Bonus properties of states of low energy

States of Low Energy (SLE) are exact Hadamard states defined on arbitrary Friedmann-Lemaitre spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian's expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann-Lemaitre spacetimes with accelerated expansion. In consequence, massless SLE are viable candidates for pre-inflationary vacua and in a soluble model are shown to entail a qualitatively correct primordial power spectrum.