Non relativistic small perturbation theory

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Perturbation theory in expanding Universe has a number of distinctive festures. Strictly speaking, this theory shloud be based within the framework of general relativity. However, if inhomogeneities are small one could neglect the effects of curvature and finite speed of interaction and use newtonian dynamics.

To describe the fluctuations of density in this approximation we need the continuity equation $$ \frac{\partial \rho} {\partial t} + \nabla \cdot \left(\rho \vec v\right) = 0 $$ and Euler equation $$ \frac{\partial \vec v} {\partial t} + \left( \vec v\nabla \right)\vec v + \frac{1} {\rho }\nabla P + \nabla \Phi = 0, $$ where Newtonian gravitational potential satisfies the Laplace equation $$ \Delta \Phi = 4\pi G\rho. $$

Problem 1

problem id: per1

Express the deviation of expansion rate from Hubble law in terms of physical and comoving coordinates.

Problem 2

problem id: per2

Obtain the equations for perturbations in linear approximation, assuming that unperturbed state is stationary gas. uniformly distributed in space.

Problem 3

problem id: per3

Demonstrate, that perturbations depend exponentially on time if unperturbed solution is stationary.

Problem 4

problem id: per4

Consider time dependent adiabatic perturbations and find the characteristic scale of instability (so-called Jeans instability).

Problem 5

problem id: per5

Using the results of previous problem, consider the cases of

  • long--wave $\lambda > \lambda _J$ and
  • short--wave $\lambda < \lambda _J$

perturbations. Cosider also the limiting case of short waves ($\lambda \ll \lambda _J$).

Problem 6

problem id: per6

Construct the equation for small relative fluctuations of density \[\delta = \frac{\delta \rho }{\rho }\] in Newtonian approximation neglecting the entropy perturbations.

Problem 7

problem id: per7

Rewrite equation from previous problem in terms of Fourier components, eliminating the Lagrangian coordinates. Estimate the order of "physical" Jeans wavelength for matter dominated Universe.

Problem 8

problem id: per8n

Obtain the dependence of fluctuations on time in flat Universe when
a) matter,
b) radition
is dominating.

Problem 9

problem id: per8

Assuming, that a particular solution to equation from prob. #per6 has the form $\delta _1\left( t \right) \sim H\left( t \right)$, construct the general solution for $\delta (t)$. Consider the flat Universe filled with the substance with $p = w\rho.$

Problem 10

problem id: per9

Demonstrate, that transverse or rotational mode in expanding Universe tends to decrease.