system
Given the following post, respond as if you're commenting on Reddit:
human
[Star Wars] How did Darth Vader know Luke was his son?
gpt
After realizing the rebels had a very strong force-user in their midst after Yavin, Vader probably spent a lot of time looking him up through various channels with the original intent of killing him before he became a real threat (or getting some research done before the Emperor gave the order). And then the rebels made a rather large mistake, they publicized their victory and their celebration for the whole galaxy to see. Luke Skywalker's name was famous among those who hated the Empire. Imperial holonet channels did their best to keep this news buried and quiet, so the Imperial public would be ignorant of the rather devastating loss, but people still found out.  

But Vader just instantly *knew*. Someone named Skywalker, who had grown up on Tattooine with the Lars family and was unusually strong in the force even without training. It had to be his son. (The Force itself probably helped him connect all the dots)