Comparison between the Logotropic and $Λ$CDM models at the cosmological scale

Pierre-Henri Chavanis, Suresh Kumar

Published 2016-12-04Version 1

We perform a detailed comparison between the Logotropic model [P.H. Chavanis, Eur. Phys. J. Plus {\bf 130}, 130 (2015)] and the $\Lambda$CDM model in an attempt to favor one over the other in an era of precision cosmology. These two models behave similarly at large (cosmological) scales up to the present. Differences appear only in the far future, in about $25\, {\rm Gyrs}$, when the Logotropic Universe becomes phantom while the $\Lambda$CDM Universe enters 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 free parameter. It is therefore important to see if one can detect small differences between the Logotropic and $\Lambda$CDM models at the cosmological scale where these are very close to each other. 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. We analytically derive the statefinders of the Logotropic model, and obtain $(q_0,r_0,s_0)=(-0.5516,1.011,-0.003518)$ instead of $(q_0,r_0,s_0)=(-0.5427,1,0)$ for the $\Lambda$CDM model.