Bows definitely go back until 8-9000 BC, with possible evidence going back to nearly 13,000 BC or even 40,000BC. That's plenty of time for it to have diffused all over Eurasia from a single origin. I mean, as soon as someone saw one they'd be like "wow, that's hot, I have to show the rest of the tribe." So it may only have been invented once or twice in human history, there's no real way to know.