Factual error: When Izzy urges Tom to watch through the telescope on the rooftop, he just looks and sees the nebula. From space observations done by myself, I know that the earth rotation is so fast that even for this amount of time, a manual correction must be done to center the view again.
Factual error: The lower portion of the constellation Orion is incorrectly oriented above the horizon in the Mayan scenes. In northern latitudes, as when Tom and Izzy view it from their rooftop, it can be seen vertically as depicted, but in the equatorial vicinity where the Mayans lived, the constellation would be seen - whether in 1500 or 2000, whether setting or rising - turned 90 degrees or so on its side. Also, the upper portion of Orion, including the bright stars Belegeuse and Bellatrix, is never seen, and at the second timecode it is conspicuously absent when the view in the frame becomes wider. A few of ther dimmer stars, notably Sigma Orionis - near Alnitak and the Horsehead Nebula - are out of place as well.
Factual error: Izzi refers to the Great Orion Nebula, also known as M42 and M43, as Shabalba, the Mayan word for it. Although I could not verify if this is what they called it, and since this is fiction, one might easily ignore it. But the nebula is never referred to as the Great Orion Nebula or M42. Visually, it is seen roughly where it should be, the middle portion of the three "stars" that make up Orion's sword (below his 3-star belt), but through Izzi's telescope or from Tom's spacecraft it does not look very much like the nebula does either visually or photographically. Visually it looks gray, and in photographs it is reddish, as can be seen from my own photographs or any made by amateurs posted online. The shape is wrong, too. The biggest mistake about this nebula is that Izzi calls it a dying star when in fact it is a stellar nursery - a birthplace for stars. Since the themes of the film involve death and rebirth this fact should have equally worked in the script. To Aronofsky's credit, the nebula does appear more like a supernova remnant, such as the Crab Nebula (M1), which is a star which has exploded, or died.