This investigation employs a computer code, initially constructed for modeling the evolution of a hot dense intergalactic medium (IGM), to study some properties of a cold dense IGM photoionized by QSOs. Within the observational constraints set by flux measurements in the far-ultraviolet, night sky brightness, and hard X-ray region, and the optical depth limits on various QSO spectra, it determines some allowable scenarios for a range of QSO spectral indices (alpha) and cutoff energies. With H(0) = 50 km/s-Mpc, closure density of an IGM (composed of H and He in their cosmic ratio) is possible only if alpha is greater than or equal to 0.1, a circumstance not likely to prevail; the most probable value, alpha = -0.7, implies an upper limit density about 0.35 of closure, which is somewhat larger than previous estimates. These estimates are substantially independent of whether or not QSOs produce the observed diffuse extragalactic X-ray background; such QSO spectral details are critical, however, in determining the ionization of heavy elements that are expected to contaminate a pure primeval plasma.