Polytropes, logotropes, the universal value of the surface density of dark matter halos, and the value of the cosmological constant

Abstract

We discuss the connection between logotropes and polytropes in astrophysics and cosmology. The logotropic equation of state $P=A\ln(ρ/ρ_P)$ may be seen as a degenerate form of the polytropic equation of state $P=Kρ^γ$ in the limit $γ\rightarrow 0$, $K\rightarrow\infty$ with $A=Kγ$ fixed. The logotropic distribution function corresponds to the polytropic distribution function of index $γ=0$ for which the density is finite but the pressure diverges logarithmically. We show that the polytropic and logotropic distribution functions can be obtained in the nondegenerate limit of the Lynden-Bell theory of violent relaxation for a particular distribution of phase levels given by the $χ$-squared distribution. This provides a justification of the Tsallis entropy from the Lynden-Bell entropy. The logotropic distribution function presents a power-law energy tail decreasing as $ε^{-5/2}$. Interestingly, this ``universal'' power-law tail is predicted by recent kinetic theories of collisionless relaxation based on the coarse-grained Vlasov equation and on the secular dressed diffusion equation. When coupled to gravity, the associated density profile decreases as $r^{-1}$. This may explain the universal surface density of dark matter halos, or account for an effective NFW density cusp. This also accounts for the universal gravitational acceleration felt by a test particle and for the Tully-Fisher relation. The logotropic model can thus provide an alternative to the modification of Newtonian dynamics (MOND) theory. We recall how the logotropic model leads to a very accurate expression of the cosmological constant $Λ={G^2m_e^6}/{α^6\hbar^4}=1.36\times 10^{-52}\, {\rm m^{-2}}$ in terms of the mass of the electron and the fundamental constants of physics.

Figures (3)

• Figure 1: Normalized density profile of a DM halo according to Eqs. (\ref{['inter4']}) and (\ref{['inter5']}). It is given by $f(x)=1$ if $x\le 1/4$ and $f(x)=(1+\lambda)^2/[4x(1+\lambda x)^2]$ if $x\ge 1/4$ (approximately), where $f(x)=\rho(r)/\rho_0$, $x=r/r_h$ and $\lambda=r_h/r_s$. It is compared to the cored Burkert profile $f_{B}(x)=1/[(1+x)(1+x^2)]$ and to the cuspy NFW profile $f_{\rm NFW}(x)=(1+\lambda)^2/[4x(1+\lambda x)^2]$ for all $x$. The pure logotropic profile is approximately given by $f_{L}(x)=1$ if $x\le 1/4$ and $f_{L}(x)=1/(4x)$ if $x\ge 1/4$. We have taken $\lambda=0.01$ for illustration.
• Figure 2: Normalized circular velocity profile of a DM halo according to Eq. (\ref{['logomodel12']}) with Eqs. (\ref{['inter4']}) and (\ref{['inter5']}). It is given by $V^2(x)=\frac{1}{x}\int_0^x f(y)y^2\, dy$, where $V(x)=v(r)/\sqrt{4\pi G\rho_0 r_h^2}$. It is compared to the cored Burkert profile $V_{B}^2(x)=\frac{1}{4x}\lbrack -2\tan^{-1}(x)+2\ln(1+x)+\ln(1+x^2)\rbrack$ and to the cuspy NFW profile $V_{\rm NFW}^2(x)=\frac{(1+\lambda)^2}{4\lambda^2 x}\lbrack -1+\frac{1}{1+\lambda x}+\ln(1+\lambda x)\rbrack$. The pure logotropic profile is approximately given by $V_{L}^2(x)=x^2/3$ if $x\le 1/4$ and $V_{L}^2(x)=\frac{1}{8x}(x^2-\frac{1}{48})$ if $x\ge 1/4$. We have taken $\lambda=0.01$ for illustration.
• Figure 3: Evolution of the scale factor (radius) of the Universe from the early inflation to the late accelerating expansion (dark energy). In between, the Universe undergoes a radiation era and a matter era. This model is based on a quadratic equation of state aipuniversevacuumon or, equivalently, on nonlinear electrodynamics gbi. In Model I, the initial density is identified with the Planck density $\rho_P$. In Model II, the initial density is identified with the electron density $\rho_e$. The final density corresponds to the cosmological density $\rho_\Lambda$.