We study the physical properties of the Lyman- \alpha forest by applying the inversion method described by Pichon et al .
( 2001 ) to the high resolution and high S/N ratio spectrum of the z _ { em } = 2.40 quasar HE 1122 - 1628 obtained during Science Verification of UVES at the VLT .
We compare the column densities obtained with the new fitting procedure with those derived using standard Voigt profile methods .
The agreement is good and gives confidence in the new description of the Lyman- \alpha forest as a continuous field as derived from our method .
We show that the observed number density of lines with log N > 13 and 14 is , respectively , 50 and 250 per unit redshift at z \sim 2 .
We study the physical state of the gas , neglecting peculiar velocities , assuming a relation between the overdensity and the temperature . T = { \overline { T } } ( \rho ( x ) / { \bar { \rho } } ) ^ { 2 \beta } .
There is an intrinsic degeneracy between the parameters \beta and { \overline { T } } .
We demonstrate that , at a fixed \beta , the temperature at mean density , { \overline { T } } , can be uniquely extracted , however .
While applying the method to HE 1122 - 1628 , we conclude that for 0.2 < \beta < 0.3 , 6000 < { \overline { T } } < 15000 K at z \sim 2 .
We investigate the small-scale structure of strong absorption lines using the information derived from the Lyman- \beta , Lyman- \gamma and C iv profiles .
Introducing the Lyman- \beta line in the fit allows us to reconstruct the density field up to \rho / { \bar { \rho } } \sim 10 instead of 5 for the Lyman- \alpha line only .
The neutral hydrogen density is of the order of \sim 2 \times 10 ^ { -9 } cm ^ { -3 } and the C iv /H i ratio varies from about 0.001 to 0.01 within the complexes of total column density N ( H i ) \sim 10 ^ { 15 } cm ^ { -2 } .
Such numbers are expected for photo-ionized gas of density n _ { H } \sim 10 ^ { -4 } cm ^ { -3 } and [ C/H ] \sim - 2.5 .
There may be small velocity shifts ( \sim 10 km s ^ { -1 } ) between the peaks in the C iv and H i density profiles .
Although the statistics is small , it seems that C iv /H i and n _ { HI } are anti-correlated .
This could be a consequence of the high sensitivity of the C iv /H i ratio to temperature .
The presence of associated O vi absorption , with a similar profile , confirms that the gas is photo-ionized and at a temperature of T \sim 10 ^ { 5 } K .