Dwarf galaxies united by dark bosons

Low-mass galaxies in the local group are dominated by dark matter and comprise the well-studied “dwarf spheroidal” (dSph) class, with typical masses of ${10}^9–{10}^{10}{M}_{\odot }$ and also the equally numerous “ultrafaint dwarfs” (UFDs), discovered recently, that are distinctly smaller and denser with masses of only ${10}^7–{10}^8{M}_{\odot }$. This bimodality amongst low-mass galaxies contrasts with the scale-free continuity expected for galaxies formed under gravity, as in the standard cold dark matter model for heavy particles. Within each dwarf class we find the core radius $R_c$ is inversely related to velocity dispersion $\sigma$, quite the opposite of standard expectations, but indicative of dark matter in a Bose-Einstein state, where the uncertainty principle requires $R_c\times \sigma$ is fixed by Planck’s constant, $h$. The corresponding boson mass, $m_b=h/R_c\sigma$, differs by one order of magnitude between the UFD and dSph classes, with ${10}^{-21.4}\mathrm{eV}$ and ${10}^{-20.3}\mathrm{eV}$, respectively. The two-boson species is reinforced by parallel relations seen between the central density and radius of UFD and dSph dwarfs, respectively, each matching the steep prediction, ${\rho }_c\propto R_c^{-4}$, for soliton cores in the ground state. Furthermore, soliton cores accurately fit the stellar profiles of UFD and dSph dwarfs where prominent, dense cores appear surrounded by low-density halos, as predicted by our simulations. Multiple bosons may point to a string theory interpretation for dark matter, where a discrete mass spectrum of axions is generically predicted to span many decades in mass, offering a unifying “axiverse” interpretation for the observed “diversity” of dark matter dominated dwarf galaxies.

Figure 1

(Left panel) Density vs half-light radius: Here, we plot the central density, $(\sigma /R_h{)}^2$, for each local dwarf (named on the plot), as reported by various groups (refer to the Appendixes pp1 and pp2). These are color-coded by luminosity, revealing a clear distinction between the UFD and dSph classes. The data forms two parallel power-law fits shown in blue. (Right panel) Density vs core radius: Here, we depict the density within our fitted core radius for each dwarf, $(\sigma /R_c{)}^2$, utilizing the soliton form for the core (refer to the Appendix pp2 for all individual dwarf fits to $\psi \mathrm{DM}$ and Plummer profiles). This approach results in sharper parallel relationships between the UFD and dSph dwarfs, showing a good fit to the slope $d\log {\rho }_c/d\log R_c=-4$. This slope corresponds to the time-independent soliton solution of the Schrodinger-Poisson relation, where a higher soliton mass leads to a narrower core. The slopes of the blue lines are constrained to be $-4$, as predicted by wave dark matter. For the left panel, the slope is $-3$ due to the $R_{\mathrm{half}}/R_c$ relation. Core densities reported for the Milky Way, DF44, and Antlia-2 have been added and are seen to be consistent with the lighter boson, aligning with the dSph class. We have included core densities reported for the Milky Way, DF44, and Antlia-2, found to be consistent with the lighter boson, aligning with the dSph class. It is important to note that $R_c$ cannot be as efficiently constrained for these cases, and thus they have been added in an illustrative manner (without accounting for the calculations), colored in white. Their values are presented without ensuring their reasonability as with the other cases.

Figure 2

(Top panel) Dwarf spheroidal galaxies: The mean star count profile, scaled to the mean core radius of all dSph dwarfs listed in Table 2, reveals prominent cores in both dSph dwarfs relative to the halo, which extends to several times the core radius. While a standard Plummer profile (red dashed curve) roughly fits the core region, it falls significantly short at larger radii. In contrast, the $\psi \mathrm{DM}$ profile, with its inherent core-halo structure, accurately fits the core from the soliton component and extends to the halo when azimuthally averaged over the excited states approximating the NFW form. The soliton profile has only one free parameter, the boson mass, $m_b$, determining the scale radius of the soliton. The sharp density drop between the core and the halo, by a factor of $\simeq 30$, is a characteristic feature of $\psi \mathrm{DM}$ at a transition radius marked by the vertical orange band. The best-fit MCMC profile parameters are tabulated in Table 1. (Lower panel) Ultrafaint dwarfs: The mean profile, averaged over all resolved profiles of ultrafaint dwarfs listed in Table 3, exhibits the predicted $\psi \mathrm{DM}$ core-halo structure. This includes a marked transition in density between the core (highlighted in orange), with the best-fit MCMC profile parameters tabulated in Table 1.

Figure 3

Velocity dispersion vs core radius: Here, we plot the observed velocity dispersion against the core radius for all dSph and UFD dwarfs, comparing them with the inverse relation required by the uncertainty principle. The best-fit blue lines to the UFD and dSPh dwarfs separately are indicated, with the corresponding boson masses of Wave-DM derived from the normalization shown in the legend. The slopes of the blue lines are constrained to be $-1$, as predicted by wave dark matter. Additionally, the CDM-related prediction [31] is shown as a thin red curve, where galaxies with NFW profiles are larger with increasing mass, contrary to the behavior expected in $\psi \mathrm{DM}$.

Figure 4

Representative filament from a two-boson simulation: Low-mass galaxies mainly form within the filaments and are dominated by the heavier boson (bottom-right profile) as the relatively large de Broglie scale of the light boson prevents light boson fragmentation along filaments. The more massive halos are observed to form at the filament nodes with a mix of bosons (bottom-left profile), reflecting the equal proportion of light and heavy bosons chosen for this simulation. The boson mass ratio $m_1/m_2=5$ here and $M_{10}$ denote the mass enclosed within 10 kpc of each halo.

Figure 5

DM density vs core radius: Expanding upon the left panel of Fig. 1, we now incorporate color to represent metallicity. It is noteworthy that the ultrafaint galaxies consistently exhibit lower metallicity compared to the dSph class, thus reinforcing the empirical distinction between these two classes of dwarf galaxies, which is based on luminosity.

Figure 6

DM density vs core radius: Expansion of the right panel of Fig. 1.

Figure 7

DM density vs half-light radius: Expansion of the left panel of Fig. 1.

Figure 8

All dSph: Classical dwarfs mean profile (Fig. 2 top panel): correlated distributions of the free parameters. As can be seen the core radius and transition radius are well-defined despite wide Gaussian priors, indicating a reliable result. The contours represent the 68%, 95%, and 99% confidence levels. The best-fit parameter values are the medians (with errors), represented by the dashed black ones, and tabulated in Table 1.

Figure 9

All UFD: Ultrafaint dwarfs mean profiles (Fig. 2 lower panel): correlated distributions of the free parameters. As can be seen the core radius and transition radius are well-defined despite wide Gaussian priors, indicating a reliable result. The contours represent the 68%, 95%, and 99% confidence levels. The best-fit parameter values are the medians (with errors), represented by the dashed black curve, and tabulated in Table 1.

Figure 10

Dwarf spheroidal galaxies: This figure presents star count profiles versus dwarf galaxy radius for well-studied dSph dwarf galaxies in the local group, as listed in Table 2. In most cases, an extended halo of stars is observable, stretching to approximately $\simeq 2\mathrm{kpc}$, most prominently visible on the linear scale of the left-hand panel. Prominent cores are also evident on a scale of less than 1 kpc in each dwarf. A standard Plummer profile (red dashed curve) roughly fits the core region but falls significantly short at larger radii. Our predictions for the dSph class ($\simeq {10}^{-22}\mathrm{eV}$) in $\psi \mathrm{DM}$ are depicted in green, where the distinctive soliton profile provides an excellent fit to the observed cores and the surrounding halo of excited states that azimuthally average to an approximately NFW-like profile beyond the soliton radius. The accuracy of the core fit to the soliton is best observed on a log scale in the right panels, while the left panel in linear scale shows the extent of the halo. This includes the characteristic density drop of about a factor of $\simeq 30$ predicted by $\psi \mathrm{DM}$ between the prominent core and tenuous halo at a radius of approximately 1 kpc, indicated by the vertical orange band. The best-fit MCMC profile parameters are tabulated in the Appendix pp2, and references to the data in this figure are as follows: Tucana [51], Cetus [52], and Aquarius [53].

Figure 11

Dwarf spheroidal galaxies: Similar to Fig. 10, this figure includes three additional dSph galaxies. References to the data for these galaxies are as follows: Draco [54], Leo I [55], and Phoenix [56].

Figure 12

Dwarf spheroidal galaxies: In line with Fig. 10, this figure features three additional dSph galaxies. References for the data are as follows: Sextans [57], Andromeda XXI [27], and Crater II [58]. It is noteworthy to highlight that Andromeda XXI exhibits the same $\psi \mathrm{DM}$ core-halo structure as the dSph satellites of the Milky Way, further supporting the “universality” of this profile for dwarfs.

Figure 13

Dwarf spheroidal galaxies: As in Fig. 10, this figure includes three additional galaxies. References for the data are as follows: Carina [59], Sculptor [59], and Ursa Minor [60].

Figure 14

Dwarf spheroidal galaxies: Similar to Fig. 10, this figure showcases three additional galaxies. References for the data are Canes Venatici [61] and Leo II [62].

Figure 15

Dwarf spheroidal galaxies: Similar to Fig. 10, this figure includes three additional galaxies. Notably, these Andromeda galaxies exhibit the same $\psi \mathrm{DM}$ core-halo structure as UFD galaxies in the Milky Way, underscoring the universality of the $\psi \mathrm{DM}$ profile for dwarfs. References for the data are as follows: Andromeda I [63], Andromeda III [64], and Andromeda V [64].

Figure 16

Dwarf spheroidal galaxies: As in Fig. 10, this figure includes three additional galaxies. References for the data are Andromeda IX [64], Andromeda XIV [64], and Andromeda XV [64].

Figure 17

Dwarf spheroidal galaxies: Similar to Fig. 10, this figure includes three additional galaxies. References for the data are Andromeda XVIII [64], Andromeda XXIII [64], and Andromeda XXV [64].

Figure 18

Ultrafaint dwarfs: This figure presents the star count profiles versus dwarf galaxy radius for the “ultrafaint” dwarf galaxies in the local group, as listed in Table 3. Many UFD dwarfs show clear evidence of extended halos stretching to approximately $\simeq 0.5\mathrm{kpc}$, most notably visible on the linear scale of the left-hand panel. Cores are also apparent on a scale of less than 0.1 kpc in these UFD dwarfs. A standard Plummer profile (red dashed curve) is observed to roughly fit the core region but falls significantly short at large radius. The soliton profile, normalized to the mean boson mass estimated for these dwarfs ($\simeq {10}^{-21}\mathrm{eV}$), is shown in green. The distinctive soliton profile provides an excellent fit to the observed cores, surrounded by a halo of excited states that azimuthally average to an approximately NFW-like profile beyond the soliton radius. The cores align well with the predicted form of the soliton profile, as best seen on a log scale in the right panels. The best-fit MCMC profile parameters are tabulated in the Appendix pp2. References for the data are as follows: Phoenix II [94], Segue I [62], and Pegasus III [62].

Figure 19

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Wilman I [62], Horologium I [62], and Pisces II [62].

Figure 20

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Coma Berenices [62], Reticulum II [95], and Hydrus I [96].

Figure 21

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Grus I [96], Leo IV [97], and Canes Venatici II [96].

Figure 22

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Bootes I [62], Tucana II [98], and Tucana IV [62]. It’s worth noting that Chiti et al. [26] claimed a surprisingly extended halo of stars and dark matter, extending to 1 kpc in extent for Tucana II.

Figure 23

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Andromeda X [64], Andromeda XI [64], and Andromeda XII [64].

Figure 24

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Andromeda XIII [64], Andromeda XVI [64], and Andromeda XVII [99].

Figure 25

Ultrafaint dwarf galaxies: Similar to Fig. 15, this figure includes three additional galaxies. References for the data are as follows: Andromeda XX [64], Andromeda XXII [100], and Andromeda XXVI [64].

Figure 26

Like Fig. 2 but just for Milky Way’s satellites.

Figure 27

Velocity dispersion vs core radius: Like Fig. 3 but just for Milky Way’s satellites.

Figure 28

DM density vs core radius: Like Fig. 1 just for Milky Way’s satellites.

Figure 29

DM density vs core radius: Like Fig. 5 but just for Milky Way’s satellites.

Figure 30

Milky Way’s UFD mean (Fig. 26 lower panel): Correlated distributions of the free parameters. As can be seen the core radius and transition radius are well-defined despite the wide Gaussian priors, indicating a reliable result. The contours represent the 68%, 95%, and 99% confidence levels. The best-fit parameter values are the medians (with errors), represented by the dashed black ones, and tabulated in Table 1.

Figure 31

Milky Way’s dSph mean (Fig. 26 top panel): Correlated distributions of the free parameters. As can be seen the core radius and transition radius are well-defined despite the Gaussian input priors, indicating a reliable result. The contours represent the 68%, 95%, and 99% confidence levels. The best-fit parameter values are the medians (with errors), represented by the dashed black ones, and tabulated in Table 1.

Figure 32

Like Fig. 2 but just for Andromeda’s satellites.

Figure 33

Velocity dispersion vs core radius: Like Fig. 3 but just for Andromeda’s satellites.

Figure 34

DM density vs core radius: Like Fig. 1 but just for Andromeda’s satellites.

Figure 35

DM density vs core radius: Like Fig. 5 but just for Andromeda’s satellites.

Figure 36

Andromeda’s dSph mean (Fig. 26 lower panel): Correlated distributions of the free parameters. As can be seen the core radius and transition radius are well-defined despite the Gaussian input priors, indicating a reliable result. The contours represent the 68%, 95%, and 99% confidence levels. The best-fit parameter values are the medians (with errors), represented by the dashed black ones, and tabulated in Table 1.

Figure 37

Andromeda’s UFD mean (Fig. 26 lower panel): Correlated distributions of the free parameters. As can be seen the core radius and transition radius are well-defined despite the Gaussian input priors, indicating a reliable result. The contours represent the 68%, 95%, and 99% confidence levels. The best-fit parameter values are the medians (with errors), represented by the dashed black ones, and tabulated in Table 1.