NEQNET: Non-equilibrium Phenomena

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

Posted by Dmitry

at April 23, 2008

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 O(d,1) ) can be understood as the geometry of the hypersurface

$-X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+\ldots+X_{d}^{2}=1$

embedded into (d+1) -dimensional Minkowski space with metric given by

$ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+\ldots+dX_{d}^{2}=$
$=-dX_{0}^{2}+dX_{1}^{2}+\ldots$
$+dX_{d-1}^{2}+$
$\left(d(1+X_{0}^{2}-X_{1}^{2}-\ldots-X_{d-1}^{2})^{-1/2}\right).$ (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 d=2 (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 $(\tau,\phi)$ are found by choosing the following parametrization for $X_{0,1,2}$ :

$X_{0}={\rm sinh}\,\tau,$ (2.1)
$X_{1}=\cos\phi{\rm cosh}\tau$ (2.2)
$X_{2}=\sin\phi{\rm cosh}\tau,$ (2.3)

where $0\le\phi\le2\pi$ and $-\infty<\tau<+\infty$ . The linear element (1) for this parametrization is given by

$ds^{2}=-d\tau^{2}+{\rm cosh}^{2}\tau\cdot d\phi^{2}.$

Let us take a closer look on the following picture representing 2-dimensional de Sitter space
dS_2 :

de Sitter in global coordinates

As you can see, surfaces $\tau={\rm Const}$ are flat, and the crossection of $dS_{2}$ by such a plane is given by a circle with $\phi$ 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

${\rm cosh}\tau=\frac{1}{{\rm cos}T}$ ,

where $-\frac{\pi}{2}<\frac{\pi}{2}$ . In new coordinates the linear element acquires the form

$ds^{2}=\frac{1}{{\rm cos}^{2}T}\left(dT^{2}+d\phi^{2}\right),$

so that de Sitter space becomes conformally equivalent to Minkowski space in new coordinates $(T,\phi)$ . Null geodesics are defined by the equation

$T={\rm Const}\pm\phi$

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

Penrose diagram of de Sitter space

(note that the light ray starting at $\tau=-\infty$ at $\phi=\pi$ only reaches $\phi=0$ at $\tau=+\infty$ , so that the actual, complete, Penrose diagram of $dS_{2}$ consists of two squares like the one presented above: one with $0<\phi<\pi$ and another — with $\pi<\phi<2\pi$ ).

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 $\phi=\pi$ 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 $\phi=\frac{\pi}{2}$ )
observer sitting at $\phi=\pi$ 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.

de Sitter in static coordinates

Static coordinate system covering $dS_{2}$ is obtained by setting the parametrization

$X_{0}=\sqrt{1-r^{2}}{\rm sinh}t,$ (3.1)
$X_{1}=r,$ (3.2)
$X_{2}=\sqrt{1-r^{2}}{\rm cosh}t$ , (3.3)

so that the linear element acquires the form

$ds^{2}=-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}.$

The name “static” comes from the fact that the vectors $\frac{\partial}{\partial t}$ are Killing vectors of the $dS_{2}$ in this coordinate system, and it looks like nothing intersting dynamically happens with QFT in $dS_{2}$ 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 r=1 represent horizon for an observer living at $\phi=\pi$ , and the modes of quantum fields in static coordinate system strongly oscillate in the vicinity of r=1 . However, geometrically, we see that r=1 is just a single point on the $dS_{2}$ hyperboloid corresponding to $X_{0}=X_{2}=0$ , so the singularity of the QFT Green functions here is unphysical (it is due to the fact that a single point $X_{0}=X_{2}=0,\, X_{1}=1$ 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

$X_{0}=\frac{1}{2}\left(e^{t}-e^{-t}(1+x^{2})\right),$

$X_{1}=xe^{-t},$

$X_{2}=\frac{1}{2}\left(e^{t}+e^{-t}(1-x^{2})\right)$ ,

so that the linear element in planar patch is given by

$ds^{2}=dt^{2}-e^{-2t}dx^{2}.$ (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 $t\to$ .

$t={\rm Const}$ hypersurfaces are planes, $t=+\infty$ is again a coordinate singularity where quantum modes diverge:

de Sitter in planar coordinates

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Lectures on de Sitter spacetime, Quantum field theory, String theory

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24. Inflation: description in terms of hydrodynamics (Inflationary perturbations 4)

Posted by Dmitry

at April 23, 2008

This post continues my lecture notes on large scale structure, cosmological perturbations and inflation. The last post in this series was about the problem of initial conditions in FRW cosmology. As we have found out, standard Hot Big Bang model with power law expansion $a\sim t^p$ is unable to explain present degree of homogeneity and isotropy of the Universe, so power law expansion of the Universe cannot be complete picture.

Let us recall that both homogeneity and spatial flatness problems of the initial condition are related to the fact that the ratio $\frac{\dot{a}_{0}}{\dot{a}_{i}}$ is extremely small. Therefore, unnaturalness of the initial condition can be avioded if we suppose that the earlier stage of the evolution of the Universe was characterized by $\frac{\dot{a}_{0}}{\dot{a}_{i}}\gg1$ or in the other words, $\ddot{a}>0$ , i.e., that the Universe was initially expanding with acceleration. How to construct a model of accelerated Universe?

Let us write down the Friedmann equation

$\frac{\ddot{a}}{a}=-\frac{4\pi}{3M_{P}^{2}}(\rho+3p)$ (1).

As we see, it is only possible to have $\ddot{a}>0$ if the pressure is negative: $p\lesssim-\frac{1}{3}\rho$ . This should correspond to the Universe filled with very special matter, with a built-in, intrinsic instability - the more pressure we put on the matter, the more it wants to collapse and vise versa. This kind of instability is naively very dangerous: for example, if one considers only the class of hydrodnamic models, where pressure $p=w\rho$ is proportional to the energy density, than negative pressure would mean negative w and imaginary speed of sound $c_{s}$ .

As we will see later, this paradox can be easily resolved if one recalls when description in terms of hydrodynamic degrees of freedom is adequite: it only works at length scales much larger than the mean free path $l_{{\rm MFP}}$ for elementary (or collective) excitations in matter; only in this case excitations in such a matter can be described as sound waves. For matter with negative pressure the mean free path is always larger than the horizon scale $l_{H}$ , so that the description of such matter in terms of hydrodynamics breaks down, and one has to use microscopic language of the quantum field theory instead.

Let us forget about this issue for a moment and try to learn something about accelerated Universe at the level of Eq. (1). First of all, we notice that the domination of positive cosmological constant would automatically provide a stage of accelerated expansion. Indeed, the energy-momentum tensor for the cosmological constant is

$T_{\alpha}^{\beta}\sim\Lambda\delta_{\alpha}^{\beta},$

so that the cosmological constant term represents a matter with negative pressure $p=-\rho=-\Lambda$ . From the Friedmann equation we immediately see that

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\Lambda,$

so that the Universe expands exponentially rapidly:

$a(t)=a_{i}\exp(\Lambda^{1/2}t)=a_{i}\exp(Ht)=a_{i}\exp(N)$ ,

where is the number of e-folds (number of steps such that the Universe expands e times during each step). Such an exponentially growing solution of the Einstein equations is known as de Sitter or inflationary universe (we will discuss it in much more details later). If there is an inhomogeneity in the de Sitter universe with characteristic scale $\lambda_{i}$ present at $t=t_{i}$ , it will be exponentially stretched to the size $\lambda=\frac{a(t)}{a_{i}}\lambda_{i}$ at time , and $\lambda$ will exponentially rapidly cross the length scale $H^{-1}\sim\Lambda^{-1/2}$ for any given $\lambda_{i}$ . On the other hand, the exponentially rapid stretching of scales does not contradict causality since the particle horizon size $l_{p}$ is much larger than the scale $H^{-1}$ in the de Sitter universe -

$l_{p}=a(t)\int_{a_{i}}^{a(t)}\frac{da}{\dot{a}a}=H^{-1}(\exp(H(t-t_{i}))-1)$

- so that it grows as rapidly as any physical length scale $\lambda$ in dS.

When accelerated stage of expansion ends giving rise to the FRW universe with deccelerated expansion, the particle horizon rapidly shrinks to the scale $H^{-1}$ . It is therefore possible to explain homogeneity of the Universe at superhorizon scales adding a stage of accelerated expansion in the history of early Universe. Similarly, isitropy problem of the standard Big Bang cosmology is solved: the amplitude of vector modes behaves as $a^{-1}$ in the expanding Universe and therefore exponentially quickly decays in a de Sitter-like universe. Spatial flatness of the post-de Sitter universe can be explained by the fact that the r.h.s. of the Eq. (1) decreases exponentially quickly during de Sitter stage so that in the end $\Omega\sim1$ with exponential precision. Finally, entropy problem is solved by considering the thermalization of the decay products of the effective cosmological constant $\Lambda$ .

In principle, why should accelerated expansion stage end? Clearly, eternally expanding de Sitter universe as it is cannot be suitable for the description of observable Universe simply beause it is necesserily empty and cold: in the de Sitter universe the energy density of non-relativistic matter behaves as $\rho_{{\rm dust}}\sim a^{-3}\sim\exp(-3Ht)$ , energy density of ultrarelativistic matter - as $\rho_{{\rm ur}}\sim a^{-4}\sim\exp(-4Ht)$ , while temperature of hot plasma - as $T\sim a^{-1}\sim\exp(-Ht)$ . Therefore, cannot be constant permanently: if the de Sitter epoch was realized in the very early Universe, cosmological constant $\Lambda$ should somehow decay. Just from the definition of the Hubble parameter one has

$\frac{\ddot{a}}{a}=H^{2}+\dot{H}$ ,

so the Sitter stage can be realized if $|\dot{H}|\ll H^{2}$ (this condition is related to the slow roll condition, and it is of extreme importance in cosmology). Since $H^{2}$ is always positive, accelerated expansion will end to give rise to the deccelerated one if and only if the running of the Hubble parameter $\dot{H}$ is negative. When de Sitter stage is getting closer to its end, $|\dot{H}|$ approaches the value of $H^{2}$ , and finally, when $|\dot{H}|\sim H^{2}$ , acceleration $\ddot{a}$ changes its sign.