  We explore the cosmological implications of a mechanism found in several
approaches to quantum-gravity, whereby the spectral dimension of spacetime runs
from the standard value of 4 in the infrared (IR) to a smaller value in the
ultraviolet (UV). Specifically, we invoke the picture where the phenomenon is
associated with modified dispersion relations. With minimal assumptions, we
find that UV behaviour leading to 2 spectral dimensions results in an exactly
scale-invariant spectrum of vacuum scalar and tensor fluctuations, regardless
of the equation of state. The fluctuation production mechanism is analogous to
the one known for varying speed of sound/light models and, unlike in inflation,
the spectrum is already scale-invariant before leaving the horizon, remaining
so after freeze-in. In the light of Planck's recent results we also discuss
scenarios that break exact scale-invariance, such as the possibility that the
spectral dimension runs down to a value slightly higher than 2, or runs down to
2 but with an extremely slow transient. We further show that the tensor to
scalar ratio is fixed by the UV ratio between the speed of gravity and the
speed of light. Not only does our model not require inflation, but at its most
minimal it seems incompatible with it. In contrast, we find that running
spectral dimensions can improve the outlook of the cyclic/ekpyrotic scenario,
solving the main problems present in its simplest and most appealing
realisations.
