Suppose we put a shell around the solar system that displays a 2D image of the rest of the universe on its inner surface. What year would we have had telescopes powerful enough to determine that this was the case.
I think it would probably come down to the resolving power vs lack of parallax but I’m not sure what the orders of magnitude are for either of those.
The answer depends entirely on the sophistication of the simulation.
The answer by AlexP suggests that the screen could be discovered in the 1600s or 1700s by parallax, but that's only true if the designers of the screen failed to account for parallax—and why would they? Compared to simulating a lawfully behaving universe outside the sphere, handling parallax is trivial. You just have to render your image from the perspective of the Earth's location in the sphere, instead of the perspective of some fixed point like the center.
The only conceivable reason they might not do that is if they wanted people to discover the sphere at that particular level of technological sophistication, as in The Sentinel. That's possible, but it's just as possible that they would design the sphere to be detected at an earlier or later time. Whenever you want it to be detected for the purposes of your story is probably fair game here.
Assuming that the screen is electromagnetically "perfect", in that it reproduces exactly the radiation we'd expect to observe, including parallax AND spectra, with the required emission and absorption lines, all appropriately redshifted (quite a difficult challenge if you ask me), the answer is probably...
Although as @benrg correctly pointed out, we had been detecting cosmic rays a hundred years before this, and neutrinos from the SN 1987A supernova in drum roll 1987, we'd be definitely suspecting something by this point.*
This is the date Voyager 1 crossed the heliopause and took direct measurements of the interstellar medium. This is the area dominated by particles coming from the other parts of our galaxy, which would behave noticeably differently, were the Sun to be found not to be moving at breakneck speed through that medium, creating a bow shock and everything. In fact Voyager 1 data would've indicated that something was amiss years earlier, when passing the termination shock layer.
But even if we assume they also installed massive interstellar fans to simulate the particle flow, we've got...
This is the date of the first non-electromagnetic measurement we've made of our universe, when we successfully detected the gravitational waves from a black hole merger.
Now the game is up, that screen may simulate any electromagnetic or particle radiation, gravitational waves are a whole different ball game. There is no known way to fake them with a projection device.
Of course if they went to the length of installing a screen so vast it'd weigh more than most stars, why wouldn't they do this too...
* There is an interesting Earth-based corollary to this. We have some fairly credible hypotheses about how cosmic rays are seeding lightning in storm clouds. If they are correct, we'd be seeing a difference in how thunderstorms behave on our planet. Of course we wouldn't have anything to compare it to, so we probably wouldn't discover this clue until after some other clues were put together but I thought it was worth mentioning.
One light year is 0.3 parsecs.
One parsec is by definition the distance at which a distance one astronomical unit subtends an angle of 1 arc-second.
A star a distance of one light-year would show an annual parallax of a massive 6 arc-seconds.
Stellar parallax is the apparent shift in the position of a star as seen from Earth over the course of a year. It is measured relative to Earth, not relative to other stars.
Even early-1600s Galilean telescopes might have been good enough to measure this, but probably not. On the other hand, early 1700s telescopes were certainly good enough to measure such a massive annual parallax; we know this because in the 1720s James Bradley FRS discovered and measured the aberration of light, which, at about 20 arc-seconds, is of comparable magnitude and appears as a collective annual parallax of all heavenly bodies.
The first successful stellar parallax measurements were done by Thomas Henderson in Cape Town South Africa in 1832-1833, where he measured parallax of one of the closest stars, alpha Centauri. Few years later, 1835-1836. followed Friedrich Georg Wilhelm von Struve at Dorpat (nowadays Tartu) university observatory, who measured the distance of Vega and published his results in 1837. — Wikipedia
Bonus: the modern-day Hubble telescope has an angular resolution of about 0.4 arc-seconds, which means that it could resolve two light-sources one astronomical unit from one another at a distance of about 8 light-years.
By 1901, Annie Jump Cannon would have noticed that the spectral composition of light suggests that all stars are made of the exact same material, and moreover that they are all made of something entirely unlike our sun...
Assuming the universe exists outside the shell:
The Oort cloud is said to be larger than 1 light year in radius. Some of its objects may have eccentric orbits that should cross (hit) the shell at some point.
This is not an issue for objects coming from outside the shell, since its outer surface can withstand any kind of impact, while still displaying inside the shell, the projected object's trajectory, like if it hadn't been completely destroyed one second ago. Perfect illusion.
Even if destroyed, those objects can now be simulated for millions of years, virtually entering and exiting the shell.
On the other hand this is an issue for real objects coming from inside trying to reach their aphelion that's located outside the shell. This time impacting the shell's inner surface cannot be hidden by any kind of displaying technique since there is now stuff between the eyes and the screen.
Also, the solar system has a tail, so space dust trying to quit would accumulate more on one side, not to mention dust loves screens.
The answer depends on when did the shell appear, and how humans are able to detect exiting objects when they hit the screen.
Frame challenge: if you're supposing the aliens are capable of building a shell of radius 1 LY, then clearly they have very advanced technology. There is no reason they would build such a shell and then equip it with flat screens which work like our contemporary display technology, by showing the same image in all directions. If these aliens want to maintain the illusion, then it would be well within their technical capabilities for the screen to emit the correct light in the correct directions so as to account for parallax even at the scale of two observers at different places on Earth.
Contemporary human technology is already able to do the inverse of this ─ light field cameras capture not just the light intensity but also its direction. (This lets you do some neat things like emulate different lenses in software.) So imagine a "light field display" which can emit exactly the right photons with the right wavelengths and directions as needed, and there is literally no optical way to tell that apart from a real universe. In fact, the shell needn't even be so big.
Here are some ways we could figure it out:
There will be any number of string theorists, cosmologists, mathematicians, etc, working very hard to integrate the curious effect into normal known theories of the universe, with success always just a few years away.
Gravitational waves? Our theories of relativity are wrong, that's all.
Neutrinos? Dark matter must be absorbing them. Etc.
People postulating a hard shell will run into the hard faith position that is materialism. A hard shell? Can you imagine announcing such a thing? You would make the trials of Wegener or the quasicrystal guy look like nothing.
You mean, a giant shell built by someone? There may be a few eccentric astronomers, or creationist Christians who posit such a thing, but they will be professionally excoriated.
With just 1 $ly$ radius sphere, we can detect infrared light coming from what appears to be the pitch blackness of space (assuming the sphere has non-zero temperature).
When it smashes.
The problem with this idea is that we're in a galaxy, filled with stars and stuff.
These things will continue moving when you put your screen up.
This insanely large screen has effectively no mass measured from the inside, because it's a sphere, so won't affect the movement of anything within it, but will affect paths of stuff outside, which will in turn affect the orbits of stuff inside.
Eventually (as in, fairly quickly), something will cross that barrier, and the screens will either smash, or need to be moved aside to allow a solar system through, either of which one would expect to have some visible effects.
But if the "screen" is actually a more complex multilayered system, then you could move the layers about enough to permit stars to pass between, while still showing the same thing to the earth.
Downside of that is then... what's the point of the screen? Is it really just a filter to prevent people on Earth from seeing all the alien activity outside the screen? This is also a problem of the movie The Truman Show: there's no real point walling it all off just to get a reality TV show.
At that point, you're back to detecting parallax, and when you get to detect that depends on whether the screens were displaying data at the sun (in which case annular parallax would change) or at the Earth (the more likely case, in which case, it'd change the display over the year, and there'd be no visible annular parallax problem; only parallax from the orbital diameter of space telescopes).
There's also the possibility that the screen is better than a mere 2D raster device, in which case even that parallax could disappear.
