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.

solution

In this case the first Friedman equation takes the form \[\left(\frac{\dot a}{a}\right)^2 =\frac{1}{a^2}\] therefore $a\sim t$ and thus $\rho(t) =\rho_{0}( t_{0}/t )^3$.

Problem 2: early Universe

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

solution

The contribution of curvature to $H^{2}$ (the first Friedman equation) is $\sim a^{-2}$, that of non-relativistic matter is $\sim a^{-3}$, of radiation $\sim a^{-4}$. Therefore for sufficiently small $a$, i.e. close enough to the Big Bang, the curvature term can be neglected.

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 $H_{0}$.

solution

The first Friedman equation is often written as \[H^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}.\] Dividing both sides by $H^2$ and rewriting it in terms of \[\Omega = \frac{\rho }{\rho _{cr}}, \:\rho _{cr} = \frac{8\pi G}{3H^2},\] we get \[\frac{k}{a^2 H^2}=\Omega - 1.\] As $a^2H^2>0$ and $k$ can only, by definition, take values $-1, 0, 1$, \[k = \mbox{sign}\left(\Omega-1\right).\] On the other hand, taking the absolute value of the same equation, for the present moment we get \begin{equation}\label{a-H-Om} a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_0 -1|}}. \end{equation}

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 \[-0.0178<(1-\Omega)<0.0063.\]

solution

The result of the previous problem (\ref{a-H-Om}), on restoring the speed of light $c$ by dimensionality, straightforwardly gives us \begin{align*} a_{0}&>7.5\cdot cH_{0}^{-1} \quad\text{for}\quad k<0;\\ a_{0}&>12.5\cdot cH_{0}^{-1} \quad\text{for}\quad k>0. \end{align*} There is no upper bound, as the observational data does not exclude (or rather, tends to imply) the possibility of a spatially flat Universe, with $a_{0}\to\infty$.

Problem 5: curvature dynamics

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

a) radiation,

b) matter.

solution

The first Friedman equation can be expressed as \[\Omega - 1 = \frac{k}{a^2 H^2}.\] As for a constant EoS parameter we have $a(t)\sim t^{2/[3(1+w)]}$, then $H\sim 1/t$ and \[\Omega-1 \sim k\;t^{\frac{2}{3}\frac{1+3w}{1+w}}.\] Finally, a) \[\Omega-1\sim k t^{2/3};\] b) \[\Omega-1\sim k t.\]

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).

no solution yet

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.

solution

Dividing the first Friedman equation by the second one, we obtain the condition for the accelerated expansion in the form \[\frac{1}{1 + 3w} \left(1 - \frac{k}{8\pi G a^{2}\rho}\right) < 0.\] Then we immediately see that
1) For a spatially flat or open Universe ($k\leq 0$) accelerated expansion corresponds to $w\leq -1/3$.
2) For a spatially closed Universe the expansion is accelerating if the following conditions hold: \[\left\{\begin{array}{l} 1+3w>0;\\ \rho>(8\pi G a^2)^{-1}; \end{array}\right.\qquad\text{or}\qquad \left\{\begin{array}{l} 1+3w<0\\ \rho<(8\pi G a^2)^{-1}. \end{array}\right.\] In both variants the first inequality is the restriction on pressure, and the second one on the spatial curvature.