How did Vader know that Obi-Wan was helping Luke?

Question

Vader: He's just a boy. Obi-Wan can no longer help him (Empire Strikes Back)

How does Vader know that Obi-Wan helped Luke before?

• He doesn't know that Luke grew up on Tatooine
• even if he guessed, he doesn't know Luke met Obi-Wan there
• from what I can tell, Vader wasn't aware that Luke was in the Millennium Falcon when the Death Star captured it after blowing up Alderaan

Answer 1

Vader probably knew Luke was on the Death Star because Vader killed Obi-Wan in sight of Luke.

Here's the relevant clip for easy reference:

Luke screams out "No!" when he sees Obi-Wan die, and he does it loudly enough to alert the stormtroopers watching the fight. Vader might have heard that. Even if he didn't, Vader likely caught a glimpse of Luke before the blast door closed on him. And it would be surprising if Vader didn't detect the Force presence of his son in such close proximity, which he later felt at the Battle of Yavin in the X-Wing that destroyed the Death Star.

Furthermore, Obi-Wan was obviously covering the escape of his friends by delaying Vader. Vader surely looked at all the security camera feeds (there was probably a camera trained on the Millenium Falcon) and saw Luke's escape. Luke's emotional reaction to Obi-Wan's death means that Luke and Obi-Wan had spent some time together.

Between Luke's Force presence with Obi-Wan on the Death Star, and the same Force presence at the Battle of Yavin, it's not difficult to put these facts together and guess that Obi-Wan -- one of the few remaining Jedi -- had helped train Luke. And since Obi-Wan died helping Luke on the Death Star, he can no longer help Luke.

Answer 2

As DVK suggested in point 2, the inference is fairly strong when you consider the entire progression of the storyline of ANH and TESB up to that point.

A New Hope

• Leia sends the Death Star plans via droids in an escape pod down to Tatooine, Vader sends a detachment down to get them
• At Mos Eisley, Ben Kenobi uses a blue lightsaber in the bar fight, witnessed by a lot of unsavory characters
• C-3PO and R2D2, the droids that the stormtroopers are looking for, are seen in the company of Luke and Ben in Mos Eisely, Vader knew both of those droids extremely well
• Luke and Ben openly went by their own names on Tatooine for years, Luke as Luke Skywalker and Kenobi as Ben Kenobi (Luke knew him by this name, though perhaps not everyone did). Vader doing a little research after the fact would easily find that out, that Kenobi had been there off and on for years, and that he would have quite plausibly known about Luke and his parents
• Ben and Luke were seen talking to Han Solo in Mos Eisley and disappeared at the same time as him, Kenobi reappearing with the Falcon to rescue Leia
• The spy Garindan (the anteater guy with the black goggles in ANH) discovered that it was Luke Skywalker that sold his landspeeder (tying him back to the Lars farm) and goes and tells the stormtroopers who try and intercept him and Solo who both leave in the Millenium Falcon thus tying Skywalker to the Millenium Falcon even perhaps before he knew his actual name
• The stormtroopers were already probably looking for Skywalker, they'd just wasted his parents Owen and Beru, after learning about them from the Jawas but didn't find the droids at their home
• Vader knew of both Owen and Beru Lars from his time on Tatooine trying to rescue his mother Shmi in AOTC
• Obi-wan shows up at the same time as the Millenium Falcon, which is recognised as the one blasting its way out of Mos Eisely, after it is captured by the Empire; the Falcon escapes later on during the rescue of Leia and where Obi-wan faces Vader holding him off
• In the control room Han without his helmet on calls him 'Luke', and in Leia's jail cell Luke takes his helmet off and says "I'm Luke Skywalker, I'm here to rescue you" and later "I'm here to rescue you. I've got your R2-unit. I'm here with Ben Kenobi."
• The aforementioned Millenium Falcon then stops Vader shooting down Skywalker, whom Vader senses somewhat "the force is strong with this one", and then Skywalker goes and destroys the Death Star

The Empire Strikes Back

• By this time Vader has used Garindan and no doubt others to do intel work on Tatooine to supplement what he already knew, and by then he knew it was Luke Skywalker who was responsible for the destruction of the Death Star, as he and Sidious discuss it

I don't think that Luke being strong in the Force necessarily implies that it was Ben who trained him, since a lot of Jedi survived the Purge/Order 66. But Vader, by that point, would definitely have known that Luke and Ben were together on Mos Eisley (the anteater guy saw them), so the most logical inference would be that Ben had trained Luke.

Answer 3

I'm not aware of canon information of how Vader knew, but there are two likely ways he guessed:

1. Luke carried Anakin's old lightsaber. According to new Disney canon (Marvel Star Wars comic, #2), Vader confronts Luke prior to the question's chat between Darth Vader and the Emperor; and recognizes Luke's lightsaber as Anakin's... which was lost during his Mustafar duel with Obi-Wan and therefore most likely was acquired by Luke from Obi-Wan.

   To borrow a screenshot from this SFF answer:

   

2. Obi-Wan shows up JUST as Luke shows up, pretty much the same day (and based on blowing up the Death Star, Luke is already strong in the Force, so someone must have trained him!). Seems like too much of a coincidence. Anakin is no dummy.

3. Also, while not proof, I have a sneaky suspicion that Vader may have felt Luke in the Force on board the Death Star, or for that matter, found him on security cam footage after the Millenium Falcon's "escape". Remember that Luke took off his helmet in Leia's cell, to prove he was not a stormtrooper.