The paper discusses a new cosmological problem for models that solve the strong CP puzzle using an invisible axion, distinct from the domain wall problem. The authors identify that the energy density stored in the oscillations of the classical axion field does not dissipate rapidly due to the axion's weak coupling. This leads to a constraint on the axion decay constant \( f_a \): if \( f_a \geq 10^{12} \) GeV, the axion energy density can exceed the critical density needed to close the universe. If this bound is saturated, axions could constitute the dark matter of the universe. The paper also explores the implications of this constraint in the context of inflationary cosmology and particle production, concluding that the axion energy density remains significant even after the universe cools. The authors suggest that axions with \( f_a \sim 10^{12} \) GeV could form a nondissipative, pressureless gas, potentially explaining dark matter.The paper discusses a new cosmological problem for models that solve the strong CP puzzle using an invisible axion, distinct from the domain wall problem. The authors identify that the energy density stored in the oscillations of the classical axion field does not dissipate rapidly due to the axion's weak coupling. This leads to a constraint on the axion decay constant \( f_a \): if \( f_a \geq 10^{12} \) GeV, the axion energy density can exceed the critical density needed to close the universe. If this bound is saturated, axions could constitute the dark matter of the universe. The paper also explores the implications of this constraint in the context of inflationary cosmology and particle production, concluding that the axion energy density remains significant even after the universe cools. The authors suggest that axions with \( f_a \sim 10^{12} \) GeV could form a nondissipative, pressureless gas, potentially explaining dark matter.