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arXiv 2201.05908gr-qc
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一个新的对数势复标量场的对数熵模型

A new logotropic model based on a complex scalar field with a logarithmic potential

Pierre-Henri Chavanis

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摘要：

我们引入了一个新的对数势的基础上的复杂的标量场，统一暗物质和暗能量的logotropic模型。该标量场满足一个非线性波动方程，该方程在相对论区域推广了克莱因-戈登方程，在非相对论区域推广了薛定谔方程。该模型具有内在的量子性质，并在经典极限$\hbar\rightarrow 0$中返回$Λ$ CDM模型。它涉及到一个新的物理学基本常数$A/c^2=2.10\times 10^{-26}\, {\rm g}\, {\rm m}^{-3}$，它负责宇宙后期的加速膨胀，并取代爱因斯坦宇宙学常数$Λ$。在大（宇宙学）尺度下，logotropic模型与$Λ$ CDM模型几乎没有区别，但在小（银河系）尺度下解决了CDM危机。它也解决了模糊暗物质模型的问题。事实上，它导致了具有普遍表面密度$Σ_0^{\rm th}=5.85\,\left ({A}/{4πG}\right )^{1/2}=133\, M_{\odot}/{\rm pc}^2$的核心暗物质晕。这个普遍的表面密度预测从logotropic模型没有可调参数，原来是接近的观测值$Σ_0^{\rm obs}=141_{-52}^{+83}\, M_{\odot}/{\rm pc}^2$。我们还认为，在$Λ$ CDM模型中，通常被解释为暗物质和暗能量的比例的量$Ω_{\rm dm,0}$和$Ω_{\rm de,0}$等于$Ω_{\rm dm,0}^{\rm th}=\frac{1}{1+e}(1-Ω_{\rm b,0})=0.2559$和$Ω_{\rm de,0}^{\rm th}=\frac{e}{1+e}(1-Ω_{\rm b,0})=0.6955$，与测量值$Ω_{\rm dm,0}^{\rm obs}=0.2589$和$Ω_{\rm de,0}^{\rm obs}=0.6911$非常吻合（它们的比值$2.669$接近纯数$e=2.71828...$）.

英文摘要：

We introduce a new logotropic model based on a complex scalar field with a logarithmic potential that unifies dark matter and dark energy. The scalar field satisfies a nonlinear wave equation generalizing the Klein-Gordon equation in the relativistic regime and the Schrödinger equation in the nonrelativistic regime. This model has an intrinsically quantum nature and returns the $Λ$CDM model in the classical limit $\hbar\rightarrow 0$. It involves a new fundamental constant of physics $A/c^2=2.10\times 10^{-26}\, {\rm g}\, {\rm m}^{-3}$ responsible for the late accelerating expansion of the Universe and superseding the Einstein cosmological constant $Λ$. The logotropic model is almost indistinguishable from the $Λ$CDM model at large (cosmological) scales but solves the CDM crisis at small (galactic) scales. It also solves the problems of the fuzzy dark matter model. Indeed, it leads to cored dark matter halos with a universal surface density $Σ_0^{\rm th}=5.85\,\left ({A}/{4πG}\right )^{1/2}=133\, M_{\odot}/{\rm pc}^2$. This universal surface density is predicted from the logotropic model without adjustable parameter and turns out to be close to the observed value $Σ_0^{\rm obs}=141_{-52}^{+83}\, M_{\odot}/{\rm pc}^2$. We also argue that the quantities $Ω_{\rm dm,0}$ and $Ω_{\rm de,0}$, which are usually interpreted as the present proportion of dark matter and dark energy in the $Λ$CDM model, are equal to $Ω_{\rm dm,0}^{\rm th}=\frac{1}{1+e}(1-Ω_{\rm b,0})=0.2559$ and $Ω_{\rm de,0}^{\rm th}=\frac{e}{1+e}(1-Ω_{\rm b,0})=0.6955$ in very good agreement with the measured values $Ω_{\rm dm,0}^{\rm obs}=0.2589$ and $Ω_{\rm de,0}^{\rm obs}=0.6911$ (their ratio $2.669$ is close to the pure number $e=2.71828...$).

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