  Getting signatures of quantum gravity is one of the topical lines of research
in modern theoretical physics and cosmology. This short review faces this
challenge under a novel perspective. Instead of separating quantum-gravity
effects of a specific model between UV and IR regimes, we consider a general
feature, possibly common to many frameworks, where all scales are affected and
spacetime geometry is characterized by a complex critical exponent. This leaves
a log-oscillating modulation pattern in the cosmic microwave background
spectrum and gives a unique opportunity, illustrated with the example of a
multi-fractional theory, to test quantum gravities at cosmological scales.
