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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., Comment: 64 pages 16 Figures. In print on Frontiers in Astronomy and Space Sciences, for the Research Topic: "When Planck, Einstein and Vera Rubin Meet. Dark Matter: What is it ? Where is it?"

In the eighties, Schroder studied a quantum mechanical model where the stationary states of Schrodinger's equation obey nonlocal boundary conditions on a circle in the plane. For such a problem, we perform a detailed one-loop calculation for three choices of the kernel characterizing the nonlocal boundary conditions. In such cases, the zeta(0) value is found to coincide with the one resulting from Robin boundary conditions. The detailed technique here developed may be useful for studying one-loop properties of quantum field theory and quantum gravity if nonlocal boundary conditions are imposed., Comment: 17 pages, Revtex4. In the final version, the presentation in section 5 has been improved, and important References have been added

The static spherically symmetric traversable wormholes are analysed in the Einstein- Cartan theory of gravitation. In particular, we computed the torsion tensor for matter fields with different spin S = 0; 1/2; 1; 3/2. Interestingly, only for certain values of the spin the torsion contribution to Einstein-Cartan field equation allows one to satisfy both faring-out condition and Null Energy Condition. In this scenario traversable wormholes can be produced by using usual (non-exotic) spinning matter., Comment: 13 pages

In this paper, we develop an iterative approach to span the whole set of exotic matter models able to drive a traversable wormhole. The method, based on a Taylor expansion of metric and stress-energy tensor components in a neighbourhood of the wormhole throat, reduces the Einstein equations to an infinite set of algebraic conditions, which can be satisfied order by order. The approach easily allows the implementation of further conditions linking the stress-energy tensor components among each other, like symmetry conditions or equations of state. The method is then applied to some relevant examples of exotic matter characterised by a constant energy density and that also show an isotropic behaviour in the stress-energy tensor or obeying to a quintessence-like equation of state., Comment: 9 pages. Version published in EPJP

The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integrals for another choice of Kasner parameters. For generic values of the three Kasner parameters, the solution of the Cauchy problem is built through a pair of integral operators, where the amplitude and phase functions in the integrand solve a coupled system of partial differential equations. The first is the so-called transport equation, whereas the second is a nonlinear equation that reduces to the eikonal equation if the amplitude is a slowly varying function. Remarkably, the analysis of such a coupled system is proved to be equivalent to building first an auxiliary covariant vector having vanishing divergence, while all nonlinearities are mapped into solving a covariant generalization of the Ermakov-Pinney equation for the amplitude function. Last, from a linear set of equations for the gradient of the phase one recovers the phase itself. This is the parametrix construction that relies upon Fourier-Maslov integral operators, but with a novel perspective on the nonlinearities in the dispersion relation. Furthermore, the Adomian method for nonlinear partial differential equations is applied to generate a recursive scheme for the evaluation of the amplitude function in the parametrix., Comment: 29 pages, Latex. Sections V and VI contain new original work on bicharacteristics and on the parametrix of the wave equation. The last misprints have been amended

The field equations of noncommutative gravity can be obtained by replacing all exterior products by twist-deformed exterior products in the action functional of general relativity, and are here studied by requiring that the torsion 2-form should vanish, and that the Lorentz-Lie-algebra- valued part of the full connection 1-form should be self-dual. Other two conditions, expressing self-duality of a pair 2-forms occurring in the full curvature 2-form, are also imposed. This leads to a systematic solution strategy, here displayed for the first time, where all parts of the connection 1-form are first evaluated, hence the full curvature 2-form, and eventually all parts of the tetrad 1-form, when expanded on the basis of {\gamma}-matrices. By assuming asymptotic expansions which hold up to first order in the noncommutativity matrix in the neighbourhood of the vanishing value for noncommutativity, we find a family of self-dual solutions of the field equations. This is generated by solving first a inhomogeneous wave equation on 1-forms in a classical curved spacetime (which is itself self-dual and solves the vacuum Einstein equations), subject to the Lorenz gauge condition. In particular, when the classical undeformed geometry is Kasner spacetime, the above scheme is fully computable out of solutions of the scalar wave equation in such a Kasner model., Comment: 37 pages, Revtex. Appendix A is a recollection of mathematical tools used in the paper. In the final version, Appendix C and some valuable References have been added. arXiv admin note: text overlap with arXiv:hep-th/0703014 by other authors. Misprints in Eq. (10.23) and (10.25) have been amended, as well as their propagation in Sec. X

This paper studies non-linear constitutive equations for gravitoelectromagnetism. Eventually, the problem is solved of finding, for a given particular solution of the gravity-Maxwell equations, the exact form of the corresponding non-linear constitutive equations., Comment: In the final version, the title has been changed, and only original calculations have been presented, removing the review part

In this paper the Seiberg-Witten map is first analyzed for non-commutative Yang-Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg-Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space-time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, when the Seiberg-Witten map is eventually inserted. In the attempt of understanding whether or not this is a peculiar property of gravity, in the second part of the paper, the Seiberg-Witten map is considered in the simpler case of Maxwell theory in vacuum in the absence of charges and currents. Once more, no obvious solution of the non-commutative field equations is found, unless the electromagnetic potential depends in a very special way on the wave vector., Comment: Misprints corrected. References added

The Seiberg--Witten map is a powerful tool in non-commutative field theory, and it has been recently obtained in the literature for gravity itself, to first order in non-commutativity. This paper, relying upon the pure-gravity form of the action functional considered in Ref. 2, studies the expansion to first order of the non-commutative Einstein equations, and whether the Seiberg--Witten map can lead to a solution of such equations when the underlying classical geometry is Schwarzschild., Comment: 6 and 1/2 pages, based on talk prepared for the Friedmann Seminar, May-June 2011. In the final version, the presentation has been improved, including a better notation

This paper applies the first-order Seiberg-Witten map to evaluate the first-order non-commutative Kerr tetrad. The classical tetrad is taken to follow the locally non-rotating frame prescription. We also evaluate the tiny effect of non-commutativity on the efficiency of the Penrose process of rotational energy extraction from a black hole., Comment: 14 pages. The original calculations are completely new

A modification of the standard product used in local field theory by means of an associative deformed product is proposed. We present a class of deformed products, one for every spin S=0,1/2,1, that induces a nonlocal theory, displaying different form for different fields. This type of deformed product is naturally supersymmetric and it has an intriguing duality., Comment: 23 pages, Latex

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. Relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, we have considered from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes has been shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F have been derived which are compatible with the adiabatic approximation., Comment: 8 pages, Latex file with IOP macros, prepared for the QFEXT07 Conference, Leipzig, September 2007

Recent work in the literature has studied a version of non-commutative Schwarzschild black holes where the effects of non-commutativity are described by a mass function depending on both the radial variable r and a non-commutativity parameter theta. The present paper studies the asymptotic behaviour of solutions of the zero-rest-mass scalar wave equation in such a modified Schwarzschild space-time in a neighbourhood of spatial infinity. The analysis is eventually reduced to finding solutions of an inhomogeneous Euler--Poisson--Darboux equation, where the parameter theta affects explicitly the functional form of the source term. Interestingly, for finite values of theta, there is full qualitative agreement with general relativity: the conformal singularity at spacelike infinity reduces in a considerable way the differentiability class of scalar fields at future null infinity. In the physical space-time, this means that the scalar field has an asymptotic behaviour with a fall-off going on rather more slowly than in flat space-time., Comment: 19 pages, Revtex4, 7 figures

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of noncommutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. The present paper, relying on the recently obtained noncommutativity effect on a static, spherically symmetric metric, considers from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes is shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited. Approximate formulae for the particle emission rate are also obtained within this framework., Comment: 14 pages, 2 figures, Latex macros. In the final version, section 5 has been amended, the presentation has been improved, and References 21-24 have been added. Last misprints amended in Section 5 and Ref. 24

This paper investigates the possible cosmological implications of the presence of an antisymmetric tensor field related to a lack of commutatitivity of spacetime coordinates at the Planck era. For this purpose, such a field is promoted to a dynamical variable, inspired by tensor formalism. By working to quadratic order in the antisymmetric tensor, we study the field equations in a Bianchi I universe in two models: an antisymmetric tensor plus scalar field coupled to gravity, or a cosmological constant and a free massless antisymmetric tensor. In the first scenario, numerical integration shows that, in the very early universe, the effects of the antisymmetric tensor can prevail on the scalar field, while at late times the former approaches zero and the latter drives the isotropization of the universe. In the second model, an approximate solution is obtained of a nonlinear ordinary differential equation which shows how the mean Hubble parameter and the difference between longitudinal and orthogonal Hubble parameter evolve in the early universe., Comment: 25 pages, Revtex file, 4 figures in attachment