Is that inside a cabinet where people won't see the piping when doors are closed? You may need to open up the wall and look behind it.
Do you know for certain that there is no vent? Or can you access the plumbing through another wall without breaking the tile?
If there is no vent, changing the pipe size won't help. You will need to run the trap arm about 4" minimum distance to a vent and could put a vent inside the wall. Or, if you can't add a vent in the wall (and your code allows it) you could add an Air Admittance Valve (AAV) under the sink. Trap arm would still have to be at least 4" long and the AAV would have to be in a place with air circulation that is accessible for when it needs to be replaced.