Higher-order Gravity Theories Research Articles

On the stability of the cosmological solutions in f(R, G) gravity

Modified gravity is one of the most promising candidates for explaining the current accelerating expansion of the Universe, and even its unification with the inflationary epoch. Nevertheless, the wide range of models capable of explaining the phenomena of dark energy imposes that current research focuses on a more precise study of the possible effects of modified gravity on both cosmological and local levels. In this paper, we focus on the analysis of a type of modified gravity, the so-called f(R, G) gravity, and we perform a deep analysis on the stability of important cosmological solutions. This not only can help to constrain the form of the gravitational action, but also facilitate a better understanding of the behavior of the perturbations in this class of higher order theories of gravity, which will lead to a more precise analysis of the full spectrum of cosmological perturbations in future.

Canonical formulation of the curvature-squared action in the presence of a lapse function

Lapse function appears as a Lagrange multiplier in the Einstein–Hilbert action and its variation leads to the equation of Einstein, which corresponds to the Hamiltonian constraint equation. In higher order theory of gravity the situation is not that simple. Here, we take up the curvature-squared (R2) action being supplemented by an appropriate boundary term in the background of Robertson–Walker minisuperspace metric, and show how to identify the constraint equation and formulate the Hamiltonian without detailed constraint analysis. The action is finally expressed in the canonical form , where the lapse function appears as the Lagrange multiplier, once again. Canonical quantization yields a Schrödinger-like equation, with nice features. To show that our result is not an artifact of having reduced the theory to a measure zero subset of its configuration space, the role of the lapse function as the Lagrangian multiplier has also been investigated in Bianchi-I, Kantowski–Sachs and Bianchi-III minisuperspace metrics. Classical and semiclassical solutions have finally been presented.

GRAVITATIONAL CHERENKOV RADIATION FROM EXTENDED THEORIES OF GRAVITY

We linearize the field equations for higher order theories of gravity that contain scalar invariants other than the Ricci scalar. We find that besides a massless spin-2 field (the standard graviton), the theory contains also spin-0 and spin-2 massive modes with the latter being, in general, ghost modes. The rate at which such particles would emit gravitational Cherenkov radiation is calculated for some interesting physical cases.

FURTHER GRAVITATIONAL WAVE MODES FROM HIGHER ORDER GRAVITY

It is shown that linearizing higher order theories of gravity, further gravitational massive modes emerge. Besides massless spin-2, also spin-0 and spin-2 massive and ghost fields have to be considered. We investigate the possible detectability of such additional modes on the stochastic cosmic background of gravitational waves by ground-based and space interferometric detectors and calculate the detectable energy density of the spectrum. In conclusion, these massive modes could be of interest for direct detection by the forthcoming LISA experiment.

Energy conditions in modified Gauss-Bonnet gravity

In considering alternative higher-order gravity theories, one is liable to be motivated in pursuing models consistent and inspired by several candidates of a fundamental theory of quantum gravity. Indeed, motivations from string/M-theory predict that scalar field couplings with the Gauss-Bonnet invariant, G, are important in the appearance of non-singular early time cosmologies. In this work, we discuss the viability of an interesting alternative gravitational theory, namely, modified Gauss-Bonnet gravity or f(G) gravity. We consider specific realistic forms of f(G) analyzed in the literature that account for the late-time cosmic acceleration and that have been found to cure the finite-time future singularities present in the dark energy models. We present the general inequalities imposed by the energy conditions and use the recent estimated values of the Hubble, deceleration, jerk and snap parameters to examine the viability of the above-mentioned forms of f(G) imposed by the weak energy condition.

Static solutions for fourth order gravity

The Lichnerowicz and Israel theorems are extended to higher order theories of gravity. In particular it is shown that Schwarzschild is the unique spherically symmetric, static, asymptotically flat, black-hole solution, provided the spatial curvature is less than the quantum gravity scale outside the horizon. It is then shown that in the presence of matter (satisfying certain positivity requirements), the only static and asymptotically flat solutions of General Relativity that are also solutions of higher order gravity are the vacuum solutions

GAUGE THEORY OF GRAVITY WITH DE SITTER SYMMETRY AS A SOLUTION TO THE COSMOLOGICAL CONSTANT PROBLEM AND THE DARK ENERGY PUZZLE

We propose a solution to the longstanding cosmological constant (CC) problem which is based on the fusion of two existing concepts. The first is the suggestion that the proper description of classical gravitational effects is the gauge theory of gravity in which the connection instead of the metric acts as the dynamical variable. The resulting field equation does not then contain the CC term. This removes the connection between the CC and the quantum vacuum energy, and therefore addresses the old CC problem of why quantum vacuum energy does not gravitate. The CC-equivalent in this approach arises from the constant of integration when reducing the field equation to the Einstein equation. The second is the assumption that the universe obeys de Sitter symmetry, with the observed accelerating expansion as its manifestation. We combine these ideas and identify the constant of integration with the inverse-square of the radius of curvature of the de Sitter space. The origin of dark energy (DE) is therefore associated with the inherent spacetime geometry, with the smallness of DE protected by symmetry. This addresses the new CC problem, or the DE puzzle. This approach, however, faces major challenges from quantum considerations. These are the ghost problem associated with higher order gravity theories and the quantum instability of the de Sitter spacetime. We discuss their possible remedies.

On the existence of anisotropic cosmological models in higher order theories of gravity

We investigate the behaviour on approach to the initial singularity in higher order extensions of general relativity by finding exact cosmological solutions for a wide class of models in which the Lagrangian is allowed to depend nonlinearly upon the three possible linear and quadratic scalars built from the Riemann tensor: R, RabRab and RabcdRabcd. We present new anisotropic vacuum solutions analogous to the Kasner solutions of general relativity and extend previous results to a much wider range of fourth-order theories of gravity. We discuss the implications of these results for the behaviour of the more general anisotropic Bianchi type VIII and IX cosmologies as the initial singularity is approached. Furthermore, we also consider the existence conditions for some other simple anisotropic Bianchi I vacuum solutions in which the expansion in each direction is of exponential, rather than power-law behaviour and their relevance for cosmic ‘no-hair’ theorems.

Gravitational instantons in R4-gravity for open inflation

In a Euclidean gravity we study the creation of an inflationary universe in a higher order theory of gravity making use of the Hawking–Turok instantons. Here we consider a nonlinear γR4 term to the Einstein gravitational action including a cosmological constant to obtain gravitational instantons relevant for describing the early universe. It is found that the theory permits gravitational instantons even in the absence of a cosmological constant. To explore instantons other than the de Sitter instanton in the framework of higher order theory of gravity, we use the conformal transformation. The higher order theory is converted into a theory of self-interacting scalar field minimally coupled to Einstein gravity. The singular instantons obtained in the conformally equivalent case is taken up to obtain an open inflationary scenario. The probability of creation of singular and non-singular instantons is evaluated here. We note that a non-singular instanton solution is permitted with a positive cosmological constant when γ < 0 and that with a negative cosmological constant when γ > 0. It is also noted that such instanton solutions are permitted even in the absence of a cosmological constant. The singular instanton solution relevant for the inflationary universe is obtained for . The potential obtained in the conformally equivalent case permits enough inflation necessary for a consistent cosmological model.

Higher-order theories of gravity matched with large-scale structure and cosmological observations

The so-called f(R)-gravity could, in principle, explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter, but more simply extending the general relativity by generic functions of the Ricci scalar. However, as a part of several phenomenological models, there is no final f(R)-theory capable of fitting all the observations and addressing all the issues related to the presence of dark energy and dark matter. Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large-scale structures and cosmic accelerating expansion. Up to now, no experimental evidence has been found, at a fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of general relativity.

Covariant effective action for loop quantum cosmology from order reduction

Loop quantum cosmology (LQC) seems to be predicting modified effective Friedmann equations without extra degrees of freedom. A puzzle arises if one decides to seek for a covariant effective action which would lead to the given Friedmann equation: The Einstein-Hilbert action is the only action that leads to second order field equations and, hence, there exists no covariant action which, under metric variation, leads to a modified Friedmann equation without extra degrees of freedom. It is shown that, at least for isotropic models in LQC, this issue is naturally resolved and a covariant effective action can be found if one considers higher order theories of gravity but faithfully follows effective field theory techniques. However, our analysis also raises doubts on whether a covariant description without background structures can be found for anisotropic models.

Some remarks on the dynamical systems approach to fourth order gravity

Building on earlier work, we discuss a general framework for exploring the cosmological dynamics of Higher Order Theories of Gravity. We show that once the theory of gravity has been specified, the cosmological equations can be written as a first-order autonomous system and we give several examples which illustrate the utility of our method. We also discuss a number of results which have appeared recently in the literature.

Gravitational collapse of homogeneous perfect fluids in higher order gravity theories

This paper investigates the evolution of collapsing Friedmann–Robertson–Walker models with a scalar field having the potential which arises in the conformal frame of high order gravity theories, coupled to matter described by a perfect fluid with energy density ρ and pressure p, obeying a barotropic equation of state. The solutions are shown to evolve generically to a singular state in a finite time and they are used as sources for radiating objects undergoing complete gravitational collapse. Although these singularities may be naked in some special cases, it is shown that generically a black hole forms.

Slice energy in higher-order gravity theories and conformal transformations

We study the generic transport of slice energy between the scalar field generated by the conformal transformation of higher-order gravity theories and the matter component. We give precise relations for this exchange in the cases of dust and perfect fluids. We show that, unless we are in a stationary spacetime where slice energy is always conserved, in non-stationary situations contributions to the total slice energy depend on whether or not test matter follows geodesics in both frame representations of the dynamics, that is on whether or not the two conformally related frames are physically indistinguishable.

Recent Developments in the Dynamical Systems Approach to Alternative Gravity

We discuss the general framework for exploring the cosmological dynamics of Higher Order Theories of Gravity presented in Carloni et al. (2007).

Exact Friedmann solutions in higher-order gravity theories

We find the general behaviour of homogeneous and isotropic cosmological models in some fourth-order theories of gravity. Explicit, exact, general solutions are given for both empty universes and those filled with a perfect fluid. For the vacuum case, solutions are found with closed, open and flat geometries, whilst the perfect fluid solutions are all spatially flat. Both early- and late-time limits are studied, and attractor behaviour towards simple power-law expansion is identified. Multiple solutions to the same theories, with the same matter content and topology are found. It is shown that these solutions exhibit a great variety in their evolution.

A dynamical system approach to higher order gravity

The dynamical system approach has recently acquired great importance in the investigation on higher order theories of gravity. In this talk I review some of the main results obtained with this method and I consider briefly a number of further developments.

THE VARIATIONAL APPROACH TO ALTERNATIVE THEORIES OF GRAVITY

This paper is based on lectures given in Ladek-Zdroj, Poland during the 42nd edition of the Karpac winter school on Theoretical Physics. The aim of these lectures was to give an introduction to a rigorous, mathematically well based, variational approach to alternative theories of Gravity, in view of their very interesting recent applications in the framework of Cosmology. We discuss alternative theories of Gravity both in the metric, purely affine and Palatini formalism, stressing differences and analogies both from a physical and mathematical viewpoint. Moreover we give and introduction to almost all the alternative theories of Gravity, that have been recently studied in view of their physical interest (higher order theories of Gravity, including Lovelock and Chern-Simons Lagrangians). Finally we skip to one of the most striking recent applications of alternative theories of Gravity in the framework of cosmological models: alternative theories of Gravity have been proven to provide a consistent model for accelerating universes.

Further exact cosmological solutions to higher-order gravity theories

We investigate the effect of deviations from general relativity on approach to the initial singularity by finding exact cosmological solutions to a wide class of fourth-order gravity theories. We present new anisotropic vacuum solutions of modified Kasner type and demonstrate the extent to which they are valid in the presence of non-comoving perfect-fluid matter fields. The infinite series of Mixmaster oscillations seen in general relativity will not occur in these solutions, except in unphysical cases.

Phenomenological model for inflationary quintessence

We present a phenomenologically motivated model which is able to give rise both to an inflationary epoch and to the present day cosmic acceleration. We introduce an approach where the energy density depends on the scale factor $a$ in such a way that a smooth transition from the inflation to the matter dominated era and then to the nowadays accelerated expansion is achieved. We use the dimensionless coordinate distance $y(z)$ to the Gold Type Ia Supernovae dataset and to a sample comprising 20 radio galaxies, the shift parameter ${\cal{R}}$ and the acoustic peak parameter ${\cal{A}}$ to test whether the model is in agreement with observations and to constrain its main parameters. As an independent cross check, we also compare model predictions with the lookback time to galaxy clusters and the age of the universe. Although phenomenologically inspired, the model may be theoretically motivated either resorting to a scenario with a scalar and phantom fields (eventually interacting) or as an effective description arising from higher order gravity theories.