NEQNET: Non-equilibrium Phenomena

9. Introduction into perturbation theory in general relativity 2 (Inflationary perturbations 3)

Posted by Dmitry

at March 27, 2008

Today I will continue our discussion of the perturbation theory in general relativity (please see the previous post here); in particular, I will focus my attention on the issue of gauge invariance.

1. Gauge transformations

Let us consider how the metric modes we have constructed in the previous post change under coordinate transformations. We introduce an infinitesimal (i.e., $\xi^{\alpha}\to0$ ) coordinate transformation

$\tilde{x}^{\alpha}=x^{\alpha}+\xi^{\alpha}(x^{\beta}),$ (1)

under which the metric tensor is transformed as

$\tilde{g}_{\alpha\beta}(\tilde{x})=\frac{\partial x^{\gamma}}{\partial\tilde{x}^{\alpha}}\frac{\partial x^{\delta}}{\partial\tilde{x}^{\beta}}g_{\gamma\delta}(x)\approx$

$\approx g_{\alpha\beta}^{(0)}(x)+\delta g_{\alpha\beta}-g_{\alpha\delta}^{(0)}\xi_{;\beta}^{\delta}-g_{\beta\delta}^{(0)}\xi_{;\alpha}^{\delta}=$

$=\tilde{g}_{\alpha\beta}^{(0)}(\tilde{x})+\delta\tilde{g}_{\alpha\beta}(\tilde{x}).$

Therefore,

$\delta\tilde{g}_{\alpha\beta}=\delta g_{\alpha\beta}-g_{\alpha\beta,\gamma}^{(0)}\xi^{\gamma}-g_{\alpha\delta}^{(0)}\xi_{;\beta}^{\delta}-g_{\beta\delta}^{(0)}\xi_{;\alpha}^{\delta}.$

Exercise 2.4. What is the origin of the second term in the right hand side?

Let us represent the 3-component of the 4-vector $\xi^{\alpha}$ in the form

$\xi^{i}=\xi_{{\rm df}}^{i}+\zeta^{,i},$

where $\xi_{{\rm df}}^{i}$ is the divergence free part of $\xi^{i}$ . Than, one has

$\delta\tilde{g}_{00}=\delta g_{00}-2a(a\xi^{0})',$

$\delta\tilde{g}_{0{}i}=[/\delta g_{0i}+a^{2}(\xi_{{\rm df}\,;i}'+(\zeta'-\xi_{0})_{,i}),$

$\delta\tilde{g}_{ij}=\delta g_{ij}+a^{2}\left(2\frac{a'}{a}\delta_{ij}\xi_{0}+2\zeta_{;ij}+(\xi_{{\rm df}\, i;j}+\xi_{{\rm df}\, j;i})\right),$

where prime denotes derivative with respect to conformal time $\eta$ . Now one can understand how various modes present in the metric tensor are transformed.

We start with scalar modes. Their contribution into the overall perturbed metric is given by

$ds^{2}=a^{2}\left((1+2\phi)d\eta^{2}+2B_{,i}d\eta dx^{i}-\left((1-2\psi)\delta_{ij}-2E_{;ij}\right)dx^{i}dx^{j}\right),$

so one finds find that

$\tilde{\phi}=\phi-\frac{1}{a}(a\xi^{0})',$

$\tilde{\psi}=\psi+\frac{a'}{a}\xi^{0},$

$\tilde{B}=B+\zeta'-\xi^{0},$

$\tilde{E}=E+\zeta$

Exercise 2.5. Check these transformation rules.

Exercise 2.6. Construct scalars which do not change under coordinate transformations (1).

Similarly, vector modes give the following contributions into the overall metric:

$ds^{2}=a^{2}\left(d\eta^{2}+2S_{i}d\eta dx^{i}-\left(\delta_{ij}-F_{i;j}-F_{j;i}\right)dx^{i}dx^{j}\right),$

and one has for the vector perturbations

$\tilde{S}_{i}=S_{i}+\xi_{{\rm df}\, i}',$

$\tilde{F}_{i}=F_{i}+\xi_{{\rm df}\, i}.$

Exercise 2.7. Construct vector which does not change under coordinate transformations (1).

Finally, one can find that the tensor mode $h_{ij}$ does not change under coordinate transformations (1).

2. Important gauges

Discussing various possible gauges, we will focus our attention on scalar perturbations, because these are the ones the most influenced by the choice of the gauge.

a) Synchronous gauge

Since in 4-dimensional world we are allowed to do four coordinate transformations, by these transformations we can always choose

$g_{00}=1,$ $g_{0i}=0,$ (2)

which can be recast in the form $\phi=0,$ B=0 using scalar modes we introduced above.

The resulting metric will look like

$ds^{2}=dt^{2}-\gamma_{ij}(t,{\bf x})dx^{i}dx^{j},$

where $dt=ad\eta$ .

The gauge defined by these conditions is called synchronous, since in this gauge clocks everywhere in the universe (or, more accurately, in a given Hubble patch) are synchronized. Conditions (2) do not fix the gauge completely, because the freedom remains to make arbitrary 3-dimensional coordinate transformations.

b) Longitudinal (or newtonian) gauge.

This gauge is fixed by the conditions B=0, E=0 , so that the metric looks like

$ds^{2}=a^{2}(1+2\phi)d\eta^{2}-a^{2}(1-2\psi)\delta_{ij}dx^{i}dx^{j}.$ If $T_{i}^{j}\propto\delta_{i}^{j}$ , the scalar modes $\phi$ and $\chi$ coincide. The variable \phi in this gauge reduces to the Newtonian potential $\varphi$ in the non-relativistic limit.

In the next post I will discuss gauge invariant variables and their dynamics.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft

11. Introduction into perturbation theory in general relativity 4 (Inflationary perturbations 3)

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

7. Newtonian perturbation theory 2 (Inflationary perturbations 2)

52. Introduction to non-gaussianities (Inflationary perturbations 6)

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

Posted by Dmitry

at March 25, 2008

Before turning to the discussion of the perturbation theory in general relativity, let me briefly remind the outline of the two previous posts (1 and 2). As we have found, Newtonian perturbation theory works well at relatively small scales (much less than hundred of MPcs), but predicts exponential growth of infrared adiabiatic modes. We have concluded that a) our description of IR physics is incomplete, and b) nonlinear interactions between Fourier modes should be taken into account at $t\to\infty$ . Nonlinearities stop exponential growth of IR adiabatic modes at later times, even if we describe IR physics by means of Newtonian perturbation theory. However, in this case IR modes $\delta\rho_{k}$ become large too rapidly.

At the next step we took expansion of the Universe into account (the later leads to a noticeable change of physics only at large scales - the ones we were exactly after) and found that the exponential growth of IR adiabatic modes is replaced by the much slower polynomial growth. Thus, we, from one hand, somewhat cured tachyonic instability of the Newtonian linear perturbation theory. On the other hand, we have qualitatively explained the LSS of the Universe at subhorizon scales (see the previous post).

Nevertheless, even taking the expansion of the universe into account, we did not treat the physics of IR modes in completely satisfactory manner: by heart, we know that general relativity is necessary to properly explain the large scale structure of the Universe but not a single bit of it was used in the discussion so far. So, where was the flow in our logic?

In fact, there were several flows. As we will see later, all of them are interrelated.

1. the physical volume of the Universe shrinks to zero at $t\to0$ (indeed,physical coordinate is ${\bf x}=a(t){\bf q}$ and $a\to 0$ at $t\to0$ . It means that in the very beginning of its evolution the Universe was supposed to be very small or, in other words, its curvature was very large.

2. If one interpolates the present scale of inhomogeneities to the past, one finds that in the early universe $\frac{\delta\rho}{\rho_{0}}$ was very small. Why this was the case realized in our Universe? To explain all the features in the observable Universe, one has not only to construct equations describing the evolution of the perturbations, but to find and explain initial conditions for them.

3. Knowing GR, we remember that the curvature scale of the space-time is $R\sim H^{2}$ . Therefore, at scales $l\gg l_{H}\sim H^{-2}$ or, in the other words, at superhorizon scales effects of GR should be taken into account.

4. Dynamics of superhorizon modes is not completely decoupled from the dynamics of subhorizon ones. The reason is that in the universe filled with a fluid (having the equation of state $p_{0}=w\rho_{0}$ with w<1 ) superhorizon modes reenter the horizon. Indeed, the horizon size behaves as $H^{-1}\sim t$ , while the physical wave length of the cosmological perturbations behaves as $\frac{a}{k}\sim t^{\frac{2}{3(1+w)}}$ .

Exercise 2.1. Try to construct the cosmological model (universe filled with an ideal fluid) when the opposite happens, and subhorizon modes leave the horizon. What is the effective equation of state of the fluid?

As we see, to properly treat the IR superhorizon modes, one needs a description in terms of general relativity. It is clear that analysis of perturbation theory in GR is going to be much more complicated. Indeed, in the Newtonian analysis (and its extension for the case of expanding Universe) the gravitational field was only described by the potential $\varphi$ . Instead, in general relativity metric describing behavior of space-time is 2nd rank tensor, containing several possible modes. There is another important issue explaining why it is hard to get as clear picture in the general relativity as the one emerged from the Newtonian perturbation theory analysis: the issue of gauge invariance.

Let us suppose that we live in a unperturbed universe filled with perfect fluid with energy density $\rho_{0}(t)$ . If one makes a coordinate transformation $t'=t+\delta t(t,{\bf x})$ , one finds that the energy density profile changes as $\rho_{0}(t'-\delta t(t,{\bf x}))\approx\rho_{0}(t')-\frac{\partial\rho_{0}(t')}{\partial t'}\delta t(t',{\bf x})$ - and it looks like we have an inhomogeneity in the new coordinate system. This inhomogeneity is however nothing but a gauge artifact — the fluid remains homogeneous as seen in the properly chosen coordinate system (original one in our case).

To find physical modes of cosmological perturbations, we need both metric and matter degrees of freedom. Let us now describe them in more details.

A. Classifying modes of gravitational perturbations

In the linear approximation, the disturbed invariant interval can be written as

$ds^{2}=(g_{\mu\nu}^{(0)}(\eta)+\delta g_{\mu\nu}(\eta,{\bf x}))dx^{\mu}dx^{\nu},$

where the background contribution is defined by the Friedman-Robertson metric

$ds_{0}^{2}=g_{\mu\nu}^{(0)}(\eta)dx^{\mu}dx^{\nu}=a^{2}(\eta)(d\eta^{2}-d{\bf x}^{2})$

(note that we use conformal time $\eta$ related to the world time according to $dt=a(\eta)d\eta)$ . The background is invariant with respect to 3-dimensional coordinate transformations (rotations, rescalings, etc.), and it is convenient to use this invariance to classify the possible modes of metric perturbations.

First of all, we notice that $\delta g_{00}(\eta,{\bf x})$ is transformed as a 3-scalar with respect to the 3-dimensional coordinate transformations, so one can introduce the scalar mode $\phi$ present in the perturbed metric according to $\delta g_{00}(\eta,{\bf x})=2a^{2}(\eta)\phi(\eta,{\bf x})$ .Therefore, there is one degree of freedom in $\delta g_{00}(\eta,{\bf x})$ .

Mixed components of the metric $\delta g_{0i}(\eta,{\bf x})$ contain both 3-scalar mode B and 3-vector mode $S_{i}$ defined according to the prescription $g_{0i}(\eta,{\bf x})=a^{2}(\eta)(B_{,i}+S_{i})$ . It is convenient to choose the vector perturbation $S_{i}$ to be divergence free, i.e., $S_{;i}^{i}=0$ . If $S_{i}$ is not divergence free, it is always possible to represent it in the form $S_{i}=\tilde{S}_{i}+\chi_{,i}$ , where $\tilde{S}_{i}$ is divergence free and $\chi$ is a 3-scalar, which can be absorbed into . Thus, there are 3 degrees of freedom in $\delta g_{0i}(\eta,{\bf x})$ : one scalar mode and two components of the vector mode.

Finally, the spatial components of the metric $\delta g_{ij}(\eta,{\bf x})$ contain two scalar modes $\psi$ and , vector mode $F_{i}$ and tensor mode $h_{ij}$ defined as

$\delta g_{ij}(\eta,{\bf x})=a^{2}(\eta)(2\psi(\eta,{\bf x})\delta_{ij}+$

$+2E_{,i;j}(\eta,{\bf x})+F_{i;j}(\eta,{\bf x})+F_{j;i}(\eta,{\bf x})+h_{ij}(\eta,{\bf x})).$

Again, it is convenient to choose the 3-vector $F_{i}$ to be divergence free (otherwise, the remaining scalar part can be absorbed into the definition of $\psi$ ) and the 3-tensor $h_{ij}$ to satisfy the constraints $h_{i}^{i}=0$ , $h_{ij}^{;j}=0$ .The first constraint means that the tensor mode is traceless (if it is not, we can always make it traceless by the redefinition of $\psi$ ) and the second constraint - that the tensor mode is transverse (if it is not, we can make it transverse redefining the 3-vector ). As we see, spatial components $\delta g_{ij}(\eta,{\bf x}$ ) of the metric tensor contain 6 independent modes: 2 scalars, 2 vector modes and 2 tensor modes.

The overall number of independent modes is 10, as it should be for a metric tensor describing a general 4-dimensional space-time. However, we know that not all of these modes are physical, since one can choose a coordinate system she likes. We will return to the discussion of physical modes in the next Sections.

Exercise 2.2. Reproduce the same analysis for the perturbations near d-dimensional FRW background. How many scalar, vector and tensor modes exist in this case?

Exercise 2.3. Consider a 10-dimensional space-time with metric

$ds^{2}=e^{2A(r)}(-dt^{2}+d{\bf x}^{2})+e^{-2A(r)}(dr^{2}+r^{2}d\Omega_{5}^{2}),$

where $(t,{\bf x})$ are coordinates of 4-dimensional world and $d\Omega_{5}^{2}$ is the metric of a 5-dimensional sphere. Metrics of this form appear in warped compactifications of the string theory. Classify possible metric perturbations in this system. What are the possible modes of cosmological perturbations for an observer leaving in the 4-dimensional slice of the 10-dimensional world?

At the linear level of the perturbation theory scalar, vector and tensor modes do not interact with each other and can be treated separately. Scalar modes are the most interesting ones since, similar to the Newtonian perturbation theory, they lead to IR instabilities. Vector modes rapidly decay (as $a^{-1}$ ) in the expanding Universe and influence its dynamics only at later stages, when perturbations enter the nonlinear regime. In particular, vector perturbations are responsible for the rotation of galaxies. Finally, tensor perturbations describe gravitational waves and do not affect the large scale structure of the Universe.

Next time we will discuss the issue of gauge invariance in the theory of cosmological perturbations.

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Cosmological perturbations and large scale structure, Cosmology, Master Mason

9. Introduction into perturbation theory in general relativity 2 (Inflationary perturbations 3)

11. Introduction into perturbation theory in general relativity 4 (Inflationary perturbations 3)

7. Newtonian perturbation theory 2 (Inflationary perturbations 2)

52. Introduction to non-gaussianities (Inflationary perturbations 6)

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9. Introduction into perturbation theory in general relativity 2 (Inflationary perturbations 3)

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

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9. Introduction into perturbation theory in general relativity 2 (Inflationary perturbations 3)

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