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({\frac {3}{2}}{\sqrt {\Omega _{v}}}H_{0}t\right)\right]^{\frac {1}{3}}

Putting (for reasons that will emerge later)

A=\left[{\frac {\Omega _{m}}{4\Omega _{v}}}\right]^{\frac {1}{3}}

and

\alpha ={\sqrt {\Omega _{v}}}H_{0}

,

the Lambda-CDM scale factor may be rewritten as

a(t)=A\left[2\sinh \left({\frac {3}{2}}\alpha t\right)\right]^{\frac {2}{3}}

a(t)=A\left[e^{{\frac {3}{2}}\alpha t}-e^{-{\frac {3}{2}}\alpha t}\right]^{\frac {2}{3}}

a(t)=Ae^{\alpha t}\left[1-e^{-3\alpha t}\right]^{\frac {2}{3}}

Formally expanding the binomial and simplifying gives

a(t)=Ae^{\alpha t}\left[1-{\frac {2}{3}}e^{-3\alpha t}-{\frac {1}{9}}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{aligned}\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{aligned}}

(Mp = megaparsec, Ga = gigayear).

These give

A=\left[{\frac {\Omega _{m}}{4\Omega _{v}}}\right]^{\frac {1}{3}}\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

HTML VERSION:

<head> <meta http-equiv=Content-Type content="text/html; charset=us-ascii"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 10"> <meta name=Originator content="Microsoft Word 10"> <link rel=File-List href="Penrose300427(Gödel)(Analysis)-01(Georgia11)_files/filelist.xml"> <title>Dear Professor Penrose, </title> <style> </style> </head>

An Apparent Contradiction<o:p></o:p>

<o:p> </o:p>

The following refers to "The Emperor's New Mind", OUP 1989(99), chapter 4, section 3 (Gödel’s Theorem). <o:p></o:p>

<o:p> </o:p>

On page 140, there is a statement, derived on the previous pages: <o:p></o:p>

<o:p> </o:p>

~$x[Õ_x proves P_k(k)] = P_k(k)<o:p></o:p>

<o:p> </o:p>

With the simple substitution P_k(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? <o:p></o:p>

<o:p> </o:p>

The following uses a modified version of the scheme used in the OU course “Number Theory & Mathematical Logic”. <o:p></o:p>

<o:p> </o:p>

Line<o:p></o:p>	Statement<o:p></o:p>	Derivation/Comments<o:p></o:p>	Assumptions used<o:p></o:p>
<o:p> </o:p>	<o:p> </o:p>	<o:p> </o:p>	<o:p> </o:p>
00<o:p></o:p>	~$x [Õ_x proves S] = S<o:p></o:p>	Assumption<o:p></o:p>	00<o:p></o:p>
01<o:p></o:p>	[Õ_x proves S] = [Õ_x. [Õ_xÞ S]]<o:p></o:p>	Assumption<o:p></o:p>	01<o:p></o:p>
02<o:p></o:p>	"A $x[A = Õ_x]<o:p></o:p>	Assumption<o:p></o:p>	02<o:p></o:p>
03<o:p></o:p>	~$x[Õ_x. [Õ_xÞ S]] = S<o:p></o:p>	Subs (01®00)<o:p></o:p>	00,01<o:p></o:p>
04<o:p></o:p>	~S<o:p></o:p>	Assumption<o:p></o:p>	04<o:p></o:p>
05<o:p></o:p>	$x[Õ_x. [Õ_xÞ S]]<o:p></o:p>	Subs/Taut (03,04)<o:p></o:p>	00, 01, 04<o:p></o:p>
06<o:p></o:p>	Õ_y. [Õ_yÞ S]<o:p></o:p>	Quant’r Removal (05)<o:p></o:p>	00, 01, 04<o:p></o:p>
07<o:p></o:p>	S<o:p></o:p>	Taut (06)<o:p></o:p>	00, 01, 04<o:p></o:p>
08<o:p></o:p>	~SÞ S<o:p></o:p>	Proof (04,07)<o:p></o:p>	00, 01<o:p></o:p>
09<o:p></o:p>	S<o:p></o:p>	Taut (08)<o:p></o:p>	00, 01<o:p></o:p>
10<o:p></o:p>	[~$x [Õ_x proves S] = S] Þ S<o:p></o:p>	Proof (00,09)<o:p></o:p>	01<o:p></o:p>
11<o:p></o:p>	$x[[~$x [Õ_x proves S] = S] = Õ_x]<o:p></o:p>	Quant’r Removal (02)<o:p></o:p>	02<o:p></o:p>
12<o:p></o:p>	[~$x [Õ_x proves S] = S] = Õ_y<o:p></o:p>	Quant’r Removal (11)<o:p></o:p>	02<o:p></o:p>
13<o:p></o:p>	Õ_y<o:p></o:p>	Subs (12®00)<o:p></o:p>	00, 02<o:p></o:p>
14<o:p></o:p>	Õ_y Þ S<o:p></o:p>	Subs (12®10)<o:p></o:p>	01, 02<o:p></o:p>
15<o:p></o:p>	Õ_{y .}[Õ_y Þ S]<o:p></o:p>	Taut (13,14)<o:p></o:p>	00, 01, 02<o:p></o:p>
16<o:p></o:p>	$x[Õ_x. [Õ_xÞ S]]<o:p></o:p>	Quant’r Insertion (15)<o:p></o:p>	00, 01, 02<o:p></o:p>
17<o:p></o:p>	~S<o:p></o:p>	Taut (16,03)<o:p></o:p>	00, 01, 02<o:p></o:p>

<o:p> </o:p>

</body>

</html> --Dendropithecus (talk) 23:14, 21 May 2010 (UTC)