**4.1 Summary of conventions and of some previous results**

The Einstein equations with a cosmological constant, or vacuum energy term, are written

$$R\_{\mu\nu} - \frac{1}{2} \mathcal{g}\_{\mu\nu} R + \Lambda \mathcal{g}\_{\mu\nu} = -\frac{8\pi G}{c^4} T\_{\mu\nu} \tag{28}$$

The corresponding action (without the boundary term) is

$$S\_E = -\frac{c^4}{8\pi G} \int d^4 \mathbf{x} \sqrt{\mathcal{g}} \, R + \frac{\Lambda c^4}{8\pi G} \int d^4 \mathbf{x} \sqrt{\mathcal{g}} \tag{29}$$

In this paper with use metric signature (+,-,-,-). With this convention, the cosmological (repulsive) background experimentally observed is of the order of *c*4/*G*=-10-9 J/m3.

In perturbative quantum gravity on a flat background, this value of corresponds to a small real graviton mass (Datta *et al*., 2003, and ref.s). Actually, in the presence of a curved background the flat space quantization must be replaced by a suitable curved-space quantization (Novello & Neves, 2003). The limit *m*0 of a theory with massive gravitons is tricky, so this global value of still represents a challenge for quantum gravity (besides the need to explain its origin; compare Sect. 2.1).

In our previous work we introduced the idea that at the *local* level, the coupling of gravity with certain coherent condensed-matter systems could give an effective local positive

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 17

macroscopic wavefunction. In order to obtain high-frequency oscillations in time, one can induce in the material Josephson currents (Modanese & Junker, 2007). Spatial variations have a typical scale of 1 nm, so we take this as the size of the volume *V* where the perturbation is spatially constant and the transition probability is computed. For this reason

> <sup>1</sup> (, ) (, ) () <sup>8</sup> *H t gt t*

Fig. 6. A time-dependent -term can be quite efficient in exciting transitions + -, because it enters the matrix elements to first order in . The denomination "Volume coupling" refers

For the evaluation of the density of final states we refer to (Modanese, 2011) and give here only the final result on the probability of transitions +- under the action of an external perturbation with oscillation frequency in resonance with the energy difference (20). Given the large number of available states, the resonance occurs for any frequency, and also

In accordance with the Fermi rule and considering a volume *V*10-27 m3 and a frequency

<sup>2</sup> <sup>1</sup> 34 38 27 23 1 ( ) 10 10 10 10 s *dP H E*

This implies that the excitation time of the zero-modes in the presence of a suitable local term is very short (10-23 s). It is likely, actually, that this excitation process is limited by the

( ) *a a aa*

 

(35)

*S m g x dx dx*

 

(34)

to its mathematical form and to the fact that it is due to de-localized coherent matter

described by a macroscopic wavefunction.

if the perturbation is not exactly monochromatic.

107 Hz (compare Sect. 3), we obtain

*dt*

The action of free incoherent particles is

energetic balance rather than by the transition probability.

**4.3 Comparison with the effect of incoherent matter** 

 

*a*

*<sup>G</sup>* **x x** (33)

we shall leave only the time dependence in and write henceforth

contribution to the cosmological constant and therefore generate instabilities in the field (imaginary graviton mass (Modanese, 1996)). This early argument was not very compelling, but was reinforced after considering the effects of the -term on the weak-field dipolar fluctuations mentioned in Sect. 2.1. Still such effects were predicted to be very weak and dependent on the sign and value of the background at the scale of interest. After the discovery of the strong-field zero-modes of the action, in (Modanese & Junker, 2007) we computed the effect of a -term on such configurations, but it still turned out to be very small.
