Testing dark matter and modifications to gravity using local Milky Way observables

Galactic rotation curves are often considered the first robust evidence for the existence of dark matter. However, even in the presence of a dark matter halo, other galactic-scale observations, such as the baryonic Tully-Fisher relation and the radial acceleration relation, remain challenging to explain. This has motivated long-distance, infrared modifications to gravity as an alternative to the dark matter hypothesis as well as various dark matter theories with similar phenomenology. In general, the standard lore has been that any model that reduces to the phenomenology of modified Newtonian dynamics (MOND) on galactic scales explains essentially all galaxy-scale observables. We present a framework to test precisely this statement using local Milky Way observables, including the vertical acceleration field, the rotation curve, the baryonic surface density, and the stellar disk profile. We focus on models that predict scalar amplifications of gravity, i.e., models that increase the magnitude but do not change the direction of the gravitational acceleration. We find that models of this type are disfavored relative to a simple dark matter halo model because the Milky Way data requires a substantial amplification of the radial acceleration with little amplification of the vertical acceleration. We conclude that models which result in a MOND-like force struggle to simultaneously explain both the rotational velocity and vertical motion of nearby stars in the Milky Way.

Figure 1

Illustrative plot presenting the potential ability of dark matter and a MOND-like model to predict the measured values of the circular velocity at the Solar position, $v_c(R_0)$, and the vertical velocity dispersion at some reference height above the midplane, ${\sigma }_z(z_{\mathrm{ref}})$. For this figure, the baryonic profile has been fixed such that it is consistent with measurements of the stellar disk and bulge, as well as the gas disk. The black point marks the measurements taken from Refs. [22] and [30] (see text for details). The blue solid curve is the prediction for dark matter with a spherical NFW profile and the dashed blue curve is the prediction for dark matter with a slightly prolate profile. The red solid curve is the prediction for a MOND-like force using Eqs. (2) and (3). The MOND-like model requires equal enhancements of the radial and vertical accelerations and cannot simultaneously fit both. On the other hand, both dark matter scenarios are able to accommodate the measurements. The prolate halo does slightly better because it increases the enhancement of the radial acceleration relative to the vertical acceleration, a feature that is the exact opposite of the MOND-like behavior.

Figure 2

Marginalized two-parameter correlations for the disk and bulge parameters $h_{*,R},M_{*,\mathrm{disk}}$, and $M_{*,\text{bulge}}$. Shaded regions correspond to 68% and 95% containment regions for the MOND-like model (red) and dark matter (blue) models. Left panel: Correlation between stellar disk mass, $M_{*,\mathrm{disk}}$, and scale radius, $h_{*,R}$. The black dashed curve shows the range of parameters consistent with the measured stellar surface density ${\mathrm{\Sigma }}_{*,\mathrm{obs}}^{1.1}=31.2M_{\odot }{\mathrm{pc}}^{-2}$ [54], as in Eq. (18). Central panel: Correlation between stellar bulge mass, $M_{*,\text{bulge}}$, and stellar disk mass, $M_{*,\mathrm{disk}}$. The red dashed curve corresponds to the correlation that reproduces the measured local circular velocity assuming Eq. (18) and a constant value of $\nu (a_{\mathrm{N}}/a_0)=1.3$ for a MOND-like model. The blue dashed curve is the analogous estimate for the dark matter model using ${\tilde{\rho }}_{\mathrm{DM}}=2.8M_{\odot }{\mathrm{pc}}^{-3}$ and $\alpha =0.5$. Right panel: Correlation between $h_{*,R}$ and $M_{*,\text{bulge}}$. The red and blue dashed curves are analogous to those in the central panel. The open/filled black points correspond to two separate bulge measurements: $M_{*,\mathrm{bulge},\mathrm{obs}}=1.50\pm 0.38M_{\odot }$ from microlensing observations [60] (filled black circle) and $M_{*,\mathrm{bulge},\mathrm{obs}}=0.62\pm 0.31M_{\odot }$ (open black circle) from photometric observations [61]. We take $h_{*,R,\mathrm{obs}}=2.6\pm 0.5\mathrm{kpc}$ [40] for both. Compared to the MOND-like model, the dark matter region has greater overlap with the favored parameter space.

Figure 3

Posterior distributions for the local acceleration amplification, $\nu (a_N(R_0))$, for two interpolation functions. The red histogram corresponds to a Taylor expansion interpolation function and a prior of $\nu (a_{\mathrm{N}})>1$. The grey histogram corresponds to the RAR interpolation function. The prior for the RAR function is plotted in the inset, together with the posterior of $a_0$, which is a free parameter of this scan. The posteriors of both interpolation functions push to low values of acceleration amplification, close to unity.

Figure 4

Discrepancy of each walker step in the MCMC with measurements of the stellar disk scale length and bulge mass for the dark matter (blue lines) and MOND-like (red lines) models. The discrepancy is calculated taking the scale length of Eq. (19) and $M_{*,\mathrm{bulge},\mathrm{obs}}=1.50\pm 0.38M_{\odot }$ from Ref. [60] (solid lines) and $M_{*,\mathrm{bulge},\mathrm{obs}}=0.62\pm 0.31M_{\odot }$ from Ref. [61] (dashed lines). In both cases, the distribution tends to peak at smaller discrepancies for the dark matter model than the MOND-like model. However, both distributions contain very long tails toward high discrepancies. Additionally, we note that the sharp drops in the distributions that can be seen near $\sim 4\sigma$ ($\sim 2.5\sigma$) in the solid (dashed) lines are due to the prior $M_{*,\mathrm{bulge}}>0$.

Figure 5

Correlation of observables that highlight the main kinematic difference between the MOND-like and dark matter scenarios. The axes correspond to the ratio of the observed acceleration to the Newtonian (baryon-only) acceleration in the vertical and radial directions. The accelerations are evaluated at the Solar radius, $R_0$. Models satisfying Eq. (2) identically predict that observations must satisfy $a_z/a_{\mathrm{N},z}=a_R/a_{\mathrm{N},R}$ (red line), which is not satisfied by a spherically symmetric dark matter halo. The blue shaded regions correspond to 68 and 95% containment for the dark matter analysis at reference positions $z=300$ and 1200 pc.

Figure 6

Marginalized two-parameter correlations for the parameters of the dark matter and MOND-like models analyzed in this study. Shaded regions correspond to 68% and 95% containment. Left panel: Parameters of the MOND-like model. The analysis very tightly constrains the model-independent values ${\nu }_0$ and ${\nu }_1$ (red shaded region). Given any interpolation function, its consistency with the results of this analysis can be tested by computing ${\nu }_0$ and ${\nu }_1$ from Eq. (3), taking $a_{\mathrm{N},\mathrm{ref}}$ to be distributed according to the inset distribution shown, and then comparing with the red shaded region. As an example we perform this procedure for the RAR interpolation function defined in Table 1, which is shown by the gray shaded region. Our analysis disfavors this interpolation function. Right panel: Parameters of the dark matter model using a modified NFW profile with inner profile slope parameter, $\alpha$, and local density, ${\rho }_{\mathrm{DM}}(R_0)$. The fully marginalized value of the local density, ${\rho }_{\mathrm{DM}}(R_0)=0.29\pm 0.06\mathrm{GeV}{\mathrm{cm}}^{-2}$, is consistent with previous measurements. The posterior values of $\alpha$ are consistent with a standard cusped NFW profile as well as a more cored profile with $\alpha$ close to zero. In particular, we find that $\alpha <1.1$ at 90% confidence.

Figure 7

A schematic diagram illustrating the geometry described in the text. $m_{\mathrm{pert}}$ is a small mass perturbation to a smooth background acceleration field, ${\mathit{a}}_{\mathrm{BG}}$, which is approximately constant in the vicinity of the perturbation. Choosing the coordinate system as depicted in the diagram, the MONDian potential around the perturbation is given by Eq. (a6).

Figure 8

The posterior distributions and two-parameter correlations for the MOND-like model parameters.

Figure 9

Same as Fig. 8 except for the dark matter model.

Figure 10

Autocorrelation lengths ($\widehat{\tau }$) computed as a function of the number of post burn-in MCMC samples for the MOND-like (left) and dark matter (right) analyses. The black dashed line representing $\widehat{\tau }=N/50$ is shown as a benchmark of convergence.