For example, if I've written a character that is supposed to be smart and uses a scientific theory as the basis for doing something interesting; does that theory have to be cited in the story. I don't think I've seen this happen, but I'd really would like to know. I know in certain research papers, there are certain theories that have to be cited, but for this I don't know.
I'm an Atheist, but the bible somtimes makes a good point:
"Even fools are thought wise when they keep silent; with their mouths shut, they seem intelligent."- Proverbs 17:27-28
Highly intelligent characters always have the downside of being limited to the intelligence of their writers, i am not calling you dumb or something but doubt you are a rocket scientist.
So the best thing to do is keep a bit of a blank area on the details because this is a classic example of more being less.
For example in a crisis situation SC (Smart Character) does something smart in the field of engineering and saves the group from certain death. The SC would not naturally say "O yea i got the idea from this book/paper and it works like this bla bla bla" because that sounds rather douche-baggie/arrogant as if he thinks himself superior because he had that bit of knowledge stored. It would be more organic if he just stays silent about it and somebody else asks how he did it with the SC giving the base explanation (no more then a sentence or two) followed up with "I read it somewhere". The other characters who probably don't have the intelligence to Truly understand it wont spend to much time thinking about it.
A good character example would be Eugene from The Walking Dead, he's a smart character but whenever he tries to explain something most characters get annoyed (and he comes over rather douchy) and just go like "Yea ok, i got other stuff to do".
The key point is for the reader to believe the motivation for your character's actions. It therefore comes down to how you explain that motivation.
Is your character rational? [Or not - consider the "mad scientist" trope, or genuine madness such as John Nash.] Are they a creative genius who invents something amazing, or the dull plodding type whose painstaking research reveals a scientific anomaly that points to looming catastrophe?
In other words, to what extent does your character need to explain themselves? Are they credible when they say "I've discovered how to travel faster than light - trust me, it's too complicated to explain"? Or does the story context require greater justification - for example, your character has to convince their university that their research or discovery is worth investing in?
Bottom line: is the scientific theory merely a plot device or vehicle for character development (in which case, citing a reference is unnecessary and could be a distraction), or is it a core component you need to bring the reader up to speed on?
Consider also that what was once readily acceptable in science is now seen as ignorant. Jules Verne firmly based his novels on scientific theory: for example, From the Earth to the Moon was surprisingly accurate on many of his estimates of trajectory etc, and the "science" underpinning the plot was readily accepted by his readers. A century later, readers would have been less accepting of the core premise - sending people to the Moon by putting them inside a projectile and shooting it out of a huge gun barrel!
The only requirement in your story is that the reader willingly suspends belief. After all, if it's fiction then by definition it simply isn't true, so there's already a suspension of belief! How you maintain the reader in that state comes down to the credibility of your characters and the situation they find themselves in.
In science fiction, for instance, almost no one bothers to explain faster-than-light travel or anti-gravity or tractor beams or teleportation or time travel or antimatter weapons, even though at human scale these are all scientifically impossible.
In your story, can the character be believed when they say "Professor Schmizbrick's calculations show that we can do X" (or that terrible thing Y might happen)? Or is it an essential component of the story arc that your character explores the science in more detail?
You can base the whole plot around a scientific theory without having to provide anything more than a glib pseudo-science explanation. The Earth is imperilled because the magnetic poles have changed and we need to send a team to the molten core of the planet to fix it. A superstorm is going to trigger a period of intense global glaciation so we have to evacuate half the USA. A super volcano is going to erupt. An asteroid is going to hit. A virus is going to escape. A new energy source has been discovered. I got irradiated and now have a super-power. I shrank the kids. Robots run amok. And so on, without any need to support the story by citing "real" science.