Revision as of 18:45, 18 June 2012

Friedman equations

Problem 1.

Starting from the Einstein equations, derive the equations for the scale factor $a(t)$ of the FLRW metric---the Friedman equations: \begin{align}\label{FriedmanEqI} \Big(\frac{\dot a}{a}\Big)^2& =\; \frac{8\pi G}{3}\rho -\frac{k}{a^2};\\ \label{FriedmanEqII} \frac{\ddot a}{a} &=- \frac{4\pi G}3\big(\rho+3p). \end{align} Consider matter an ideal fluid with $T^{\mu}_{\nu}=diag(\rho,-p,-p,-p)$ (see problem).

Problem 2.

Derive the Friedman equations in terms of conformal time.

Problem 3.

Show that the first Friedman equation is the first integral of the second one.

Problem 4.

Show that in the weak field limit of General Relativity the source of gravity is the quantity $(\rho + 3p)$.

Problem 5.

How does the magnitude of pressure affect the expansion rate?

Problem 6.

Consider the case $k = 0$ and show that the second Friedman equation can be presented in the form \[HH' = - 4\pi G\left(\rho + p\right),\] where $H' \equiv \frac{dH}{d\ln a}.$

Problem 7.

Are solutions of Friedman equations Lorentz-invariant?

Problem 8.

The critical density corresponds to the case of spatially-flat Universe $k=0$. Determine its actual value.

Problem 9.

Show that the first Friedman equation can be presented in the form \[\sum\limits_i\Omega_i=1,\] where $\Omega_i$ are relative densities of the components, \[\Omega_i\equiv\frac{\rho_{i}}{\rho_{cr}}, \quad \rho_{cr}=\frac{3H^{2}}{8\pi G}, \quad \rho_{curv}=-\frac{3}{8\pi G}\frac{k}{a^2},\] and $\rho_{curv}$ describes the contribution to the total density of the spatial curvature.

Problem 10.

Express the scale factor $a$ in a non-flat Universe through the Hubble's radius and total relative density $\rho/\rho_{cr}$.

Problem 11.

Show that the relative curvature density $\rho_{curv}/\rho_{cr}$ in a given region can be interpreted as a measure of difference between the average potential and kinetic energies in the region.

Problem 12.

Prove that in the case of spatially flat Universe \[\dot{H} = - 4\pi G (\rho + p).\]

Problem 13.

Obtain the Raychadhuri equation \[H^2 + \dot H = - \frac{4\pi G}{3}(\rho + 3p).\]

Problem 14.

Starting from the Friedman equations, obtain the conservation equation for matter in an expanding Universe: \[\dot{\rho}+3H(\rho+p)=0.\] Show that it can be presented in the form \[\frac{d\ln\rho}{d\ln a}+3(1+w)=0,\] where $w=p/\rho$ is the state parameter for matter.

Problem 15.

Obtain the conservation equation in terms of the conformal time.

Problem 16.

Starting from the energy momentum conservation law $\nabla_{\mu}T^{\mu\nu}=0$, obtain the conservation equation for the expanding Universe.

Problem 17.

Show that among the three equations (two Friedman equations and the conservation equation) only two are independent, i.e. any of the three can be obtained from the two others.

Problem 18.

For a spatially flat Universe rewrite the Friedman equations in terms of the $e$-foldings number for the scale factor \[N(t)=\ln\frac{a(t)}{a_0}.\]

Problem 19.

Express the conservation equation in terms of $N$ and find $\rho(N)$ for a substance with equation of state $p=w\rho$.

Problem 20.

For the case of spatially flat Universe express pressure in terms of the Hubble parameter and its time derivative.

Problem 21.

Show that for a spatially flat Universe the Friedman equations are invariant\footnote{V. Faraoni. A symmetry of the spatially flat Friedmann equations with barotropic fluids. arXiv:1108.2102v1.} under the change of variables to a new scale factor \[a \to \alpha=\frac{1}{a}\] and a the new equation of state: \[(w+1)\to (\omega+1)=-(w+1).\]

Problem 22.

Evaluate the derivatives of the state parameter $w$ with respect to cosmological and conformal times.

Problem 23.

Consider an FLRW Universe dominated by a substance, such that Hubble parameter depends on time as $H=f(t)$, where $f(t)$ is an arbitrary differentiable function. Find the state equation for the considered substance.

Problem 24.

Find the upper bound for the state parameter $w$.

Problem 25.

Show that for non-relativistic particles the state parameter $w$ is much less unity.