Exploring black holes with multiple photon spheres by interferometric signatures

TL;DR

This work addresses how hairy Schwarzschild black holes (hSBHs), as constructed via gravitational decoupling, imprint interferometric signatures on black-hole images when multiple photon spheres are present. By coupling thin-disk emission with thin-ring models and exact numerical ray-tracing, the authors derive analytical expressions for the complex visibility $V(u)$ and demonstrate that single photon rings produce damped oscillations, while double photon spheres can generate beat patterns in $|V(u)|$ depending on the relative heights of the inner and outer photon-sphere potentials. The key findings show that a higher inner photon-sphere peak leads to a detectable beat in the visibility with period $oldsymbol{ extDelta u} obreak= obreak 1/(eta_{ ext{ph}}^{ ext{out}}-eta_{ ext{ph}}^{ ext{in}})$, offering a potential observational probe of hair and photon-sphere structure with future very-long-baseline interferometry. These results advance the prospect of using interferometric measurements to test GR and the nature of black-hole hair, while outlining avenues for incorporating more realistic accretion physics and rotating/hairy backgrounds.

Abstract

In this paper, we investigate the interferometric signatures of hairy Schwarzschild black holes (hSBHs) that have either single or double photon spheres. Our interest mainly stems from two considerations: (i) the photon ring structure in black hole images produces strong and universal interferometric signatures on long baselines, enabling precision measurements of black hole parameters and testing gravitational theory; (ii) the hSBH describes the deformation of standard Schwarzschild black hole (SBH) induced by additional sources, and they can feature double photon spheres within certain parameter regimes. Using both analytical and numerical methods, we find that for a hSBH with a single photon sphere, the complex visibility amplitude of the image exhibits damped oscillations. A similar behavior appears in the double photon sphere case when the inner photon sphere has lower effective potential than the outer one, as the photons near the inner photon sphere remain trapped by gravity. However, when the inner potential is higher, a beat pattern rises. Our findings reveal that the complex visibility amplitude can encode the signature of the photon sphere structure of the central black hole.

Figures (4)

• Figure 1: The effective potential $V_{\text{eff}}$ of the hSBH with single (Left) and double (Middle and Right) photon spheres. We choose $\alpha=7$, $\alpha=7.5$ and $\alpha=7.8$, respectively, with fixed $l_o=1$ and $M=1$.
• Figure 2: The hSBH with a single photon sphere. Here we fixed $\alpha=7$, $l_o=1$ and $M=1$, with the effective potential shown in the left panel of Fig.\ref{['figVeff']}. The angle of photon sphere is $\beta_{\text{ph}}=0.05$. Top-left: different observed intensities originated from the direct (black), lensed ring (gold) and photon ring (red) intensities, respectively. Top-middle: the observed intensities as a function of angle $\beta$. Top-right: optical appearance: the intensity distribution across a two-dimensional plane. Bottom: total visibility amplitude $|V(u)|$ (green) and its components contributed by direct (black), lensed ring (gold) and photon ring (red) intensities, respectively.
• Figure 3: The hSBH with double photon spheres. Here we fixed $\alpha=7.5$, $l_o=1$ and $M=1$, with the effective potential shown in the middle panel of Fig.\ref{['figVeff']}. Top-left: different observed intensities originated from the direct (black), lensed ring (gold) and photon ring (red) intensities, respectively. Top-middle: the observed intensities as a function of angle $\beta$. Top-right: optical appearance: the intensity distribution across a two-dimensional plane. Bottom: total visibility amplitude $|V(u)|$ (green) and its components contributed by direct (black), lensed ring (gold) and photon ring (red) intensities, respectively.
• Figure 4: The hSBH with double photon spheres. Here we fixed $\alpha=7.8$, $l_o=1$ and $M=1$, with the effective potential shown in the right panel of Fig.\ref{['figVeff']}. The angles of double photon spheres are $\beta_{\text{ph}}^{\text{in}}=0.0271$ and $\beta_{\text{ph}}^{\text{out}}=0.0497$. Top-left: different observed intensities originated from the direct (black), lensed ring (gold) and photon ring (red) intensities, respectively. Top-middle: the observed intensities as a function of angle $\beta$. Top-right: optical appearance: the intensity distribution across a two-dimensional plane. Bottom: total visibility amplitude $|V(u)|$ (green) and its components contributed by direct (black), lensed ring (gold) and photon ring (red) intensities, respectively. For the sufficiently long baseline, the beat pattern will emerge. The widths of two adjacent beat patterns match well with $1/(\beta_{\text{ph}}^{\text{out}}-\beta_{\text{ph}}^{\text{in}})=44.25$.