Photon rings, gravitational lensing, and ISCOs of exotic compact objects in Einstein-scalar-Maxwell theories

Abstract

In Einstein-scalar-Maxwell theories with a coupling between the scalar field $φ$ and the electromagnetic field strength $F$ of the form $μ(φ) F$, we investigate the existence of exotic compact objects (ECOs) and their observational signatures in photon and massive-particle dynamics. For $μ(φ)$ diverging at the origin while all physical quantities remain finite, we demonstrate the existence of electrically charged ECOs with a shell-like structure whose density peaks at an intermediate radius. We compute their mass and radius, together with the scalar and vector field profiles, on a static and spherically symmetric background. We then examine the existence of photon rings and place bounds on a model parameter by requiring the absence of a linearly stable photon ring. Under this condition, photon echoes from ECOs are absent. We also compute the gravitational-lensing deflection angle $Ψ$ and show that it attains a maximum for an impact parameter of the same order as the ECO radius. Finally, we study the parameter space in which innermost stable circular orbits of massive particles exist.

Figures (8)

• Figure 1: The left panel shows the scalar-field derivative $\varphi={\rm d}\bar{\phi}/{\rm d}x$ and the electric field $B_0 \equiv {\rm d}\bar{A}_0/{\rm d}x$ as functions of $x=r/r_0$ for $m=2$, $\lambda=1$, and $\alpha=0.6835$. The right panel displays $\bar{\rho}$ and $-\bar{P}_r$ as functions of $x$ for the same set of parameters. We perform the integration outward starting from $r = 10^{-4} r_0$.
• Figure 2: Numerical solutions for $h$, $N$, $\bar{\phi}-\bar{\phi}_0$, and ${\cal M}/M_0$ as functions of $r/r_0$ for $m=2$ ($p=3$), $\alpha=0.6835$, and $\lambda=1$ (left panel), and for $m=4$ ($p=5/2$), $\alpha=0.15$, and $\lambda=1$ (right panel). In both panels, we choose values of $\alpha$ slightly smaller than $\alpha_{p}$, where $\alpha_{p}$ denotes the critical value below which photon rings are absent. The two cases are qualitatively similar, particularly regarding the positions of the minima of $h$. The asymptotic values of $\bar{\phi}$ at spatial infinity differ slightly between the two cases.
• Figure 3: Plot of $\bar{\rho}$ (left) and $-\bar{P}_r$ (right) as functions of $r/r_0$ for $m = 2, 4,$ and $10$, with $\lambda = 1$ in all cases. For each value of $m$, we fix $\alpha$ close to $\alpha_p$, namely $\alpha = 0.6835$ for $m = 2$, $\alpha = 0.15$ for $m = 4$, and $\alpha = 0.0261$ for $m = 10$.
• Figure 4: The mass-radius relation for $m=2$ is shown. In the left panel, we vary $\alpha$ in the range $10^{-5} \leq \alpha \leq 0.6835$ while fixing $\lambda=1$, and define the shell thickness $\Delta r = r_{\rm out} - r_{\rm in}$, where $\rho(r_{\rm out}) = \rho(r_{\rm in}) = \rho_{\rm max}/100$. In the right panel, we instead fix $\alpha = 0.6835$ and vary $\lambda$ in the range $10^{-2} \leq \lambda \leq 10^3$. Here, the radius $r_s$ is defined as the value of $r$ satisfying $\mathcal{M}(r_s/r_0)= (90\%, 99\%)\,M$.
• Figure 5: Left panel: Example with two photon rings, whose locations correspond to the zeros of $2 f - r f'$ (blue points). The shaded regions indicate the sign of $2 f - r^2 f"$: green denotes linearly unstable photon rings, while red denotes linearly stable ones. Right panel: Example of the marginal case where $\alpha$ is close to the upper limit $\alpha_p$, above which two photon rings appear. For $\alpha < \alpha_p$, the function $2 f - r f'$ remains positive for all $r$, and no photon rings are present.