World Science Scholars
4.2 The Flatness Problem Review
summary
According to General Relativity, the 'flatness' of the universe is related to its relative mass density.

• “Flat” means Euclidean, as opposed to non-Euclidean curved spaces, which are allowed by General Relativity.
• The value of the relative mass density, $\Omega$, is defined as $$\Omega = \frac{\textrm{Actual mass density}}{\textrm{Critical mass density}}$$ where the critical density depends on the expansion rate of the universe.
• The geometry of the universe is related to the value of $\Omega$. $$\array{\Omega = 1 & \longrightarrow & \textrm{Flat} \\ \Omega > 1 & \longrightarrow & \textrm{Closed} \\ \Omega < 1 & \longrightarrow & \textrm{Open}}$$

Initially, Ω was measured to be far less than 1.

• Until 1998, astronomers measuring $\Omega$ found its value to be approximately 0.2, implying a universe with an open geometry.
• The missing ingredient was dark energy, a material with negative pressure.
• When the contributions of dark energy are included, the observed value of $\Omega$ is very close to 1.
• The latest observations, measured by the Planck satellite, indicate $\Omega$ to be approximately $1.0010 \pm 0.0065$.

It turns out that the value of the relative mass density is approximately equal to 1.

• This poses a puzzle—in the standard Big Bang theory, the value $\Omega=1$ is an unstable equilibrium point.
• Small deviations from this value would have significant effects on the nature and curvature of the universe.
• If $\Omega$ had been exactly equal to 1, it would be 1 forever. But since we measure it to be almost equal to 1 today, the instability implies it must have been substantially closer to 1 in the early universe.

Cosmologists wondered how the initial density came to be so closely fine-tuned to this 'special' value.

• For the value of $\Omega$ to be anywhere near 1 today, its value would have to be tuned fantastically close to 1 in the early universe.
• At one second after the Big Bang, $\Omega$ would have had to equal 1 to an accuracy of 15 decimal places.
• Inflation proposes a solution for this “fine-tuning” problem. Since inflation makes gravity repulsive, the evolution of $\Omega$ also changes.
• $\Omega$ is driven rapidly toward 1 in the inflationary model. It is driven so rapidly that its initial value could have been almost anything. Yet, this mechanism almost always overshoots in accuracy, predicting a much flatter universe than we measure today.

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