6. Newtonian perturbation theory 1 (Inflationary perurbations 2)

Posted by Dmitry
March 19, 2008

This is the 4th post in the series based on my lectures on inflationary perturbations at the University of Helsinki.

I decided to postpone the basics for the next time and rush a bit; today instead we will start to develop the theory explaining the large scale sctructure of the Universe as described here. The subject of this post is directly related to patterns in the large scale structure observed at distances less than 100 MPcs.

The appearance of such features in the large scale structure of the Universe as filaments of matter and voids is due to the gravitational instability.
The essence of the latter can be seen already at the lowest possible level of analysis - at the level of Newtonian perturbation theory. Let us suppose that we live in the Minkowski space-time, which is filled with incompressible hydrodynamic matter (dust), having the equation of state p=0. The main characteristics of matter is its energy density \rho({\bf x},t). If due to the thermal fluctuation there appears an inhomogeneity \delta\rho in some particular point of space, this inhomogeneity starts to attract near-by matter towards the point it is located in, and according to the Newton law the attracting force itself is proportional to \delta\rho. Therefore, \ddot{\delta\rho}\sim M_{P}^{-2}\delta\rho, and we see that the exponential instability develops —the amplitude of the fluctuation \delta\rho exponentially grows until it becomes of the same order as the background energy density \rho_{0}.

As we will find out later, taking into account the expansion of the universe will change the character of instability but will not remove it. Before turning to the case of expanding universe let us first discuss how exactly the gravitational instability develops at different length scales in the static universe.

For this purpose we need to construct the linear Newtonian perturbation theory of the hydrodynamic matter, which is completely described by its energy density \rho, pressure p, velocity {\bf v}, and entropy density S. One also has to take into account the gravitational potential \varphi. Equations determining the dynamics of matter and gravitational potential are the following:

\dot{\rho}+\nabla\cdot(\rho{\bf v})=0, (1)

\dot{{\bf v}}+({\bf v}\cdot\nabla){\bf v}+\frac{1}{\rho}\nabla p+\nabla\varphi=0, (2)

\triangle\varphi=\frac{4\pi}{M_{P}^{2}}\rho, (3)

p=p(\rho,S), (4)

\dot{S}+({\bf v}\cdot\nabla)S=0. (5)

The Eq. (5) means that dissipation in the fluid is absent, while the Eq. (4) determines the equation of state.

Exercise 1.1: Try to determine what are the possible sources of non-linearities in these equations.

The next step is to determine the behavior of linear perturbations. One can always choose the reference frame moving together with the fluid, so that its background velocity v=0. It is also convenient to consider \rho_{0}=\rho-\delta\rho to be constant. Although it is inconsistent with the Poisson equation (3), one can always introduce an additional cosmological constant term and discuss the case of the static Einstein universe.

The equations defining the dynamics of linear perturbations are

\ddot{\delta\rho}-c_{s}^{2}\triangle\delta\rho-\frac{4\pi}{M_{P}^{2}}\rho_{0}\delta\rho=\sigma\triangle\delta S, (6)

\dot{\delta}S\=0, (7)

\delta p=c_{s}^{2}\delta\rho+\sigma\delta S, (8)

\triangle\delta\varphi=\frac{4\pi}{M_{P}^{2}}\delta\rho, (9)

where c_{s}^{2}=\bigl(\frac{\delta p}{\delta\rho}\bigr)_{|_{S}} is the speed of sound squared.

Exercise 1.2. Derive the Eqs. (6)-(9) by perturbing quantities in the Eqs. (1)-(5). Hint: take a divergence of the linearized version of the Eq. (2).

Since there is a translational invariance in the problem, one can perform the Fourier transformation

\delta\rho({\bf x},t)=\int e^{i{\bf k}\cdot{\bf x}}\delta\rho_{k}(t)

\delta{\bf v}({\bf x},t)=\int e^{i{\bf k}\cdot{\bf x}}\delta{\bf v}_{k}(t),

\delta S({\bf x},t)=\int e^{i{\bf k}\cdot{\bf x}}\delta S_{k}(t),

\delta\varphi({\bf x},t)=\int e^{i{\bf k}\cdot{\bf x}}\delta\varphi_{k}(t)

and focus on the behavior of separate Fourier modes. Let us start our analysis of the Eqs. (6)-(9) with finding vector modes - the modes of velocity \delta{\bf v}_{k}. To do that, we set \delta\rho=0, \delta S=0, so that the equations for the modes are reduced to

\frac{\partial\delta{\bf v}_{k}}{\partial t}=0,

\nabla\delta{\bf v}_{k}=0,

i.e., for a given {\bf k} two independent vector modes exist perpendicular one to the other and to the direction of the vector {\bf k}, and the amplitude of both modes does not change with time. At the linear level they are decoupled from the energy density and entropy modes.

Now we turn to scalar modes, and that is where the effect of gravitational instability appears. It is convenient to classify scalar perturbations according to the following prescription. The modes with \delta S=0 are denoted as adiabatic (the entropy does not change, so the processes involving only these modes are adiabiatic), while we will call the modes with \delta S\ne0 and \dot{\delta\rho}=0 entropy perturbations. This classification in fact will be with us during the whole course, for example, when the perturbation theory in GR and perturbations generated from inflation will be discussed.

One can immediately see that in Minkowski space-time entropy perturbations do not grow, while adiabatic perturbations do change in time (and are sourced by entropy perturbations). Pure short wavelength adiabatic modes exponentially grow. Namely, perturbations with
k\ll k_{J} where k_{J}=\bigl(\frac{4\pi\rho_{0}}{M_{P}^{2}c_{s}^{2}}\bigr)^{1/2} grow as \delta\rho_{k}(t)\sim e^{\omega_{k}t}, where the growth rate is

\omega_{k}\sim\left(\frac{4\pi}{M_{P}^{2}}\rho_{0}\right)^{1/2},

while short wavelength modes (with k\gg k_{J}) oscillate with frequency \omega_{k}=c_{s}\sqrt{k^{2}-k_{J}^{2}}.

The length scale l_{J}=k_{J}^{-1} is called the Jeans length. It characterizes the development of gravitational instability in the static universe filled with ideal fluid.

Exercise 1.3. What is the characteristic time scale t_{J} for the infrared perturbations of the energy density to become of the same order of magnitude as \rho_{0}?

When \delta\rho_{k}\sim\rho_{0}, it is clear that our linear analysis breaks down. It means that nonlinearities should start playing important role at later time t\gg t_{J} and/or our description of gravitational perturbations is not valid at all for infrared modes. We will analyze these possibilities in the following subsections but before let us briefly discuss the behavior of entropy perturbations.

Since \delta\dot{\rho}=0 for the entropy perturbations, one can immediately find that

\delta\rho_{k}=-\frac{\sigma k^{2}\delta S_{k}}{k^{2}c_{s}^{2}-\frac{4\pi}{M_{P}^{2}}\rho_{0}}\to{\rm Const},

at k\to\infty. It is necessary to remember that the entropy modes can only appear in the multicomponent fluids (so they are for example absent in inflationary models where inflation is driven by a single scalar field).

Next time we will discuss how the gravitational instability develops in the presence of expansion.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft

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8. Introduction into perturbation theory in general relativity (Inflationary perturbations 3)
111. Talk in Munich. One interesting infrared scale in inflationary cosmology
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