3 Fluctuations of the gravitational potential

Since we assume that fluctuations spread with the speed of light, there can be no gravitational potential outside the bubble, because there is no way to know that it exists, while inside the bubble we assume that the gravitational potential is just the (Newtonian) potential of a sphere of uniform energy density $\rho$:

$$\Phi (r)=-\frac{4\pi G\rho }{3c^{2}}\left( \frac{3}{2}R_{0}^... ...c}{2c^{2}R_{0}^{2}}\left( 3-\frac{r^{2}}{% R_{0}^{2}}\right) .$$

The gravitational potential at the observation point ${\vec{x}_{0}}$ due to the fluctuations created at distance $r$ at an earlier time $t^{\prime}$ is

$$\mathrm{d} ^{2}\Phi = -{\frac{{G\hbar c} }{{2c^4(t-t^{\prime})^2}}}\left( {3-{% \frac{{... ...^{\prime})^2}}}} \right)\cdot 4\pi r^2 n_{0} \mathrm{d} r \mathrm{d} t^{\prime}$$

$$= -{\frac{{2\pi G\hbar c n_{0} } }{{c^2}}}\left( {3-{\frac{{r^2} }{... ...cdot {\frac{{r^2} }{{c^2(t-t^{\prime})^2}}} \mathrm{d} r \mathrm{d} t^{\prime}.$$

Integrating, as we already did in previous section for the energy density, we can find the total average potential

$$\Phi _{{\mathrm{tot}}} = -{\frac{{2\pi G\hbar n_{0}}}{{c}}} \int\limits_{r_{{\mathrm{min}}... ...\prime })^{2}}}}}\right) \cdot {\frac{{r^{2}}}{{% c^{2}(t-t^{\prime })^{2}}}}}}$$

$$= -{\frac{{2\pi G\hbar n_{0}}}{{c}}} \int\limits_{r_{{\mathrm{min}}... ...thrm{max}} }}% {\mathrm{d} r\left( {{\frac{{3r}}{c}}-{\frac{r}{{3c}}}}\right) }$$

$$\approx -{\frac{{16\pi G\hbar n_{0}}}{{3c^{2}}}}r^{2}_{{\mathrm{max}} }$$

$$= -{\frac{{16\pi G\hbar n_{0}}}{{3c^{2}}}}R^{2}_{U}$$ (5)

and its variance

$$\sigma _{\Phi }^{2} = {\frac{{\pi G^{2}\hbar ^{2}n_{0}}}{{c^{2}}}} \int\limits_{r_{{\ma... ...me })^{2}}}}}\right) ^{2}\cdot {% \frac{{r^{2}}}{{c^{4}(t-t^{\prime })^{4}}}}}}$$

$$= {\frac{{68\pi G^{2}\hbar ^{2}n_{0}}}{{35c^{3}}}}\ln {\frac{{% r_{{\mathrm{max}} }}}{{r_{{\mathrm{min}} }}}}$$

$$= {\frac{{68\pi G^{2}\hbar ^{2}n_{0}}}{{35c^{3}}}}\ln {\frac{{ R_{U}}}{{l_{\mathrm{P}} }}}.$$

Just as it happened previously with the energy density, both the average potential and the variance do not depend either on the observation point ${\vec{x}_{0}}$ or on time, because of the space-time translational invariance of the stochastic process. Notice also that we can express the last result in terms of the Planck length and of the total average energy density, so that we can write

$$\sigma _{\Phi} ^{2} = \frac{68 \pi}{35} G \rho _{{\mathrm{tot}}} l _{\mathrm{P}} ^{2} .$$ (6)

Once again the variance 6 is related to the local fluctuations, even though not exclusively and rather more weakly than the variance of the energy density, while the average potential 5 is determined solely by the large scale behaviour of the Universe.