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ABSTRACT  We study the evolution of phase‐space density during the hierarchical structure formation of Λ cold dark matter (CDM) haloes. We compute both a spherically averaged surrogate for phase‐space density  ( Q =ρ/σ 3 )  and the coarse‐grained distribution function   f (  x  ,   v  )  for dark matter (DM) particles that lie within ∼2 virial radii of four Milky Way sized dark matter haloes. The estimated   f (  x  ,   v  )  spans over four decades at any radius. DM particles that end up within 2 virial radii of a Milky Way sized DM halo at   z = 0  have an approximately Gaussian distribution in  log ( f )  at early redshifts, but the distribution becomes increasingly skewed at lower redshifts. The value  f  peak  corresponding to the peak of the Gaussian decreases as the evolution progresses and is well described by   f  peak ( z ) ∝ (1 + z ) 4.5   for   z  &gt; 1 . The highest values of  f  (responsible for the skewness of the profile) are found at the centres of dark matter haloes and subhaloes, where  f  can be an order of magnitude higher than in the centre of the main halo. We confirm that   Q ( r )  can be described by a power law with a slope of  −1.8 ± 0.1  over 2.5 orders of magnitude in radius and over a wide range of redshifts. This   Q ( r )  profile likely reflects the distribution of entropy  ( K ≡σ 2 /ρ 2/3  DM ∝ r  1.2 ) , which dark matter acquires as it is accreted on to a growing halo. The estimated   f (  x  ,   v  ) , on the other hand, exhibits a more complicated behaviour. Although the median coarse‐grained phase‐space density profile   F ( r )  can be approximated by a power law,  ∝ r  −1.6±0.15  , in the inner regions of haloes  (&lt;0.6  r  vir ) , at larger radii the profile flattens significantly. This is because phase‐space density averaged on small scales is sensitive to the high‐ f  material associated with surviving subhaloes, as well as relatively unmixed material (probably in streams) resulting from disrupted subhaloes, which contribute a sizable fraction of matter at large radii.

Evolution of the dark matter phase‐space density distributions of ΛCDM haloes