[Free]

Suppose the auction starts:
1♣ - (3♠) - Dbl - (4♠)
...

Now, what's do you expect when the auction continues (opps quiet):
1. Dbl *
2. 4NT - 5♦ - 5♥
3. 4NT - 5♦ - 5♠
4. 4NT - 5♦ - 5NT
5. 4NT - 5♦ - 6♥
6. 5♥
7. 5♠
8. 5NT
9. 6♥
*. If you take Dbl as takeout, what do opener's rebids 5♥/5♠/5NT/6♥ mean?

Lots of ways to show various hands, I wonder if there's an "expert standard" and what it is.

[FrancesHinden]

I doubt there is an expert standard.
I acknowledge that responder's double does not promise four hearts, so strain is a real issue. Thus much of the auction is about finding the right strain.

I play

Dbl = "take-out" i.e. extra values, no long suit, nothing particular in spades
4NT = EITHER both minors OR a good 5♥ bid

Working backwards:

6♥: a desire to play in 6♥
5NT: No idea. Either grand slam force in hearts or offering choice of 6C and 6H. I'd say the latter, except that I have a general agreement that we only play choice-of-slams when a grand slam is impossible.
5♠: extremely good hand, first round spade control. Most likely to have long clubs (i.e. 6C by responder is a sign-off)
5♥: a desire to play in 5♥

4NT - 5♦ - 5♥ = good 5♥ bid (general slam try in hearts)
4NT - 5♦ - 5♠ = grand slam try in diamonds
4NT - 5♦ - 5NT = GSF in diamonds
4NT - 5♦ - 6♥ = grand slam try in hearts, probably doesn't have first round spade control (that would bid 5♠ - 6♥)

4NT - 5♦ - 6♣ = choice of minor suit slam