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I'm currently reading a cosmology textbook, looking at inflation in particular. I have been seeing "scales smaller than the horizon [Hubble length]" or "horizon crossing" in the context of inflation. I'm just wondering what those phrases mean. What are they using as a scale? What does it mean to 'cross' the horizon? Is this all related to the causality between two events (say, two different regions of the CMB - or in this case, the 'future' CMB) within the observable universe at that time? Or am I getting all this completely wrong?

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You're on the right track. Inflation solves, amongst other things, the homogeneity/horizon problem: the problem that distinct patches of the sky are

  1. causally disconnected; but
  2. the same temperature.

To solve this, the Universe underwent a period of inflation. 'Scales', that is, distances between patches of the Universe, inflated during inflation. Before inflation, the patches were causally connected and equilibrated. After inflation, those scales expanded, crossed the horizon and thus the patches now appear to have been causally disconnected. At late times, those scales may re-enter the horizon.

Note that smaller scales are the last to exit and first to re-enter the horizon.

enter image description here

In this figure, taken from these notes, 'scale' relative to the Hubble radius is shown on the y-axis (which decreases up the page, as the Hubble radius grows with time) and time is shown on the x-axis (which increases to the right of the page). You can see both horizon crossing (first exiting, then re-entering once inflation has finished).

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