Quintessential early dark energy

Abstract: We introduce a unified model of early and late dark energy. We call it {\it quintessential early dark energy} model where early and late dark energy are explained by a single scalar field {\it i.e.}, two different energy scales are related by a single scalar field potential. To achieve this we introduce the modified steep exponential potential, which is chosen phenomenologically. This potential has a hilltop nature during the early time which consists of a flat region followed by a steep region. This nature of the potential plays a crucial role in achieving early dark energy solution. During recent time, the potential can almost mimic the cosmological constant which can result into late time acceleration. But, at the perturbation level the potential shows significant difference with the $\Lambda$CDM model. We also constrain and compare the models for steep exponential, modified steep exponential, axionlike and power law potentials by using the available background cosmological data from CMB, BAO (including DESI DR1 2024), supernovae (Pantheon$+$, DESY5 and Union3) and Hubble parameter measurements. Even after the presence of required EDE solution in all four potentials we don't get any significant improvement in the value of $H_0$. The maximum improvement we get in the present value of Hubble parameter compared to the standard $\Lambda$CDM model is for the axionlike potential. For other potentials the constraints are similar to the $\Lambda$CDM model. We also see that the data prefers $\Lambda$CDM model over the considered scalar field models at least for the data combinations with Pantheon$+$ and Union3.