  Dark matter comprised of axion-like particles (ALPs) generated by the
realignment mechanism in the post-inflationary scenario leads to primordial
isocurvature fluctuations. The power spectrum of these fluctuations is flat for
small wave numbers, extending to scales accessible with cosmological surveys.
We use the latest measurements of Cosmic Microwave Background (CMB) primary
anisotropies together with CMB lensing, Baryonic Acoustic Oscillations (BAO)
and Sunyaev Zel'dovich (SZ) cluster counts to measure the amplitude and tilt of
the isocurvature component. We find preference for a white-noise isocurvature
component in the CMB primary anisotropies; this conclusion is, however,
weakened by current large-scale structure (LSS) data. Interpreting the result
as a conservative upper limit on the isocurvature component, the combined bound
on the ALP mass from all probes is $m_{a} \gtrsim 10^{-19}$ eV, with some
dependence on how $m_{a}$ evolves with temperature. The expected sensitivity of
cosmic shear and galaxy clustering from future LSS experiments and CMB lensing
suggests improved bounds of $m_{a} \gtrsim 10^{-18}$-$10^{-13}$ eV, depending
on scale cuts used to avoid non-linearities and the ALP mass-temperature
dependence.
