# Friedman equations

### Problem 1: derivation of Friedman equations

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 $T^{\mu}_{\nu}=diag(\rho,-p,-p,-p)$.

### Problem 2: reformulation in terms of conformal time

Derive the Friedman equations in terms of conformal time.

### Problem 3: relations between equations

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

### Problem 4: source of gravity in GR in the weak field limit

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

### Problem 5: expansion and pressure

How does the magnitude of pressure affect the expansion rate?

### Problem 6: second equation for $k=0$

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: Lorentz invariance

Are solutions of Friedman equations Lorentz-invariant?

### Problem 8: critical density

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

### Problem 9: relative densities

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: scale factor via observables

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

### Problem 11: Newtonian interpretation

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: $\dot H$

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

### Problem 13: Raychaudhuri equation

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

### Problem 14: conservation equation

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: conservation in terms of conformal time

Obtain the conservation equation in terms of the conformal time.

### Problem 16: conservation equations for the expanding Universe

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

### Problem 17: relations between the three equations

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: e-foldings number

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: conservation in terms of e-foldings number

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

### Problem 20: pressure in terms of H

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

### Problem 21: EoS parameter in terms of H

Express the state parameter $w = p/\rho$ in terms of the Hubble parameter and its time derivative.

### Problem 22: hidden symmetry of Friedman equations

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

### Problem 23: relation between the total pressure and the deceleration parameter

Find the relation between the total pressure and the deceleration parameter for a flat Universe.

### Problem 24: evolution of the state parameter

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

### Problem 25: EoS via the expansion dynamics

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 26: upper bound on w

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

### Problem 27: non-relativistic state parameter

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

### Problem 28: condition $\dot X = 0$

The Friedman equation can be regarded as a constraint of the form $X \equiv H^2 + \frac{k}{a^2} - \frac{8\pi G}{3}\rho = 0$ Show that the conservation equation $\dot \rho + 3H(\rho + p)$ represents the condition $\dot X = 0$.