"Spine," as we use the term today, is simply the relative stiffness of an arrow. Archers use the term improperly, but that's what they're talking about.
How much does arrow diameter matter? A larger diameter arrow has to be spine matched to the bow a bit better than a smaller diameter arrow. And the opposite is true also, a smaller diameter arrow will shoot well from a bow, even if it's spine is really a bit too stiff for the bow.
You can see quite a difference in how the arrow sits on the bow's shelf when you look how a 2512 compares to a skinny little Easton Axis. It doesn't take a lot of effort to see that the 2512 is pointing way off to the side and will have to paradox a lot to bring it's point of impact over to your point of aim. The skinny Axis isn't going to have to paradox much to get it flying down the middle.
A lot is being said these days about how a small diameter hunting arrow penetrates game better than a large diameter arrow. Maybe in certain circumstances, but I remain unconvinced. Doesn't seem to me like it matters whether the arrow is 1/4" diameter or 3/8", it's following a broadhead that has already cut at least a 1" hole. It's a non issue, in my book.
So, what diameter you need to shoot from your bow is a moot point. The critical thing you need to do is find the arrow that spines best for you, regardless of diameter.
I like my hunting arrows to be a little long in order to keep the broadhead away from my fingers. I know some guys who cut their arrows exactly to where the back of the broadhead's blade touches their index finger at full draw. They use it as sort of a draw check. I don't like something that sharp, with that much force behind it, anywhere near that close to any part of my body that tends to bleed when cut.
Of course, arrow length also has a bearing on spine. So, the first thing you must do is determine what length arrow you intend to shoot before you begin sorting out what arrows to get. I suggest going to Easton's website and using their Legacy arrow selector program. The standard charts will suggest an arrow that is many sizes too stiff. The Legacy chart will get you in the ball park.
Getting the correct arrow spine is half the battle. You've got one devil of a starter bow, that's for sure. With that Widow, you've got no excuses.[&:]