The role of curvature in the dynamics of the Universe

From Universe in Problems
Jump to: navigation, search

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

a) radiation,

b) matter.

Problem 6: curvature in the first three minutes

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).

Problem 7: conditions for acceleration

Derive and analyze the conditions of accelerated expansion for a one-component Universe of arbitrary curvature with the component's state parameter $w$

S.Kumar, arXiv:1109.6924.