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A recent experiment [J.L. Garrett et al., Phys. Rev. Lett {\bf 120}, 040401(2018)] measured for the first time the gradient of the Casimir force betweentwo gold spheres at room temperature. The theoretical analysis of the data wascarried out using the standard Proximity Force Approximation (PFA). A fit ofthe data, using a parametrization of the force valid for the sphere-plategeometry, was used by the authors to place a bound on deviations from PFA.Motivated by this work, we compute the Casimir force between two gold spheresat finite temperature. The semi-analytic formula for the Casimir force that weconstruct is valid for all separations, and can be easily used to interpretfuture experiments in both the sphere-plate and sphere-sphere configurations.We describe the correct parametrization of the corrections to PFA for twospheres that should be used in data analysis.

Beyond-PFA Casimir Force between two spheres at finite temperature