Testing bosonic dark matter through white dwarf mass measurements

This paper proposes that discrepancies between electromagnetic and gravitational mass measurements of white dwarfs can be explained by the presence of a gravitationally coupled, electromagnetically invisible bosonic scalar field, which constitutes 5–15% of the stellar mass and allows for new constraints on ultralight dark matter particles.

The Gist

The Invisible Weight: How White Dwarfs Might Be Hiding Dark Matter

Imagine you are trying to weigh a heavy suitcase. You have two different ways to do it:

1. The "Look and Guess" Method (Electromagnetic): You look at the suitcase, measure its size, check the material it's made of, and use a formula to guess how heavy it should be based on how it looks.
2. The "Pull" Method (Gravitational Redshift): You measure how much the suitcase's gravity bends the light coming from a lamp behind it. This tells you exactly how much mass is actually pulling on that light, regardless of what the suitcase looks like.

Usually, these two methods agree. But for some White Dwarfs (the dense, dead cores of stars like our Sun), they don't. The "Pull" method says the star is 5% to 15% heavier than the "Look and Guess" method predicts. It's as if the suitcase is pulling harder on the light than its visible material should allow.

This paper proposes a solution: The suitcase has a hidden, invisible weight inside it.

The Invisible Guest: Bosonic Dark Matter

The authors suggest that these White Dwarfs aren't just made of normal "dead star" material (fermions). Instead, they might be hosting a secret guest: a cloud of bosonic dark matter (specifically, a scalar field).

Think of the White Dwarf as a solid chocolate cake.

• The Cake: This is the normal star material. It glows, it has a surface, and we can measure its size and temperature.
• The Hidden Syrup: This is the dark matter. It doesn't glow, it doesn't reflect light, and it's invisible to our telescopes. However, it has mass.

When we use the "Look and Guess" method, we only see the chocolate cake. We calculate the weight based on the cake's size and density. But when we use the "Pull" method (gravity), we feel the weight of both the cake and the hidden syrup. The extra weight comes from the syrup, which explains why the star seems heavier than it looks.

Building the Star: A Cosmic Hybrid

The researchers didn't just guess; they built a mathematical model of these "hybrid stars." They created a simulation where a normal White Dwarf is mixed with a cloud of these invisible particles.

They found that if the dark matter particles have a specific, very tiny mass (around $10^{-11}$ electron-volts), they can form a stable cloud that fits perfectly inside or around the star.

• The "Compact Core" Scenario: In some models, the dark matter forms a tiny, dense ball right in the center of the star, like a heavy marble inside a hollow ball.
• The "Diffuse Cloud" Scenario: In other models, the dark matter spreads out, filling the whole star like a mist, making the whole object slightly heavier without changing its shape much.

The Results: A Perfect Match

The team tested their models against real data from famous White Dwarfs, including Sirius B (the brightest star in our night sky).

• The Problem: For Sirius B, the "Pull" method said it was about 1.12 times the mass of our Sun, but the "Look" method said it was only 1.02. That 10% difference was a mystery.
• The Solution: When the researchers added their "invisible syrup" (dark matter) to their model, the math worked out perfectly. The model showed that if about 9% to 10% of the star's total mass was this invisible dark matter, the "Pull" and "Look" measurements would finally agree.

They found that a specific type of dark matter particle (with a mass of roughly $10^{-10}$ eV) was the most likely candidate to explain the data. Their statistical analysis showed that it is 50 times more likely that these stars contain this hidden dark matter than that the measurements are just wrong or that the stars are pure normal matter.

Why This Matters (According to the Paper)

The paper argues that this isn't just a math trick. It offers a physical explanation for why some stars seem to have a "mass bias."

• The "Capture" Problem: Usually, stars can't just "catch" enough dark matter from space to become this heavy. The authors suggest these stars must have formed with this dark matter already inside them, or the dark matter collapsed into the star before the star finished forming.
• The "Silent" Nature: Because this dark matter is "bosonic" and doesn't interact with light, it doesn't change the star's color or temperature. It only adds weight. This is why we didn't notice it before—we were only looking at the "chocolate cake" and ignoring the "syrup."

The Bottom Line

This paper suggests that some White Dwarfs are actually composite objects: a normal star wrapped in or filled with a cloud of invisible, heavy particles. This hidden mass explains why gravity measurements say the stars are heavier than their appearance suggests. It's a way of "weighing" the invisible universe by looking at the stars that hold it.

Technical Summary

Technical Summary: Testing Bosonic Dark Matter through White Dwarf Mass Measurements

Problem Statement
White dwarf (WD) mass estimates derived from electromagnetic (EM) methods (spectroscopy and photometry) frequently differ from those obtained via gravitational redshift (GRS) measurements. While large-scale surveys generally validate the canonical mass–radius relation, significant discrepancies persist in a non-negligible subset of objects, with offsets ranging from a few percent to approximately 15%. Standard explanations involving thermal effects, atmospheric modeling uncertainties, or calibration errors fail to fully account for the magnitude of these discrepancies in extreme cases. This paper investigates an alternative hypothesis: that WDs may contain a gravitationally coupled, electromagnetically invisible bosonic scalar field component (dark matter) that contributes to the total gravitational mass without altering the spectroscopic atmospheric profile.

Methodology
The authors construct stationary, static mixed configurations consisting of a fermionic WD coupled to a complex scalar field, forming a composite WD–boson star system.

• Theoretical Framework: The system is modeled using the coupled Einstein–Klein–Gordon equations and general relativistic hydrodynamics. The fermionic component is treated as a perfect fluid obeying a polytropic equation of state ($p = K\rho^\Gamma$) with an index $n=1.5$ (adjusted to $n=1.48$ for specific high-mass cases). The scalar field is described by a massive complex field with a quadratic potential $V(|\Phi|^2) = \mu^2|\Phi|^2$, minimally coupled to gravity.
• Numerical Approach: The authors solve the modified Tolman-Oppenheimer-Volkoff (TOV) equations using a shooting method to find equilibrium configurations. They employ a fourth-order Runge-Kutta (RK4) integrator to determine the eigenvalue frequency of the scalar field that ensures regularity and decay at large radii.
• Mass Decomposition: To quantify the dark matter fraction, the authors utilize the Komar mass integral to separate the total gravitational mass ($M_{tot}$) into the fermionic component ($M_{WD}$) and the bosonic scalar field component ($M_{BS}$).
• Data Analysis: The study compares these theoretical models against a sample of WDs with independent EM and GRS mass measurements (including Sirius B, DB, DBA, and Chandra objects). A Bayesian model selection framework is employed to evaluate the statistical preference for the mixed WD model over a standard pure WD model ($M_{GRS} = M_{EM}$) and to infer constraints on the boson mass $\mu$.

Key Contributions and Results

1. Reproduction of Observational Discrepancies: The study demonstrates that composite WD models can reproduce the observed mass differences ($\Delta M = M_{GRS} - M_{EM}$) by attributing the excess gravitational mass to a scalar field component. The models successfully account for discrepancies in the range of $f_{DM} \sim 5\%–15\%$ of the total mass.
2. Boson Mass Constraints: The authors explore three specific boson mass scales: $\mu = 10^{-10}$ eV, $5.04 \times 10^{-11}$ eV, and $1.68 \times 10^{-11}$ eV.
   • $\mu = 10^{-10}$ eV: Results in a compact scalar field core (radius $\sim$ hundreds of km) that significantly distorts the internal pressure profile of the fermionic component.
   • $\mu = 1.68 \times 10^{-11}$ eY: Produces a more diffuse scalar field distribution comparable to the size of the WD, resulting in pressure profiles that closely resemble pure WDs with minimal structural disruption.
3. Bayesian Model Selection: Using a dataset of WDs with available mass and radius measurements, the authors compute the likelihood ratio between the mixed WD model and the standard pure WD model. The mixed model is found to be approximately 50 times more probable than the pure WD model ($p_\mu(d)/p_0 \approx 56$). The analysis suggests that the hypothesis $M_{GRS} = M_{EM}$ is rejected with a probability $p \geq 0.99$ for boson masses in the range $[10^{-11}, 10^{-9}]$ eV.
4. Handling Sirius B: The most massive object in the sample, Sirius B, initially presented a challenge where standard polytropic models predicted a radius larger than observed. By slightly adjusting the polytropic index to $n=1.48$, the authors achieved a fit consistent with both the observed mass and radius, yielding a scalar field contribution of $\sim 9\%$.

Significance and Claims
The paper claims to provide a physically motivated explanation for the persistent mass bias between EM and GRS measurements in white dwarfs. By introducing a gravitationally coupled scalar field, the models offer a mechanism where the total gravitational mass exceeds the luminous (fermionic) mass, mimicking a redshift excess without modifying the atmospheric parameters used in EM mass estimates.

The authors emphasize that their results place preliminary constraints on the mass and compactness of the scalar field configuration, specifically favoring ultralight bosons in the range of $10^{-11}$ to $10^{-10}$ eV. They note that while the polytropic approximation is a simplification, the models remain consistent with established mass–radius relations and luminosity evolution. The study suggests that future high-precision surveys, asteroseismology, and searches for environmental correlations (e.g., in globular clusters) could further test this proposal. The authors conclude that while the scenario requires specific formation conditions (primordial or co-evolutionary presence of the scalar field rather than standard capture), it offers a viable alternative to standard physics for explaining the observed mass discrepancies.