Cosmological signatures of ultralight dark matter with an axionlike
  potential
  Nonlinearities in a realistic axion field potential may play an important
role in the cosmological dynamics. In this paper we use the Boltzmann code
CLASS to solve the background and linear perturbations evolution of an axion
field and contrast our results with those of CDM and the free axion case. We
conclude that there is a slight delay in the onset of the axion field
oscillations when nonlinearities in the axion potential are taken into account.
Besides, we identify a tachyonic instability of linear modes resulting in the
presence of a bump in the power spectrum at small scales. Some comments are in
turn about the true source of the tachyonic instability, how the parameters of
the axionlike potential can be constrained by Ly-$\alpha$ observations, and the
consequences in the stability of self-gravitating objects made of axions.
