Why is special relativity not enough to explain the seemingly accelerating expansion of the universe? I have only recently studied Special Relativity, and have yet to learn anything regarding General Relativity, so I may be asking this due to a lack of knowledge - please educate me.
I just read about the discovery that the expansion of the universe is surprisingly accelerating instead of decelerating, and this effect is credited to some "Dark Energy".
Though if I understand correctly, according to Special Relativity, space will keep compressing around a relatively accelerating observer, so the opposite case is that space will expand in an accelerating manner around a relatively decelerating observer.
Am I correct? And if so, why isn't it enough to explain this phenomenon and why do we still need Dark Energy to explain it?
 A: Let me start by saying that it's not quite correct to say space compresses around an accelerating observer and expands around a decelerating observer. But I do see what you mean and, since I'm not a mathematician, I'm willing to accept your wording and work within that framework to explain the answer to your question.
Point the first:
We are not decelerating. We consider ourselves to be in mostly inertial motion. We are also accelerating slowly towards the Andromeda galaxy. Given this fact, your premise would not be enough to explain the expansion we see, let alone the acceleration of that expansion.
Point the second:
The expansion/compression of space as a special relativity phenomenon is not isotropic. That is, it does not affect your perception of spacetime the same way in every single direction. Space is "compressed"/"expanded" more along the direction of your motion than along the sides. Also, things beside you tend to bend behind you as you accelerate. You end up with a very polar view of space with your destination ahead of you and pretty much everything else behind you. However, the expansion of space we see is isotropic; it looks the same in every direction.
Number three:
Special relativity can only be applied relative to an observer. If we are decelerating relative to one observer, then we are actually accelerating relative to another. Why is this the case? Why can general relativity be applied to all of spacetime but special relativity cannot? Because of the very thing that separates one from the other. Special relativity is quite literally the special case of general relativity where spacetime is assumed to be flat, boring, and unchanging. One can describe effects on space and time within it but only when referencing another observer's point of view. General relativity, on the other hand, allows for there to be things with mass, energy, and stress. It allows for spacetime to be something other than flat and boring when in the context of one inertial observer. So, were we to use special relativity to explain expansion, there would also be another observer/frame of reference to which space would be contracting. By using general relativity, we can have space expand to people in all reference frames by having some non-boring structure of spacetime onto which we add the effects of various types of energy.
Side note: I know this all seems a bit hand-wavy. I'm trying to explain it without going into all the math. At your current level of study, I fear the math would very much overwhelm you. Problem is, this topic is literally all math. I mean the actual answer to your question would sound like "As you can see in step 634, this term here in the equation represents dark energy. This cannot be a special relativity phenomenon because, if you recall from step 182, we derived that no special relativistic phenomenon could produce an expansion that looks similar to the data our observations....." You get the idea. It's gross and a lot. So again, sorry for being really vague but my answer options are "this is simple and vague because trust me" or "big words, math stuff, big words, look how important I think I am". I'm going with the one that might be helpful.
Final point (and probably the most important):
We know it's not a special relativity effect because we have carefully calculated all possible effects of SR and removed them from the data. The best data we have for a lot of cosmology are maps of the cosmic microwave background (CMB). The maps we have show it all looking the same from every direction. However, Earth is actually moving relative to the CMB. When we take photos, there's a dipole in the data. One side is blue-shifted and the other is red-shifted due to our movement. Our very diligent physicists spend years working on all of this, figuring it from every angle, and arguing with each other about who is more correct incessantly. They have calculated all the SR effects and when they remove them, we're left with a beautiful image of space that is isotropically expanding. Since we've already removed all of the SR effects, what remains must be a GR effect to explain why the universe expands. Naturally, that the expansion is accelerating is a further detail within that GR formulation.
A: Measurements of accelerating expansion are done by generally comparing velocity of distant galaxies by their redshift up against measurement of distances using Type 1A Supernovae.
Not a collection of measured distances between two objects over time. As would have related to Special Relativity.
