Flatness Problem

Overview #

In the Big Bang model there are multiple parameters which appear to be “fine tuned” in that, small changes to a given parameter have massive effects on the evolution of the universe as a whole.

Wrt the flatness problem itself, the questionable parameter is the Density Parameter (\(\Omega\)), which is defined as

\begin{equation} \Omega \equiv \frac{\rho}{\rho_{c}}\tag{1} \end{equation}

where the subscript \(c\) is in reference to the “critical value” of density (\(\rho\)). Which, today, is measured to be–roughly–five atoms of Hyrodgen per cubic meter. \(\rho_{c}\) is defined:

\begin{equation} \rho_{c} = \frac{3H^{2}}{8\pi G}\tag{2} \end{equation}

Remembering, of course, that density (\(\rho\)) is defined as:

\begin{equation} \rho = \frac{m}{V}\tag{4} \end{equation}

which implies

\begin{equation} \Omega \equiv \frac{\rho}{\rho_{c}} = \frac{8\pi G m}{3H^{2}V}\tag{3} \end{equation}

This value \(\Omega\) determines the shape (curvature) of the universe, i.e.:

\begin{equation} \Omega < 1\tag{a} \end{equation}

\begin{equation} \Omega = 1\tag{b} \end{equation}

\begin{equation} \Omega > 1\tag{c} \end{equation}

For each case:

1. the universe will be open (hyperbolic), will expand forever
2. the universe will be flat
3. the universe will be closed (spherical), will eventually stop expanding then collapse

What’s the problem? #

If \(\Omega\) is not equivalent/near unity, the universe will undergo expansion or contraction. This is problematic because taking eq. \(3\), above, we have a variable volume (due to the contraction or expansion), which shows that any deviation from unity in the density parameter will have acclerating changes in the shape of the universe over the course of the universe’s timeline. Working backwards given the universe’s age, we find that the original density must have only deviated from \(\rho_{c}\) by less than one part in \(10^{62}\). This, naturally, leads one to ask,

Why is the universe’s density sitting so close to this critical value?