Simple test with rotation curves for the multistate scalar field dark matter model

We use the concept of coadded rotation curves of Salucci et al. to investigate the properties of axisymmetric multistate scalar field dark matter (SFDM) halos in low surface brightness (LSB) galaxies and dwarf disc galaxies. To this end, we expand the wave function of the scalar field into its multistates and solve the field equations numerically. We fit their rotation curves in two-state configurations finding that a two-state configuration fits better than the single ground-state (soliton) configuration in the dwarf disc and in the smaller of the LSB galaxies. We obtain a SFDM mass $\mu =2.38\pm 0.12\times {10}^{-23}\mathrm{eV}/c^2$ for the dwarf disc galaxies and $\mu =1.05\pm 0.14\times {10}^{-23}\mathrm{eV}/c^2$ for the smaller of the LSB galaxies. For the larger of the LSB galaxies a mass of the order of $\mu =O({10}^{-24})-O({10}^{-25})\mathrm{eV}/c^2$ is obtained, where the mass $\mu$ is the effective mass measured by an outside observer due to the finite temperature of the SFDM in the galaxy.

Figure 1

Family of solutions of the multiSFDM $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{210})$. The excited state radial function ${\psi }_{210}$ (upper), first function $V_{00}$ (middle panel) and second function $V_{20}$ (next panel) in the expansion of the potential $V$, and the enclosed mass $N(r)$ (bottom pannel). In the color scale, the total mass $N_T=N(r_f)$ of each of the solutions in the family is shown.

Figure 2

Projection in the $(x,z)$ plane of the dimensionless mass density as a function of the ($x$, $y$, $z$) dimensionless cartesian coordinates for the multiSFDM $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{210})$. The left panel shows the solution with $N_T=2.0$ where the monopole term ${\psi }_{100}$ dominates over the dipole term ${\psi }_{210}$, and the right panel is the solution with $N_T=5.5$ where the excited state ${\psi }_{210}$ dominates. In color scale, the mass density is shown.

Figure 3

Family of solutions of the multiSFDM $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{200})$. The wave function ${\psi }_{200}$ (upper panel) and the potential $V$ (bottom panel). In the color scale, the total mass $N_T$ of each of the solutions in the family is shown.

Figure 4

Upper panel: Circular velocity measurements of 36 dwarf disc galaxies; middle panel: the normalized rotation curves; and bottom panel: the coadded rotation curves of dwarf disc galaxies. Data from [62].

Figure 5

Coadded rotation curves for each of the five bins of the low surface brightness galaxies. Data from [61].

Figure 6

Circular velocity $v_h$ for the $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{200})$ family (upper panel), the $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{210})$ family (bottom panel). In the color scale, the total mass $N_T$ of each of the solutions in the family is shown.

Figure 7

Upper panel: Dwarf Disc galaxies coadded rotation curve. The disc, HI disc, and dark matter contributions are also shown. Dark matter is in the multiSFDM $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{210})$. The best-fit parameters are shown in Table 4. The horizontal line is the disc characteristic length $a_d$. Bottom panel: We show the posterior distribution of parameters, as an example, in this case. Particle mass $\tilde{\mu }$ is in 1/kpc units and disc mass $M_d$ is in ${10}^{10}{M}_{\odot }$ units.

Figure 8

LSB bin 1 galaxies coadded rotation curve. The disc and dark matter contributions are also shown. Dark matter is in the multiSFDM $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{210})$. The best-fit parameters are shown in Table 5. The horizontal line is the disc characteristic length $a_d$.

Figure 9

Three-state $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{200},{\mathrm{\Psi }}_{300})$ multistate configuration. The three wave functions and the gravitational potential are shown.

Figure 10

LSB bin 5 fit with a three-state $({\mathrm{\Psi }}_{100},{\mathrm{\Psi }}_{200},{\mathrm{\Psi }}_{300})$ spherically symmetric multistate configuration. The scalar dark matter mass $\mu =(1.24\pm 0.06)\times {10}^{-24}\mathrm{eV}/c^2$ becomes bigger than for a two-states configuration.

Figure 11

The baryonic fraction $M_d/Mh$ as a function of the halo mass $M_d$ for dwarf disc and LSB bins 1 to 5. The blue line is the relation of [74].

Figure 12

Dark matter central surface density ${\mathrm{\Sigma }}_0={\rho }_0{r}_c$ as a function of the optical velocity $v_{\mathrm{opt}}$. Red markers are the quantities calculated from the dwarf disc and LSB bins, and blue markers are calculated for the individual galaxies belonging to each bin.

Figure 13

Core radius, dark matter central density, and virial radius as a function of the disc scale length of the dwarf disc and all LSB bins. The blue line is the logarithmic fit.