[Paper Review] Constraints on ADM tetrad gravity parameter space from S2 star in the center of the Galaxy and from the Solar System

Mattia Villani|arXiv (Cornell University)|Feb 24, 2015

Geophysics and Gravity Measurements5 citations

TL;DR

This paper constrains the parameter space of ADM tetrad gravity—a Hamiltonian reformulation of General Relativity that interprets dark matter as an inertial effect from spacetime slicing—using orbital data from the S2 star near the Galactic Center and planetary perihelion precessions in the Solar System. It derives tight bounds: $4.2 \times 10^{-4}\ \text{AU} \lesssim \delta \lesssim 4.6 \times 10^{-4}\ \text{AU}$ for the Yukawa coupling strength and $\mu \lesssim 3.5 \times 10^{-6}\ \text{AU}^{-1}$ for the inverse length scale, indicating minimal deviation from standard gravity.

ABSTRACT

ADM tetrad gravity is an Hamiltonian reformulation of General Relativity which gives new insight to the Dark Matter Problem. We impose constraints on the parameter space of ADM tetrad gravity with a Yukawa-like ansatz for the trace of the extrinsic curvature of the 3D hypersurfaces by fitting the orbit of the S2 star around the Black Hole in the Galactic center and using the perihelia of some of the planets of the Solar System. We find very thight constraints on the \emph{strength} of the coupling, $4.2 \, imes \, 10^{-4} \, ext{AU}\,\lesssim \, δ\, \lesssim \, 4.6 \, imes \, 10^{-4} \, ext{AU}$, and an upper limit for the (inverse) scale length, $μ\, \lesssim \, 3.5 \, imes \, 10^{-6} \, ext{AU}^{-1}$.

Motivation & Objective

• To constrain the free parameters of ADM tetrad gravity, a Hamiltonian formulation of General Relativity that reinterprets dark matter as a geometric inertial effect.
• To test whether the 0.5 Post-Newtonian correction from the trace of extrinsic curvature in ADM tetrad gravity improves orbital fits compared to Newtonian gravity.
• To combine constraints from the S2 star's orbit around the Galactic Center black hole and planetary perihelion precessions in the Solar System.
• To derive upper limits on the Yukawa coupling strength $\delta$ and inverse scale length $\mu$ in a phenomenological ansatz for the extrinsic curvature.

Proposed method

• Proposes a Yukawa-like ansatz for the first-order trace of extrinsic curvature: ${}^{3}\widetilde{\mathcal{K}}_{(1)} = c t \delta \frac{1}{r} \exp(-\mu r)$, with linear time dependence.
• Uses a 4th-order Runge-Kutta numerical integration to simulate the S2 star's orbit under the modified equation of motion derived from ADM tetrad gravity.
• Performs a $\chi^2$ minimization between simulated and observed S2 star positions, with errors estimated via the Fisher matrix method.
• Derives a formula for the 0.5 PN correction to orbital precession, expressed as an integral over radial coordinate using the variable $z = (1 + e \cos\varphi - 1)/e$.
• Applies the precession formula to observed perihelion shifts of Mercury, Venus, Earth, Mars, and Saturn to constrain $\delta$ and $\mu$.
• Combines constraints from both S2 star fitting and Solar System precession data to derive a joint, tighter parameter space.

Experimental results

Research questions

• RQ1 Can ADM tetrad gravity, with a Yukawa-like correction from the trace of extrinsic curvature, provide a better fit to the S2 star's orbit than Newtonian gravity?
• RQ2 What are the upper bounds on the Yukawa coupling strength $\delta$ and inverse scale length $\mu$ that are consistent with planetary perihelion precession data?
• RQ3 How do constraints from the Galactic Center S2 star and the Solar System combine to further restrict the parameter space of ADM tetrad gravity?
• RQ4 Does the 0.5 PN correction in ADM tetrad gravity lead to measurable deviations in orbital precession that are consistent with observational limits?

Key findings

• The fit of ADM tetrad gravity to the S2 star orbit yields a reduced $\chi^2$ of 1.516890, slightly better than the Newtonian Keplerian fit ($\chi^2 = 1.5477$), but only marginally so.
• The combined constraints from the S2 star and planetary precessions yield a tight bound on the Yukawa coupling strength: $4.2 \times 10^{-4}\ \text{AU} \lesssim \delta \lesssim 4.6 \times 10^{-4}\ \text{AU}$.
• An upper limit of $\mu \lesssim 3.5 \times 10^{-6}\ \text{AU}^{-1}$ is established for the inverse scale length of the Yukawa potential.
• Mercury's perihelion precession provides the tightest constraint among the planets, followed by Venus and Earth, while Jupiter's large error renders it ineffective for constraint.
• The final allowed parameter space is significantly reduced, with the combined constraint forming a narrow, dark green region in the parameter space, as shown in Figure 4.
• The results indicate that ADM tetrad gravity must closely resemble standard General Relativity in its low-energy limit, with only very small deviations allowed by current observations.