A Lower Bound on Local Source Density from the Observed Flux and the Propagation Horizon

The observed flux of ultra-high-energy cosmic rays, combined with the propagation horizon, fixes the power the local universe must supply. Taking the energy density above 1e19 eV as 5.5e-22 erg per cubic centimetre and the loss time as the light-travel time across a 100 megaparsec horizon, 3.262e+08 years, the required luminosity density is 4.956e+43 erg per cubic megaparsec per year. If that power comes from sources of 1e42 erg per second each, their number density must exceed 1.570e-06 per cubic megaparsec; at 1e40 erg per second per source, the required density rises to 1.570e-04 per cubic megaparsec. Above 1e20 eV, where the horizon shrinks to about 15 megaparsecs, the required luminosity density is 3.304e+44 erg per cubic megaparsec per year. These bounds are independent of any acceleration theory and of the density-bound framework; they are bookkeeping that every source model must satisfy, and several popular source classes satisfy them only marginally. Simulated and computed from stated assumptions; not a measurement.