Waiting...
If it was sent out on the 4th, definitely no need to call yet.
The shortest time I have heard of in terms of commitment was like 9 days. I think my wife's took about a week and mine took somewhere along those lines.
It is not likely that you can speed up the process.
My surgeon's office said (Aetna is my insurance) that the insurance company goes in order of when the surgeries are scheduled. Do you have a tentative date scheduled?
I know it is hard to plan like this, but it is part of the process and assuming you have jumped through all the hoops, it will happen.
I am with Cigna. Before I proceeded I called them and was told they would pay. Later when the hospital checked for the approval process which was under 2 wk prior to surgery they said they would not approve unless I got a 2nd opinion from another doctor. They rushed me down to see another doctor and I was approved about 4 days later in time for my surgery.