Eddington Inversion Method

• Eddington inversion method is a mathematical technique that reconstructs a system's phase-space distribution function using its density profile and gravitational potential.
• It employs the inversion of Abel integrals and is extended to handle anisotropic velocity distributions and boundary effects, ensuring physical consistency by maintaining a non-negative distribution function.
• The method is key in modeling dark matter halos and stellar systems, refining kinematic predictions and impacting both direct and indirect dark matter detection strategies.

The Eddington inversion method (EIM) is a rigorous mathematical approach for reconstructing the phase-space distribution function (DF) of a collisionless, steady-state, spherically symmetric system using only its mass-density profile and gravitational potential. This formalism provides a physicality filter on models by requiring positivity of the DF everywhere and is central to interpreting the internal structure and velocity distribution of dark-matter halos and stellar systems. It has seen extensive application in dark-matter phenomenology, notably in the context of direct and indirect detection, and in recent work to infer dark matter properties of dwarf galaxies from photometric data alone (Almeida et al., 2024, Lacroix et al., 2018).

1. Mathematical Foundation of the Eddington Inversion Method

The Eddington inversion is derived for a spherically symmetric, isotropic stellar or particle system with density profile $\rho(r)$ and gravitational potential $\Phi(r)$. The relative potential is $\Psi(r) = \Phi_0 - \Phi(r)$, with $\Psi \to 0$ as $r \to \infty$. The phase-space DF, $f(\mathcal{E})$, depends on the relative energy $\mathcal{E} = \Psi(r) - \frac{1}{2}v^2$. The relation between $\rho(r)$ and $f(\mathcal{E})$ in an isotropic system is

$$\rho(r) = 4\pi\sqrt{2} \int_{0}^{\Psi(r)} f(\varepsilon)\sqrt{\Psi(r)-\varepsilon}\,d\varepsilon$$

Differentiating $\rho$ with respect to $\Psi$ leads to the Abel integral equation:

$$\frac{d\rho}{d\Psi} = 2\pi\sqrt{2} \int_{0}^{\Psi} \frac{f(\varepsilon)}{\sqrt{\Psi-\varepsilon}}\,d\varepsilon$$

The classical Eddington formula provides the inversion:

$$f(\varepsilon) = \frac{1}{\sqrt{8}\pi^2} \frac{d}{d\varepsilon} \int_{0}^{\varepsilon} \frac{d\rho}{d\Psi} \frac{d\Psi}{\sqrt{\varepsilon-\Psi}}$$

or equivalently, after integrating by parts,

$$f(\varepsilon) = \frac{1}{\sqrt{8}\pi^2} \int_{0}^{\varepsilon} \frac{d^2\rho}{d\Psi^2} \frac{d\Psi}{\sqrt{\varepsilon-\Psi}}$$

A physically viable system has $f(\varepsilon)\geq 0$ everywhere. This criterion rejects unphysical combinations of density profiles and potentials (Almeida et al., 2024, Lacroix et al., 2018).

2. Extensions: Anisotropy and Boundary Effects

The isotropic Eddington inversion can be generalized using Osipkov-Merritt models for anisotropic velocity distributions. In this formalism, $f(\mathcal{E},L) = f_\mathrm{OM}(Q)$ with $Q = \mathcal{E} - L^2/(2r_a^2)$. The "augmented density" is $\rho_{\mathrm{OM}}(r) = (1 + r^2/r_a^2)\rho(r)$. The inversion proceeds as in the isotropic case, applied to $(Q, \rho_{\mathrm{OM}})$.

Finite halo boundaries introduce a divergent term $\sim 1/\sqrt{\mathcal{E}}$ in the DF as $\mathcal{E}\to 0$, producing an unphysical high-velocity spike. If removed manually, the self-consistency of the reconstruction is lost. Multiple regularization schemes (density regularization, King-type truncation, or modified DF subtraction) cure this divergence and maintain the physicality of $f$ in bounded domains (Lacroix et al., 2018).

3. Practical Implementation: Application to Galactic Systems

Recent work (Almeida et al., 2024) applies EIM to $\sim$100 low-mass galaxies ($M_\star = 10^6$-$10^8\,M_\odot$) using only observed photometric profiles. The steps are:

• Fit observed surface brightness $\Sigma(R)$ to analytic models (polytropes or double-slope $\rho_{abc}$ profiles).
• Assume a spherically symmetric DM potential (e.g., NFW, cored, or flexible forms).
• Compute the best-fit spatial density, project to deprojected $\rho(r)$, and map $\rho(r)\to\rho(\Psi)$.
• Use the Eddington inversion to calculate $f(\varepsilon)$ for each potential and check for $f<0$ as an exclusion criterion.
• Construct diagnostic diagrams over core/cusp parameters, quantifying the fraction of galaxies consistent with various DM halos.

Empirically, polytropic (cored) fits to starlight profiles exclude all NFW-like DM potentials (i.e., those with inner slope $c_p \gtrsim 0.1$) for the sample, and floating-slope models reveal that $40$-$70\%$ of galaxies favor cored DM distributions inconsistent with collisionless CDM halos (Almeida et al., 2024).

4. Model Fitting, Projection, and Profile Families

Projection from $\rho(r)$ to $\Sigma(R)$ and vice versa is essential for using photometric data. Analytic density models used include:

Model Family	Functional Form	Inner Slope
Polytrope	$\rho_m(r)=\rho_0\big[1+(r/r_s)^2\big]^{-m/2}$	Core ($c=0$)
General slope	$$\rho_c(r) = \frac{\rho_s}{(r/r_s)^c[1+(r/r_s)^2]^{(5-3c)/(2-c)}}$$	Variable $c$
Double-slope	$$\rho_{abc}(r) = \frac{\rho_s}{(r/r_s)^c[1+(r/r_s)^a]^{(b-c)/a}}$$	Variable $c$, $b$

• Special cases include Plummer ($a{=}2, b{=}5, c{=}0$) and NFW ($a{=}1, b{=}3, c{=}1$) profiles.
• Profiles are fit to observed $\Sigma(R)$ via Levenberg-Marquardt minimization; best-fit parameters are transferred into EIM computations (Almeida et al., 2024).

Systematic uncertainties from the choice of analytic forms and projection effects (e.g., PSF, ellipticity, component multiplicity) are mitigated by discarding poor fits and validating via core-dominated globular cluster benchmarks.

5. Physical Implications and Applications

The ability of EIM to exclude NM-compatible DM potentials using only photometric data demonstrates the method’s diagnostic power. For low-mass dwarfs, where baryonic feedback cannot flatten an NFW cusp, the prevalence of inferred cores points to alternatives such as self-interacting, fuzzy, or warm dark matter (Almeida et al., 2024). In indirect and direct detection, the accurate reconstruction of the velocity distribution $f_v(v, r)$ and its moments (e.g., mean speed, $\eta(v_\mathrm{min})$, $\langle v_r^n \rangle$ for annihilation rates) is critical:

• Deviations from the standard Maxwellian halo velocity distribution (SHM) reach $20$-$50\%$ in detection rates and may shift flux predictions by $O(1)$-$O(10)$ in the Galactic center or dwarfs (Lacroix et al., 2018).
• Microlensing constraints on primordial black holes (PBHs) depend on low-velocity tails, with EIM producing $20$-$100\%$ differences in predicted event rates.
• EIM reliably quantifies these uncertainties and corrects for unphysical predictions from SHM or non-self-consistent modeling.

6. Assumptions, Limitations, and Robustness

Key assumptions of EIM include spherical symmetry and velocity isotropy ($\beta = 0$). Real systems may be elliptical or triaxial, but generalizations (Lynden-Bell extension) indicate the core versus NFW incompatibility persists in axisymmetric cases. Anisotropy (radial, $\beta>0$; tangential, $\beta<0$) impacts the core requirement: radial anisotropy strengthens the exclusion of cusps, while tangential orbits can mask it but require fine-tuning and are generally contrived (Almeida et al., 2024).

Systematic bias may arise from errors in deprojection, model fitting, or profile mis-specification. The methodology in (Almeida et al., 2024) controls for these via fit selection, multiple profile families, and code validation. Stability of the DF ($df/d\mathcal{E}>0$ everywhere; Antonov’s law) and regularization at the halo boundary are essential for physical results (Lacroix et al., 2018).

7. Theoretical and Observational Impact

The Eddington inversion method has become foundational for modeling the internal kinematics of dynamical systems from limited observational data. Its no-negativity constraint on the phase-space DF serves as a robust test of DM halo models and enables photometric inference of DM properties where kinematics are inaccessible. Systematic application in current and future surveys can sharply constrain the nature of DM in low-mass galaxies and refine astrophysical predictions for DM detection across the electromagnetic and particle spectrum (Almeida et al., 2024, Lacroix et al., 2018).

The Eddington inversion framework complements simulation-based halo modeling and offers analytic transparency, flexibility in profile selection, and rigorous physicality criteria for theoretical and observational studies in galactic dynamics and dark matter astrophysics.