Cosmological observations are used to test for imprints of an ultra-light axion-like field ( ULA ) , with a range of potentials V ( \phi ) \propto [ 1 - \cos ( \phi / f ) ] ^ { n } set by the axion-field value \phi and decay constant f .
Scalar field dynamics dictate that the field is initially frozen and then begins to oscillate around its minimum when the Hubble parameter drops below some critical value .
For n = 1 , once dynamical , the axion energy density dilutes as matter ; for n = 2 it dilutes as radiation and for n = 3 it dilutes faster than radiation .
Both the homogeneous evolution of the ULA and the dynamics of its linear perturbations are included , using an effective fluid approximation generalized from the usual n = 1 case .
ULA models are parameterized by the redshift z _ { c } when the field becomes dynamical , the fractional energy density f _ { z _ { c } } \equiv \Omega _ { a } ( z _ { c } ) / \Omega _ { tot } ( z _ { c } ) in the axion field at z _ { c } , and the effective sound speed c _ { s } ^ { 2 } .
Using Planck , BAO and JLA data , constraints on f _ { z _ { c } } are obtained .
ULAs are degenerate with dark energy for all three potentials if 1 + z _ { c } \lesssim 10 .
When 3 \times 10 ^ { 4 } \gtrsim 1 + z _ { c } \gtrsim 10 , f _ { z _ { c } } is constrained to be \lesssim 0.004 for n = 1 and f _ { z _ { c } } \lesssim 0.02 for the other two potentials .
The constraints then relax with increasing z _ { c } .
These results strongly constrain ULAs as a resolution to cosmological tensions , such as discrepant measurements of the Hubble constant , or the EDGES measurement of the global 21 cm signal .