# The role of curvature in the dynamics of the Universe

### Problem 1: curvature domination

Derive $\rho(t)$ in a spatially open Universe filled with dust for the epoch when the curvature term in the first Friedman equation is dominating.

### Problem 2: early Universe

Show that in the early Universe the curvature term is negligibly small.

### Problem 3: curvature and matter

Show that $k =\text{sign}(\Omega-1)$ and express the current value of the scale factor $a_{0}$ through the observed quantities $\Omega_{0}$ and $H_{0}$.

### Problem 4: lower bound on $a_0$

Find the lower bound for $a_{0}$, knowing that the Cosmic background (CMB) data combined with SSNIa data imply $-0.0178<(1-\Omega)<0.0063.$

### Problem 5: curvature dynamics

Fnd the time dependence of $\left|\Omega-1\right|$ in a Universe with domination of

Estimate the upper bound of the curvature term in the first Friedman equation during the electroweak epoch ($t\sim 10^{-10}$~s) and the nucleosynthesis epoch ($t\sim 1-200$~s).
Derive and analyze the conditions of accelerated expansion for a one-component Universe of arbitrary curvature with the component's state parameter $w$