Cosmic strings are predicted by many Standard Model extensions involving the
cosmological breaking of a symmetry with nontrivial first homotopy group and
represent a potential source of primordial gravitational waves (GWs). Present
efforts to model the GW signal from cosmic strings are often based on minimal
models, such as, e.g., the Nambu-Goto action that describes cosmic strings as
exactly one-dimensional objects without any internal structure. In order to
arrive at more realistic predictions, it is therefore necessary to consider
nonminimal models that make an attempt at accounting for the microscopic
properties of cosmic strings. With this goal in mind, we derive in this paper
the GW spectrum emitted by current-carrying cosmic strings (CCCSs), which may
form in a variety of cosmological scenarios. Our analysis is based on a
generalized version of the velocity-dependent one-scale (VOS) model, which, in
addition to the mean velocity and correlation length of the string network,
also describes the evolution of a chiral (light-like) current. As we are able
to show, the solutions of the VOS equations imply a temporarily growing
fractional cosmic-string energy density, ${\mathrm{\Omega}}_{\mathrm{c}\mathrm{s}}$ . This results in an
enhanced GW signal across a broad frequency interval, whose boundaries are
determined by the times of generation and decay of cosmic-string currents. Our
findings have important implications for GW experiments in the Hz to MHz band
and motivate the construction of realistic particle physics models that give
rise to large currents on cosmic strings.