Optical Images of Mini Boson Stars in Palatini $f(R)$ Gravity

TL;DR

The paper investigates optical appearances of mini boson stars in Palatini $f(R)$ gravity with $f(R)=R+ξR^2$ by solving modified field equations and applying backward ray-tracing for spherical sources and thin accretion disks. Through an Einstein-frame mapping and numerical solutions, it shows these objects lack photon rings ($V_{ ext{eff}}'(r) eq 0$ for all $r$) and instead display images dominated by direct emission, with Einstein rings arising from lensing rather than photon orbits, and the results depend sensitively on the central scalar field $ψ_0$ and coupling $ξ$. Polarization patterns reveal internal polarization vectors, differentiating boson-star images from those of black holes and providing an accessible observational signature. The work underscores Palatini $f(R)$ gravity as a viable framework to test strong-field gravity using high-resolution imaging and motivates extensions to rotation and more realistic disk physics for future observational tests.

Abstract

We investigate the optical properties of mini boson stars within the framework of Palatini $f(R)$ gravity, adopting a quadratic form $f(R) = R + ξR^2$, where $ξ$ is the gravitational coupling constant. By deriving the modified scalar Lagrangian and solving the field equations numerically, we explore photon trajectories and the resulting optical images under spherical light sources and thin accretion disks. Unlike Schwarzschild black holes (BHs), boson stars lack stable photon rings due to the positive second derivative of their effective potential. Consequently, their images are dominated by direct emissions from photons completing a single orbit. The study examines the dependence of the optical characteristics on the initial scalar field $ψ_0$ and the coupling parameter $ξ$. Numerical results include effective potentials, redshift maps, and detailed imaging of boson stars, providing insights into distinguishing boson stars from black holes using high-resolution astronomical observations.

Figures (14)

• Figure 1: Radial variation of the scalar field for different boson stars, with $\xi=0.05$.
• Figure 2: Metric components $-g_{tt}$ (left) and $g_{rr}$ (right) for different boson stars and the Schwarzschild black hole, with $\xi=0.05$. The mass of the Schwarzschild black hole is set to $1$. The black solid line represents the Schwarzschild black hole, while the dotted lines with different colors represent different boson stars.
• Figure 3: Numerical results and fitting functions of the metric components $-g_{tt}$ (left) and $g_{rr}$ (right) for different boson stars, with $\xi=0.05$. The dotted lines represent the numerical results, while the solid lines represent the fitting functions.
• Figure 4: Derivative of the effective potential. The red, blue, green, and orange solid lines correspond to $\psi_0 = 0.2$, $0.16$, $0.13$, and $0.09$, with $\xi = 0.05$ fixed. The black solid line represents a Schwarzschild black hole with mass $1$.
• Figure 5: Optical images and Einstein rings corresponding to different boson stars under a spherical light source, with $\xi=0.05$, $\alpha_{\text{fov}}=45^\circ$, $r_{\text{obs}}=50m$ ($m$ denotes the mass of the boson star), and the inclination angle of the observer is $\theta=45^\circ$.