The motivation comes from one of the deepest puzzles in modern physics. The standard model of particle physics, built on the symmetry group SU(3) x SU(2) x U(1), describes the strong, weak and electromagnetic interactions with spectacular success, while general relativity handles gravity. Yet many theorists suspect that this edifice is incomplete. One popular possibility is an extra U(1) symmetry, often called a dark U(1), whose mediator would be an extremely light spin-1 boson resembling a generalized dark photon. Such a particle would generate an extraordinarily feeble new force whose couplings to ordinary matter would involve combinations of baryon number B, lepton number L, or their difference B minus L. A spin-0 mediator could produce a similar effect. Because these forces would couple differently to different materials, they could produce apparent violations of the equivalence principle, the exact identity between inertial and gravitational mass that underlies general relativity and was famously tested by Baron Eötvös more than a century ago.

The sharpest current probe of such effects is the MICROSCOPE satellite mission, which monitored the relative acceleration of two freely falling test masses, one of titanium and one of platinum, aboard a drag-free spacecraft orbiting at an average altitude of roughly 710 kilometers. The mission found no deviation, constraining the Eötvös parameter, which measures the relative difference in accelerations, to a level of a few parts in ten to the fifteen. Translating this null result into limits on the strength of a hypothetical new force, however, requires knowing exactly how the Earth’s extended mass distribution sources that force, and this is where the new mathematics enters.

For a new interaction mediated by a particle of mass m, the force has a finite range lambda equal to the inverse mass in natural units. If the range greatly exceeds the Earth’s radius, the potential outside the planet is essentially the same as that of a point charge at the center. But as the range shrinks toward the satellite altitude, the potential at a given point is increasingly generated by the parts of the Earth closest to it, down to a minimum distance equal to the altitude itself. Fayet shows that the outside potential can always be written as the Yukawa potential of a pointlike source multiplied by a correction factor Phi(x), where x is the ratio of the Earth’s radius to the force range. This factor, always greater than or equal to one, is what he calls the hyperbolic form factor.

The definition is elegantly simple: the hyperbolic form factor is the bilateral Laplace transform of the density distribution, expressible as the average of the hyperbolic cosine of the dot product of the wave vector and position vector, or equivalently as the average of the hyperbolic sine of kr divided by kr over the body. Remarkably, this quantity is related by a duality transformation, or by analytic continuation through the substitution of imaginary for real wave numbers, to the ordinary form factor familiar from scattering theory, which involves the ordinary sine and cosine. The two are two faces of the same underlying function, and the connection extends to a compact formula in which the product of the wave number and the form factor appears as the bilateral Laplace transform of the quantity two-pi times r times the density. An inversion formula even allows one to reconstruct the density profile itself from an analytic continuation of the form factor, closing the loop between structure and potential.

Fayet also introduces a companion concept, the effective density. For each force range, one can ask what uniform sphere would generate the same outside potential as the real, layered Earth. The answer is a density that decreases as the range shortens, sliding from the global average density for very long-range forces down toward the density near the surface for short ranges. This makes physical sense: a short-ranged force is sourced preferentially by the outer shells of the planet, which are less dense than the iron core. The effective density thus encodes, in a single curve, how the sensitivity of an experiment migrates inward or outward as the hypothesized mediator mass changes.

The practical payoff comes from a surprising discovery about the Earth itself. Rather than needing a detailed numerical model with many layers, Fayet finds that remarkably simple analytic density profiles reproduce the form factor of a realistic five-shell Earth model, which distinguishes inner and outer cores, inner and outer mantles, and crust, with shell boundaries at radii of 1121, 3480, 5701, 6341 and 6371 kilometers. A density falling off as one over the radius yields a form factor equal to the square of the hyperbolic sine of x over 2 divided by x over 2, accurate to within about one percent up to x equal to 4. Even better, a hybrid profile combining the one-over-r behavior with a linear decrease gives a closed-form expression that matches the five-shell calculation to within 0.7 percent all the way up to x equal to 64, corresponding to force ranges above 100 kilometers or mediator masses below roughly two times ten to the minus twelve electron-volts. The power series expansion of this analytic expression, beginning with one plus 0.0833 x squared, sits extraordinarily close to the five-shell result, and the tiny differences that do exist are in fact mathematically mandatory, since identical form factors would imply identical densities through the inversion formula.

These approximations matter because the experimental limits on new-force couplings scale with the form factor. For a mediator mass of ten to the minus twelve electron-volts, the coupling limits derived from MICROSCOPE are weakened by a factor of about 34 compared with the massless case. Concretely, the analysis yields upper limits of roughly 3.6 times ten to the minus twenty-four on the magnitude of a coupling to B minus L for a spin-1 mediator, and about 2.6 times ten to the minus twenty-three for a coupling to baryon number alone, with slightly different numbers, differing by a factor of about 1.2, in the spin-0 case. The limits grow rapidly with mediator mass, and the analytic and numerical approaches agree to better than 0.4 percent across the relevant mass range, meaning future analyses can replace layered numerical models with a single compact formula without loss of accuracy. An even simpler approximation, in which the square root of the form factor is just the hyperbolic sine of x over 2 divided by x over 2, suffices to within about 2.3 percent for ranges down to a tenth of the Earth’s radius.

Beyond the immediate application to equivalence-principle tests, the framework has broad reach. The same hyperbolic form factors can be computed for Gaussian density distributions, where they reduce to simple exponentials, and for the electron cloud of a hydrogen atom in its ground state, where the ordinary form factor reproduces the well-known result governing electron scattering. The formalism also illuminates curious resonant behavior: for certain density profiles, the ordinary form factor vanishes at specific values of the wave number, corresponding to internal solutions of the field equations with vanishing outside potential, the static analogue of a stationary wave confined within the sphere. And because the coupling limits depend only weakly on the fine details of the Earth’s deep interior, further refinements of the internal structure are unlikely to change the conclusions. What began as an exercise in redefining a form factor may thus sharpen one of the most sensitive searches for physics beyond the standard model, turning the whole planet into a precisely calibrated instrument for detecting forces too faint to imagine.

Subject of Research: Hyperbolic form factors describing Yukawa potentials of extended bodies and their application to constraining new finite-range forces using Earth's density distribution

Article Title: Hyperbolic form factors for Yukawa interactions, and applications to the Earth

Article References: Fayet, P. (2026). Hyperbolic form factors for Yukawa interactions, and applications to the Earth. The European Physical Journal C, 86(9), Article 1125. https://doi.org/10.1140/epjc/s10052-026-15933-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-15933-4

Keywords: Yukawa potential, hyperbolic form factor, dark photon, equivalence principle, MICROSCOPE mission, new forces, Earth density, dark sector, fifth force, Laplace transform, particle physics, geophysics

The starting point is a fact that has been measured directly both on Earth and across the Solar System: matter dilates time in its surroundings. The canonical description comes from the Schwarzschild metric, the general relativistic solution for the spacetime around an uncharged, non-rotating mass. In that solution, the rate of a clock at radial distance r from a mass is slowed relative to a clock infinitely far away by a factor involving the mass’s Schwarzschild radius, which is proportional to the mass itself. Pound and Rebka measured such dilation in a laboratory stairwell in 1959, and Shapiro delay measurements with the Cassini spacecraft confirmed the effect to exquisite precision. What Petarius and Nemiroff add is a cosmological twist: mass-energy anywhere, even at distances of billions of light years, contributes to the dilation experienced by an observer on Earth.

A crucial subtlety is that gravitational time dilation is a scalar quantity and does not sum the way gravitational forces do. The gravitational pull inside a spherical shell of matter is exactly zero, a classic Newtonian result, yet the time dilation produced by that shell at its center is not zero. This follows from Birkhoff’s theorem, which states that the interior of an empty spherically symmetric shell is flat, described by the Minkowski metric, while the exterior is Schwarzschild. Because the metric is flat inside the shell, there is no gradient of gravitational potential and hence no relative time dilation between two interior points. But compared with an external reference clock, the entire interior sits at a constant, non-zero dilation. It is this global, comparative effect, not the local redshift, that the new work tallies.

The authors build up the calculation with a striking thought experiment. Place the Earth one Schwarzschild radius from a black hole’s center, and time at the Earth effectively freezes compared with a distant observer, an infinite dilation. Now split the black hole in two and place the halves on opposite sides of the Earth. Each fragment has half the mass, so the Earth sits two Schwarzschild radii from each. In the weak-field approximation where the individual potentials simply add, the total dilation is found by multiplying the two factors, and the result is that an external observer would see Earth clocks ticking at exactly half speed, even though the total mass is unchanged. Keep splitting the mass into ever more fragments distributed around the Earth, and the dilation weakens: the more fragmented and distributed the surrounding mass, the smaller the effect. In the limit of a thin spherical shell built from infinitely many pieces, the cumulative factor converges to an exponential form, exp of minus the shell’s effective Schwarzschild radius over twice its distance, which is precisely the expression that arises independently in the Newtonian gauge of cosmological perturbation theory. The authors are careful to note this agreement is a consistency check on their multiplicative approximation, not a derivation of it.

With that machinery in place, the team first considers a purely Newtonian universe filled uniformly with mass. There, the amount of mass enclosed grows with the cube of distance while the potential contribution of each shell falls only as the inverse of distance, so the cumulative gravitational potential, and with it the cumulative time dilation, diverges as the considered radius grows to infinity. This divergence is the mathematical heart of the paper’s central puzzle. The authors point out a famous coincidence that emerges if the divergence is capped at the radius of the observable universe: using the present critical density and a comoving radius of about 46.5 billion light years, the enclosed mass has a Schwarzschild radius roughly equal to that radius, a coincidence long noted in discussions of whether our universe could be the interior of a black hole. Taken at face value in that capped scenario, the cumulative dilation factor would be on the order of ten to the minus four, meaning cosmic mass-energy alone would slow Earth clocks by roughly that fraction relative to a hypothetical empty reference.

The relativistic treatment is more subtle. In Friedmann-Lemaître-Robertson-Walker cosmologies, the authors work in the Newtonian gauge, where the metric component g00 takes the form that yields a clock rate of approximately one plus the gravitational potential over c squared, matching the weak-field expansion of the Schwarzschild result. They model the universe as a series of concentric comoving shells centered on the Earth and compute the dilation of each shell relative to the same FLRW universe with that shell emptied of mass-energy, then multiply the factors, or equivalently add their logarithms, across all shells out to the particle horizon. Because every location in a homogeneous universe sees the same total dilation, the calculation is inherently comparative: there is no external clock at infinity to reference, so the filled-versus-empty-shell construction is what isolates the gravitational contribution. The authors stress that this is exploratory basic science rather than a measurement, and that general relativity’s non-linearity means the summation is not exact.

The headline result is that the fate of the cumulative dilation depends dramatically on what the universe is made of. When plotted as a function of redshift, the cumulative time dilation converges to finite values in universes dominated by matter, by radiation, or by the concordance mixture of roughly 68.5 percent dark energy, 31.5 percent matter, and a trace of radiation that matches current Planck satellite parameters. The reason is geometric: in those cosmologies, the comoving distance to a shell flattens out at high redshift, so integrating over all redshifts only ever includes a finite amount of mass-energy. The lone exception is a universe continually dominated by dark energy alone, where the comoving distance itself diverges with redshift, allowing shells from arbitrarily far away to enter the sum and driving the cumulative dilation to infinity. Meanwhile, as a function of comoving distance rather than redshift, the dilation generally diverges in several flat cosmologies, echoing the Newtonian result.

The authors also extend the framework to inhomogeneous universes by redefining the baseline. In a perfectly uniform universe, any overdensity slows time relative to the mean while any underdensity, a void, accelerates it. They show that dividing each spherical shell into n pieces, concentrating all its mass into one dense clump and leaving the rest empty, produces a net relative time dilation that grows as the universe becomes more concentrated, with the curves for different concentration levels converging toward a common behavior at high redshift. This connects their work to the classic Rees-Sciama Swiss cheese picture, in which masses surrounded by voids produce their own characteristic redshift signatures, though the present calculation focuses on the global dilation at the Earth rather than redshifts along specific lines of sight.

Why does any of this matter beyond conceptual housekeeping? The motivation traces back to tests of the weak equivalence principle using photons from distant gamma-ray bursts and fast radio bursts. Those tests compare the arrival times of photons of different energies that have crossed billions of light years of gravitational potential, and they are limited by how much of the universe’s mass distribution is included in the calculation. Many previous analyses used only the potential of the Milky Way, or extended to the Laniakea supercluster, precisely because including more distant mass seemed to invite divergences that researchers worked around rather than confronted. Petarius and Nemiroff argue that facing those divergences head-on is a necessary stepping stone: cumulative time dilation, unlike gravitational force, accumulates monotonically with enclosed mass-energy and is unusually sensitive to the distant universe. The work also distinguishes observable gravitational time dilation from a kind of raw potential that persists even when redshift effects between two locations cancel, raising the possibility that if different forms of matter or radiation respond to that raw potential differently, the large-scale universe itself could become a laboratory for testing the equivalence principle. The authors caution that their infinities may not be physical, since they are sensitive to conditions in the early universe where the underlying assumptions of FLRW cosmology, and perhaps general relativity itself, may break down. But by mapping exactly where the divergences appear and where they vanish, the study turns a long-ignored embarrassment into a tractable question, one that future measurements with cosmic transients may finally be able to answer.

Subject of Research: Cumulative gravitational time dilation from cosmological mass-energy distributions in Newtonian and FLRW universe models

Article Title: Cumulative gravitational time dilations across the universe

Article References: Petarius, R., III, & Nemiroff, R. J. (2026). Cumulative gravitational time dilations across the universe. Astrophysics and Space Science, 371(10), Article 113. https://doi.org/10.1007/s10509-026-04645-6

Image Credits: AI Generated

DOI: 10.1007/s10509-026-04645-6

Keywords: gravitational time dilation, general relativity, Schwarzschild metric, FLRW cosmology, Newtonian gauge, cosmological redshift, dark energy, equivalence principle, gamma-ray bursts, fast radio bursts, comoving distance, cosmology