- #1
roam
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Homework Statement
(a)[/B] From the continuity equation show that if ##P=\omega \rho## and ##\omega > -1/3##, show that an expanding universe will eventually reach a maximum size. Take ##k=1## (closed universe).
(b) Show that if ##\omega <-1##, the energy density ##\rho## will increase as the universe expands.
Homework Equations
Continuity equation: ##\dot{\rho}+3 \frac{\dot{a}}{a} (\rho +P)=0##
First Friedmann equation:
##\left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2}##
The Attempt at a Solution
(a)[/B] The hint says I must first solve for ##a## when ##\dot{a}=0## the first Friedmann equation and then consider ##\ddot{a}##. So solving that equation yields:
##a=\sqrt{\frac{3k}{8 \pi G \rho}}=\sqrt{\frac{3}{8 \pi G \rho}}##
To find ##\ddot{a}##, I substitute this into the second Friedmann equation:
##\frac{\ddot{a}}{a}= - \frac{4 \pi G}{3} (\rho + 3P) \implies \ddot{a} = \frac{-4 \pi G}{3} (\rho +3P). \sqrt{\frac{3}{8\pi G \rho}}##
Using the equation of state ##\rho=P/\omega##
##\ddot{a} = \frac{-4 \pi G}{3} ((P/\omega)+3P). \sqrt{\frac{3}{8\pi G (P/\omega)}}##
Do I need to show that we have positive acceleration (i.e. ##\ddot{a}>0##)? I think the equation above shows that acceleration is reduced as the universe expands (##\ddot{a}## decreases as P decreases).
Also, how can I use the continuity equation here?
(b) Do I need to be looking at the expression I found previously with negative P?
Any corrections or explanation is greatly appreciated.