152. Volume of the Universe after inflation

Back to work after a way too short Xmas break Since we were recently a bit into black hole complementarity and information loss paradox, maybe it is also worth discussing a bit the physics of de Sitter space.

In a sense, de Sitter is (not so close) relative of the Schwarzschild black hole: metric of the latter looks somewhat similar to the one of de Sitter space in the static patch. As a consequence, an observer living in dS (namely, in the southern diamond (SD) – see the Penrose diagram of de Sitter space below) experiences a kind of Hawking radiation from the horizon of de Sitter space with temperature related to the area of the dS horizon. The fact that the spectrum of this radiation is thermal is directly related to the causal structure of de Sitter space – one gets thermal density matrix tracing out degrees of freedom beyond dS horizon (more accurately, the ones in the northern diamond (ND)).

Existence of thermal radiation from dS horizon forces many people to think that de Sitter space may evaporate in the same fashion as a (small) black hole evaporates constantly emitting quanta of Hawking radiation. Also, there is a smaller amount of people around thinking in this respect of de Sitter complementarity – since Schwarzschild BH is so remarkably similar to static de Sitter patch, and we expect BH complementarity to exist, why wouldn’t we also expect that some analogue of complemetarity in de Sitter space like de Sitter space may exist?

Contrary to the picture of the dS static patch fovoured by string theorists, in cosmology we don’t deal with static de Sitter space – rather, we are working with quasi-de Sitter space in planar patch, describing inflationary Universe. Not so much is left of the picture of Hawking radiation and evaporating de Sitter space in the planar patch. Instead, one has constant generation of superhorizon fluctuations of the inflaton field. As a result, the causal structure of the de Sitter spacetime becomes very non-trivial and differs quite seriously from the one presented on the picture above. For example, the resulting Penrose diagram does contain self-reproducing (fractal) structure of triangles , attached to the future infinity . While an observer still lives inside the causally connected southern diamond region, it is not quite clear that tracing out of superhorizon degrees of freedom will give a thermal density matrix.

How to match these planar and static pictures? The answer to this question is probably far from trivial, the problem remains open for decades, but people are working on it. For example, Sergey Dubovsky, Leonardo Senatore and Giovanni Villadoro have just published a paper about the volume of the Universe, de Sitter entropy and dS complementarity, and I would like to briefly review it maybe testing and stretching their ideas a bit.

In the paper, Sergey, Leonardo and Giovanni they would like to argue that

1. the expectation value of the number of efoldings is finite away of the regime of eternal inflation (namely, it is peaked in the vicinity of the number of efoldings you get from the equation of motion for the inflaton).

2. In the regime of eternal inflation, probability to have finite number of efoldings rapidly (exponentially) decreases. The transition between eternal and non-eternal regimes is at

(1)

3.(using their own words)

The probability for slow-roll inflation to produce a finite volume larger than where  is de Sitter entropy at the end of the inflationary stage, is suppressed below the uncertainty due to non-perturbative quantum gravity effects,

the latter statement is supposed to provide a link between the pictures of eternal inflation and dS complementarity from the authors’ point of view.

How do Sergey, Leonardo and Giovanni come to the conclusions 1 and 2? They want to calculate reheating volume of the Universe

, (2)

where is the time when the Universe achieved reheating in a given Hubble patch (at the point – we coarse grain at Hubble scale, and every Hubble volume is just a point in 3d space for us). The probability   to measure a given value of the scalar field in a given Hubble patch is determined from the inflationary Fokker-Planck equation, and one can construct a similar (but much more non-trivial, that is, non-linear) equation for the volume distribution function (2). Authors argue that as long as the inflation is in the deterministic regime, the reheating volume (2) is finite, but it quickly jumps to infinity when inflation becomes eternal.

How robust is this conclusion and how physically relevant is the quantity (2)? All calculations that authors perform are done for the model of a single scalar field in the static de Sitter spacetime. Technically, there is no backreaction of the scalar field on the background geometry – they have a large (classical) inflaton VEV and weak quantum inflaton fluctuations near the quasi-classical condensate.

As we know, exactly in the regime of eternal inflation the latter cease to be small. Moreover, the backreaction of quantum fluctuations on the geometry is important for eternal inflation – it essentially defines all the  asymptotic probability distributions such as in (1), makes it normalizable and independent of initial conditions for eternal inflation. As the result, the expectation number of efoldings as well as higher moments are necessarily finite, even in the regime of eternal inflation, and the distribution function for the number of efoldings is always normalizable. (See our paper on IR divergences in non-gaussianities as well as original Starobinsky’s paper about stochastic inflation in this respect, where it is explained how to calculate things like .) Note that probability for the given total number of efoldings is the quantity quite similar to  the probability for the Universe to have a given 3-volume (2), since

.

Based on this simple observation, you can kindly make your conclusions yourselves, but it does not quite look to me like that planar (cosmology) and static (string theory) de Sitter pictures became any closer to each other than they were before

Let us now test the authors’ conclusion 3 a bit. Basically, what that they say is that the probability for inflation to produce a very large but finite reheating volume is extremely small (it is virtually impossible to produce finite reheating volume corresponding to the total number of efoldings larger than de Sitter entropy), and in order to calculate this probability accurately we would actually have to modify the Fokker-Planck equation in order to take quantum gravity effects into account.

I think that the nature of the conclusion 3 is the following. Their interpretation of the bound (1) is that with the increase of the threshold of eternal inflation becomes more easily achievable, and when the inflaton crosses the threshold, the reheating volume suddenly becomes infinite, so it is just impossible to get a very large but finite reheating volume out of inflation. Since we agreed that the reheating volume is not so well defined quantity as long as backreaction of inflaton fluctuations is properly taken into account, we need another interpretation of the bound (1). What if the expectation value of the number of efoldings is necessarily finite (this is true indeed) and is probably bounded from above by some number related somehow to the entropy of de Sitter space? To make a quick test of this idea, we will take a model with potential

,

where is so large that all other two terms can be almost always considered small w.r.t. the first one. Corresponding de Sitter entropy is given by .

If

,

there is a potential barrier, and as a result, the expectation number of efoldings is extremely large in the limit . Namely, one has

,

i.e., a famous Hawking-Moss exponent. It is clear that is almost always larger than any de Sitter entropy taken in almost any point of the potential if we consider the weak coupling limit . I would actually expect that for almost any potential with barrier (such that Hawking-Moss instanton with sufficiently large exponent exists) the bound (1) will be violated.

Finally, what about “quantum gravity effects” and complementarity? Actually, I did not find any quantum gravity calculation in the paper (if you’ll find it, kindly let me know in comments) – as I said, they don’t even take the backreaction of inflaton fluctuations on the geometry and inflaton background into account, which would certainly produce some finite effects in the Fokker-Planck equation.

So, I think that the authors’ claim

Taking seriously the similarity between the causal structures of de Sitter and Schwarzschild causes a serious doubt on the validity of the global semiclassical picture of the eternally inflating Universe.

should be really transformed into

Taking seriously the global semiclassical picture of the eternally inflating Universe causes a serious doubt on the validity of analogies between quasi-de Sitter and Schwarzschild causes.

Update: As Instanton pointed out to me, Leonardo Senatore has recently gave a talk in Perimeter Institute about volume of the Universe and de Sitter entropy. You can check out if you missed some important points of the paper.