Evolution of Universe
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 $$ q = \frac{1}{2}\frac{\Omega _{m0}(1 + z)^3- 2\Omega _{\Lambda 0} } {\Omega _{m0}(1 + z)^3+\Omega_{\Lambda 0}} $$ $$ q(z \to \infty ) = \frac{1}{2},\;q(z \to - 1) = - 1 $$
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Dependence of the deceleration parameter on the redshift.
