Large-N and Mean-Field Limits

The large-$N$ limit makes the distinction between a mean source and its connected fluctuations quantitative. With $N$ independent matter species and $NG$ fixed, the expectation value that curves the mean metric remains finite, while the rms stochastic correction is suppressed by $N^{-1/2}$. This is a controlled mean-field limit only for specified states, observables, smearing scales, and frequencies; it is not a license to ignore collective correlations or modes above the gravitational cutoff.

Required background. Einstein Equations supplies the field equation whose source and metric response are counted in powers of $1/N$.

Helpful background. Quantum Fields in Curved Spacetime supplies renormalized stress operators and admissible states, while Black-Hole Geometry and Mechanics supplies the stationary backgrounds used in later applications.

Species scaling of the mean source

Take $N$ identical free fields $\phi_i$ that interact only through gravity. Write

$$T^{\mathrm{tot}}_{\mu\nu}=\sum_{i=1}^N T^{(i)}_{\mu\nu}, \qquad G=\frac{\bar G}{N},$$

and keep $\bar G$ fixed as $N\to\infty$. For a product state with the same one-field marginal,

$$\langle T^{\mathrm{tot}}_{\mu\nu}\rangle =N\langle T^{(1)}_{\mu\nu}\rangle, \qquad 8\pi G\langle T^{\mathrm{tot}}_{\mu\nu}\rangle =8\pi\bar G\langle T^{(1)}_{\mu\nu}\rangle.$$

Thus matter-loop backreaction is order unity. A graviton propagator carries $G=O(N^{-1})$, but each closed matter loop carries a species sum $N$. Chains of matter polarization bubbles therefore survive at leading order, whereas a graviton loop has no compensating species factor and is down by another power of $1/N$. This diagrammatic argument is given in Hu and Verdaguer 2008, § 3.3.1, Eqs. (3.19)–(3.20).

For the whole effective action to scale as $N$, keep the rescaled curvature couplings $\alpha/N$ and $\beta/N$ fixed as well as $\bar G=NG$ and $\Lambda$. Then $G\alpha$ and $G\beta$, which enter the normalized field equation, remain finite. This completes the scaling beyond its Einstein term; see Anderson, Molina-París, and Mottola 2003, § II, Eq. (2.7), PDF.

The result is a self-consistent mean equation, not a fixed-background approximation. The quantum state and its renormalized stress tensor are evaluated on the same metric that they source. Boundary conditions, renormalization conditions for the rescaled couplings, and the causal contour remain fixed data throughout the counting.

Connected fluctuations count differently

Define $t^{(i)}_{\mu\nu}=T^{(i)}_{\mu\nu}-\langle T^{(i)}_{\mu\nu}\rangle$. Independence removes cross-species connected correlators, so

$$N^{\mathrm{tot}}_{\mu\nu\rho\sigma}(x,y) =\frac12\left\langle \left\{t^{\mathrm{tot}}_{\mu\nu}(x),t^{\mathrm{tot}}_{\rho\sigma}(y)\right\} \right\rangle =N N^{(1)}_{\mu\nu\rho\sigma}(x,y).$$

The mean source scales as $GN=O(1)$, while its rms fluctuation scales as $G\sqrt N=O(N^{-1/2})$. If the retarded response remains bounded uniformly in this limit, the induced metric two-point function is therefore $O(N^{-1})$ on the stated time interval. A pole approaching the observation band, or evolution over an $N$-dependent secular time, can defeat that conclusion. At this same order, stochastic gravity reproduces the symmetrized quantum metric two-point function when its homogeneous initial covariance is also matched to the quantum metric state Hu and Verdaguer 2008, § 3.2, Eqs. (3.16)–(3.18).

The frequently quoted ratio $\sqrt{\langle T^2\rangle_c}/\lvert\langle T\rangle\rvert\sim N^{-1/2}$ is meaningful only after tensor components have been contracted with an observer or test tensor and smeared. If the mean happens to vanish, that ratio diverges even when the absolute induced metric fluctuation is tiny. The response-weighted metric correlator is then the relevant diagnostic.

Correlations, occupations, and renormalization

Species number does not by itself guarantee factorization. If $\operatorname{Cov}(T^{(i)},T^{(j)})=C$ for every $i\ne j$, then

$$\operatorname{Var}(T^{\mathrm{tot}})=N V+N(N-1)C.$$

A positive correlation $C=O(1)$ destroys the relative suppression. Retaining the usual $N^{-1/2}$ rate requires $C=O(1/N)$ or smaller; any positive $C\to0$ can still give suppression, possibly more slowly. These covariances cannot be arbitrary: positivity of the equal-covariance matrix requires $-V/(N-1)\le C\le V$. Highly squeezed states can similarly enhance particular smeared components. Large occupation can sharpen some relative observables, but it may also increase backreaction or introduce short wavelengths, so it is a separate state-dependent limit.

Renormalization does not upset the counting when performed consistently. Local identity-valued counterterms cancel from $t_{\mu\nu}$, while their finite parts reside in the gravitational couplings. Ward identities require both the mean stress tensor and the separated-point noise kernel to be conserved. A species-dependent shift of those couplings must be held fixed by renormalization conditions before comparing different $N$.

Independent checks and domain

Two limits discriminate the interpretation. At fixed $N$, taking $G\to0$ removes both mean backreaction and induced metric fluctuations. At fixed $\bar G=NG$, taking $N\to\infty$ retains the mean equation but removes the leading rms metric fluctuation. Those limits would be indistinguishable if one incorrectly treated $\langle T T\rangle$ as the square of $\langle T\rangle$.

The large-$N$ expansion controls matter-induced metric correlations at leading nontrivial order, but not graviton self-interactions, non-Gaussian higher cumulants at arbitrary order, or trans-cutoff response poles. Anderson, Molina-París, and Mottola show why causal, gauge-invariant response still has to be tested after the counting is established Anderson, Molina-París, and Mottola 2003, §§ III–IV. Fixed-background state construction remains outside this backreaction analysis, and microscopic quantum-gravity degrees of freedom remain separate.

There is also a useful normalization check. Replacing each one-field stress tensor by $aT^{(i)}$ multiplies the total mean by $a$ and the noise kernel by $a^2$, so its rms value again scales linearly with $a$. Any derivation in which these two quantities scale differently has confused a connected correlator with an uncentered second moment. Conservation provides a tensorial check independent of this statistical one.

Exercise

Suppose every species has nonzero mean $\mu$, fixed variance $V\gt 0$, and cross-covariance $C_N=c/N^p$ after the same smearing, with $0\lt c\le V$ and $p\ge0$. With $G=\bar G/N$, for which values of $p$ does the relative rms metric source vanish as $N\to\infty$? Why is extending this fixed-$V$ family to $p\lt 0$ inadmissible?

Solution

The total variance is $NV+N(N-1)c/N^p\sim NV+cN^{2-p}$. The metric-source variance multiplies this by $G^2=\bar G^2/N^2$, giving terms of order $N^{-1}$ and $N^{-p}$. The mean metric source is $\bar G\mu=O(1)$, so the relative rms fluctuation behaves as the larger of $N^{-1/2}$ and $N^{-p/2}$. It vanishes for every $p\gt 0$ and remains order one for $p=0$. Exact independence gives the usual $N^{-1/2}$ rate. For $p\lt 0$, the covariance eventually exceeds $V$, violating $|C_N|\le\sqrt{V V}=V$; that formal scaling does not describe a realizable fixed-variance family.

References

• Anderson, P. R., Molina-París, C., and Mottola, E. 2003. “Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space.” Physical Review D 67, 024026. Open PDF.
• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.