The Physical State of the Intergalactic Medium or Can We Measure Y?

We present an argument for a {\it lower limit} to the Compton-$y$ parameter describing spectral distortions of the cosmic microwave background (CMB). The absence of a detectable Gunn-Peterson signal in the spectra of high redshift quasars demands a high ionization state of the intergalactic medium (IGM). Given an ionizing flux at the lower end of the range indicated by the proximity effect, an IGM representing a significant fraction of the nucleosynthesis-predicted baryon density must be heated by sources other than the photon flux to a temperature $\go {\rm few} \times 10^5\, K$. Such a gas at the redshift of the highest observed quasars, $z\sim 5$, will produce a $y\go 10^{-6}$. This lower limit on $y$ rises if the Universe is open, if there is a cosmological constant, or if one adopts an IGM with a density larger than the prediction of standard Big Bang nucleosynthesis.