We consider the possibility that the universe is made of a single dark fluid described by a logotropic equation of state P = A \ln ( \rho / \rho _ { * } ) , where \rho is the rest-mass density , \rho _ { * } is a reference density , and A is the logotropic temperature .
The energy density \epsilon is the sum of two terms : a rest-mass energy term \rho c ^ { 2 } that mimics dark matter and an internal energy term u ( \rho ) = - P ( \rho ) - A that mimics dark energy .
This decomposition leads to a natural , and physical , unification of dark matter and dark energy , and elucidates their mysterious nature .
In the early universe , the dark fluid behaves as pressureless dark matter ( P \simeq 0 , \epsilon \propto a ^ { -3 } ) and , in the late universe , it behaves as dark energy ( P \sim - \epsilon , \epsilon \propto \ln a ) .
The logotropic model depends on a single parameter B = A / \rho _ { \Lambda } c ^ { 2 } ( dimensionless logotropic temperature ) where \rho _ { \Lambda } = 6.72 \times 10 ^ { -24 } { g } { m } ^ { -3 } is the cosmological density .
For B = 0 , we recover the \Lambda CDM model with a different justification .
For B > 0 , we can describe deviations from the \Lambda CDM model .
Using cosmological constraints , we find that 0 \leq B \leq 0.09425 .
We consider the possibility that dark matter halos are described by the same logotropic equation of state .
When B > 0 , pressure gradients prevent gravitational collapse and provide halo density cores instead of cuspy density profiles , in agreement with the observations .
The universal rotation curve of logotropic dark matter halos is consistent with the observational Burkert profile up to the halo radius .
It decreases as r ^ { -1 } at large distances , similarly to the profile of dark matter halos close to the core radius [ Burkert , arXiv:1501.06604 ] .
Interestingly , if we assume that all the dark matter halos have the same logotropic temperature B , we find that their surface density \Sigma = \rho _ { 0 } r _ { h } is constant .
This result is in agreement with the observations [ Donato et al . , MNRAS 397 , 1169 ( 2009 ) ] where it is found that \Sigma _ { 0 } = 141 M _ { \odot } / { pc } ^ { 2 } for dark matter halos differing by several orders of magnitude in size .
Using this observational result , we obtain B = 3.53 \times 10 ^ { -3 } .
Then , we show that the mass enclosed within a sphere of fixed radius r _ { u } = 300 { pc } has the same value M _ { 300 } = 1.93 \times 10 ^ { 7 } M _ { \odot } for all the dwarf halos , in agreement with the observations [ Strigari et al . , Nature 454 , 1096 ( 2008 ) ] .
Finally , assuming that \rho _ { * } = \rho _ { P } , where \rho _ { P } = 5.16 \times 10 ^ { 99 } { g } { m } ^ { -3 } is the Planck density , we predict B = 3.53 \times 10 ^ { -3 } , in perfect agreement with the value obtained from the observations .
We approximately have B \simeq 1 / \ln ( \rho _ { P } / \rho _ { \Lambda } ) \simeq 1 / [ 123 \ln ( 10 ) ] where 123 is the famous number occurring in the ratio \rho _ { P } / \rho _ { \Lambda } \sim 10 ^ { 123 } between the Planck density and the cosmological density .
This value of B is sufficiently low to satisfy the cosmological bound 0 \leq B \leq 0.09425 and sufficiently large to differ from CDM ( B = 0 ) and avoid density cusps in dark matter halos .
It leads to a Jeans length at the beginning of the matter era of the order of \lambda _ { J } = 40.4 { pc } which is consistent with the minimum size of dark matter halos observed in the universe .
Therefore , a logotropic equation of state is a good candidate to account both for galactic and cosmological observations .
This may be a hint that dark matter and dark energy are the manifestation of a single dark fluid .
If we assume that the dark fluid is made of a self-interacting scalar field , representing for example Bose-Einstein condensates , we find that the logotropic equation of state arises from the Gross-Pitaevskii equation with an inverted quadratic potential , or from the Klein-Gordon equation with a logarithmic potential .
We also relate the logotropic equation of state to Tsallis generalized thermodynamics and to the Cardassian model ( motivated by the existence of extra-dimensions ) .