  Quantum gravity can determine the dependence of gauge couplings in a scalar
field, which is related to possible fifth forces and time varying fundamental
"constants". This prediction is based on the scaling solution of functional
flow equations. For momenta below the field-dependent Planck mass the quantum
scale invariant standard model emerges as an effective low energy theory. For a
small non-zero value of the infrared cutoff scale the time-variation of
couplings or apparent violations of the equivalence principle turn out to be
negligibly small for the present cosmological epoch, unless some further
quantum scale symmetry violation beyond the standard model comes into play.
More sizable effects of quantum scale symmetry violation are expected during
nucleosynthesis or before. Scaling solutions relate field dependence and
dependence on the renormalization scale. We find asymptotically free gauge
couplings for the standard model coupled to gravity, while for grand unified
models the gauge couplings are asymptotically safe with non-zero values at the
ultraviolet fixed point. The scaling solution of asymptotically safe metric
quantum gravity yields restrictions for model building, as limiting the number
of scalars in grand unified theories.
