Two-body relaxation in the EMRI-TDE disk model for Quasi Periodic Eruptions

This study presents the first end-to-end quantitative calculation of the expected Quasi Periodic Eruption (QPE) rate within the impact model, demonstrating that the predicted number density of QPEs—ranging from $10^{-12}$to$10^{-6} \rm Mpc^{-3}$ depending on orbital parameters—depends critically on the interplay between tidal disruption events and extreme mass-ratio inspirals interacting with accretion disks.

The Gist

The Big Picture: Cosmic "Popcorn" in Black Hole Kitchens

Imagine the center of a galaxy as a giant, busy kitchen. In the middle sits a Supermassive Black Hole (MBH), which is like a massive, hungry chef. Around this chef, there is a swirling soup of stars and smaller black holes.

Recently, astronomers have noticed something strange happening in these galactic kitchens: Quasi-Periodic Eruptions (QPEs). These are bright flashes of X-ray light that happen like clockwork—every few hours or days. They are like popcorn kernels popping in a pan, but instead of heat, the "pop" is caused by something crashing into a disk of gas.

This paper asks a simple but difficult question: How common are these "pops" in the universe, and what is actually causing them?

The Theory: The "Disk Impact" Model

The authors focus on a specific theory called the "Disk Impact Model." Here is how it works:

1. The Accretion Disk (The Soup): When a star gets too close to the central black hole, the black hole's gravity rips the star apart. This debris forms a flat, spinning disk of gas (like a pizza dough) around the black hole. This is called a Tidal Disruption Event (TDE).
2. The Orbiter (The Diver): Somewhere else in the galaxy, a smaller object (either a Stellar Black Hole or a Star) is orbiting the central black hole. Over millions of years, gravitational interactions with other stars nudge this object into a very tight, elliptical orbit. This is called an EMRI (Extreme Mass-Ratio Inspiral).
3. The Crash (The Pop): As this orbiting object swings around, it periodically dives through the gas disk.
   • If it's a Star: It punches a hole in the gas, creating a bubble that expands and flashes brightly.
   • If it's a Black Hole: It acts like a vacuum cleaner, sucking up gas as it passes through, which also creates a flash.

Every time the object completes its orbit and dives through the disk again, we get another "pop" (eruption).

The Problem: Is the Recipe Too Specific?

The problem is that for this "pop" to look like the QPEs we actually see, the orbiting object has to be very specific:

• It can't be too tilted (it must dive through the disk almost straight on).
• It can't be too stretched out (the orbit can't be too oval-shaped).

Think of it like trying to hit a bullseye on a dartboard. If the rules say you have to hit the center and the dart must be thrown from a specific angle, it becomes very hard to hit the target. The authors wanted to know: Are there enough of these "perfectly aimed" objects in the universe to explain how many QPEs we are seeing?

The Experiment: A Cosmic Simulation

To answer this, the authors built a super-computer simulation. They didn't just look at one galaxy; they modeled seven different types of galaxies, ranging from those with small black holes to those with massive ones.

Inside each simulation, they populated the galaxy with:

• 1 million stars (like our Sun).
• Two types of stellar black holes (light ones and heavy ones).

They then let the simulation run for 10 billion years (the age of the universe). They watched how gravity and random collisions (called "two-body relaxation") nudged these objects into the tight orbits needed to crash into the disk.

The Results: Stars vs. Black Holes

Here is what they found, broken down simply:

1. The "Star" Channel (The Easy Target)
When the object crashing into the disk is a normal star, the math works out beautifully.

• Analogy: Imagine throwing a beach ball through a trampoline. It's big and easy to hit.
• Result: The number of these events predicted by the simulation matches the number of QPEs we actually see in the sky. It seems very likely that QPEs are caused by stars crashing into gas disks.

2. The "Black Hole" Channel (The Hard Target)
When the object is a small black hole, the situation is much more difficult.

• Analogy: Now imagine trying to hit the same trampoline with a tiny marble. It's much harder to get the angle and speed just right.
• Result: Because black holes need very specific orbits to produce the right kind of flash, the simulation predicts 1,000 times fewer of these events than we see.
• The Catch: If we relax the rules (allowing the black holes to hit the disk at weird angles or with very stretched-out orbits), the numbers go up. But current observations suggest the orbits should be neat and tidy, which makes the black hole theory less likely.

The Conclusion

The paper concludes that while both scenarios are possible, the "Star" explanation is the most probable.

• Stars crashing into gas disks happen often enough to explain the "popcorn" we see in the universe.
• Black holes crashing into disks are likely too rare to be the main cause, unless our understanding of how they move is incomplete.

Why does this matter?
If we can confirm that QPEs are caused by stars or black holes, it helps us understand how galaxies evolve. Furthermore, if these are indeed black holes, they might be the "electromagnetic twins" of the gravitational waves that the LISA space observatory will detect in the future. It's like finding the sound of a drum (the X-ray flash) to match the vibration of the drum skin (the gravitational wave).

In short: The universe is full of cosmic "pops," and it looks like stars are the ones doing the popping, not the black holes.

Technical Summary

1. Problem Statement

Quasi-Periodic Eruptions (QPEs) are luminous, soft X-ray bursts observed in galactic nuclei, recurring on timescales of hours to weeks. While the "impact model" posits that QPEs arise from a compact object (a stellar-mass black hole, sBH, or a main-sequence star) on an Extreme Mass-Ratio Inspiral (EMRI) orbit piercing a Tidal Disruption Event (TDE) accretion disk, a critical gap remains: it is unclear if such specific physical configurations are sufficiently common to explain the observed number density of QPEs.

Previous studies have modeled the phenomenology of individual events but lacked an end-to-end quantitative calculation of the cosmic volumetric abundance (number density per unit volume) of QPEs derived from first principles of stellar dynamics.

2. Methodology

The authors developed a comprehensive framework to calculate the expected QPE rate and abundance by combining stellar dynamics, gravitational wave (GW) physics, and accretion disk theory.

A. Physical Model

• The Mechanism: QPEs are generated when a compact orbiter (CO) intersects a TDE accretion disk. The impact strips mass, creating expanding bubbles that cool and emit radiation.
• Orbital Constraints:
  • sBHs: To reproduce observed luminosities, the Bondi radius (interaction cross-section) must be large, requiring low relative velocities. This imposes strict constraints: eccentricity $e \lesssim 0.5$ and inclination $i \lesssim 20^\circ$ relative to the disk.
  • Stars: The constraint is primarily avoiding tidal disruption before the EMRI phase, requiring the pericenter $r_p > r_t$ (tidal radius). No strict inclination constraint is applied, though stars must avoid disruption.
• Dynamical Formation: The authors assume QPE sources form via two-body relaxation in Nuclear Star Clusters (NSCs). Relaxation drives objects into the "loss cone" (low angular momentum orbits), where GW emission dominates, leading to an EMRI.

B. Simulation Framework

• Code: The study utilizes PhaseFlow, a public code solving the 1D ergodic Fokker-Planck equation for spherical, isotropic stellar systems.
• System Setup:
  • MBH Masses: Seven systems spanning $10^5 M_\odot$to$10^8 M_\odot$.
  • Stellar Populations: Three components per system:
    1. Main-sequence stars ($1 M_\odot$).
    2. Light sBHs ($10 M_\odot$).
    3. Heavy sBHs ($40 M_\odot$).
  • Mass Ratios: The stellar mass is $20 \times M_{BH}$; sBHs follow a Kroupa IMF-derived mass ratio ($5/7$for$10 M_\odot$,$2/7$for$40 M_\odot$).
• Timescales: Simulations run for $10^{10}$ years (approx. age of the universe).
• Rate Calculation:
  • TDE Rate ($\Gamma_{TDE}$): Calculated from the flux of stars entering the tidal radius. The number of active disks is $N_{disk} = \Gamma_{TDE} \times \tau_{disk}$ (assuming $\tau_{disk} \approx 10$ years).
  • EMRI Rate ($\Gamma_{EMRI}$): Calculated from the flux of objects entering the GW-dominated region ($t_{GW} < t_{relax}$).
  • Selection Factor ($\mu_{T,e,i}$): The fraction of EMRIs that satisfy the specific orbital constraints (period $T_{QPE}$, eccentricity $e$, inclination $i$) required to produce observable QPEs.

C. Abundance Calculation

The cosmic volumetric abundance $\langle n_{QPE} \rangle$ is computed by integrating the per-galaxy abundance over the local Black Hole Mass Function (BHMF):
$$\langle n_{QPE} \rangle = \sum_i \langle N_{QPE}(M_i) \rangle \times \int_{\Delta M_i} \Phi(M_\bullet) d\log M_\bullet$$
Where $N_{QPE} = N_{disk} \times N_{CO}$, and $N_{CO}$ is the number of compatible EMRIs surviving to coalescence within the constraints.

3. Key Contributions

1. First End-to-End Calculation: This is the first study to quantitatively predict the cosmic number density of QPEs based on the EMRI-TDE impact model, moving beyond phenomenological fitting to dynamical prediction.
2. Differentiation of Channels: The study explicitly compares the viability of stellar EMRIs vs. sBH EMRIs as QPE progenitors under the same dynamical framework.
3. Parameter Space Exploration: The authors systematically vary the maximum orbital period ($T_{QPE} \in \{2, 9, 20, 48\}$ hours) and eccentricity thresholds ($e_{max} \in \{0.05, 0.2, 1.0\}$) to assess sensitivity.
4. Incorporation of Inclination Constraints: The study rigorously applies the $i < 20^\circ$ constraint for sBHs, a critical factor often overlooked in simplified rate estimates.

4. Key Results

A. Selection Factors ($\mu_{T,e,i}$)

• The fraction of EMRIs compatible with QPEs is highly sensitive to eccentricity and period constraints.
• Stellar EMRIs: Have a much higher selection factor because they lack the strict inclination constraint and are lighter (smaller GW-dominated region, allowing more time in the QPE-compatible phase).
• sBH EMRIs: The selection factor is suppressed by $\sim 3$ orders of magnitude compared to stars when strict constraints ($e < 0.5, i < 20^\circ$) are applied.

B. Cosmic Volumetric Abundance ($\langle n_{QPE} \rangle$)

• Range: The predicted abundance spans $10^{-12} \text{ Mpc}^{-3}$to$10^{-6} \text{ Mpc}^{-3}$, heavily dependent on the assumed constraints.
• Stellar Channel: Stellar-driven QPEs can reach number densities of $\sim 10^{-7} - 10^{-6} \text{ Mpc}^{-3}$, which is compatible with current observational estimates (e.g., Arcodia et al. 2024).
• sBH Channel:
  • With strict constraints ($e < 0.5, i < 20^\circ$): Abundance is $\sim 10^{-9} - 10^{-10} \text{ Mpc}^{-3}$, roughly 1,000 times lower than stellar QPEs and inconsistent with observations.
  • With relaxed constraints (removing $e$ and $i$ limits): sBH QPEs become compatible with the lower bound of observational estimates.

C. Eccentricity vs. Observations

• Two-body relaxation naturally produces a population of highly eccentric EMRIs.
• However, timing models of observed QPEs suggest low eccentricity ($e \lesssim 0.2$).
• Imposing low-eccentricity constraints drastically reduces the predicted rates for both stars and sBHs, potentially pushing theoretical predictions below observed values.

5. Significance and Implications

• Viability of the Impact Model: The study confirms that the EMRI-TDE disk impact model is a viable explanation for QPEs, provided the progenitors are main-sequence stars.
• Challenge for sBHs: The sBH channel is strongly disfavored under standard two-body relaxation dynamics due to the stringent orbital constraints required for the emission mechanism. This suggests that if sBHs are the progenitors, alternative formation channels (e.g., binary deposition, migration in AGN disks) that produce low-eccentricity, low-inclination orbits must be dominant.
• Multimessenger Astronomy: The results highlight the tension between dynamical predictions (high eccentricity) and timing observations (low eccentricity). Resolving this could identify QPEs as electromagnetic counterparts to future LISA gravitational wave sources.
• Future Directions: The authors suggest that refining galactic models, including MBH spin effects, and exploring non-relaxation formation channels are necessary to fully constrain the nature of QPEs.

In conclusion, the paper provides a rigorous dynamical test of the QPE impact model, strongly favoring stellar progenitors over sBHs under standard relaxation scenarios, while identifying the eccentricity constraint as the primary bottleneck for matching theoretical rates to observations.