Peculiar dark matter halos inferred from gravitational lensing as a manifestation of modified gravity

Abstract

If modified gravity holds, but the weak lensing analysis is done in the standard way, one finds that dark matter halos have peculiar shapes, not following the standard Navarro-Frenk-White profiles, and are fully predictable from the distribution of baryons. Here we study in detail the distribution of the apparent dark matter around point masses, which approximate galaxies and galaxy clusters, and their pairs for the QUMOND MOND gravity, taking an external gravitational acceleration $g_e$ into account. At large radii, the apparent halo of a point mass $M$ is shifted against the direction of the external field. When averaged over all lines-of-sight, the halo has a hollow center, and denoting the by $a_0$ the MOND acceleration constant, its density behaves like $ρ(r)=\sqrt{Ma_0/G}/(4πr^2)$ between the galacticentric radii $\sqrt{GM/a_0}$ and $\sqrt{GMa_0}/g_e$, and like $ρ\propto r^{-7}G^2M^3a_0^3/g_e^5$ further away. Between a pair of point masses, there is a region of a negative apparent dark matter density, whose mass can exceed the baryonic mass of the system. The density of the combined dark matter halo is not a sum of the densities of the halos of the individual points. The density has a singularity near the zero-acceleration point, but remains finite in projection. We compute maps of the surface density and the lensing shear for several configurations of the problem, and derive formulas to scale them to further configurations. In general, for a large subset of MOND theories in their weak field regime, for any configuration of the baryonic mass $M$ with the characteristic size of $d$, the total lensing density scales as $ρ({\vec{x}})=\sqrt{Ma_0/G}d^{-2}f\left(\vecα,\vec{x}/d,g_ed/\sqrt{GMa_0}\right)$, where the vector $\vecα$ describes the geometry of the system. Distinguishing between QUMOND and cold dark matter seems possible with the existing instruments.

Figures (26)

• Figure 1: Distribution of the PDM density around a point mass in a homogeneous external gravitational field. The mass of the point source, located in the center of the coordinate system, is $10^{11}$ M$_\sun$. The external field has the value of $0.02\,a_0$ and points up. The left and right color scales indicate the positive and negative densities of PDM, respectively. The dotted half-circle marks the theoretical estimate of the radius $r_\mathrm{ef}\xspace$ below which the effects of the external field are negligible.
• Figure 2: Stacked point masses in randomly oriented external fields. From top to bottom: Row 1: Density of PDM. Row 2: Surface densities obtained by the projection of the volume densities above. Row 3: Ratio of the surface density of the PDM halo and the surface density of the approximating isothermal sphere. Row 4: Tangential shear. The different lines correspond to the indicated various intensities of the external field. The small horizontal line on each curve marks the corresponding radius $r_\mathrm{ef}$. For the given models, 1 kpc corresponds to 1.03.
• Figure 3: Detail of the central profile of the surface density of PDM.
• Figure 4: Comparison of a PDM halo to an NFW halo. The point mass has a baryonic mass of $4\times 10^{10}\,M_\sun$ and lies at the distance of 200 Mpc. Row 1: Difference of densities (the curves for $g_e = 0.005\,a_0$ and $0.01\,a_0$ nearly coincide). Row 2: Difference of surface densities. Row 3: Difference of tangential shears. The sources are assumed to be at infinity. The vertical dashed line indicates the MOND transitional radius $r_\mathrm{M}$.
• Figure 5: Distribution of the PDM around two point masses (zero external field assumed) with various mass ratios indicated above each column. The mass of $M_1$ is $10^{11}\,M_\sun$ and the two point masses are 200 kpc apart. The mass ratio is the same for each column. The top row shows the whole system, in the bottom row we can see magnified the region of the negative PDM. Note that the vertical ranges of the plots in the bottom row are different for each plot. In each plot, the $r$ and $z$ axes have the same scale.