Lifetime of the universe - FRW

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Homework Statement

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(a) Find the value of A and $\Omega(\eta)$ and plot them.
(b) Find $a_{max}$, lifetime of universe and deceleration parameter $q_0$.

Homework Equations

Unsolved problems: Finding lifetime of universe.

The Attempt at a Solution

Part(a)[/B]
FRW equation is given by
$$\left( \frac{\dot a}{a}\right)^2 = H_0^2 \Omega_{m,0} a^{-3} - \frac{kc^2}{a^2}$$
Subsituting and using $dt = a d\eta$, I find that $A = \frac{H_0^2}{c^2}\Omega_{m,0}$.
Using $\\Omega_m = \Omega_{m,0}a^{-3}$, I find that $\Omega_m = \frac{kc^2}{H_0^2 sin^2(\frac{\sqrt{k} c \eta}{2})}$.

Part(b)
Maximum value of normalized scale factor is
$$a_{max} = \frac{A}{k} = \frac{H_0^2}{kc^2}\Omega_{m,0}$$
Deceleration parameter is given by
$$q_0 = -\frac{\ddot a_0 a_0}{\dot a_0^2}$$
This can be found by using $\sqrt {k} c \eta = sin (\sqrt {k} c \eta)$.

How do I find the lifetime of the universe? Is it simply $\int_0^\infty t d\eta$? If I can solve for the lifetime, I can compare it to its current age and see if that is feasible.

 

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bumpp

 

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bumpp

 

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bumpp - What is the expression for lifetime of a universe? Is it simply$\int_t^{t_0} dt = \int_0^\eta a(\eta) d\eta$?

 

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bumpp

 

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bumpp

 

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Is the lifetime simply $\int dt = \int \frac{1}{aH} da$? If so, what are the limits of integration?

 

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If it is a closed universe, curvature eventually dominates and $a \rightarrow 0$? so the limits would be from $1$ to $0$?

 

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limits anyone?

 

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solved.