Angular-frequency Planck energy density ($u_{\omega }=\hslash {\omega }^3/[{\pi }^2{c}^3(e^{\hslash \omega /(k_{B}T)}-1)]$)

The bulk photon mode density per volume per angular frequency is ${\omega }^2/({\pi }^2{c}^3)$, including two transverse polarizations. Multiply it by the thermal oscillator energy $\hslash \omega /(e^{\hslash \omega /(k_{B}T)}-1)$ to obtain the displayed spectrum. Its integral is $u={\pi }^2{k}_B^4{T}^4/(15{\hslash }^3{c}^3)$. This is volume energy density rather than radiance or surface exitance; isotropic equilibrium radiation has outward black-surface flux $cu/4$. zero-point energy is excluded from this thermal spectrum.