A scenario on the power-law total mass-density profile of the inner regions of early-type-galaxies

We naturally obtain a power-law total mass-density profile for the inner regions (within several effective radius) of early-type galaxies(ETGs) from space-time structures, which are described by the static spherically-symmetric solutions of Einstein equations with the perfect fluid matters. We point out that, in the inner region of an early type galaxy, the average value of the equation of state $\omega$ determines the power-law index $\gamma$ of the total mass-density profile given by \cite{2006ApJ...649..599K}. We then use 119 galaxy-galaxy strong gravitational lensing systems from SLACS, BELLS, BELLS GALLERY and SL2S, to constrain the equation of state $\omega$ for the inner regions of ETGs. We find that the average value of $\omega$ for totally 119 ETGs is $\omega=-0.315\pm0.073$ (68\% CL), which is very close to $-1/3$. If $\omega=-1/3$ is the best value to describe the equation of state of the inner regions of the dark matter in galaxies, the deviation from $-1/3$ could represent the mixture of baryonic matter and dark matter in galaxies.