Is the expansion of the universe really accelerating? Or does it just look like it? I pulled this quote from an article on the Hubble Constant:
"...for example, if the Hubble Constant was determined to be 50 km/s/Mpc, a galaxy at 10 Mpc, would have a redshift corresponding to a radial velocity of 500 km/s."
In this illustration provided from the article, if this was the measured observation from earth, this describes a constant rate of expansion, not an acceleration. It only appears to be an acceleration because of our somewhat fixed vantage in the universe. If the expansion is actually accelerating, we would expect to see velocity in addition to the constant every time you measured out another 10 Mpc (to stay within the framework of the provided example). So if the constant rate of expansion was 50 km/s/Mpc , an expansion that was also authentically accelerating would yield 50 km/s/Mpc +X ... with X equaling the velocity added by the acceleration of expansion. One number describing a velocity does not give any indication of acceleration, what is needed is a ratio that shows an increase in velocity over distance beyond what one would expect to observe from a constant expansion.
Hubble's constant is how we know the universe is expanding, but I am not clear how it shows acceleration, if it indeed does.
So... I already know that my observation is not common knowledge because it is never brought up in public explanations about the expansion or conversations about dark energy. There is often a mention of how the expansion is happening everywhere at once, and this is illustrated in several different ways to help confused people understand why it seems we are in the "middle" of the big bang... I get all that, no need to re-hash. What I am saying, is there is a corollary observation of "acceleration" to the expansion that is equally confusing. A universe expanding at a steady rate won't look much different from a universe expanding at an accelerating rate. An authentic acceleration may indeed be occurring, and perhaps it is simply not explained well, because no one has come up with good pictures to paint to describe the difference without thick equations... I dunno. That's what I'm trying to get to the bottom of.
What is the rate of "acceleration" and how are we differentiating it from the observed "acceleration" of distant objects we would expect to observe given a constant rate of expansion?
 A: This seems to be the main problem you have :

A universe expanding at a steady rate won't look much different from a universe expanding at an accelerating rate.

And the answer to this depends on a key point : light travels at a constant speed and, on the scale of the universe, a speed that lets us see into the past.
When you look at very distant objects in the universe you are seeing them in the past.  Say an object appears to be 5 billion light years, then the light we saw it by took five billion years to reach us and the object has moved since then.
We can tell they are moving because we can measure the wavelengths of certain lines in their spectrum and we know the value they would be if they were at rest and any difference tells us the velocity of the objects relative to us.
And from that we see a very interesting thing : the further away objects are (on a cosmic scale) the faster they are going.  This data is consistent in all directions and gives rise to Hubble's Law.
That tells us that there is an acceleration going on and it relates to the distance objects are from each other.
So we reached the point of explaining that from our "snapshot" of the universe we see now, we can tell not only that it is expanding, but that there is an acceleration of that expansion.
Now you don't want mathematics, so all I can do is tell you that after the general theory of relativity (GR) was discovered a very significant theoretical finding was made called the FLRW metric.  This metric predicts something non-relativistic physics cannot - that the universe can expand in the way we see.
More recently we have discovered that the universe is not simply expanding, but that the acceleration is faster than our early theory explained.  This has led us to consider that something called "dark energy" exists that "super accelerates" (for want of a better expression) the expansion.
A: I am assuming you would prefer not to have an answer with math. If I am mistaken about this, please let me know so I can present a more "correct" explanation about the acceleration.
You show that you accept (in the manner of your example) that the universe is expanding according to the Hubble Law (please see
https://en.wikipedia.org/wiki/Hubble's_law ).
speed of a distant object = Hubble constant times distance.
It was well known since the 1920s that the Hubble constant is not a constant, but its value changes over time.
An object moving with a speed relative to an observer (less than the speed of light), either away from or toward the observer, is also subject to the possibility that it can accelerate or decelerate. Before about twenty yeas ago, it was understood that the universe was decelerating. That is, the Hubble constant was getting smaller since the universe beginning. The rate of deceleration of the universe was in terms of the deceleration parameter
$$q = -(1 + \frac {dH/dt} {H^2} ).$$
About twenty years ago, after quite a collection of astronomical data was collected, it was determined that the value of q changed from negative (decelerating) to positive (accelerating).
There is an equation which related the value of he Hubble constant to what is called the "scale factor", which is a scale of the size of the universe. The Hubble constant is the rate of change of the scale factor divided by the scale factor. The Friedmann equation equation describes how the Hubble constant changes as the scale factor changes. This equation also predicts that in the distant future the universe acceleration will slow down, and the Hubble parameter will become an actual non-changing constant value.
