10. Introduction into perturbation theory in general relativity 3 (Inflationary perturbations 3)

Posted by Dmitry
March 28, 2008

Sorry, but today’s post is going to be rather short: the end of this week is a lot busier than usual, since I am trying to finish a couple of papers. One of them is about intrinsic instability of de Sitter spacetime (became a hot subject recently; also, it is one of the most important problems in cosmology and its relation to string theory from my point of view). Another paper is related to the physics of eternal inflation on the landscape. As long as I’ll finish these two both subjects will be discussed extensively in this blog :-)

Now let me introduce a solution for one of the problems I offered in the previous post; namely, I asked whether it is possible to construct scalar and vector modes which are gauge-invariant, i.e., they do not change under infinitesimal coordiate transformations

x_i'=x_i + \xi_i. (1)

Gauge invariant perturbations

The answer is positive: it is easy to see from the transformation law for the metric tensor that the scalars

\Phi=\phi+\frac{1}{a}\left((B-E')a\right)',

\Psi=\psi-\frac{a'}{a}(B-E')

and the vector

V_{i}=S_{i}-F_{i}'

remain unchanged under the infinitesimal coordinate transfomations (1) and are therefore gauge invariant (the modes \Phi and \Psi are called sometimes Bardeen potentials).

Why should we bother about gauge invariant perturbations in cosmology? The reason is that gauge invariant modes describe physical degrees of freedom of the system “gravitational field + matter”, degrees of freedom that cannot be removed by a coordinate transformation.

Let me also remind you that we were dealing so far with linear perturbation theory.
If one takes non-linearities of gravity into account, expressions for the gauge invariant modes become more and more cumbersome as more and more orders of perturbation theory are taken into account. In the regime of strong coupling, when perturbations are of the order 1, it is in fact possible to show that there are no dynamical gauge invariant degrees of freedom. This statement can be also rephrased as follows: all possible gauge invariant degrees of freedom in full theory of gravity are constants. This is a reflection of the fact that there are no local observables in the full theory of gravity.

Let us go back to the linear analysis. One can check that in the longitudinal gauge \Phi=\phi and \Psi=\psi, so it is always convenient to do all calculations in the longitudinal gauge to get understanding of what happens with gauge-invariant variables.
Basically, to turn to the gauge invariant answer, one needs to do a mapping \phi \to \Phi and \psi \to \Psi in the end of calculation in longitudinal gauge. This made the longitudinal gauge extremely popular among cosmologists :-)

Let me also note that the physical meaning of the gauge-invariant scalars \Phi and \Psi is extremely clear: both of them are reduced to the Newtonian potential \varphi in the non-relativistic limit c \to \infty.

Next time we will discuss dynamics of the gravitational and matter modes in the longitudinal gauge.

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Cosmological perturbations and large scale structure, Cosmology, Master Mason

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