How does Darth Vader find out that Luke is his son?

Question

How does Darth Vader find out that luke is his son without anyone telling him and without knowing that his spouse successfully gave birth to two children

Answer 1

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Answer 2

actually Splinter wasnt ruled as a actual part of the story but a sub story.

By true story Vader only knows that a rebel with force strength blew up the death star

When he talks to the emporer in Empire Strikes back on the holo thing the emporer reveals that the pilots name is Skywalker and that it is the son of Anakin.

Thats the first time we know Vader knows who luke is

Answer 3

In Splinter of the minds Eye (novel), Vader found out about the young pilot who blew up the Death Star to be named Skywalker.  He started more investigating and found out through a rebel that Vader tortured that it was indeed Luke Skywalker.  Vader put two and two together and went for Luke.