Shape Shifting Light Dark Matter Solitons

TL;DR

This work investigates ultra-light Bose-Einstein condensate dark matter as soliton cores in galaxies hosting central masses. By solving the Schrödinger-Poisson equation with baryonic coupling and representing the soliton via a five-Gaussian basis, the authors derive analytical, $F$-dependent soliton shapes and robust scaling relations linking soliton properties to the particle mass $m_0$ and soliton fraction $F$. Comparisons with nearly fifty dSph and UFD systems suggest an approximate common particle mass around $m_0\sim10^{-22}$ eV/c$^2$, with an upper bound near $3\times10^{-22}$ eV/c$^2$, and point to interpretive possibilities involving central black holes or enhanced stellar velocity dispersion in the dark matter-dominated regime. The results offer a framework to test fuzzy dark matter against observations, address degeneracies in interpretation, and explore implications such as dark matter evaporation in galactic evolution.

Abstract

Dark matter consisting of a Bose--Einstein condensate (BEC) of ultra-light particles is predicted to have a soliton shape that shifts with the dark matter mass fraction in galaxies containing a centrally localized point mass (or black hole), consistent with previous numerical results and analytical approximations in both the cored self-gravitating and cusped hydrogenic limits. Solutions of the Schrödinger-Poisson equation with baryonic coupling are here accurately represented as a sum of five Gaussians with numerically optimized amplitudes and widths, thereby facilitating galactic predictions and observational comparisons as a function of dark matter mass fraction. The results are used to derive mass, energy and velocity scaling relations as functions of soliton mass fraction, as well as to predict dark matter halo size, mass and core density in terms of observed half-light radii and velocity dispersions by invoking observationally validated approximations relating rotational velocity and velocity dispersion. Applications of the predictions, as well as challenges associated with critically testing dark matter models, are illustrated using comparisons with dwarf spheroidal (dSph) and ultra-faint dwarf (UFD) galaxy observations, which, under the present soliton-based modeling assumptions, are found to be compatible with soliton particle masses of the order of $10^{-22}$ (eV/c$^2$), with an upper bound of approximately $3\times 10^{-22}$ (eV/c$^2$). Implications of the results are discussed, including speculations regarding the role of dark matter evaporation in galactic evolution.

Figures (5)

• Figure 1: The top two panels (a) and (b) show the potential energy (blue curves, left axis) and mass probability density (red curves, right axis) for a self-gravitating soliton composed of ultra-light particles of mass $m_0$ in a system with a total mass $M$, plotted using either (a) log-log scales (for the bottom and left axes) or (b) linear scales (for all axes). The solid, dashed, and dot-dashed curves pertain to the 5G, Schive, and Gaussian approximations, respectively. The green horizontal lines indicate the total binding energy $\epsilon$ of an ultra-light soliton particle, including the core tunneling region (dashed green line). The lower two panels (c) and (d) show the tracer rotational velocity (purple curves, left axis) and integrated soliton mass fraction (orange curves, left axis), again plotted using either (c) log-log or (d) linear scales. The arrows mark the locations of the half-density radius $r_c$ and the radius $r_{99\%}$ containing $99\%$ of the total soliton mass, as well as the radii $r_{{-}2}$ and $r_{{-}3}$ at which the logarithmic slope of the soliton density is $-2$ and $-3$, respectively. The radius $r_{{-}3}$ is also close to the radius $r_{50\%} = 3.925\,a_0$ containing $50\%$ of the total soliton mass, as well as to the location of the maximum rotational velocity.
• Figure 2: Shape shifting soliton predictions as a function of dark matter mass fraction $F$. (a) Soliton probability density, (b) potential energy, (c) tracer rotational velocity, and (d) the integrands of Eqs. \ref{['E:Vave']} and \ref{['E:Kave']}. The inset panels in (a)--(c) show the same results plotted on a logarithmic scale. The dashed and dotted $F=0$ curves pertain to the exact hydrogenic and approximate 5G predictions, respectively.
• Figure 3: Soliton property predictions as a function of the soliton mass fraction $F$, dark matter particle mass $m_0$, total galactic mass $M$, and tracer star velocity dispersion $\sigma$. (a) $F$-dependences of $r_c/a_0 = x_c$, $r_{{-}3}/a_0 = x_{{-}3}$, $r_{{-}2}/a_0 = x_{{-}2}$, $\sigma_S = \sqrt{\langle v_S^2\rangle}/v_0$, $\nu_{{-}3} = v_{\rm{rot}}(r_{{-}3})/v_0$, $\epsilon = E/\epsilon_0$, and $F \rho_0$ (multiplied by 100). (b) Dependence of $a_0$ on $M$ and $m_0$, expressed in observational units of $a_0$ (kpc), $M$ (M$_\odot$), and $m_0$ (eV/c$^2$). The right-hand axis shows the corresponding values of $r_c$ (kpc) for a self-gravitating ($F=1$) soliton. The lower three panels show the predicted dependence of (c) radius $r_c$ (kpc), (d) mass $M$ (M$_\odot$), and (e) soliton core density $\rho_\odot$ (M$_\odot$/kpc$^3$) on the observed velocity dispersion $\sigma$ (km/s) for different values of $m_0$ (eV/c$^2$). The grey lines pertain to self-gravitating solitons ($F=1$) with different $m_0$ values, and colored (and dotted) lines show $F$-dependent predictions when $m_0 = 10^{-22}$ (eV/c$^2$). The $+$ symbol pertains to the Fornax (dSph) galaxy, with $r_c = 0.93$ (kpc) SchCos14 and $\sigma = 12$ (km/s) PasAct18. The closed and open black circles pertain to other dSph and UFD galaxies, using the observed stellar $\sigma = \sigma_{{\rm los,obs}}$ and $r_c$ values PozDwa24 (and Appendix Tables I and II), and the predicted $M$ and $\rho_\odot$ obtained using the experimental $\sigma$ and $r_c$ in Eqs. \ref{['E:m0SigmaRc']}--\ref{['E:MSigma']}, as further described in the text.
• Figure 4: Comparison of the observed and predicted properties of dark matter dominated dwarf spheroidal (dSph) and ultrafaint dwarf (UFD) galaxies, obtained from the observed stellar $\sigma$, $r_c$ and $R_{1/2}$PozDwa24. (a) $M_{1/2}$ mass within $R_{1/2}$, obtained using Eq. 2 of WolAcc10 given the observed values of $\sigma$ and $R_{1/2}$PozDwa24. (b) Apparent soliton particle mass $m_0$ obtained assuming $F=0$ and $\sigma_S=\sigma_\ast$, where $m_0$ is adjusted to match the observed $M_{1/2}$. The lower two panels show predictions obtained assuming a constant (galaxy independent) values of $1\times 10^{-22} \le m_0 \le 3\times 10^{-22}$ (eV/c$^2$) obtained assuming one of the following additional constraints. (a) Assuming $\sigma_S=\sigma_\ast$ and adjusting $F$ to obtain agreement with $M_{1/2}$. (b) Assuming $F=0.999$ and adjusting $\sigma_S$ to obtain agreement with $M_{1/2}$. The colored points in panels (c) and (d) represent the masses enclosed within $R_{1/2}$. Specifically, the purple ($\odot$) points represent the central point (black hole) masses, the green $M_S(R_{1/2}$ points represent the soliton mass enclosed within $r\le R_{1/2}$ and the orange ($\ast$) points represent approximate total stellar masses $M_\ast$ ($\ast$) obtained from the observed luminosity $L_{obs}$ (L$_\odot$) assuming a mass to light ratio of 1.
• Figure 5: Comparisons of the predicted and observationally inferred properties of the Draco dSph galaxy. (a) Enclosed mass $M(r)$ and tracer rotational velocity $V_{\rm{rot}}(r)$ (inset panel). (b) Soliton density $\rho(r)$ plotted either on a linear or log (inset panel) axes. The dashed black curves show the observationally constrained predictions reported in Fig. 12 of VitHst24. The colored curves are the present predictions at $F =$ 1, 0.999, 0.95 and 0.9, obtained by adjusting both $m_0$ and $\sigma_S$ to approximately match both $M_{1/2}$ and $M(r<0.9 {\rm{kpc}}) \approx 1.2\times 10^8$ (M$_\odot$) reported in VitHst24. The latter optimal values of $m_0$ and $\sigma_S$ are $m_0 = 2.3\times 10^{-22}$ (eV/c$^2$) and $\sigma_S =14.3$ (km/s) when $F=1$ or 0.999, $m_0 = 2.0\times 10^{-22}$ (eV/c$^2$) and $\sigma_S =13.8$ (km/s) when $F=0.95$, and $m_0 = 1.6\times 10^{-22}$ (eV/c$^2$) and $\sigma_S =13.0$ (km/s) when $F=0.9$.