The correct option is C that the universe began in an explosive instant ten to twenty billion years ago and has been expanding
The sentence requires a simple past tense with respect to 'universe'. So, 'began' is correctly used in the given sentence. The past perfect 'had begun' of B implies that we are talking about some other event in the past and the universe had begun before that. "To have been" and "to have begun" of options D and E respectively are way too complicated when we need a simple past tense. C uses 'that has expanded' as if the Big Bang phenomenon is already over. Even if we do not understand the Big Bang theory, its been mentioned as something that is still going on. Also, the beginning of C uses too many words to state something simple. Thus, option A best befits the context.