Editing Appendix/Ramblings/StrongNuclearForce (section)

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[SSC/Dynamics/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

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

<tr>
  <td align="right">
<math>~\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{k(R_i, v_i)}{R^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\ddot{R}}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
where, &nbsp; &nbsp; <math>~k(R_i,v_i)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
  </td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner &amp; Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll &amp; Ostlie (2007, 2<sup>nd</sup> Edition)].  Their equations (2) and (3) are written in the following table &#8212; with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.  

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

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

<tr>
  <td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\ddot{a}}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
  </td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==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>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\ddot{R}}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
  </td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[SSC/Dynamics/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]].  Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>M_R</math>.  When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration.  But the second term cannot be treated that way.  Following our [[DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr]  R  \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3)  \biggr]^{-1}
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>.  Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{GM_R}{R^2} 
- \frac{GM_R }{R^2}  \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3)  \biggr]^{-1}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{GM_R}{R^2} 
- \frac{GM_R }{R^2}  \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3)  \biggr]^{-1}
\biggr\}  \, .
</math>
  </td>
</tr>
</table>
</div>

==Insert Dependence on (Energy) Density==

The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential.  The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime.  Generally speaking, a QGP is achieved in a very high energy-density environment.

We can mimic this behavior in our modified cosmology by assuming that the coefficient on the <math>1/r</math> term in the gravitational acceleration varies with the energy-density of the fluid.  (More simply, let's have it vary with the ''mass''-dentiy.)  We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>.  Let's try &hellip;
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\ddot{R}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>- \frac{GM}{R^2} \biggl\{ 1 + \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \frac{R}{a_T} \biggr\} \, .</math>
  </td>
</tr>
</table>
Note that the <math>~R-</math>dependent potential from which this expression for the acceleration is derived is,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>~\Phi(R)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>+ \frac{GM}{R}  - \frac{GM_r}{a_T} \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \ln \biggl(\frac{R}{a_T}\biggr)  \, .</math>
  </td>
</tr>
</table>

This expression for the gravitational acceleration has the desired properties:
<ul>
<li>
In the early universe, when <math>\rho/\rho_H \gg 1</math>, the density-dependent coefficient of the second (confining) term goes to zero; we have an ''asymptotically free regime'' in which a Coulomb-like potential dominates throughout the universe.
</li>
<li>
As the universe expands, the density will steadily drop.  For <math>\rho/\rho_H \ll 1</math>, the density-dependent coefficient of the confining term approaches unity and we retrieve our originally proposed, modified cosmology; that is, the potential is dominated by a logarithmic term for all distances greater than <math>~a_T</math>.
</li>
</ul>