I’m not a flame, but I set the sky on fire. I’m not a curtain, but I close the day. What am I?

My first two is the name for a circle's fundamental ratio, my last two are the symbol for an element, used to make signs stand out at night. Together, I'm a synonym for yearn.

Whispered oaths in dark, Bound by shadows' tight embrace, Sealed with mystic ties.

One part of my name is sour, the other has bite, I’m more social than ready to fight.

Paired up with a cat, And oh, what a sound! Four lines of her gut, All so tightly wound...