Bayesian Inference of Gravity through Realistic 3D Modeling of Wide Binary Orbits: General Algorithm and a Pilot Study with HARPS Radial Velocities

TL;DR

This work develops a fully general Bayesian framework to infer 3D wide-binary orbits and a gravity anomaly parameter $\Gamma = \log_{10} \sqrt{G/G_{\rm N}}$ from precise 3D relative positions and velocities, addressing degeneracies between orbital geometry and gravity. By applying the method to 32 Gaia binaries with HARPS RVs, the study finds Newtonian gravity is consistent in the high-acceleration regime but is disfavored in the low-acceleration regime, where a boosted gravity signal emerges and the true anomaly distribution $\Delta\phi$ strongly supports non-Newtonian gravity via Bayesian model comparison ($\Delta\mathrm{BIC}_{\Delta\phi}>10$). The consolidated $\Gamma$ distributions show $g_N>10^{-9}$ m s$^{-2}$ align with Newton, while $g_N<10^{-9}$ m s$^{-2}$ imply a significant deviation, driven largely by one system (#24) that would be unbound under Newtonian gravity but compatible with MOND-like scenarios. The pilot demonstrates the potential of the Bayesian 3D approach to quantify gravity at very low accelerations, with more decisive tests possible as larger, higher-quality samples become available.

Abstract

When 3D relative displacement $\mathbf{r}$ and velocity $\mathbf{v}$ between the pair in a gravitationally-bound system are precisely measured, the six measured quantities at one phase can allow elliptical orbit solutions at a given gravitational parameter $G$. Due to degeneracies between orbital-geometric parameters and $G$, individual Bayesian inferences and their statistical consolidation are needed to infer $G$ as recently suggested by a Bayesian 3D modeling algorithm. Here I present a fully general Bayesian algorithm suitable for wide binaries with two (almost) exact sky-projected relative positions (as in the Gaia data release 3) and the other four sufficiently precise quantities. Wide binaries meeting the requirements of the general algorithm to allow for its full potential are rare at present, largely because the measurement uncertainty of the line-of-sight (radial) separation is usually larger than the true separation. As a pilot study, the algorithm is applied to 32 Gaia binaries for which precise HARPS radial velocities are available. The value of $Γ\equiv \log_{10}\sqrt{G/G_{\rm N}}$ (where $G_{\rm N}$ is Newton's constant) is $-0.002_{-0.018}^{+0.012}$ supporting Newton for a combination of 24 binaries with Newtonian acceleration $g_{\rm N}>10^{-9}$m\,s$^{-2}$, while it is $Γ=0.134_{-0.036}^{+0.056}$ (thermal prior on eccentricity) or $0.115_{-0.028}^{+0.048}$ (flat prior) for 8 binaries with $g_{\rm N}<10^{-9}$m\,s$^{-2}$ representing $>3.7σ$ discrepancy with Newton. Moreover, the inferred orbital true anomalies clearly favor modified gravity over Newton with the difference of Bayesian information criterion $>10$. The pilot study demonstrates the potential of the algorithm in measuring and testing gravity at low acceleration with future samples of wide binaries.

Figures (11)

• Figure 1: A general 3D geometry of observing an elliptical orbit of the relative motion between the stars of a binary system whose plane makes an inclination angle of $i$ with the sky plane. Two angles $i$ and $\theta$ can represent any arbitrary orientation of the observer's direction (the $z^\prime$-axis) with respect to the $z$-axis of the orbital plane.
• Figure 2: This figure shows an example of the posterior PDFs of the parameters with $\Gamma$ free (generalized gravity). This binary is one of the few that have relatively well-measured radial separations. The fitted parameters in general have broad ranges. See Tables \ref{['tab:prmt_newton']} and \ref{['tab:prmt_general']} for the parameter values of all the binaries in Newtonian and generalized gravity.
• Figure 3: The upper and lower rows show distributions of the explicit orbit and inclination parameters as well as the mass parameter $f_M$ in Newtonian and generalized gravities. Each histogram represents the combination of all individual PDFs of the 32 binaries. The distribution of the phase $\Delta\phi(=\phi-\phi_0)$ in Newtonian gravity is in tension with the expectation in contrast to that in the generalized gravity, as the large value of $\Delta\rm{BIC}_{\Delta\phi}=\rm{BIC}_{\Delta\phi,{\rm Newton}}-\rm{BIC}_{\Delta\phi,{\rm general}}=65.3$ demonstrates. This tension is largely contributed by the subsample of 9 binaries with $g_{\rm N}<10^{-9}$ m s$^{-2}$ (indicated by red color), which alone gives $\Delta\rm{BIC}=25.5$. The posterior distribution of $\log_{10}f_M$ in Newtonian gravity indicates a mild asymmetry due to a few systems (see the text).
• Figure 4: This figure exhibits the individual PDFs of $\Delta\phi$ for all 32 binaries from the Bayesian results in the Newtonian and generalized gravity models whose combined PDFs of $\Delta\phi$ are shown in the middle column of Figure \ref{['fig:distribution']}. MLE values (see Appendix \ref{['sec:MLEtables']}) are also indicated by dashed lines. The low-acceleration ($g_{\rm N}<10^{-9}$ m s$^{-2}$) wide binaries are marked by red numbers. Two (#3 and #24) from them have the PDFs of $\Delta\phi$ very close the periastron only in the Newtonian case.
• Figure 5: Similar to Figure \ref{['fig:distribution']}, but for the Bayesian results with flat/no priors on $e$, $i$, and $\Delta\phi$. Here the generalized (i.e., allowing each wide binary to have its own PDF of $\Gamma$) gravity model of Figure \ref{['fig:distribution']} is replaced by the MOND-type toy model where $G$ is fixed for each binary, but it has a value of $G_{\rm N}$ or $1.4G_{\rm N}$ depending on $g_{\rm N}$ (see the text for details). Thus, in this case the parameters $e$, $i$, and $\Delta\phi$ are determined by the data of $\mathbf{r}$ and $\mathbf{v}$ alone.