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Direct detection of dark energy or modified gravity may finally be withinreach due to ultrasensitive instrumentation such as atom interferometry capableof detecting incredibly small scale accelerations. Forecasts, constraints andmeasurement bounds can now too perhaps be estimated from accurate numericalsimulations of the fifth force and its Laplacian field at solar system scales.The cubic Galileon gravity scalar field model (CGG), which derives from the DGPbraneworld model, describes modified gravity incorporating a Vainshteinscreening mechanism. The nonlinear derivative interactions in the CGG equationsuppress the field near regions of high density, thereby restoring generalrelativity (GR) while far from such regions, field enhancement is comparable toGR and the equation is dominated by a linear term. This feature of thegoverning PDE poses some numerical challenges for computation of the scalarpotential, force and Laplacian fields even under stationary conditions. Here wepresent a numerical method based on finite differences for solution of thestatic CGG scalar field for a 2D axisymmetric Sun-Earth system and a 3DCartesian Sun-Earth-Moon system. The method relies on gradient descent of anintegrated residual based on the normal attractive branch of the CGG equation.The algorithm is shown to be stable, accurate and rapidly convergent toward theglobal minimum state. We hope this numerical study, which can easily beextended to include smaller bodies such as detection satellites, will proveuseful to future measurement of modified gravity force fields at solar systemscales.

Robust numerical computation of the 3D scalar potential field of the cubic Galileon gravity model at solar system scales