1

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

Posted by Dmitry
at March 31, 2008

Today I am going to discuss equations driving the dynamics of cosmological perturbations. As you remember, we are considering now gravitational perturbations in general relativity, so these equations are nothing else but Einstein equations in where one keeps only first order terms w.r.t. perturbations of the metric tensor.

I will consider cosmological perturbations in longitudinal gauge since all results will be easily generalized to the gauge invariant case but simpy substituting \phi{}\to{}\Phi, \psi{}\to{}\Psi (please see the previous post).

Cosmological perturbations in longitudinal gauge in the presence of hydrodynamic matter

Let us determine dynamics of scalar modes \phi and \psi in the universe filled with ideal fluid with energy-momentum tensor T_{\alpha}^{\beta}=(p+\rho)u_{\alpha}u^{\beta}-p\delta_{\alpha}^{\beta}, where \rho is the energy density of the fluid, p - its pressure (given by the equation of state p=w\rho) and u_{\alpha} - 4-velocity.

Exercise 2.8. Derive equations of motion for the fluid in the FRW spacetime from the conservation law for the energy-momentum tensor T_{\alpha;\beta}^{\beta}=0.

As I said, this dynamics is completely determined by the Einstein equations

R_{\alpha}^{\beta}-\frac{1}{2}\delta_{\alpha}^{\beta}R=\frac{8\pi}{M_{P}^{2}}T_{\alpha}^{\beta}.

For the FRW background we have

R_{0}^{0}-\frac{1}{2}R=\frac{3(a')^{2}}{a^{4}}\equiv\frac{3{\cal H}^{2}}{a^{2}},

R_{0}^{\alpha}=0,

R_{i}^{j}-\frac{1}{2}R\delta_{i}^{j}=\frac{1}{2}(2{\cal H}'+H^{2})\delta_{i}^{j},

so that the background u_{\alpha}=(a,0,0,0) and the spatial part of the energy-momentum tensor is diagonal: T_{i}^{j}\sim\delta_{i}^{j}.

At the level of linear perturbations we find that

\psi_{;i}^{;i}-3{\cal H}(\psi'+{\cal H}\psi)=\frac{4\pi}{M_{P}^{2}}a^{2}\delta\rho,

(\psi+{\cal H}\phi)_{,i}=\frac{4\pi}{M_{P}^{2}}a(\rho_{0}+p_{0})\delta u_{i},

\left(\psi^{,,}+{\cal H}(2\psi+\phi)'+(2{\cal H}'+H^{2})\phi+\frac{1}{2}(\phi-\psi)_{;i}^{;i}\right)
\delta_{j}^{k}-\frac{1}{2}(\phi-\psi)_{;j}^{;k}=

=\frac{4\pi}{M_{P}^{2}}a^{2}\delta p\delta_{j}^{k}.

Exercise 2.9. Derive these Eqs. by perturbing the Einstein equations directly.

We immediately see that (\phi-\psi)_{;j}^{;k}=0 and \phi=\psi (as I have already explained before, this is always the case when T_{i}^{j}\sim\delta_{i}^{j}). After taking these simplifications into account, we find:

\phi_{;i}^{;i}-3{\cal H}(\phi'+{\cal H}\phi)=\frac{4\pi}{M_{P}^{2}}a^{2}\delta\rho,

\phi^{,,} +3{\cal H}\phi'+(2{\cal H}'+{\cal H}^{2})\phi=\frac{4\pi}{M_{P}^{2}}a^{2}\delta p.

The first Eq. above is the Poisson equation generalized for the case of relativistic dynamics (as can be seen by taking the limit {\cal H}\to0).

Taking into account the thermodynamic relation we can combine these two equations into one:

\phi^{,,} +3(1+c_{s}^{2}){\cal H}\phi'-c_{s}^{2}\phi_{;i}^{;i}+

+(2{\cal H}'+(1+3c_{s}^{2}){\cal H}^{2})\phi=\frac{4\pi}{M_{P}^{2}}a^{2}\sigma\delta S. (1)

Let us as usual focus on the behavior of adiabatic modes setting \delta S=0. Naturally, we will be especially interested to know the behavior of long wavelength modes, since, as we have found discussing the newtonian approximation, one may expect them to grow.

It is impossible to find exact solution of the Eq. (1) in the general case of scale-dependent c_{s}, but to have some qualitative understanding of the dynamics of its solution, let us consider a simplified case c_{s}={\rm Const.} - it corresponds to the universe filled with ideal fluid having the equation of state p=w\rho. The regime of expansion is given by

a(t)\sim\eta^{\frac{2}{1+3w}}

and

{\cal H}=\frac{2}{(1+3w)\eta}.

One can check that the third term in the left hand side of Eq. (1) vanishes, and for a given Fourier mode of the adiabatic perturbation one has

\phi_{k}^{,,}+\frac{6(1+w)}{(1+3w)\eta}\phi_{k}'+wk^{2}\phi_{k}=0.

Let us first see what happens in the deep infrared limit k\to0 where the Eq. (1) acquires the form

\phi_{k=0}^{,,}+\frac{6(1+w)}{(1+3w)\eta}\phi_{k=0}'=0,

and its general solution can be easily found: it is

\phi_{k=0}(\eta)=C_{1}+C_{2}\eta^{-\frac{6(1+w)}{1+3w}}.

As we see, deep infrared adiabatic perturbation has two modes: one of them is decaying, while the second one remains constant. Therefore, taking into account effects of general relativity cures the tachyonic instability of the newtonian perturbation theory completely - IR modes with wavelength larger than the scale of cosmological horizon do not grow.

The solution of the Eq. (1) can also be found for arbitrary values of the momentum k; it is expressed in terms of Bessel functions

\phi_{k}(\eta)=\eta^{-\nu}\left(C_{1}J_{\nu}(\sqrt{w}k\eta)+C_{2}Y_{\nu}(\sqrt{w}k\eta)\right),

where \nu=\frac{5+3w}{2(1+3w)}. In the short wavelength limit \sqrt{w}k\eta=c_{s}k\eta\gg1 adiabatic modes behave as sound waves; their amplitude decreases in the expanding universe as \eta^{-\nu-\frac{1}{2}}:

\phi_{k\to\infty}(\eta)\approx\frac{1}{\eta^{\nu+\frac{1}{2}}}\left(c_{1}e^{ic_{s}k\eta}+c_{2}e^{-ic_{s}k\eta}\right).

In this respect, taking into account effects of GR does not lead to a physical picture very different from what we have before, discussing Newtonian approximation. For any fluid there exists a Jeans length \lambda_{J}\sim c_{s}t such that adiabatic modes with \lambda\gg\lambda_{J} behave monotonically as functions of \eta, while adiabatic modes with \lambda\ll\lambda_{J} behave as sound waves. As we have said, the only important difference is that taking into account effects of GR leads to freeze out of the growing adiabiatic mode at superhorizon scales.

Finally, let us consider physically interesting limit of incompressible fluid (dust) corresponding to the choice w=0. The Jeans length is formally infinite, but the limit w\to0 is very simple: one needs to go back from the Fourier decomposition to the Eq. (1) to find that it is reduced to

\phi^{,,}+\frac{6}{\eta}\phi'=0.

The solution is trivial:

\phi(\eta,x)=C_{1}(x)+\frac{C_{2}(x)}{\eta^{5}},

(note that the constants C_{1,2} are now x-dependent) and again we see that the gravitational potential freezes at later times. Using the generalized Poison equation, we find that

\delta=\frac{\delta\rho}{\rho_{0}}=\frac{1}{6}\left((-12C_{1}(x)+\delta C_{1}(x)\eta^{2})+
\frac{1}{\eta^{5}}(18C_{2}(x)+\delta C_{2}(x)\eta^{2})\right),

where \delta C_{1}(x), \delta C_{2}(x) are variations of constants C_{1,2} over the given length scale. These variations can be neglected for superhorizon scales, where \delta\approx-2C_{1}+\frac{3C_{2}}{\eta^{5}},while for short wavelengths \delta C_{1,2}\gg C_{1,2} and

\delta\approx c_{1}\eta^{2}+c_{2}\eta^{-3}.

We conclude that superhorizon adiabatic modes of the density perturbation freeze similar to the case w\ne0.

Rate this:
2.5
These icons link to social bookmarking sites where readers can share and discover new web pages.
  • StumbleUpon
  • del.icio.us
  • bodytext
  • Reddit
  • Technorati
  • Facebook
Cosmological perturbations and large scale structure, Cosmology, Master Mason

Related posts:

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

0

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

Posted by Dmitry
at 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.

Rate this:
2.5
These icons link to social bookmarking sites where readers can share and discover new web pages.
  • StumbleUpon
  • del.icio.us
  • bodytext
  • Reddit
  • Technorati
  • Facebook
Cosmological perturbations and large scale structure, Cosmology, Master Mason

Related posts:

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

< Previous page 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Next Page >
Science Blogs - Blog Catalog Blog Directory