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 $f_{z_c} equiv Omega_a(z_c)/Omega_{rm tot}(z_c)$ 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 $3times10^4 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.
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