We study the non-adiabatic cosmological production of ultra light dark matter ( ULDM ) under a minimal set of assumptions : a free ultra light real scalar as a spectator field in its Bunch-Davies vacuum state during inflation and instantaneous reheating into a radiation dominated era .
For ( ULDM ) fields minimally coupled to gravity , non-adiabatic particle production yields a distribution function peaked at low comoving momentum \mathcal { N } _ { k } \propto 1 / k ^ { 3 } .
The infrared behavior is a remnant of the infrared enhancement of light minimally coupled fields during inflation .
We obtain the full energy momentum tensor , show explicity its equivalence with the fluid-kinetic one in the adiabatic regime , and extract the abundance , equation of state and free streaming length ( cutoff in the matter power spectrum ) .
Taking the upper bound on the scale of inflation from Planck , the ( UDLM ) saturates the dark matter abundance for m \simeq 1.5 \times 10 ^ { -5 } \mathrm { eV } with an equation of state parameter w \simeq 10 ^ { -14 } and a free streaming length \lambda _ { fs } \simeq 70 \mathrm { pc } .
Thus this cosmologically produced ( ULDM ) yields a cold dark matter particle .
We argue that the abundance from non-adiabatic production yields a lower bound on generic ( ULDM ) and axion-like particles that must be included in any assessment of ( ULDM ) as a dark matter candidate .