  The electromagnetic local density of states (LDOS) is crucial to many aspects
of photonics engineering, from enhancing emission of photon sources to
radiative heat transfer and photovoltaics. We present a framework for
evaluating upper bounds on LDOS in structured media that can handle arbitrary
bandwidths and accounts for critical wave scattering effects with no heuristic
approximations. The bounds are solely determined by the bandwidth, material
susceptibility, and device footprint, with no assumptions on geometry. We
derive an analytical expression for the maximum LDOS consistent with the
conservation of energy across the entire design domain, which upon benchmarking
with topology-optimized structures is shown to be nearly tight for large
devices. Novel scaling laws for maximum LDOS enhancement are found: the bounds
saturate to a finite value with increasing susceptibility and scale as the
quartic root of the bandwidth for semi-infinite structures made of lossy
materials, with direct implications on material selection and design
applications.
