The standard self-similar model of galaxy cluster formation predicts that the X-ray luminosity–temperature ( L _ { X } – T _ { X } ) relation of galaxy clusters should have been L _ { X } \propto T _ { X } ^ { 2 } in absence of the baryonic physics , such as radiative cooling and feedback from stars and black holes .
However , this baseline relation is predicted without considering the fact that the halo concentration and the characteristic density of clusters increases as their mass decreases , which is a consequence of hierarchical structure formation of the universe .
Here , we show that the actual baseline relation should be L _ { X } \propto T _ { X } ^ { \alpha } , where \alpha \sim 1.7 , instead of \alpha = 2 , given the mass dependence of the concentration and the fundamental plane relation of galaxy clusters .
Numerical simulations show that \alpha \sim 1.6 , which is consistent with the prediction .
We also show that the baseline luminosity–mass ( L _ { X } – M _ { \Delta } ) relation should have been L _ { X } \propto M _ { \Delta } ^ { \beta } , where \beta \sim 1.1 –1.2 , in contrast with the conventional prediction ( \beta = 4 / 3 ) .
In addition , some of the scatter in the L _ { X } – M _ { \Delta } relation can be attributed to the scatter in the concentration–mass ( c – M ) relation .
The confirmation of the shallow slope could be a proof of hierarchical clustering .
As an example , we show that the new baseline relations could be checked by studying the temperature or mass dependence of gas mass fraction of clusters .
Moreover , the highest-temperature clusters would follow the shallow baseline relations if the influences of cool cores and cluster mergers are properly removed .