  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 $\phi$, which forms a
coherently oscillating classical field and interacts with Standard Model
particles via quadratic couplings in $\phi$, produces `slow' cosmological
evolution and oscillating variations of the fundamental constants. We derive
limits on the quadratic interactions of $\phi$ with the photon, electron and
light quarks from measurements of the primordial $^4$He 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 $\phi$ with the massive vector bosons from
measurements of the primordial $^4$He abundance.
