Misconceptions about the universe - Hubble Sphere

Ok, so I've finally got to the bottom of this. Basically, the confusion stems from what it means to say the universe is accelerating so let's clear that up. The Hubble parameter is defined to be $H(t)=\dot a(t) /a(t)$ where $a(t)$ is the scale factor. When we say that the universe's expansion is accelerating, we mean that $\ddot a$ is greater than zero. This absolutely does not, however, tell us anything about $H(t)$. $H(t)$ could be increasing, it could be decreasing or it could even be constant.

The de Sitter model, for instance, describes a universe in which the Hubble parameter, $H(t)$, is actually constant in time. That is, $\dot a(t) /a(t)=H_0$ which has an exponential solution for $a(t)$ but, for the most part, $H(t)$ can be decreasing or increasing despite $\ddot a$ being greater than zero.

How can I see this?

Right, picture two cars driving apart with an infinitely elastic bit of tape between them. They have some markings on them which you can think of as galaxies (or comoving coordinates). I've drawn three sets of pictures and I will explain them each in turn.

Picture one

First consider two cars driving apart at speeds $3c$ (ignore the unrealism of that claim - I'll take $c$ to be $10$mph) with us at the centre. These cars are not accelerating. These cars are driving away at $30$mph in opposite directions. Somewhere along the tape, there's a Hubble Sphere where comoving galaxies are receding away at $c$ (or $10$mph). I've marked it in green. We can draw a chart of recession velocity, $v_{rec}$ against proper distance, $d$, and see that there is a linear relationship between the two. That is, things 2km away from us are moving twice as fast as things 1km away from us. The Hubble parameter, $H(t)$, is the gradient.

Picture 2

Keep with the idea that these cars are not accelerating and let's have a look a little while later. Remember, they're not decelerating either. Everything got pulled apart and the Hubble Sphere goes along for the ride. Think about it - if the cars are still going $30$mph then our Hubble Sphere is still one third of the way along the tape from us. Importantly, if the Hubble sphere is growing it can start capturing photons that were once outside the Hubble Sphere and, once these photons are inside, they can start their journey towards us. We will see these photons. Note that the gradient of our $v_{rec}-d$ plot has tanked as the Hubble Sphere expanded.

Picture 3

Ok, let's add acceleration or, for the mathemagicians, $\ddot a(t)>0$. What happens now? Well, it depends on how hard the cars are accelerating. In my example, I made the cars accelerate just hard enough that the Hubble Sphere maintains constant size. Notice the gradient staying constant in the chart to the right (cough, de Sitter cough). In this case, the photons outside are stuffed. They're never getting in. End of story.

But what about if the car didn't accelerate all the way to $6c$ ($60$mph)? What about if they only reached, say, $5c$. Well in that case, the slope on our plot would have dropped a bit, our $c$ marker will be one fifth of the way along and our Hubble Sphere will have moved out a bit. What was that? The Hubble sphere moved about even though the universe's expansion is accelerating? Bingo.

How do we fit into all this?

Believe it or not, we think we're in that last camp. The 2011 Nobel Prize was awarded for the discovery that the universe is (or cars are) accelerating. But it's not happening at a rate fast enough to maintain a constant sized Hubble Sphere so the Hubble Sphere is also growing. Because it's growing, we can capture more photons from galaxies that are currently receding away at superluminal velocities. These photons can then begin their journey towards us.

One final thought

Was Veritasium right? Well... he wasn't really wrong. Then again, saying "due to the accelerating expansion of space" isn't really right either. Perhaps he should have said, "due to the decreasing Hubble parameter..." instead. Oh well... who cares. This took a long time for me to figure out - I truly hope it helps someone else out.