PREPRINT

How Close Dark Matter Halos and MOND Are to Each Other: Three-Dimensional Tests Based on Gaia DR2

Yongda Zhu, Hai-Xia Ma, Xiao-Bo Dong, Yang Huang, Tobias Mistele, Bo Peng, Qian Long, Tianqi Wang, Liang Chang, Xi Jin

Submitted on 23 November 2022

Figures are fetched from the INSPIRE database at: https://inspirehep.net/literature/2514356

Rotation curves of the Milky Way, with the predicted ones of the gravitational models compared with the observations. The cyan data points with error bars ($\pm 1\sigma$) are our averaged rotation curve over spatial bins with $\Delta R = 0.5$kpc (note that we increase the bin size at a few large-$R$ bins), based on the data of \citet[][]{eilers_circular_2019,mroz_rotation_2019,chrobakova_gaia-dr2_2020}. The baryonic mass parameters is from \citet{wangMilkyWayTotal2022}. The orange, blue, green, and red curves represent the Newtonian baryon-only, DM, QUMOND, and MOG models, respectively.
Figure 1: Rotation curves of the Milky Way, with the predicted ones of the gravitational models compared with the observations. The cyan data points with error bars (±1σ) are our averaged rotation curve over spatial bins with ΔR=0.5kpc (note that we increase the bin size at a few large-R bins), based on the data of \citet[][]{eilers_circular_2019,mroz_rotation_2019,chrobakova_gaia-dr2_2020}. The baryonic mass parameters is from \citet{wangMilkyWayTotal2022}. The orange, blue, green, and red curves represent the Newtonian baryon-only, DM, QUMOND, and MOG models, respectively.
Radial Jeans-equation ($T_R$) tests of the gravitational models vs. the data at various $(R,z)$ locations, illustrated as a function of $R$ at different altitudes ($|z|$). In every panel, the dashed black line represents the quantities calculated from the data of the entire Gaia$+$LAMOST sample of red clump stars (\citealt{huang_mapping_2020-1}); Dark and light shades show 68\% and 95\% confidence intervals, respectively; The orange, blue, green, and red curves represent the Newtonian baryon-only, DM, QUMOND, and MOG models, respectively.
Figure 2: Radial Jeans-equation (TR) tests of the gravitational models vs. the data at various (R,z) locations, illustrated as a function of R at different altitudes (|z|). In every panel, the dashed black line represents the quantities calculated from the data of the entire Gaia+LAMOST sample of red clump stars (\citealt{huang_mapping_2020-1}); Dark and light shades show 68\% and 95\% confidence intervals, respectively; The orange, blue, green, and red curves represent the Newtonian baryon-only, DM, QUMOND, and MOG models, respectively.
As Figure~\ref{fig:TR-W22}, but showing the vertical Jeans-equation ($T_z$) test results.
Figure 3: As Figure~???, but showing the vertical Jeans-equation (Tz) test results.
$T_R$ test results assuming different tracer's density profiles, based on the entire Gaia$+$LAMOST sample of red clump stars (\citealt{huang_mapping_2020-1}). Three schemes are shown: the total-disk profile (weighted thin$+$thick disks, left panel), thick-disk profile (middle panel), and thin-disk profile (right panel). Denotations are the same as in Figure~\ref{fig:TR-W22}.
Figure 4: test results assuming different tracer's density profiles, based on the entire GaiaLAMOST sample of red clump stars (\citealt{huang_mapping_2020-1}). Three schemes are shown: the total-disk profile (weighted thinthick disks, left panel), thick-disk profile (middle panel), and thin-disk profile (right panel). Denotations are the same as in Figure~.
As Figure~\ref{fig:TR_tracers-W22}, but for $T_z$ test results.
Figure 5: As Figure~, but for test results.
As Figure~\ref{fig:Tz_tracers-W22}, but for $T_z$ test results using only the data of the thin-disk red clump stars chemically selected from the \citet{huang_mapping_2020-1} sample.
Figure 6: As Figure~, but for test results using only the data of the thin-disk red clump stars chemically selected from the \citet{huang_mapping_2020-1} sample.
Radial Jeans-equation ($T_R$) tests of the gravitational models vs. the data at various $(R,z)$ locations, illustrated as a function of $z$ at three radial positions. In every panel, the dashed black line represents the quantities calculated from the data of the entire Gaia$+$LAMOST sample of red clump stars (\citealt{huang_mapping_2020-1}); Dark and light shades show 68\% and 95\% confidence intervals, respectively; The orange, blue, green, and red curves represent the Newtonian baryon-only, DM, QUMOND, and MOG models, respectively. The vertical grey dotted line denotes the spatial resolution limit in the $z$ direction to the field strengths and observed acceleration ($T_R$).
Figure 7: Radial Jeans-equation () tests of the gravitational models vs. the data at various locations, illustrated as a function of at three radial positions. In every panel, the dashed black line represents the quantities calculated from the data of the entire GaiaLAMOST sample of red clump stars (\citealt{huang_mapping_2020-1}); Dark and light shades show 68\% and 95\% confidence intervals, respectively; The orange, blue, green, and red curves represent the Newtonian baryon-only, DM, QUMOND, and MOG models, respectively. The vertical grey dotted line denotes the spatial resolution limit in the direction to the field strengths and observed acceleration ().
As Figure~\ref{fig:TR-z-W22}, but showing the vertical Jeans-equation ($T_z$) test results.
Figure 8: As Figure~, but showing the vertical Jeans-equation () test results.
\QesComm{\textbf{Top row:}} Gravitational potential difference between the Newtonian baryon-only model and other models ($\Delta \Phi = \Phi_{\rm N} - \Phi_{\rm model} $). The left-hand, middle, and right-hand panels are for the QUMOND, DM and MOG cases, respectively. Generally the MOG model is not favored by our Jeans-equations tests. At present it is yet an open question to what degrees DM and MOND, respectively, represent the real gravitational field of the MW. \QesComm{\textbf{Bottom row:} The corresponding field-strength difference between the Newtonian baryon-only model and other models, $\mathbf{g}_\mathrm{model} -\gN $. The direction of the vectors is denoted by arrows, and their magnitude is color coded. Note that around the center the darker the MOG's extra gravity is (in blue and even purple) the weaker is the field strength. }
Figure 9: \QesComm{\textbf{Top row:}} Gravitational potential difference between the Newtonian baryon-only model and other models (). The left-hand, middle, and right-hand panels are for the QUMOND, DM and MOG cases, respectively. Generally the MOG model is not favored by our Jeans-equations tests. At present it is yet an open question to what degrees DM and MOND, respectively, represent the real gravitational field of the MW. \QesComm{\textbf{Bottom row:} The corresponding field-strength difference between the Newtonian baryon-only model and other models, . The direction of the vectors is denoted by arrows, and their magnitude is color coded. Note that around the center the darker the MOG's extra gravity is (in blue and even purple) the weaker is the field strength. }
The ``extra mass'' distribution translated directly from the ``extra potential'' (Figure~\ref{fig:potential}) in terms of the normal Poisson equation. {\bf Left:} The density of the effective DM (namely ``phantom dark matter'') predicted by QUMOND. {\bf Middle:} The density of the DM halo in the fiducial mass model. {\bf Right:} The ``extra-mass'' density of the MOG case. Caution that the MOG's ``extra mass'' is just a mimic in the DM paradigm, and basically useless if not viewed as merely the divergence of the ``extra gravity'' field but interpreted as ``mass'' (see the text for the detail).
Figure 10: The ``extra mass'' distribution translated directly from the ``extra potential'' (Figure~\ref{fig:potential}) in terms of the normal Poisson equation. {\bf Left:} The density of the effective DM (namely ``phantom dark matter'') predicted by QUMOND. {\bf Middle:} The density of the DM halo in the fiducial mass model. {\bf Right:} The ``extra-mass'' density of the MOG case. Caution that the MOG's ``extra mass'' is just a mimic in the DM paradigm, and basically useless if not viewed as merely the divergence of the ``extra gravity'' field but interpreted as ``mass'' (see the text for the detail).
As Figure~\ref{fig:rcsrm-W21}, but for the mass model and observational data based on $R_\odot=8~{\rm kpc}$ and $v_0 = 220~\rm km\,s^{-1}$. The cyan data points with $\pm1\sigma$ error bars are our averaged rotation curve over spatial bins with $\Delta R = 0.5$ kpc (we increase the bin size at large $R$), based on the data compiled by {\tt galkin} \citep[][]{patoGalkinNewCompilation2017}. The baryonic parameters are from the ``Weaker $R_0$ prior'' mass model of \citet{mcmillan_mass_2017}.
Figure 11: As Figure~, but for the mass model and observational data based on and . The cyan data points with error bars are our averaged rotation curve over spatial bins with kpc (we increase the bin size at large ), based on the data compiled by {\tt galkin} \citep[][]{patoGalkinNewCompilation2017}. The baryonic parameters are from the ``Weaker prior'' mass model of \citet{mcmillan_mass_2017}.
As Figure~\ref{fig:TR-W22}, but showing the $T_R$ test results for the ``Weaker $R_0$ prior'' mass model of \citet{mcmillan_mass_2017}. \QesComm{Also plotted is the results based on the RAVE sample (black dots with $\pm1\sigma$ error bars).}
Figure 12: As Figure~, but showing the test results for the ``Weaker prior'' mass model of \citet{mcmillan_mass_2017}. \QesComm{Also plotted is the results based on the RAVE sample (black dots with error bars).}
As Figure~\ref{fig:TR-M17}, but showing the $T_z$ test results.
Figure 13: As Figure~, but showing the test results.