4 Correlation functions

We have already assumed that the number fluctuations at different space-time positions are uncorrelated, i.e., we have assumed that in spacetime volumes $\Delta V_{1,2} \Delta t_{1,2}$

$$\langle \Delta \left[n({\vec{x}}_{1},t_{1})\Delta V_{1}\Delt... ...}-{\vec{x}} _{2})\delta (t_{1}-t_{2})\Delta V_{1}\Delta t_{1};$$

then

$$\left\langle {\Delta \Phi ({\vec{x}}_{1},t_{1}) \Delta \Phi ({\vec{x}}_{2},t_{2})}% \right\rangle = \left( {{\frac{{G\hbar c}}{{2c^{2}}}}}\right) ^{2}\sum\limits_{{\... ...})^{2}}}}}\right) \left( {{\frac{1}{{c^{2}(t_{1}-t^{\prime })^{2}}}}}\right) }}$$

$$\qquad \times \sum\limits_{{\vec{x}}^{\prime \prime }\in S}{\ \s... ...prime }) \Delta n({\vec{x}}^{\prime \prime },t^{\prime \prime })} \right\rangle$$

$$= \left( {{\frac{{G\hbar c}}{{2c^{2}}}}}\right) ^{2}n_{0}\int\limit... ...{2}(t_{1}-t)^{2}}}}}\right) \left( {{\frac{1}{{ c^{2}(t_{1}-t)^{2}}}}}\right) }$$

$$\qquad \times \left( {3-{\frac{r_{2}^{2}}{{c^{2}(t_{2}-t)^{2}}}}}\right) \left( {{\frac{1}{{c^{2}(t_{2}-t)^{2}}}}}\right) ,$$ (7)

where the shorthand notation $r_{1}=\left\vert {{\vec{x}}_{1}-{\vec{x}}}\right\vert$ and $r_{2}=\left\vert {{\vec{x}}_{2}-{\vec{x}}}\right\vert$ has been used, $S$ is the range of acceptable values of ${\vec{x}}_{1}$ and ${\vec{x}}_{2}$ - i.e., values such that $r_{{\mathrm{min}}}<r_{1,2}<r_{{\mathrm{max}}}$ - and $T^{\prime }$ and $T^{\prime\prime }$ are the ranges of $t^{\prime}$ and $t^{\prime \prime }$, i.e., $T^{\prime}=(-\infty,t_{1}-r_{1}/c)$ and $T^{\prime \prime}= (-\infty,t_{1}-r_{2}/c)$.

Because of the axial symmetry about the ${\vec{x}}_{1}-{\vec{x}}_{2}$ direction and because of the approximate stationarity of the random process, the integral 7 can be partly simplified and we obtain

$$\left\langle {\Delta \Phi ({\vec{x}}_{1},t_{1}) \Delta \Phi ({\vec{x}}_{2},t_{2})} \right\rangle \approx \left( {{\frac{{G\hbar }}{{2c}}}}\right) ^{2}\frac{ \pi }{4\ell }... ...c{r_{2}^{2}}{{ c^{2}(t-\tau )^{2}}}}}\right) {{\frac{1}{{c^{2}(t-\tau )^{2}}}}}$$

$$\qquad + \left. \int_{\max (-\ell ,-c\tau )}^{+\ell }\left( u^{2}... ...^{2}}{{ c^{2}(t-\tau )^{2}}}}}\right) {{\frac{1}{{c^{2}(t-\tau )^{2}}}}}\right]$$

where $\ell =\left\vert {{\vec{x}}_{1}-{\vec{x}}}_{2}\right\vert$ is the (nonzero) spatial separation between the space-time points 1 and 2, $t_{2}>t_{1}$ (so that $\tau >0)$ and the radial distances are expressed as functions of the new integration variables $u$, $v$: $r_{1}=(u+v)/2$, $r_{2}=(u-v)/2$.

We may also consider the fluctuation at a well specified position in space but at different times to obtain the time correlation function

$$R(\tau ) = \left\langle {\Delta \Phi ({\vec{x}},0)\Delta \Phi ({\vec{x}},\tau )} \right\rangle$$

$$= \left( {{\frac{{G\hbar }}{{2c}}}}\right) ^{2} \!\! n_{0} \!\! \in... ...rac{{r^{2}}}{{c^{2}(t-\tau )^{2}}}}}\right) {{\frac{1}{{c^{2}(t-\tau )^{2}} }}}$$

$$= {\frac{{G^{2}\hbar ^{2}\pi }}{{c^{2}}}n_{0}}\left\{ \frac{2}{35c^... ...ln {\frac{{r_{{\mathrm{max}} }}}{{ r_{{\mathrm{max}} }+c\tau }}}\right. \right.$$

$$\qquad \left. -\left( 50r_{{\mathrm{min}} }^{7}-84r_{{\mathrm{min... ...\ln {\frac{{r_{{\mathrm{max}} }+c\tau }} {{r_{{\mathrm{min}} }+c\tau }}}\right]$$

$$\qquad \qquad +\left( \frac{20}{7}\frac{r_{{\mathrm{max}} }^{6}}{... ...{3}{\tau }^{2}}-\frac{239}{105} \frac{r_{{\mathrm{max}} }}{c^{2}{\tau }}\right)$$

$$\qquad \qquad -\left( \frac{20}{7}\frac{r_{{\mathrm{min}} }^{6}}{... ...3}{\tau }^{2}}-\frac{239}{105} \frac{r_{{\mathrm{min}} } }{c^{2}{\tau }}\right)$$

$$\qquad \qquad \qquad \left. -\frac{\tau }{r_{{\mathrm{max}} }+c\t... ...}{3r_{{\mathrm{min}} }^{2}+6r_{{\mathrm{min}} }c\tau + 3c^{2}\tau ^{2}}\right\}$$

$$\approx \frac{68}{35}{\frac{{G^{2}\hbar ^{2}\pi }}{{c^{3}}}n_{0}}\ln { \frac{{r_{{\mathrm{max}} }+c\tau }}{{r_{{\mathrm{min}} }+c\tau }}},$$

where we have assumed, in the last step, that $r_{{\mathrm{max}} }\gg c\tau \gg r_{{\mathrm{min}} }.$ The mean square fluctuation can be computed using the Wiener-Kintchine theorem:

$$\left\langle {\left\vert \Delta \Phi (f)\right\vert }^{2}\right\rangle = \int\limits_{-\infty }^{+\infty }{R(\tau )\mathrm{e} ^{-2\pi \mathrm{i} f \tau }\mathrm{d} \tau}$$

$$= 2\int_{0}^{\infty }{R(\tau )\cos (2\pi f\tau )\mathrm{d} \tau }$$

$$\approx \frac{136}{35}{\frac{{G^{2}\hbar ^{2}\pi }}{{c^{3}}}n_{0}\int_{0}... ...} }+c\tau }}{{r_{{\mathrm{min}} }+c\tau }} \cos (2\pi f\tau )\mathrm{d} \tau }}$$

$$\approx \frac{34}{35}{\frac{{G^{2}\hbar ^{2}n_{0}\pi }}{{c^{3}}}} \frac{1}{{f}},$$ (8)

which is, notably, one of those ubiquitous $1/f$ spectra that appear again and again in many different contexts. The space correlation at equal time and $\ell \neq 0$ can also be computed with a long but straightforward integration:

$$\left\langle {\Delta \Phi ({\vec{x}}_{1},0)\Delta \Phi ({\vec{x}}_{2},0)} \right\rangle \approx \left( {{\frac{{G\hbar }}{{2c}}}}\right) ^{2}n_{0} \frac{\pi }{4\... ...}\left( {3-{\frac{r_{2}^{2}}{{c^{2}t^{2}} }}}\right) {{\frac{1}{{c^{2}t^{2}}}}}$$

$$\qquad + \left. \int_{0}^{+\ell }\left( u^{2}-v^{2}\right) \mathr... ... {3-{\frac{r_{2}^{2}}{{ c^{2}t^{2}}}}}\right) {{\frac{1}{{c^{2}t^{2}}}}}\right]$$

$$\approx \frac{68}{35}\frac{G^{2}\hbar ^{2}\pi }{c^{3}}n_{0}\ln \frac{\ell +2r_{{\mathrm{max}} }}{\ell +2r_{{\mathrm{min}} }},$$

where we have assumed, in the last step, that $r_{{\mathrm{max}} }\gg \ell \gg r_{{\mathrm{min}} }$.

The general case requires more laborious calculations, and one must also distinguish between the two cases $\ell >c\tau$ and $\ell <c\tau$: we report here only the simpler result for $r_{{\mathrm{max}} }\gg \ell >c\tau \gg r_{{\mathrm{min}} }$, i.e.,

$$\left\langle {\Delta \Phi ({\vec{x}}_{1},0)\Delta \Phi ({\vec{x}}_{2},\tau )}% \right\rangle \approx \frac{G^{2}\hbar ^{2}\pi }{c^{3}}n_{0}\left[ \frac{% \left( 5\ell... ...ll )}{(2r_{{\mathrm{max}} }+\ell )(2r_{{\mathrm{max}} }+2c\tau - \ell )}\right.$$

$$\qquad \left. +\frac{34}{35}\ln \frac{(2r_{{\mathrm{max}} }+2c\ta... ...r_{{\mathrm{max}} }+2c\tau +\ell )}{4(r_{{\mathrm{min}} }+c\tau )^{2}}\right] .$$