Lifetime of the universe - FRW

Tags:
1. Apr 18, 2015

unscientific

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
$$\left( \frac{\dot a}{a}\right)^2 = H_0^2 \Omega_{m,0} a^{-3} - \frac{kc^2}{a^2}$$
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.

2. Apr 19, 2015

unscientific

bumpp

3. Apr 20, 2015

unscientific

bumpp

4. Apr 22, 2015

unscientific

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

5. Apr 23, 2015

unscientific

bumpp

6. Apr 24, 2015

unscientific

bumpp

7. Apr 25, 2015

unscientific

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

8. Apr 26, 2015

unscientific

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

9. Apr 30, 2015

unscientific

limits anyone?

10. May 4, 2015

solved.