  Based on dynamical mean-field theory with a continuous-time quantum
Monte-Carlo impurity solver, static as well as dynamic spin and charge
susceptibilites for the phase diagram of the sodium cobaltate system
Na$_x$CoO$_2$ are discussed. The approach includes important vertex
contributions to the q-dependent two-particle response functions by means of a
local approximation to the irreducible vertex function in the particle-hole
channel. A single-band Hubbard model suffices to reveal several charge- and
spin-instability tendencies in accordance with experiment, including the
stabilization of an effective kagome sublattice close to x=0.67, without
invoking the doping-dependent Na-potential landscape. The in-plane
antiferromagnetic-to-ferromagnetic crossover is additionally verified by means
of the computed Korringa ratio. Moreover an intricate high-energy mode in the
transverse spin susceptiblity is revealed, pointing towards a strong energy
dependence of the effective intersite exchange.
