Spectral dimensions from the spectral action

The generalized spectral dimension DS(T) provides a powerful tool for comparing different approaches to quantum gravity. In this work, we apply this formalism to the classical spectral actions obtained within the framework of almost-commutative geometry. Analyzing the propagation of spin-0, spin-1 and spin-2 fields, we show that a nontrivial spectral dimension arises already at the classical level. The effective field theory interpretation of the spectral action yields plateau structures interpolating between a fixed spin-independent DS(T)=dS for short and DS(T)=4 for long diffusion times T. Going beyond effective field theory the spectral dimension is completely dominated by the high-momentum properties of the spectral action, yielding DS(T)=0 for all spins. Our results support earlier claims that high-energy bosons do not propagate.