  Noncommutative spectral geometry offers a purely geometric explanation for
the standard model of strong and electroweak interactions, including a
geometric explanation for the origin of the Higgs field. Within this framework,
the gravitational, the electroweak and the strong forces are all described as
purely gravitational forces on a unified noncommutative space-time. In this
study, we infer a constraint on one of the three free parameters of the model,
namely the one characterising the coupling constants at unification, by
linearising the field equations in the limit of weak gravitational fields
generated by a rotating gravitational source, and by making use of recent
experimental data. In particular, using data obtained by Gravity Probe B, we
set a lower bound on the Weyl term appearing in the noncommutative spectral
action, namely $\beta \gtrsim 10^{-6}$m$^{-1}$. This constraint becomes
stronger once we use results from torsion balance experiments, leading to
$\beta \gtrsim 10^{4}$m$^{-1}$. The latter is much stronger than any constraint
imposed so far to curvature squared terms.
