There's a bar in Delray (Detroit) that they filmed some outside shots at. The bar was called "Ideal" and had been closed since 1986. The old man who owned the bar had to purchase the house next door in order to have access to the cement stairway between the buildings to get to the outside basement level of the bar. The old man told everyone on the cast and crew who would listen how it took him so long to purchase the bar and to talk the old lady that owned the house to sell to him although he didn't want the house (he only wanted the cement staircase that led to the basement level). Shortly after the filming at the bar (which took place in late July 2012) the old man torched the house next door for the insurance money.
Although Jim Jarmusch wanted to shoot this movie on film and has a general distaste for digital cinematography, he had to use the Arri Alexa for budgetary reasons. He managed to work with low lighting and specific lenses in order to come up with a look he found acceptable enough to work for him.
Eve books a flight for herself and Adam under the names "Daisy Buchanan" and "Stephen Dedalus," main characters in Fitzgerald's The Great Gatsby and Joyce's A Portrait of the Artist as a Young Man, respectively.
Contrary to most viewers' assumption, Jarmusch did not name the two main characters directly after the Adam and Even of the bible; rather, he was referring to Mark Twain's satirical work The Diaries of Adam and Eve. Jarmusch told a Hollywood Reporter interviewer that "it wasn't until shooting the film that I realized everyone's first impression will be the biblical reference. Whoops. It's too late now. What can I do?" Twain's is one of the faces visible on Adam's wall of framed portraits in the movie.
The white dwarf Eve mentions to Adam while driving across the city is the star called BMP 37093, also known as Lucy (named after the Beatles song Lucy in the Sky with Diamonds). Its nuclei is in a cooling down process, which makes the star core pulsate like a giant gong.
When Eve flies to Detroit, she uses the name Fibonacci for her reservation. Tilda Swinton played the mathematician Ada Augusta Byron King in the movie "Conceiving Ada". Ada King wrote an algorithm that would calculate Fibonacci numbers.
The trivia item below may give away important plot points.
In the first draft, Eve was 2000 years old and the druid of a matriarchal Celtic tribe. Adam was 5-600 years old. Jarmusch decided that Adam and Eve would be aware of each others true age so he argued they have no reason to state it in the film.