Comparison between the Logotropic and $\Lambda$CDM models at the cosmological scale

We perform a detailed comparison between the Logotropic model [P.H. Chavanis, Eur. Phys. J. Plus 130 (2015) 130] and the $\Lambda$CDM model. These two models behave similarly at large (cosmological) scales up to the present. Differences will appear only in the far future, in about $25\, {\rm Gyrs}$, when the Logotropic Universe becomes phantom while the $\Lambda$CDM Universe enters in the de Sitter era. However, the Logotropic model differs from the $\Lambda$CDM model at small (galactic) scales, where the latter encounters serious problems. Having a nonvanishing pressure, the Logotropic model can solve the cusp problem and the missing satellite problem of the $\Lambda$CDM model. In addition, it leads to dark matter halos with a constant surface density $\Sigma_0=\rho_0 r_h$, and can explain its observed value $\Sigma_0=141 \, M_{\odot}/{\rm pc}^2$ without adjustable parameter. This makes the logotropic model rather unique among all the models attempting to unify dark matter and dark energy. In this paper, we compare the Logotropic and $\Lambda$CDM models at the cosmological scale where they are very close to each other in order to determine quantitatively how much they differ. This comparison is facilitated by the fact that these models depend on only two parameters, the Hubble constant $H_0$ and the present fraction of dark matter $\Omega_{\rm m0}$. Using the latest observational data from Planck 2015+Lensing+BAO+JLA+HST, we find that the best fit values of $H_0$ and $\Omega_{\rm m0}$ are $H_0=68.30\, {\rm km}\, {\rm s}^{-1}\,{\rm Mpc}^{-1}$ and $\Omega_{\rm m0}=0.3014$ for the Logotropic model, and $H_0=68.02\, {\rm km}\, {\rm s}^{-1}\,{\rm Mpc}^{-1}$ and $\Omega_{\rm m0}=0.3049$ for the $\Lambda$CDM model. The difference between the two models appears at the percent level.