Can I cook pasta without heating it up... ever?
Pasta (by which I infer you mean dried, Italian-style semolina pasta) is edible raw, right out of the package.  It is not, however, palatable.

If you soak it in water, it will hydrate and soften over time, but that is not the same as cooking it.  

True cooking also cooks the proteins and takes away that raw starchy taste.  There is no way to achieve that without the application of some heat above about 180 F for at least a few minutes.

See also:  

Preparing spaghetti without boiling water

Note that in one of the answers to that question, Rumtscho points out that supermarket ramen noodles are precooked (they are generally par-fried, sometimes par-baked) and are much more palatable eaten straight from the package, or simply soaked.