Simple dimensional analysis or not, there a lot of cosmology folk around who go along with the evidence that the universe does indeed appear to be expanding and gaining speed.
This, from Wikipedia should start you in the right direction for researching the issue.
Of course there are other thoughts on the subject, cosmologist who suggest the universe may not be expanding at all, but I don't recall any of them suggesting the reason lie within simple dimensional analysis.
I never denied that each galaxy is receding from each other galaxy (in general) at an accelerated rate (for any observer on those galaxies). This is not the contention.
It all boils down to the definition of Universe. If we call Universe all the space and the matter it contains, leaving out time, then yes, the Universe is expanding at a certain rate. Which translates into: space is expanding at an accelerated rate.
The problem is that we cannot leave out time. Time is not external to the Universe if relativity is true. It is not a metaphysical external stage in which the whole Universe evolves. Only space and time together (Spacetime) make sense of relativity.
So, even if we observe expansion of space, we do not observe expansion of spacetime. The very concept of spacetime expansion is inherently absurd, since things like "rate", "velocity", make sense only within it.
And if we extend the definition of Universe so that it contains time as well, as we should, then we have a 4-dimensional continuum that does not expand at all, as a whole.
A possible analogy, albeit not perfect, is to imagine yourself falling into a huge funnel starting from the funnel's neck. Now, you can consider the direction of your falling as time (something you cannot stop from increasing) and the orthogonal region encompassed by the walls of the funnel as space. You can only see this orthogonal space.
What do you experience while free falling? You will peacefully float while observing that space around you is getting bigger and bigger (because the walls of the funnel get larger as you fall down). Since you do not have any other reference point, you think that space around you is increasing at a certain rate and there must be therefore a beginning of your two dimensional space.
But if we consider the whole funnel and what contains as a whole, we will see that it is nor expanding at all. It never "began". It is you who is dragged to see regions of it with bigger surrounding space extensions.
Ciao
- viole