We demonstrate that massive fields, such as dark matter, can directly produce a cosmological evolution of the fundamental constants of Nature. We show that a scalar or pseudoscalar (axion-like) dark matter field ϕ, which forms a coherently oscillating classical field and interacts with Standard Model particles via quadratic couplings in ϕ, produces 'slow' cosmological evolution and oscillating variations of the fundamental constants. We derive limits on the quadratic interactions of ϕ with the

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... hoton, electron and light quarks from measurements of the primordial ^4He abundance produced during Big Bang nucleosynthesis and recent atomic dysprosium spectroscopy measurements. These limits improve on existing constraints by up to 15 orders of magnitude. We also derive limits on the previously unconstrained linear and quadratic interactions of ϕ with the massive vector bosons from measurements of the primordial ^4He abundance.