Editing Appendix/Ramblings/StrongNuclearForce (section)

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe.  The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll &amp; Ostlie.  (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll &amp; Ostlie.)  

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case &#8212; see, for example, our [[SSC/Dynamics/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] &#8212;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} -  \frac{r_i^2}{R^2} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 -  \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta -  \sec^4\zeta \biggr] \, .</math>
  </td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta -  \sec^4\zeta \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta -  \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
  </td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll &amp; Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 - \cos^2\zeta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\pi}{4} \, .</math>
  </td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H_0^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 -  \sec^4\zeta_0 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 -  2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
  </td>
</tr>
</table>

As our [[SSC/Dynamics/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
  </td>
</tr>
</table>
</div>