When we speak about alien encounters, we mostly have one of these two scenarios in mind: Either the aliens know nothing about us, or they know everything.
Let's explore a third option. The idea comes from the great sci-fi comedy Galaxy Quest:
Imagine an alien race which knows about us only the things displayed in the Star Trek franchise. And let's be generous with this one: Once it had the name "Star Trek" in it, these aliens know it. But they do not know anything else about us than the Star Trek franchise.
One awful hand-wave here is answering the question "How come?" But for the time being, just accept it as a fact.
The alien race spends an awful lot of time and energy to be just as the characters in Star Trek. And after finally getting the dilithium to work, they approach us.
A perfect working copy of the USS Enterprise NCC-1701-D is in Earth standard orbit right now. Open a communication channel, Mr. Sulu, we want to apply to the Federation.
But we all know the problem. There is no Federation. There are no Klingons or Romulans. But we can somehow fake it, because we at least have Captain Jean-Luc Picard.
The question is: How long can we hide the fact that the Star Trek world is non-existent?
Because, let's face it. Most of you would gladly sacrifice your left hand just for a possibility to have a ride in the real thing. So we want to fake it and get the technology from them.