We examine the evolution of the inner dark matter (DM) and baryonic density profile of a new sample of simulated field galaxies using fully cosmological, Lambda CDM, high resolution SPH + N-Body simulations. These simulations include explicit H2 and metal cooling, star formation (SF) and supernovae (SNe) driven gas outflows. Starting at high redshift, rapid, repeated gas outflows following bursty SF transfer energy to the DM component and significantly flatten the originally `cuspy central DM mass profile of galaxies with present day stellar masses in the 10^4.5 -- 10^9.8 Msolar range. At z=0, the central slope of the DM density profile of our galaxies (measured between 0.3 and 0.7 kpc from their centre) is well fitted by rhoDM propto r^alpha with alpha simeq -0.5 + 0.35 log_10(Mstar/10^8Msolar) where Mstar is the stellar mass of the galaxy and 4 < log_10 Mstar < 9.4. These values imply DM profiles flatter than those obtained in DM--only simulations and in close agreement with those inferred in galaxies from the THINGS and LITTLE THINGS survey. Only in very small halos, where by z=0 star formation has converted less than ~ 0.03% of the original baryon abundance into stars, outflows do not flatten the original cuspy DM profile out to radii resolved by our simulations. The mass (DM and baryonic) measured within the inner 500 pc of each simulated galaxy remains nearly constant over four orders of magnitudes in stellar mass for Mstar 10^9 Msolar. This finding is consistent with estimates for faint Local Group dwarfs and field galaxies. These results address one of the outstanding problems faced by the CDM model, namely the strong discrepancy between the original predictions of cuspy DM profiles and the shallower central DM distribution observed in galaxies.
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