  It is known that the value L of the vacuum energy density affects the
propagation equation for gravitons: A mass term appears in the propagation
equation, such that m^2=-L. As a consequence, the polarization states of
gravitons also change. This effect of the L-term has been confirmed by recent
calculations in a curved background, which is the only proper setting, since
solutions of the classical Einstein equations in the presence of a L-term
represent a space with constant curvature. A real value for the mass (when L<0)
will show up as a slight exponential damping in the gravitational potential,
which is however strongly constrained by astronomical data. The consequences of
an imaginary mass (for L>0) are still unclear; on general grounds, one can
expect the onset of instabilities in this case. This is also confirmed by
numerical simulations of quantum gravity which became recently available. These
properties gain a special interest in consideration of the following. (1) The
most recent cosmological data indicate that L is positive and of the order of
0.1 J/m^3. Is this value compatible with a stable propagation of gravitons? (2)
The answer to the previous question lies perhaps in the scale dependence of the
effective value of L. L may be negative at the small distance/large energy
scale at which the quantum behavior of gravitational fields and waves becomes
relevant. Furthermore, local contributions to the vacuum energy density (in
superconductors in certain states, and in very strong static electromagnetic
fields) can change locally the sign of L, and so affect locally the propagation
and the properties of gravitons. The graviton wavefunction, for different
values of the parameters, may be characterized by superluminal phase velocity
or by unitarity only in imaginary valued time.
