Fourth-order Gravity Research Articles

Weak gravitational lensing by compact objects in fourth order gravity

We discuss weak lensing characteristics in the gravitational field of a compact object in the low-energy approximation of fourth order f(R) gravity theory. The particular solution is characterized by a gravitational strength parameter $\sigma $ and a distance scale $r_{c}$ much larger than the Schwarzschild radius. Above $r_{c}$ gravity is strengthened and as a consequence weak lensing features are modified compared to the Schwarzschild case. We find a critical impact parameter (depending upon $r_{c}$) for which the behavior of the deflection angle changes. Using the Virbhadra-Ellis lens equation we improve the computation of the image positions, Einstein ring radii, magnification factors and the magnification ratio. We demonstrate that the magnification ratio as function of image separation obeys a power-law depending on the parameter $\sigma $, with a double degeneracy. No $\sigma \neq 0$ value gives the same power as the one characterizing Schwarzschild black holes. As the magnification ratio and the image separation are the lensing quantities most conveniently determined by direct measurements, future lensing surveys will be able to constrain the parameter $\sigma $ based on this prediction.

Stability of Schwarzschild black holes in fourth-order gravity revisited

We revisit the classical stability of Schwarzschild black hole in fourth-order theories of gravity. On the contrary of the stability of this black hole, it turns out that the linearized perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of five-dimensional black string. This shows clearly an instability of black hole in fourth-order gravity with $\alpha=-\beta/3$.

Isotropic universe with almost scale-invariant fourth-order gravity

We study a class of isotropic cosmologies in the fourth-order gravity with Lagrangians of the form L = f(R) + k(G) where R and G are the Ricci and Gauss-Bonnet scalars, respectively. A general discussion is given on the conditions under which this gravitational Lagrangian is scale-invariant or almost scale-invariant. We then apply this general background to the specific case L = αR2 + β Gln G with constants α, β. We find closed form cosmological solutions for this case. One interesting feature of this choice of f(R) and k(G) is that for very small negative value of the parameter β, the Lagrangian L = R2/3 + βGln G leads to the replacement of the exact de Sitter solution coming from L = R2 (which is a local attractor) to an exact, power-law inflation solution a(t) = tp = t−3/β which is also a local attractor. This shows how one can modify the dynamics from de Sitter to power-law inflation by the addition of a Gln G-term.

Atmospheric confinement of jet streams on Uranus and Neptune

The observed cloud-level atmospheric circulation on the outer planets of the Solar System is dominated by strong east-west jet streams. The depth of these winds is a crucial unknown in constraining their overall dynamics, energetics and internal structures. There are two approaches to explaining the existence of these strong winds. The first suggests that the jets are driven by shallow atmospheric processes near the surface, whereas the second suggests that the atmospheric dynamics extend deeply into the planetary interiors. Here we report that on Uranus and Neptune the depth of the atmospheric dynamics can be revealed by the planets' respective gravity fields. We show that the measured fourth-order gravity harmonic, J4, constrains the dynamics to the outermost 0.15 per cent of the total mass of Uranus and the outermost 0.2 per cent of the total mass of Neptune. This provides a stronger limit to the depth of the dynamical atmosphere than previously suggested, and shows that the dynamics are confined to a thin weather layer no more than about 1,000 kilometres deep on both planets.

Black Holes, Cosmological Solutions, Future Singularities, and Their Thermodynamical Properties in Modified Gravity Theories

Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann–Lemaˆıtre–Robertson–Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f(R) models. In addition, we revise some attempts to extend the Cardy–Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a deep study on cosmological singularities, which appear as a real possibility for some kind of modified gravity theories, is performed, and the validity of the entropy bounds is studied.

Cosmological dynamics of fourth-order gravity: A compact view

We construct a compact phase space for flat FLRW spacetimes with standard matter described by a perfect fluid with a barotropic equation of state for general f(R) theories of gravity, subject to certain conditions on the function f. We then use this framework to study the behaviour of the phase space of Universes with a non-negative Ricci scalar in R + {\alpha}R^n gravity. We find a number of interesting cosmological evolutions which include the possibility of an initial unstable power-law inflationary point, followed by a curvature fluid dominated phase mimicking standard radiation, then passing through a standard matter (CDM) era and ultimately evolving asymptotically towards a de-Sitter-like late-time accelerated phase.

Weak gravitational lensing in fourth order gravity

For a general class of analytic functions $f(R,{R}_{\ensuremath{\alpha}\ensuremath{\beta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}},{R}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}})$ we discuss the gravitational lensing in the Newtonian limit of theory. From the properties of the Gauss-Bonnet invariant it is enough to consider only one curvature invariant between the Ricci tensor and the Riemann tensor. Then, we analyze the dynamics of a photon embedded in a gravitational field of a generic $f(R,{R}_{\ensuremath{\alpha}\ensuremath{\beta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}})$ gravity. The metric is time independent and spherically symmetric. The metric potentials are Schwarzschild-like, but there are two additional Yukawa terms linked to derivatives of $f$ with respect to two curvature invariants. Considering first the case of a pointlike lens, and after the one of a generic matter distribution of the lens, we study the deflection angle and the angular position of images. Though the additional Yukawa terms in the gravitational potential modifies dynamics with respect to general relativity, the geodesic trajectory of the photon is unaffected by the modification if we consider only $f(R)$ gravity. We find different results (deflection angle smaller than the angle of general relativity) only due to the introduction of a generic function of the Ricci tensor square. Finally, we can affirm that the lensing phenomena for all $f(R)$ gravities are equal to the ones known for general relativity. We conclude the paper by showing and comparing the deflection angle and position of images for $f(R,{R}_{\ensuremath{\alpha}\ensuremath{\beta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}})$ gravity with respect to the gravitational lensing of general relativity.

An updated analysis of two classes of f(R) theories of gravity

The observed accelerated cosmic expansion can be a signature of fourth-order gravity theories, where the acceleration of the Universe is a consequence of departures from Einstein General Relativity, rather than the sign of the existence of a fluid with negative pressure. In the fourth-order gravity theories, the gravity Lagrangian is described by an analytic function f(R) of the scalar curvature R subject to the demanding conditions that no detectable deviations from standard GR is observed on the Solar System scale. Here we consider two classes of f(R) theories able to pass Solar System tests and investigate their viability on cosmological scales. To this end, we fit the theories to a large dataset including the combined Hubble diagram of Type Ia Supernovae and Gamma Ray Bursts, the Hubble parameter H(z) data from passively evolving red galaxies, Baryon Acoustic Oscillations extracted from the seventh data release of the Sloan Digital Sky Survey (SDSS) and the distance priors from the Wilkinson Microwave Anisotropy Probe seven years (WMAP7) data. We find that both classes of f(R) fit very well this large dataset with the present-day values of the matter density, Hubble constant and deceleration parameter in agreement with previous estimates; however, the strong degeneracy among the f(R) parameters prevents us from strongly constraining their values. We also derive the growth factor g = dln δ/dln a, with δ = δρM/ρM the matter density perturbation, and show that it can still be well approximated by g(z)∝ΩM(z)γ. We finally constrain γ (on some representative scales) and investigate its redshift dependence to see whether future data can discriminate between these classes of f(R) theories and standard dark energy models.

Cosmological dynamics of fourth-order gravity with a Gauss-Bonnet term

We consider the cosmological dynamics of fourth-order gravity with both f(R) and Φ(G) corrections to the Einstein gravity (G is the Gauss-Bonnet invariant). The particular case for which both terms are equally important on power-law solutions is described. These solutions and their stability are studied using the dynamic system approach. We also discuss the condition of existence and stability of a de Sitter solution in a more general situation of power-law f and Φ.

Shear-free perturbations off(R)gravity

Recently it was shown that if the matter congruence of a general relativistic perfect fluid flow in an almost FLRW universe is shear-free, then it must be either expansion or rotation-free. Here we generalize this result for a general $f(R)$ theory of gravity and show there exist scenarios where this result can be avoided. This suggests that there are situations where linearized fourth-order gravity shares properties with Newtonian theory not valid in General Relativity.

Rotation curves of galaxies by fourth order gravity

We investigate the radial behavior of galactic rotation curves by a Fourth Order Gravity adding also the Dark Matter component. The Fourth Order Gravity is a Lagrangian containing the Ricci scalar, the Ricci and Riemann tensor, but the rotation curves are depending only on two free parameters. A systematic analysis of rotation curves, in the Newtonian Limit of theory, induced by all galactic sub-structures of ordinary matter is shown. This analysis is presented for Fourth Order Gravity with and without Dark Matter. The outcomes are compared with respect to classical outcomes of General Relativity. The gravitational potential of point-like mass is the usual potential corrected by two Yukawa terms. The rotation curve is higher or also lower than curve of General Relativity if in the Lagrangian the Ricci scalar square is dominant or not with respect to the contribution of the Ricci tensor square. The curves are compared with the experimental data for the Milky Way and the galaxy NGC 3198. Although the Fourth Order Gravity gives more rotational contributions, in the limit of large distances the Keplerian behavior is present, and it is missing if we add a Dark Matter component. By modifying the theory of Gravitation consequently also the spatial description of Dark Matter could undergo a modification. At last we compare the gravitational potential by Fourth Order Gravity with respect to more used potential induced by power law of Ricci scalar.

LUKEWARM BLACK HOLES IN QUADRATIC GRAVITY

Perturbative solutions to the fourth-order gravity describing spherically-symmetric, static and electrically charged black hole in an asymptotically de Sitter universe is constructed and discussed. Special emphasis is put on the lukewarm configurations, in which the temperature of the event horizon equals the temperature of the cosmological horizon.

Restricting fourth-order gravity via cosmology

The cosmology of general fourth order corrections to Einstein gravity is considered, both for a homogeneous and isotropic background and for general tensor perturbations. It is explicitly shown how the standard cosmological history can be (approximately) reproduced and under what condition the evolution of the tensor modes remain (approximately) unchanged. Requiring that the deviations from General Relativity are small during inflation sharpens the current constraints on such corrections terms by some thirty orders of magnitude. Taking a more conservative approach and requiring only that cosmology be approximately that of GR during Big Bang Nucleosynthesis, the constraints are improved by 4 - 6 orders of magnitude.

Most general fourth-order theory of gravity at low energy

The Newtonian limit of the most general fourth order gravity is performed with metric approach in the Jordan frame with no gauge condition. The most general theory with fourth order differential equations is obtained by generalizing the $f(R)$ term in the action with a generic function containing other two curvature invariants: \emph{Ricci square} ($R_{\alpha\beta}R^{\alpha\beta}$) and \emph{Riemann square} ($R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$). The spherically symmetric solutions of metric tensor yet present Yukawa-like spatial behavior, but now one has two characteristic lengths. At Newtonian order any function of curvature invariants gives us the same outcome like the so-called \emph{Quadratic Lagrangian} of Gravity. From Gauss - Bonnet invariant one have the complete interpretation of solutions and the absence of a possible third characteristic length linked to Riemann square contribution. From analysis of metric potentials, generated by point-like source, one has a constraint condition on the derivatives of $f$ with respect to scalar invariants.

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

Fourth-order braneworld gravity

We discuss a general fourth-order theory of gravity and formulate the generalized Israel junction conditions for the theory by application of a generalized Gibbons-Hawking boundary term which is added to the action of the fourth-order gravity theory. Next, we use the generalized Israel junction condition for the construction of a higher-order gravity braneworld model.

Fourth‐order braneworld gravity

We discuss a general fourth-order theory of gravity and formulate the generalized Israel junction conditions for the theory by application of a generalized Gibbons-Hawking boundary term which is added to the action of the fourth-order gravity theory. Next, we use the generalized Israel junction condition for the construction of a higher-order gravity braneworld model.

Comparing scalar–tensor gravity and [formula omitted]-gravity in the Newtonian limit

Recently, a strong debate has been pursued about the Newtonian limit (i.e. small velocity and weak field) of fourth order gravity models. According to some authors, the Newtonian limit of f(R)-gravity is equivalent to the one of Brans–Dicke gravity with ωBD=0, so that the PPN parameters of these models turn out to be ill-defined. In this Letter, we carefully discuss this point considering that fourth order gravity models are dynamically equivalent to the O'Hanlon Lagrangian. This is a special case of scalar–tensor gravity characterized only by self-interaction potential and that, in the Newtonian limit, this implies a non-standard behavior that cannot be compared with the usual PPN limit of General Relativity. The result turns out to be completely different from the one of Brans–Dicke theory and in particular suggests that it is misleading to consider the PPN parameters of this theory with ωBD=0 in order to characterize the homologous quantities of f(R)-gravity. Finally the solutions at Newtonian level, obtained in the Jordan frame for an f(R)-gravity, reinterpreted as a scalar–tensor theory, are linked to those in the Einstein frame.

Structure growth in f(R) theories of gravity with a dust equation of state

We investigate the connection between dark energy and fourth-order gravity by analyzing the behavior of scalar perturbations around a Friedmann–Robertson–Walker background. The evolution equations for scalar perturbations are derived using the covariant and gauge invariant approach and applied to two simple, yet widely studied f(R) gravity models. This analysis differs from existing analysis by the fact that the full fourth-order equations are studied. The equations are integrated without the implementation of the usual approximations which inevitably lead to a set of second-order equations. The structure of the general fourth-order perturbation equations and the analysis of scalar perturbations lead to the identification of interesting features in the matter power spectrum within fourth-order gravity. The fact that such features exist even in the case of such simple background models indicates the need for further detailed investigation within the context of more realistic scenarios.

The Newtonian limit of metric gravity theories with quadratic Lagrangians

The Newtonian limit of fourth-order gravity is worked out discussing its viability with respect to the standard results of general relativity. We investigate the limit in the metric approach which, with respect to the Palatini formulation, has been much less studied in the recent literature, due to the higher order of the field equations. In addition, we refrain from exploiting the formal equivalence of higher-order theories considering the analogy with specific scalar–tensor theories, i.e. we work in the so-called Jordan frame in order to avoid possible misleading interpretations of the results. Explicit solutions are provided for several different types of Lagrangians containing powers of the Ricci scalar as well as combinations of the other curvature invariants. In particular, we develop the Green's function method for fourth-order theories in order to find out solutions. Finally, the consistency of the results with respect to general relativity is discussed.