What revolutionized our knowledge of the age of the universe was mapping the fluctuations in the cosmic microwave background (CMB) — leftover radiation from the early universe observed using sensitive instruments. It turns out that the precise size and distribution of tiny hot and cold temperature fluctuations in the CMB are sensitive to the age of the universe.
How do astronomers know that the universe is 13.7 billion years old?
Jose Eduardo Machado, Rio de Janeiro, Brazil