This essay discusses phenomenological aspects of the diffusion time dependence of the spectral dimension predicted by the Causal Dynamical Triangulations ( CDT ) approach to quantum gravity .
The deformed form of the dispersion relation for the fields defined on the CDT space-time is reconstructed .
Using the Fermi satellite observations of the GRB 090510 source we find that the energy scale of the dimensional reduction is E _ { * } > 6.7 \cdot 10 ^ { 10 } GeV at ( 95 \% CL ) .

By applying the deformed dispersion relation to the cosmological perturbations it is shown that , for a scenario when the primordial perturbations are formed in the UV region , the scalar power spectrum \mathcal { P } _ { S } \propto k ^ { n _ { S } -1 } where n _ { S } -1 \approx \frac { 3 r ( d _ { UV } -2 ) } { r + 48 ( d _ { UV } -3 ) } .
Here , d _ { UV } \approx 2 is obtained from the CDT value of the spectral dimension in the UV limit and r is the tensor-to-scalar ratio .
We find that , the predicted deviation from the scale-invariance ( n _ { S } = 1 ) is in contradiction with the up to date Planck and BICEP2 results .