Solar system and laboratory tests of the spectral action scale

Abstract

We derive the observational consequences of the spectral action \(S = \mathrm{Tr}\,f(D^2/\Lambda^2)\) for weak-field gravity, using the one-loop form factors computed with the full Standard Model content. The linearized field equations yield a modified Newtonian potential with two Yukawa corrections whose amplitudes (\(-4/3\) and \(+1/3\)) are fixed entirely by the spin decomposition of the graviton propagator, and whose ranges are set by calculable effective masses \(m_2 = \Lambda\sqrt{60/13}\) (spin-2) and \(m_0 = \Lambda/\sqrt{6(\xi-1/6)^2}\) (spin-0), where \(\Lambda\) is the spectral cutoff and \(\xi\) is the Higgs non-minimal coupling. The \(1/r\) singularity of the Newtonian potential is regularized: \(V(r)/V_\mathrm{N}(r) \to 0\) as \(r \to 0\), and Newton's law is recovered for \(r \gg 1/\Lambda\). We extract the post-Newtonian parameter \(\gamma(r)\) and confront it with seven gravitational experiments spanning torsion-balance, time-delay, Casimir, satellite, and lunar-ranging measurements. The strongest bound arises from the Eot-Wash experiment, yielding \(\Lambda > 2.565 \times 10^{-3}\,\mathrm{eV}\) (\(1/\Lambda < 77\,\mu\mathrm{m}\)) at 95% CL. All solar-system tests are satisfied with exponential margin. The bound is determined by the SM content at conformal coupling \(\xi = 1/6\) and depends weakly on \(\xi\) otherwise.