Heavy dark matter in rapidly evolving massive stars

This study demonstrates that heavy dark matter captured by rapidly evolving massive stars, particularly the first stars, can significantly alter their lifecycles through thermalization, annihilation equilibrium, or self-gravitating collapse into black holes, with capture rates critically dependent on stellar composition, evolutionary stage, and the local dark matter velocity distribution.

The Gist

Imagine the universe as a giant, bustling city. In this city, there are invisible ghosts called Dark Matter. We can't see them, but we know they are everywhere, drifting through space like a thick fog. Usually, these ghosts just pass right through solid objects without a scratch.

But what happens if a ghost walks into a very dense, very hot building? That's the question this paper asks, but instead of a building, they are looking at massive stars—specifically the very first stars that ever formed in the universe (called "Population III" stars).

Here is the story of their discovery, broken down into simple parts:

1. The Star as a "Cosmic Trap"

Think of a massive star as a giant, glowing net made of hydrogen and helium gas. As the invisible dark matter ghosts drift through this net, they occasionally bump into the gas atoms.

• The Old Idea: Scientists used to think of the star as a static, uniform cloud. They assumed the ghosts would bounce off a few times and get stuck.
• The New Idea: The authors realized the star is actually a dynamic, evolving city. It starts as a simple gas cloud, but as it ages, it cooks up heavier elements (like carbon and oxygen) in its core, creating a dense, heavy "city center" surrounded by a lighter "suburb."

2. The "Three-Target" Problem

Imagine you are trying to stop a speeding bullet.

• Early Stage: The star is young. It's mostly made of Hydrogen and Helium. It's like trying to stop the bullet with a net made of two types of string. This is easy to calculate.
• Late Stage: As the star ages, it builds a super-dense core of heavy metals (like Oxygen and Neon). Now, the bullet has to pass through three different types of obstacles: the outer gas, the heavy core, and the transition zone.
• The Discovery: The authors realized that if you ignore the heavy core and only look at the outer gas, you miss a huge amount of the action. For very heavy dark matter, the star's heavy core acts like a magnet, sucking in the dark matter much more efficiently than previously thought. They had to invent a new math formula to account for these "three-target" collisions.

3. The Speed Trap (The "Slow Ghost" Problem)

To get caught in the star, a dark matter ghost needs to be moving slowly enough to get stuck.

• The Old Assumption: Scientists used to guess the speed of these ghosts using a standard "bell curve" (Maxwell-Boltzmann distribution), assuming ghosts move at all speeds equally.
• The New Reality: The authors used a more sophisticated method (Eddington inversion) to look at the specific neighborhood where these stars live (right in the center of a dark matter halo). They found that there are actually fewer slow ghosts than the old models predicted.
• The Result: Because there are fewer slow ghosts, the star catches fewer of them than we thought. This corrected a previous mistake where scientists were overestimating how many ghosts get trapped.

4. The Two Possible Endings

Once the dark matter gets trapped inside the star, two things can happen, depending on the "personality" of the dark matter:

Scenario A: The "Self-Destructing" Ghost (Annihilating DM)
If the dark matter ghosts hate each other, they will bump into one another and vanish in a flash of energy (annihilation).

• The Outcome: They reach a balance. As fast as the star catches new ghosts, the old ones destroy each other. The star stays safe, and the dark matter doesn't build up enough to cause trouble.

Scenario B: The "Heavy Sleeper" (Non-Annihilating DM)
If the ghosts are shy and don't destroy each other, they just pile up in the center of the star.

• The Outcome: They get so heavy and crowded that they start pulling on each other with their own gravity. Eventually, they collapse into a tiny, super-dense Black Hole.
• The Disaster: This baby black hole starts eating the star from the inside out, like a termite eating a house from the foundation. It can swallow the entire star before the star is even ready to die naturally.

5. Why This Matters

This paper is a wake-up call for astronomers.

• It changes the rules: We can't just use simple, static models to study dark matter anymore. We have to understand how stars grow, change their composition, and evolve over time.
• It opens new doors: Even though the dark matter density in these early stars is relatively low, the new models show that heavy dark matter could still destroy these stars by turning them into black holes.
• The Big Picture: By watching how the first stars lived and died, we might finally catch a glimpse of the invisible dark matter that makes up most of the universe.

In a nutshell: The authors built a more realistic "simulation" of the first stars. They found that these stars are much better at catching heavy dark matter than we thought (because of their heavy cores), but also that the dark matter is moving slightly faster than we thought (catching fewer of them). The result? If the dark matter doesn't destroy itself, it could collapse the star from the inside, turning a giant sun into a black hole.

Technical Summary

1. Problem Statement

The paper investigates the capture, thermalization, and potential collapse of heavy Dark Matter (DM) within massive Population III (Pop. III) stars. Previous studies on DM capture in celestial bodies often relied on simplifying assumptions that limit their accuracy for massive, rapidly evolving stars:

• Constant Density/Velocity: Assuming constant stellar density and escape velocity throughout the star, rather than accounting for radial profiles.
• Single/Multi-Target Limitations: Treating capture primarily as scattering off Hydrogen and Helium, or using simplified multi-scattering formalisms that do not account for the evolving chemical composition of the star.
• Velocity Distribution: Using a Maxwell-Boltzmann (MB) distribution for DM velocities, which is known to overestimate the low-velocity tail in the deep potential wells of halos (near the center).
• Neglect of Late-Stage Evolution: Focusing only on the Zero-Age Main Sequence (ZAMS) and ignoring the formation of metal-rich cores in later evolutionary stages.

The authors aim to provide a more realistic calculation of DM capture rates for heavy DM ($10^3$ GeV to $10^{15}$ GeV) by incorporating stellar evolution simulations, realistic velocity distributions, and extended multi-scattering formalisms.

2. Methodology

A. Stellar Evolution Modeling

• Code: The authors used MESA (Modules for Experiments in Stellar Astrophysics) to simulate non-rotating, single Pop. III stars with masses of $20, 100,$ and $1000 \, M_\odot$.
• Parameters: Zero metallicity ($Z=0$), zero mass loss, evolving from ZAMS ($z \sim 10$) until core helium depletion.
• Output: The simulations provided time-dependent radial profiles for:
  • Number density of constituents ($n_i$).
  • Escape velocity ($v_{esc}$).
  • Temperature ($T$).
• Key Evolutionary Feature: As the stars evolve, they develop a dense, metal-rich core (composed of $^{12}\text{C}, ^{16}\text{O}, ^{20}\text{Ne}$) surrounded by a lighter envelope of H and He. This structural change significantly alters the capture dynamics.

B. Dark Matter Velocity Distribution

• Location: The study focuses on stars located $\sim 5$ pc from the center of a $10^5 \, M_\odot$ halo (NFW profile), where the local DM density is $\rho_\chi \approx 5.3 \, \text{GeV/cm}^3$.
• Distribution Function: Instead of the standard Maxwell-Boltzmann distribution, the authors employed the Eddington inversion method. This method derives the phase-space distribution function directly from the halo's density profile and gravitational potential.
• Result: The Eddington distribution shows a suppressed low-velocity tail compared to the MB distribution, leading to lower capture rates in the relevant regime.

C. Capture Formalism

• Interaction: Assumed scalar effective operator interaction ($\mathcal{L}_{eff} \supset \frac{c_N}{\Lambda^2} \bar{\chi}\chi \bar{N}N$).
• Response Functions:
  • Two-Target (ZAMS): Extended the formalism from Ref. [9] to handle multiple scatterings with two species (H and He).
  • Three-Target (Red Giant): Developed a new response function for three distinct nuclear species ($i, j, k$). This is crucial for the late red giant phase where the DM particle traverses the atmosphere (H/He) and the metal-rich core (C/O/Ne).
  • Hydrogen Specifics: Derived a specific response function for Hydrogen (Appendix A.2), as its trivial nuclear form factor requires a different mathematical treatment than heavier nuclei.
• Capture Rate Calculation: The total capture rate $C$ is calculated by integrating over the stellar volume, accounting for the radial dependence of $v_{esc}$, $n_i$, and the velocity distribution $f_{\chi\star}$.

D. Post-Capture Dynamics

• Thermalization: Solved the differential equation for the DM kinetic energy ratio $x = E_{kin}/T_c$ to determine thermalization time ($t_{therm}$).
• Annihilation: Modeled the competition between capture and annihilation ($\Gamma_{ann} \propto N_\chi^2$).
• Self-Gravitation & Collapse: Analyzed the condition where the DM population becomes self-gravitating ($N_\chi > N_{self}$) and subsequently collapses into a black hole (BH) if the Chandrasekhar-like limit is exceeded.
• BH Accretion: Calculated the BH growth rate via spherical accretion (Bondi accretion) to determine if the BH can consume the star before its natural end-of-life.

3. Key Contributions

1. Realistic Stellar Structure: First application of time-dependent, radially resolved stellar profiles (from MESA) to DM capture, moving beyond constant-density approximations.
2. Three-Target Formalism: Extension of the multi-scattering response function to include three target species. This is critical for accurately modeling capture in the late stages of Pop. III stars where a metal core exists.
3. Eddington Inversion: Implementation of the Eddington inversion method for DM velocity distributions in the deep potential of a halo center, correcting the overestimation of low-velocity DM inherent in MB assumptions.
4. Hydrogen Response Function: Derivation of an exact and asymptotic response function for Hydrogen scattering, filling a gap in previous literature.
5. Parameter Space Mapping: Comprehensive mapping of the $m_\chi - \sigma_{\chi p}$ plane for different stellar masses, identifying regions where DM thermalizes, self-gravitates, or destroys the star.

4. Key Results

A. Capture Rates and Evolution

• ZAMS Phase: Capture is dominated by Helium due to coherent scattering ($\sigma \propto A^4$), despite its lower abundance compared to Hydrogen.
• Red Giant Phase: The formation of a metal-rich core significantly enhances the capture rate for heavy DM ($m_\chi \gtrsim 10^9$ GeV).
  • Using a three-target model is essential. Limiting the calculation to two targets (ignoring the core's specific metal composition or treating it as a single block) underestimates the capture rate by orders of magnitude for small cross-sections.
  • The capture rate saturates first at the "core's geometric limit" (all DM passing through the core is captured) and later at the star's total geometric limit.
• Velocity Distribution Impact: The Eddington distribution yields capture rates lower than those calculated with the Maxwell-Boltzmann distribution (by up to an order of magnitude in some regimes) due to the suppression of the low-velocity tail.
• Previous Overestimation: The combination of constant-density assumptions and MB distributions in prior work leads to an overestimation of capture rates by more than an order of magnitude.

B. Thermalization and Annihilation

• Thermalization: Heavy DM ($m_\chi > 1$ TeV) generally thermalizes within the stellar core during the star's lifetime for a wide range of cross-sections.
• Annihilating DM: Rapidly reaches capture-annihilation equilibrium ($\Gamma_{ann} \approx C/2$). The accumulated DM population remains dynamically irrelevant and does not significantly alter stellar evolution.

C. Self-Gravitation and Black Hole Formation (Non-Annihilating DM)

• Self-Gravitation: For non-annihilating DM, the accumulated population can become self-gravitating ($N_\chi > N_{self}$).
• Collapse: In specific regions of the parameter space (specifically for $m_\chi \gtrsim 10^9$ GeV and specific cross-sections), the DM core collapses into a black hole.
• Stellar Destruction:
  • For $20 \, M_\odot$ and $100 \, M_\odot$ stars, the formed BH can accrete the entire star before the end of the red giant phase.
  • This destruction mechanism is possible in parameter regions not yet excluded by direct detection experiments (LZ, etc.).
  • For the $1000 \, M_\odot$ star, the BH accretion is slower relative to the star's lifetime, often failing to destroy the star before helium depletion.

5. Significance

• New Constraints: The study demonstrates that Pop. III stars act as sensitive probes for heavy DM. The potential for DM-induced black hole formation and premature stellar death provides a novel astrophysical constraint on heavy DM models, particularly for non-annihilating scenarios.
• Astrophysical Impact: It highlights that the internal composition and evolutionary stage of a star are critical factors in DM capture calculations. Ignoring metal cores or using simplified velocity distributions leads to incorrect bounds on DM properties.
• JWST Relevance: As the James Webb Space Telescope (JWST) begins observing high-redshift galaxies, understanding the potential for DM to alter the lifecycles of the first stars is crucial for interpreting observations of early stellar populations.
• Theoretical Advancement: The development of the three-target response function and the application of Eddington inversion to stellar DM capture sets a new standard for precision in this field, applicable to other compact objects and evolved stars.

In conclusion, the paper establishes that heavy dark matter can significantly impact the evolution and ultimate fate of massive primordial stars, potentially destroying them via black hole formation in parameter spaces currently unconstrained by terrestrial experiments. Accurate modeling of stellar evolution and halo dynamics is essential to derive robust limits on heavy dark matter.