  A possible connection between the energy W of the vacuum fluctuations of
quantum fields and gravity in "empty space" is conjectured in this paper using
a natural cutoff of high momenta with the help of the gravitational radius of
the vacuum region considered. We found that below some "critical" length $L = 1
mm$ the pressure $sigma$ is one third of the energy density $epsilon$, as for
dark matter, but above $1 mm$ the equation of state is $sigma = -(epsilon)$
(dark energy). In the case of a massive field, W does not depend on the mass of
the field for $L<<1 mm$ but for $L>>1 mm$ it does not depend on the Planck
constant. In addition, when the Newton constant tends to zero, W becomes
infinite. The energy density is also a function of the volume V of the vacuum
region taken into account.
