Spectral dimensions from the spectral action

The generalised spectral dimension $D_{ S}(T)$ 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. Analysing the propagation of spin-0, spin-1 and spin-2 fields, we show that a non-trivial 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 $D_{ S}(T) = d_S$ for short and $D_{ S}(T) = 4$ 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 $D_{ S}(T)=0$ for all spins. Our results support earlier claims that high-energy bosons do not propagate.