[taejo]

> What stops me from attacking that network by mining on my private chain with a double spend?

Nothing stops you mining on a private chain, but since you have a small minority of the compute power, the network doesn't care about your private chain. The same is true in Bitcoin.

> I have no opportunity cost.

You could be mining blocks that generate useful byproducts and currency.

[VMG]

> Nothing stops you mining on a private chain, but since you have a small minority of the compute power, the network doesn't care about your private chain.

Right, this is not an issue for small miners. But if you have a significant part of mining power, it's important that there is a significant opportunity cost mining on the wrong chain. That can only be ensured by using a PoW that is wasteful outside the currency you're mining.