# The role of curvature in the dynamics of the Universe

## Contents

### 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.

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.

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 $H_{0}$.

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

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.

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).

### 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 $w$

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.