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In Max Tegmark's book, Our Mathematical Universe, we can find (in chapter 5, figure 5.3) the following (horrible and poor quality) plot that is supposed to highlight the extreme sensitivity of the Friedman equations.

Plot of the scale factor, according to the Friedmann equations

"Mort thermique" means "Heat death".

But I'm not able to find any textbook that provides a detailed mathematical justification of such a sensitivity (assuming known all the Friedman equations).

Any help, idea or advice?

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    $\begingroup$ Does he not explain the figure in the text? $\endgroup$ Mar 26, 2022 at 20:03
  • $\begingroup$ He explain it with words (ie rethoric). but i'm looking for detailed mathematical proof. $\endgroup$ Mar 26, 2022 at 20:38
  • $\begingroup$ @VincentISOZ can you quote some of those words for context? Thanks, and Welcome to Stack Exchange! $\endgroup$
    – uhoh
    Mar 26, 2022 at 23:52
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    $\begingroup$ There is a Wikipedia page on it. en.m.wikipedia.org/wiki/Flatness_problem $\endgroup$
    – ProfRob
    Mar 27, 2022 at 0:21

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As shown in the Wikipedia page on the flatness problem, you can easily show from the Friedmann equations that the density parameter $\Omega$ is related to the energy density $\rho$ and scale factor $a$ by $$\left( \Omega^{-1} -1\right)\rho a^2 = -\frac{3 \kappa c^2}{8\pi G}\ ,$$ where $\kappa$ is the curvature parameter, and is zero, $+1$ or $-1$ for a flat, closed or open universal geometry.

$\Omega$ is the ratio of $\rho$ to the critical energy density - the density that would just halt the expansion of a flat universe with no dark energy. I believe the plot you show imagines such a universe and $\Omega$ determines the time evolution of the scale factor - if $\Omega >1$ the universe ends in a big crunch, if $\Omega <1$ it expands forever.

The RHS of the equation above is constant (zero in a flat universe) and therefore the LHS also must be a constant. Now in principle $\Omega$ varies with the scale factor and because the density $\rho$ is proportional to $a^{-4}$ (early, radiation-dominated universe) or $a^{-3}$ (later, matter-dominated universe) then we can either write $$\left(\Omega^{-1}-1\right) \propto a^2\ \ \ {\rm radiation}$$ $$\left(\Omega^{-1}-1\right) \propto a\ \ \ {\rm matter}$$

In either case we see that if $\Omega$ differs from 1 then that difference is amplified as $a$ becomes larger.

That is what you are seeing in the plot. An early universe with $\Omega>1$ by a tiny amount, grows into a universe with $\Omega \gg 1$ as it expands. Similarly, an early universe with $\Omega < 1$ by a tiny amount grows into a universe with $\Omega \ll 1$. Only an early universe where $\Omega$ was very, very close to 1 can grow into the universe with $\Omega \sim 1$ that we see today.

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