Ultra-compact Objects of Non-minimally Coupled Dark Matter

TL;DR

This paper introduces a relativistic model in which collisionless dark matter is non-minimally coupled to gravity, generating an anisotropic effective pressure that can counter gravity and sustain horizonless ultra-compact objects (NMC-UCOs). By formulating the action S_NMC = ∫√{-g}[ c^4/(16πG) R + L_DM + ε L^2 G_{μν} T_DM^{μν} ], deriving the modified field equations, and solving for static, spherically symmetric configurations, the authors show the existence of horizonless, highly compact DM cores for a narrow central density range; the exterior spacetime matches Schwarzschild, while the interior exhibits an effective ρ_eff and p_r with possible two light rings at high density. Geodesic analysis reveals rich orbital structures for both massive and massless particles, including stable inner light rings and characteristic precession, and ray-tracing demonstrates a pseudo-shadow—a lensing signature resembling BH shadows despite the absence of horizons. The results suggest potential observational signatures and cosmological relevance, particularly for early-universe or primordial formation scenarios, though the required densities depend on the coupling scale L and warrant further study on formation and stability.

Abstract

In the framework of a collisionless dark matter fluid which is non-minimally coupled to gravity, we investigate the existence and properties of static, spherically symmetric solutions of the general relativistic field equations. We show that the non-minimal coupling originates an (anisotropic) pressure able to counteract gravity and to allow the formation of regular, horizonless ultra-compact objects of dark matter (NMC-UCOs). We then analyze the orbits of massive and massless particles in the gravitational field of NMC-UCOs, providing some specific example and a general discussion in terms of phase portraits. Finally, we study the gravitational lensing effects around NMC-UCOs, and effectively describe these in terms of a pseudo-shadow.

Figures (5)

• Figure 1: Radial profiles of UCOs solutions: density $\rho(r)$ (top left), cumulative mass $M(<r)$ (top right), metric component $g^{rr}$ (middle left), metric component $-g_{tt}$, extrinsic curvature component $-\kappa_{tt}$ (bottom left), and effective radial pressure $p_r(r)$ (bottom right). The various solutions are parameterized by the allowed values of the central density $\rho(0)$, as specified by the color scale. The dashed black line highlights the limiting solutions featuring the maximal value of the central density.
• Figure 2: Relevant physical length-scales of UCOs solutions: boundary radius $R_{\rm b}$ (red) and light rings (orange; dashed for the inner and solid for the outer); these are measured in units of total mass $M$ of the UCO, and are plotted as a function of the UCO central density $\rho(0)$ defining the various solutions. The grey shaded area displays the typical size of spherically symmetric regular black hole solutions (e.g., black hole mimickers) in general relativity. The inset displays the compactness parameter $\sigma=1-2 G\, M/c^2\, R_{\rm b}$ (green); the dotted line is the limiting value of the density $\frac{8\pi G\,L^2}{c^2}\, \rho(0)= 0.25$ below which no physical solution exists.
• Figure 3: Relevant examples of orbital trajectories for massless (magenta) and massive (green) particles around an UCO solution with central density $\hat{\rho}_0=0.95$. Top diagram show the trajectories in polar coordinates, bottom diagrams illustrate the radial position of the test particle as a function of proper time. The dashed grey lines displays the position of the boundary, dotted grey lines that of the inner and outer light rings. Left diagrams refer to an unbound orbit, middle diagrams to a loose bound orbit for massive particles, and right diagrams to a tigthly bound orbit for both massless and massive particles; the values of the energy parameter $E$, of the angular momentum $j$, of the starting radial coordinate $r_0$ and the average proper period $\langle T\rangle$ of the orbit are reported (polar initial angle is assumed to be null).
• Figure 4: Orbital portraits diagrams for the UCO solution with central density $\hat{\rho_0}=0.95$, slightly below the maximal value. Left panel refers to a massless and right panel to a massive test particle. The radius $r$ at the radial inversion point(s) with $\dot r=0$ is shown as a function of the angular momentum $j$ ($x$-axis) and of the energy parameter $E$ (color-scale) of the test particle. The dashed black line highlights the portraits for $E=1$. Note that in both the panels, the scale of the $y$-axis starts linear at the bottom and then becomes logarithmic at the top.
• Figure 5: Pseudo-shadow of the UCO solution with central density $\hat{\rho}_0=0.95$. Left panel illustrates a ray-tracing simulation, with the observer at the black spot on the right and an illuminated screen assumed at $x=-\infty$. Solid lines are light rays traced back in time from the observer with different impact parameters $j/E$: in the remote past, cyan rays hit the illuminated screen at $x=-\infty$, and hence can be seen by the observer; orange rays are instead deflected back to $x=+\infty$ and are precluded to vision. Dashed grey lines displays the position of the boundary, dotted grey lines that of the inner and outer light rings. Right panel is a polar plot illustrating the face-on view of the UCO from the observer position: as a consequence of gravitational lensing, a multiple photon ring structure is originated around the typical radii $(c^2/G)\, r\approx 1.8-4-5\, M$.