This is one of my two biggest speculations in terms of the breadth of plot elements and themes it covers. It's also one of the few times I'll connect Deltarune and Undertale since it requires the WD Gaster content from Undertale to make sense of. Ordinarily I would only bring in evidence from Undertale to supplement what we see in Deltarune, but here I will have to rely heavily on it. However, I still think the stories of Undertale and Deltarune are entirely separate, and Gaster's appearance in Undertale is actually a retroactive easter egg since he appears to be much more important to the story of Deltarune. I will present things in an order that seems logical to me but is not chronological, so information from Undertale will be brought out in the middle of discussing Deltarune. This page will primarily focus on Gaster, so concepts like Titans and the Angel will be covered on other pages, even if they related to Dark Worlds.
A common trope of Deltarune (and Undertale) theories is connecting anything unexplained or mysterious to W.D. Gaster. I will try to avoid that here and follow what we've been shown as of chapter 4.
At the start of Chapter 3, Ralsei explains the exact nature of Dark Worlds. This ties into the hidden Entry 17 from Undertale which I will explain later.
To summarize, as light is removed from a room, objects in it become more indistinct and can appear to move. Eventually, complete darkness is reached where nothing is visible. However, if the light that is no longer there is removed, a situation where things begin to become visible again arises. That is the nature of a Dark World, and the Darkners are the objects in the original room brought to life as a kind of illusion or fantasy. The phrase "darker than dark" is used by Ralsei to describe the removal of light that isn't there.
The Gaster followers who can appear in Hotland give his backstory. From the first follower:
It makes sense why ASGORE took so long to hire a new Royal Scientist.
After all, the old one... Dr. Gaster. What an act to follow!
They say he created the CORE.
However, his life... Was cut short.
One day, he fell into his creation, and...
So Gaster was the former Royal Scientist before Alphys was appointed. He fell into "his creation" and presumably died or vanished. The second follower adds:
Alphys might work faster. But the old Royal Scientist, Doctor W.D. Gaster?
One day, he vanished without a trace.
They say he shattered across time and space.
Ha ha... How can I say so without fear?
I'm holding a piece of him right here.
The third follower says something similar to the first two, but does mention that "it's rude to talk about someone who's listening" adding to the idea that Gaster was shattered across time and space and now exists everywhere at all times.
And now we come to the connections between Gaster in Undertale and the Dark Worlds of Deltarune. In the True Lab accessed in the pacifist route in Undertale, there are several entries written by Alphys numbered 1 to 21. However, entry 17 is missing in the current release. In previous versions of the game, there was an Entry 17 written by Alphys which is still in the code, but no longer accessible in game. However, there is also another entry 17 in the game data, written in Wingdings, associated with WD Gaster:
DARK DARKER YET DARKER
THE DARKNESS KEEPS GROWING
THE SHADOWS CUTTING DEEPER
PHOTON READINGS NEGATIVE
THIS NEXT EXPERIMENT
SEEMS
VERY
VERY
INTERESTING
...
WHAT DO YOU TWO THINK
This seems to be from an experiment to take away light that wasn't there. Photons are the quantum particles that make up light, so a negative reading of them would be "darker than dark." Whatever Gaster was doing, he would have created a Dark World (and possibly a Dark Fountain) by doing so.
Putting together what we have so far, we can summarize what happened to Gaster before the events of Undertale (and Deltarune):
Gaster's exact role in Deltarune is unclear and whether we will see him directly or if it will be like Undertale where he's only ever mentioned in RNG-locked secrets remains to be seen. However, his influence seems to be even stronger in Deltarune than Undertale.
In the introduction to Deltarune, the player is asked to create a "vessel" by an unnamed voice which speaks in ALL CAPS. In response to the player's chocies, the Voice may say "VERY INTERESTING" and uses other speaking mannerisms reminiscent of Gaster's entry 17 from Undertale.
There is also another voice in this introduction, who tells the player they can't choose who they are in this world and "discards" the vessel. Based on the way they talk in Japanese, I think they might be Chara, but it is not relevant to this theory. What is clear is that they are an entirely different person than the first voice, both because they reject the vessel the first voice has you create, and because they talk and behave completely differently.
As of chapter 4, Gaster has made no direct appearances, and I think it's unlikely he ever will, but his presence is felt throughout the game. This section will collect evidence for Gaster in Deltarune beyond the introduction, but does not seek to be exhaustive.
In Undertale, a song titled mus_smile.ogg plays during Gaster's entry 17 in an inaccessible room in the game. In Deltarune, if the player attempts to use Kris' cell phone in a Dark World, 6 seconds of mus_smile.ogg plays. I believe this is Gaster attempting to communicate with Kris (he is shown communicating directly with the player already, so he doesn't need to use a phone to do that). It seems Gaster needs some kind of "connection" in order to communicate, like he says at the start of Deltarune in the goner maker section.
It is implied that Spamton was communicating with Gaster on his phone, and possibly able to understand him. An Addison in chapter 2 says the following:
He must've left in the middle of a conversation,
Because I could still hear someone on the other end...
...
But when I put the receiver to my ear...
There was nothing but garbage noise.
I believe the shadow crystals are related to Gaster, and may be a part of him. They are distinct from the Black Shard(s), which come from the Knight or their sword, but are handed out by the Knight to the shadow crystal bosses. The text when the item is used is how I support this hypothesis; it seems like it might show an alternate universe where things go much worse than what happens in the game. Examples are:
Another possibility for the shadow crystals is that they are (like) human SOULS. Since there are seven chapters and presumably one shadow crystal per chapter, along with the text in chapter 4 when you try to drop a crystal making it sound like they are a SOUL, this is possible, but I still connect them with Gaster. Since Gaster was "shattered across time and space," it is feasible to believe that parts of him would touch alternate universes, and you could see those alternate realities through those parts, sort of like an Aleph.
Gaster/the Voice is almost certainly connected to the Shelter. When you approach the Shelter in the Light World, a slowed down version of mus_smile.ogg plays, and at the end of Chapter 3 the darkness from a Dark World can be seen emerging from the doors of the Shelter. It's possible Gaster was able to influence somebody to set up a research lab or equipment in the Shelter, or he can manifest there in a way that is more direct than what we have seen so far. The Shelter itself may warrant another page for how many mysteries surround it, but it needed to be mentioned here. Alphys in Chapter 4, when asked about the Shelter, states that it's "place for people to survive an emergency, right?" but then goes on to suggest that maybe it's locked to keep something inside:
The shelter... it's... a place for people to survive an emergency, right?
But, ever since I've moved here, it's been... closed for some reason.
It makes me wonder if... even if there was some disaster...
Whatever's inside there... might be worse...?
These are the questions I still have revolving around Gaster and Dark Worlds. I do not have a good solution for them, so I'll just write them down and wait for chapter 5.