Non-Parametric Gravitational Imaging

• Non-parametric gravitational imaging is defined as a data-driven approach that reconstructs spatial mass distributions and gravitational potentials using minimal assumptions.
• It employs computational methods such as regularized least-squares, kernel and spline interpolation, level-set parametrization, and Bayesian bin modeling to capture localized features.
• Applications span astrophysics, geophysics, gravitational-wave studies, and quantum metrology, enabling enhanced sensitivity and robust uncertainty quantification.

Non-parametric gravitational imaging comprises a set of methodologies aimed at reconstructing spatial distributions of mass or gravitational fields directly from observational data, without enforcing restrictive parametric models for the underlying mass or potential. These frameworks are notable for their data-driven approach, computational flexibility, and applicability across scales spanning planetary interiors, astrophysical lensing, geophysical surveys, and gravitational-wave population studies. Several key domains exemplify non-parametric gravitational imaging: direct inversion techniques for static and dynamic fields, hybrid kernel-based approaches for subsurface tomography, splined and binned models for population inference, level-set techniques for interior mapping, quantum interferometric measurement, and Bayesian grid-based imaging for lensing detection.

1. Non-Parametric Gravitational Imaging Principles

Non-parametric gravitational imaging seeks to reconstruct mass distributions, gravitational potentials, or related population properties by minimizing assumptions about physical structure. Instead of adopting analytic forms (e.g., power laws, Gaussians, spheres), these methods operate either by expanding the solution in discrete spatial bins, employing functional regularizers, or using flexible interpolants to directly encode observational constraints.

In standard gravitational lensing, for example, pixel-based corrections to the lensing potential are optimized against the observed image data rather than enforcing a fixed profile for substructure (Powell et al., 8 Oct 2025). In waveform population studies, the distribution of binary black hole properties is reconstructed using Gaussian processes over binned parameter spaces or via flexible (spline or autoregressive) expansions (Ray et al., 2023, Rinaldi et al., 6 Jun 2025, Fabbri et al., 28 Jan 2025). In geophysics and planetary science, density anomalies are parameterized by non-explicit boundaries (level-sets) and the joint inversion is calibrated against measured gravitational fields (Caldiero et al., 2023).

The haLLMark of these techniques is that each degree of freedom (e.g., a mass bin, a spatial cell, a grid point for lens potential) absorbs empirical corrections, allowing localized features and correlations to be detected. The methodologies are robust against model-induced bias and are systematically improvable by increasing bin resolution, regularization order, or iterating over random initializations.

2. Computational Methodologies and Mathematical Formulations

Several computational strategies are employed:

• Regularized Least-Squares and Bayesian Pixel Grid Inversion: In lensing applications, the imaging operator connecting observed visibilities or images to sky brightness and lens potential is augmented by block columns corresponding to pixelated perturbations, and solved by iterative Bayesian optimization (Powell et al., 8 Oct 2025). The system is typically of the form

$$[(DL)^\mathrm{T} C (DL) + R]\cdot \mathrm{MP} = (DL)^\mathrm{T} C d$$

with $DL$ the combined Fourier and lensing operators, $C$ the noise covariance, and $R$ the regularization.

• Kernel and Spline Interpolation: In geophysical imaging or image gravimetry, interpolants are constructed using spheroidal splines defined over georeferenced satellite imagery (Kiani, 2020). The system of equations is sparse:

$$\sum_{l=-2}^{2} d_{l,k} W(\phi_{i+1}, \lambda_{j+k}) = 0$$

where $d_{l,k}$ are determined by the spheroidal geometry.

• Level-Set Parametrization of Density Anomalies: Internal planetary structure is modeled via piecewise constant regions, each defined implicitly by the zero-level surface of a scalar field $\phi_j(x)$ (Caldiero et al., 2023). The inversion iteratively updates $\phi_j$ and density contrast $\rho_j$ by minimizing a Tikhonov-regularized objective with linearized residuals, maintaining computational tractability:

$$\Psi(\delta t) = \| y - J\,\delta t \|_2^2 + \lambda^2 \| \delta t \|_2^2 + \Psi_c(\delta t)$$

where $J$ is the Jacobian of the predicted gravity coefficients with respect to all density and level-set parameters.

• Gaussian Process and Bayesian Bin Modelling: The distribution of population parameters (e.g., mass, spin, redshift) is expanded over bins with flexible regularization using Gaussian Processes, enabling reconstruction of correlations and redshift dependencies (Ray et al., 2023, Afroz et al., 29 Sep 2025). Typical priors on the logarithmic rate densities include covariance kernels:

$$S_{gg'} = \sigma^2 \exp\left( -\frac{(c_g - c_{g'})^2}{2\lambda^2} \right)$$

and population likelihoods constructed as:

$$p(d|n) = \exp(-N_{\text{obs}}(n)) \prod_{i=1}^{N_{\text{obs}}} p_i(n)$$

3. Applications: Astrophysical, Geophysical, and Quantum Regimes

Non-parametric gravitational imaging is widely applied:

• Detection of Sub-Galactic Structures and Dark Matter Subhalos: High-resolution VLBI observations, coupled with pixelized lensing potential corrections, allow detection of million-solar-mass objects far below the mass threshold of prior lensing detections—e.g., a $1.13\pm0.04\times10^6 M_\odot$ subhalo at cosmological distance, with a statistical significance of $26\sigma$ and fractional uncertainty of $3.3\%$ (Powell et al., 8 Oct 2025).
• Geophysical Tomography and Subsurface Imaging: Joint inversions of muon tomography and gravimetry employ acquisition and resolving kernels to link measured signals to volume density, overcoming the limitations of geometric coverage by combining complementary spatial sensitivities (Jourde et al., 2014). Spheroidal modelling and spline interpolants enhance the stability and precision of deposit characterization (Sizikov, 2015, Kiani, 2020).
• Gravitational-Wave Population Inference: Hierarchical, non-parametric models reconstruct distributions of black hole masses, ratios, and redshifts from GW catalogs, allowing the identification of features such as subpopulations, redshift-dependent clustering, and possible correlations (Ray et al., 2023, Rinaldi et al., 6 Jun 2025, Afroz et al., 29 Sep 2025). Bayesian approaches correct for selection effects via injection/recovery statistics.
• Quantum Sensing and Atom Interferometry: Quantum imaging-inspired setups measure gravitational phases via atom-photon entanglement, enabling readout through photon interferometry, with repeated measurements to isolate the gravitational potential and inertial acceleration (Cepok et al., 5 Sep 2024). Differential atom interferometers extract curvature and gradient information non-parametrically by measuring phase differences proportional to higher-order spatial derivatives (Werner et al., 5 Sep 2024).

4. Model Selection, Validation, and Uncertainty Quantification

Robustness and interpretability are achieved by:

• Comparative Model Reconstruction: Frameworks allow detailed population fits to be “described” non-parametrically then “interpreted” post hoc by reconstructing parametric fits using loss-minimization (Jeffreys divergence) on precomputed hyperposterior samples, streamlining the analysis pipeline and providing internal model diagnostics (Fabbri et al., 28 Jan 2025).
• Feature Robustness and Significance: Detection of mass perturbations (e.g., lensing subhalos) is validated by repeating reconstructions with varied regularization schemes and grid resolutions, assessing significance by noise thresholds and Bayesian evidence gains (Powell et al., 8 Oct 2025).
• Uncertainty and Degeneracy Handling: Inversion ensembles generated from random initial conditions are clustered to estimate feature significance and map uncertainty; local density standard deviations quantify confidence in recovered anomalies (Caldiero et al., 2023). Bayesian posterior samples over functional coefficients (e.g., Taylor expansion in redshift) directly encode both detection significance and degeneracy (Afroz et al., 29 Sep 2025).
• Limitations and Biases: While non-parametric methods minimize model-induced bias, they require careful correction for selection effects (e.g., GW detector sensitivity), and their interpretability in terms of physical formation channels is contingent on subsequent parametric mapping. Certain regimes (e.g., symmetric mass ratio binaries $q>0.7$) remain poorly constrained and their apparent features may be prior-driven rather than data-driven (Rinaldi et al., 6 Jun 2025).

5. Impact on Scientific Discovery and Future Prospects

Non-parametric gravitational imaging has enabled:

• Extension of Sensitivity to New Regimes: Detection of ultralow-mass subgalactic objects at cosmological distances opens tests of galaxy formation and dark matter models far beyond previous capability (Powell et al., 8 Oct 2025).
• Enhanced Population Characterization: Joint mass-redshift reconstructions uncover correlations and evolutionary trends (e.g., positive linear redshift evolution for $m\gtrsim50\,M_\odot$ BHs, consistent quadratic term), informing models of cosmic history and formation channels (Afroz et al., 29 Sep 2025, Ray et al., 2023).
• Flexible and Efficient Data Analysis: Separation of data description from interpretation via robust non-parametric fits allows astrophysical testing of multiple parametric hypotheses without repeated raw data reanalysis, an advance critical for large datasets from future GW catalogs (Fabbri et al., 28 Jan 2025).
• Novel Quantum Metrological Applications: Implementation of quantum imaging for gravity measurement via atom-photon entanglement and atom interferometric curvature probing exemplifies the intersection of atomic physics and gravitational imaging (Cepok et al., 5 Sep 2024, Werner et al., 5 Sep 2024).

A plausible implication is that the further development of non-parametric methods—especially those capable of three-dimensional, time-dependent, and multi-sensor inversions—will deepen understanding of both astrophysical and geophysical processes, resolving features not accessible using constrained parametric frameworks.

6. Summary Table: Representative Non-Parametric Gravitational Imaging Applications

Paper/Method	Main Target	Essential Feature
(Powell et al., 8 Oct 2025) GI lensing	Cosmological halo	Pixel grid correction to lens potential
(Ray et al., 2023, Afroz et al., 29 Sep 2025) GP binning GW pops	BBH populations	Flexible mass-redshift correlations
(Jourde et al., 2014) Resolving kernels	Volcano interior	Hybrid muon-gravimetry kernel inversion
(Caldiero et al., 2023) Level-set inversion	Small body interior	Piecewise-constant density via implicit shapes
(Cepok et al., 5 Sep 2024) Quantum imaging	Local gravity	Atom-photon interferometry phase mapping

These diverse techniques share the property of minimal or functionally flexible assumptions regarding the underlying mass or potential, enabling high-fidelity reconstruction across domains from planetary interiors to cosmological substructures.