Deceleration parameter
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Observational cosmology problems
Problem 1
problem id: 150_dp0
Show that the fact that in the present time $t_0$ the relation $H_0t_0\approx1$ holds unavoidably follows existance of the accelerated expansion stage of the Universe evolution.
The combination of the definitions \[H=\frac{\dot a}{a}=\frac{d\ln a}{dt},\quad q(t)=-\frac{\ddot a a}{\dot a^2}\] give the equation \[\dot H+(1+q)H^2=0.\] Integrating this equation between the limits $x=0$ and $x=1$, $x\equiv t/t_0$, and taking into account that the present value $H_0t_0\approx1$ we obtain \[0\approx\int\limits_0^1q(x)dx.\] At a very high redshift $q(x)>0$. There must be periods (or a period) of accelerated expansion in the history of the universe in order to turn the integral $\int_0^1q(x)dx$ to zero.
Problem 2
problem id: 150_dp2-problem-1
Suppose that $dq/dt=f(q)$. Find the Hubble parameter in terms of $q$.
\[1+q=\frac d{dt}\left(\frac1H\right),\quad \frac{dq}{dt}=f(q)\Rightarrow dt=\frac{dq}{f(q)},\] \[\frac1{H(q)}=\int\frac{1+q}{f(q)}dq.\]
Problem 3
problem id: 150_dp2-problem-2
Find connection between the scalar curvature and deceleration parameter for the flat Universe.
Using the relation \[R=-6\left(\frac{\ddot a}a+H^2\right)\] and definition of the deceleration parameter \[q\equiv-\frac{\ddot a}{aH^2},\] one finds \[R=-6H^2(1-q).\]
Problem 4
problem id: 150_dp3
Show that the deceleration parameter $q$ relates the density of the Universe $\rho$ to the critical density $\rho_{cr}$ through \[q=\frac12(1+3w)\frac\rho{\rho_{cr}}.\]
\[H^2=\frac{8\pi G}3\rho-\frac k{a^2}\Rightarrow1+\frac k{a^2H^2}=\frac{\rho}{\rho_{cr}}.\] Using \[q=\frac12(1+3w)\left(1+\frac k{a^2H^2}\right),\] we obtain \[q=\frac12(1+3w)\frac\rho{\rho_{cr}}.\]
Problem 5
problem id: 150_dp4
Solve the previous problem using the second Friedmann's equation.
\[-\frac{\ddot a}a=\frac{4\pi G}3(\rho+3p)\Rightarrow-\frac{\ddot a}{aH^2}=\frac{4\pi G}{3H^2}(1+w)\rho,\] \[q=\frac12(1+3w)\frac\rho{\rho_{cr}}.\]