Non-minimal light-curvature couplings and black-hole imaging

TL;DR

The work investigates how non-minimal light-curvature couplings, exemplified by the Horndeski vector-tensor term with coupling $\alpha$, modify photon propagation around black holes via polarization-dependent effective metrics. By computing the $n=1$ lensing band with backward ray-tracing in the two polarization sectors, it demonstrates that for $|\alpha|/M^2$ of order unity the bands can be substantially deformed and polarization-dependent, potentially producing non-overlapping regions with GR. Applying the framework to M87*, the authors derive a bound $-0.3 \lesssim \alpha/M^2_{\rm M87^*} \lesssim 0.3$, translating to $\sqrt{|\alpha|} \lesssim 5.34 \times 10^9$ km, illustrating how photon-ring observations can constrain non-minimal propagation effects. The results emphasize the importance of disentangling propagation-induced signatures from spacetime-geometry changes in black-hole imaging and motivate extending the analysis to rotating black holes and other non-minimal couplings.

Abstract

Non-minimal couplings between the electromagnetic field strength and the spacetime curvature are part of the effective field theory of gravity and matter. They alter the local propagation of light in a significant way if the ratio of spacetime curvature to the non-minimal coupling is of order one. Spacetime curvature can become appreciable around black holes, and yet the effect of non-minimal couplings on electromagnetic observations of black holes remains underexplored. A particular feature of the non-minimal coupling between the electromagnetic field-strength and the Riemann tensor is that it generates two distinct photon rings for different polarizations. Working within the paradigm of lensing bands and focusing on the $n = 1$ lensing band, we illustrate by which diagnostics a modified light propagation may be distinguished from a modified spacetime geometry and how constraints on the value of the non-minimal coupling can be obtained

Figures (7)

• Figure 1: We show the $n=1$ lensing band (which, for near edge-on inclinations, fills nearly the entire lower-half of the image) for the PPL polarization for three different inclinations $i$ and distinct coupling strengths $\alpha/M^2$. The lensing band for the Schwarzschild metric without non-minimal coupling is shown in black, and the lensing band with non-minimal coupling is overlaid in light pink. Wherever these regions overlap, the overlap is indicated in reddish brown. The lensing bands expand inwards (outwards) for negative (positive) value of the coupling. This behaviour is reversed for the PPM polarization.
• Figure 2: Area of the overlap between the $n=1$ lensing bands with non-minimal coupling and GR (reddish brown regions in Fig. \ref{['fig:PPL_lensing_bands_vs_Schwarzschild']} for the PPL case) in terms of the coupling strength $\alpha/M^2$, for four different inclinations $i$. The total field of view of the image is $20 M \times 20M$. For all inclinations, generically, the PPL lensing bands have a larger overlap with the GR lensing band, compared to the PPM case, for negative coupling, and vice-versa for positive couplings. For large inclinations, the overlap area can plateau because the lower-half of the image is filled by the lensing band both in GR and with non-minimal couplings.
• Figure 3: Ratio of the total area of the $n=1$ lensing bands with non-minimal light-curvature couplings (LCC) and GR in terms of the coupling strength $\alpha/M^2$ for four different inclinations $i$. This figure compares how much larger or smaller, compared to GR, is the region in the image plane where the $n=1$ photon ring can be located. The total area of the image is $20 M \times 20M$. For negative values of the coupling, there is a larger disparity between the two polarizations for small inclinations, making them more favorable to constrain the coupling.
• Figure 4: Top: Ratio of the area of the $n=1$ lensing bands for the PPM and PPL polarizations in terms of the coupling strength $\alpha/M^2$, for four different inclinations $i$. Bottom: Ratio of the area of the overlap between the $n=1$ lensing bands for the PPM and PPL polarizations and the area of the PPL $n=1$ lensing band in terms of the coupling strength $\alpha/M^2$, for four different inclinations $i$. The total area of the image is $20 M \times 20M$. For large inclinations, the overlap area plateaus because both polarizations fill a large portion of the lower-half plane of the image, and no overlap exists in the upper-half plane of the image. For sufficiently small inclinations, and large absolute values of the coupling, there is no overlap between the lensing bands, as increasing $|\alpha|$ pushes them in different directions.
• Figure 5: We show the $n=1$ lensing bands (which, for near edge-on inclinations, nearly fill the entire lower-half plane of the image) for the PPL and PPM polarizations for the largest inclination $i = 89^{\circ}$ and coupling strength $\alpha/M^2 = 0.9$. The total area of the image is $20 M \times 20M$. The lensing band for the PPL polarization is shown in light pink, and the lensing band for the PPM polarization is overlaid in blue. The region where these lensing bands overlap is indicated in purple. In the upper-half plane $y \geq 0$, the PPL lensing band is everywhere located outward from the PPM lensing band, while the PPL and PPM lensing bands greatly overlap in the lower-half plane $y < 0$, filling it almost entirely.