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In Barbara Ryden's Intro to Cosmology, in chapters 4.1 and 6.1, she states that $\dot{a}$ (or equivalently, $H(t)$) enters only as $\dot{a}^2$ in the Friedmann equation. She then deduce that a contraction phase is just the time reversal of the expansion phase. She then continues the chapter with taking only the positive root of $\dot{a}^2$.

A few questions:

  1. What is the explanation that $\dot{a}^2$ in an equation implies that contraction is the time reversal of expansion?

  2. Given that, why can we ignore the negative root of the Friedmann equation?

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  1. If you find a solution with da/dt > 0 then this solution will describe an expanding universe. If you then take that solution and replace t by -t everywhere you can see that the sign of da/dt changes so da/dt < 0 so this solution will describe a contracting universe. Then if you plug this solution into the Friedmann equation you find that this is also a solution because da/dt squared is the same as da/d(-t) squared.

  2. You don't have to ignore the negative root, but as we are more interested in the expanding universe from the big bang initial conditions then it is the positive root she wants to focus on.

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