Polytropes, logotropes, the universal value of the surface density of dark matter halos, and the value of the cosmological constant
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| Format: | Preprint |
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2026
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| _version_ | 1866914413659815936 |
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| author | Chavanis, Pierre-Henri |
| author_facet | Chavanis, Pierre-Henri |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polytropes, logotropes, the universal value of the surface density of dark matter halos, and the value of the cosmological constant Chavanis, Pierre-Henri General Relativity and Quantum Cosmology Astrophysics of Galaxies 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. |
| title | Polytropes, logotropes, the universal value of the surface density of dark matter halos, and the value of the cosmological constant |
| topic | General Relativity and Quantum Cosmology Astrophysics of Galaxies |
| url | https://arxiv.org/abs/2603.21302 |