Dark ages bounds on non-accreting massive compact halo objects

This paper establishes new, tighter cosmological upper bounds on the fraction of dark matter composed of massive compact halo objects (MACHOs) with masses between $10^3$ and $10^7$ solar masses by analyzing distortions in the global 21-cm signal during the cosmic dark ages, a method that avoids astrophysical uncertainties associated with star formation.

The Gist

The Big Picture: Hunting for the "Invisible Ghosts" of the Universe

Imagine the universe is a giant, dark ocean. We know there is a lot of "stuff" in this ocean that we can't see, called Dark Matter. It makes up about 85% of all matter, but we've never actually seen a single piece of it.

Scientists have two main theories about what this invisible stuff is:

1. The "Ghost" Theory: It's made of tiny, invisible particles (like a fog).
2. The "Rock" Theory: It's made of massive, compact objects (like invisible boulders or black holes) floating around. These are called MACHOs (Massive Compact Halo Objects).

This paper is about testing the "Rock" theory. The authors ask: Could the dark matter be made of giant, invisible rocks?

The Detective Tool: The "Cosmic Radio" Signal

To find these invisible rocks, the authors don't use telescopes that look at light (because the rocks are invisible). Instead, they listen to the 21-cm signal.

Think of the early universe as a giant, cold room filled with hydrogen gas. This gas has a specific "hum" or radio frequency (21 cm) that it emits.

• The Standard Story: In the normal universe, this gas cools down very predictably as the universe expands. It creates a specific "hum" pattern that scientists can calculate.
• The Twist: If giant invisible rocks (MACHOs) are swimming through this gas, they act like a speedboat cutting through water.

The Mechanism: The "Speedboat" Effect

Here is the core physics, explained simply:

1. The Speedboat: Imagine a massive MACHO (a giant invisible rock) moving through the gas of the early universe.
2. The Wake: Just as a speedboat creates a wake (a trail of disturbed water) behind it, the MACHO creates a gravitational wake in the gas.
3. The Friction: As the MACHO pulls the gas along, there is a tug-of-war. The gas drags back on the MACHO, slowing it down. This is called dynamical friction.
4. The Heat: When you rub your hands together, they get hot. Similarly, as the MACHO slows down, the energy it loses doesn't disappear; it turns into heat and warms up the surrounding gas.

Why does this matter?
If the gas gets warmer than it should be, the "hum" (the 21-cm signal) changes. It might turn from a "whisper" (absorption) into a "shout" (emission), or the whisper might get quieter.

The Two Time Periods: "The Dark Ages" vs. "The Cosmic Dawn"

The authors looked at two different eras in the universe's history:

1. The Cosmic Dawn (The Messy Era): This is when the first stars were turning on. It's like a construction site. There are stars, explosions, and X-rays heating things up. It's very hard to tell if the gas is warm because of a giant invisible rock or because of a nearby star. It's too noisy to be a good detective tool.
2. The Dark Ages (The Quiet Era): This happened before the first stars existed. The universe was a quiet, empty room with just gas and the cosmic background radiation. There were no stars to mess up the signal.
   • The Analogy: If you are trying to hear a pin drop, you do it in a silent library, not at a rock concert. The "Dark Ages" are the silent library.

The Findings: What the Authors Discovered

The authors ran computer simulations to see what would happen if the universe was filled with different amounts of these giant rocks.

• The Test: They checked if the "heat" from the rocks would distort the 21-cm signal too much.
• The Rules: They set strict limits. If the signal changed by more than a tiny bit (50 millikelvin at one time, 15 millikelvin at another), it would mean the rocks are too heavy or too numerous.
• The Result:
  • They found that if the dark matter were made of rocks with masses between 1,000 and 10 million times the mass of our Sun, they would have heated the gas too much.
  • Therefore, dark matter cannot be made mostly of these specific sizes of rocks.
  • The "Dark Ages" signal gave them much stricter limits than looking at the "Cosmic Dawn" because the Dark Ages were free from the "noise" of star formation.

The Different "Rock" Shapes

The authors didn't just assume all rocks were the same size. They tested three scenarios:

1. Monochromatic: All rocks are exactly the same size (like a bag of identical marbles).
2. Log-Normal: Most rocks are a certain size, but there are some smaller and some larger (like a bag of mixed nuts).
3. Critical Collapse: A specific distribution where the sizes follow a steep curve (like a pyramid of rocks).

The Surprise: The "mixed nut" scenarios (extended distributions) actually gave tighter limits. Even if most rocks were small, the few giant ones in the mix would still heat up the gas enough to be noticed. This means the universe is even more restrictive on these invisible rocks than we thought.

The Conclusion: A Clearer Window

The paper concludes that the "Dark Ages" of the universe provide a cleaner, more powerful window to hunt for dark matter than the "Cosmic Dawn."

• Why? Because in the Dark Ages, there were no stars to confuse the signal.
• What does it mean? We can now say with high confidence that dark matter is not made of massive compact objects (like giant black holes) in the mass range of 1,000 to 10 million suns.

In a nutshell: The universe is like a quiet room. If giant invisible rocks were swimming through it, they would have made a "splash" (heat) that would have changed the room's temperature. Since the room is still at the expected temperature, those giant rocks probably aren't there. The dark matter must be something else (likely the tiny "ghost" particles).

Technical Summary

1. Problem Statement

The nature of Dark Matter (DM) remains a fundamental open question in cosmology. While the standard $\Lambda$CDM model assumes collisionless cold DM, alternative scenarios suggest DM could consist of Massive Compact Halo Objects (MACHOs), such as primordial black holes (PBHs), axion stars, or quark nuggets.

Existing constraints on MACHOs come from:

• Gravitational Microlensing: Effective for low masses ($M \lesssim 10 M_\odot$) but limited by survey duration and coverage.
• Dynamical Heating: Constraints from globular cluster disruption and stellar heating in dwarf galaxies.
• Accretion: Constraints based on gas accretion onto MACHOs, which affects the Cosmic Microwave Background (CMB) and 21-cm signals.

The Gap: Most previous 21-cm studies focused on the Cosmic Dawn ($z \sim 17$), where the signal is heavily contaminated by uncertain astrophysical processes (star formation efficiency, X-ray heating, Lyman-$\alpha$ coupling). There is a need for a complementary probe that is free from these astrophysical uncertainties. The Dark Ages ($z \gtrsim 30$, specifically $z \sim 89$) offer a pristine window where the universe is homogeneous, and the 21-cm signal is determined solely by standard physics unless perturbed by new physics.

This paper investigates the constraints on non-accreting MACHOs ($M \gtrsim 10^3 M_\odot$) using the global 21-cm signal during the Dark Ages, where dynamical friction heats the intergalactic medium (IGM).

2. Methodology

A. Physical Mechanism: Dynamical Heating

The authors model MACHOs moving through the post-recombination baryonic fluid.

• Dynamical Friction: As MACHOs move with a bulk relative velocity ($v_{bc} \sim 30$ km/s) relative to the gas, they create gravitational wakes. The interaction between the MACHO and its wake creates a drag force ($F_{DF}$), dissipating kinetic energy into the IGM.
• Energy Dissipation: This heating raises the gas temperature ($T_{gas}$), potentially distorting the 21-cm differential brightness temperature ($T_{21}$).
• Non-Accreting Assumption: The study explicitly considers objects that are too diffuse to accrete matter significantly (e.g., axion stars with radii $> 3 \times$ Schwarzschild radius). Thus, heating is driven solely by dynamical friction, not accretion.

B. Mass Distributions

The authors analyze three types of mass distributions for MACHOs:

1. Monochromatic: All MACHOs have a single mass $M_c$.
2. Log-Normal: A distribution centered at $M_c$ with width $\sigma$.
3. Critical Collapse: A power-law distribution with an exponential cutoff, typical of PBH formation scenarios.

C. Simulation Framework

The authors solve a coupled system of differential equations:

1. Velocity Evolution: The deceleration of MACHOs due to drag ($dv_{bc}/dz$).
2. Thermal Evolution: The evolution of the IGM gas temperature ($T_{gas}$), incorporating:
   • Adiabatic cooling.
   • Compton scattering with CMB photons.
   • Dynamical heating (the new term derived from the drag force).
   • X-ray heating (included for the Cosmic Dawn comparison).
3. Ionization Evolution: Tracking the free electron fraction ($x_e$).
4. 21-cm Signal Calculation: Computing the global brightness temperature $T_{21}$ using the spin temperature ($T_s$), which depends on $T_{gas}$, CMB temperature ($T_\gamma$), and Lyman-$\alpha$ coupling (for Cosmic Dawn).

D. Constraints and Criteria

The authors derive upper bounds on the MACHO fraction of dark matter ($f_M$) by imposing three exclusion criteria:

1. Cosmic Dawn ($z \sim 17$): The deviation in the absorption signal $\Delta T_{21}$ must not exceed 50 mK (consistent with EDGES/SARAS tensions).
2. Dark Ages ($z \sim 89$): The deviation $\Delta T_{21}$ must not exceed 15 mK (based on projected sensitivity of future lunar experiments like LuSee-Night).
3. High Redshift ($z \gtrsim 300$): No emission signal should appear (as the standard model predicts $T_{21} \approx 0$ in this regime).

3. Key Contributions

• Novel Probe: This is the first study to systematically derive constraints on non-accreting MACHOs specifically using the Dark Ages global 21-cm signal, providing a cleaner probe than Cosmic Dawn.
• Extended Mass Spectra: Unlike many previous works that assume a monochromatic mass distribution, this paper rigorously incorporates Log-Normal and Critical Collapse distributions, showing how the integrated effect of a mass spectrum alters constraints.
• Astrophysical Independence: By focusing on $z \sim 89$, the constraints are largely independent of uncertain star formation parameters (efficiency, X-ray emissivity) that plague Cosmic Dawn analyses.
• Refined Velocity Modeling: The authors account for the transition of MACHOs from supersonic to subsonic regimes, which significantly affects the drag force and heating efficiency.

4. Key Results

A. Heating Efficiency and Mass Dependence

• Monochromatic Case:
  • Low Mass ($M < 10^3 M_\odot$): Heating occurs at late redshifts, mildly reducing the Cosmic Dawn absorption amplitude.
  • Intermediate Mass ($10^4 \lesssim M \lesssim 10^6 M_\odot$): Heating is most efficient during the Dark Ages ($30 \lesssim z \lesssim 600$), potentially turning the absorption trough into an emission signal.
  • High Mass ($M > 10^7 M_\odot$): These objects decelerate rapidly into the subsonic regime before recombination, suppressing drag and leaving negligible imprint at $z < 1100$.
• Extended Distributions:
  • Log-Normal and Critical Collapse distributions produce stronger heating than monochromatic distributions for a fixed characteristic mass $M_c$.
  • This is because extended distributions include significant populations of masses both above and below $M_c$ that contribute to the integrated energy dissipation.
  • Critical Collapse models yield the tightest constraints at intermediate masses ($M \sim 10^5 M_\odot$).

B. Derived Upper Bounds on $f_M$

The study establishes new upper limits on the fraction of dark matter residing in MACHOs ($f_M$):

• Tightest Constraint: For a monochromatic distribution, the strongest bound is $f_M \lesssim 0.075$ at $M_c \sim 2 \times 10^5 M_\odot$, driven by the Dark Ages criterion ($z \sim 89$).
• Comparison with Previous Work:
  • The Dark Ages constraints are tighter (by a factor of $\sim 0.2$ in $f_M$) than previous bounds derived solely from Cosmic Dawn signals (e.g., Ref [68]) for masses $M_c \lesssim 4 \times 10^4 M_\odot$.
  • For $M_c \gtrsim 10^5 M_\odot$, the Dark Ages signal provides a qualitatively different and stronger probe because it is insensitive to star formation history.
• Extended vs. Monochromatic: Extended distributions generally yield tighter bounds than monochromatic ones. For example, a Log-Normal distribution with $\sigma=2$ produces the strongest overall constraints due to its extended low-mass tail.

C. Comparison with Other Probes

The derived bounds are:

• Stronger than microlensing constraints (ICARUS), globular cluster disruption limits, and milli-lensing of radio sources for masses $\lesssim 10^5 M_\odot$.
• Complementary to constraints derived from the half-light radii of ultrafaint dwarf galaxies.

5. Significance and Future Outlook

• Clean Cosmological Window: The paper demonstrates that the Dark Ages 21-cm signal is a superior probe for massive compact objects because it bypasses the "astrophysical fog" of the Cosmic Dawn.
• Future Experiments: The constraints are projected for future lunar-based experiments (e.g., LuSee-Night, REACH, FARSIDE) which aim to measure the global 21-cm signal with uncertainties as low as 5–15 mK.
• Implications: If future observations detect a deviation from the standard Dark Ages signal, it could provide direct evidence for the existence of non-accreting MACHOs or rule out specific mass ranges for primordial black holes.
• Limitations: The study assumes a uniform bulk velocity and neglects accretion effects (which may be relevant for the most massive objects). Future work will aim to include spatial variance in heating and accretion physics.

In conclusion, this paper establishes that the Dark Ages global 21-cm signal provides the most stringent and astrophysically robust constraints to date on non-accreting MACHOs in the mass range $10^3 - 10^7 M_\odot$, significantly narrowing the allowed parameter space for these dark matter candidates.