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

Posted by Dmitry

March 31, 2008

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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, – 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 ; 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 $w\ne0$ .

Cosmological perturbations and large scale structure, Cosmology, Master Mason

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