Rationality of spectral action for Robertson-Walker metrics

A bstract 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 2 n 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.