SPARC Data Selection:

Flat rotation curves at the outer galactic radii are among the observational
evidence of dark matter (DM) on the galactic scale. The standard cold darkmatter (CDM) model successfully explained the formation of the large-scale
structures in the universe. However, many issues arise on the galactic scale(small-scale problems), such as the core-cusp problem, i.e., the discrepancy
between the observed cored DM profile of dwarf galaxies and the predictedcuspy Navarro-Frenk-White (NFW) profile based on CDM simulations. Thisproblem had motivated astronomers to propose empirical DM density profile thatbetter matches various observations, e.g., the Burkert profile.
𝜌𝑠 : characteristic density
𝑟𝑠 : characteristic radius𝜌𝑠𝑟𝑟 2
𝑟𝑠 1 + 𝑟𝑠

𝜌Burkert (𝑟) =𝑟 2
1+ 𝑟𝑠

Another issue is the rotation curves diversity problem, i.e., the scatter in theinner shape of rotation curves of galaxies of similar masses is much broader thanpredicted by CDM simulations.
One commonly proposed solution is to use alternative DM models, e.g., fuzzy
dark matter and self-interacting dark matter.

𝑀∗ ≤ 1010 𝑀⊙

𝑉flat ≠ 0𝑅min < 0.5 kpc• The rotation curves are decomposed into disk, gas, and DM halo using theMarkov Chain Monte Carlo (MCMC) method in Python (the emcee package).
We assumed that the DM halos have spherical symmetry.
• We followed Khelashvili et al. (2023), Ren et al. (2019) and Zentner et al.
(2022) for deriving the velocity profile of the FDM and SIDM halo models.
• The free parameters used for each model are shown in the table below. Weused a constant 𝜎/𝑚𝜒 for the SIDM model (3 cm2 /g).
• For model comparison, we calculated BIC &ModelParameters
ΔBIC value from each fitting.

𝜌𝑠𝑟1+𝑟𝑠

Data points > 524 sample galaxiesBurkert

NFW
𝜌NFW (𝑟) =Q=1

NFW

𝑉200 , 𝐶200 , Υ∗

Burkert

𝑉200 , 𝐶200 , Υ∗

FDM

𝑉200 , 𝑚22 , 𝛼, 𝛿, Υ∗

SIDMΓ0 , 𝜎𝑣0 , Υ∗

𝑉200 : DM velocity at 𝑅200 , 𝐶200 : DM concentration at 𝑅200 , 𝑅200 : radius at which the spherically averaged DM density
reached 200ρcrit , Υ∗ : stellar disk mass-to-light ratio, 𝛼: 𝑟𝑎 /𝑟𝑠𝑜𝑙 , 𝛿: uncertainty (up to 50%) of FDM halo-soliton massscaling relation, Γ0 : SIDM particles scattering rate, 𝜎𝑣0 : 1-D DM velocity dispersion

BIC = −2 ln 𝐿෠ + 𝑘 ln 𝑁
ΔBIC = BICalternative − BICbest
𝐿෠ : maximum likelihood, 𝑘: number of free parameters,
𝑁: number of data points

Rotation Curve Fits
• For many galaxies, the reduced chi-squared (𝜒𝜈2 ) values obtained from therotation curves fitting were not representing the fit qualities well, due to thewide uncertainties on the data points. Therefore, the BIC value was chosen formodel comparison.
• The fits using the FDM halo model are strongly penalized by the BIC, probably
due to the high complexity of the model. In this work, we used 5 freeparameters (𝑘 = 5) for the FDM model, contrast to the three other models with
only 3 free parameters (𝑘 = 3). As from the formula, BIC penalize models with ahigher number of free parameters (𝑘). Using a more simplified model of FDM
might be useful for a “fairer” comparison, if the constraints on the model had
been more stringent.
FDM Particle Mass
• The constraints on FDM particle mass of this work overlap with the results fromKhelashvili et al. (2023) (𝑚𝐹𝐷𝑀 ≈ 10−23 eV). There are some differences on the95% credible intervals for some overlapping galaxies, due to the additional freeparameters used in the latter work (distance and inclination of the galaxies).

Fuzzy Dark Matter (FDM)• Consists of ultralight DM particles (10−24 ≲like particles (ALPs).
𝜆𝑑𝐵 =ℎ𝑚𝐹𝐷𝑀 𝑣

𝑚𝐹𝐷𝑀
≲ 10−19 ), e.g., axion and axioneV

Rotation Curve Fits

• FDM exhibit wave-like behavior on galactic scales,
described by a Schrödinger-Poisson equation. Its deBroglie wavelength is in the order of ~1 kpc.

• On the scale 𝜆 < 𝜆𝑑𝐵 , quantum pressure
provides stability against gravitational
collapse, forming a constant density
soliton core (~0.3 – 1.6 kpc) at the halo
center, which alleviates the core-cuspproblem.
• On the outside, quantum pressure
become less significant, and FDM
shows interference patterns in the form
FDM (DM-only) simulation by Schive et al. (2014)of granules and fringes which shows• FDM halo density profile can bethe DM distribution.
modeled as a soliton core on the0.019 𝑀⨀
inside, transitioning to an NFW halo𝜌𝑠𝑜𝑙,0 =on the outside.
𝑟𝑠𝑜𝑙 4 pc 32𝑚22FDM
𝜌 𝑟 ,𝜌𝐹𝐷𝑀 𝑟 = ቊ 𝑠𝑜𝑙
𝜌𝑁𝐹𝑊 𝑟 ,𝜌𝑠𝑜𝑙 𝑟 =𝑟 ≤ 𝑟𝑎𝑟 ≥ 𝑟𝑎𝜌𝑠𝑜𝑙,01 + 0.091 𝑟

2 8𝑟

𝑠𝑜𝑙

𝑟𝑎 = 𝛼𝑟𝑠𝑜𝑙

𝜒𝜈2 = 1.785

𝜒𝜈2 = 0.046

𝜒𝜈2 = 0.077

FDM Particle Masskpc

𝑚22 =𝜒𝜈2 = 0.099

𝜒𝜈2 = 0.363

𝜒𝜈2 = 0.967

𝜒𝜈2 = 0.242

𝜒𝜈2 = 0.914

𝜌𝑠𝑜𝑙,0 : soliton core density
𝑟𝑠𝑜𝑙 : soliton characteristic radius
𝑟𝑎 :halo transition radius

GalaxySchive et al. (2014)

𝜌 𝑟 ,𝜌SIDM 𝑟 = ቊ iso
𝜌NFW 𝑟 ,𝜌iso 𝑟 = 𝜌0 exp𝑟 ≤ 𝑟1𝑟 ≥ 𝑟1

Φtot 0 − Φtot 𝑟
2𝜎v0

𝜌𝑖𝑠𝑜 (𝑟) is obtained by solving the Poisson’s equation:∇2 Φtot 𝑟 = 4𝜋𝐺 𝜌iso 𝑟 + 𝜌bar 𝑟𝑡age : age of the halo𝑣rel : relative scattering velocity
𝜎v0 : 1D velocity dispersion
Φtot : total potential gravity
𝜌bar : baryon density profile
𝜌0 : isothermal core central density
Ren et al. (2019) & Zentner et al. (2022)

The profile transition occurs at a radius where:

𝜎𝑣rel
Number of≈𝜌iso 𝑟1 𝑡age ≈ 1𝑚𝜒
interactions
For spherical halos:

𝜎𝑣rel = 𝜎4𝜎v0𝜋• Bernal et al. (2018) fitted the rotation curves of 18 LSB galaxies and 6 NGCgalaxies (4 of them are from the SPARC catalog). They also found that theconstraints from each galaxy did not overlap. Based on individual galaxies
analysis, they obtained a constraint in the range:
2.12 × 10−24 < 𝑚𝐹𝐷𝑀 /eV < 2.7 × 10−22
Based on combined analysis (all of the galaxies are fitted simultaneously), theyobtained a best-fit value:𝑚𝐹𝐷𝑀 = 5.54 × 10−24 eV

MAP (Maximum A Posteriori) values and 95% credible intervals of log 𝑚22 from the 24 sample galaxies.

𝜒𝜈2 = 0.98

𝜒𝜈2 = 0.407

𝜒𝜈2 = 0.394

𝜒𝜈2 = 0.475−22𝑚𝐹𝐷𝑀 1𝜎 = 0.322+1.141
eV−0.251 × 10−22 eV
𝑚𝐹𝐷𝑀 2𝜎 = 0.322+6.324−0.307 × 10

• Based on the rotation curves fitting, we obtained a constraint on
the FDM particle mass as above. The 95% credible intervals of themass are in tension for different individual galaxies, which agrees
with the results of a previous work (Khelashvili et al., 2023).

Self-Interacting
Matter(SIDM)
Self-InteractingDark
Dark Matter(SIDM)• SIDM suggests interactions among DM particles with a large scattering crosssection (𝜎/𝑚𝜒 ) due to short-range interactions or weak interactions mediatedby light particles exchange.
• DM self-interactions allow thermalization to occur in the innermost halo region,
leading to the formation of a constant density isothermal core (~0.5 – 1 kpc),
solving the core-cusp problem. The transition radius is defined as radius 𝑟1 at
which the DM particles have only interacted once in the halo’s lifetime.
• Thermalization ties the core sizes and shapes of DM halos to the stars’ spatial
distribution, which explained the rotation curves diversity problem.

MAP (Maximum A Posteriori) values and 95% credible intervals of log 𝑚22 of 42 LSB galaxies from SPARC(Khelashvili et al., 2023)

Conclusions:• For the standard CDM model, the Burkert model fits most galaxies better thanNFW. Based on the BIC value, Burkert is the most supported model (10galaxies), while NFW was only supported by 5 galaxies.
• The FDM model produces the best fits for most of the galaxies. However, it isrejected by all galaxies based on the ΔBIC criterion, due to penalties on the BIC
value because of the high complexity of the model (5 free parameters).
• For most galaxies, the SIDM model produces similar fits to those of the Burkert
model. Based on the BIC value, it received the most support after Burkert (9galaxies). It can explain the rotation curves diversity problem in the samples.
• Based on the fitting results, we obtained a constraint on the FDM particle mass
which overlaps with the results of Bernal et al. (2018) and Khelashvili et al.
(2023). However, it contradicts various astrophysical observations.

𝑚𝐹𝐷𝑀
10−22 eV• The mass of FDM particle is currently have not been well constrained. The
constraints from previous works are in tension with each other and are strongly
inconsistent with many cosmological constraint. Further observations andsimulations have to be done to get a more stringent constraint.

SIDM• Based on the BIC values, 10 galaxies support the Burkert model, 9galaxies support the SIDM model, and 5 galaxies support the NFWmodel.
• Despite the remarkable fits, none of the galaxies in the sample
supports the FDM model.
• Based on the 𝚫BIC values, there are positive evidence and strong
evidence to reject the FDM model from 15 and 9 galaxies,respectively.
• There is only one positive evidence to reject each of the NFW,
Burkert, and SIDM model, and one strong evidence to reject theNFW model.

𝜒𝜈2 = 2.472

𝜒𝜈2 = 1.212

𝜒𝜈2 = 0.222

𝜒𝜈2 = 1.459

The strongest constraint(Hayashi et al., 2021)• Generally, NFW halo (average 𝜒𝜈2 of 2.146) produces total rotation curves model that are toohigh in the inner radii due to the cuspy nature of the halo.
•radii than NFW. However, for some galaxies it fits quite worse at the outer radii.

• FDM halo (average 𝜒𝜈2 of 0.529) produces the best fits for most of the galaxies. Somegalaxies that have a “sharp” turn on their rotation curves can be fitted well using the FDM
halo by the presence of soliton-NFW transition. The soliton’s characteristic radius (𝑟𝑠𝑜𝑙 ) andtransition radius (𝑟𝑎 ) have an average of 1.89 kpc and 3.62 kpc, respectively.
• Visually, SIDM halo (average 𝜒𝜈2 of 1.16) produces similar fits as the Burkert halo in mostcases. For many galaxies, it produces an isothermal core-NFW transition far at the outerradii, even outside of the SPARC’s data points, which agrees with the results of a previouswork (Loizeau et al., 2021). In these cases, it indicates that the observational data can be
sufficiently fitted using a pure isothermal halo model. The transition radius (𝑟1 ) have anaverage of 10.64 kpc.

small-scale problems
solution

Rotation curve fits of 4 of the sample galaxies using NFW, Burkert, FDM, and SIDM halo models.

Burkert halo (average 𝜒𝜈2 of 1.092) produces fits that matches the data better in the innerorder of ~10−22 eV:constraints (inclusionrange) from this work
(1𝜎 and 2𝜎), Bernal et al.
Colored region: exclusion range
Point & error bar: inclusion range(2018), and Khelashvili
et al. (2023)

FDM particle mass constraints from various astrophysical observations (Ferreira,2021), this work, Khelashvili et al. (2023), and Bernal et al. (2018).

• The constraint set from this work is strongly inconsistent withvarious astrophysical observations. It is also inconsistent with the−19 eV), which is obtained
strongest constraint to date (1.1+8.3−0.7 × 10
from dynamical analysis of an ultrafaint dwarf, Segue I (Hayashi etal., 2021).

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