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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##.

**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.