Difference between revisions of "Evolution of Universe"
(Created page with "2") |
|||
Line 1: | Line 1: | ||
[[Category:Standard Cosmological Model|2]] | [[Category:Standard Cosmological Model|2]] | ||
+ | |||
+ | <div id="SCM_24"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 28 === | ||
+ | Find the redshift dependence of the deceleration parameter. Analyze the limiting cases. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;"> | ||
+ | $$ | ||
+ | \begin{gathered} | ||
+ | q \equiv - \frac{a\ddot a}{\dot a^2} = - \frac{\ddot a}{a}\frac{1}{H^2}; \\ | ||
+ | \frac{\ddot a}{a} = - \frac{4\pi G}{3}\sum\limits_i \left( \rho _i + 3p_i \right) = - \frac{1}{2}H_0^2\sum\limits_i \Omega _{i0}\left(1 + 3w_i\right)(1 + z)^{3\left(1 + w_i\right)} ; \\ | ||
+ | H^2 = H_0^2\sum\limits_i \Omega _{i0}(1 + z)^{3\left( 1 + w_i \right)}; \\ | ||
+ | q = \frac{3}{2}\frac{\sum\limits_i \Omega _{i0}\left( 1 + w_i \right)(1 + z)^{3\left( 1 + w_i \right)}} | ||
+ | {\sum\limits_i \Omega _{i0}(1 + z)^{3\left( 1 + w_i\right)} } - 1\\ | ||
+ | \end{gathered} | ||
+ | $$ | ||
+ | In the SCM | ||
+ | $$ | ||
+ | q = \frac{1}{2}\frac{\Omega _{m0}(1 + z)^3- 2\Omega _{\Lambda 0} } | ||
+ | {\Omega _{m0}(1 + z)^3+\Omega_{\Lambda 0}} | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | q(z \to \infty ) = \frac{1}{2},\;q(z \to - 1) = - 1 | ||
+ | $$ | ||
+ | |||
+ | <gallery widths=600px heights=500px> | ||
+ | File:12_24.jpg| | ||
+ | </gallery> | ||
+ | Dependence of the deceleration parameter on the redshift. | ||
+ | </p> | ||
+ | </div> | ||
+ | </div></div> |
Revision as of 08:51, 4 October 2012
Problem 28
Find the redshift dependence of the deceleration parameter. Analyze the limiting cases.
$$ \begin{gathered} q \equiv - \frac{a\ddot a}{\dot a^2} = - \frac{\ddot a}{a}\frac{1}{H^2}; \\ \frac{\ddot a}{a} = - \frac{4\pi G}{3}\sum\limits_i \left( \rho _i + 3p_i \right) = - \frac{1}{2}H_0^2\sum\limits_i \Omega _{i0}\left(1 + 3w_i\right)(1 + z)^{3\left(1 + w_i\right)} ; \\ H^2 = H_0^2\sum\limits_i \Omega _{i0}(1 + z)^{3\left( 1 + w_i \right)}; \\ q = \frac{3}{2}\frac{\sum\limits_i \Omega _{i0}\left( 1 + w_i \right)(1 + z)^{3\left( 1 + w_i \right)}} {\sum\limits_i \Omega _{i0}(1 + z)^{3\left( 1 + w_i\right)} } - 1\\ \end{gathered} $$ In the SCM $$ q = \frac{1}{2}\frac{\Omega _{m0}(1 + z)^3- 2\Omega _{\Lambda 0} } {\Omega _{m0}(1 + z)^3+\Omega_{\Lambda 0}} $$ $$ q(z \to \infty ) = \frac{1}{2},\;q(z \to - 1) = - 1 $$
- 12 24.jpg
Dependence of the deceleration parameter on the redshift.