Auto- and cross-correlations for multiple images of corotating hotspots in accretion disks

TL;DR

This work tackles how to extract black hole geometry from variability in Sgr A* by analyzing the spatiotemporal auto- and cross-correlations of multiple corotating hotspot images, leveraging a Schwarzschild-based ray-tracing framework that captures up to eighth-order photon images. It demonstrates that light-curve profiles are highly sensitive to hotspot morphology and thus unreliable for BH geometry inference, whereas correlations—particularly those involving higher-order images—encode inclination and BH parameters in a morphology-insensitive way. The study reveals band-like correlation structures in (Δt,ΔΦ) with slopes set by apparent hotspot rotation and a width that grows with inclination, and shows fixed-point intersections across different rotation speeds, implying a robust pathway to BH parameter estimation. Overall, the results advocate using spatial-temporal correlations of high-order images as a robust probe of BH geometry in horizon-scale observations of Sgr A*.

Abstract

Due to the short gravitational timescale of Sgr A*, variable emissions near the galactic center are expected in the Very-long-baseline interferometry observations. Phenomenologically, the high-flux variable emissions could be interpreted as occasional events from hotspots within accretion disks. It provides a probe of black hole (BH) geometry and accretion matter in the strong-field regime of gravity. In this study, we find that light curve profile alone is not proper for distinguishing BH geometries, as our results show that the profiles, even including those from higher-order images, are dependent on hotspot shapes, which are known in practice as amorphous. To alleviate this situation, we examine the spatial-temporal correlations between multiple images of the corotating hotspots. Our results find that the correlations, particularly those from higher-order images, could serve as a robust observable to reflect the inclination angles and BH geometries, because i) the correlated band structure is independent of the hotspot shapes, and ii) the correlations from higher-order images could encode BH geometries and exhibit no overlap with observational signatures from the lower-order ones. We present a comprehensive study on correlations from primary the eighth-order images with various orbital configurations and inclination angles, and show its observational signatures. It is expected that BH geometries can be inferred via the spatial-temporal correlation analysis.

Figures (11)

• Figure 1: The multiple images of corotating point-like hotspots for inclination angles $4\pi/9$ (left panel) and $\pi/9$ (right panel). The numbers represent the image orders. For example, the no.1 represents the primary image. The solid curves represent the apparent tracks of hotspots, and the black grids denote apparent region of the event horizon. We have introduced $(\text{X},\text{Y})\equiv(\tan\Psi \cos\Phi, \tan\Psi \sin\Phi)$, which is a gnomonic projection of the celestial coordinates $(\Phi,\Psi)$.
• Figure 2: Schematic diagrams for emission models of hotspots. Left panel: hotspots are distributed on the surface of thin accretion disks. Right panels: hotspots are in a shape of sphere.
• Figure 3: Normalized light curves, $\bar{F}_\nu^{(n,O)}[\equiv F_\nu^{(n,O)}/F_{\nu,\text{max}}^{(0,\text{A})}]$, from primary to eighth-order images. We consider the hotspot undergoing Keplerian motion with $r_\text{s}=10r_g$. Left panel: the emission sources distributed on the outer surface of accretion disks. Right panel: the emissions are in the shape of sphere. We consider a long-lived hotspot and set $J$ to be a constant.
• Figure 4: Dimensionless auto-correlations of observed intensity, $\bar{\mathcal{C}}(\Delta t,\Delta \Phi)[\equiv{\mathcal{C}}(\Delta t,\Delta \Phi)/{\mathcal{C}}(0,0)]$. We set inclination angles to be $\pi/3$ and $\pi/9$ in the top and bottom panels, respectively. The hotspot undergoes Keplerian motion with $r_\text{s}=10r_g$. The gamma value is set to be 0.08.
• Figure 5: Dimensionless auto- and cross-correlations, $\bar{\mathcal{C}}^{(n_1,O_1,n_2,O_2)}[\equiv{\mathcal{C}^{(n_1,O_1,n_2,O_2)}}/{\mathcal{C}}(0,0)]$, for inclination angle $\pi/3$. The hotspot undergoes Keplerian motion with $r_\text{s}=10r_g$. The gamma value is set to be 0.05.