Is my positional pawn sacrifice justified? Today I have played a game and the following position was reached:

[fen "rn1qk2r/p4ppp/1p2bb2/1Q1pp3/8/3P2P1/PP2PPBP/RN2K1NR b - - 0 1"]

1... Nd7 2. Bxd5 a6 3. Qb3 Nc5 4. Bc6+ Ke7 5.Qa3 Rc8 6. Bg2 Qd6 7. Nd2 e4


I quickly thought that covering the check with the knight would hang a pawn, but it seems that there is so much to it.
After 10. ...Nd7 11. Bxd5 a6 12. Qb3 Nc5 13. Bc6+ Ke7 14.Qa3 Rc8 15. Bg2 Qd6 16. Nd2 e4, Black is still a pawn down but apparently it is compensation enough for Black.

However, I am unable to see why the pawn sacrifice is justified for a small lead in development which White can finish in at most 2 moves with Nf3 and O-O.
Word of advice regarding the screenshot of lichess you had posted in the OP: Whenever you're checking with an engine, try to limit the number of best variations to 1-2 lines, as the more variations you choose the more distributed the allocated resources of the engine become and the less accurate the displayed evaluation.

Black here is clearly better thanks to the dominant central control, the bishop pair (both developed too) and the extra piece white develops for you with their Qb5+ check. So even before Bxd5, as things stand, black has the lead in time (development) and in space. Therefore, white's best bet is to at least try and win a pawn, in order to claim some sort of compensation.


  However, I am unable to see why the pawn sacrifice is justified for a
  small lead in development which White can finish in at most 2 moves
  with Nf3 and O-O.


If this was really the case then indeed white would be safely getting away with the snapped pawn, but therein lies the problem: if white plays it correctly, their best bishop retreat option is to e4 and not g2, and although black has lost their castling right they've gained a huge initiative on the kingside starting from g5 (aimed for g4) to boot the knight to a passive square and launch the kingside pawn avalanche. Let's just see some of these variations:

 [title "Analysing variations after 11.Bxd5"]
 [fen "r2qk2r/p2n1ppp/1p2bb2/1Q1pp3/8/3P2P1/PP2PPBP/RN2K1NR w KQkq - 2 11"]
 [startflipped "1"]

 1.Bxd5 a6 {our aim is to boot the queen away from the defense of the bishop} 2.Qb3 Nc5 3.Bc6+ {intermezzo check: only move to not lose material immediately} Kf8 {the king is even well placed on f8 as there are literally no foreseeable threats against it.} 4.Qd1 Rc8 5.Bg2 {a poorer choice than Be4, as the continuation shows white will be losing material very quickly if black's e-pawn is not blocked} (5.Be4 {white's best chance, to maintain the position somewhat closed and block the e-pawn} g5 {important positional move, to aim for g4 as quickly as possible exploiting the bishop and knight's awkward placement on e4-f3. With the knight booted away from f3, black is ready to launch an all out attack on the kingside with h5-g4 if white were to castle} 6.Nf3 g4 7.Nfd2 h5 8.O-O h4 {next to our bishops, our rook on h8 is perfectly placed to assist in the attack. The conversion from here is technically simple, and as soon as white retreats with Bg2 we have e4 ready. So there's no easy consolidation for white's king.}) e4 6.Nc3 (6.dxe4 Qxd1+ 7.Kxd1 Bxb2 {wins trivially}) exd3 {white's center and king safety are becoming further comprised, white has 5 active and well posted pieces in an open position against a king stuck in the center.} 7.Nf3 (7.exd3 Nxd3+ 8.Kf1 Bxc3 9.bxc3 Bc4 10.Ne2 Qf6 11.f4 Nxf4 12.gxf4 Qxf4+ 13.Ke1 Re8 14.Bf1 Qf3 15.Rg1 Qxc3+ 16.Kf2 Qe3+ 17.Kg2 Bxe2 {and the win is trivial from here.}) dxe2 8.Qxd8+ Rxd8 9.Kxe2 Bc4+ 10.Ke1 Nd3+ {and black wins trivially as further loss of material is unavoidable thanks to the comprised king position}


So you see, black is comfortably cashing in their lead in development and space in all the variations ensuing from Bxd5, and converting them into winning positions. And in most cases, the moves sort of play themselves considering how natural black's initiative and attack are as we saw in the diagram. Thus, all in all, we consistently come to the conclusion that the pawn sacrifice starting from Nd7 is perfectly sound. I encourage you to practice playing these lines and the ones we haven't considered yet against the computer.