Fourth order gravity and experimental constraints on Eddington parameters

PPN-limit of higher order theories of gravity represents a still controversial matter of debate and no definitive answer has been provided, up to now, about this issue. By exploiting the analogy between scalar-tensor and fourth-order theories of gravity, one can generalize the PPN-limit formulation. By using the definition of the PPN-parameters $\gamma$ and $\beta$ in term of the $f(R)$ derivatives, we show that a family of third-order polynomial theories, in the Ricci scalar $R$, turns out to be compatible with the PPN-limit and the deviation from General Relativity theoretically predicted agree with experimental data.Comment: 7 pages, 3 figure

A general solution in the Newtonian limit of f(R)- gravity

We show that any analytic $f(R)$-gravity model, in the metric approach, presents a weak field limit where the standard Newtonian potential is corrected by a Yukawa-like term. This general result has never been pointed out but often derived for some particular theories. This means that only $f(R)=R$ allows to recover the standard Newton potential while this is not the case for other relativistic theories of gravity. Some considerations on the physical consequences of such a general solution are addressed.Comment: 5 page

The Newtonian Limit of F(R) gravity

A general analytic procedure is developed to deal with the Newtonian limit of $f(R)$ gravity. A discussion comparing the Newtonian and the post-Newtonian limit of these models is proposed in order to point out the differences between the two approaches. We calculate the post-Newtonian parameters of such theories without any redefinition of the degrees of freedom, in particular, without adopting some scalar fields and without any change from Jordan to Einstein frame. Considering the Taylor expansion of a generic $f(R)$ theory, it is possible to obtain general solutions in term of the metric coefficients up to the third order of approximation. In particular, the solution relative to the $g_{tt}$ component gives a gravitational potential always corrected with respect to the Newtonian one of the linear theory $f(R)=R$. Furthermore, we show that the Birkhoff theorem is not a general result for $f(R)$-gravity since time-dependent evolution for spherically symmetric solutions can be achieved depending on the order of perturbations. Finally, we discuss the post-Minkowskian limit and the emergence of massive gravitational wave solutions.Comment: 16 page

Comparing scalar-tensor gravity and f(R)-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 $\omega_{BD} = 0$, so that the PPN parameters of these models turn out to be ill defined. In this paper, 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 $\omega_{BD} = 0$ in order to characterize the homologous quantities of $f(R)$-gravity. Finally the solutions at Newtonian level, obtained in the Jordan frame for a $f(R)$-gravity, reinterpreted as a scalar-tensor theory, are linked to those in the Einstein frame.Comment: 9 page

The post-Minkowskian limit of f(R)-gravity

We formally discuss the post-Minkowskian limit of $f(R)$-gravity without adopting conformal transformations but developing all the calculations in the original Jordan frame. It is shown that such an approach gives rise, in general, together with the standard massless graviton, to massive scalar modes whose masses are directly related to the analytic parameters of the theory. In this sense, the presence of massless gravitons only is a peculiar feature of General Relativity. This fact is never stressed enough and could have dramatic consequences in detection of gravitational waves. Finally the role of curvature stress-energy tensor of $f(R)$-gravity is discussed showing that it generalizes the so called Landau-Lifshitz tensor of General Relativity. The further degrees of freedom, giving rise to the massive modes, are directly related to the structure of such a tensor.Comment: 9 page

Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach

We search for spherically symmetric solutions of f(R) theories of gravity via the Noether Symmetry Approach. A general formalism in the metric framework is developed considering a point-like f(R)-Lagrangian where spherical symmetry is required. Examples of exact solutions are given.Comment: 17 pages, to appear in Class. Quant. Gra

Spherical symmetry in f(R)f(R)-gravity

Spherical symmetry in $f(R)$ gravity is discussed in details considering also the relations with the weak field limit. Exact solutions are obtained for constant Ricci curvature scalar and for Ricci scalar depending on the radial coordinate. In particular, we discuss how to obtain results which can be consistently compared with General Relativity giving the well known post-Newtonian and post-Minkowskian limits. Furthermore, we implement a perturbation approach to obtain solutions up to the first order starting from spherically symmetric backgrounds. Exact solutions are given for several classes of $f(R)$ theories in both $R =$ constant and $R = R(r)$.Comment: 13 page

Jumping from Metric f(R) to Scalar-Tensor Theories and the relations between their post-Newtonian Parameters

We review the dynamical equivalence between $f(R)$ gravity in the metric formalism and scalar-tensor gravity, and use this equivalence to deduce the post-Newtonian parameters $\gamma$ and $\beta$ for a $f(R)$ theory, obtaining a result that is different with respect to that known in the literature. Then, we obtain explicit expressions of these paremeters in terms of the mass of the scalar field (or, differently speaking, the mass of the additional scalar degree of freedom associated to a $f(R)$ theory) which can be used to constrain $f(R)$ gravity by means of current observations.Comment: 10 pages, 1 table, no figures Accepted for publication in CQ

The Cauchy problem of f(R) gravity

The initial value problem of metric and Palatini f(R)gravity is studied by using the dynamical equivalence between these theories and Brans-Dicke gravity. The Cauchy problem is well-formulated for metric f(R)gravity in the presence of matter and well-posed in vacuo. For Palatini f(R)gravity, instead, the Cauchy problem is not well-formulated.Comment: 16 latex pages, to appear in Class. Quantum Grav; typographical errors corrected, new references adde

Modified-Source Gravity and Cosmological Structure Formation

One way to account for the acceleration of the universe is to modify general relativity, rather than introducing dark energy. Typically, such modifications introduce new degrees of freedom. It is interesting to consider models with no new degrees of freedom, but with a modified dependence on the conventional energy-momentum tensor; the Palatini formulation of $f(R)$ theories is one example. Such theories offer an interesting testing ground for investigations of cosmological modified gravity. In this paper we study the evolution of structure in these ``modified-source gravity'' theories. In the linear regime, density perturbations exhibit scale dependent runaway growth at late times and, in particular, a mode of a given wavenumber goes nonlinear at a higher redshift than in the standard $\Lambda$CDM model. We discuss the implications of this behavior and why there are reasons to expect that the growth will be cut off in the nonlinear regime. Assuming that this holds in a full nonlinear analysis, we briefly describe how upcoming measurements may probe the differences between the modified theory and the standard $\Lambda$CDM model.Comment: 22 pages, 6 figures, uses iopart styl