Comparison between the Logotropic and ΛCDM models at the cosmological scale

Abstract

We perform a detailed comparison between the Logotropic model [P.H. Chavanis, Eur. Phys. J. Plus, 130 (2015)] and the ΛCDM 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 Gyrs, when the Logotropic Universe becomes phantom while the ΛCDM Universe enters in the de Sitter era. However, the Logotropic model differs from the ΛCDM 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 ΛCDM model. In addition, it leads to dark matter halos with a constant surface density Σ0=ρ0 rh, and can explain its observed value Σ0=141 M⊙/pc2 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 ΛCDM 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 H0 and the present fraction of dark matter Ωm0. Using the latest observational data from Planck 2015+Lensing+BAO+JLA+HST, we find that the best fit values of H0 and Ωm0 are H0=68.30 km s−1 Mpc−1 and Ωm0=0.3014 for the Logotropic model, and H0=68.02 km s−1 Mpc−1 and Ωm0=0.3049 for the ΛCDM model. The difference between the two models is at the percent level. As a result, the Logotropic model competes with the ΛCDM model at large scales and solves its problems at small scales. It may therefore represent a viable alternative to the ΛCDM model. Our study provides an explicit example of a theoretically motivated model that is almost indistinguishable from the ΛCDM model at the present time while having a completely different (phantom) evolution in the future. We analytically derive the statefinders of the Logotropic model for all values of the logotropic constant B. We show that the parameter s0 is directly related to this constant since s0=−B/(B+1) independently of any other parameter like H0 or Ωm0. For the predicted value of B=3.53× 10−3, we obtain (q0,r0,s0)=(−0.5516,1.011,−0.003518) instead of (q0,r0,s0)=(−0.5427,1,0) for the ΛCDM model corresponding to 0B=.