Editing Appendix/Ramblings/NonlinarOscillation (section)

====Near the Maximum Mass====
We seek a power-series expression for,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(1-\frac{\epsilon_+}{3}\biggr)^{-2} -1 \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(1-\frac{\epsilon_-}{3}\biggr)^{-2} -1 \, .
</math>
  </td>
</tr>
</table>
</div>

Via the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl(1-\frac{\epsilon_\pm}{3}\biggr)^{-2} - 1
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2}{3}\biggr)\epsilon_\pm + \biggl( \frac{1}{3} \biggr) \epsilon_\pm^2
+ \biggl(\frac{2^2}{3^3} \biggr) \epsilon_\pm^3 + \biggl( \frac{5}{3^4} \biggr)\epsilon_\pm^4 + \mathcal{O}(\epsilon_\pm^5) \, .
</math>
  </td>
</tr>
</table>
</div>

So, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\epsilon_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\pm 6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
\pm \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{3} \biggl\{ 
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{3} \biggl\{ 
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2^2}{3^3} \biggl\{ 
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
\biggr\}^3 
+ \frac{5}{3^4} \biggl\{ 
6^{1 / 2}  \mu
\biggr\}^4 
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^2}{3^2} \biggr)^{1 / 2} (2\cdot 3)^{1 / 2}  \mu
- \biggl( \frac{2}{3} \biggr) \biggl(\frac{2^2}{3}\biggr)\mu^2  
+ \biggl( \frac{2}{3} \biggr) \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl( \frac{2}{3} \biggr) \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{6}{3} \mu^2 \biggl\{ 1
- (2\cdot 3)^{-1 / 2} \biggl(\frac{2^2}{3}\biggr)\mu
+ (2\cdot 3)^{-1 / 2} \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^2  
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2^2}{3^3} (2\cdot 3)^{3 / 2}  \mu^3 \biggl[ 1 
- (2\cdot 3)^{-1 / 2}\biggl(\frac{2^2}{3}\biggr)\mu  
\biggr]^3 
+ \frac{5}{3^4} \biggl[ 
(2\cdot 3)^{2}  \mu^4
\biggr]
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
- \biggl(\frac{2^3}{3^2}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{3^5}\biggr) \mu^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 \mu^2 \biggl\{ 1 + \biggl[
- \biggl(\frac{2^3}{3^3}\biggr)^{1 / 2}\mu 
+ \biggl( \frac{5\cdot 23}{2^3 \cdot 3^3} \biggr) \mu^2  \biggr]
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{2^7}{3^3} \biggr)^{1 / 2}  \mu^3 \biggl[ 1 
- \biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu  
\biggr]^3 
+ \biggl( \frac{2^2 \cdot 5}{3^2} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+  2\mu^2 
- \biggl(\frac{2^3}{3^2}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{2^7}{3^3}\biggr)^{1 / 2} \mu^3 
+ \biggl( \frac{2^7}{3^3} \biggr)^{1 / 2}  \mu^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{5\cdot 23}{2 \cdot 3^3} \biggr) \mu^4  
+ \biggl(\frac{2^4}{3^3}\biggr) \mu^4 
- \biggl(\frac{191}{3^5}\biggr) \mu^4
- \biggl( \frac{2^{10}}{3^4} \biggr)^{1 / 2}  \mu^4  
+ \biggl( \frac{2^2 \cdot 5}{3^2} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+  \biggl[ 2 
- \biggl(\frac{2^3}{3^2}\biggr) \biggr]\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}}   
\biggr] \mu^3 
+ \biggl[ \frac{5\cdot 23}{2 \cdot 3^3} 
+ \frac{2^4}{3^3}
- \frac{191}{3^5}
- \frac{2^{5}}{3^2}    
+ \frac{2^2 \cdot 5}{3^2} \biggr] \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>
</table>
</div>
Similarly, we have deduced that,

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

<tr>
  <td align="right">
<math>~\Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
- \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, 

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

<tr>
  <td align="right">
<math>~\Delta C_2 + \Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[
- \biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
- \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{20}{9}\biggr) \mu^2  
+ \biggl( \frac{293}{3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^6) \, ,
</math>
  </td>
</tr>
</table>
</div>

and, 

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

<tr>
  <td align="right">
<math>~\Delta C_2 - \Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[
- \biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
- \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5}{3} \biggr)^{1 / 2}\mu
+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>

so we have,

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

<tr>
  <td align="right">
<math>~\mathfrak{x}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{8}( \Delta C_2 - \Delta C_1 )\biggl[ 
(3-2\beta^2)   + \frac{1}{8}\biggl( \Delta C_2 + \Delta C_1 \biggr) (8\beta^4 - 15 )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{8}\biggl\{ \biggl( \frac{2^5}{3} \biggr)^{1 / 2}\mu
+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
 \biggr\}
\biggl\{ 
(3-2\beta^2)   + \frac{1}{8}\biggl[ \biggl(\frac{20}{9}\biggr) \mu^2  
+ \biggl( \frac{293}{3^5} \biggr) \mu^4
\biggr] (8\beta^4 - 15 )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl( \frac{1}{2 \cdot 3} \biggr)^{1 / 2}\mu
+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3)^{7 / 2}} \biggr] \mu^3 
 \biggr\}
\biggl\{ 
(3-2\beta^2)   + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr] (8\beta^4 - 15 )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\mu^2}{2 \cdot 3} \biggr)^{1 / 2} (3-2\beta^2) 
\biggl[ 1 + \biggl( \frac{5\cdot 23}{6^3} \biggr) \mu^2 \biggr]
\biggl\{ 
1   + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr] \frac{(8\beta^4 - 15 )}{(3-2\beta^2)}
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

Finally, remembering that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(4 -3~m_\xi^{2/3})^{-1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (3-2\beta^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3(4 -3~m_\xi^{2/3}) - 2}{(4 -3~m_\xi^{2/3})} = \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3} } \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

and defining an overall normalization,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{x}_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{5}{2} \biggr) \mathfrak{x}\biggr|_{\beta = 1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2}~\biggl( \frac{\mu^2}{2 \cdot 3} \biggr)^{1 / 2}  
\biggl[ 1 + \biggl( \frac{5\cdot 23}{6^3} \biggr) \mu^2 \biggr]
\biggl\{ 
1   -7 \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr]\biggr\} 
\, ,
</math>
  </td>
</tr>
</table>
</div>
such that the normalized amplitude is always <math>~\tfrac{2}{5}</math> at the surface  &#8212; that is, at <math>~m_\xi = 1</math> and, hence, at <math>~\beta = 1</math> &#8212; we have,

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

<tr>
  <td align="right">
<math>~x_\mathfrak{x} \equiv \frac{\mathfrak{x}}{\mathfrak{x}_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3} } \biggr] 
\biggl\{ 
1   + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr] \frac{(8\beta^4 - 15 )}{(3-2\beta^2)}
\biggr\}\biggl\{ 
1   -7 \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr]\biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
To leading order &#8212; in which case, 
<div align="center">
<math>~\mathfrak{x}_\mathrm{norm} = \biggl( \frac{5^2}{2^3\cdot 3} \biggr)^{1 / 2} \mu \, ,</math>
</div>
&#8212; this exactly matches the [[#Analytic.2C_Marginally_Unstable_Eigenfunction|analytically derived eigenfunction]] for the marginally unstable model, namely,

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

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3} } \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>