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Cosmography represents an important branch of cosmology which aims to describe the universe without the need of postulating a priori any particular cosmological model. All quantities of interest are expanded as a Taylor series around here and now, providing in principle, a way of directly matching with cosmological data. In this way, cosmography can be regarded a model-independent technique, able to fix cosmic bounds, although several issues limit its use in various model reconstructions. The main purpose of this review is to focus on the key features of cosmography, emphasizing both the strategy for obtaining the observable cosmographic series and pointing out any drawbacks which might plague the standard cosmographic treatment. In doing so, we relate cosmography to the most relevant cosmological quantities and to several dark energy models. We also investigate whether cosmography is able to provide information about the form of the cosmological expansion history, discussing how to reproduce the dark fluid from the cosmographic sound speed. Following this, we discuss limits on cosmographic priors and focus on how to experimentally treat cosmographic expansions. Finally, we present some of the latest developments of the cosmographic method, reviewing the use of rational approximations, based on cosmographic Padé polynomials. Future prospects leading to more accurate cosmographic results, able to better reproduce the expansion history of the universe, are also discussed in detail.

The neutrino oscillations in the field of a rotating deformed mass is investigated. The phase shift is evaluated in the case of weak field limit, slow rotation and small deformation. To this aim the Hartle-Thorne metric is used, which is an approximate solution of the vacuum Einstein equations accurate to second order in the rotation parameter $a/M$ and to first order in the mass quadrupole moment $q$. Implications on atmospheric, solar and astrophysical neutrinos are discussed., 16 pages; to appear in PLA

We use cosmography to present constraints on the kinematics of the Universe, without postulating any underlying theoretical model. To this end, we use a Monte Carlo Markov Chain analysis to perform comparisons to the supernova Ia Union 2 compilation, combined with the Hubble Space Telescope measurements of the Hubble constant, and the Hubble parameter datasets. We introduce a sixth order cosmographic parameter and show that it does not enlarge considerably the posterior distribution when comparing to the fifth order results. We also propose a way to construct viable parameter variables to be used as alternatives of the redshift $z$. These can overcome both the problems of divergence and lack of accuracy associated with the use of $z$. Moreover, we show that it is possible to improve the numerical fits by re-parameterizing the cosmological distances. In addition, we constrain the equation of state of the Universe as a whole by the use of cosmography. Thus, we derive expressions which can be directly used to fit the equation of state and the pressure derivatives up to fourth order. To this end, it is necessary to depart from a pure cosmographic analysis and to assume the Friedmann equations as valid. All our results are consistent with the $\Lambda$CDM model, although alternative fluid models, with nearly constant pressure and no cosmological constant, match the results accurately as well., Comment: 23 pages. 1 reference added. Minor corrections

One of the main goals of modern cosmology remains to summon up a self consistent policy, able to explain, in the framework of the Einstein's theory, the cosmic speed up and the presence of Dark Matter in the Universe. Accordingly to the Holographic principle, which postulates the existence of a minimal size of a physical region, we argue, in this paper, that if this size exists for the Universe and it is accrued from the independent geometrical second order invariants, it would be possible to ensure a surprising source for Dark Matter and a viable candidate for explaining the late acceleration of the Universe. Along the work, we develop low redshift tests, such as Supernovae Ia and kinematical analysis complied by the use of Cosmography and we compare the outcomes with higher redshift tests, such as CMB peak and anisotropy of the cosmic power spectrum. All the results indicate that the models presented here can be interpreted as unified models that are capable to describe both the dark matter and the dark energy., 12 figures, revtex style

A remarkable property of naked singularities in general relativity is their repulsive nature. The effects generated by repulsive gravity are usually investigated by analyzing the trajectories of test particles which move in the effective potential of a naked singularity. This method is, however, coordinate and observer dependent. We propose to use the properties of the Riemann tensor in order to establish in an invariant manner the regions where repulsive gravity plays a dominant role. In particular, we show that in the case of the Kerr-Newman singularity and its special subcases the method delivers plausible results., Comment: to appear in the proceedings of MG12, Marcel Grossman Meeting, Paris, France, 12-18 Jul 2009

An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with arbitrary mass quadrupole moment and is specified by three parameters, the mass $M$, the angular momentum per unit mass $a$ and the quadrupole parameter $q$. It reduces to the Kerr spacetime in the limiting case $q=0$ and to the Erez-Rosen spacetime when the specific angular momentum $a$ vanishes. The geometrical properties of such a solution are investigated. Causality violations, directional singularities and repulsive effects occur in the region close to the source. Geodesic motion and accelerated motion are studied on the equatorial plane which, due to the reflection symmetry property of the solution, turns out to be also a geodesic plane., 24 pages, IOP macros used, eps figures n. 28; to appear on Classical and Quantum Gravity, 2009

We investigate a general class of equations of state reproducing the dark energy effects in terms of geometric considerations on thermodynamic interaction. We infer cosmological solutions by combining thermodynamics with contact manifold and Riemannian geometry, showing that the standard ΛCDM model can be treated as a limiting case of a more general approach, providing early time departures as the universe expands. Thus, we interpret the microscopic nature of dark energy through the mathematical formalism of geometrothermodynamics (GTD). In particular, we investigate the thermodynamic nature of a class of cosmological models which reproduce how the universe is currently speeding up. To do so, we aim to describe thermodynamic equilibrium states of the universe through a particular equilibrium space, where the Riemannian metric g becomes a thermodynamical ruler between different states. The particular assumption we made is to consider the metric structure to be invariant under precise Legendre transformations. It turns out that any thermodynamic interaction is determined once the scalar curvature of the equilibrium manifold is known. The consequence of our recipe leads to a class of dark energy equations of state which relates standard pressure to the volume occupied by the fluid itself. In our picture, dark energy is thus determined from constant thermodynamic interaction on the manifold of GTD.

The problem of the cosmic acceleration is here revisited by using the fact that the adiabatic speed of sound can be assumed to be negligible small. Within the context of general relativity, the total energy budget is recovered under the hypothesis of a vanishing speed of sound by assuming the existence of one fluid only. We find a cosmological model which reproduces the main results of the ΛCDM paradigm at late-times, showing an emergent cosmological constant, which is not at all related with the vacuum energy term. As a consequence, the model presented here behaves as a unified dark energy (DE) model.

In this paper, we demonstrate that the variation of acceleration, namely the jerk parameter j, could give hints for determining the dark energy equation of state (EoS). In particular, it is possible to show that a viable cosmological model is compatible with a constant jerk, here conventionally rewritten as j = 1+ϵ, with ϵ > 0 representing a departure from the ΛCDM model. This suggests that the cosmological constant could be seen as a limiting case of a more general dark energy model. We use the most recent union 2.1 compilation of supernovae Ia, showing at 1σ confidence level, that j is compatible with the condition j ≥ 1. In doing so, we infer a corresponding cosmological model, viable with a negative acceleration parameter, in the observed range -1 < q0 < 0.

Motivated by recent works on the role of the Holographic principle in cosmology, we relate a class of second order Ricci invariants to the IR cutoff characterizing the holographic Dark Energy density. The choice of second order invariants provides an invariant way to account the problem of causality for the correct cosmological cutoff, since the presence of event horizons is not an \emph{a priori} assumption. We find that these models work fairly well, by fitting the observational data, through a combined cosmological test with the use of SNeIa, BAO and CMB. This class of models is also able to overcome the fine-tuning and coincidence problems. Finally, to make a comparison with other recent models, we adopt the statistical tests AIC and BIC., Comment: 7 pages, 4 figures

We consider the neutrino fluxes and photon lensing from Gamma Ray Bursts and galaxy shocks, in order to predict measurable quantum corrections at late times. In particular, changes of the Einstein shell condition are expected to modify the time delay of neutrinos and photons at the Planck scale. Hence, we study the consequences of the Lorentz invariance violation through the use of such photons and neutrinos. The Lorentz invariance violation is expected to depend on the cosmological model, considered a priori in the Einstein equations; therefore, we focus on the so-called Hořava model which has recently attracted great interest. In addition, we compare our results with the ΛCDM and modified Chaplygin gas models, respectively. We infer several theoretical constraints which could reveal such quantum gravity effects.

In the present study, we consider an extended form of teleparallel Lagrangian $f(T,\phi,X)$, as function of a scalar field $\phi$, its kinetic term $X$ and the torsion scalar $T$. We use linear perturbations to obtain the equation of matter density perturbations on sub-Hubble scales. The gravitational coupling is modified in scalar modes with respect to the one of General Relativity, albeit vector modes decay and do not show any significant effects. We thus extend these results by involving multiple scalar field models. Further, we study conformal transformations in teleparallel gravity and we obtain the coupling as the scalar field is non-minimally coupled to both torsion and boundary terms. Finally, we propose the specific model $f(T,\phi,X)=T + \partial_\mu \phi\ \partial^\mu \phi +\xi T \phi^2$. To check its goodness, we employ the observational Hubble data, constraining the coupling constant, $\xi$, through a Monte Carlo technique based on the Metropolis-Hastings algorithm. Hence, fixing $\xi$ to its best-fit value got from our numerical analysis, we calculate the growth rate of matter perturbations and we compare our outcomes with the latest measurements and the predictions of the $\Lambda$CDM model., Comment: 17 pages, 4 figures, accepted for publication in Physical Review D

In the context of $f(\mathcal{R})$ gravity, dark energy is a geometrical fluid with negative equation of state. Since the function $f(\mathcal{R})$ is not known \emph{a priori}, the need of a model independent reconstruction of its shape represents a relevant technique to determine which $f(\mathcal{R})$ model is really favored with respect to others. To this aim, we relate cosmography to a generic $f(\mathcal R)$ and its derivatives in order to provide a model independent investigation at redshift $z \sim 0$. Our analysis is based on the use of three different cosmological distance definitions, in order to alleviate the duality problem, i.e. the problem of which cosmological distance to use with specific cosmic data sets. We therefore consider the luminosity, $d_L$, flux, $d_F$, and angular, $d_A$, distances and we find numerical constraints by the Union 2.1 supernovae compilation and measurement of baryonic acoustic oscillations, at $z_{BAO}=0.35$. We notice that all distances reduce to the same expression, i.e. $d_{L;F;A}\sim\frac{1}{\mathcal H_0}z$, at first order. Thus, to fix the cosmographic series of observables, we impose the initial value of $H_0$ by fitting $\mathcal H_0$ through supernovae only, in the redshift regime $z1$. Finally, we infer the functional form of $f(\mathcal{R})$ by means of a truncated polynomial approximation, in terms of fourth order scale factor $a(t)$., 10 pages, 4 figures, to appear in Annalen der Physik