User:Dendropithecus/Sandbox

(Work on the Lambda-CDM metric)

The FLRW metric with two spatial dimensions suppressed is
 * $$ds^2 = c^2 dt^2 - a(t)^2 dx^2$$

Ignoring the effects of radiation in the early universe and assuming k = 0 and w = −1, the Lambda-CDM scale factor is
 * $$a(t) = \left[ \frac{\Omega_m}{\Omega_v} \sinh^2 \left( \frac32 \sqrt{\Omega_v} H_0 t \right) \right]^{\frac13}$$

Putting (for reasons that will emerge later)
 * $$A = \left[ \frac{\Omega_m}{4\Omega_v} \right]^{\frac13} $$

and
 * $$\alpha = \sqrt{\Omega_v} H_0 $$,

the Lambda-CDM scale factor may be rewritten as


 * $$a(t) = A \left[ 2 \sinh \left( \frac32\alpha t \right) \right] ^{\frac23} $$
 * $$a(t) = A \left[ e ^{ \frac32\alpha t } - e ^{- \frac32\alpha t } \right] ^{\frac23} $$
 * $$a(t) = A e ^ { \alpha t } \left[ 1 - e ^{ -3\alpha t } \right] ^{\frac23} $$

Formally expanding the binomial and simplifying gives
 * $$a(t) = A e ^ { \alpha t } \left[ 1 - \frac23 e ^ { -3 \alpha t } - \frac19 e ^ { -6 \alpha t } - \frac{4}{81} e ^ { -9 \alpha t } - \frac{7}{243} e ^ { -12 \alpha t } ...   + \frac{ \text{-2.1.4.7....(3n-5)} }{ 3^n n!} e ^{ -3n \alpha t } - ...  \right] $$

Ratio of successive terms = $$\frac{ \text{(3n-5)} }{ 3n} e ^{ -3 \alpha t }$$ which tends to $$e ^{ -3 \alpha t }$$ as n tends to infinity.

Best Current Numerical Values

The WMAP five-year report gives
 * $$\begin{align} \Omega_m & \approx 0.279 \\ \Omega_v & \approx 0.721 \\ H_0 & \approx 70.1\ \text{km}\ \text{s}^{-1}\ \text{Mp}^{-1} \approx 0.0717\ \text{Ga}^{-1} \end{align}$$

(Mp = megaparsec, Ga = gigayear).

These give
 * $$A = \left[ \frac{\Omega_m}{4\Omega_v} \right]^{\frac13} \approx 1.15_7 $$

and
 * $$\alpha = \sqrt{\Omega_v} H_0 \approx 0.0609\ \text{Ga}^{-1}$$

The path of the light ray satisfies $$dx/dt = c/a(t)$$.

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An Apparent Contradiction (Unfortuanately this Word file contains special characters that won't print here. )

The following refers to "The Emperor's New Mind", OUP 1989(99), chapter 4, section 3 (Gödel’s Theorem). On page 140, there is a statement, derived on the previous pages: ~x[x proves Pk(k)] = Pk(k) With the simple substitution Pk(k) = S made to simplify the analysis and since k doesn’t feature explicitly in what follows, this is my assumption 00. There are two other assumptions: 01 and 02. From these three assumptions a contradiction emerges on lines 09 and 17. The question is: where and why does the contradiction arise? The following uses a modified version of the scheme used in the OU course “Number Theory & Mathematical Logic”.

Line	Statement	Derivation/Comments	Assumptions used 00	~x [x proves S] = S	Assumption	00 01	[x proves S] = [x. [x  S]]	Assumption	01 02	A x[A = x]	Assumption	02 03	~x[x. [x  S]] = S	Subs (0100)	00,01 04	~S	Assumption	04 05	x[x. [x  S]]	Subs/Taut (03,04)	00, 01, 04 06	y. [y  S]	Quant’r Removal (05)	00, 01, 04 07	S	Taut (06)	00, 01, 04 08	~S  S	Proof (04,07)	00, 01 09	S	Taut (08)	00, 01 10	[~x [x proves S] = S]  S	Proof (00,09)	01 11	x~x [x proves S] = S] = x]	Quant’r Removal (02)	02 12	[~x [x proves S] = S] = y	Quant’r Removal (11)	02 13	y	Subs (1200)	00, 02 14	y  S	Subs (1210)	01, 02 15	y . [y  S]	Taut (13,14)	00, 01, 02 16	x[x . [x  S	Quant’r Insertion (15)	00, 01, 02 17	~S	Taut (16,03)	00, 01, 02

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     Dear Professor Penrose,

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 An Apparent Contradiction

  

  

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> The following refers to &quot;The Emperor's New Mind&quot;, OUP 1989(99), chapter 4, section 3 (G&ouml;del&#8217;s Theorem). <o:p></o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> <o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> On page 140, there is a statement, derived on the previous pages: <o:p></o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> <o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> ~ <span style='font-size:11.0pt;font-family:Symbol;mso-ascii-font-family:Georgia; mso-hansi-font-family:Georgia;mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type:symbol;mso-symbol-font-family:Symbol'>$ <span style='font-size:11.0pt;font-family:Georgia'>x[ <span style='font-size: 11.0pt;font-family:Symbol;mso-ascii-font-family:Georgia;mso-hansi-font-family: Georgia;mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type:symbol;mso-symbol-font-family:Symbol'>&Otilde;  <span style='font-size:11.0pt;font-family:Georgia'>x <span style='font-size:11.0pt;font-family:Georgia'> proves Pk(k)] = Pk(k)<o:p></o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> <o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> With the simple substitution Pk(k) = S made to simplify the analysis and since k doesn&#8217;t feature explicitly in what follows, this is my assumption 00.<span style='mso-spacerun:yes'> There are two other assumptions: 01 and 02. From these three assumptions a contradiction emerges on lines 09 and 17. <span style='mso-spacerun:yes'> The question is: where and why does the contradiction arise? <o:p></o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> <o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;text-align:justify;tab-stops: 467.4pt'> The following uses a modified version of the scheme used in the OU course &#8220;Number Theory &amp; Mathematical Logic&#8221;. <o:p></o:p>

<p class=MsoPlainText style='margin-right:.35pt;tab-stops:467.4pt'><span style='font-size:11.0pt;font-family:Georgia'><o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;tab-stops:467.4pt'><span style='font-size:11.0pt;font-family:Georgia'><o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;tab-stops:467.4pt'><span style='font-size:11.0pt;font-family:Georgia'><o:p> </o:p>

<p class=MsoPlainText style='margin-right:.35pt;tab-stops:467.4pt'><span style='font-size:11.0pt;font-family:Georgia'><o:p> </o:p>

--Dendropithecus (talk) 23:14, 21 May 2010 (UTC)