Is SSN for a newborn mandatory?

My child was born a few days ago, and I plan to fly outside the US after less than two months, so once I got the birth certificate, I applied for the child's passport on the same day before receiving the SSN while it is stated that if you don't have a SSN, fill with zeros.
Will that cause a problem or a delay?
If yes, how long could this delay be?
And if I received the baby's SSN in the following days, can I contact some place to update the application with it?
Thank you.