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$$\left(\frac{\dot{a}}{a}\right)^2 + \frac{k}{a^2} = \frac{8 \pi G}{3} \rho$$ $$\frac{d}{da}(\rho a^3) = -3 w\rho a^2$$ I am trying to integrate the Friedmann equations to solve for the scale factor $a(t)$. I have two questions.

  1. I understand that the scale factor $a(t)$ gives me the size of the universe at a given time. If $a(t)>a(t')$, then the universe is bigger at time $t$ than $t'$. However, what does the absolute value of the scale factor tell me about the universe at a given point in time? For example, if the scale factor is $a=1$, how does one interpret this?

  2. I simulated the universe and got the solution curve $a(t)$ for values of $k=0,-1,+1$. The curve only deviated from each other. But as I understand $k=0,-1,+1$ implies very different kinds of universe. $k>0$ is finite volume but $k=0$ is infinite volume. How does one interpret $a(t)$ in the three cases and only a deviation in the $a(t)$ that was observed in the simulation?

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  1. The scale factor is an arbitrary measure of the size of the universe. We never worry about the scale factor directly, but instead work with ratios of the scale factor (e.g. ask what was the scale factor back then compared to today). By analogy, think about redshift. A photon from a distant galaxy at z=1 has doubled it's wavelength by the time it reaches us. It doesn't matter if $\lambda=400$ nm or $\lambda=400$ nm. In this analogy, you have think of the scale factor as the photon's wavelength. Often, for simplicity we set a=1 at present day (although nothing will change if a different convention is used).

  2. I don't quite follow what you are asking in your second question, but universes with different curvature will grow and evolve differently (e.g. different evolution of $a(t)$ since you have a different differential equation to solve).

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