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On the orbital eccentricities of primordial black hole binaries inside and outside of dark matter halos

This paper simulates the orbital evolution of primordial black hole binaries to predict that while isolated systems circularize before entering most gravitational wave bands, those within dark matter halos retain higher eccentricities due to dynamical interactions, offering a pathway for future space-based observatories like LISA and DECIGO to constrain primordial black hole abundance by detecting these residual eccentricities.

Original authors: Muhsin Aljaf, Ilias Cholis

Published 2026-07-28

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CC BY 4.0

Original authors: Muhsin Aljaf, Ilias Cholis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic dance floor where invisible partners are constantly spinning, colliding, and sometimes crashing into each other. Some of these partners are "Primordial Black Holes" (PBHs), mysterious heavyweights that might have formed right at the very beginning of time, like cosmic dust settling into dense clumps. When two of these black holes get close enough, they start orbiting one another, creating a gravitational waltz that ripples through space itself. These ripples are called "gravitational waves," and they are like the sound of the dance floor shaking. Scientists have built giant ears, called detectors, to listen for these sounds. The big question is: do these black holes dance in perfect circles, or do they wobble in wild, oval-shaped paths? The shape of their orbit, known as "eccentricity," acts like a fingerprint. If we can hear a wobble, it tells us exactly how the black holes met and where they have been hiding. This matters because finding these wobbles could prove that black holes are ancient leftovers from the Big Bang, rather than the children of dead stars.

In this paper, two researchers named Muhsin Aljaf and Ilias Cholis decided to simulate a massive cosmic dance party to see what kind of music these primordial black holes would make. They didn't just look at one type of dance; they tracked billions of potential black hole pairs from the moment they were born until they crashed together. They looked at two main groups: the "loners" who danced alone in the empty space between galaxies, and the "party-goers" who got stuck inside crowded dark matter halos (which are like invisible, fuzzy clouds of extra gravity that surround galaxies).

The researchers ran huge computer simulations to track how these orbits changed over time. They found that the "loner" pairs are very boring dancers. Even if they started with a wild, oval orbit, the friction of their own gravitational waves smoothed them out into perfect circles long before they reached the ears of our current detectors. By the time these pairs are ready to be heard by ground-based detectors like LIGO, they are spinning in perfect circles, making them impossible to tell apart from regular black holes.

However, the story gets much more exciting for the "party-goers." When black holes get trapped inside those dense dark matter clouds, they don't just dance with their partner; they get bumped and jostled by other black holes nearby. These "binary-single interactions" are like a crowded mosh pit where dancers keep getting pushed. This chaos prevents the orbits from smoothing out. The researchers found that these pairs can keep their wild, oval shapes (high eccentricity) all the way until they are about to merge.

The paper uses simulations to predict what our future listening devices will hear. They found that ground-based detectors (like LIGO, ET, and CE) will likely only hear the smooth, circular dances of the loners. But space-based detectors, which are like super-sensitive ears listening to lower-pitched sounds (specifically LISA and DECIGO), might catch the wild, wobbly dancers. The authors estimate that LISA and DECIGO could detect around 100 of these eccentric pairs if primordial black holes exist in the numbers we currently suspect. If these space telescopes look and don't find any wobbly dancers, it would be a huge clue that primordial black holes are much rarer than we think, tightening the rules on how much of the universe they can make up.

In short, the paper suggests that if we want to find these ancient black holes, we shouldn't just listen for the crash; we need to listen for the wobble. While the ground-based detectors might miss the signal because the dancers have already calmed down, the space-based detectors might catch them mid-wobble, offering a unique way to prove these cosmic ghosts exist. If we don't hear that wobble, the paper suggests we might have to rewrite our understanding of how much of the universe is made of these primordial black holes.

Technical Summary: Orbital Eccentricities of Primordial Black Hole Binaries Inside and Outside of Dark Matter Halos

Problem Statement
Primordial black holes (PBHs) in the stellar mass range remain a viable candidate for a fraction of the universe's dark matter and a potential source of gravitational wave (GW) signals detected by observatories like LIGO. While PBH binaries are theorized to form with high orbital eccentricities, their subsequent evolution depends heavily on their environment. Early-formed binaries evolving in isolation are expected to circularize via GW emission long before entering the frequency bands of ground-based detectors. However, binaries residing within dark matter halos may undergo stochastic binary-single interactions with neighboring PBHs, potentially preserving or enhancing their eccentricity. Furthermore, binaries formed via direct GW captures in dense environments may retain extreme eccentricities. A detailed understanding of the eccentricity distribution of these binaries across multiple GW frequency bands (from space-based to ground-based detectors) is currently missing, particularly regarding how environmental interactions affect detectability and the ability to distinguish PBH origins from standard astrophysical black hole binaries.

Methodology
The authors simulate large samples of stellar-mass PBH binaries ( $m_1 = m_2 = 30 M_\odot$ ) to track their full orbital evolution from formation at redshift $z=3400$ up to their merger or the present day. The study categorizes PBH binaries into three distinct evolutionary pathways:

Unperturbed Binaries: Binaries that formed early and evolved solely via GW emission, remaining outside dark matter halos.
Perturbed Binaries (Binary-Single Interactions): Early-formed binaries that fell into dark matter halos and underwent stochastic interactions with single PBHs.
Direct Captures: Binaries formed at any redshift within dark matter halos when two unbound PBHs lose energy via GW emission during a close encounter.

The orbital evolution of the semi-major axis ( $a$ ) and eccentricity ( $e$ ) is modeled using coupled differential equations that account for both GW emission (Peters-Mathews equations) and environmental interactions (characterized by density $\rho_{env}$ and velocity dispersion $v_{disp}$ ). The simulations utilize adaptive Runge-Kutta solvers for GW-dominated phases and Euler integration for environmental interaction phases within concentric spherical shells of dark matter halos ranging from $10^4 M_\odot$ to $10^{15} M_\odot$ .

To assess detectability, the authors compute the characteristic strain ( $h_c$ ) and signal-to-noise ratio (SNR) for each binary, considering the peak harmonic of the GW emission which shifts with eccentricity. They evaluate detectability against the noise curves of current and future observatories: LISA, DECIGO, Einstein Telescope (ET), Cosmic Explorer (CE), and aLIGO. The expected number of detectable events ( $N_{det}$ ) is calculated by integrating the merger rates over the comoving volume up to the detector's horizon redshift ( $z_{max}$ ), assuming PBH abundance limits consistent with current LVK constraints ( $f_{PBH} \lesssim 10^{-2}$ ).

Key Contributions and Results

Circularization of Unperturbed Binaries: The study confirms that PBH binaries evolving in isolation completely circularize ( $e < 10^{-4}$ ) before their GW signals enter the frequency bands of ground-based detectors (ET, CE, aLIGO). Even for LISA, residual eccentricities are small ( $e \sim 10^{-2}$ ).
Impact of Binary-Single Interactions: Binaries residing in dark matter halos, particularly in denser inner shells, experience binary-single interactions that can harden orbits and maintain higher eccentricities. While massive halos ( $>10^9 M_\odot$ ) behave similarly to unperturbed cases due to low interaction rates, smaller halos ( $10^4 - 10^8 M_\odot$ ) significantly alter the eccentricity distribution.
Direct Capture Contribution: Despite the potential for high eccentricity, the direct capture channel contributes negligibly to the total number of detectable eccentric binaries given current constraints on PBH abundance.
Detectable Eccentric Binaries:
- Ground-based (ET, CE, aLIGO): The vast majority of detectable PBH binaries will appear effectively circular ( $e \approx 0$ ), making eccentricity an ineffective discriminator against astrophysical binaries for these instruments.
- Space-based (LISA, DECIGO): These observatories can probe binaries at earlier inspiral stages where eccentricity is non-negligible. The authors predict that LISA and DECIGO combined could detect approximately $O(10^2)$ PBH binaries with eccentricity $e > 0.01$ . Specifically, DECIGO is projected to detect $\sim 1.1 \times 10^4$ such events, while LISA may detect $\sim 68$ events from the binary-single channel (totaling $\sim 76$ when including unperturbed channels).
- High Eccentricity Events: A small fraction of these space-based detections may exhibit $e > 0.1$ .

Significance
The paper establishes that while ground-based GW observatories will likely detect PBH binaries, they will not be able to use orbital eccentricity to distinguish them from standard astrophysical black hole binaries, as environmental interactions and GW emission will have circularized the orbits by the time of merger. However, the study highlights a critical opportunity for space-based observatories (LISA and DECIGO). If these future missions can detect binaries with $e > 0.01$ , it would provide strong evidence for the PBH origin of some compact object binaries. Conversely, the authors claim that if LISA and DECIGO exclude the existence of such eccentric binaries, the upper limits on the abundance of stellar-mass PBHs could be improved by an order of magnitude. This work provides the necessary theoretical framework and predicted event rates to guide the interpretation of future GW data regarding the nature of dark matter and binary formation channels.

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