The accurate determination of the absolute energy scale in cosmic ray
measurements is both a challenging and fundamentally important task. We
present how measurements of radio pulses from extensive air
showers with the Auger Engineering Radio Array, combined with per-event
simulations of radio emission using the CoREAS extension of CORSIKA, allow us
to determine the energy scale of cosmic rays between $3\cdot 10^{17}$ eV and
several $10^{18}$ eV.
Our analysis accounts for many factors, each of which is controlled on the 5%
level or better. The absolute calibration of the antennas and the entire
analog signal chain builds on a Galactic calibration in combination with a
detailed understanding of the antenna-gain patterns. Additional key elements
include compensation for temperature-dependent signal amplification,
continuous detector health monitoring, an active veto for thunderstorm
conditions, an unbiased event reconstruction, and per-event atmospheric
modeling in the simulations. The analysis benefits from a high-statistics
dataset of over 800 measured cosmic ray showers.
We describe our analysis method, perform multiple cross-checks, and evaluate
systematic uncertainties. We find that absolute energies determined with AERA are 12% higher
than those established with the Auger Fluorescence Detector, a result well in
agreement within systematic uncertainties and thus a strong independent confirmation
of the absolute energy scale of the Pierre Auger Observatory.