25. Geometry and causal structure of de Sitter space (Inflationary perturbations 4)

Since we are interested so much to understand physics in de Sitter space, let us take a closer look on its geometry and causal structure.

Geometry of -dimensional de Sitter space (spacetime of constant positive curvature with maximal allowed symmetry which is ) can be understood as the geometry of the hypersurface

embedded into -dimensional Minkowski space with metric given by

(1)

The latter expression for the linear element in de Sitter space is horrable and in order to simplify it one can introduce several coordinate parametrizations of the de Sitter space. Let us focus on the case (so that there are only two independent coordinates, one is timelike and another is spacelike) and review them.

A. Global coordinates and Penrose diagram

Global coordinates are found by choosing the following parametrization for :

(2.1)
(2.2)
(2.3)

where and . The linear element (1) for this parametrization is given by

Let us take a closer look on the following picture representing 2-dimensional de Sitter space
:

As you can see, surfaces are flat, and the crossection of by such a plane is given by a circle with parametrizing it.

The global coordinate system covers the whole de Sitter space (d-dimensional generalization of the parametrization (2) is straightforward), that is why this coordinate system is called “global”

Causal structure of the de Sitter space can be understood after making a substitution

,

where . In new coordinates the linear element acquires the form

so that de Sitter space becomes conformally equivalent to Minkowski space in new coordinates . Null geodesics are defined by the equation

so that the Penrose diagram has the form represented in the following picture:

(note that the light ray starting at at only reaches at , so that the actual, complete, Penrose diagram of consists of two squares like the one presented above: one with and another — with ).

What’s important on the picture above for good understanding of the QFT in de Sitter space?

• as you can see, no single observer can access entire de Sitter space
• observer sitting at is only able to exchange signals with somebody inside the southern diamond (it will take infinite time for him to receive feedback from another observer sitting at )
• observer sitting at can only send signals to southern diamond + future triangle

B. Static coordinates

This is the most favorite string theorist’s coordinate system; in a moment you will understand why.

Static coordinate system covering is obtained by setting the parametrization

(3.1)
(3.2)
, (3.3)

so that the linear element acquires the form

The name “static” comes from the fact that the vectors are Killing vectors of the in this coordinate system, and it looks like nothing intersting dynamically happens with QFT in in this coordinate system (and that is why string theorists like it we will see that life is much more complicated though that this static picture).

As one can see from the representation (3) and the Fig. above, static coordinate system covers only quarter of the de Sitter space.

As it follows from the Penrose diagram, lines represent horizon for an observer living at , and the modes of quantum fields in static coordinate system strongly oscillate in the vicinity of . However, geometrically, we see that is just a single point on the hyperboloid corresponding to , so the singularity of the QFT Green functions here is unphysical (it is due to the fact that a single point corresponds to a single value of and arbitrary value of ).

C. Planar coordinates

This one is in turn the cosmologist’s most favorite patch of de Sitter space. Planar coordinates are determined as parametrization

,

so that the linear element in planar patch is given by

(4)

This coordinate system covers only half of de Sitter (on the Penrose diagram it corresponds to the past triangle + southern diamond). Another half is described by the metric (4) with .

hypersurfaces are planes, is again a coordinate singularity where quantum modes diverge: