
non-unitary time evolution of the state vector, where gravity plays a
prominent role, has been put forward by a number of authors, on different
grounds [232, 233, 234]. Thus two main questions arise, which require an
answer: how to reconcile the basic processes of QM, i.e. the Schrödinger
deterministic evolution and the non-unitary process associated with the act of
measurement, and how to explain the transition to classicality.

These issues appear to be deeply connected with quantum gravity and, in
particular, with the way in which gravitational fields are generated by
quantum matter. It is still widely debated if gravity has to be quantized at
all or it is intrinsically classic and should be properly treated. In the
latter case a proposed route is the semi-classical gravity, based on the
requirement that the energy-momentum tensor appearing in Einstein equations is
taken as an expectation on the quantum state. This prescription, together with
the Newtonian limit, leads to the Newton-Schrödinger equation [235]. While its
solutions for a lump of matter show interesting properties, this equation
cannot be a valid candidate to give self-gravity at low energies because the
non-linearity allows for superluminal communications.

The effective low-energy model of Newtonian gravity introduced by S. De
Fliippo twenty years ago, termed Non-unitary Newtonian Gravity (NNG) (see e.g.
[236, 237, 238], references therein), belongs to this line of research. It is
obtained as the non-relativistic limit of a non-unitary version of higher
derivative gravity, which is classically stable and allows in principle for
regularization of gravitational collapse singularities [238], leading to
results in agreement with Bekenstein-Hawking entropy [239, 240]. Among its
appealing features, besides a non-Markov evolution, one finds that the
ordinary Newton’s action-at-a-distance interaction is recovered at a
macroscopic level. Furthermore it exhibits a a mass threshold for dynamical
self-localization, which amounts to $10^{11}$ proton masses for ordinary
matter densities [236, 238, 241]. As a consequence, even for closed systems,
macroscopic coherent superpositions of states evolve into ensembles of pure
states thanks to the peculiar dynamics of NNG model. Thus, as a further bonus,
density matrix emerges as the fundamental description of physical reality,
characterized by a non-Markovian evolution even from a pure state.

Due to the above features the NNG model is among the best candidates to
address the controversial and still unsolved issue of the quantum foundations
of the Second Law of thermodynamics [242, 243]. Indeed, as elucidated in
[244], a microscopic derivation of the Second Law of thermodynamics for a
closed system could be obtained only via a non-unitary quantum evolution and
in correspondence of suitable initial conditions. This feature is also shared
by the process of formation and evaporation of black holes [245]. In this
respect a prominent, unifying role is played by the concept of von Neumann
entropy as entanglement entropy with hidden degrees of freedom.

Preliminary steps in demonstrating the ability of NNG to give rise to a
gravity-induced relaxation towards thermodynamic equilibrium even for a
perfectly isolated system have been already performed in Refs. [246] and
[247]. As a first case study, a system has been considered, built out of two
particles trapped in a harmonic potential, interacting via delta-like and
gravitational interaction [246]. By choosing an eigenstate of the physical
Hamiltonian as initial condition, numerical simulations have been carried out
in order to study the time evolution of von Neumann entropy. The picture is
consistent with the interpretation of von Neumann entropy as an entanglement
entropy with some hidden degrees of freedom, which is clearly reminiscent of
well known black hole entropy calculations [248]. As a further result, entropy
fluctuations have been found to take place, thanks to the non-unitary part of
gravitational interactions, with the initial pure state evolving into a
mixture.

These promising results have been confirmed by switching to a more involved
system, i.e., a harmonic nanocrystal within a cubic geometry [247]. In this
case a numerical simulation has been carried out as well, but the choice of
initial conditions is much more involved. By following the procedure outlined
in Refs. [249, 250], an initial pure state with mean energy $E$ has been
selected, built up uniformly at random as a superposition of a huge number of
energy eigenstates within the energy interval $\Delta E$ around $E$. The von
Neumann entropy as a function of time exhibits a sharp monotonic increase,
followed by a stabilization at late times. These findings clearly show that a
micro-canonical ensemble grew up within the initial energy levels, as a
consequence of a non-unitary gravity-induced relaxation towards thermal
equilibrium.

Now a further step has to be performed in order to confirm the above scenario,
which implies the simulation of a real crystal. Of course, it is the
fundamental non-unitarity of the NNG model which is responsible of a net
entropy growth for the system as a whole. Thus a microscopic derivation of the
Second Law of thermodynamics can be self-consistently carried out, which makes
the NNG model the first low-energy gravity model leading in a natural way to
the emergence of Thermodynamics even in a closed system [251].

A second topical issue, concerning causality violations in Newton-Schrödinger
equation [235], has been fully addressed within the NNG model as well [252,
253]. Here the single particle Newton-Schrödinger equation has been obtained
as the mean-field approximation of an equation of $N$ identical copies of the
particle, interacting each other via gravitational interaction, when $N$ goes
to infinity [254]. The starting point is a general $N$-copy model, which is a
fully consistent quantum theory without causality violation problems thanks to
the validity of the no-signaling condition [255, 256]. Furthermore, while
superluminal communications are suppressed, new strange communications among
Everett branches of the wave function [253] are shown to appear, in close
agreement with previous findings by Polchinski [256]. Within NNG model the
density matrix fully characterizes the state of a system. This may suggest to
assume the Everett Many World Interpretation as the most natural conceptual
framework of that theory. But, at odds with other non-linear approaches to
quantum mechanics, here the possibility of constructing an Everett phone
between different branches of the wave function appears to be strongly
inhibited.

The implementation of an Everett phone deserves further investigations and is
currently under study [257] together with the search for a general proof of
the existence of gravity induced correlations between the different branches
of the wave function within a path integral formulation of the NNG model a’ la
Feynman-Vernon [258]. We strongly believe that this issue will help to clarify
the role of gravity induced de-coherence in wave function collapse and
stimulate to look for the answer to a further question: how to discriminate in
principle this kind of fundamental de-coherence against the usual
environmental de-coherence and, finally, how to experimentally detect such a
difference. To consider astrophysical consequences of NNG model is premature
but one can forecast non- GR violation of Newtonian law at interesting scales.

##  13 Physics meets Cosmology: Turbulence in SPH simulations of galaxy
clusters

The author [259] presented results from a large set of N-body/SPH
hydrodynamical cluster simulations aimed at studying the statistical
properties of turbulence in the ICM (InterGalactic Cluster Medium). The
cosmological model is a flat CDM model, with vacuum energy density
$\Omega_{\Lambda}=0.7$, matter density parameter $\Omega_{m}=0.3$ and Hubble
constant $h=0.7=H_{0}/(100\ km\ sec^{-1}\ Mpc^{-1})$. $\Omega_{b}=0.019\
h^{-2}$ is the value of the baryonic density.

The numerical hydrodynamical scheme employs a SPH formulation in which
gradient errors are strongly reduced by using an integral approach [260]. The
ensemble of hydrodynamical cluster simulations has been constructed by
performing a set of individual runs, with initial conditions for each cluster
extracted from a cosmological N-body simulation with only dark matter. We
consider both adiabatic and radiative simulations. For the cooling runs, the
modeling of the gas incorporates radiative cooling, star formation and energy
feedback from supernovae.

A gas particle is eligible to form a star particle if the gas flow is locally
convergent and the gas density exceeds a given threshold. If these conditions
are satisfied, star formation will occur with a characteristic dynamical time-
scale. We refer to [261] for a detailed description of the recipes
implemented.

We construct clusters subsamples according to the cluster dynamical status or
gas physical modeling. We quantify the cluster dynamical state by using, as a
morphological indicator, the power ratio method [259]. According to this
method, the projected X-ray surface brightness $\Sigma_{X}(\vec{x})$ is the
source term of the pseudo potential $\Psi(\vec{x})$ which satisfies the 2-D
Poisson equation. A multipole expansion of the solution then gives the moments
which are used to provide an unambiguous detection of asymmetric structures.

To analyze the turbulent velocity field of the simulated clusters we introduce
a filtering procedure, aimed at decomposing the fluid velocity into a large-
scale component and a small-scale part. We use an iterative multifilter
approach, in which mean velocities are estimated locally using an adaptive
filtering scheme [262]. We then extract from cluster subsamples small-scale
turbulent velocities, obtained by applying to cluster velocities the
multiscale filtering method. We also quantify the statistical properties of
the small-scale turbulent velocity field by estimating the velocity power
spectrum $E(k)$.

The main results of that work can be summarized as follows [259]. The velocity
power spectra of non-radiative relaxed clusters are mostly solenoidal and
exhibit a peak at wavenumbers set by the injection scales $\simeq R_{200}/10$.
The radius $R_{\Delta}$ is defined such that

$M_{\Delta}=(4\pi/3)\,\Delta\,\rho_{\mathrm{c}}\,R_{\Delta}^{3}$ (20)

denotes the mass contained in a sphere of radius $R_{\Delta}$ with mean
density $\Delta$ times the critical density $\rho_{\mathrm{c}}$.

In the high wavenumber regime, the spectral behavior is steeper than in the
Kolmogorov case. Radiative simulations are characterized by a shallower
wavenumber dependency, which is due to the injection of turbulence at small
scales. This in turn originates from the interaction of compact cool gas cores
with the ICM. This small scale driving source acts in addition to the usual
large scale injection mechanisms, thus showing that there are multiple
injection scales which operate to generate turbulence in galaxy clusters. For
cluster dynamically at equilibrium, the enstrophy profiles of cooling
simulations show a power-law dependency over a large range of radii, and a
very small turbulent-to-thermal energy ratio.

For an highly relaxed cluster we find in the core very low gas velocities, as
seen by Hitomi. Similarly, the radial profile of the sloshing oscillation
period is found in accord with Fornax observations. In particular, the
associated Froude number $Fr$ satisfies $Fr\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1$ within $r/R_{200}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1$. The number $Fr$ is introduced
to assess the importance of buoyancy forces relative to to stirring motions,
so that very small values indicate that stratification is important.

Our findings suggest that in cluster cores ICM turbulence approaches a
stratified anisotropic regime, with weak stirring motions dominated by gravity
buoyancy forces and strongly suppressed along the radial direction. In accord
with similar findings [263], we conclude that turbulent heating cannot be
considered the main heating source in cluster cores and thus provide a viable
solution to solve the so-called cooling flow problem. The center of relaxed
clusters is often characterized by the presence of cool dense cores with
cooling times much shorter than the age of the Universe. This implies
radiative losses which will lead to an inward motion producing a ‘cooling
flow’ [264] and large mass accretion rates. This is not observed, and some
heating sources [265] must be operating in the cluster cores to regulate the
cooling flows.

In a forthcoming work, we are planning to investigate the emergence of the
cool-core/non-cool-core (CC/NCC) dichotomy in simulations of merging clusters.
Specifically, we want to investigate the impact of radiative cooling on the
survival of CC clusters in a merging environment.

##  14 A new challenge for the $\Lambda$CDM scenario: the extremely dense
environments around Clusters of Galaxies

Galaxy clusters, the most massive gravitationally bound cosmic structures, are
not isolated systems. Theoretical models within the framework of the
$\Lambda$CDM cosmological model predict that they form in the highest-density
regions of the underlying dark matter density field [266]. This is widely
supported by cosmological numerical simulations that show a detailed picture
of the so-called cosmic web, with massive galaxy clusters located in the
highest-density nodes. The spatial correlation between the dark matter halos
(hosting galaxy clusters) and the matter density field is described by the
halo bias $b_{h}$ [267]:

$b_{h}\,=\,\frac{P_{h}(k)}{P_{m}(k)}$ (21)

where $P_{h}$ and $P_{m}$ are the power spectrum of the dark matter halo
distribution and the dark matter density field, respectively. Both quantities
are direct function of the scale $k$. Hence, dark matter halos are not
randomly distributed, but their clustering is enhanced relatively to the
general mass distribution. Theoretical models also show that the matter
density beyond the virial region is strongly correlated to the dark matter
halo mass [267]. Recent observational works have confirmed this picture. In
particular, Refs. [268] and [269] have confirmed the predicted relation
$b_{h}(M)$ between the halo bias and the mass $M$ of the dark matter halo (the
so-called 1-halo term). These works used the technique known as stacked weak
gravitational lensing [270] (see also [271, 272]). Weak gravitational lensing,
i.e., the coherent and weak distortion of the image of background sources by
means of a foreground gravitational field,is a fundamental tool to measure the
mass distribution in cosmic structures. However, the typical small signal-to-
noise due to the correlated matter distribution beyond the galaxy cluster
virial radius is not large enough to allow the measurement of the halo bias in
individual systems. This problem can be addressed by using stacked samples of
galaxy clusters, where the signal-to-noise is enhanced by averaging the weak
lensing signal over homogeneous samples, where clusters are grouped according
to some observable quantity (such as, e.g., the optical richness). Therefore,
stacked weak lensing provides an effective approach to test the statistical
predictions of the theoretical formation scenarios, as the direct
investigation of the correlated matter distribution around individual systems
is hampered by the lack of deep and wide astronomical surveys around massive
galaxy clusters.

However, the work in [273] recently reported the first detection of an
extremely dense environment around a massive galaxy cluster, PSZ2
G099.86+58.45, located at redshift $z=0.616$, from the PSZ2LenS [274] cluster
sample. PSZ2LenS is a small but homogeneous and complete sample of 35 galaxy
clusters, detected by the Planck mission in the sky regions observed by the
CFHTLenS (Canada–France–Hawaii Telescope Lensing Survey) [275] and RCSLenS
(Red Cluster Sequence Lensing Survey) [276]. Hence PSZ2LenS is an unbiased
sub-sample of the Sunyaev-Zeldovich sources detected by Planck (PSZ2
catalogue). PSZ2 G099.86+58.45 is a massive cluster (the cluster velocity
dispersion is $\sigma_{v}=1040\pm 110\,{\rm km/s}$) and the most distant
system in the PSZ2LenS sample. [273] performed the weak lensing analysis by
using about $150\times 10^{3}$ background galaxies, over a sky region about
$2.5\times 2.5$ square degrees. Remarkably, PSZ2 G099.86+58.45 is unique in
PSZ2LenS, as its large angular diameter distance ($D\simeq 0.98\,{\rm
Gpc}\,h^{-1}$) makes it possible to determine the weak lensing signal out to
about 30 Mpc from the cluster centre, well beyond the virial radius. The weak
lensing analysis presented in [273] reveals a galaxy cluster mass in broad
agreement with previous measurements based on a dynamical study of the
cluster’s galaxies motions and on the X-ray temperature profile:
$M_{200}=(8.2\pm 3.5)\,\times 10^{14}M_{\odot}\,h^{-1}.$ However, beyond about
10 Mpc, the environment matter density largely exceeds the cosmological mean.
Within the $\Lambda$CDM paradigm, the expected halo bias is $b_{h,{\rm\Lambda
CDM}}=11.1\pm 2.8$ [273]. We found a much larger signal from the second halo
term: $b_{h}\,=\,72\pm 20\,,$ see Fig. 12 for a comparison with the
theoretical predictions from [267]. We note that the measured bias for the
whole PSZ2LenS sample is well in agreement with the theoretical predictions,
see Fig. 12. The observational results obtained by [273] present us with a new
dilemma: is PSZ2 G099.86+58.45 a rare case, unexpectedly found in a small (yet
complete) sample of galaxy clusters? Or, is this system the first example of a
larger population of clusters whose existence requires a modification of the
formation scenario within the $\Lambda CDM$ model? In the latter case, the
immediate implication is that enhancing mechanisms around high-mass halos
could be much more effective than previously thought.

In the next few years, the aim will be at determining the statistics of the
halo bias values around single massive galaxy clusters, by completing a
systematic study of the correlated matter distribution around a large sample
of galaxy clusters, using recent and future data from wide surveys, such as
KiDS and Euclid. KiDS (Kilo Degree Survey) is a public survey at the VLT
Survey Telescope (Chile) which just completed the observations over about 1500
deg2 in the Southern hemisphere. Euclid, the forthcoming ESA space telescope,
will survey about 15 thousand square degrees of the sky, detecting about
$10^{5}$ galaxy clusters up to redshift $z\sim 3$. This wealth of data will
allow both to trace the evolution of the correlated matter distribution and to
robustly quantify the occurrence of rare systems such as PSZ2 G099.86+58.45.
The statistics of these peculiar systems will tell us that our Cluster
formation theory requires a critical revision.

Figure 12: Comparison between the measured values of the halo bias and the
theoretical predictions, as a function of the virial mass. Theoretical
predictions are computed assuming a flat $\Lambda$CDM cosmological model with
$\sigma_{8}=0.83$. Black points represent the measured values in the stacked
sample from Covone et al. (2014) [269], at $z=0.36$. The blue point represents
the single system at $z\simeq 0.62$.

##  15 Merging rates of compact binaries in galaxies and Stellar BHs.

With the recent detections of gravitational waves (GWs) by the LIGO/Virgo
collaborations, the era of multimessenger astronomy has begun. The future
AdvLIGO/Virgo improved sensitivity configurations, together with the advent of
the forthcoming detectors as Einstein Telescope (ET), will provide an enormous
number of informations in many different fields: from astrophysics to
cosmology and even to fundamental physics (e.g., [277, 278]). In the last
years the authors focused on forecasting the merging rates of compact binaries
and the associated detection rates with different detectors as a function of
redshift [279]. This issue depends on various astrophysical processes
happening on different spatial and time scales: we must correctly model single
stellar and binary evolution phenomena, GW physics and the environment in
which binaries are formed; therefore also galaxy formation and evolution must
be kept into account. In the approach of reference [279] the authors have
exploited the most recent observations of galaxies’ properties (i.e., their
UV+far-IR/submillimeter/radio luminosity functions, their spectral energy
distribution, the mass metallicity relationships) and combined them with
stellar evolution simulations outcomes. This approach can remarkably provide
joint probability distribution functions of the host galaxy properties (star
formation rate, stellar mass, metallicity, etc.) and the properties of the GW
signal.

In order to compute the merging rates as a functions of redshift we rely on
three main ingredients: (i) a statistics of galaxies, coming from their UV and
IR luminosity functions. (ii) a model able to reproduce the observations of
the chemical enrichment history of each individual galaxy, (iii) the outcome
of stellar and binary evolution simulations.

The first ingredient, given the relation between the luminosity and the star
formation rate (SFR) ([280]), can be translated into a star formation rate
function at different redshifts; it provides the number density of galaxies
producing stars at different star formation rates $\psi$ at various cosmic
times/redshifts $\rm t$: $\rm dN/\rm d\psi\,dV$ (see [281], [279]). The second
ingredient allows to associate a metallicity to galaxies with different
properties (SFR, mass, age or morphological type). The metallicity ($\rm Z$)
stellar winds, supernova kicks, direct collapse, common envelope effects all
depend on it. Using a very simple model, in agreement with observational data,
featuring a rapid linear increase of the metallicity with the galactic age up
to a saturation value, the authors of [279] are able to assign a probability
distribution function of metallicities at given SFR: $\rm dp/\rm dZ\,(\rm
Z|\psi)$ (see [282, 283, 284, 285, 286, 287]). Finally, the outcomes of
stellar evolution simulations can provide three crucial factors: the number of
merging binaries in an Hubble time per unit of star forming mass ($\rm dN/\rm
dM_{\rm SFR}\,(Z)$), the chirp mass probability distribution ($\rm dp/\rm
dM_{\rm chirp}\,(\rm M_{\rm chirp}|Z)$) and a time delay distribution between
the formation of the binary and the merging ($\rm dp/\rm dt_{\rm delay}\,(\rm
t_{\rm delay}|Z)$) (see e.g. [288, 289, 290, 291, 292, 293]).

The ingredients described above can be combined to compute the merging
Koushiappasrates per bin of redshift and chirp mass as follows:

$\begin{split}\frac{\rm d\dot{N}_{\rm merging}}{\rm dz\,dM_{\rm
chirp}}(z,M_{\rm chirp})=&\frac{\rm dV}{\rm dz(1+z)}\int\rm dt_{\rm
delay}\int\rm d\psi\psi\frac{\rm dN}{\rm d\psi\,dV}(\psi,z_{t-t_{\rm
delay}})\times\\\ &\times\int\rm dZ\frac{\rm dp}{\rm dZ}(Z|\psi)\frac{\rm
dp}{\rm dt_{\rm delay}}(t_{\rm delay}|Z)\frac{\rm dN}{\rm dM_{\rm
SFR}}(Z)\frac{\rm dp}{\rm dM_{\rm chirp}}(M_{\rm chirp}|Z)\end{split}$ (22)

where $\rm dV/dz$ is the differential comoving volume, dependent on the
cosmological parameters, the $1/(1+z)$ factor keeps into account the
cosmological time dilation and $z_{t-t_{\rm delay}}$ is the redshift computed
at the time of formation of the binary (i.e., the cosmic time at merging minus
the delay time $t-t_{\rm delay}$). This equation can be easily convolved with
the sensitivity curve of a GW detector (e.g., AdvLIGO/Virgo or Einstein
Telescope) to get the detected rate per redshift bin by that detector:

$\frac{\rm d\dot{N}_{\rm detected}}{\rm dz}(z)=\int\rm dM_{\rm chirp}\frac{\rm
d\dot{N}_{\rm merging}}{\rm dzdM_{\rm chirp}}(z,M_{\rm chirp})\int_{\rho_{\rm
th}}^{\infty}\rm d\rho\frac{\rm dp}{\rm d\rho}(\rho|z,M_{\rm chirp})$ (23)

where $\rho$ is the signal to noise ratio (SNR), $\rho_{\rm th}$ is the SNR
threshold of detectability and $\rm dp/d\rho$ depends on the properties of the
GW signal and on the characteristics of the detector. The result of equation
(23) is shown in Fig. 13 for the ET detector.

Figure 13: GW event rate per unit redshift expected for ET. Solid lines refer
to BH-BH events, dashed lines to NS-NS events, and dotted lines to BH-NS
events. Black lines refer to the total contribution from all galaxies, while
blue lines represent only the contribution of disk galaxies at $z<2$. The
orange lines refer to galaxy-scale gravitational lensing of GWs with
magnification $\mu>10$.

The approach pursued in [279] allows not only to compute the merging and
detection rates, but also to relate the properties of the GW signal (chirp
mass or SNR) to the properties of the host galaxy. First of all, since late
type galaxies and early type galaxies show a completely different history of
star formation and chemical enrichment, by using the approach of [279], one
can understand, depending on the characteristics of the signal, the galaxy
type from which a GW event is likely to come. Moreover, one can take equation
(22) and not marginalizing over the star formation rate, obtain a joint
probability distribution for the chirp mass and the SFR: $\rm
d\dot{N}/dz\,dM_{\rm chirp}\,d\psi$. This quantity allows to relate a property
directly measured from the GW signal, like the chirp mass, to a property of
the host galaxy, i.e. the star formation rate, which can be converted into
stellar mass or luminosity. This can result a powerful tool for cosmological
investigation through GWs since alleviates the problem of the host galaxy
association.

We can also perform tomographic cross correlations between galaxies and
gravitational wave signals, as performed by the authors of [294]. The outcome
of the cross correlations can give useful informations both on astrophysics
and cosmology. Finally the approach described above can be also exploited to
compute the number of lensed gravitational wave signals detected by different
instruments. Lensed GWs, especially if they have an electromagnetic
counterpart, can be a very strong tool to investigate both cosmology and the
theory of gravity itself.

##  16 The Nature of Dark Matter

The existence of the ”Dark Matter Phenomenon” is fully proven by the presence
of a large number of (extremely strong) physical anomalies in the behaviour of
both the luminous matter and the radiation. These include, among others: the
large-scale structures and the expansion rate of the Universe, the rotational
speeds of galaxies (see Fig 14), the weak and strong gravitational lensing of
background objects, the extraordinary cosmological object called the Bullet
Cluster [295], the temperature radial distribution of the hot gas in galaxies
and clusters of galaxies and the pattern of anisotropies in the cosmic
microwave background (CMB) radiation as detected by Planck. Furthermore, the
theory of Big Bang nucleosynthesis (BBN), by accurately predicting the
cosmological abundance of the lightest chemical elements, indicates that most
of the matter in the universe, so as in every galaxy, cannot be made by
baryons [296].

Remarkably, the dark matter relates with the properties of the entire Universe
supporting the case for which a modification of GR cannot explain, alone, all
the above observational evidences.

Figure 14: The observed Universal Rotation Curve (blue surface) of Spirals and
its Luminous Matter component (red surface). $x\equiv R/R_{D}$, with $R_{D}$
the disk lenght-scale, $\tilde{V}$ is the double normalized URC:
$\tilde{V}\equiv V(x,M_{h})/V(3,M_{h})$ with $M_{h}$ the galaxy halo virial
mass and: $M_{h}=10^{11+n/5}M_{\odot}$. The discrepancy implies the presence
of a dark massive component (see [332]).

An accurate discussion on various DM particle candidates can be found in
(e.g.[297]), here we give a brief account of them. It is well known that, for
this elusive massive component, there is a favoured scenario: the
$\Lambda$-cold dark matter ($\Lambda$CDM) one [298]; the dark particle is non-
relativistic and can be described as a collisionless fluid; it interacts
gravitationally with itself and with other massive particles and very weakly
with the Standard Model (SM) particles ([299, 300]). In detail, the Weakly
Interacting Massive Particles (WIMPs) cross section with the SM ones is taken
to be that of the Weak interactions: $\sigma\lesssim 3\ 10^{-26}cm^{2}$;
particles in the (1-200) GeV mass range that interact via the electroweak
force, with such self-annihilation cross section, imply a relic density of the
same order of the observed matter density $\rho_{c}\Omega_{m}$. WIMPs are the
relic particle from the early Universe, when all particles were in a state of
thermal equilibrium. Being the temperatures of the Universe $T\gg
m_{{}_{WIMP}}$ the WIMP particles and their antiparticles were both created
and annihilated into lighter Standard Model particles ($DM\ particle+DM\
particle\rightleftharpoons SM\ particle+SM\ particle$). As the Universe
expanded and cooled down ($T\lesssim m_{{}_{WIMP}}$), the average thermal
energy of the lighter particles decreased and eventually became too small to
form a dark matter particle-antiparticle pair. The annihilation of the dark
matter particle-antiparticle pairs did however continue and the number density
of the DM particles started to decrease exponentially ($\propto
exp[-m_{{}_{WIMP}}/T]$) so that the number density became so low that the DM
particle-antiparticle interaction vanished. Since then, the number of dark
matter particles has remained constant during the continued expansion of the
Universe. Because of their large mass, WIMPs move non-relativistically since
the Decoupling Time and are candidates of the CDM scenario. Noticeably, they
clump together, from small structures to the largest ones according to the
bottom-up scenario. They have a particular power spectrum of perturbations
which guarantees unique initial conditions. It is relevant the fact that
Supersymmetric extensions of the standard model of particle physics have new
particles with the above properties including the “WIMP miracle”(e.g. [301,
298]). Crucially, these particle, as any other which is collisionaless at
galactic scale, lead to DM halos having an Universal density profile that
behaves as: $\rho_{DM}\propto r^{-1}$ in its innermost regions [302].

The Ultralight axion (ULA) with $m\sim 10^{-22}\,eV$ is a scalar field ([303,
304]), that, at large scales mimics the behaviour of a cold particle, while in
(small) galaxies, where the inter-particle distance is much smaller than its
de Broglie wave length, behaves in a collective way acquiring a Bose-Einstein
condensates (BEC) equation of state that leads to cored dark halo density
configurations, like those observed.

In self-interacting dark matter (SIDM) the DM particles do self-interact with
a large scattering cross-section, but with a very small annihilation or
dissipation. The relative cross-section is likely due to strong, short-range
interactions, similar to neutron-neutron scattering at low-energies, or to
weak interactions mediated by the exchange of light particles [305]. Then, the
dark matter particles, inside the originally formed halo cusp, scatter
elastically among themselves so that they are heated up and leave the region;
this effect transforms the original cuspy density profile into a cored one.
The SIDM collision rate is negligible during the early Universe epochs and
during the period in which the cosmological structures form, so the relative
cosmological scenario retains the large-scale successes of that of
$\Lambda$CDM one , affecting the dark structures only at later times and at
small scales. See [306, 307].

The sterile neutrino is a lepton particle beyond the SM of particle physics.
(e.g. [308, 309]). Its existence is motivated also by arguments on the
chirality of fermions and on the possibility to explain in a natural way the
small active neutrino masses by means of the seesaw mechanism (e.g. [310]. The
mass of this particle when in the keV range (e.g. [311]), being so a warm dark
matter (WDM) particle. Created in the early Universe ([312, 313]) it decouples
from the cosmological plasma when it is still mildly relativistic. WDM
candidates may account for the various issues at small scales affecting the
collisionless CDM scenario; in detail, the fermionic nature of this particle
is be cosmologically crucial. In fact, for a $\sim$ keV mass the particle de-
Broglie scale length is of the order $\sim$ tens kpc, i.e. the spirals’s disks
sizes; then,in the latter, a quantum pressure emerges balancing the
gravitational force and shaping the inner DM density profile into a cored
distribution.([314])The different power spectrum of this particle might
account for the observed lack of small mass halos in the local Universe with
respect to the outcome of $\Lambda$ CDM N-Body simulations.

Recently, there appeared an unexpected new candidate for the Cold Dark Matter
scenario that could raise an amazing connection between the DM in galaxies and
the gravitational waves produced by the merging of stellar-mass black holes
and detected by LIGO-Virgo experiments. The inferred masses of the merging
Black Holes $\sim 50M_{\odot}$, infact, seems to be too high to be one of the
dead ends of the evolution of the stars in galaxies, and these GW instead may
signal the existence primordial black holes created in the very early Universe
and working since then as collisionless Macroscopic particles The most
interesting case is that in which they provide all the required galactic Dark
Matter. Then, a reasonable number of detections of primordial black hole
binaries could resolve the nature of this mysterious component [315]. Let us
stress that these objects had continuous merging since recombination and this
violent process could have generated a stochastic background of gravitational
waves, likely detectable by LISA and PTA. [316] Of course, just substituting
WIMPs with primordial BHs does not immediately relieve the severe tension with
the observations at galactic scales that any cold particle has, independently
of its mass.

The current status of the Universe features the crucial lack of any detection
of Dark particles, expecially the WIMP ones, in a direct or indirect way or
via super-collider experiment, allied with the lack of the distinctive central
cusp in the DM halo density (e.g.) [317, 318] a set of observed scaling laws
among the structure properties of the dark and the luminous matter components
in galaxies that are too refined to arise from two of them that just share the
same gravitational field (see [5]). This challenges the 30-year-old paradigm,
that, resting on a priori knowledge of the DM nature, has led us to a quite
small number of scenarios led by the collisionless Cold Dark Matter one.
Motivated by such observational evidence, it is on the table the idea of
resolving the dark matter mystery and its related not understood observations
by following a new Paradigm: the nature of DM must be guessed/derived by
deeply analyzing the properties of the dark and luminous mass distribution at
any scales independently on whether the emerging scenario look to us ”main
stream” or ”exotic”.

An application of this paradigm leads one to propose the existence of a direct
interaction between Dark and Standard Model particles which has finely shaped
the inner regions of galaxies [319]. Furthermore, other ”exotic” DM
candidates, including among many others e.g. the Mirror Dark Matter (e.g
[320]) and the Strongly Interacting Dark Matter, (e.g. [321]), neutral dark
atoms of composite dark matter [322] appear now reasonable and not anymore
crushed by the existence of the WIMP particle a-priori nominated as the actual
one.

##  17 Rotation Curves and Dark Matter in galaxies at high redshifts

###  17.1 Dark halos and the RCs at high z

We can fairly understand the properties of Dark Matter (DM) in the local
Universe (redshift $z\approx 0$) and, by means of the benefits of advancement
in the instrumentation, we can now explore the DM properties at $z\sim 1$ to
interpret the evolution of Dark and Luminous matter. The latter will certainly
contribute to give us crucial information on the nature of DM. In detail, we
have the KMOS-Redshift One Spectroscopic Survey (KROSS) data for investigating
and understanding the Dark matter scenario at redshift one ($z\sim 1$) in
rotation dominated galaxies (disk-type galaxies). KROSS is aimed to study the
gas kinematics of redshift one Star-Forming Galaxies (SFGs). Details of the
observations can be found in first and foremost papers of KMOS (e.g., [323],
[324], [325]).

[326] investigated the samples in [324] (hereafter H17) for the above
explained purpose. The sample includes 409 objects with an integrated
$H\alpha$ flux $F_{H\alpha}>1.5\times 10^{-17}ergs^{-1}cm^{-2}$, i.e. with an
acceptable Signal-to-noise (S/N) and an identified $H\alpha$ emission line.
This sample has median redshift $z=0.85^{+0.13}_{-0.04}$, $H\alpha$ luminosity
$log(L_{H\alpha}[erg\ s^{-1}])=41.47\pm 0.4$ and effective radii
$log(R_{e}[kpc])=0.45^{+0.23}_{0.33}$. (see TableA1 of H17) Moreover, each
galaxy is not an AGN and contains high quality $H\alpha$ spatially resolved
data. Effective radii ($R_{e}$), position angles (PA), inclination angles
($\theta_{i}$), absolute H-band magnitudes ($M_{H}$), $H\alpha$ Luminosities
($L_{H\alpha}$) and redshifts ($z$) and stellar masses ($M_{*}$) are also
known.

They have modelled the kinematics of the samples under investigation with
${}^{3D}BAROLO$ code ([327]) that incorporates the instrumental and
atmospheric effects (i.e. related to the point spread function (PSF) which
determines the spatial resolution and the line spread function (LSF) which
corresponds to spatial broadening, in combination these effects are known as
beam-smearing.) This is done in a dynamical environment, differently from the
2D maps modelling which is strongly beam-smearing dominated in the inner
region of galaxies. By using 3D-Barolo, it is possible to flawlessly compare
the data and model in 3D observation space, thus correct for the beam-smearing
and other effects dynamically ([327, 328]).

The modelling of 3D-Barolo requires three geometrical parameters, i.e., co-
ordinate of galaxy centre in the datacube ($x_{0},y_{0}$), inclination angle
($\theta_{i}$), position angle (PA) and three kinematic parameters, i.e.,
redshift ($z$), rotation velocity ($v_{c}$) and dispersion velocity from
ionized gas ($\sigma_{H\alpha}$). In their modelling, they precisely define
the central co-ordinates of galaxy ($x_{0},y_{0}$), $\theta_{i}$, $z$ and keep
the $v_{c}$, $\sigma_{H\alpha}$ and PA free to fit. This approach allows us to
estimate the free parameters in annulii of increasing distance from the galaxy
centre without making any assumption on their evolution with radius, yielding
a reliable approach on kinematic modelling of 3D datacube.

Figure 15: 1) $H\alpha$ image from chopped cube; 2) moment0 map; 2) moment1
map; 3) moment2 map; 4) Rotation Curve, black shade is data, red contour is
the model and orange square is the best fit $v_{rot}$; 5) Dispersion curve,
black shade is data, red contour is model and orange square is best fit,
$\sigma_{H\alpha}$ of ionized gas

Fig. 15 shows a sample of Barolo outputs. From left to right: $H\alpha$ image
, total $H\alpha$ intensity maps, $H\alpha$ velocity field, $H\alpha$
dispersion field, P-V diagrams in comparison with data and model. Maps are
plotted on pixel coordinates (x, y-axis). The black cross shows the kinematic
centre of the galaxy, which is in agreement with central photometric
coordinates of the galaxy. The grey tilted line is kinematic position angle,
red contour in P-V diagram (RC) is model; black shaded area represents the
data, and orange squares with error bars are best-fit velocity measurements.

Figure 16: Comparison of $z\sim 1$ (hexagon connected dashed line curves) RCs
and those of local spirals derived from the URC ([330, 332]) (solid line
curves). Colour code is given in the legend. Numbers in the bracket indicate
the total number of RCs used in each bin. Dotted and solid vertical lines show
$R_{out}$ and $R_{opt}$ for each RC (colour coded as the velocity bins)

They have obtained four co-added and binned RCs built out of from 201
individual RCs. Such a statistical approach has remarkable advantages: 1) it
gives us a smooth distribution of RC without being affected by the random
fluctuations that arise in each RC from bad data points, i.e., it virtually
enhances the S/N in data; 2) it allows mass decomposition of galaxies of
similar velocities but having different spatial sampling in the velocity
field. This kind of approach in RC studies has been used for decades,
pioneered by [329] and later developed in several works ([330, 331, 332, 333,
334, 335, 336]). In Fig. 16 the results of co-added and binned RCs, in
comparison with local RCs are shown. Within $2Kpc$, there is not data due to
the low spatial and kinematical resolution, that affect also the first measure
at $2kpc$. check this The RCs emerge surprisingly flattish from
$R_{opt}=3.2R_{D}$ to 4 times $R_{opt}$, and the dark matter starts dominating
within $R_{opt}$. The comparison with local RCs shows that in physical units
(i.e. $V(R/kpc)/(kms^{-1})$ the $z\sim 1$ RCs are very similar to the local
RCs, though only the mass modelling may clarify this.

###  17.2 The Rotation curves profiles and the Nature of Dark matter

The careful study of RCs both at low and high redshifts opens the way to
pinpoint the nature of the dark particle and to follow the evolution of the DM
halos during the galaxy formation era.

Among other recent works, by studying the RC of a sample of 30 dwarf irregular
galaxies, it has been shown that these objects represent new targets for DM
indirect searches in gamma rays: in fact, they are DM dominated object with
astrophysical gamma-ray emission negligible with respect to the gamma-ray flux
expected by DM annihilation events [337]. This claim have been already
investigated not only at GeV energy scale, yet for heavy DM candidates at TeV
energy scales, e.g. by the High Altitude Water Cherenkov (H.A.W.C) observatory
[338]. Although (multi-)TeV DM candidates are not favoured by benchmark models
in particle physics - e.g. Super Symmetry, after decades of searching no
experimental evidence of WIMP DM at GeV energy scale has been found at
colliders so far. This fact has generated a crisis in physics and requires an
epochal paradigm shift. In view of the new era of TeV observatories the study
of particle physics nature of TeV DM candidate is a well-timed challenge and
represents a new frontier in physics.

There are few multi-TeV DM candidates available on a theoretical side. Among
others, branons are WIMPs originated by a symmetry breaking mechanism in the