General analytic solutions of scalar field cosmology with arbitrary potential

We present the solution space for the case of a minimally coupled scalar field with arbitrary potential in a Friedmann-Lemaître-Robertson-Walker metric. This is made possible due to the existence of a nonlocal integral of motion corresponding to the conformal Killing field of the two-dimensional minisuperspace metric. Both the spatially flat and nonflat cases are studied first in the presence of only the scalar field and subsequently with the addition of noninteracting perfect fluids. It is verified that this addition does not change the general form of the solution, but only the particular expressions of the scalar field and the potential. The results are applied in the case of parametric dark energy models where we derive the scalar field equivalence solution for some proposed models in the literature. © 2016 American Physical Society.