We present 40 fully hydrodynamical numerical simulations of the intergalactic gas that gives rise to the Ly \alpha forest .
The simulation code , input and output files are available at http : //www.cosmos.ucsd.edu/g̃so/index.html .
For each simulation we predict the observable properties of the H I absorption in QSO spectra .
We then find the sets of cosmological and astrophysical parameters that match the QSO spectra .
We present our results as scaling relationships between input and output parameters .
The input parameters include the main cosmological parameters \Omega _ { b } , \Omega _ { m } , \Omega _ { \Lambda } , H _ { 0 } and \sigma _ { 8 } ; and two astrophysical parameters \gamma _ { 912 } and X _ { 228 } .
The parameter \gamma _ { 912 } controls the rate of ionization of H I , He I and He II and is equivalent to the intensity of the UV background .
The second parameter X _ { 228 } controls the rate of heating from the photoionization of He II and can be related to the shape of the UVB at \lambda < 228 Å .
We show how these input parameters ; especially \sigma _ { 8 } , \gamma _ { 912 } and X _ { 228 } ; effect the output parameters that we measure in simulated spectra .
These parameters are the mean flux \bar { F } , a measure of the most common Ly \alpha line width ( b -value ) b _ { \sigma } , and the 1D power spectrum of the flux on scales from 0.01 – 0.1 s/km .
We compare the simulation output to data from Kim et al .
( 2004 ) and Tytler et al .
( 2004 ) and we give a new measurement of the flux power from HIRES and UVES spectra for the low density IGM alone at z = 1.95 .

We find that simulations with a wide variety of \sigma _ { 8 } values , from at least 0.8 – 1.1 , can fit the small scale flux power and b -values when we adjust X _ { 228 } to compensate for the \sigma _ { 8 } change .
We can also use \gamma _ { 912 } to adjust the H I ionization rate to simultaneously match the mean flux .
When we examine only the mean flux , b -values and small scale flux power we can not readily break the strong degeneracy between \sigma _ { 8 } and X _ { 228 } .

We can break the degeneracy using large scale flux power or other data to fix \sigma _ { 8 } .
When we pick a specific \sigma _ { 8 } value the simulations give X _ { 228 } and hence the IGM temperature that we need to match the observed small scale flux power and b -values .
We can then also find the \gamma _ { 912 } required to match the mean flux for that combination of \sigma _ { 8 } and X _ { 228 } .
We derive scaling relations that give the output parameter values expected for a variety of input parameters .
We predict the line width parameter b _ { \sigma } with an error of 1.4 % and the mean amount of H I absorption to 2 % , equivalent to a 0.27 % error on \bar { F } ar z = 1.95 .
These errors are four times smaller than those on the best current measurement .
We can readily calculate the sets of input parameters that give outputs that match the data .
For \sigma _ { 8 } = 0.9 , with \Omega _ { b } = 0.044 , \Omega _ { m } = 0.27 , \Omega _ { \Lambda } = 0.73 , h = 0.71 and n = 1.0 we find X _ { 228 } = 1.26 and \gamma _ { 912 } = 1.00 , equivalent to \Gamma _ { 912 } = 1.33 \times 10 ^ { -12 } ionizations per H I atom per second .
If we run an optically thin simulation with these input parameter values in a box size of 76.8 Mpc comoving with cells of 18.75 kpc comoving we expect that the simulated spectra will match Ly \alpha forest data at z = 1.95 .
The rates predicted by correspond to \gamma _ { 912 } = 1 and X _ { 228 } = 1 .
We are in accord for \gamma _ { 912 } while the larger X _ { 228 } is reasonable to correct for the opacity that is missing from the optically thin simulations .
To match data for smaller \sigma _ { 8 } , structure is more extended and we need a smaller X _ { 228 } corresponding to a cooler IGM .
We also need a larger \gamma _ { 912 } to stop the neutral fraction from increasing at the lower temperatures .