7. Newtonian perturbation theory 2 (Inflationary perturbations 2)

Posted by Dmitry
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 k), 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 1, 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

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6. Newtonian perturbation theory 1 (Inflationary perurbations 2)

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