The preferred perturbative dynamics of early dark energy

The tension between direct measurements of the current expansion rate, H0, and the value predicted from observations of early universe physics (from either the cosmic microwave background (CMB) or measurements of the light element abundances from Big Bang nucleosynthesis (BBN)), can be resolved by positing the existence of a scalar field which contributes about 10% of the total energy density around the time of matter/radiation equality. The increased energy density leads to a decrease in the sound horizon which, in turn, leads to an increase in the CMB/BBN predicted value of H0. Several studies have shown that measurements of the CMB anisotropies are sensitive to the detailed perturbative dynamics of this scalar field. The scalar field perturbations are predominately controlled by the shape of its potential. We explore constraints to a scalar field potential which consists of one power law around its minimum and another power law far from its minimum. We find that the flatter the potential is far from its minimum, the higher the inferred value of H0. We discuss how this result can be understood in terms of an effective sound speed for the scalar field. These results can be generalized in order to give guidelines for building models to address this tension.