The framework, developed by Gustav Stålhammar of St. Erik Eye Hospital and the Karolinska Institutet in Stockholm, rests on one deceptively simple hypothesis. Ordinary matter is modeled as a collection of localized emitters, each radiating waves of an additional scalar field with random, mutually uncorrelated phases. In neutral matter, a bound atom acts as one emitter; in fully ionized plasma, each free charged particle counts as an independent source. No individual emitter is observable, and the frequencies involved are arbitrary. What matters is only the statistics of the summed field, which by the central limit theorem behaves as a Gaussian random field regardless of the underlying spectrum.

When the waves from roughly 10^57 emitters in a star like the Sun are added together at any point in space and the intensity of the total field is averaged over time, three kinds of terms appear, and each maps onto one aspect of gravitation. The first is the set of self-terms: each wave multiplied by itself, always positive, adding incoherently like the masses of the sources. The framework identifies ordinary Newtonian gravity with this stationary part, taking Newton’s constant from measurement and leaving the tensor sector of gravity exactly that of general relativity. The hypothesized waves are not gravitational waves; they belong to an additional scalar degree of freedom of the kind long studied in scalar-tensor theories.

The second and third sectors arise from the cross-terms in the double sum. Waves arriving from all matter within the causal horizon combine into a fluctuating background whose variance sets a characteristic acceleration scale. Using the cosmic mean baryon density multiplied by the Hubble length, and a geometric factor of 3 over 8π, the framework derives an acceleration scale g_w of about 7.9 × 10^-11 meters per second squared, fixed entirely by measured cosmological parameters rather than fitted. This is tantalizingly close to the acceleration scale near cH_0, the speed of light times the Hubble constant, at which galaxy dynamics are known to depart from Newtonian expectations, a coincidence that modified Newtonian dynamics, or MOND, has long treated as fundamental but never explained.

The crucial new ingredient is the interference between the locally generated field and this cosmic background. The signed average of the cross-term is zero, so nothing would happen if the medium responded to the instantaneous signed product. The framework’s central assumption, labeled A2, is that the material medium responds instead to the product of the two components’ envelope amplitudes, their absolute magnitudes, independent of phase. Under this rectified response, the mean of the product for two independent Gaussian envelopes is exactly π/2 times the product of their standard deviations, a result verified by Monte Carlo simulation to within 0.12 percent. The result is an additional attractive acceleration proportional to the geometric mean of the Newtonian acceleration and the cosmic scale, directed along the local field, precisely the square-root behavior that characterizes dark-matter phenomenology in the weak-field regime.

Three phenomenological functions modulate this term. A screening factor suppresses the effect where a system’s own column density exceeds the cosmic source column, which is what keeps the Solar System clean. A transport saturation factor describes how the rectified response accumulates over a density-dependent length anchored at the local Galactic disk value. And an ionized-gas enhancement reflects the emitter-counting rule: ionization multiplies the number of independent emitters per unit mass, so plasma environments should feel a stronger effect. Three constants are fixed in advance from cosmology and local Galactic measurements; only one coefficient, the plasma coefficient γ_p, is calibrated on data, using the X-COP sample of twelve massive clusters, and then frozen for every other test.

The performance on galaxy rotation curves is the headline result. Applied with no galaxy-fitted parameters whatsoever to 3,187 weak-field points in 175 galaxies from the SPARC database, the framework achieves a root-mean-square error of 0.228 dex in log-acceleration, with a 95 percent confidence interval of 0.204 to 0.253 dex. For comparison, general relativity with baryons alone yields 0.398 dex, MOND at its canonical acceleration constant yields 0.270 dex, and MOND with the constant re-fitted to the same selection yields 0.213 dex. The framework thus outperforms canonical MOND and baryons-only gravity while landing between the two MOND variants, all without a single parameter tuned to galaxies. Notably, its prediction shifts by only 0.005 dex when the stellar mass-to-light assumptions are changed, whereas the fitted MOND constant moves by a factor of 2.4 under the same variation.

On the cluster side, the story is more mixed but structurally interesting. Leave-one-cluster-out validation on X-COP yields an error of just 0.072 dex, and the calibrated plasma coefficient of 0.69 lies within a factor of 1.6 of an independent, parameter-free ionization estimate of 1.08. What distinguishes the framework from MOND here is not fit quality but internal consistency: MOND requires an acceleration constant 5.5 times larger to fit clusters than to fit galaxies, whereas the same kernel with the same fixed constants spans both regimes. However, applied unchanged to 22 clusters from the eROSITA sample, the framework roughly halves the central shortfall of baryons alone but retains a radially structured under-prediction of 0.435 dex overall, comparable in magnitude to MOND’s own central failure. Four of five dwarf spheroidal galaxies are also under-predicted, bracketing the model’s domain of validity at low baryonic density.

The framework survives the obvious local tests through its own screening structure. Predicted fractional anomalies in the Solar System range from 8.5 × 10^-14 at Mercury to 4.0 × 10^-8 at Neptune, between three and more than eight orders of magnitude below the Cassini sensitivity limit, with a steep radial increase that future ephemeris measurements at Neptune-to-Kuiper distances could probe directly. Wide-binary predictions depend on a stated transport convention for point-pair systems; under the ambient-disk reading, the predicted velocity excesses remain below current Gaia sensitivity, but measurements beyond 100 kAU would discriminate sharply between the two readings. As a conditional cosmological check, a matter-like scalar background reproduces the baryon-acoustic-oscillation sound horizon to within 0.04 percent under a matched drag-epoch prescription, though the full cosmological sector remains an open problem.

The author is explicit about what remains unproven. The rectified response of Assumption A2 has no microscopic derivation, and the Hubble-length causal column is an imposed simplification. The screening and transport functions are phenomenological, with demonstrated functional-form freedom at the 0.01 to 0.06 dex level. Yet the framework makes concrete, falsifiable predictions: the plasma coefficient must re-calibrate to the same value on independent cluster samples, ultra-wide-binary velocities beyond 100 kAU must match the stated curve, matched-radius lensing and dynamical masses must agree, and the scalar background’s energy density must evolve like non-relativistic matter through the drag epoch. All numerical results were produced with a single released analysis pipeline, making every modeling choice auditable. Whether the universe really generates its missing gravity from the interference statistics of ordinary matter is now a question that measurement, not philosophy, will decide.

Subject of Research: An emergent scalar-field model that derives Newtonian gravity and dark-matter phenomenology from random-phase waves emitted by ordinary matter

Article Title: Newtonian gravity and dark-matter phenomenology from an emergent scalar field sourced by ordinary matter

Article References: Stålhammar, G. (2026). Newtonian gravity and dark-matter phenomenology from an emergent scalar field sourced by ordinary matter. Frontiers in Astronomy and Space Sciences, 13, Article 1904894. https://doi.org/10.3389/fspas.2026.1904894

Image Credits: AI Generated

DOI: 10.3389/fspas.2026.1904894

Keywords: dark matter, scalar field, modified gravity, MOND, galaxy rotation curves, galaxy clusters, cosmology, SPARC, X-COP, baryonic matter, gravitation, emergent gravity