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

Ok, friends, I feel that the time has come to let you know about things I am currently involved in – namely, understanding of intrinsic stability of the de Sitter space.

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.