<p>As shown in the <a href="https://en.m.wikipedia.org/wiki/Flatness_problem#Energy_density_and_the_Friedmann_equation" rel="noreferrer">Wikipedia page</a> on the flatness problem, you can easily show from the Friedmann equations that the density parameter <span class="math-container">$\Omega$</span> is related to the energy density <span class="math-container">$\rho$</span> and scale factor <span class="math-container">$a$</span> by
<span class="math-container">$$\left( \Omega^{-1} -1\right)\rho a^2 = -\frac{3 \kappa c^2}{8\pi G}\ ,$$</span>
where <span class="math-container">$\kappa$</span> is the curvature parameter, and is zero, <span class="math-container">$+1$</span> or <span class="math-container">$-1$</span> for a flat, closed or open universal geometry.</p>
<p><span class="math-container">$\Omega$</span> is the ratio of <span class="math-container">$\rho$</span> to the critical energy density - the density that would <em>just</em> halt the expansion of a flat universe with no dark energy. I believe the plot you show imagines such a universe and <span class="math-container">$\Omega$</span> determines the time evolution of the scale factor - if <span class="math-container">$\Omega &gt;1$</span> the universe ends in a big crunch, if <span class="math-container">$\Omega &lt;1$</span> it expands forever.</p>
<p>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 <span class="math-container">$\Omega$</span> varies with the scale factor and because the density <span class="math-container">$\rho$</span> is proportional to <span class="math-container">$a^{-4}$</span> (early, radiation-dominated universe) or <span class="math-container">$a^{-3}$</span> (later, matter-dominated universe) then we can either write
<span class="math-container">$$\left(\Omega^{-1}-1\right) \propto a^2\ \ \ {\rm radiation}$$</span>
<span class="math-container">$$\left(\Omega^{-1}-1\right) \propto a\ \ \ {\rm matter}$$</span></p>
<p>In either case we see that if <span class="math-container">$\Omega$</span> differs from 1 then that difference is <em>amplified</em> as <span class="math-container">$a$</span> becomes larger.</p>
<p>That is what you are seeing in the plot. An early universe with <span class="math-container">$\Omega&gt;1$</span> by a tiny amount, grows into a universe with <span class="math-container">$\Omega \gg 1$</span> as it expands. Similarly, an early universe with <span class="math-container">$\Omega &lt; 1$</span> by a tiny amount grows into a universe with <span class="math-container">$\Omega \ll 1$</span>. Only an early universe where <span class="math-container">$\Omega$</span> was very, very close to 1 can grow into the universe with <span class="math-container">$\Omega \sim 1$</span> that we see today.</p>

CC BY-SA 4.0
2022-03-27T09:21:55.003Z
2022-03-27T09:21:55.003Z
ProfRob
<p>In Max Tegmark's book, <em>Our Mathematical Universe</em>, 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.</p>
<p><a href="https://i.sstatic.net/6iyUZ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6iyUZ.jpg" alt="Plot of the scale factor, according to the Friedmann equations" /></a></p>
<p>&quot;Mort thermique&quot; means &quot;Heat death&quot;.</p>
<p>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).</p>
<p>Any help, idea or advice?</p>

CC BY-SA 4.0
2022-03-26T20:03:02.327Z
Does he not explain the figure in the text?
Daddy Kropotkin
CC BY-SA 4.0
2022-03-26T20:38:19.780Z
He explain it with words (ie rethoric). but i'm looking for detailed mathematical proof.
Vincent ISOZ
CC BY-SA 4.0
2022-03-26T23:52:56.707Z
@VincentISOZ can you quote some of those words for context? Thanks, and Welcome to Stack Exchange!
uhoh
CC BY-SA 4.0
2022-03-27T00:21:37.800Z
There is a Wikipedia page on it. https://en.m.wikipedia.org/wiki/Flatness_problem
ProfRob
<cosmology><expansion><big-bang-theory><friedmann-equations>
Strange plot in Max Tegmark book, Our Mathematical Universe
Vincent ISOZ
# Strange plot in Max Tegmark book, Our Mathematical Universe

(Asked by: username_0 on 2022-03-26)

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](https://i.sstatic.net/6iyUZ.jpg)](https://i.sstatic.net/6iyUZ.jpg)

"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?

**Tags:** cosmology, expansion, big-bang-theory, friedmann-equations

> **Comments:**
> **username_1**: Does he not explain the figure in the text?  
> **username_0**: He explain it with words (ie rethoric). but i'm looking for detailed mathematical proof.  
> **username_2**: @username_0 can you quote some of those words for context? Thanks, and Welcome to Stack Exchange!  
> **username_3**: There is a Wikipedia page on it. https://en.m.wikipedia.org/wiki/Flatness_problem  

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## 1 answers

### Answer by username_3 on 2022-03-27. Score: 6 | Accepted answer

As shown in the [Wikipedia page](https://en.m.wikipedia.org/wiki/Flatness_problem#Energy_density_and_the_Friedmann_equation) 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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