Although usual hierarchical clustering scenarios agree with the main observed properties of Lyman- \alpha clouds and galaxies , they seem to fail to reproduce the cluster temperature - X-ray luminosity relation .
A possible explanation is that the IGM is reheated by supernovae or quasars before cluster formation .
In this article , using a unified analytic model for quasars , galaxies , Lyman- \alpha absorbers and the IGM , we obtain the redshift evolution of the temperature and the entropy of the gas and the corresponding cluster temperature - X-ray luminosity relation .
We consider three scenarios : 1 ) no energy source in addition to photoionization heating , 2 ) heating from supernovae and 3 ) heating from quasars , in an open universe \Omega _ { m } = 0.3 , \Omega _ { \Lambda } = 0 , with a CDM power-spectrum .
We show that although quasars can easily reheat the IGM and raise its entropy up to the level required by current cluster observations the energy provided by supernovae is unlikely to be sufficient .
Indeed , the efficiency factor needed for the supernova scenario is of order unity ( \alpha _ { SN } \simeq 1.7 ) while for quasars we get \alpha _ { Q } \simeq 0.008 .
Thus the IGM is more likely to have been reheated by quasars .
Moreover , we find that if both scenarios are normalized to present observations the reheating due to quasars occurs somewhat earlier ( z _ { S } \sim 2 ) than for supernovae ( z _ { S } \sim 0.4 ) because of the sharp drop at low z of the quasar luminosity function .
We also show that the Compton parameter y induced by the IGM is well below the observed upper limit in all cases .
Finally , we note that such a reheating process may partly account for the decline at low redshift of the comoving star formation rate and of the quasar luminosity function .
In particular , we show that the contradictory requirements arising from clusters ( which require a large reheating to affect the relation T - L _ { X } ) and galaxies ( which require a small reheating so that galaxy and star formation are not too much inhibited ) provide strong constraints on such models .
Thus , the IGM should be reheated at low redshift z \la 2 up to T \sim 5 10 ^ { 5 } K. On the other hand , the reionization process of the universe is almost not modified by these entropy sources which means that our predictions for the former should be quite robust .