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.
