  High-resolution hydrodynamics simulations of the Ly-alpha forest in cold dark
matter dominated cosmologies appear to predict line widths that are
substantially narrower than those observed. Here we point out that Compton
heating of the intergalactic gas by the hard X-ray background (XRB), an effect
neglected in all previous investigations, may help to resolve this discrepancy.
The rate of gain in thermal energy by Compton scattering will dominate over the
energy input from hydrogen photoionization if the XRB energy density is
0.2x/<epsilon> times higher than the energy density of the UV background at a
given epoch, where x is the hydrogen neutral fraction in units of 1e-6 and
<epsilon> is the mean X-ray photon energy in units of m_ec^2. The numerical
integration of the time-dependent rate equations shows that the intergalactic
medium approaches a temperature of about 1.5e4 K at z>3 in popular models for
the redshift evolution of the extragalactic background radiation. The
importance of Compton heating can be tested experimentally by measuring the
Ly-alpha line-width distribution as a function of redshift, thus the
Lyman-alpha forest may provide a useful probe of the evolution of the XRB at
high redshifts.
