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

Let us now show that QFT of a massive scalar field in de Sitter space features instabilities if the number of dimensions is odd. The expression for the two-point function found in the previous post will be of no help, so we will have to switch to the language of Bogolyuov coefficients and modes.

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