  We use pseudodifferential calculus and heat kernel techniques to prove a
conjecture by Chamseddine and Connes on rationality of the coefficients of the
polynomials in the cosmic scale factor $a(t)$ and its higher derivatives, which
describe the general terms $a_{2n}$ in the expansion of the spectral action for
general Robertson-Walker metrics. We also compute the terms up to $a_{12}$ in
the expansion of the spectral action by our method. As a byproduct, we verify
that our computations agree with the terms up to $a_{10}$ that were previously
computed by Chamseddine and Connes by a different method.
