Ultralight scalars as cosmological dark matter

Many aspects of the large-scale structure of the Universe can be described successfully using cosmological models in which $27\pm 1\%$ of the critical mass-energy density consists of cold dark matter (CDM). However, few—if any—of the predictions of CDM models have been successful on scales of $\sim 10\mathrm{kpc}$ or less. This lack of success is usually explained by the difficulty of modeling baryonic physics (star formation, supernova and black-hole feedback, etc.). An intriguing alternative to CDM is that the dark matter is an extremely light ($m\sim {10}^{-22}\mathrm{eV}$) boson having a de Broglie wavelength $\lambda \sim 1\mathrm{kpc}$, often called fuzzy dark matter (FDM). We describe the arguments from particle physics that motivate FDM, review previous work on its astrophysical signatures, and analyze several unexplored aspects of its behavior. In particular, (i) FDM halos or subhalos smaller than about $10^7(m/{10}^{-22}\mathrm{eV}{)}^{-3/2}$ $M_{\odot }$ do not form, and the abundance of halos smaller than a few times $10^{10}(m/{10}^{-22}\mathrm{eV}{)}^{-4/3}$ $M_{\odot }$ is substantially smaller in FDM than in CDM. (ii) FDM halos are comprised of a central core that is a stationary, minimum-energy solution of the Schrödinger-Poisson equation, sometimes called a “soliton,” surrounded by an envelope that resembles a CDM halo. The soliton can produce a distinct signature in the rotation curves of FDM-dominated systems. (iii) The transition between soliton and envelope is determined by a relaxation process analogous to two-body relaxation in gravitating N-body systems, which proceeds as if the halo were composed of particles with mass $\sim \rho {\lambda }^3$ where $\rho$ is the halo density. (iv) Relaxation may have substantial effects on the stellar disk and bulge in the inner parts of disk galaxies, but has negligible effect on disk thickening or globular cluster disruption near the solar radius. (v) Relaxation can produce FDM disks but a FDM disk in the solar neighborhood must have a half-thickness of at least $\sim 300(m/{10}^{-22}\mathrm{eV}{)}^{-2/3}\mathrm{pc}$ and a midplane density less than $0.2(m/{10}^{-22}\mathrm{eV}{)}^{2/3}$ times the baryonic disk density. (vi) Solitonic FDM subhalos evaporate by tunneling through the tidal radius and this limits the minimum subhalo mass inside $\sim 30\mathrm{kpc}$ of the Milky Way to a few times $10^8(m/{10}^{-22}\mathrm{eV}{)}^{-3/2}$ $M_{\odot }$. (vii) If the dark matter in the Fornax dwarf galaxy is composed of CDM, most of the globular clusters observed in that galaxy should have long ago spiraled to its center, and this problem is resolved if the dark matter is FDM. (viii) FDM delays galaxy formation relative to CDM but its galaxy-formation history is consistent with current observations of high-redshift galaxies and the late reionization observed by Planck. If the dark matter is composed of FDM, most observations favor a particle mass $\gtrsim {10}^{-22}\mathrm{eV}$ and the most significant observational consequences occur if the mass is in the range $1–10\times {10}^{-22}\mathrm{eV}$. There is tension with observations of the Lyman-$\alpha$ forest, which favor $m\gtrsim 10–20\times {10}^{-22}\mathrm{eV}$ and we discuss whether more sophisticated models of reionization may resolve this tension.

Figure 1

The lifetime of a stellar system in the ground state of the Schrödinger-Poisson equation when a spherically symmetric tidal field is present. The vertical axis gives the dimensionless lifetime $\tau$ in units of the orbital period [Eq. (c7)] and the horizontal axis gives the ratio of the central density to the mean density of the host averaged over the orbital radius.

Figure 2

The dynamical friction force on a test object of mass $m_{\mathrm{cl}}$ traveling at speed $v$ through FDM with density $\rho$ is given by Eq. (d6), with the dimensionless function $C$ plotted in this figure. The horizontal axis is $kr$ where $k=mv/\hslash$ is the wave number of the FDM particles in the rest frame of the test object, and $r$ is an upper cutoff to the distance from the test object that approximately represents the smaller of the radius of the orbit and the size of the host system. Each curve is for a fixed value of $\mathrm{\Lambda }=v^{2}r/G{m}_{\mathrm{cl}}$, which by the virial theorem is approximately the ratio of the mass of the host to the mass of the test object; from bottom to top $\mathrm{\Lambda }=3$, 10, 30, 100, 300. The dashed red line shows the asymptotic behavior in the limit $\mathrm{\Lambda }\to \infty$ [Eq. (d14)].

Figure 3

Density evolution in a self-similar FDM solution, according to Eq. (e9). The dimensionless time $\tilde{t}$ and distance $\tilde{x}$ are defined in Eq. (e3), where $x_0$ is an arbitrary length scale. The overall normalization of the density $\rho$ is also arbitrary. With the choice $\mathcal{A}=\mathcal{B}$ made here, the ratio of the asymptotic density at $x\to \pm \infty$ is $(\mathcal{A}+\mathcal{B}{)}^2/{\mathcal{B}}^2=4$.