Welcome
Here is a link to back pain
http://neurotalk.psychcentral.com/forum22.html
I haven't had SI ligament problems but I have had low back pain. Back pain is miserable pain. Pulled ligaments can be very painful. They can take several weeks. Some heat may help. Although heat is comforting, it can cause swelling on a microscopic basis. Sometimes ice can do you good, 10 to 15 mntues. Maybe alternate heat and ice. Over the counters like BenGay, BioFreeze (maybe at physical therapy and chiros), Activ-On, ThermaCare Heat Wraps. Massages, hot tub, jacuzzi, shower, soak in bathtub. Just a few suggestions to try for a little more comfort, if that's possible.
I found oversized tennis balls at Wal-Mart sportings good and put that in my low back spine. You'll need several if it helps you. Keep them at different places so you don't have to carry your balls around all the time. It gave me great relief. I also had to sleep on the couch, in a 45 degree angle, in a nest of pillows with those balls, and my back kinda against the back of the couch.
Sometimes rest is the best. Sometimes stretching, yoga or pool therapy can do a lot of good too.
There's lots of things to try.
Just a few things for thought.