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Lifetime of the universe - FRW

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1. The problem statement, all variables and given/known data

(a) Find the value of A and ##\Omega(\eta)## and plot them.
(b) Find ##a_{max}##, lifetime of universe and deceleration parameter ##q_0##.
2013_B5_Q3.png


2. Relevant equations

Unsolved problems: Finding lifetime of universe.

3. The attempt at a solution

Part(a)

FRW equation is given by
[tex]\left( \frac{\dot a}{a}\right)^2 = H_0^2 \Omega_{m,0} a^{-3} - \frac{kc^2}{a^2} [/tex]
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
[tex]a_{max} = \frac{A}{k} = \frac{H_0^2}{kc^2}\Omega_{m,0}[/tex]
Deceleration parameter is given by
[tex]q_0 = -\frac{\ddot a_0 a_0}{\dot a_0^2}[/tex]
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 - 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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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.
 

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