Testing multiflavored ultralight dark matter models with SPARC

We perform maximum likelihood estimates for single and double flavor ultralight dark matter (ULDM) models using the Spitzer Photometry and Accurate Rotation Curves database. These estimates are compared to maximum likelihood estimates for several commonly used cold dark matter (CDM) models. By comparing various CDM models we find, in agreement with previous studies, that the Burkert and Einasto models tend to perform better than other commonly used CDM models. We focus on comparisons between the Einasto and ULDM models and analyze cases for which the ULDM particle masses are fixed or free to vary. For each of these analyses, we perform fits assuming the soliton and halo density profiles are summed together or matched at a given radius. Letting the particle masses vary and assuming the summed models, we find a negligible preference for any particular range of particle masses within ${10}^{-25}\mathrm{eV}\le m\le {10}^{-19}\mathrm{eV}$. For the matched models, we find that almost all galaxies prefer particle masses in the range ${10}^{-23}\mathrm{eV}\lesssim m\lesssim {10}^{-20}\mathrm{eV}$. For both double flavor models we find that most galaxies prefer approximately equal particle masses. Fixing the particle masses, we find the best fit results for the particle mass $m={10}^{-20.5}\mathrm{eV}$, assuming the single flavor models, and $m_1={10}^{-20.5}\mathrm{eV}$, $m_2={10}^{-20.2}\mathrm{eV}$, assuming the double flavor, matched model.

Figure 1

Particle masses free—analysis (1) in Table 1. Difference in the BIC statistics for Einasto and ULDM assuming the model: single flavor, profiles summed [SS(1)] (top left); single flavor, profiles matched [SM(1)] (top right); double flavor, profiles summed [DS(1)] (middle and bottom left); double flavor, profiles matched [DM(1)] (middle and bottom right). The middle panel corresponds to $m_1$, and the bottom panel to $m_1/m_2$. Black points correspond to each of the galaxies analyzed, the black dashed line corresponds to $\mathrm{\Delta }\mathrm{BI}\mathrm{C}=0$, blue lines correspond to $|\mathrm{\Delta }\mathrm{BIC}|=2$, red lines to $|\mathrm{\Delta }\mathrm{BIC}|=6$, and green lines to $|\mathrm{\Delta }\mathrm{BIC}|=10$. Inset is the fraction of galaxies that fall within a given range for $\mathrm{\Delta }\mathrm{BIC}$ where the Einasto model is taken first in the difference. The points are shaded corresponding to the approximate probability density, with darker points corresponding to denser regions.

Figure 2

Particle masses free—analysis (1) in Table 1. Reduced chi-square for the Einasto model vs reduced chi-square for the ULDM models: SS(1) (top left); SM(1) (top right); DS(1) (bottom left); DM(1) (bottom right). The black dashed lines correspond to ${\chi }_{\nu ,\text{Einasto}}^2={\chi }_{\nu ,\mathrm{ULDM}}^2$ while the blue horizontal lines show where ${\chi }_{\nu ,\text{Einasto}}^2=1$ and the blue vertical lines show where ${\chi }_{\nu ,\mathrm{ULDM}}^2=1$. The points are shaded corresponding to the approximate probability density, with darker points corresponding to denser regions.

Figure 3

Particle masses free—analysis (1) in Table 1. ${\log }_{10}(M_{\mathrm{sol}}/M_{\mathrm{sol},\mathrm{SH}})$ vs ${\log }_{10}m$ where $M_{\mathrm{sol},\mathrm{SH}}$ refers to the soliton mass assuming the SH relation given by Eq. (2.18) for the assumed models: SS(1) (top left), SM(1) (top right), DS(1) (middle and bottom left), and DM(1) (middle and bottom right) with the middle row corresponding to $m_1$ and the bottom to $m_1/m_2$. We plot each galaxy with a given marker depending on the error measurements for the particle and soliton masses. The error calculated can either be ($-$) nonexistent or larger than the best fit parameter or ($+$) smaller than the best fit parameter. Black points correspond to galaxies with ($+$) for both the soliton and particle mass; red squares correspond to galaxies with ($+$) for the soliton mass and ($-$) for the particle mass; blue triangles correspond to galaxies with ($-$) for the soliton mass and ($+$) for the particle mass; green Xs correspond to galaxies with ($-$) for both the soliton and particle masses.

Figure 4

Particle masses fixed and scanned—analysis (2) in Table 1. $f({\chi }_{\nu }^2)\equiv 1-\sum {\chi }_{\nu ,\mathrm{ULDM}}^2/\sum {\chi }_{\nu ,\text{Einasto}}^2$ vs particle mass for the assumed models: SS(2) (top left), SM(2) (top right), DS(2) (bottom left), and DM(2) (bottom right).

Figure 5

Particle masses fixed and scanned—analysis (2) in Table 1. ${\log }_{10}(M_{\mathrm{sol}}/M_{\mathrm{sol},\mathrm{SH}})$ vs ${\log }_{10}m$ (top and bottom panels) where $M_{\mathrm{sol},\mathrm{SH}}$ refers to the soliton mass assuming the SH relation given by Eq. (2.18). The top row corresponds to the SS(2) (left) and SM(2) (right) models. The middle and bottom rows correspond to the DS(2) (left) and DM(2) (right) models. The middle row corresponds to the particle mass fixed at $m={10}^{1.5}{m}_{22}$ and shows the histogram of ${\log }_{10}(M_{\mathrm{sol}}/M_{\mathrm{sol},\mathrm{SH}})$ for this particular mass. The bottom row corresponds to the particle mass that is varied in the double flavor models. For the top and bottom rows, the black points (Xs) correspond to the mean (median) value of ${\log }_{10}(M_{\mathrm{sol}}/M_{\mathrm{sol},\mathrm{SH}})$.

Figure 6

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Difference in the BIC statistics for Einasto and ULDM assuming the model: SS (top left), SM (top right), DS (bottom left), DM (bottom right). The black dashed line corresponds to $\mathrm{\Delta }\mathrm{BI}\mathrm{C}=0$, blue lines correspond to $|\mathrm{\Delta }\mathrm{BIC}|=2$, red lines to $|\mathrm{\Delta }\mathrm{BIC}|=6$, and green lines to $|\mathrm{\Delta }\mathrm{BIC}|=10$. Inset is the fraction of galaxies that fall within a given range for $\mathrm{\Delta }\mathrm{BIC}$ where the Einasto model is taken first in the difference.

Figure 7

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Reduced chi-square for the Einasto model vs reduced chi-square for the ULDM models: SS (top left), SM (top right), DS (bottom left), DM (bottom right). The black dashed lines correspond to ${\chi }_{\nu ,\text{Einasto}}^2={\chi }_{\nu ,\mathrm{ULDM}}^2$ while the blue horizontal lines show where ${\chi }_{\nu ,\text{Einasto}}^2=1$ and the blue vertical lines show where ${\chi }_{\nu ,\mathrm{ULDM}}^2=1$. The points are shaded corresponding to the approximate probability density, with darker points corresponding to denser regions.

Figure 8

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). $M_{\mathrm{sol}}/M_{\mathrm{sol},\mathrm{SH}}$ where $M_{\mathrm{sol},\mathrm{SH}}$ refers to the soliton mass assuming the soliton halo relation given by Eq. (2.18). The top row corresponds to the SS (left) and SM (right) models. The middle and bottom rows correspond to the DS (left) and DM (right) models. The top and middle row corresponds to the particle mass fixed at $m={10}^{1.5}{m}_{22}$, and the bottom row to the particle mass fixed at $m={10}^{1.8}{m}_{22}$. The points are marked in the same way as in Fig. 3.

Figure 9

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Reduced chi-square ${\chi }_{\nu }^2$ for the assumed models: single, summed (top, left); single, matched (top, right); double, summed (bottom, left); double, matched (bottom, right).

Figure 10

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Distributions for $c_{200}$ (top row), $V_{200}$ (second from top row), ${\tilde{\mathrm{\Upsilon }}}_d$ (third from top row), and ${\tilde{\mathrm{\Upsilon }}}_b$ (bottom row) for the single, summed (left column), single, matched (second column from the left), double, summed (third column from the left), and double, matched (right column) models.

Figure 11

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Rotation curves for galaxies NGC5055 (top row) and NGC3109 (bottom row) for the assumed models: single, summed (left); double, summed (right).

Figure 12

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Rotation curves for galaxies NGC5055 (top row) and NGC3109 (bottom row) for the assumed models: single, matched (left); double, matched (right).

Figure 13

Particle masses fixed ($m_1={10}^{1.5}{m}_{22}$, $m_2={10}^{1.8}{m}_{22}$). Empirical relations for the single, summed (left column), single, matched (second column from the left), double, summed (third column from the left), and double, matched (right column) models. Top row: gravitational RARs. Blue points correspond to Eq. (a1) for the MOND value $g_{†}=1.2\times {10}^{-10}\mathrm{m}/{\mathrm{s}}^2$, and orange points to Eq. (a1) with the best fit $g_{†}$. Second row from the top: ${\log }_{10}{c}_{200}$ vs ${\log }_{10}{M}_{200}$. Black solid lines correspond to values calculated from Eq. (a3) and black dashed lines to values calculated from Eq. (a2). Third row from the top: ${\log }_{10}{M}_{\text{baryons}}$ (i.e., $M_{*}+M_{\mathrm{gas}}$) vs ${\log }_{10}{V}_f$. Black solid lines correspond to values calculated from Eq. (a4). Bottom row: ${\log }_{10}{M}_{*}$ vs ${\log }_{10}{M}_{200}$. Black solid lines correspond to values calculated from Eq. (a5). For the second from the top, third from the top and bottom rows, the points are marked the same as in Fig. 3.

Figure 14

Particle mass free. Reduced chi-square ${\chi }_{\nu }^2$ for the assumed models: single, summed (top, left); single, matched (top, right); double, summed (bottom, left); double, matched (bottom, right).

Figure 15

Particle mass free. Distributions for $c_{200}$ (top row), $V_{200}$ (second from top row), ${\tilde{\mathrm{\Upsilon }}}_d$ (third from top row), and ${\tilde{\mathrm{\Upsilon }}}_b$ (bottom row) for the single flavor summed (left column), single flavor matched (second column from the left), double flavor summed (third column from the left), and double flavor matched (right column) models.

Figure 16

Particle mass free. Rotation curves for galaxies NGC5055 (top row) and NGC3109 (bottom row) for the assumed models: single, summed (left); double, summed (right).

Figure 17

Particle mass free. Rotation curves for galaxies NGC5055 (top row) and NGC3109 (bottom row) for the assumed models: single, matched (left); double, matched (right).

Figure 18

Particle mass free. Empirical relations for the single, summed (left column), single, matched (second column from the left), double, summed (third column from the left), and double, matched (right column) models. Top row: gravitational RARs. Blue points correspond to Eq. (a1) for the MOND value $g_{†}=1.2\times {10}^{-10}\mathrm{m}/{\mathrm{s}}^2$, and orange points to Eq. (a1) with the best fit $g_{†}$. Second row from the top: ${\log }_{10}{c}_{200}$ vs ${\log }_{10}{M}_{200}$. Black solid lines correspond to values calculated from Eq. (a3) and black dashed lines to values calculated from Eq. (a2). Third row from the top: ${\log }_{10}{M}_{\text{baryons}}$ (i.e., $M_{*}+M_{\mathrm{gas}}$) vs ${\log }_{10}{V}_f$. Black solid lines correspond to values calculated from Eq. (a4). Bottom row: ${\log }_{10}{M}_{*}$ vs ${\log }_{10}{M}_{200}$. Black solid lines correspond to values calculated from Eq. (a5). For the second from the top, third from the top and bottom rows, the points are marked the same as in Fig. 3.

Figure 19

Distributions of the difference between BIC statistics for different priors cases. We check five different priors cases: (1) uniform priors and finite parameter ranges for all parameters; (2) case (1) with the change $0<c_{200}<\infty$; (3) case (1) with the change $0<V_{200}<\infty$; (4) case (1) with the change $0<{\tilde{\mathrm{\Upsilon }}}_d<\infty$; (5) case (1) with the change $0<{\tilde{\mathrm{\Upsilon }}}_b<\infty$. The difference between BIC statistics is given by as $\mathrm{\Delta }\mathrm{BIC}={\mathrm{BIC}}_{(1)}-{\mathrm{BIC}}_{(i)}$, where $i=2$ corresponds to the top row, $i=3$ to the second row from the top, $i=4$ to the third row from the top, and $i=5$ to the bottom row. The Burkert model corresponds to the left column, the DC14 model to the second column from the left, the Einasto model to the third column from the left, and the NFW model to the right column.

Figure 20

BIC statistics for Burkert vs DC14 (top left), Burkert vs Einasto (top middle), Burkert vs NFW (top right), DC14 vs Einasto (bottom left), DC14 vs NFW (bottom middle), and Einasto vs NFW (bottom right). Black points correspond to each of the galaxies analyzed, the black dashed line corresponds to $\mathrm{\Delta }\mathrm{BI}\mathrm{C}=0$, blue lines correspond to $|\mathrm{\Delta }\mathrm{BIC}|=2$, red lines to $|\mathrm{\Delta }\mathrm{BIC}|=6$, and green lines to $|\mathrm{\Delta }\mathrm{BIC}|=10$. Inset is the fraction of galaxies that fall within a given range for $\mathrm{\Delta }\mathrm{BIC}$.

Figure 21

Reduced chi-square ${\chi }_{\nu }^2$ for the Burkert (top left), DC14 (top right), Einasto (bottom left), and the NFW (bottom right) models.

Figure 22

Distributions for $c_{200}$ (top row), $V_{200}$ (second from top row), ${\tilde{\mathrm{\Upsilon }}}_d$ (third from top row), and ${\tilde{\mathrm{\Upsilon }}}_b$ (bottom row) for the Burkert (left column), DC14 (second from left column), Einasto (third from left column), and the NFW (right column) models.

Figure 23

Rotation curves for galaxies NGC5055 (top row) and NGC3109 (bottom row) for the Burkert (left column), DC14 (second from left column), Einasto (third from left column), and the NFW (right column) models.

Figure 24

Galaxies for which the best fit ${\tilde{\mathrm{\Upsilon }}}_d$ is strongly degenerate (top row) and those for which the degeneracy is weaker (bottom row).

Figure 25

Empirical relations for the Burkert (left column), DC14 (second column from the left), Einasto (third column from the left), and NFW (right column) models. Top row: gravitational RARs. Blue points correspond to Eq. (a1) for the MOND value $g_{†}=1.2\times {10}^{-10}\mathrm{m}/{\mathrm{s}}^2$, and orange points to Eq. (a1) with the best fit $g_{†}$. Second row from the top: ${\log }_{10}{c}_{200}$ vs ${\log }_{10}{M}_{200}$. Black solid lines correspond to values calculated from Eq. (a3) and black dashed lines to values calculated from Eq. (a2). Third row from the top: ${\log }_{10}{M}_{\text{baryons}}$ (i.e., $M_{*}+M_{\mathrm{gas}}$) vs ${\log }_{10}{V}_f$. Black solid lines correspond to values calculated from Eq. (a4). Bottom row: ${\log }_{10}{M}_{*}$ vs ${\log }_{10}{M}_{200}$. Black solid lines correspond to values calculated from Eq. (a5). For the second from the top, third from the top and bottom rows, the points are marked the same as in Fig. 3.