**4.2 Time-dependent and zero-modes transitions**

A substantial progress was made in (Modanese, 2011), where we showed that the effect of a high-frequency (t)-term can be quite large and independent from its sign and from the background . This was obtained considering the *interactions* between the zero-modes, as we detail in the following. Our latest computations also allow us to recognize more clearly the difference between the gravitational effect of coherent matter mediated by the -term and the (negligible) gravitational effect of the classical *T* of the same matter. After writing the total Lagrangian *L*grav+*L*matter, we split *L*matter into an "incoherent particles" part (Sect. 4.3) and a coherent matter part, described by a scalar field . Only the latter part contains a nonlinear factor *g* , which can have non-vanishing matrix elements already to first order in .

We suppose that the scalar field which describes the coherent matter has in flat space an action of the standard form

$$S\_{\phi} = \int d\mathbf{x} L\_{\phi} = \int d\mathbf{x} \left( \frac{1}{2} \hat{\boldsymbol{\varepsilon}}^{\mu} \phi \hat{\boldsymbol{\varepsilon}}\_{\mu} \phi - \frac{1}{2} m\_{\phi}^{2} \phi^{2} + k \phi^{4} \right) \tag{30}$$

The gravitational coupling introduces a *g* volume factor. The dynamics of is driven by external forces, so this coupling can be regarded as an external perturbation *H* , a local vacuum energy density term due to the presence of coherent matter described by a macroscopic wavefunction equivalent to a classical field:

$$H\_{\Lambda}(t, \mathbf{x}) = \frac{1}{8\pi G} \sqrt{g(t, \mathbf{x})} \Lambda(t, \mathbf{x}) = \frac{1}{8\pi G} \sqrt{g(t, \mathbf{x})} L\_{\phi}(t, \mathbf{x}) \tag{31}$$

The term in coherent matter turns out to be much larger than the cosmological background: for instance, one typically has *c*4/*G*=106-108 J/m3 in superconductors, depending on the type, while the currently accepted value for the cosmological background is of the order of *c*4/*G*=10-9 J/m3. The value above for superconductors is the result of a complex evaluation of the relativistic limit of the Ginzburg-Landau Lagrangian, which yields the following expression for in terms of the pairs density (Modanese, 2003):

$$\Lambda(t, \mathbf{x}) = -\frac{1}{2m} \left[ \hbar^2 (\nabla \rho)^2 + \hbar^2 \rho \nabla^2 \rho - m\beta \rho^4 \right] \tag{32}$$

where is the second Ginzburg-Landau coefficient and *m* is the Cooper pair mass. This energy density has strong variations in space and time, following the behaviour of the 16 Quantum Gravity

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

A substantial progress was made in (Modanese, 2011), where we showed that the effect of a high-frequency (t)-term can be quite large and independent from its sign and from the background . This was obtained considering the *interactions* between the zero-modes, as we detail in the following. Our latest computations also allow us to recognize more clearly the difference between the gravitational effect of coherent matter mediated by the -term and the (negligible) gravitational effect of the classical *T* of the same matter. After writing the total Lagrangian *L*grav+*L*matter, we split *L*matter into an "incoherent particles" part (Sect. 4.3) and a coherent matter part, described by a scalar field . Only the latter part contains a nonlinear

factor *g* , which can have non-vanishing matrix elements already to first order in .

We suppose that the scalar field which describes the coherent matter has in flat space an

*S dxL dx m k* 

2 2

 

The gravitational coupling introduces a *g* volume factor. The dynamics of is driven by external forces, so this coupling can be regarded as an external perturbation *H* , a local vacuum energy density term due to the presence of coherent matter described by a

1 1 (, ) (, ) (, ) (, ) (, ) 8 8 *H t gt t gt L t G G*

The term in coherent matter turns out to be much larger than the cosmological background: for instance, one typically has *c*4/*G*=106-108 J/m3 in superconductors, depending on the type, while the currently accepted value for the cosmological background is of the order of *c*4/*G*=10-9 J/m3. The value above for superconductors is the result of a complex evaluation of the relativistic limit of the Ginzburg-Landau Lagrangian, which yields the following expression for in terms of the pairs density (Modanese, 2003):

> <sup>1</sup> 2 2 22 4 (, ) ( ) <sup>2</sup> *t m*

where is the second Ginzburg-Landau coefficient and *m* is the Cooper pair mass. This energy density has strong variations in space and time, following the behaviour of the

 

*m*

 

1 1 22 4

 

 **<sup>x</sup>** (32)

 

**x xx x x** (31)

(30)

**4.2 Time-dependent and zero-modes transitions** 

macroscopic wavefunction equivalent to a classical field:

 

small.

action of the standard form

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 we shall leave only the time dependence in and write henceforth

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 to its mathematical form and to the fact that it is due to de-localized coherent matter described by a macroscopic wavefunction.

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 if the perturbation is not exactly monochromatic.

In accordance with the Fermi rule and considering a volume *V*10-27 m3 and a frequency 107 Hz (compare Sect. 3), we obtain

$$\frac{dP}{dt} = \frac{1}{\hbar} \left| \left\langle \boldsymbol{\nu}^{+} \left| \boldsymbol{H}\_{\Lambda} \left| \boldsymbol{\nu}^{-} \right\rangle \right|^{2} \right\rangle \rho(E) \approx 10^{34} 10^{-38} 10^{27} \approx 10^{23} \,\mathrm{s}^{-1} \tag{34}$$

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 energetic balance rather than by the transition probability.
