The phrase "big asteroid" refers to a large rock floating in outer space. It is a term used to describe a space object that is much larger than most asteroids. Full definition
According to the study, that would ensure that Earth would be re-seeded with life, following big, cataclysmic events like the Late Heavy Bombardment which happened approximately 4 billion years ago when the space environment around the Earth was dominated by untold numbers of big asteroids in the aftermath of its formation. (americaspace.com)
The authors suggest that searches for these impact ejecta layers will be more fruitful for determining how many times Earth was hit by big asteroids than searches for large craters. (sciencedaily.com)
But with big asteroids, it doesn't matter much where they land. (discovermagazine.com)