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, but above 1 mm the equation of state is sigma = -epsilon as for dark energy). In the case of a massive field, W does not depend on the mass of the field for L << 1mm but for L >> 1 mm it does not depend on h. 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.
UV cutoff; Schwarzschild radius; Dark energy; Holographic principle; Quantum gravity
