Asymptotically Schwarzschild solutions in f(R) extension of General Relativity

section Milan We consider the gravitational field outside a static, spherically symmetric source in the context of a general f(R) extension of General Relativity. We study the modified Einstein equations (EE), which involve the two free potentials of the metric together with f(R) and its derivative ϕ = ∂f ∂R , without making any preliminary assumption on f as a function of the scalar curvature. Instead, we do require complete agreement with the usual Schwarzschild solution far from the source and minimal regularity of both the potentials and ϕ as functions of the coordinates. Under these conditions we are able to perturbatively solve the modified EE, explicitly compute the leading correction to the Schwarzschild line element and retrieve a posteriori the corresponding f(R). This is non analytical in R = 0 and depends on two parameters: a universal coupling c1 and an integer number n ≥ 2, which determines the order of the correction. In the second part of the work, we firstly compute the parametrized post Newtonian parameters for the modified Schwarzschild line element: while γ agrees by construction with the strict Cassini bound for every n, the constraint on β from the precession of Mercury places a severe upper bound on c1 for n = 2. We then compute the leading correction to the gravitational redshift and use observations of the sunlight gravitational redshift to set numerical upper bounds on c1 at varying n. The corrections to the bending of light from a distant star by the Sun, to the precession of Mercury and to the Shapiro delay are also computed. The result is a class of f(R) theories built from a purely bottom-up approach and compatible with the local tests. This result can also help constraining exact f(R) models working in Cosmology, since it provides the correct local limit.