  We theoretically analyze the response properties of ultracold bosons in
optical lattices to the static variation of the trapping potential. We show
that, upon an increase of such potential (trap squeezing), the density
variations in a central region, with linear size of >~ 10 wavelengths, reflect
that of the bulk system upon changing the chemical potential: hence measuring
the density variations gives direct access to the bulk compressibility. When
combined with standard time-of-flight measurements, this approach has the
potential of unambiguously detecting the appearence of the most fundamental
phases realized by bosons in optical lattices, with or without further external
potentials: superfluid, Mott insulator, band insulator and Bose glass.
