Towards well-posed and versatile numerical solutions of scalar-tensor theories of gravity with screening mechanisms: applications at sub-Solar system scales

Kurzfassung

Scalar-tensor theories of gravity are among the most compelling, resilient and phenomenologically-rich alternatives to General Relativity. Viable models make use of screening mechanisms in order to be consistent with local tests of gravity whilst still retaining physical relevance. The hunt for such hypothetical scalar fields therefore hinges on the design of sophisticated model-dependent experiments. Alas, this task is greatly hampered by the difficulty of accurately modeling fifth force effects in realistic setups. Indeed, the latter requires solving semi-linear partial differential equations in the presence of complex matter distributions, for which analytical approaches are clearly insufficient. In this perspective, the present PhD work tackles this issue by developing a versatile numerical tool devoted to obtaining well-posed solutions to the nonlinear Klein-Gordon equations arising in such modified gravity models. The tool, called femtoscope, builds on the finite element method which allows one to deal with arbitrarily complex geometries and multi-scale problems through local mesh refinement. Nonlinearities, on the other hand, are handled via Newton's method. The novelty and most important feature of femtoscope is its careful treatment of asymptotic boundary conditions - i.e. when the field's behavior is only known infinitely far away from the sources - which is often essential to obtain physically meaningful numerical solutions. This is achieved by employing the inverted finite element method. We then make use of femtoscope to investigate screened scalar-tensor gravity at sub-Solar system scales. Using a realistic model of the Earth, we address the question of the detectability of a putative chameleon fifth force in orbit by means of GRACE-FO-like space geodesy missions. The influence of the atmosphere as well as the backreaction of spacecraft on the scalar field are both considered. We find that, although the fifth force has a supposedly measurable effect on the dynamics of a point-like spacecraft, the imperfect knowledge of the mass distribution inside the Earth gives rise to degeneracies, which in turn severely limit the constraining power of such space missions. These degeneracies can in principle be lifted by performing the experiment at two different altitudes. Finally, we open up new perspectives by exploring the possibility of testing screened scalar-tensor theories with atomic clocks, exploiting the distinctive imprint of the scalar field on the gravitational redshift with respect to General Relativity. It is emphasized that such experiments are profoundly different in nature from fifth force searches.