Strong lensing by fermionic dark matter in galaxies

It has been shown that a self-gravitating system of massive keV fermions in thermodynamic equilibrium correctly describes the dark matter (DM) distribution in galactic halos (from dwarf to spiral and elliptical galaxies) and that, at the same time, it predicts a denser quantum core towards the center of the configuration. Such a quantum core, for a fermion mass in the range of $50\mathrm{keV}\lesssim m{c}^2\lesssim 345\mathrm{keV}$, can be an alternative interpretation of the central compact object in Sgr A*, traditionally assumed to be a black hole (BH). We present in this work the gravitational lensing properties of this novel DM configuration in nearby Milky-Way-like spiral galaxies. We describe the lensing effects of the pure DM component both on halo scales, where we compare them to the effects of the Navarro-Frenk-White and the nonsingular isothermal sphere DM models, and near the galaxy center, where we compare them with the effects of a Schwarzschild BH. For the particle mass leading to the most compact DM core, $m{c}^2\approx 10^2\mathrm{keV}$, we draw the following conclusions. At distances $r\gtrsim 20\mathrm{pc}$ from the center of the lens the effect of the central object on the lensing properties is negligible. However, we show that measurements of the deflection angle produced by the DM distribution in the outer region at a few kpc, together with rotation curve data, could help to discriminate between different DM models. In the inner regions $10^{-6}\lesssim r\lesssim 20\mathrm{pc}$, the lensing effects of a DM quantum core alternative to the BH scenario becomes a theme of an analysis of unprecedented precision which is challenging for current technological developments. We show that at distances $\sim 10^{-4}\mathrm{pc}$ strong lensing effects, such as multiple images and Einstein rings, may occur. Large differences in the deflection angle produced by a DM central core and a central BH appear at distances $r\lesssim 10^{-6}\mathrm{pc}$; in this regime the weak-field formalism is no longer applicable and the exact general-relativistic formula has to be used for the deflection angle which may become bigger than $2\pi$. An important difference in comparison to BHs is in the fact that quantum DM cores do not show a photon sphere; this implies that they do not cast a shadow (if they are transparent). Similar conclusions apply to the other DM distributions for other fermion masses in the above-specified range and for other galaxy types.

Figure 1

Density profiles for MW-type galaxies for different particle mass values that satisfy the boundary conditions: $M_{\mathrm{DM}}(r=20\mathrm{kpc})=9\times 10^{10}{M}_{\odot }$ [or equivalently $M_{\mathrm{DM}}(r=40\mathrm{kpc})=2\times 10^{11}{M}_{\odot }$], with a DMCC of mass $M_c=4.2\times 10^6{M}_{\odot }$. The set of initial conditions ${\theta }_0=32.2$; ${\beta }_0=2.7\times 10^{-7}$; $W_0=58.5$, ${\theta }_0=37.1$; ${\beta }_0=1.04\times 10^{-5}$; $W_0=65$, 20, ${\theta }_0=45.8$; ${\beta }_0=4.0\times 10^{-3}$; $W_0=76.049$ for $m=10\mathrm{keV}$, $m=48\mathrm{keV}$ and $m=345\mathrm{keV}$, respectively, have been here adopted from the original solutions computed in [7].

Figure 2

Distribution of DM in MW-type galaxies predicted by the RAR model. The solid line refers to the most compact (MC) solution for $m=345\mathrm{keV}$ (“inos MC” in the legend). For comparison, we show with the dotted brown line the solution for $m=48\mathrm{keV}$, which we refer to as “inos” in the legend. We also show the NFW and NSIS profiles given by the formulas (1) and (2), respectively. The free parameters in these profiles were taken from [30] and [31], respectively, satisfying the same (total) rotation curve data as in the RAR case, with the corresponding considerations of bulge and disk counterparts.

Figure 3

Convergence for the NFW, NSIS and RAR density profiles. The condition of strong lensing is achieved for the RAR profile (inos MC) below $10^{-4}\mathrm{pc}$ while the halo part is characterized by weak lensing effects in a way similar to the other profiles.

Figure 4

Deflection angle for the NFW, NSIS and RAR density profiles. It can be seen the relation between the deflection angle and the rotation curve as it was found in [34]. See also Ref. [5] for the rotation curve behavior.

Figure 5

Magnification factor for all the profiles listed in the legend and computed by Eq. (20).

Figure 6

Comparison between BH and fermionic DM quantum core of the inos MC configuration. The vertical line indicates the core radius of the DMCC: $r_c\approx 9GM/c^2$.

Figure 7

Gravitational potential comparison between the central BH and the fermionic DM configuration. The vertical line indicates the core radius of the DMCC: $r_c=9GM/c^2$.

Figure 8

Comparison between the BH lensing contribution along the entire galaxy as well as the most compact solution for the inos profile and the NFW profile.