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The Power-Law Cosmology - Revision history
2026-09-21T11:08:19Z
Revision history for this page on the wiki
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http://universeinproblems.com/index.php?title=The_Power-Law_Cosmology&diff=2071&oldid=prev
Cosmo All at 01:22, 23 May 2014
2014-05-23T01:22:13Z
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Cosmo All
http://universeinproblems.com/index.php?title=The_Power-Law_Cosmology&diff=2069&oldid=prev
Cosmo All: Created page with "10 <div id="PWL_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> === Problem 1 === <p style= "color: #999;font-size: 11px">problem id: PW..."
2014-05-23T01:11:01Z
<p>Created page with "<a href="/index.php/Category:Dark_Energy" title="Category:Dark Energy">10</a> <div id="PWL_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> === Problem 1 === <p style= "color: #999;font-size: 11px">problem id: PW..."</p>
<p><b>New page</b></p><div>[[Category:Dark Energy|10]]<br />
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=== Problem 1 ===<br />
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Show that for power law $a(t)\propto t^n$ expansion slow-roll inflation occurs when $n\gg1$.<br />
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<p style="text-align: left;">Slow-roll inflation corresponds to \[\varepsilon\equiv-\frac{\dot H}{H}\ll1.\] For power law expansion $H=n/t$ so that $\varepsilon=n^{-1}$. Consequently, slow-roll inflation occurs when $n\gg1$.</p><br />
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=== Problem 2 ===<br />
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Show that in the power-law cosmology the scale factor evolution $a\propto\eta^q$ in conformal time transforms into $a\propto t^p$ in physical (cosmic) time with \[p=\frac{q}{1+q}.\]<br />
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=== Problem 3 ===<br />
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Show that if $a\propto\eta^q$ then the state parameter $w$ is related to the index $q$ by the following \[w=\frac{2-q}{3q}=const.\]<br />
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<p style="text-align: left;">\[\bar H=\frac q t,\quad \bar H'=-\frac q{t^2}.\]<br />
Using<br />
\[\bar H'=-\frac{1+3w}{2}\bar H^2,\] <br />
one obtains<br />
\[w=\frac{2-q}{3q}.\]</p><br />
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Cosmo All