NEQNET: Non-equilibrium Phenomena

7. Newtonian perturbation theory 2 (Inflationary perturbations 2)

Posted by Dmitry

at March 20, 2008

This is the 5th post in the series based on my lectures on inflationary perturbations at the University of Helsinki.

Today I will continue to discuss Newtonian perturbation theory (the first post on this topic can be found here). Let me remind you that my ultimate goal is to explain the features (filaments and voids) in the large scale structure of the Universe which appear at scales less than 100 MPcs. Our hypothesis was that these features are due to the intrinsic instability of gravitation, and we started to check this hypothesis in Newtonian perturbation theory.

Last time we considered ideal fluid consisting of particles interacting with each other gravitationally. This is a very good approximation for the description of matter at cosmological distances, since only gravitational interaction survives at these huge scales. We found that if gravitation is described in old-fashioned way introduced by Newton (i.e., we suppose that there is a gravitational attractive force between two particles with masses m_1 and m_2 with radius vectors $\vec{r_1}$ and $\vec{r_2}$ which behaves as

$\vec{F}=G\frac{m_1{}m_2(\vec{r_1}-\vec{r_2})}{|\vec{r_1}-\vec{r_2}|^3}$ ,

that is, according to the usual inverse square law), than the long wavelength perturbations in matter grow exponentially.

From the point of view of a field theorist that means that the non-relativistic theory of gravitation is tachyonically unstable, and there should exist some way to cure this tachyonic instability. Not only long wavelength perturbations rapidly go out of the linear regime of the perturbation theory, they do it too fast (namely, exponentially fast). Therefore, the infrared physics of the non-relativistic theory of gravitation should be somehow changed.

One simple way to change it is to take expansion of the Universe into account. How to do it without turning to the general relativity? One simple way to do it is to model the expansion of background by setting the background velocity of the fluid as

${\bf v}_{0}=H(t)x=\frac{\dot{a}}{a}x$ (1)

(in a sense, this is exactly what we see on the sky — due to the Hubble expansion, objects at larger distances from us run away from us faster). The expression (1) defines what is called Hubble flow. In an expanding universe the background energy density does satisfy the Poisson equation, which gives

$\dot{H}+H^{2}=-\frac{4\pi}{M_{P}^{2}}\rho_{0},$ (2)

i.e., the Friedmann equation.

Exercise 1.5. Derive it taking the divergence of the Euler equation from the previous post, the Poisson equation and taking into account the fact that physical distances are scaled as ${\bf r}=a(t){\bf x}$ . How did it happen that we derived the Friedmann equation without using general relativity and solving Einstein equations?

Let us again start by analyzing the behavior of vector modes $\delta{\bf v}_{k}$ . In the absence of energy density and entropy perturbations ( $\delta\rho=0,$ $\delta S=0$ ) one has

$\nabla\delta{\bf v}=0,$ $\dot{\delta{\bf v}_{k}}+\frac{\dot{a}}{a}\delta{\bf v}_{k}=0.$

The first equation means that the peculiar velocity is perpendicular to the momentum (so we again have two independent vector modes perpendicular to each other and to the direction of momentum ), while the second equation shows that $\delta{\bf v}_{k}\sim a^{-1}$ , i.e., vector modes decay in the expanding universe.

As for scalar perturbations, it is more convenient to study the behavior of the relative density fluctuations $\delta(t,{\bf x})=\frac{\rho-\rho_{0}}{\rho_{0}}$ at the linearized level rather than the fluctuation of the energy density itself (the reason is that the background energy density decreases itself in the expanding universe, and we want to know how does the perturbation look like on this decreasing background). One finds that $\delta$ and the gravitational potential perturbations are governed by the equations

$\ddot{\delta}+2H\dot{\delta}-\frac{c_{s}^{2}}{a^{2}}\triangle\delta-\frac{4\pi}{M_{P}^{2}}\rho_{0}\delta=\frac{\sigma}{\rho_{0}a^{2}}\delta S,$ (3)

$\triangle\delta\varphi=\frac{4\pi}{M_{P}^{2}}a^{2}\rho_{0}\delta,$ (4)

and the equation for entropy conservation.

Exercise 1.6. Derive the Eq. (3).

Exercise 1.7. What happens with the entropy density in the expanding universe?

Exercise 1.8. What is the Jeans scale for the expanding universe filled with radiation ( $a(t)\sim t^{1/2}$ )? with incompressible fluid ( $a(t)\sim t^{2/3}$ )?

We see that the qualitative picture is similar to the one we found in the case of Minkowski space-time. One can divide energy density modes into adiabatic and entropy ones. The latter seed the adiabatic modes, while the adiabatic ones grow at scales larger than the Jeans scale

$\lambda_{J}=\frac{2\pi a}{k_{J}}=c_{s}\sqrt{\frac{\pi M_{P}^{2}}{\rho_{0}}}$

(note that $\rho_{0}$ is changing itself due to the expansion!) and oscillate at smaller scales, while the amplitude of these oscillations decreases with time. The growth rate of long wavelength adiabatic modes however is weaker than exponential. Indeed, in the limit $k\to{}0$ one gets

$\delta_{k}=C_{1}H\int\frac{dt}{a^{2}H^{2}}+C_{2}H,$

i.e., for the universe filled with non-relativistic matter (this corresponds to $a(t)\sim t^{2/3}$ ) this gives the polynomial growth

$\delta_{k}(t)=C_{1}t^{2/3}+C_{2}t^{-1}$

instead of exponential one (!) as in the case of Minkowski space-time.

The polynomial instability means that if we want to have large inhomogeneities today ( $\delta_{k}>1$ ), they should be already rather large at early times. The physics of the instability developement is the following. The Hubble flow stretches linear perturbations, their relative amplitude grows, although the overall energy density decreases. When the relative amplitude $\delta$ becomes of order of , the gravitational interaction is so effective that it overcomes the Hubble expansion, and the perturbation drops drops out of the Hubble flow. We see that is possible to cure somewhat the exponential instability of Newtonian perturbation theory by taking expansion into account, but that the instability does not disappear, so that it is necessary to take non-linearities into account at late times $t\gg t_{J}$ .

Finally, one has to note that the physics at time scales and wavelengths larger than $H^{-1}$ cannot be treated by the method we used here. The reason is that the general relativity effects should be taken into account at these scales ( H^2 is the scale of the curvature of space-time itself). We will turn to the discussion of the perturbation theory in GR soon but before I will explain in the first approximation what happens with perturbations in the nonlinear regime.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft

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

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

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

6. Newtonian perturbation theory 1 (Inflationary perurbations 2)

Posted by Dmitry

at 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 . 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

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

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

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

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

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7. Newtonian perturbation theory 2 (Inflationary perturbations 2)

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7. Newtonian perturbation theory 2 (Inflationary perturbations 2)

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