Are joins on an FK faster than joins without an FK?

Question

Say I have two tables, a and b:

a {
 pk as int
 fk as int
 ...
}

b {
 pk as int
 ...
}

I want to join a and b in a query like so:

FROM a
JOIN b on a.fk = b.pk

Which of the following scenarios will be faster?

1. a.fk is set up to be a foreign key on b.pk - b.pk is indexed
2. a.fk is set up to be a foreign key on b.pk - b.pk is not indexed
3. there is no relationship between the tables - b.pk is indexed
4. there is no relationship between the tables - b.pk is not indexed

Bonus question - how much faster/slower will each of these scenarios be?

If you could back up your answer with a reference then that'd be awesome. Thank you!

Answer 1

Best practice

1. Foreign Keys are a relational integrity tool, not a performance tool. You should always create indexes on FK columns to reduce lookups. SQL Server does not do this automatically.
2. As stated here Foreign keys boost performance

Logically, this gives following ranking performance wise

1. a.fk is set up to be a foreign key on b.pk - b.pk is indexed
2. there is no relationship between the tables - b.pk is indexed
3. a.fk is set up to be a foreign key on b.pk - b.pk is not indexed
4. there is no relationship between the tables - b.pk is not indexed

Answer 2

The performance differnces would be greatest between the indexed and non indexed versions, however whether it would be faster or slower would depend on whether it was a select or an insert. Having indexes and foreign key constraints slow down inserts but speed up selects (the index) or make the data more reliable (the FK). Since generally most inserts are not noticably slowed (unless you are doing large bulk inserts), it is usually in your best interests to have the FK and the index.