  We investigate a method to determine the temperature-density relation of the
intergalactic medium (IGM) at z~2-4 using quasar absorption line systems. Using
a simple model combined with numerical simulations we show that there is a
lower cutoff in the distribution of column density (NHI) and line width (b
parameter). The location of this cutoff can be used to determine the
temperature-density relation (under certain conditions). We describe and test
an algorithm to do this. The method works as long as the amplitude of
fluctuations on these scales (~100 kpc) is sufficiently large. Models with less
power can mimic higher temperatures. A preliminary application is made to data
from two quasar lines-of-sight, and we determine an upper limit to the
temperature of the IGM. Finally, we examine the full distribution of
b-parameters and show that this is completely specified by just two parameters:
the temperature of the gas and the amplitude of the power spectrum. Using the
temperature upper limit measured with the NHI-b cutoff method, we derive an
upper limit to the amplitude of the power spectrum.
