If we launched a rocket at them with our garbage, or maybe our political leaders.

Humans have shown time and time again in their history that they are expansionist, imperialistic assholes. For aliens to be spooked, we only need to show that tendency on a cosmic scale. If we start sending out probes or even manned missions to far away places that are barely likely to support life, we will instantly label ourselves as greedy conquerors. As soon as a colony of ours starts sending out new missions to create even more colonies, that's when all civilizations, even the less advanced, realize that we're the first stage of a cosmic cancer that must be nipped in the bud at all costs.

This reminds me of the six possibilities for extra terrestrial life, one of which is everyone is listening and no one is broadcasting.

They’re probably waiting for us to get past the Great Filter.