Salvatore Capozziello

• Full Professor in Astronomy and Astrophysics
• Professor (Full) at University of Naples Federico II

We investigate the impact of dark matter halos on the gravitational lensing produced by electrically charged, spherically symmetric black holes in the strong-field regime. The study focuses on two dark matter models: the Universal Rotation Curve Model and the cold dark matter model. We derive the coefficients for the strong deflection limit and num...

Noether’s symmetries play a prominent role in any branch of physics. Here, we discuss their role in Quantum Cosmology. In particular, in this Chapter, we consider minisuperspace cosmological models, showing that the existence of symmetries, and then conserved quantities, not only allows to obtain exact cosmological solutions but also gives a select...

The emergence of the Ultra-Diffuse Galaxies in recent years has posed a severe challenge to the galaxy formation models as well as the Extended Theories of Gravity. The existence of both dark matter lacking and dark matter dominated systems within the same family of astrophysical objects indeed requires the gravity models to be versatile enough to...

The effects of cosmological expansion on quantum bosonic states are investigated, using quantum information theory. In particular, a generic Bogoliubov transformation of bosonic field modes is considered and the state change on a single mode is regarded as the effect of a quantum channel. Properties and capacities of this channel are thus explored...

We investigate a Schwarzschild metric exhibiting a signature change across the event horizon, which gives rise to what we term a Lorentzian-Euclidean black hole. The resulting geometry is regularized by employing the Hadamard partie finie technique, which allows us to prove that the metric represents a solution of vacuum Einstein equations. In this...

We discuss spherically symmetric dynamical systems in the framework of a general model of $f({\cal R})$ gravity, i.e. $f({\cal R})={\cal R}e^{\zeta {\cal R}}$, where $\zeta$ is a dimensional quantity in squared length units [L$^2$]. We initially assume that the internal structure of such systems is governed by the Krori-Barua ansatz, alongside the...

We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function F(R,□−1R), where R denotes the Ricci curvature scalar and the operator □−1 in...

We intend to construct these geometries using a slightly modified Hilbert’s axioms system in the same way as it is done in [36, 37, 138, 185]. An interesting thing is related to the fact that it exists as a common part for Euclidean and non-Euclidean Geometry, the so-called Absolute Geometry. Roughly speaking, the Absolute Geometry consists in all...

We are ready to present the two big pictures of non-Euclidean geometry models.

In the previous discussions, we presented General Relativity as a theory essentially based on metric as the main object capable of being related to physical observables. Thanks to the Equivalence Principle, the Einstein formulation is a “metric theory of gravity” where causal structure (related to metric and light cones) and geodesic structure (rel...

Einstein’s point of view written above may seem malicious. But almost all the mathematics we present in this chapter is related to the work by Tullio Levi-Civita.

We intend to present some basic facts related to the Differential Geometry of a surface in a 3D-space in the simplest form we imagined. All readers must know basic Calculus and they have to accept from the beginning that we work with functions which are smooth, that is they are indefinitely differentiable functions in one or several variables at ea...

General Relativity is not the end of the story. Several issues, ranging from Quantum Gravity to the Dark Side of the Universe need to be addressed in a self-consistent theory. Here we want to summarize some of these approaches. In particular, we want to show that the same theory, General Relativity, can be represented in different ways. This fact i...

Newtonian mechanics is a branch of Physics which studies the way in which the bodies are changing in time their position in space. The space in which the objects are at rest (or they change their position) is the Euclidean three-dimensional space \(E^3\). All objects, regardless of size, can be identified as points with a given mass in the previous...

The gravitational field in General Relativity is described by the Einstein field equations.

An imaginary discussion between Newton and Einstein could be the following. Isaac Newton: Dear Prof. Einstein, my Universe is very simple. I can describe it using vectors and calculus. Between any two objects, a gravitational force is acting and, according to the masses of objects and the distance between them, the gravitational force law is \(F=G\...

In Chapter 1, we found out that there exist different geometries in a plane. It depends on the axioms one chooses if Euclidean Geometry or Non-Euclidean Geometry is described. But how these geometries look like? In this chapter we present an algebraic model for Euclidean Geometry discussing some important theorems. We obtain a visual representation...

This chapter is devoted to a first model of Non-Euclidean Geometry. To construct this model, we need to deal with one of the most important transformations of the Euclidean plane, the geometric inversion. We still need some other acquirements, therefore we meet the Projective Geometry. An invariant described by a special projective map of a circle...

In seventeenth-century, Newton considered light as a collection of particles, now called photons according to Quantum Mechanics, traveling through space. Reflection and refraction of light were explained in a satisfactory way interpreting light rays as trajectory of photons.

There are a lot of similarities between de Sitter and Anti-de Sitter space–times; therefore, we want to present here some geometric aspects which are not highlighted in the previous chapter. Before developing a geometric theory of gravity which can be extended in Minkowski \(M^{(p,q)}\) spaces and before reaching the subject of Anti-de Sitter space...

We want to conclude this book considering a gravity theory without masses which can be constructed in Minkowski spaces using a geometric Minkowski potential. From the point of view of this book, this can be considered a full geometric realization of the relativistic approach.

A critical discussion on the H0 Hubble constant tension is presented by considering both early and late-type observations. From recent precise measurements, discrepancies emerge when comparing results for some cosmological quantities obtained at different redshifts. We highlight the most relevant measurements of H0 and propose potential ideas to so...

The GINGER (gyroscopes in general relativity) project foresees the construction of an array of large frame ring laser gyroscopes, rigidly connected to the Earth. Large frame ring laser gyroscopes are high-sensitivity instruments used to measure angular velocity with respect to the local inertial frame. In particular, they can provide sub-daily vari...

Physical evolution of cosmological models can be tested by using expansion data, while growth history of these models is capable of testing dynamics of the inhomogeneous parts of energy density. The growth factor, as well as its growth index, gives a clear indication of the performance of cosmological models in the regime of structure formation of...

This paper presents a study of the dissociation energy of the N–S poles of a quantum magnetic particle, carried out from both classical and quantum mechanical perspectives. A simple model of a harmonic oscillator is employed to estimate the dissociation energy of the N–S poles, as well as the corresponding breakdown temperature and internal pressur...

Gravitational radiation-reaction phenomena occurring in the dynamics of inspiralling compact binary systems are investigated at the first post-Newtonian order beyond the quadrupole approximation in the context of Einstein-Cartan theory, where quantum spin effects are modeled via the Weyssenhoff fluid. We exploit balance equations for the energy and...

Although the Λ Cold Dark Matter model is the most accredited cosmological model, information at high redshifts (z) between type Ia supernovae (z=2.26) and the Cosmic Microwave Background (z=1100) is crucial to validate this model further. To this end, we have discovered a sample of 1132 quasars up to z=7.54 exhibiting a reduced intrinsic dispersion...

We consider static and spherically symmetric wormhole solutions in extended metric-affine theories of gravity supposing that stability and traversability of these objects can be achieved by means of the more geometric degrees of freedom with respect to general relativity. In particular, we concentrate on f(R) metric, f(T) teleparallel, and f(Q) sym...

We outline the scientific objectives, the experimental layout, and the collaborations envisaged for the GINGER (Gyroscopes in general relativity) project. The GINGER project brings together different scientific disciplines aiming at building an array of ring laser gyroscopes (RLGs), exploiting the Sagnac effect, to measure continuously, with sensit...

Large frame ring laser gyroscopes, based on the Sagnac effect, are top sensitivity instrumentation to measure angular velocity with respect to the fixed stars. GINGER (Gyroscopes IN GEneral Relativity) project foresees the construction of an array of three large dimension ring laser gyroscopes, rigidly connected to the Earth. GINGER has the potenti...

The Gauss–Bonnet topological scalar is presented in metric-teleparallel formalism as well as in the symmetric and general teleparallel formulations. In all of the aforementioned frameworks, the full expressions are provided explicitly in terms of torsion, non-metricity and Levi-Civita covariant derivative. The number of invariant terms of this form...

Physical evolution of cosmological models can be tested by using expansion data, while growth history of these models is capable of testing dynamics of the inhomogeneous parts of energy density. The growth factor, as well as its growth index, gives a clear indication of the performance of cosmological models in the regime of structure formation of...

We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend th...

We consider static and spherically symmetric wormhole solutions in extended metric-affine theories of gravity supposing that stability and traversability of these objects can be achieved by means of the geometric degrees of freedom. In particular, we consider $f(R)$ metric, $f(T)$ teleparallel, and $f(Q)$ symmetric teleparallel models where curvatu...

The Gauss-Bonnet topological scalar is presented in metric-teleparallel formalism as well as in the symmetric and general teleparallel formulations. In all of the aforementioned frameworks, the full expressions are provided explicitly in terms of torsion, non-metricity and Levi-Civita covariant derivative. The number of invariant terms of this form...

The longstanding search for the cosmological model that best describes the Universe has been made more intriguing since the recent discovery of the Hubble constant, H0, tension observed between the value of H0 from the Cosmic Microwave Background and from type Ia supernovae (SNe Ia). Hence, the commonly trusted flat Λ cold dark matter (ΛCDM) model...

We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend th...

The longstanding search for the cosmological model that best describes the Universe has been made more intriguing since the recent discovery of the Hubble constant, $H_{0}$, tension observed between the value of $H_{0}$ from the Cosmic Microwave Background and from type Ia supernovae (SNe Ia). Hence, the commonly trusted flat $\Lambda$CDM model is...

In the framework of metric-affine gravity, we consider the role of the boundary term in Symmetric Teleparallel Gravity assuming $f(Q,B)$ models where $f$ is a smooth function of the non-metricity scalar $Q$ and the related boundary term $B$. Starting from a variational approach, we derive the field equations and compare them with respect to those o...

The elusive physical nature of Horndeski gravity is elucidated in a new approach depicting this class of theories as a dissipative effective fluid. Requiring the constitutive equations of the latter to be those of a Newtonian fluid restricts the theory to only two disconnected subclasses of “viable” Horndeski gravity. Therefore, a stress-energy ten...

In the framework of scalar-tensor gravity, we consider non-flat interacting quintessence cosmology where a scalar field is interacting with dark matter. Such a scalar field can be a standard or a phantom one. We use the Noether Symmetry Approach to obtain general exact solutions for cosmological equations and to select scalar-field self-interaction...

Cosmological models and their parameters are widely debated, especially about whether the current discrepancy between the values of the Hubble constant, H 0 , obtained by Type Ia supernovae (SNe Ia) and the Planck data from the cosmic microwave background radiation could be alleviated when alternative cosmological models are considered. Thus, combi...

f ( Q ) symmetric-teleparallel gravity is considered in view of quantum cosmology. Specifically, we derive cosmological equations for f ( Q ) models and then investigate the related energy conditions. In the Minisuperspace formalism, the point-like f ( Q ) Hamiltonian is taken into account. In this framework, we obtain and solve the Wheeler–De Witt...

We probe four cosmological models which, potentially, can solve the Hubble tension according to the dark energy equation of state. In this context, we demonstrate that the Einstein Telescope is capable of achieving a relative accuracy below 1% on the Hubble constant independently of the specific dark energy model. We firstly build mock catalogs con...

We consider a model-independent approach to constrain the equivalence redshift, $z_{eq}$, at which dark energy, baryons and cold dark matter equate their magnitudes. To this aim, in the context of a homogeneous and isotropic universe, we first consider a generic model where the dark energy contribution is provided by an unknown function of barotrop...

Currently, the Λ cold dark matter model, which relies on the existence of cold dark matter and a cosmological constant Λ, best describes the universe. However, we lack information in the high-redshift ( z ) region between Type Ia supernovae (SNe Ia; up to z = 2.26) and the cosmic microwave background ( z = 1100), an interval crucial to test cosmolo...

Different astrophysical methods can be combined to detect possible deviations from General Relativity. In this work, we consider a class of f ( R ) gravity models selected by the existence of Noether symmetries. In this framework, it is possible to determine a set of static and spherically symmetric black hole solutions, encompassing small departur...

In the framework of scalar-tensor gravity, we consider non-flat interacting quintessence cosmology where a scalar field is interacting with dark matter. Such a scalar field can be a standard or a phantom one. We use the Noether Symmetry Approach to obtain general exact solutions for cosmological equations and to select scalar-field self-interaction...

Currently, the $\Lambda$ Cold Dark Matter model, which relies on the existence of cold dark matter and a cosmological constant $\Lambda$, best describes the Universe. However, we lack information in the high-redshift ($z$) region between Type Ia Supernovae (SNe Ia) (up to $z=2.26$) and the Cosmic Microwave Background ($z=1100$), an interval crucial...

Cosmological models and their parameters are widely debated, especially about whether the current discrepancy between the values of the Hubble constant, $H_{0}$, obtained by type Ia supernovae (SNe Ia), and the Planck data from the Cosmic Microwave Background Radiation could be alleviated when alternative cosmological models are considered. Thus, c...

The effects of dark matter spike in the vicinity of the supermassive black hole, located at the center of M87 (the Virgo A galaxy), are investigated within the framework of the so-called Bumblebee Gravity. Our primary aim is to determine whether the background of spontaneous Lorentz symmetry breaking has a significant effect on the horizon, ergo-re...

Different astrophysical methods can be combined to detect possible deviations from General Relativity. In this work, we consider a class of $f(R)$ gravity models selected by the existence of Noether symmetries. In this framework, it is possible to determine a set of static and spherically symmetric black hole solutions, encompassing small departure...

f(Q)$ symmetric-teleparallel gravity is considered in view of Quantum Cosmology. Specifically, we derive cosmological equations for $f(Q)$ models and then investigate the related energy conditions. In the minisuperspace formalism, the point-like $f(Q)$ Hamiltonian is taken into account. In this framework, we obtain and solve the Wheeler-De Witt equ...

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a...

Teleparallel geometry offers a platform on which to build up theories of gravity where torsion rather than curvature mediates gravitational interaction. The teleparallel analogue of Horndeski gravity is an approach to teleparallel geometry where scalar-tensor theories are considered in this torsional framework. Teleparallel gravity is based on the...

We disclose a close correspondence between Verlinde's Emergent Gravity (VEG) theory and the non-local gravity theories. Such non-local effects can play crucial role at small distances as well as in large scale structures. In particular, we argue that the emergent gravity effectively is a manifestation of the entanglement entropy and can modify Newt...

The effects of dark matter spike in the vicinity of the supermassive black hole, located at the center of M87 (the Virgo A galaxy), are investigated within the framework of the so-called Bumblebee Gravity. Our primary aim is to determine whether the background of spontaneous Lorentz symmetry breaking has a significant effect on the horizon, ergo-re...

The recent ∼4 σ Hubble constant, H0, tension is observed between the value of H0 from the Cosmic Microwave Background (CMB) and Type Ia Supernovae (SNe Ia). It is a decade since this tension is excruciating the modern astrophysical community. To shed light on this problem is key to consider probes at intermediate redshifts between SNe Ia and CMB an...

The recent $\sim 4 \, \sigma$ Hubble constant, $H_{0}$, tension is observed between the value of $H_{0}$ from the Cosmic Microwave Background (CMB) and Type Ia Supernovae (SNe Ia). It is a decade since this tension is excruciating the modern astrophysical community. To shed light on this problem is key to consider probes at intermediate redshifts b...

Type Ia Supernovae (SNe Ia) are considered the most reliable \textit{standard candles} and they have played an invaluable role in cosmology since the discovery of the Universe's accelerated expansion. During the last decades, the SNe Ia samples have been improved in number, redshift coverage, calibration methodology, and systematics treatment. Thes...

One of the most crucial areas of gravity research, after the direct observation of gravitational waves, is the possible modification of general relativity at ultraviolet and infrared scales. In particular, the possibility of parity violation should be considered in strong field regime. The Chern-Simons gravity takes into account parity violation in...

We show that the so-called chaos bound, proposed by Maldacena, Shenker, and Stanford, can be violated in $f(T)$ teleparallel gravity. In particular, it is possible to select a new gravitational Lyapunov parameter, controlling chaotization of circular trajectories, in black hole photo-sphere, that can exceed the Maldacena, Shenker and Stanford therm...

One of the most crucial areas of gravity research, after the direct observation of gravitational waves, is the possible modification of General Relativity at ultraviolet and infrared scales. In particular, the possibility of parity violation should be considered in strong field regime. The Chern-Simons gravity takes into account parity violation in...

We show that the so-called chaos bound, proposed by Maldacena, Shenker & Stanford, can be violated in f(T) teleparallel gravity. In particular, it is possible to select a new gravitational Lyapunov parameter, controlling chaotization of circular trajectories, in black hole photo-sphere, that can exceed the Maldacena, Shenker & Stanford thermal boun...

The Hubble tension is investigated taking into account the cosmological look-back time. Specifically, considering a single equation, widely used in standard cosmology, it is possible to recover both values of the Hubble constant $H_0$ reported by the SHOES and Planck collaborations: the former is obtained through cosmological ladder methods (e.g. C...

We investigate how the cosmological equation of state can be used for scrutinizing extended theories of gravity, in particular, the Palatini $$f({{\mathcal {R}}})$$ f ( R ) gravity. Specifically, the approach consists, at first, in investigating the effective equation of state produced by a given model. Then, the inverse problem can also be conside...

We show that the Einstein field equations with a negative cosmological constant can admit black hole solutions whose thermodynamics coincides with that of logotropic fluids, recently investigated to heal some cosmological and astrophysical issues. For this purpose, we adopt the Anton–Schmidt equation of state, which represents a generalized version...

Cosmological models and their parameters are widely debated because of theoretical and observational mismatches of the standard cosmological model, especially the current discrepancy between the value of the Hubble constant, H 0 , obtained by Type Ia supernovae (SNe Ia), and the cosmic microwave background radiation (CMB). Thus, considering high-re...

We construct a holographic model for dark energy in the Brans–Dicke cosmology by using the holographic principle considering the Barrow entropy instead of the standard Bekenstein–Hawking one. The former arises from the effort to account for quantum gravitational effects in black hole physics and, according to the gravity–thermodynamic conjecture, i...

The Local Group (LG) of galaxies, modeled as a two body problem, is sensitive to cosmological contributions like those related to the presence of a cosmological constant $\Lambda$ into dynamics. Here we study the LG dynamics in the context of Extended Theories of Gravity like $f(R)$ gravity considered as dark energy and dark matter contributions. I...

Teleparallel geometry offers a platform on which to build up theories of gravity where torsion rather than curvature mediates gravitational interaction. The teleparallel analogue of Horndeski gravity is an approach to teleparallel geometry where scalar-tensor theories are considered in this torsional framework. Being teleparallel gravity of lower o...

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a...

Alternative cosmological models have been under deep scrutiny in recent years, aiming to address the main shortcomings of the $\Lambda$CDM model. Moreover, as the accuracy of cosmological surveys improved, new tensions have risen between the model-dependent analysis of the Cosmic Microwave Background and lower redshift probes. Within this framework...

Alternative cosmological models have been under deep scrutiny in recent years, aiming to address the main shortcomings of the ΛCDM model. Moreover, as the accuracy of cosmological surveys improved, new tensions have risen between the model-dependent analysis of the Cosmic Microwave Background and lower redshift probes. Within this framework, we rev...

Following results presented in Spallicci et al. (Eur Phys J Plus 137, 2022) by the same authors, we investigate the observed red shift z, working under the hypothesis that it might be composed by the expansion red shift \(z_{\mathrm{C}}\) and an additional frequency shift \(z_{\mathrm{S}}\), towards the red or the blue, due to Extended Theories of...

We investigate how the cosmological Equation of State can be used for scrutinizing extended theories of gravity, in particular, the Palatini $f(R)$ gravity. Specifically, the approach consists, at first, in investigating the effective Equation of State produced by a given model. Then, the inverse problem can also be considered in view of determinin...

The cosmographic technique is a powerful model-independent tool for distinguishing between competing cosmological scenarios. The key strengths and weaknesses of standard cosmography are discussed in view of healing the convergence problem endangering the high-redshift expansions of cosmological distances. We focus especially on rational cosmographi...

Following results presented in Eur. Phys. J. Plus, {\bf 137} (2022) 253 by the same authors, we investigate the observed red shift $z$, working under the hypothesis that it might be composed by the expansion red shift $z_{\rm C}$ and an additional frequency shift $z_{\rm S}$, towards the red or the blue, due to Extended Theories of Electromagnetism...

Cosmological models and their parameters are widely debated because of theoretical and observational mismatches of the standard cosmological model, especially the current discrepancy between the value of the Hubble constant, $H_{0}$, obtained by Type Ia supernovae (SNe Ia), and the Cosmic Microwave Background Radiation (CMB). Thus, considering high...

Discrepancies between observations at early and late cosmic epochs, and the vacuum energy problem associated with the interpretation of cosmological constant, are questioning the $\Lambda$CDM model. Motivated by these conceptual and observational facts, extensions of Einstein's gravity are recently intensively considered in view of curing unsolved...

We consider Non-local Gravity in view to obtain stable and traversable wormhole solutions. In particular, the class of Non-local Integral Kernel Theories of Gravity, with the inverse d'Alembert operator in the gravitational action, is taken into account. We obtain constraints for the null energy condition and derive the field equations. Two special...