Cosmological horizons - Revision history

Cosmo All at 22:03, 12 November 2012

2012-11-12T22:03:44Z

Igor at 12:30, 14 October 2012

2012-10-14T12:30:33Z

Igor at 12:28, 14 October 2012

2012-10-14T12:28:17Z

Show changes

Igor: /* Problem 13. */

2012-08-10T10:59:45Z

‎Problem 13.

Cosmo All: /* Problem 8. */

2012-08-07T17:56:10Z

‎Problem 8.

Igor: /* Problem 13. */

2012-07-23T18:08:15Z

‎Problem 13.

Igor: /* Problem 13. */

2012-07-23T18:07:36Z

‎Problem 13.

Igor: /* Problem 13. */

2012-07-23T18:05:39Z

‎Problem 13.

Cosmo All: /* Problem 1. */

2012-07-23T16:37:31Z

‎Problem 1.

Cosmo All at 16:34, 23 July 2012

2012-07-23T16:34:35Z

← Older revision		Revision as of 22:03, 12 November 2012
Line 1:		Line 1:
−	[[Category:Dynamics of the Universe in the Big Bang Model\|5]]	+	[[Category:Dynamics of the Universe in the Big Bang Model\|6]]
	__TOC__		__TOC__

@@ Line 2: / Line 2: @@
 __TOC__
 <div id="dyn47"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
@@ Line 148: / Line 149: @@
 </div></div>
+== FLRW in terms of proper distances ==
 The following five problems are based on work by F. Melia ([http://arxiv.org/abs/0711.4181 arXiv:0711.4181], [http://arxiv.org/abs/0907.5394 arXiv:0907.5394]).

@@ Line 281: / Line 281: @@
 \[R_{h}=\frac{1}{H_0}.\]
-'''b)''' an equation of state $w=-1/3$ is the only one for which the current age, $t_0$, of the Universe can equal the light-crossing time, $t_{h}=R_h$. In this case
+'''b)''' the equation of state $w=-1/3$ is the only one for which the current age, $t_0$, of the Universe can equal the light-crossing time, $t_{h}=R_h$. In this case
 \begin{align}
 d{s^2}& = \Phi {\Big[ {cdt + \big(\frac{R}{t}\big)

@@ Line 152: / Line 152: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">We use again the expression for the radius of the $3$-sphere ($ a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_0 -1|}}$) through observables in form $a_{0}=(H_{0}\sqrt{|\Omega_{curv}|})^{-1}$, and the one for its volume $V=2\pi^{2}a^{3}$ ([[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ22|problem of chapter 2]]). Then neglecting curvature on the scales of the Hubble sphere (see problem  (\ref{dyn56})), we obtain
+     <p style="text-align: left;">We use again the expression for the radius of the $3$-sphere ($ a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_0 -1|}}$) through observables in form $a_{0}=(H_{0}\sqrt{|\Omega_{curv}|})^{-1}$, and the one for its volume $V=2\pi^{2}a^{3}$ ([[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ22|problem of chapter 2]]). Then neglecting curvature on the scales of the Hubble sphere (see [[#dyn56|problem]]), we obtain
 \[\frac{V_H}{V_U}
 	\approx\frac{(4\pi/3)R_{H}^3}{2\pi^2 a_{0}^{3}}

@@ Line 294: / Line 294: @@
 a(t)& = {\left( {2{H_0}t} \right)^{1/2}},
 \quad f(t) = \frac{1}{2}\ln \left( {2{H_0}t} \right),
-\quad \dot f = \frac{1}{{2t}}\\
+\quad \dot f = \frac{1}{2t}\\
 d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{2t}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
 \Phi & = 1 - \left( {\frac{R}{2t}} \right)^2.

@@ Line 284: / Line 284: @@
 \begin{align}
 d{s^2}& = \Phi {\Big[ {cdt + \big(\frac{R}{t}\big)
-	{\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2}
+	{\Phi ^{ - 1}}dR} \Big]^2} - \Phi^{-1}d{R^2}
-	- {R^2}d{\Omega ^2},\\
+	- R^{2}d{\Omega ^2},\\
-\Phi& = 1 - {\left( {\frac{R}{{ct}}} \right)^2}
+\Phi& = 1 - \left(\frac{R}{ct}\right)^2
 \end{align}
 The cosmic time $dt$ diverges for a measurable line element as $R\to R_h =t$.<br/>
@@ Line 295: / Line 295: @@
 \quad f(t) = \frac{1}{2}\ln \left( {2{H_0}t} \right),
 \quad \dot f = \frac{1}{{2t}}\\
-d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{{2t}}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
+d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{2t}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
-\Phi & = 1 - \left( {\frac{R}{{2t}}} \right)^2.
+\Phi & = 1 - \left( {\frac{R}{2t}} \right)^2.
 \end{align*}
 Thus, measurements made at a fixed $R$ and $t$ still produce a gravitationally-induced dilation of $dt$ as $R$ increases, but this effect never becomes divergent within that portion of the Universe (i.e., within $t_0$) that remains observable since the Big Bang.<br/>
@@ Line 302: / Line 302: @@
 '''d)''' Matter domination:
 \begin{align*}
-a(t) &= {\left( {3/2{H_0}t} \right)^{2/3}},\quad f(t) = \frac{2}{3}\ln \left( {3/2{H_0}t} \right),\quad \dot f = \frac{2}{{3t}}\\
+a(t) &= {\left( {3/2{H_0}t} \right)^{2/3}},\quad f(t) = \frac{2}{3}\ln \left( {3/2{H_0}t} \right),\quad \dot f = \frac{2}{3t}\\
-d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{{3t/2}}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
+d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{3t/2}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
-\Phi  &= 1 - {\left( {\frac{R}{{3t/2}}} \right)^2}
+\Phi  &= 1 - {\left( {\frac{R}{3t/2}} \right)^2}
 \end{align*}
 The situation is similar to that for a radiation dominated universe, in that $R_h$ always recedes from us faster than lightspeed. Although dilation is evident with increasing $R$, curvature alone does not produce a divergent redshift.</p>

@@ Line 264: / Line 264: @@
 Using FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$, consider specific cosmologies:
-'''a)''' the De Sitter Universe ;
+'''a)''' the De Sitter Universe;
 '''b)''' a cosmology with  $R_h =t$, ($w=-1/3$);
@@ Line 274: / Line 274: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">''''a)''' De Sitter
+     <p style="text-align: left;">
 \[H=H_{0}=const,\quad a(t)=e^{H_{0}t},\quad
 	f=\ln a(t)=H_{0}t.\]

@@ Line 26: / Line 26: @@
 \[a(t)\sim t^{\frac{2}{3}(w+1)^{-1}}.\]
 Then
-\[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\]
+\[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\]</p>
-\end{description}</p>
    </div>
 </div>

@@ Line 1: / Line 1: @@
-[[Category:Dynamics of the Universe in the Big Bang Model]]
+[[Category:Dynamics of the Universe in the Big Bang Model|5]]
 __NOTOC__
 <div id="dyn47"></div>