Bose-Einstein Condensate dark matter with logarithmic nonlinearity

TL;DR

This paper investigates dark matter as a low-temperature Bose-Einstein condensate described by a Gross-Pitaevskii equation with a logarithmic self-interaction. It derives an ideal-gas-like equation of state p = (b/m) ρ_m, and, under rotation, reduces the problem to a generalized Lane-Emden equation solved via the Laplace-Adomian Decomposition Method with Padé refinements to obtain density, mass, and tangential-velocity profiles. The authors fit the resulting rotation-curve predictions to SPARC data for both bulgeless and bulge-containing galaxies, using MCMC to constrain central density ρ_m0, rotation Ω, and the combination B = sqrt(b/m). They find generally good fits across a large sample, providing constraints on the self-interaction parameter and suggesting logarithmic BEC dark matter as a viable alternative to standard halo models. The work connects microscopic logarithmic nonlinearity to macroscopic galactic dynamics, offering a testable framework with clear observational signatures and parameter distributions.

Abstract

If dark matter is composed of massive bosons, a Bose-Einstein Condensation process must have occurred during the cosmological evolution. Therefore, galactic dark matter may be in a form of a self-gravitating condensate, in the presence of self-interactions. We consider the possibility that the self-interacting potential of the condensate dark matter is of the logarithmic form. In order to describe the condensate dark matter we use the Gross-Pitaevskii equation with a logarithmic nonlinearity, and the Thomas-Fermi approximation. With the use of the hydrodynamic representation of the Gross-Pitaevskii equation we obtain the equation of state of the condensate, which has the form of the ideal gas equation of state, with the pressure proportional to the dark matter density. The basic equation describing the density distribution of the static condensate is derived, and its solution is obtained in the form of a series solution, constructed with the help of the Adomian Decomposition Method. To test the model we consider the properties of the galactic rotation curves in the logarithmic Bose-Einstein Condensate dark matter scenario, by using a sample from the Spitzer Photometry and Accurate Rotation Curves (SPARC) data. The fit of the theoretical predictions of the rotation curves with the observational data indicate that the logarithmic Bose-Einstein Condensate dark matter model gives an acceptable description of the SPARC data, and thus it may be considered as a possible candidate for the in depth understanding of the dark matter properties.

Figures (12)

• Figure 1: Comparison between the full numerical solution of the generalized Lane-Emden equation for $W(\theta)$ (solid curve), with the $W[8/8](\theta)$ Padé approximant of the Adomian series solution (dotted curve) (left panel), and between the full numerical density distribution $\rho (\theta)/\rho _{m0}=\exp [-W(\theta)]$ with the Padé approximant expression $\exp \left[-W[8/8](\theta)\right)$ (dotted curve) (right panel), for $\omega =0$ (red curves) and $\omega =0.10$ (blue curves), respectively.
• Figure 2: Comparison between the full numerical solution of the generalized Lane-Emden equation for $W(\theta)$ (solid curve), with the $W[2/2](\theta)$ Padé approximant of the Adomian series solution (dotted curve) (left panel), and between the full numerical density distribution $\rho (\theta)/\rho _{m0}=\exp [-W(\theta)]$ with the Padé approximant expression $\exp \left[-W[2/2](\theta)\right)$ (dotted curve) (right panel), for $\omega =0$ (red curves) and $\omega =0.10$ (blue curves), respectively.
• Figure 3: Comparison between the logarithmic Bose-Einstein Condensate density profile (\ref{['rho2']}), $\Pi (\theta)=\exp \left]-10\theta ^2/\left(3\theta ^2+60\right)\right]$ (solid blue line), the Gaussian profile (\ref{['Chav']}), $\Pi (\theta)=e^{-0.01\theta ^2}$ (orange dashed curve), the super-Gaussian profile $\Pi (\theta)=e^{-0.02\theta ^{2.4}}$ (green dashed line), the power law potential (\ref{['Schive']}), $\Pi (\theta)=\left(1+0.091\theta ^2\right)^{-8}$ (brown dashed curve), and the solitonic core potential (\ref{['Che']}) in the presence of dark matter self interaction, $\Pi (\theta)=\left[1+\left(2^{1/\beta}-1\right)\theta ^\alpha\right]^{-\beta}$ with $\beta=7.8$ and $\alpha=2.4$ (red dashed curve), respectively.
• Figure 4: Variations of the dimensionless mass $M(\theta)/M_0$ of the logarithmic BEC halo (left panel), and of the tangential velocity $V(\theta)$ of a massive particle in rotational motion around the galactic center (right panel), for different values of $\omega$: $\omega =0$ (solid curve), $\omega =0.15$ (dotted curve), $\omega =0.17$ (short dashed curve), $\omega =0.18$ (dashed curve), and $\omega =0.19$ (long dashed curve), respectively.
• Figure 5: Rotation curves for 32 SPARC galaxies without bulge velocity component. The observed data points are shown with their respective uncertainties, while solid lines represent the best-fit model rotation curves based on SPARC data, including the $1\sigma$ confidence interval. The dashed curves represent the contribution from logarithmic BEC dark matter, whereas the dotted curves indicate the contribution from baryonic matter.