On the Shape of Early Dark Energy

Early dark energy-- a scalar field, φ, which is initially held fixed by Hubble friction and becomes dynamical around matter/radiation equality and then redshifts faster than matter-- has emerged as one of the more successful cosmological models to resolve the Hubble tension. A detailed implementation of this model entails specifying the shape of the scalar field potential and the initial scalar field value. The use of an axion-like potential, which scales as φ⁶ around the minimum and flattens at larger field values, is commonly used. When fitting to current cosmological data (in particular observations of the cosmic microwave background from the Planck satellite), the initial field value is tightly constrained to be close to the flatter part of the potential. Using a more flexible shape for the potential we establish that this preference is driven by a strong constraint on the effective sound-speed of the scalar field perturbations on scales which enter the horizon around the time the field becomes dynamical. This constraint, in turn, dictates both the initial field value and shape of the potential away from its minimum. These results provide clear guidance when trying to build scalar field models which address the Hubble tension.