The phrase "chases down" essentially means to pursue with the intention of catching. There are numerous synonyms that can convey the same meaning and add variety to your writing. Alternative phrases to consider using include "hunts down," "tracks," "trails," or "follows relentlessly." Other possible synonyms include "goes after," "runs after," "hounds," "stalks," or "pursues vigorously." Remember that each synonym can have a slightly different connotation or tone, so consider the context of your sentence and choose the appropriate word accordingly. Using synonyms can elevate your writing and keep your readers engaged by avoiding repetition.