The explicit density (rho) dependence in the coupling coefficients of the non-relativistic nuclear energy-density functional (EDF) encodes effects of three-nucleon forces and dynamical correlations. The necessity for a coupling coefficient in the form of a small fractional power of rho is empirical and the power often chosen arbitrarily. Consequently, precision-oriented parameterisations risk overfitting and loss of predictive power. Observing that the Fermi momentum kF~rho^1/3 is a key variable in Fermi systems, we examine if a power hierarchy in kF can be inferred from the properties of homogeneous matter in a domain of densities which is relevant for nuclear structure and neutron stars. For later applications we want to determine an EDF that is of good quality but not overtrained. We fit polynomial and other functions of rho^1/3 to existing microscopic calculations of the energy of symmetric and pure neutron matter and analyze the fits. We select a form and parameter set which we found robust and examine the parameters naturalness and the resulting extrapolations. A statistical analysis confirms that low-order terms like rho^1/3 and rho^2/3 are the most relevant ones. It also hints at a different power hierarchy for symmetric vs. pure neutron matter, supporting the need for more than one rho^a terms in non-relativistic EDFs. The EDF we propose accommodates adopted properties of nuclear matter near saturation. Importantly, its extrapolation to dilute or asymmetric matter reproduces a range of existing microscopic results, to which it has not been fitted. It also predicts neutron-star properties consistent with observations. The coefficients display naturalness. Once determined for homogeneous matter, EDFs of the present form can be mapped onto Skyrme-type ones for use in nuclei. The statistical analysis can be extended to higher orders and for different ab initio calculations.
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