Dimensional reduction in momentum space and scale-invariant cosmological fluctuations

We adopt a framework where quantum gravity's dynamical dimensional reduction of spacetime at short distances is described in terms of modified dispersion relations. We observe that by subjecting such models to a momentum-space diffeomorphism one obtains a dual picture with unmodified dispersion relations, but a modified measure of integration over momenta. We then find that the UV Hausdorff dimension of momentum space which can be inferred from this modified integration measure coincides with the short-distance spectral dimension of spacetime. This result sheds light into why scale-invariant fluctuations are obtained if the original model for two UV spectral dimensions is combined with Einstein gravity. By studying the properties of the inner product we derive the result that it is only in two energy-momentum dimensions that microphysical vacuum fluctuations are scale invariant. This is true ignoring gravity, but then we find that if Einstein gravity is postulated in the original frame, in the dual picture gravity switches off, since all matter becomes conformally coupled. We argue that our findings imply that the following concepts are closely connected: scale invariance of vacuum quantum fluctuations, conformal invariance of the gravitational coupling, UV reduction to spectral dimension two in position space, and UV reduction to Hausdorff dimension two in energy-momentum space.