  Recent observations suggest that many old pulsar wind nebulae (PWNe) are
bright TeV gamma-ray sources without a strong X-ray counterpart. In this paper,
we study the spectral evolution of PWNe taking into account the energy which
was injected when they were young for old PWNe. We model the evolution of the
magnetic field and solve for the particle distribution inside a uniformly
expanding PWN. The model is calibrated by fitting the calculated spectrum to
the observations of the Crab Nebula at an age of a thousand years. We find that
only a small fraction of the injected energy from the Crab Pulsar goes to the
magnetic field, consistent with previous studies. The spectral evolution model
of the Crab Nebula shows that the flux ratio of TeV gamma-rays to X-rays
increases with time, which implies that old PWNe are faint at X-rays, but not
at TeV gamma-rays. The increase of this ratio is primarily because the magnetic
field decreases with time and is not because the X-ray emitting particles are
cooled more rapidly than the TeV gamma-ray emitting particles. Our spectral
evolution model matches the observed rate of the radio flux decrease of the
Crab Nebula.
