In The Scarlet Letter, why does Chillingworth ask Hester to keep his identity a secret?

When the
cold, sinister, and aptly named , Hester's husband, arrives and finds her in disgrace as an
adulteress, he says:

Thou and thine, , belong to me. My
home is where thou art and where he is. But betray me not!

He explains that doesn't want his own reputation damaged by being associated with her,
a fallen woman. Nobody here knows they are connected, and he wants it to stay that way, which is
what he means by "betray me not." Second, he plans to find out who she committed
adultery with, and can best do that if his identity is unknown. He threatens both her and her
lover with retribution if she should reveal the secret of who he is.

Chillingworth is a particularly devious and creepy figure. Hester is unhappy at his
presence and asks him if he is like the "Black Man," or devil, who haunts the forest.
She wonders if he is out to destroy her soul. He assures it is not her soul
he is after.