Constraints on f(RijklRijkl) gravity: evidence against the covariant resolution of the Pioneer anomaly

We consider corrections in the form of ΔL(RijklRijkl) to the Einstein-Hilbert Lagrangian. Then we compute the corrections to the Schwarzschild geometry due to the inclusion of this general term to the Lagrangian. We show that ΔL3=α1/3(RijklRijkl)1/3 gives rise to a constant anomalous acceleration for objects orbiting the Sun, toward the Sun. This leads to the conclusion that α1/3=(13.91±2.11)x10-26(1/meters)2/3 would have covariantly resolved the Pioneer anomaly if this value of α1/3 had not contradicted other observations. We note that the experimental bounds on ΔL3 grow stronger in case we examine the deformation of the spacetime geometry around objects lighter than the sun. We therefore use the high precision measurements around the Earth (LAGEOS and LLR) and obtain a very strong constraint on the corrections in the form of ΔL(RijklRijkl) and in particular ΔL = αn(RijklRijkl)n. This bound requires α1/3≤6.12x10-29(1/meters)2/3. Therefore, it refutes the covariant resolution of the Pioneer anomaly.