# Static Einstein's Universe

### Problem 1

Find the static solution of Friedman equations with cosmological constant and non-relativistic matter (static Einstein's Universe).

### Problem 2

Show that the static Einstein's Universe must be closed. Find the total volume and mass of this Universe.

### Problem 3

Find the parameters of static Einstein's Universe filled with cosmological constant and radiation.

### Problem 4

Find the parameters of static Einstein's Universe under the assumption that both matter and radiation are absent.

### Problem 5

Estimate the radius of static Einstein's Universe if the zero-point energy of electromagnetic field is cut off at the classical electron radius.

### Problem 6

Show that Einstein's Universe is unstable.

### Problem 7

What are the concrete mechanisms that drive the instability of the static Einstein's Universe?

### Problem 8

What is the most unsatisfactory peculiarity of the static Einstein's model of the Universe (besides the instability)?

### Problem 9

Construct the effective one-dimensional potential $V(a)$ for the case of flat Universe filled with non-relativistic matter and dark energy in the form of cosmological constant.

### Problem 10

Show that the static Einstein's Universe may be realized only in the maximum of the effective potential $V(a)$ of the previous problem.

### Problem 11

Problem #DE32 can be considered in more general setup.
Assuming arbitrary values for the contributions of the cosmological constant $\lambda$, matter $\mu$, radiation $\gamma$ and curvature $\varkappa$ respectively, present the first Friedman equation $$\label{Friedman4k+} H^2\equiv\left(\frac{\dot a}{a}\right)^2=\lambda-\frac{\varkappa}{a^2}+\frac{\mu}{a^3}+\frac{\gamma}{a^4}$$ in the form of the energy conservation law $p^2/2 + U(q) = E.$