Extended Dark Objects: Definition & Bounds

• EDOs are non-luminous, macroscopic dark-matter structures with extended, diverse density profiles that defy the point-lens approximation.
• Their observational signatures include anomalous microlensing light curves, modified gravitational wave inspiral frequencies, and distinct dynamical heating effects.
• Advanced analysis techniques, such as machine learning and multi-messenger surveys, are employed to constrain their properties and explore dark-sector physics.

An Extended Dark Object (EDO) is a macroscopic, non-luminous structure whose physical radius is comparable to or exceeds its gravitational Einstein radius, and whose internal mass profile may arise from a variety of dark-sector mechanisms. Unlike compact point-mass candidates such as primordial black holes (PBHs), EDOs can have spatially extended and physically diverse density distributions, including ultracompact minihalos, boson stars, Q-balls, and axion miniclusters. Their observational phenomenology is determined by gravitational interactions—primarily microlensing, dynamical friction, accretion—and by their impacts on baryonic systems across a wide mass and size spectrum (Croon et al., 25 Nov 2025, Romão et al., 2024, Kim et al., 25 Aug 2025, Croon et al., 2024, Croon et al., 2024).

1. Physical Definition, Classification, and Mass Profiles

An EDO is defined by its mass $M$ and physical radius $R_{\rm EDO}$, where $R_{\rm EDO} \gtrsim R_E$ (with $R_E$ the lens-plane Einstein radius for mass $M$). The key distinction is the breakdown of the point-lens approximation for gravitational interactions (Croon et al., 25 Nov 2025, Romão et al., 2024). The EDO's nature is further specified by its internal density profile $\rho(r)$. Relevant physical realizations and their characteristic properties include:

EDO Type	Mass Range ($M_\odot$)	Radius ($R_\odot$)	Density Profile $\rho(r)$
Ultracompact Minihalo (UCMH)	$10^{-6}$ – $10^3$	$10^2$ – $10^5$	$\propto (1+r/r_c)^{-\alpha}$, UCMH: $\propto r^{-9/4}$
Axion Minicluster	$10^{-12}$ – $10^{-3}$	$10^2$ – $10^4$	$\propto r^{-9/4}$ (core/cutoff)
Boson Star	$10^{-10}$ – $10$	$10^1$ – $10^3$	$\exp(-r^2/R^2)$ (solitonic, ground state)
NFW Subhalo	$10^{-12}$ – $10^6$	$10^3$ – $10^7$	$\propto [r/r_s][1+r/r_s]^2$ (NFW)
Dark Fermion Star	$1$ – $5$	$3$ – $10$ km	Gaussian (TOV equilibrium)
Q-ball	Model-dependent	$10$ – $10^3\,R_S$	Uniform inside $R_Q$
Dressed PBH (dPBH)	$1$ – $10^4$	$\gg R_S$	PBH+DM halo: $\rho(r)\propto r^{-9/4}$

Here, $R_S$ is the Schwarzschild radius, and $R_\odot \approx 7\times 10^{10}$ cm.

Standardized mass profiles, as used in computational repositories, include the NFW, boson soliton, uniform-sphere, and UCMH profiles (Croon et al., 2024). Each profile is parameterized for consistency when comparing constraints across different astrophysical probes.

2. Gravitational Microlensing by EDOs

The extended nature of EDOs alters their microlensing signatures relative to point-mass lenses. In the lens equation framework, for a 2D projected mass distribution $\Sigma(\theta)$, the lensing potential is

$$\psi(\boldsymbol \theta) = \frac{1}{\pi}\int d^2\theta'\, \Sigma(\boldsymbol\theta') \ln|\boldsymbol\theta - \boldsymbol\theta'| + \frac{1}{2}\bar\kappa |\boldsymbol\theta|^2 - \frac{1}{2}\bar\gamma(\theta_1^2-\theta_2^2)$$

with macrolens convergence $\bar\kappa$ and shear $\bar\gamma$ in strong lensing fields (the "Chang–Refsdal" approximation) (Croon et al., 25 Nov 2025).

The normalized lens size parameter,

$\tau_m \equiv \frac{R_{\rm EDO}}{R_E}$

governs the phenomenological regime. For $\tau_m \lesssim 1$, EDOs act as point mass lenses; for $1 \lesssim \tau_m \lesssim \sqrt{\mu_t}$ (with $\mu_t$ the tangential macro-magnification), additional narrow caustics appear, generating multi-peaked or plateaued light curves not reproducible by point lenses (Croon et al., 25 Nov 2025, Romão et al., 2024). For $\tau_m \gg \sqrt{\mu_t}$, the microlensing effect is washed out.

Machine learning pipelines (such as MicroLIA) can classify EDO-induced microlensing signatures, distinguishing solitonic (boson-star–like) and NFW-like EDOs from point-lens light curves, even under realistic survey cadence (Romão et al., 2024). The identification efficiency is highest for $0.8 \lesssim \tau_m \lesssim 3$.

3. Constraints from Gravitational Waves and Dynamical Heating

Coalescing EDO binaries source gravitational waves distinct from black hole mergers. The main finite-size effect is a lowered ISCO frequency,

$$f_{\rm ISCO}(M, R) \simeq \frac{(G M/R)^{3/2}}{3^{3/2}\pi (2M)}$$

which truncates the observable inspiral earlier than for PBHs of equal mass. Advanced LIGO (aLIGO) has sensitivity to EDOs with $R \sim 10$–$10^3$ km and mass $1$–$10^2\,M_\odot$ if they compose $\gtrsim1\%$ of the local dark matter density (Croon et al., 2022). For larger radii ($R \gtrsim 10^3$ km), terrestrial detectors lose sensitivity, but space-based laser interferometers can probe into this region.

Dynamical heating in stellar systems by EDO flybys, calculated from the velocity-kick per passage, also constrains the EDO parameter space. Excess heating can be compared to observed stellar velocity dispersions in ultra-faint dwarfs and the Galactic disk (Croon et al., 2024).

4. Accretion, Gas Heating, and CMB Constraints

EDOs accrete baryonic gas in the early and late universe; the associated energy deposition affects both the interstellar medium and the cosmic microwave background (CMB)—depending on the mass, size, and internal structure of the EDO (Kim et al., 25 Aug 2025, Croon et al., 2024). The classical Bondi rate,

$$\dot M_{\rm Bondi} = 4 \pi \lambda (G M)^2 \frac{\rho_b}{c_s^3}$$

applies for compact EDOs, but for extended profiles, effective accretion rates are profile-dependent. For CMB constraints, energy injection from EDO-induced accretion and subsequent collisional ionization and Bremsstrahlung emission perturb the ionization fraction $x_e(z)$, which is measured via CMB anisotropies. The tightest bounds exclude $f_{\rm EDO}\gtrsim10^{-3}$ for $R\sim10^3$–$10^4$ AU and $M\sim10^2$–$10^4\,M_\odot$ (Croon et al., 2024).

In dwarf galaxies like Leo T, the combined effects of dynamical friction and accretion-driven heating bound the allowed EDO fraction, with compact configurations (e.g., dressed PBHs, dark fermion stars, small Q-balls) most constrained: $f \lesssim 10^{-2}$–$10^{-3}$ for $M\sim1$–$10^4\,M_\odot$ (Kim et al., 25 Aug 2025).

5. Catalogued Observational Bounds and Methodology

A unified computational repository enables cross-comparison of EDO constraints from microlensing, gravitational waves, CMB, and stellar dynamics for standardized mass profiles (NFW, boson soliton, uniform, UCMH). Each bound is tabulated as $(M, R_{90}, f_{\rm DM})$ arrays, with methodology tailored to the relevant observable (Croon et al., 2024). For microlensing, the effective lens mass $M_\text{eff}(u)$ is mapped to observed event rates; for GW, the inspiral cutoff sets the detection window; for CMB, the profile-dependent accretion and energy deposition determine $x_e(z)$ and thus the bound.

The main constraint categories are:

Bound Type	Observable	Parameter Sensitivity
Microlensing	Light-curve shape, event rate	$M$, $R_{90}$, profile; $R_{90} \sim 0.5$–$10^3\,R_\odot$
Gravitational Wave	Inspiral cutoff $f_{\rm ISCO}$	$M \gtrsim1\,M_\odot$, $R_{90}\lesssim10^3$ km
CMB-Accretion	$x_e(z)$, $C_\ell$ shifts	$M\gtrsim10\,M_\odot$, $R\sim1$–$10^4$ AU
Dynamical Heating	Stellar velocity dispersion	$M\gtrsim10^5\,M_\odot$, $R\sim10^2$–$10^4\,R_\odot$

All data products and plotting scripts are released; users can generate custom exclusion plots for any combination of EDO profile, radius, and observational bound (Croon et al., 2024).

6. Case Studies: Astrophysical and Cosmological EDO Candidates

Structural discoveries such as the ultra-low surface brightness galaxy Nube exemplify EDOs on galactic scales: $M_\star\approx4\times 10^8\,M_\odot$, $R_e=6.9$ kpc, with an effective stellar surface density $\Sigma_{\rm eff}\sim1\,M_\odot\,\mathrm{pc}^{-2}$ (Montes et al., 2023). Its structure is best fitted by a soliton core of $r_c \approx6.6$ kpc, consistent with fuzzy dark matter models (ultralight axion of $m_B= (0.8^{+0.4}_{-0.2})\times10^{-23}$ eV). The feedback-inefficient and dark-matter–dominated regime mapped by such objects offers a sensitive laboratory for the microphysical nature of EDOs.

In the context of transient events, simulations of thermonuclear supernovae with extended DM cores yield dimmer, broader light curves, enhanced neutrino emission, and compact remnants mimicking sub-solar-mass black holes in gravitational lensing and microlensing (Chan et al., 2020). Such signatures can be differentiated from standard scenarios by light curve properties, neutrino signals, and remnant demographics.

7. Implications, Complementarity, and Future Directions

EDOs provide a parametric framework to interpolate between PBHs, subhalo clumps, and more exotic composite objects, characterized by their extended structures and nontrivial internal physics. The synergy between microlensing, gravitational wave, CMB, and astrophysical surveys probes radial scales from $\sim 10\,R_\odot$ up to $\sim10^7\,R_\odot$ (and in galactic cases, kpc scales), extending sensitivity orders of magnitude beyond classical PBH searches (Croon et al., 25 Nov 2025, Croon et al., 2024, Croon et al., 2024).

Forthcoming surveys (LSST, Roman, JWST for caustic-crossing events; next-generation gravitational-wave detectors; and deeper HI and optical imaging) will vastly expand the parameter space explored, enabling differentiation among bosonic, fermionic, and composite EDO models, and potentially linking properties of the invisible sector to observable transient and structural features at multiple cosmological epochs.

For robust, model-independent limits, standardizing density profiles, careful characterizations of macrolens environments, and multi-messenger observational strategies are essential (Croon et al., 2024). The EDO paradigm thus encompasses both a broad class of dark-matter sectors and an operational methodology for constraining dark microphysics via gravitational phenomena.