April 26, 2007

Effective Field Theory and Gravity

I’ve written, before, about the fact that, at energies below the Planck scale, there’s a perfectly nice effective field theory of gravity. But what is its range of validity? The naïve answer is that it should be trustworthy when curvatures are small, and particle momenta are sub-Planckian. That’s clearly true, but are there other regimes when the effective field theory analysis is naïvely valid but, in fact, breaks down?

Arkani-Hamed et al have a proposal for such a breakdown.

The motivating example for suspecting that there must be a breakdown of effective field theory, in regimes where it should naïvely be valid is blackhole evaporation.

The “nice slice” argument says that one can foliate the spacetime of an evaporating blackhole by spatial slices on which the Riemann curvature is small and whose extrinsic curvatures are all small. Only at relatively late times, when the blackhole has shrunk to near-Planckian size, do we necessarily encounter regions of high curvature. Prior to that time, our spacelike slices intersect all of the infalling matter that formed the blackhole and nearly all of the outgoing Hawking radiation. As the blackhole evaporates, the entanglement entropy of the state outside the horizon grows monotonically. By the time our spatial slices start to encounter regions of high curvature, almost all of the mass of the blackhole has radiated away, and there is not enough remaining energy to carry off the information and restore a pure state on $\mathcal{I}^+$.

For unitarity to be restored, effective field theory must break down earlier and, over the years, various arguments have been put forward that this is the case.

A rather intuitive one, presented by Arkani-Hamed et al is that the “nice” slices tend to pile up inside the horizon. If we want local clocks to be able to resolve the time interval between successive slices, we need heavier and heavier clocks, which are ultimately limited by the mass of the blackhole. If one demands that the time-slicing outside the horizon be at least as fine as the inverse-frequency of the outgoing Hawking radiation, one finds that one can cover only a time interval of order the evaporation time $t_{\text{ev}} \sim M R_S^2 \sim M^{\frac{d-1}{d-3}} M_{\text{pl}}^{-2\left(\frac{d-2}{d-3}\right)}$ At this time, the blackhole has emitted $S_{\text{BH}} \sim {(M/M_{\text{pl}})}^{\frac{d-2}{d-3}}$ quanta and its mass has decreased by a factor of $M(t_{\text{ev}}) = {\left(\frac{2}{d-1}\right)}^{\frac{d-3}{d-1}} M(0)$ Curvatures are still small, and one might think that effective field theory should still be adequate. But the large number of Hawking quanta vitiates this expectation. As argued by Maldacena, the “exact” $N$-point functions in the outgoing Hawking state differ from those calculated in effective field theory by terms that go like $e^{-(S_{\text{BH}}-N)}$. This is exponentially tiny (and the Hawking radiation looks precisely thermal) until $N\sim S_{\text{BH}}$. At $t\sim t_{\text{ev}}$, these corrections become $O(1)$.

This is the story that Arkani-Hamed et al wish to generalize.

The idea, again, is to look for situations where the deviations from effective field theory ought to be very small (for $N$-point functions with fixed $N$), but this is overwhelmed by the presence of $O(S)$ quanta.

De Sitter space shares many features of the blackhole case. There is, again, a horizon and an apparently thermal spectrum of radiation, as seen by any individual observer. The finite entropy

(1)$S_{\text{dS}}= \frac{\pi}{H^2 G}$

has led some to speculate that de Sitter might be described by a finite dimensional Hilbert Space (of dimension $e^{S_{\text{dS}}}$). But, unlike the blackhole case, there’s a global picture of de Sitter space. The degrees of freedom that have exited a particular observer’s horizon may be inaccessible to him, but it’s not so clear that there’s a sense in which we can omit them from the Hilbert space. And what the heck are the observables in de Sitter, anyway?

Arkani-Hamed et al decided to regulate the issue by making (approximate) de Sitter just a temporary phase of an inflating universe. Eventually, inflation ends, and we end up in a flat FRW universe. In this case, the super-horizon modes of the de Sitter (inflationary) phase eventually re-enter the horizon, and are accessible to late-time observers. At least in this setup, they are definitely ‘part of the Hilbert space.’

Does this mean that late-time observers can, in fact, access more than $e^{S_{\text{dS}}}$ degrees of freedom?

One might think that the effective field theory would be an effective tool for analyzing this situation since, semiclassically, the curvature is always small. But Arkani-Hamed et al argue that, for this to be the case, there must be an upper bound on the duration of the period of inflation.

In slow roll inflation, the expansion rate changes slowly in time $\dot{H} = - (4\pi G) \dot{\phi}^2$ so the increase in the entropy (1) per e-folding is

(2)$\frac{d S_{\text{dS}} }{d N} = 8\pi^2 {\left(\frac{\dot{\phi}}{H^2}\right)}^2 \sim {\frac{\delta \rho}{\rho}}^{-2}$

For this description to be valid, however, we need

(3)$\left\vert \frac{\delta\rho}{\rho}\right\vert \lt 1$

throughout. Integrating (2), we find the total number of e-foldings

(4)$N_{\text{tot}} \lesssim S_{\text{end}}$

where $S_{\text{end}}$ is the de Sitter entropy at the end of inflation. Since the number of e-foldings is bounded, so is the number of modes that re-enter the horizon, and a late-time observer never sees more than $e^{S_{\text{dS}}}$ degrees of freedom.

It’s perfectly possible to make the inflaton potential as flat as we want, and thereby ensure an arbitrarily large number of e-foldings. But when we do so, the magnitude of the fluctuations in the inflaton field grows. When $\delta\phi/\dot{\phi} \sim \delta\rho/\rho$ becomes of order 1, the region of spacetime containing the unfortunate late-time observer collapses into a blackhole.

$\delta\phi/\dot{\phi} \sim O(1)$ is the characterization of eternal inflation which, we see, is not reliably treated within the realm of validity of the effective field theory. More generally, Arkani-Hamed et al argue that

(5)$\frac{|\dot{H}|}{G H^4}$

is the relevant quantity for any inflationary model satisfying the null energy condition (not just the slow-roll model of the above analysis). When (5) is $\ll 1$, the effective field theory analysis is valid and (4) is satisfied. When (5) becomes $O(1)$, we have eternal inflation, and the effective field theory description breaks down.

1 Note that there is a limit in which the effective field theory remains valid for all time. Send $M_{\text{pl}}\to\infty$, $M\to\infty$, holding the temperature $T_{\text{BH}} \sim R_s^{-1} \sim M^{-\frac{1}{d-3}} M_{\text{pl}}^{\left(\frac{d-2}{d-3}\right)}$ fixed. But in this limit, $t_{\text{ev}}\to\infty$, the blackhole never evaporates, and the information that went into it really is lost forever.

Posted by distler at April 26, 2007 11:21 PM

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