On the basis of a large scale ’ adiabatic ’ , namely non-radiative and non-dissipative , cosmological smooth particle hydrodynamic simulation we compare the entropy profiles of the gas and the dark matter ( DM ) in galaxy clusters .
We employ the quantity K _ { g } = 3 k _ { B } T _ { g } \rho { { } _ { g } ^ { -2 / 3 } } / ( \mu m _ { p } ) = \sigma ^ { 2 } _ { g% } \rho { { } _ { g } ^ { -2 / 3 } } as measure for the entropy of the intra-cluster gas .
By analogy the DM entropy is defined as K _ { DM } = \sigma { { } ^ { 2 } _ { DM } } \rho { { } _ { DM } ^ { -2 / 3 } } ( \sigma { { } ^ { 2 } _ { DM } } is the 3D velocity dispersion of the DM ) .
The DM entropy is related to the DM phase space density by K _ { DM } \propto Q { { } _ { DM } ^ { -2 / 3 } } .
In accord with other studies the radial DM phase space density profile follows a power law behaviour , Q _ { DM } \propto r ^ { -1.82 } , which corresponds to K _ { DM } \propto r ^ { 1.21 } .
The simulated intra-cluster gas has a flat entropy core within ( 0.8 \pm 0.4 ) R _ { s } , where R _ { s } is the NFW scale radius .
The outer profile follows the DM behaviour , K _ { g } \propto r ^ { 1.21 } , in close agreement with X-ray observations .
Upon scaling the DM and gas densities by their mean cosmological values we find that outside the entropy core a constant ratio of K _ { g } / K _ { DM } = 0.71 \pm 0.18 prevails .
By extending the definition of the gas temperature to include also the bulk kinetic energy the ratio of the DM and gas extended entropy is found to be unity for r \gtrsim 0.8 R _ { s } .
The constant ratio of the gas thermal entropy to that of the DM implies that observations of the intra-cluster gas can provide an almost direct probe of the DM .