# What is the relation between comoving particle horizon $d_{hor}^c(t)$ and comoving Hubble radius $1/(aH)$ for arbitrary expansion laws?

I was learning Inflation by reading Hannu's lecture. But on page 4 of this lecture (Chapter 7 Inflation, 7.2.1 Accelerated expansion), there are some things I don't understand. The content is shown in figure.

Everything is OK except content of the red box.

Here are my questions:

1. What does it mean that "if $$\mathcal{H}^{-1}$$ is increasing with time, the comoving distances traveled at earlier 'epochs' are shorter"? Why shorter?
2. What does it mean that "if $$\mathcal{H}^{-1}$$ is shrinking, then at earlier epochs light was traveling longer comoving distances"? Why longer?
3. What is the relation between his statement (content in the figure) and the conclusion $$d_{hor}^c(t)\gtrsim\mathcal{H}(t)^{-1}$$ ? I don't get it.

1. If $$\mathcal H^{-1} \equiv \dot a^{-1}$$ is increasing, that means the growth rate of the scale factor is shrinking. If you convert the distance traveled by light at early times to a comoving distance corresponding to physical distance today, then compared to the case in which $$\dot a$$ remained constant, that comoving distance will be smaller since the scale factor hasn't grown as much between then and now.
2. If $$\mathcal H^{-1}$$ is decreasing, the growth rate is speeding up, so distances frozen in early on will be larger in comoving coordinates than if the growth rate had remained constant.
3. $$\mathcal H^{-1}$$ defines the comoving distance beyond which new signals cannot reach an observer assuming $$H$$ remains constant. So
$$d^c_{hor}(t) \gtrsim \mathcal H^{-1}(t)$$
means that the comoving distance light could travel from the Big Bang to today is greater than that distance. This is because the signal from the Big Bang would have already crossed the comoving distance $$\mathcal H^{-1}$$ at an earlier time, and would already be inside the Hubble volume, able to propagate to the observer. I'm not sure if this is intended to follow from the red box, or it's just being stated in addendum.