# Exactly Integrable n-dimensional Universes

### Problem 1

problem id: gnd_1

Derive Friedmann equations for the spatially n-dimensional Universe.

### Problem 2

problem id: gnd_2

Obtain the energy conservation law for the case of n-dimensional Universe.

### Problem 3

problem id: gnd_3

Obtain relation between the energy density and scale factor in the case of a two-component n-dimensional Universe dominated by the cosmological constant and a barotropic fluid.

### Problem 4

problem id: gnd_4

Obtain equation of motion for the scale factor for the previous problem.

To integrate (\ref{14}), we recall Chebyshev's theorem:

For rational numbers $p,q,r$ ($r\neq0$) and nonzero real numbers $\alpha,\beta$, the integral $\int x^p(\alpha+\beta x^r)^q\,d x$ is elementary if and only if at least one of the quantities $$\label{cd} \frac{p+1}r,\quad q,\quad \frac{p+1}r+q,$$ is an integer.

Another way to see the validity of the Chebyshev theorem is to represent the integral of concern by a hypergeometric function such that when a quantity in (\ref{cd}) is an integer the hypergeometric function is reduced into an elementary function. Consequently, when $k=0$ or $\Lambda=0$, and $w$ is rational, the Chebyshev theorem enables us to know that, for exactly what values of $n$ and $w$, the equation (\ref{14}) may be integrated.

### Problem 5

problem id: gnd_4_0

Obtain analytic solutions for the equation of motion for the scale factor of the previous problem for spatially flat ($k=0$) Universe.

### Problem 6

problem id: gnd_5

Use the analytic solutions obtained in the previous problem to study cosmology with $w>-1$ and $a(0)=0$.

### Problem 7

problem id: gnd_6

Use results of the previous problem to calculate the deceleration $q=-a\ddot a/\dot a^2$ parameter.

### Problem 8

problem id: gnd_4_1

Obtain the exact solvability conditions for the case $\Lambda=0$ in equation (\ref{14}) (see problem #gnd_4 ).

### Problem 9

problem id: gnd_4_2

Obtain the explicit solutions for the case $n=3$ and $w=-5/9$ in the previous problem.

### Problem 10

problem id: gnd_10

Rewrite equation of motion for the scale factor (\ref{14}) in terms of the conformal time.

### Problem 11

problem id: gnd_11

Find the rational values of the equation of state parameter $w$ which provide exact integrability of the equation (\ref{48}) with $k=0$.

### Problem 12

problem id: gnd_12

Obtain explicit solutions for the case $w=\frac1n-1$ in the previous problem.

### Problem 13

problem id: gnd_13

Obtain exact solutions for the equation (\ref{48}) with $\Lambda=0$.

### Problem 14

problem id: gnd_14

Obtain inflationary solutions using the results of the previous problem.