Cosmo All: /* Problem 9 */

2012-12-01T13:55:36Z

‎Problem 9

Cosmo All: /* Problem 1 */

2012-12-01T13:54:38Z

‎Problem 1

Cosmo All at 13:52, 1 December 2012

2012-12-01T13:52:34Z

Cosmo All at 13:51, 1 December 2012

2012-12-01T13:51:41Z

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Cosmo All at 13:36, 1 December 2012

2012-12-01T13:36:07Z

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Cosmo All: Created page with "2"

2012-12-01T13:19:50Z

Created page with "2"

New page

[[Category:Dark Energy|2]]

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 '''2.''' Open or flat ($k\le0$) Universe: $\rho_\Lambda\ge0$.
-<div class="NavFrame collapsed">
+<!--<div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;"></p>
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+</div>--></div>

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">We know that (see problem \ldots of chapter 2)
+     <p style="text-align: left;">We know that (see problems of chapter 2)
 \[q=\frac{\Omega_{tot}}{2}+\frac32\sum\limits_i w_i\Omega_i.\]
 The sign of deceleration parameter at any given epoch depends upon the equation of state and not on alone. While in the presence of a cosmological constant ($w=-1$), $\Omega_{tot}$ no longer determines the ultimate fate of the Universe.</p>
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 <div id="DE18_2"></div>
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 === Problem 2 ===
 Show that decoupling between $\Omega_{tot}$ and destiny of the Universe is due to violation of strong energy condition.

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 ''Presence of the cosmological constant reevaluates standard notions about the connection between geometry and the fate of the Universe. The traditional philosophy of General Relativity is that "Geometry is Destiny", with "geometry" in this context implying openness or closure of the three-space of constant cosmological time.  If energy content is provided by "ordinary" matter (nonrelativistic matter or radiation) this slogan transforms into "Density is Destiny". If the density of matter is less or equal than the critical value (and the Universe is open), then the destiny of Universe is eternal expanding; if the density is greater, and the Universe is closed, then the destiny is recollapse. If the Universe contains cosmological constant (or any energy component with $w<-1/3$) the situation changes radically: an open Universe can recollapse, while a closed Universe can expand forever. Geometry no longer determines the fate of the Universe.''
@@ Line 28: / Line 29: @@
 <div id="DE18_2"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 2 ===
 Show that decoupling between $\Omega_{tot}$ and destiny of the Universe is due to violation of strong energy condition.
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 <div id="DE18_3"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 3 ===
 Find the maximum value of scale factor for a hypothetical flat Universe with $\Omega_{\Lambda0}<0$.
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 <div id="DE18_4"></div>
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-=== Problem 1 ===
+=== Problem 4 ===
 For the Universe considered in the previous problem, find the analytical solution $t(a)$ and time of the collapse from $a=a_{\max}$ back down to $a=0$.
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 <div id="DE18_5"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 5 ===
 In the flat Universe with $\Omega_{m0}<1$ and positive cosmological constant find the late time asymptotic for the scale factor.
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 <div id="DE18_6"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 6 ===
 Show that eternal expansion is inevitable if and only if \[\Omega_\Lambda>4\Omega_m\left\{\cos\left[\frac13\arccos(\Omega_m^{-1}-1)+\frac{4\pi}{3}\right]\right\}.\]
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 <div id="DE18_7"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 7 ===
 Show that if $\Omega_{m0}+\Omega_{\Lambda0}>1$ (positive spatial curvature) and $\Lambda>0$, then it is possible to have a Universe that expands at late times, but without the initial Big Bang ($a=0,\ t=0$).
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 <div id="DE18_8"></div>
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-=== Problem 1 ===
+=== Problem 8 ===
 Show that collapse of the Universe is possible if the following conditions are satisfied:
 <br/>
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 <div id="DE18_9"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 9 ===
 Show that eternal expansion of the Universe is possible if the following conditions are satisfied
 <br/>

Geometry and Destiny - Revision history

Cosmo All: /* Problem 9 */

Cosmo All: /* Problem 1 */

Cosmo All at 13:52, 1 December 2012

Cosmo All at 13:51, 1 December 2012

Cosmo All at 13:36, 1 December 2012

Cosmo All: Created page with "2"