We propose a heuristic unification of dark matter and dark energy in terms of a single “ dark fluid ” with a logotropic equation of state P = A \ln ( \rho / \rho _ { P } ) , where \rho is the rest-mass density , \rho _ { P } = 5.16 \times 10 ^ { 99 } { g } { m } ^ { -3 } is the Planck density , and A is the logotropic temperature .
The energy density \epsilon is the sum of a rest-mass energy term \rho c ^ { 2 } mimicking dark matter and an internal energy term u ( \rho ) = - P ( \rho ) - A mimicking dark energy .
The logotropic temperature is approximately given by A \simeq \rho _ { \Lambda } c ^ { 2 } / \ln ( \rho _ { P } / \rho _ { \Lambda } ) \simeq \rho _ { \Lambda } c ^ { % 2 } / [ 123 \ln ( 10 ) ] , where \rho _ { \Lambda } = 6.72 \times 10 ^ { -24 } { g } { m } ^ { -3 } is the cosmological density and 123 is the famous number appearing in the ratio \rho _ { P } / \rho _ { \Lambda } \sim 10 ^ { 123 } between the Planck density and the cosmological density .
More precisely , we obtain A = 2.13 \times 10 ^ { -9 } { g } { m } ^ { -1 } { s } ^ { -2 } that we interpret as a fundamental constant .
At the cosmological scale , this model fullfills the same observational constraints as the \Lambda CDM model ( they will differ in about 25 Gyrs when the logotropic universe becomes phantom ) .
However , the logotropic dark fluid has a nonzero speed of sound and a nonzero Jeans length which , at the beginning of the matter era , is about \lambda _ { J } = 40.4 { pc } , in agreement with the minimum size of the dark matter halos observed in the universe .
At the galactic scale , the logotropic pressure balances gravitational attraction and solves the cusp problem and the missing satellite problem .
The logotropic equation of state generates a universal rotation curve that agrees with the empirical Burkert profile of dark matter halos up to the halo radius .
In addition , it implies that all the dark matter halos have the same surface density \Sigma _ { 0 } = \rho _ { 0 } r _ { h } = 141 M _ { \odot } / { pc } ^ { 2 } and that the mass of dwarf galaxies enclosed within a sphere of fixed radius r _ { u } = 300 { pc } has the same value M _ { 300 } = 1.93 \times 10 ^ { 7 } M _ { \odot } , in remarkable agreement with the observations .
It also implies the Tully-Fisher relation M _ { b } / v _ { h } ^ { 4 } = 44 M _ { \odot } { km } ^ { -4 } { s } ^ { 4 } .
We stress that there is no free parameter in our model ( we predict the values of \Sigma _ { 0 } , M _ { 300 } and M _ { b } / v _ { h } ^ { 4 } in terms of fundamental constants ) .
We sketch a justification of the logotropic equation of state in relation to the Cardassian model ( motivated by the existence of extra-dimensions ) and in relation to Tsallis generalized thermodynamics .
We also develop a scalar field theory based on a Gross-Pitaevskii equation with an inverted quadratic potential , or on a Klein-Gordon equation with a logarithmic potential .