In the mean time, when we keep waiting for the final outcome of negotiations, let us continue our small study of dS/CFT. Last time we have concluded that it is hard to realize dS/CFT, since correlations between bulk physics and degrees of freedom living on infinities should be absent due to the presence of horizon.

On the other hand, as we have argued, horizon really means absence of correlations only if we prepare (quasi) de Sitter space by some kind of time evolution. So, what kind of correlation between the bulk physics and the physics at infinities we might expect? In what follows, we will mostly focus on the simplest case of the three-dimensional de Sitter space .

1. How to generally define correlations with boundary?

Formal way to do that is to construct bulk-boundary correlation function . What do I exactly mean by this Green function?

Suppose that we have a free massive scalar field theory with e.o.m.

. (1)

and we have found the general solution of the eq. (1). It is written in terms of mode expansion

(2)

Every mode has an asymptotic behavior

, .

In overall, we define the bulk-boundary Green function as

. (3)

We can write down the explicit expression for the bulk-boundary correlation function as follows. Starting from the e.o.m. (1), we write for the bulk Green function (namely, Feynman propagator)

.

Then,

.

Integrating twice by parts we have

.

Taking an appropriate limit (vicinity of boundary for a given spacetime), we can then find the bulk-boundary Green function.

2. The case of

Let us see how these considerations work for the particular case of spacetime and calculate bulk-boundary Green function for the .

The metric of in the global coordinates is

,

where are complex coordinates on the sphere (it is especially convenient to use them, since the topology of infinities is the one of a sphere as we discussed).

We again consider massive free scalar field . Near the infinity it has the following asymptotic behavior:

,

where

(4)

(we measure mass in units of the Hubble scale) and is determined from the boundary conditions.

Now, using formula for the boundary-bulk Green function we find that it has the following form for (the limit is taken):

,

where

is invariant measure on the sphere,

or,

after substituting asymptotic behavior and some subsequent algebra,

,

where

. (5)

We are almost ready to formulate the correspondence. Namely, we would like to interpret (5) as the correlation function of a conformal field theory operators that live on a 2d sphere and have a conformal dimension (this is very much analogous to AdS/CFT).

There is one interesting observation to be made, though. The conformal weights (4) become complex if the field is sufficiently heavy (namely, ), so in order to make sense of dS/CFT correspondence we would probably want to stick to the case of light scalars.

On the other hand, heavy scalars are the only ones for which we can even define in- and out- vacuum states (as well as Allen-Mottola vacuum states which are linear combinations of the latter!) in the de Sitter (since the asymptotic behavior at infinities should be oscillatory – we want to separate negative frequency modes from positive frequency ones).

So naively it seems like we either have nice well defined QFT in de Sitter space or the dS/CFT correspondence

To be continued.

Post from: NEQNET: Non-equilibrium Phenomena

271. Continuing dS/CFT – correspondence. Part 2

According to Bousso, Maloney and Strominger, the correspondence works as follows.

First of all, de Sitter space does not have a nice spatial infinity like AdS space (where dual field theory degrees of freedom might live) – but it has and infinities, in the global coordinates

(1)

corresponding to and (see the Figure above). As follows from the form of the metric (1), topology of space at is the one of a sphere .

We naturally would like to map physics in the bulk to the physics localized on infinities . It is actually hard to do that, since for an observer in the bulk, say, located near the throat of de Sitter space both infinities at are behind horizon.

It is especially clear to see in the static coordinate system

. (2)

On the Penrose diagram above our bulk observer is located at . Infinities in turn correspond to and, as we see, they are separated from the bulk observer by event horizon at . There is no way physics at I can influence physics at . From my opinion, this is the heart of the problem, the reason why all attempts to construct dS/CFT correspondence failed so far.

However, the relation between physics at and in the bulk is not that meaningless. Let us consider a free massive scalar field in de Sitter space and calculate its two point Green function. It should be de Sitter invariant and can therefore only depend on the invariant interval separating two points in de Sitter space. We can show that there are generally two linearly independent contributions into this Green function: one has UV singularity usual for all field theories, even in Minkowski space, another – has singularity when one takes antipodal points in de Sitter. However, for any observer, his antipodal point in de Sitter space is behind horizon!

How can it be so? The resolution of the puzzle comes from introducing time dependence into the problem. If we want to somehow prepare de Sitter space (say, smoothly evolve it from Minkowski by increasing the expectation value of the inflaton field ), there is no way the second linearly independent term with antipodal singularity might appear (it contradicts causality). Therefore, we simply have to put the coefficient in front of this term to zero by hands.

On the other hand, suppose that de Sitter always existed Then, antipodal singularity was always there, it always influenced the physics inside horizon. We cannot remove the term with antipodal singularity by hands and, as a result, acquire the whole family of Allen-Mottola de Sitter invariant vacua. The same logic is applied to relation between physics in the bulk and at .

To be continued.

Post from: NEQNET: Non-equilibrium Phenomena

254. Continuing dS/CFT – the correspondence. Part 1

As you probably remember, I finished last time pointing out the discrepancy between non-perturbative and perturbative values of the dimension of the dS Hilbert space. Namely, since the entropy of dS space is finite, the dimension of of the Hilbert space is finite as well since

.

On the other hand, perturbation theory in dS space is defined as expansion w.r.t. the small parameter , where is the value of the cosmological constant. The total number of states is clearly infinite at (since this situation corresponds to just Minkowski space). Naively, it seems to be impossible to make the number of states finite making infinitely small but non-zero.

How to reconcile non-perturbative value of (finite) with infinite perturbative value? This question is clearly reduced to the question how the number of states behaves as a function of the perturbation theory parameter .

A satisfying answer would be that the number of states behaves as

,

where is some constant – if , then entropy diverges. As we will see, this is indeed the case (well, you know what is the entropy of de Sitter space ), and we will explicitly calculate the value of the constant .

But to get ready for the quantitative analysis, let us first quickly go through the paper of the most prominent criticizers of dS/CFT – Dyson, Lindesay and Susskind.

1. Why DLS are unhappy with dS/CFT?

The reason is of course the finite number of states that dS supports. Let us recall how the AdS/CFT correspondence works. If the curvature of the background is small (gauge coupling is large), good degrees of freedom are strings (attached to branes) living in the bulk. If the curvature of the background is large, the only relevant degrees of freedom are gauge fields (corresponding to Chan-Paton factors of strings attached to -branes; if there are N coincident -branes, the symmetry group of the gauge fields is ). Since the physics of the latter is the physics of CFT, that is, quantum field theory (sic! stupid statement), then the number of states is infinite in the vicinity of horizon.

The same should hold for dS geometry, if duality holds and the physics of gravitational degrees of freedom on dS background can be effectively described by QFT. The number of short wave length modes in the vicinity of dS horizon is diverging – so is the associated entropy. However, as we have found out, non-perturbatively entropy is supposed to be finite! So, DSL conclude, bulk physics of dS cannot be dual to near-horizon QFT physics.

The argument goes deeper than that. According to proponents of dS/CFT, one point correlation functions in the static patch of dS should exponentially decay. DSL show that this is impossible in the situation when the number of states that the theory supports is finite.

2. What DLS analysis is based on?

It is based on the key proposition of dS complemetarity – there are no independent degrees of freedom, no non-trivial physics beyond dS horizon. In a sense, dS can be considered as closed thermal cavity. This is closely related to so called membrane paradigm – dS horizon is considered as impenetrable thermal membrane that can absorb, thermalize and then reemit the information via Hawking radiation.

Trying to revive the dS/CFT correspondence, we will have to understand where these complementarity considerations can go wrong.

Both ideas (complemetarity and membrane paradigm) come from black hole physics. Although there are many similarities between physics of BHs and de Sitter space, there is also a crucial difference. BH horizon is “physical” (several billion years ago the gas in the center of our galaxy collapsed and formed a black hole – many observers living in different corners of the Milky way can certainly detect it by, say, studying behavior of nearby stars in the vicinity of the galactic center). On the other hand, dS horizon is observer-dependent.

To be continued.

Post from: NEQNET: Non-equilibrium Phenomena

234. Continuing dS/CFT. Why it is so hard to prove?

First of  all, what is the motivation for us to look for gauge theory-gravity dualities? The reason is that a general confining gauge theory at strong couplings  looks effectively like a string theory – flux tubes connecting fermions charged over the gauge group do behave like relativistic strings. In the case of QCD we still have no idea which one in particular is this string theory describing confining asymptotically free QCD at strong coupling, but from 1970s (thanks to ‘t Hooft) we are pretty sure that this string theory is free if the number of colors in the gauge theory is very large (that is, if the gauge group is , should be taken to infinity).

Although it proved to be extremely hard to get any other information about string theory dual to QCD, one important piece of information became available in the end of 1990s -  as it turns out,  supersymmetric YM theory (which is a conformal field theory) is dual to IIB string theory on background. That is, we generally have to expect that if duality between a string theory and a gauge theory exists, string theory should live on a curved background spacetime.

1. AdS/CFT

More precisely, this particular duality works as follows.

Let us first focus on its gravity side. The near horizon geometry of coincident -branes is exactly . Strings attached to these branes are relevant degrees of freedom that we are interested in. As we know, spectrum of the string contains graviton and at low energies we can describe dynamics of the string simply by using Einstein-Hilbert action (with matter fields). More precisely, this is possible when the radius of the curvature of the background is much larger than the string length (in other words, the curvature of the background is sufficiently small), i.e., .

(Note: Well, I am note completely honest here. If you say that , honestly it is only guaranteed that the SUGRA approximation holds for the stringy degrees of freedom, but the effective action is not necessarily the Einstein-Hilbert one. To suppress higher curvature corrections, you need in addition to take the limit .)

On the other hand, if the curvature of the background is large, the only relevant degrees of freedom are the ones that live in the world volume of – brane. The dynamics of these degrees of freedom is governed by gauge theory with supersymmetry, and thus we come to the field theory side of the correspondence. Description of the system in terms of the supersymmetric gauge field theory holds when the curvature of the background is very large, i.e., when .

Therefore, dynamics of the super Yang-Mills theory at strong couplings is equal to the dynamics of gravity + matter degrees of freedom on asymptotically background. Does this fact give us a lot of information? Yes and no.

Yes, because the AdS/CFT duality is established extremely well. For example, has the global – symmetry, and this fact is reflected on the gravity side – extra 5 dimensions of the 10-dimensional space are compactified in the sphere with the group of symmetry . 4-dimensional conformal symmetry of the SYM is , and the symmetry of the space is also .

(Note: By the way, why do we need supersymmetry so much? Bosonic, that is, non-supersymmetric string contains a tachyon in its spectrum, so we cannot expect that the dual gauge theory is stable.)

No, because the duality is a kind of boring. First of all, dynamics of conformal (this is maximal possible amount of supersymmetry in 4 dimensions!) at strong coupling is much more trivial than the dynamics of QCD at strong coupling that we are really interested to learn about. Second, dual description in terms of gravity really works for large and large ‘t Hooft coupling , while planar QCD confining string is the string in the regime of large N and small (even worse, the low energy regime of QCD is the regime with relatively small and small , since does not become terribly larger than 1 in the real life).

So, the second question one really comes to is…

2. What if we deform AdS/CFT

Naturally, we would like to deform the duality and see what happens – does it hold? How much should I deform the theory to break the gauge-gravity duality? For example, we could try to deform it on the CFT side – say, we take a slightly non-conformally invariant theory and try to establish whether it is possible to find its dual gravitational description describing behavior of the theory in the regime of strong coupling (gravity on the background which is slightly deformed ). This is actually the route we will really have to pursuit if we want to construct a gravity dual to QCD. Indeed, the latter is asymptotically free, that is, has negative beta-function while a conformal field theory corresponds to a situation with precisely zero beta-function.

This way however is a hard one, so we can try to deform instead the gravity side of the duality – say, by deforming the asymptotic background. As it turns out, many other CFTs (than SYM) exist which correspond at strong coupling to a gravity with some non-trivial asymptotic behavior. For example, we can try to factor asymptotic AdS space with a discrete group with compact fundamental domain or discuss, for example, . As it seems, both cases correspond to some conformal field theories, but what are the theories exactly is not quite clear yet.

(Note: Some of string theorists (who maybe really like to exxagerate) say that every single CFT that can be found in Nature (and even those that cannot)  has a gravity dual.)

But what if we will do something even more radical – say, take de Sitter space time as a background instead of Anti-de Sitter? 5-dimensional de Sitter space has group – that is, Euclidean conformal group, not really weaker than the conformal group of symmetry of .

Can gravity on an asymptotically de Sitter 5-dimensional background be dual to a Euclidean gauge theory? As it turns out, we come to many obstacles trying to test this idea. First of all …

3. Hilbert space of dS is finite dimensional and de Sitter space has entropy associated with it

Indeed, no single observer can have access to the whole spacetime – its Hubble colume is separated from the rest of the spacetime by the horizon (I discussed the causal structure of dS space in details before, but if you remain dissatisfied after reading my post, I can advise you to take a look at Hawking, Ellis book, Chapter 5). Similar to the case of a black hole emitting Hawking radiation, the observer detects thermal radiation which comes from the horizon (the difference is that thermal radiation is emitted inside the horizon, while in the case of BH – outside) with temperature proportional to the Hubble scale .

Since the region outside is not accessible, we have to average over quantum modes with wavelength larger than the horizon – we are unable to discreminate between them and the constant zero mode. This is a source of entropy associated with de Sitter space. This entropy can be calculated to be

,

where is the area of the de Sitter horizon in exact analogy with Bekenstein-Hawking entropy of BHs.

So, why the Hilbert space of dS is supposed to be finite-dimensional? The reason is that string theory really teaches us to interpret gravitational entropy like any other entropy – as the logarithm of the number of states in the causally connected region. Finite entropy therefore means the finite number of states:

.

4. Dimension of the Hilbert space in perturbation theory

Suppose that we start from a flat spacetime (say, by setting the cosmological constant to zero), where the dimension of de Sitter space is infinite, and then slowly increase the value of the cosmological constant. We then conclude that the dimension of the Hiblert space is infinite for the flat space time, i.e., the case , while for any arbitrarily small positive value of the dimension is infinite. If we go vise versa, from the case of small positive to the case of exactly zero , we have a jump from finite values of the total number of states to infinite .

As we see, one interesting question is how to reconcile the fact that Hilbert space of dS is finite non-perturbatively with the fact that it is infinite perturbatively? I am going to discuss the answer to this question in the next few days.

Update: I have found a terrific book on Amazon which contains almost all TASI lectures related to string theory (in particular, it contains Klebanov’s lectures on AdS/CFT).

Post from: NEQNET: Non-equilibrium Phenomena

223. Starting dS/CFT: Hilbert 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.

Post from: NEQNET: Non-equilibrium Phenomena

152. Volume of the Universe after inflation

So, here we go.

On Page 2. It took some more months before I added to my work the abelian theory of quark confinement based on the idea of instantons. The result was quite stunning – in 3d the instantons ( which were magnetic monopoles) lead to the formation of the electric string for all  couplings, while in 4d the instantons where the closed loops of the monopoles trajectories and the confinement occurred after the coupling exceeded the critical one. A little later Gerard t’Hooft and Stanley Mandelstam arrived at the qualitative picture of dual superconductors , which is of course equivalent to the one I just described.

Comment: although nonabelian instantons were discovered soon (Belavin-Polyakov-Schwartz-Tyupkin), the picture did not work so well in nonabelian case. Abelian instantons are weakly interacting and rigid gas approximation works well for them. To sum over instantons in the partition function, you basically need to invent an analogue of Debye theory for the corresponding plasma – the mass gap that appears in the abelian confining theory is inverse Debye length in the plasma of abelian instantons. In the nonabelian theory the situation is much more complicated – instantons represent the liquid, i.e. strongly interact with each other. Situation is again simplified in SUSY theories, where instantons again start to interact weakly (=> Seiberg-Witten theory).

On pages 2-3. It made its appearance already in the Wilson work on the lattice gauge theory, in which the strong coupling expansion was described as a sum over random surfaces. These surfaces were the result of propagation of one dimensional objects- electric fluxes. The major difficulty was to find the continuous limit of this picture. But already on the qualitative level I found the picture very useful. It helped me to predict the deconfining transition, leading to the quark- gluon plasma. This transition takes place simply because the strings are melting, as can be seen from the Peierls argument.

Comment: One (wrong, but a kind of appealing) picture I have in mind is the following. Short strings with quarks on its ends can be considered as dipoles. As seen from large distances, dipoles are neutral, so the overall system is color neutral. At large temperatures dipoles want to dissociate, and color degrees of freedom become accessible. Why is it wrong?  Because in plasma of dipoles correlation length is infinite in the neutral phase, while it is finite in dissociation phase. Here it is vise versa: mass gap exits at low temperatures and is absent (deconfinement) at large temperatures. So, let me erase this wrong picture from my head zzzz…. done.

And proceed to correct one (I hope I did not confuse you too much) He thinks in terms of instantons of the abelian theory. In 2+1 compact QED, these instantons are point-like, while in 3+1 they become lines. We again construct a kind of Debye theory but for lines instead of particles now. The line of a length L gives the contrbution

to the partition function (here is the coupling) and, if the coupling is strong enough, combinatoric factor overcomes the action factor, and instanton lines should dissociate. That’s Peierls argument he mentions and that’s what he means here by strings -  instanton world lines.

On page 3. This picture of the strings describing the flux lines is often confused with the t’ Hooft picture which suggests that the string world sheet appears because the lines of Feynman diagrams become dense. In the normal gauge theory this certainly doesn’t happen.

Comment: He saying is that strings in AdS/CFT are flux tubes, not those “strings” that appear from the planar expansion in the YM perturbation theory, with worldsheets given by planar Feyman diagrams (and people confuse these two things even in the reviews of AdS/CFT!). He explains that planar diagrams only become dense (that is when we can interpret them as worldsheets of strings) at YM complex coupling. Recall that he is using this idea in this paper to find a physical meaning of dS/CFT.

On page 3. I also thought that the string representation may help to solve the 3d Ising model
by reducing it to the free fermionic strings.

Comment: See his book. 3D Ising is related to gauge theory through the Kramers-Wannier duality, and strong coupling limit of the latter should be described in terms of strings (as usual for the gauge theory). By the way, now I (a kind of) see how to make sense of AdS/CFT applications to condensed matter systems I critisized so bluntly

On page 4. But when I quantized it there was a surprise – an extra, longitudinal mode, which appears due to the quantum ? thickening? of the string. This new field is called the Liouville
mode.

Comment: He is talking about Polyakov’s action, where all are integrated out and only Liouville mode survives due to anomaly non-cancellation in non-critical number of dimensions. I’ve heard many times that Liouville mode corresponds to thickening of the string, but honestly was never able to understand why. Liouville mode comes from the overall rescaling of the metric on the worldsheet by a factor

,

i.e., it corresponds to dilatation of the world sheet. If the Liouville mode is not cancelled, world sheet either wants to stretch or to collapse, doesn’t it? Dear Smart Reader, if you understand this topic, I beg you, please explain it to me.

On page 4. Dynamics of 2d gravity is very rich and even now not completely explored. One of the problems was the field -dependent cut-off which one must use in order to preserve general covariance on the world sheet.

Comment: Naturally, you want to introduce a covariant UV cutoff to your theory (otherwise, if general covariance is broken in the UV, nothing will prefent it from breaking in the IR, too). Therefore, what you need to cut is the interval

,

i.e., you have to introduce the minimal . Here comes the field dependence from.

On page 5. Soon after the Liouville mode was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by
the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one.

Comment: Oh, now he is talking about AdS/CFT and basically says that the Liouville field is the fifth coordinate of the AdS. How can it correspond to thickening of the string then??? An infinitely thin string is finely propagating on , vibrating, rotating and creating gravity in the target space

On page 5. In ‘96 I came to the conclusion that in order to describe gauge theories this five dimensional space must be warped. The logic was as following. In gauge/ string duality the open strings describe the Wilson loop and the only allowed vertex operators in the open string sector are the ones corresponding to gluons ( and extra fields, if present). At the same time, in the closed string sector we have infinite number of states. So, all massive modes of the open string must go away. This can happen only if the ends of the open strings lie either at singularity or infinity and the metric is such that this region has infinite blue shift with respect to the bulk. In this case the masses of all but massless open string states go to infinity.

Comment: This is for Moshe Rozali, who cannot find where infinite blueshift is in the AdS space

To be continued.

Update: Peter Woit also discusses the preprint here but covers mostly the “historical” part.

Post from: NEQNET: Non-equilibrium Phenomena

125. From quarks to strings. On Liouville mode, instantons and confinement in abelian theories

1. formalism

The idea of  formalism is quite simple. If we take into account that curvature perturbation gives the following contribution into the overall metric of spacetime:

( includes all three spatial coordinates) and recall the definition of the number of efoldings

,

we immediately find that

, (1)

where is the inflaton and prime denotes derivative w.r.t. the value of the inflaton field.

Simple formula (1) allows us to easily construct correlation functions of the curvature perturbation, if correlation functions of the inflaton are known. Namely, keeping only the first order on the expansion (1), one has

(2)

(here we have used the logarithmic  divergence of the inflaton correlator discussed in details in one of the previous posts). What is the scale ? What is the physical effect responsible for its value? Here I would like to argue that this effect is eternal inflation.

2. How to calculate curvature perturbation from stochastic formalism

This is a bit complicated, but let me still try to explain it. Trivially, one has

, (3)

where is the total number of efoldings achieved in a given inflationary model.

What is not so trivial is the question how to calculate (3) in the stochastic formalism. Inflation will surely come to the end in a given Hubble patch when the inflaton expectation value drops below the boundary corresponding to the scale of selfreproduction, and the evolution of the inflaton field becomes deterministic. However, although the motion of the inflaton is nearly deterministic, there are still stochastic fluctuations of the inflaton field which may lead to a sudden change of in a given Hubble patch from a value to the value , where the slow roll conditions break down.

The probability distribution for the stochastic moment of the end of inflation or, in other words, the total number of e-folds can be determined from the probability distribution in the Fokker-Planck equation according to

(4)

The reason is that probability is conserved along the path

(5)

as long as inflation became deterministic, so we actually can calculate distribution functions like – they are not completely meaningless

Generally, it is very hard to calculate the distribution function (4) directly. Instead, one can calculate the moments

(6)

directly related to this distribution function. The moments (6) satisfy the following Fokker-Planck-type recursive set of equations:

(7)

while the very first equation in this set is

(8)

It is actually possible to find the solution of the equations (7), (8) in the closed form (please see our paper on the subject), but I don’t want to present it here (it is enormously long).

The bottom line is that all the integrals in this solution I did not present are well behaved since the probability distribution is well-behaved normalizable function of and , as we agreed in the post about correlators of the inflaton. Therefore, all the correlation functions of the total number of efoldings (3) and, automatically, correlators of the curvature perturbation are infrared finite.

Actually, to get a reasonable answer (or at least order of magnitude estimation) for the curvature perturbation correlation function (2) in the formalism, one just needs to keep the first term in the expansion and put the scale of eternal inflation as the IR cutoff scale present in the logarithm

3. So what?

Ok, all these correlation functions (both inflaton and curvature perturbation) are infrared finite, so what? The message is that eternal inflation generally predicts very high level of non-gaussianity. Indeed, we can use the formalism developed above to calculate , etc. just to find that they are all nonzero and generally very large. Is there a chance to see this non-gaussianity? The answer is positive only if you long enough to approach the scale of eternal inflation For us, human beings, there is maybe only a chance (also, very low) to see the running in the correlation functions of non-gaussianity that will give a hint about the scale of eternal inflation.

This is somewhat similar to what happens in the continuum limit of the lattice QFT. At large scales, only renormalizable terms survive and we are unable to get any information about physics that shows up at the scales of the order of the lattice spacing.

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124. Talk in Munich. Regularizing correlators of curvature perturbation

As I have explained in the post about leading logs, naive perturbation theory (expanding correlators at small couplings) seems to break down at scales

, (1)

where  is the self-interaction coupling for the scalar field. The scale (1) is actually quite a bit longer than the scale of eternal inflation, and not surprisingly serious doubts appear that eternal inflation may lead to the regularization of the naively diverging perturbation series. Nevertheless, as I will explain below, this is indeed the case, and to understand how this regularization happens, one has to learn how stochastic formalism for eternal inflation works.

1.  Langevin and Fokker-Planck equations in stochastic formalism

The EoM for the Heisenberg operator for the scalar field (neglecting renormalizations) is

. (2)

Also this equation is simple (at least, it is differential), the Schwinger-Dyson equations for the correlation functions are extremely complicated ? they are integro-differential. It would be stupid to act by brute force and construct their solution in terms of expansion in powers of ? this would just give us seeming IR singularity in the correlation functions and breakdown of the perturbation theory at the scale (1). Instead, to understand IR behavior of the correlation functions of the scalar field, let us actually exploit the following peculiar property of the scalar QFT in de Sitter space: as we know, the modes of the scalar field freeze as long as they cross the de Sitter horizon scale.

We will decompose the Heisenberg operator in (2) into IR (superhorizon) and UV (subhorizon) parts as

(3)

where the second term in the r.h.s. satisfies EoM for a free scalar field, i.e.,

and is a small number.

Substituting the decomposition (3) into (2), we find that

(4)

where

(5)

Although (5) is a very complicated operator equation, all the terms in (4) actually commute with each other, since creation and annihilation operators for a single mode with momentum enter (5) in a single combination. We can therefore consider (4) as a classical equation of motion for the classical quantity . On the other hand, we cannot prescribe any particular numeric value to the quantity (5). Calculating its pair correlation function in the Bunch-Davies vacuum, we find

(6)

i.e., the equation (4) is actually the Fokker-Planck equation describing the random Gaussian walk of the quantity in time.

By Stratonovich prescription, we can immediately derive the corresponding Fokker-Planck equation. It has the form:

. (7)

Note that does not enter the Eq. (7), so it is perfectly applicable for a QFT in fixed de Sitter background.

Equations (6) and (7) are the essence of stochastic formalism for eternal inflation developed by A. Starobinsky back in 1980s.

2. Finiteness of inflaton correlation functions in the infrared. Genesis of leading logs

How to calculate inflaton correlation functions using this formalism? Well, having found the distribution function from the Fokker-Planck eq. (7), we can simply write

. (8)

If the distribution function is finite and normalizable everywhere, then the correlation functions (7) remain finite (integral (8) converges).

The solution for is given by

, (9)

where and are eigenfunctions and the eigenvalues of the operator

and .

The operator (9) is supersymmetric, and all its eigenvalues are positive definite (this shows us that  can be indeed interpreted as probability associated with random walk of ).

Therefore, it is almost guaranteed  that the correlation function (8) is finite.

To understand how leading logs appear in stochastic formalism, we need to

a) recall that and

b) expand the exponents in (9) into power series.

3. Why does stochastic formalism actually keep track of IR divergences?

Or even of IR physics? Indeed, we consider correlation functions of the form (8) where all the points where the operators are taken at are the same. Does not it correspond to UV limit instead?

The answer is negative. When we derive the Langevin equation (4), (5) in the stochastic formalism, we actually perform coarse graining with the scale . The physical reason for that is simple: once inflation has ended, we have a causal contact with our  Hubble patch only. All superhorizon modes are frozen and we are unable to distinguish them from the quasiclassical condensate.

On the other handm, studying inflationary perturbations, we are naturally interested in correlation functions

(10)

with x and x’ taken within our Hubble volume. Divergences in perturbative expansion (i.e., in loop corrections to (10)) come from the superhorizon region.

On the other hand, all correlation functions (10) are the same for the stochastic formalism due to coarse graining that we have used to derive it. Thus, stochastic formalism should correctly capture the most interesting IR physics encoded in the correlation functions of the scalar field .

Some relevant papers

1. While it is rather hard to find the original paper by Starobinsky (the one where stochastic formalism is derived) nowadays, a good technical review of the stochastic formalism is given in the paper by Starobinsky and Yokoyama. It is also explained there why stochastic formalism should reproduce leading logs in the perturbative expansion for a self-interacting scalar field on dS background.
2. Relation between leading logs expansion and stochastic formalism is carefully studied in Prokopec, Tsamis and Woodard, see also references therein to earlier Woodard’s papers.
3. I could certainly recommend you to read my own paper about IR divergences
4. If you want a book where stochastic formalism is presented in understandable way, you will find none. The closest to the understandable one though is the book “Particle physics and inflationary cosmology” by A. Linde. At least, you will learn from there how inflationary random walk works physically.

Update: I have actually found the volume where Starobinsky’s paper is published at, it is even rather cheap. I would seriously recommend reading the very first, original paper, it is actually so beautiful and full of ideas that even an experienced physicist will be able to reread it many times without a slightest chance of getting bored.

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120. Talk in Munich. Regularizing inflaton correlation functions

.

Let me now show how these divergences appear. First of all, let us forget about the interaction term and show that the correlation function for the free field on dS background diverges in the IR.

We can write

, (1)

where

(2)

are the eigenmodes of the free scalar field on de Sitter background.

Keeping the leading term in the expansion (2) of the Hankel function, we find that the integral (1) diverges logarithmically on both upper and lower limits. The UV cutoff for the integral is given by the scale , where the Hankel function starts to oscillate rapidly. The IR cutoff for (1) is given by the value of scale factor at the very beginning of inflation, when all modes were still subhorizon: 

.

We therefore find

(3)

If you want to see a more accurate derivation, then I can recommend to take a look at the famous book by Andrei Linde. What I am going to talk about next is not written in any book yet

How to estimate behavior of this correlator in the interacting theory in de Sitter background?

Let us take renormalizations into account and write the effective Lagrangian as

,

where and . The equation of motion for the Heisenberg operator is

. (4)

We just considered the r.h.s. of (4) as a source term and want to write the solution of this equation in the symbolic integral form:

, (5)

where is the solution of the EOMs for , i.e., for the case of free field.

The Eq. (5) is called Yang-Feldman equation and we can easily construct its solution by iterations. It is easy to see that the corresponding expansion for the two point correlator has the following general (and again symbolic ) form:

(6)

If we note that the most divergent contribution comes from in the integral in (5) and take (3) into account, we conclude that the small parameter in the expansion (6) is actually instead of .

This expansion is called the leading log expansion. It was first introduced by Woodard as far as I know (see for example this paper and references therein).

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115. Talk in Munich. Leading logs

, (1)

where is the slow roll parameter. On the other hand, during the same amount of time the fluctuations of are generated with characteristic amplitude

. (2)

The latter become important for the evolution of the inflaton background, when (1) becomes of the same order as (2). We have then

,

which coincides with the condition for to become of the order 1, as I derived in the previous post (see the expression after Eq. (3) there). In other words, both and are given by the scale of self-reproduction.

Let me now mention two other infrared scales which are of interest for cosmologists.

First one appears if we analyze the loop expansion for the correlators of in the theory on de Sitter background. It turns out that every term in the loop expansion diverges in the infrared, and if we keep only leading IR divergences, we find

. (3)

As we note, the actual small parameter of the perturbation theory is and not . Also, it seems that the expansion breaks down when

,

i.e., the breakdown happens at the scale

[Unparseable or potentially dangerous latex formula. Error 2 ].

This scale is quite a bit longer than the scale of self-reproduction .

What actually happens at this scale? Does the series (3) diverge or not? I will answer to this question in the next post.

Finally, analyzing loop expansion in the formalism in order to calculate corrections to the correlation functions of the curvature perturbation , we find

,

where is the number of efoldings, is the power spectrum of the inflaton field, dots denote terms of higher power w.r.t. the parameter , and is some infrared scale where we need to cut the logarithm.

Experts in formalism call this scale the size of the box. What is the size of the box, is it arbitrary and if not, how can it и related to other infrared scales we discussed? Does the expansion diverge? This was the subject of my talk in Munich and will be the subject of a couple of the next posts.

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112. Talk in Munich. Other two interesting infrared scales

For those of you how were unable to come to the Sommerfeld Center in Munich to hear my talk and for those of you who were there but did not understand it ? I decided to put the outline of my talk on the blog.

The talk was about different infrared divergences seemingly present in correlators we calculate in inflationary perturbation theory and how to properly regularize them.

As a kind of introduction, let us first to identify some interesting infrared scales present in inflating Universe. In what follows, I shall particularly focus on chaotic inflation with quartic potential .

We find one such scale when look at the behavior of the correlation function of the scalar perturbation (Newtonian potential) . As we know (see for example the excellent book by Mukhanov),

. (1)

The physical meaning of this result is very simple: the spectrum of inflationary perturbations has slightly red tilt, i.e., while we look at larger and larger scales, we find that characteristic amplitude of cosmological perturbations grows more and more.

Imagine that we live in the radiation-dominated Universe. The horizon size slowly grows, more and more inflationary modes reenter our Hubble volume. Since the spectrum tilt is red, we will see inevitably at some point that cosmological perturbations became of the order 1. Which scale does this happen at?

From (1) we immediately conclude that the corresponding physical scale (we will call it ) is

.(2)

This scale is huge compared to the present size of the cosmological horizon. Indeed, recall that from the Cobe normalization, so we have something like

.

Also observe that the value of the scale is non-perturbative w.r.t. the coupling .

Let us now take the perturbation of the curvature of 3?dimensional slice instead of the scalar potential and estimate its mean square. We find:

. (3)

At which scale  does become of the order 1?

From (3) we have

,

where is the slow roll parameter. Taking into account that

and

,

we find that thу scale  where the perturbation of 3?dimensional curvature slice also becomes of the order 1 actually coincides with , i.e.,

Could you guess before I continue what physical situation or process does this scale correspond to?

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111. Talk in Munich. One interesting infrared scale in inflationary cosmology

One subject of the authors’ study is the interplay between stochastic and eternal N-flation. Let us recall what is N-flation (or assisted inflation). Suppose that we have a large number of scalar fields with equal potentials and relatively large vacuum expectation values. How large? We want to have the ordinary slow roll inflation in this setup. While one needs an expectation value of the inflaton to be superplanckian in single field inflationary models in order for slow roll conditions to hold, expectation values of any one among N scalar fields can be rather small and still the inflationary slow roll regime will not be spoiled. Indeed, we have

,

and is sufficiently large even for small . Now, we notice that generally during one Hubble time there are fluctuations of the field with the wavelength of the quasi de Sitter horizon and the characteristic amplitude of the order of . Again, we see that there is one more amplification effect in N-flation compared to the single field inflation: the quantum fluctuations of individual scalar fields in N-flation are generally stronger than in a single field inflation (the amplitude amplification is proportional to ). As a result, the regime of stochastic inflation (see my lectures on stochastic inflation) for an individual field is generally achieved earlier for N-flation than for a single field inflation.

All this is rather clear, but the authors indicate the following thing which does not yet belong to the general lore: for N-flation, stochastic inflation of an individual field is not the same as the regime of eternal inflation! Indeed, in this setup eternal inflation would mean self-reproduction regime such that the classical displacement of the “mean field”

during one Hubble time is of the same order as the characteristic amplitude of the quantum fluctuation . Clearly, this regime is established at much higher values of than the regime of stochastic inflation for a single individual field .
In other words, for a single field inflation the onset of stochastic regime is synonymous to the amplitude of density fluctuations exceeding unity. For N-flation, stochastic regime is turned on at much lower amplitudes of density fluctuations.

Note that for both cases amplitude of density fluctuations exceeding unity=eternal inflation. This is a very generic and model-independent but (unfortunately) not very well known statement.

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99. Eternal inflation with many light scalar fields

In the global coordinates the metric of the space is given by

(1)

where is the metric of -dimensional sphere. Let us consider a free massive scalar field on the de Sitter background. General solution of the Klein-Gordon equation

(2)

can be represented as a sum over spherical harmonics

(3)

where the functions satisfy the equation

(4)

Let us introduce

(5)

In this post, we only consider the case when is large enough for the expression above to be real (i.e., the case of heavy scalar field).

By substitution

(6)

this equation is transformed into the hypergeometric form

(7)

In-modes (corresponding to the absence of particles at , i.e., at ) are given by

(8)

At one has

(9)

so that in-modes are positive frequency modes at .

Since the equation (4) is symmetric with respect to the transformation , out-modes (corresponding to the absence of particles at , i.e., at ) can be immediately identified as

(10)

so that

(11)

At one has

(12)

so that out-modes are positive frequency modes at . Normalization of both in- and out-modes is easily found to be

. (13)

As one may notice, both in- and out-modes are divergent at : according to the Raabe criterion, hypergeometric series defining the functions

(14)

diverge there for any and . Therefore, strictly speaking, we are not allowed to calculate matrix elements between in- and out-modes directly; instead, we have to introduce some modes regular at to calculate the Bogolyubov coefficients and separately. As we see, something interesting happens near the throat of the de Sitter hyperboloid.

(Note that Strominger, Bousso and Maloney calculate Bogolyubov coefficients between and directly, which is not fare way to do the caclulation from my point of view.)

To show how particles are created between and infinities, any choice of modes is suitable. However, to demonstrate the physical essence of instability in de Sitter space, we choose Euclidean modes as , since they have a remarkable property of CPT-invariance

(compare it with the condition (10).

Normalized Euclidean modes have the form

(11)

where

. (12)

Using properties of the hypergeometric functions, we immediately find that

, (13)

where the Bogolyubov coefficients are

(14)

(15)

where

(16)

and the scalar product is defined as usual:

(17)

(we integrated over angles of and used orthogonalitity of spherical harmonics).

The Bogolyubov coefficients between Euclidean and out-modes

(18)

are simply related to (14), (15). Indeed, one finds

(19)

After a trivial calculation we conclude that

. (20)

Therefore, there is no particle production in de Sitter space (in- and out-vacua coincide) if . (21)

From the expressions for Bogolyubov coefficints we immediately see that

(22)

Therefore, if is odd, interference between and parts of de Sitter is desctructive, in- and out-vacua coincide (there is no overall particle production), and de Sitter space is stable. On the other hand, if d is even, interference between and part of de Sitter is constructive, and de Sitter space should be unstable. The distribution of particles produced in the throat is independent of the angular momentum and is given by

, (23)

so that the total number of produced particles strongly diverges. It would be nice to see the instability of odd-dimensional de Sitter space at the level of Green’s functions though, and we will show it but not this time

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58. Stability of de Sitter space: dS as a perfect interferometer

The reason why I am so much excited about the subject that I was quiet for the whole week (or so?) is this paper by Alexander Polyakov.

We all know that SYM at large ‘t Hooft coupling  is in one-to-one correspondence to SUGRA on . AdS curvature is related to the ‘t Hooft coupling as

.

On the other hand, Coulomb interaction between two heavy quarks in SYM plasma is proportional to .

Suppose we want to do an analytic continuation from AdS to dS on the SUGRA side. Most probably, we will loose the duality but let us for a moment suppose that it is not so, and there exists some gauge theory dual to SUGRA on dS background. Since

,

one has

i.e., at continuation from AdS to dS square root of the ‘t Hooft coupling changes sign, and so does the Coulomb interaction between two heavy particles in the dual gauge theory plasma! Therefore, we expect Dyson instability in such a plasma, dual gauge theory is unstable (non-unitary, recall what different people spoke about during the end of dS/CFT era).  This instability should show itself on the SUGRA side, too. In particular, one may expect that QFT on dS background should be unstable.

Taking into account that, as Polyakov proposed, QFT on dS background should be dual to the ‘t Hooft’s planar limit of the YM with complex coupling, we really want to understand the nature of this instability and describe it.

(Note that this is not what cosmologists working with QFT in dS space are aware of – in cosmology, say, in inflationary models, QFT matrix elements are calculated in Bunch-Davies vacuum and naively no sign of dS instability is seen anywhere. In a couple of post I will show what is the ultimate reason for this.)

So, let us forget about these duality considerations and focus on QFT (namely, free massive scalar field theory) in de Sitter space. First of all, we will want to work in global coordinate system that covers de Sitter completely. Let us set curvature radius of the de Sitter space to 1.

Without much hustle one can show that a general two point function for the massive scalar field in the d-dimensional de Sitter space has the form:

. (1)

Let me explain different terms in this expression. First of all, is of course the hypergeometric function “Conformal weights” are defined according to

(so that we are talking about sufficiently light scalar fields in de Sitter space)

and , where is a geodesic distance between the points and .

There are two terms in the expression (1) – the values of the constants depend on which of Allen-Mottola vacua you choose. For example, the choice  corresponds to the choice of favorite cosmologist’s “Euclidean” Bunch-Davis vacuum etc.

Expression (1) is especially interesting because of its sungularities. Namely, at (coincident points, ) it has usual singularity similar to the one QFT in Minkowski spacetime has. There is however new singularity: the second term in (1) gets singular at , i.e., for antipodal points. 

What is the physical meaning of this singularity? Usually people prefer to say that it should be unphysical since it is beyond dS horizon for any given observer. To cut this singularity off, string theorists even invented the notion of elliptic de Sitter space. As we will see, this is not the smartest way to deal with this singularity (in particular, when one calculates linear response, one finds that both singularities – standard UV and antipodal give contributions into the final answer).

Second interesting thing is that the general two point correlation function has brunch cuts at and . Physical meaning of them should be also quite unclear to you at this point :-)

Finally, as it turns out, for odd (odd-dimensional de Sitter space) a very complicated expression (1) can be expressed in terms of elementary functions. Namely, for one has

.

Does the fact that (1) can be expressed in so simple form have some physical meaning? As we will see, it most definitely does.

Although these are important questions we will want to answer, the most interesting issue – the issue of instability of the even-dimensional de Sitter space – is not seen at the level of Green’s function (1) and I will have to formulate this part of the problem separately.

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57. Stability of de Sitter space: statement of the problem 1

As we know, there are two descriptions of the Brownian motion: in terms of the Langevin equation and in terms of the Fokker-Planck equation describing diffusion of the probability distribution to find a randomly moving particle at given in a given moment of time . This Fokker-Planck equation has the form

. (1)

The probability distribution describes how the values of are distributed among different Hubble patches in the multiverse and how they are correlated with each other in different Hubble patches.

The general solution to the Fokker-Planck equation (1) is given by

, (2)

where and are respectively the eigenfunctions and the eigenvalues of the effective Hamiltonian

. (3)

Here

(4)

is a functional of the scalar field potential . It is often denoted as the superpotential due to its “supersymmetric” form: the Hamiltonian (3) can be rewritten as

(5)

where

(6)

with

(7)

The eigenfunctions and eigenstates of the Hamiltonian (3) satisfy the Schrodinger equation

(8)

Its solutions have the following features:

1. The eigenvalues of the Hamiltonian (3) are all positive definite due to the supersymmetric form of . Assuming normalizable wavefunctions , the ground state corresponds to the zero eigenvalue and defines the steady state solution of the Fokker-Planck equation. One can easily check that the ground state has the form

(9)

where is defined from the normalization condition.

2. The contributions from eigenfunctions of excited states to the solution (2) become exponentially quickly damped with time.

Arbitrary correlation functions of the inflaton field can be easily found given the probability distribution . Namely,

(10)

where .

In the limit only the ground state in the solution (2) survives, and one can write

(11)

This expression in principle completely defines the asymptotic structure of the distribution of in spacetime (and the structure of the spacetime itself). If we are interested in finite behavior, the situation becomes more complicated because the constants in the general solution (2) are defined as convolutions of with the distribution function determining the structure of the spacetime in the initial moment of time, and we can hardly determine such a thing from observations.

Finally, let me make some concluding remarks. First of all, we notice that the stochastic formalism works well exactly in the regime when quantum fluctuations of the inflaton field become stronger than the effect of the classical force – i.e., in the regime where stochastic force in the Langevin equation is stronger than the lassical force. Therefore, stochastic approach is suitable for the description of physics where the standard inflationary perturbation theory breaks down. It also describes the structure of spacetime at superlarge scales. This structure according to the expression (11) is rather special – the hypersurface of constant in the spacetime is strongly fluctuating at very long scales and the distribution of these fluctuations is very far from being Gaussian (compare it to what happens at smaller scales where quantum fluctuations of the inflaton are suppressed with respect to the effect of the classical force – at the present horizon scale fluctuations are almost exactly gaussian!). In fact, one can show that the hypersurface of the constant value of the inflaton field is fractal, but this is beyond the scope of our present discussion.

Finally, let me introduce a couple of exercises which will be a good warm-up if you are willing to learn the physics of eternal inflation better.

Problem: chaotic inflation with quadratic potential

Consider a chaotic inflationary model with potential , derive the corresponding Lanfevin and Fokker-Planck equations. Hint: instead of the world time use the number of efolds .

Problem: average length of inflationary stage

Calculate the average number of efoldings for the model with potential . Note that inflation ends in the regime when stochastic random force f has negligible effect on the evoluton of the inflaton field.

Problem: non-gaussianities from superlarge scales

According to the formalism, the curvature perturbation can be defined as . Show how to calculate the arbitrary one-point correlation function of the form using the stochastic approach.

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55. Eternal inflation: stochastic approach 3 (Inflationary perturbations 7)