White dwarfs in minimal dilatonic gravity

Abstract: We study static, spherically symmetric white dwarfs in minimal dilatonic gravity (MDG) -- a Brans-Dicke theory with fixed coupling $α^2=1/3$ and free Compton length $λ<em>Φ$. Solving the full relativistic structure equations we find that MDG white dwarfs are strictly sub-Chandrasekhar for every $λ</em>Φ$: the maximum mass falls from $1.425\,M_\odot$ in GR to $1.27\,M_\odot$ at $200$ km and $1.09\,M_\odot$ at $500$ km. The ratio by which MDG reduces the maximum mass is robust against the equation of state. Including rigid rotation near mass shedding, consistency with the most massive observed white dwarfs restricts $λ<em>Φ \lesssim 300$ km. An independent gravitational-redshift bound gives $λ</em>Φ \lesssim 720$ km. Because the dilaton mass is density-independent the same coupling produces a percent-level altitude dependence of the Kepler-inferred $GM_\oplus$, constraining it from the Solar-System side. To tackle the problem, we introduce a free-boundary collocation method with a linearized-exterior Robin condition that eliminates the exponential stiffness of shooting methods and gives access to the screened regime in double precision.

• The paper develops a free-boundary collocation method with a linearized exterior Robin condition, eliminating exponential shooting instabilities while achieving approximately 10⁻⁶ relative accuracy at 20 times lower computational cost.
• The paper finds that minimal dilatonic gravity always suppresses white-dwarf maximum masses, reducing the Chandrasekhar-EOS maximum from 1.425 solar masses in GR to 0.996 solar masses at a 700 km dilaton range and approaching a 0.6495 unscreened-mass ratio.
• The paper uses massive-white-dwarf, gravitational-redshift, and geodetic observations to constrain the dilaton range to roughly 300 km, while identifying composition, rotation, screening, and gravity-model independence as important remaining uncertainties.

The model and its motivation

Minimal dilatonic gravity (MDG) is a Brans–Dicke theory with $\omega_{\rm BD}=0$ — fixing the scalar coupling to $\alpha^2=1/3$ — supplemented by a "withholding" cosmological potential $U(\Phi)$ that confines the dilaton $\Phi$ and admits a unique stationary point at the GR value $\Phi=1$ (2608.13236). Linearizing about this minimum turns the dilaton into a Klein–Gordon field of Compton length $\lambda_\Phi$, making MDG a one-parameter extension of GR: only the range of the fifth force can be varied, not its strength. This is a structurally restrictive setting for confronting modified gravity with compact-star data, since every observable depends on a single length scale.

The paper computes, for the first time, fully relativistic white-dwarf structure in this theory. White dwarfs are an attractive probe because Gaia-era samples now test the mass–radius relation at the 6% ($1\sigma$) level via gravitational redshifts in common-proper-motion pairs, independently of any theoretical $M(R)$ relation. The relevant dilaton window, $\lambda_\Phi\sim10^2$–$10^5$ km, is complementary to neutron-star constraints.

Numerical methodology

The static problem is a genuine two-point boundary-value problem: the central value $\alpha^2=1/3$0 is an eigenvalue fixed by requiring the exterior "disphere" solution to decay toward $\alpha^2=1/3$1 at the de Sitter horizon. The traditional shooting approach fails exponentially, since errors in $\alpha^2=1/3$2 are amplified by $\alpha^2=1/3$3; double precision breaks down for $\alpha^2=1/3$4.

The paper's methodological contribution is a free-boundary collocation scheme: the interior is solved as a BVP with $\alpha^2=1/3$5 free, and the exterior is replaced by its linearized Yukawa solution condensed into a Robin condition at the surface. Because the growing mode never enters the computation, the exponential stiffness disappears entirely. The collocation solver reproduces shooting results to $\alpha^2=1/3$6 relative accuracy at roughly 20× lower cost, extends access to $\alpha^2=1/3$7, and converges toward the analytic in-matter equilibrium $\alpha^2=1/3$8 as expected in the deep-screening limit. Verification includes recovery of TOV/Lane–Emden limits to $\alpha^2=1/3$9 and controlled surface-cut truncation errors ($U(\Phi)$0).

Sub-Chandrasekhar maximum masses

The central physical result is that MDG white dwarfs are strictly sub-Chandrasekhar for every $U(\Phi)$1. With the Chandrasekhar EOS ($U(\Phi)$2), the maximum mass falls from $U(\Phi)$3 in GR to:

$U(\Phi)$4 [km]	$U(\Phi)$5 [$U(\Phi)$6]	ratio to GR
100	1.346	0.944
200	1.268	0.889
300	1.201	0.843
500	1.090	0.765
700	0.996	0.699

This sign is opposite to what super-luminous Type Ia supernova phenomenology would require, and contrasts with perturbative $U(\Phi)$7 models that admit both branches depending on the sign of the curvature correction. Physically, the fixed-coupling dilaton adds an attractive Yukawa force within one Compton length, steepening the pressure gradient; part of the deficit is stored in the gravitating exterior disphere, which recovers only 0.02%–1.7% of it for $U(\Phi)$8–$U(\Phi)$9 km. In the unscreened limit the Newtonian reduction is analytically fixed by homology: $\Phi$0 gives $\Phi$1, matching the computed $\Phi$2 at $\Phi$3 km.

A key robustness claim is that the suppression factor $\Phi$4 is nearly EOS-independent: including the Salpeter Coulomb correction shifts absolute GR masses by 2–5% but changes the suppression factor by at most ~0.26 percentage points. Bounds on $\Phi$5 therefore do not hinge on microphysical details.

Observational bounds

Because MDG suppresses masses, the observed population constrains $\Phi$6 directly. The most massive known white dwarf, ZTF J1901+1458 with $\Phi$7–$\Phi$8, sits close to the GR limit; consistency requires $\Phi$9–170 km for that object non-rotating. Allowing rigid rotation near mass shedding (which raises maxima by only 4–6%, computed via a Hartle slow-rotation treatment cross-checked against a Hachisu self-consistent-field code to $\Phi=1$0), the paper quotes a conservative bound of $\Phi=1$1 km. An independent gravitational-redshift bound from the mass–radius test gives $\Phi=1$2 km ($\Phi=1$3), or $\Phi=1$4 km at $\Phi=1$5. A useful discriminator emerges: MDG suppresses the moment of inertia, whereas $\Phi=1$6 gravity enhances it by 30–40% for neutron stars.

Solar-system consistency

Since $\Phi=1$7 is density-independent, there is no chameleon screening: the same coupling operates in Earth orbit. Failure of the shell theorem for a massive scalar makes the Kepler-inferred $\Phi=1$8 altitude-dependent, producing predicted GRACE–LAGEOS inconsistencies of $\Phi=1$9 at $\lambda_\Phi$0 km and $\lambda_\Phi$1 at 700 km — percent-level signals exceeding plausible cross-mission precision for $\lambda_\Phi$2–500 km. Light deflection places no constraint (the Yukawa can be absorbed identically into null geodesics), but perihelion residuals give $\lambda_\Phi$3 km. Notably, laboratory torsion-balance tests with $\lambda_\Phi$4 already require $\lambda_\Phi$5 m, so the stellar-scale dilaton masses used here should be regarded as effective low-density values; the paper's stated purpose is to close the astrophysical window in which MDG could still be visible.

Limitations and open questions

Several caveats are explicit. The composition, finite temperature, and magnetization of individual objects such as ZTF J1901+1458 are not modeled, so absolute maxima serve only as upper limits and the measured mass relies on an assumed mass–radius relation. The Earth-orbit argument is deliberately not converted into a formal bound, because contemporary analyses share GRACE-derived gravity models whose mutual independence has not been established. Rotation is restricted to rigid rotation; differential rotation, which in GR supports substantially super-Chandrasekhar configurations, is left open. The conclusion is specific to the withholding potential: chameleon-screened realizations with density-dependent scalar mass, and Vainshtein-screened theories outside the $\lambda_\Phi$6 family, evade the pincer between white-dwarf and terrestrial constraints entirely.

Conclusion

This work establishes that MDG cannot produce super-Chandrasekhar white dwarfs for any parameter choice, reducing maximum masses by up to 16% (23.5% relative to the same-composition GR value) within the observationally allowed range, and confines the theory to $\lambda_\Phi$7 km through mutually consistent stellar and geodetic probes. Its lasting technical contribution is the stiff-free collocation method with linearized-exterior Robin condition, which transfers directly to screened scalar–tensor models and to the relativistic treatment of partially unscreened extremely-low-mass white dwarfs identified here as the remaining viable population.