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In Sean M Carroll's Introduction to General Relativity: Spacetime and Geometry, he introduces the Hubble parameter as the 'rate of change of expansion': $$H = \frac{\dot{a}}{a},$$ where $a$ is the scaling factor in the Robertson-Walker metric.

My understanding is that $\dot{a}$ is the time derivative of the scaling factor, and is a measure of 'expansion', that is, how the scaling of two coordinates in comoving coordinates changes with time. If $\dot{a}$ is greater than zero, then the universe is expanding; else, it is contracting. The Hubble parameter, on the other hand, measures how the expansion $(\dot{a})$ varies with the scaling $a$. In a way, it is the expansion per unit scaling; the incremental expansion per scaling factor at every instant of time. So, $\dot{a} > 0$ and $\dot{H}<0$ means that the universe is expanding, but at a decelerating rate.

A couple of questions:

  1. The Friedmann equations in terms of the Hubble constant can be written: (see page 338 of the book) $$H^2 = \frac{8\pi G}{3}\sum_{i(c)}\rho_i,$$ where we sum over the various energy density contributions and also the fictitious curvature density. He argues that if all the densities are strictly positive, then the universe is expanding, and will never transition from expanding to contracting. I do not understand how this conclusion is reached, given that the left hand side depends on $H^2$, and not $H$.

  2. He argues that the time derivative of the scale factor $\dot{a}$ and the Hubble parameter $H$ are answers to two different questions. In particular, the Hubble parameter tells us by how much two test particles at a fixed initial distance are separated after a short time. The change in the scale factor tells us how a fixed source moves away from us in time. I do not understand how this conclusion is reached. Also, if it is possible for $\ddot{a}>0$ and $\dot{H}<0$ simultaneously, that would mean that two sources appear to move away from us at an accelerating rate while simultaneously expanding away from each other at a decelerating rate. How does that make sense?

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if all the densities are strictly positive, then the universe is expanding, and will never transition from expanding to contracting.

Continuous $H$ can only change sign if at some time $H=0$, which cannot occur if the density is positive.

[$H$] tells us by how much two test particles at a fixed initial distance [emphasis mine] are separated after a short time. The change in the scale factor tells us how a fixed source moves away from us in time.

These are respectively descriptions of relative and absolute rates of change in distance. Carroll regards an FID as a "unit" distance. At the time this initial distance occurs, we can regard $a$ as $1$ so $H=\dot{a}$, but of course $H$ doesn't change if we adopt a different convention.

$\ddot{a}>0$ and $\dot{H}<0$... would mean that two sources appear to move away from us at an accelerating rate while simultaneously expanding away from each other at a decelerating rate. How does that make sense?

The expansion would be superlinear but subexponential in time. Since $\dot{H}=\frac{a\ddot{a}-\dot{a}^2}{a^2}$, your scenario is $0<a\ddot{a}<\dot{a}^2$. You can verify this works, for example, if $a\propto t^n$ with $n>1$.

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