We derive for applications to isolated systems-on the scale of the Solar System-the first relativistic terms in the 1/c expansion of the space time metric g for metric f(R) gravity theories, where f is assumed to be analytic at R=0. For our purpose it suffices to take into account up to quadratic terms in the expansion of f(R), thus we can approximate f(R)=R+aR2 with a positive dimensional parameter a. In the nonrelativistic limit, we get an additional Yukawa correction with coupling strength G/3 and Compton wave length 6a to the Newtonian potential, which is a known result in the literature. As an application, we derive to the same order the correction to the geodetic precession of a gyroscope in a gravitational field and the precession of binary pulsars. The result of the Gravity Probe B experiment yields the limit a5×1011m2, whereas for the pulsar B in the PSR J0737-3039 system we get a bound which is about 104 times larger. On the other hand the Eöt-Wash experiment provides the best laboratory bound a10-10m2. Although the former bounds from geodesic precession are much larger than the laboratory ones, they are still meaningful in the case some type of chameleon effect is present and thus the effective values could be different at different length scales.

For a general class of analytic f(R)-gravity theories, we discuss the weak field limit in view of gravitational lensing. Though an additional Yukawa term in the gravitational potential modifies dynamics with respect to the standard Newtonian limit of General Relativity, the motion of massless particles results unaffected thanks to suitable cancellations in the post-Newtonian limit. Thus, all the lensing observables are equal to the ones known from General Relativity. Since f(R)-gravity is claimed, among other things, to be a possible solution to overcome for the need of dark matter in virialized systems, we discuss the impact of our results on the dynamical and gravitational lensing analyses. In this framework, dynamics could, in principle, be able to reproduce the astrophysical observations without recurring to dark matter, but in the case of gravitational lensing we find that dark matter is an unavoidable ingredient. Another important implication is that gravitational lensing, in the post-Newtonian limit, is not able to constrain these extended theories, since their predictions do not differ from General Relativity.

We investigate the gravitational radiation emitted by an isolated system for gravity theories with Lagrange density f(R)=R+aR2. As a formal result, we obtain leading order corrections to the quadrupole formula in general relativity. We make use of the analogy of f(R) theories with scalar-tensor theories, which in contrast to general relativity feature an additional scalar degree of freedom. Unlike general relativity, where the leading order gravitational radiation is produced by quadrupole moments, the additional degree of freedom predicts gravitational radiation of all multipoles, in particular, monopoles and dipoles, as this is the case for the most alternative gravity theories known today. An application to a hypothetical binary pulsar moving in a circular orbit yields the rough limit a1.7×1017m2 by constraining the dipole power to account at most for 1% of the quadrupole power as predicted by general relativity. AbstractWe investigate the gravitational radiation emitted by an isolated system for gravity theories with Lagrange density f (R) = R + aR 2 . As a formal result we obtain leading order corrections to the quadrupole formula in General Relativity. We make use of the analogy of f (R) theories with scalar-tensor theories, which in contrast to General Relativity feature an additional scalar degree of freedom. Unlike General Relativity, where the leading order gravitational radiation is produced by quadrupole moments, the additional degree of freedom predicts gravitational radiation of all multipoles, in particular monopoles and dipoles, as this is the case for the most alternative gravity theories known today. An application to a hypothetical binary pulsar moving in a circular orbit yields the rough limit a 1.7 · 10 17 m 2 by constraining the dipole power to account at most for 1% of the quadrupole power as predicted by General Relativity.

We study the effect of a cosmological constant Λ on the propagation and detection of gravitational waves. To this purpose we investigate the linearised Einstein's equations with terms up to linear order in Λ in a de Sitter and an anti-de Sitter background spacetime. In this framework the cosmological term does not induce changes in the polarization states of the waves, whereas the amplitude gets modified with terms depending on Λ. Moreover, if a source emits a periodic waveform, its periodicity as measured by a distant observer gets modified. These effects are, however, extremely tiny and thus well below the detectability by some twenty orders of magnitude within present gravitational wave detectors such as LIGO or future planned ones such as LISA.

Law invariant convex risk measures, Robust representation, Variational methods, 91G99, D81, G22,