Gravitational lensing by black holes in Einsteinian cubic gravity

Mohammad Bagher Jahani Poshteh and Robert B. Mann

Phys. Rev. D 99, 024035 – Published 25 January 2019

Abstract

We investigate the predictions of Einsteinian cubic gravity (ECG) for the lensing effects due to supermassive black holes at the center of the Milky Way and other galaxies. Working in the context of spherical symmetry, we obtain the metric function from a continued fraction method and find that both time delays and the angular positions of images considerably deviate from general relativity, as large as milliarcseconds. This suggests that observational tests of ECG are indeed feasible.

Received 28 October 2018

DOI:https://doi.org/10.1103/PhysRevD.99.024035

Physics Subject Headings (PhySH)

Alternative gravity theories Classical black holes

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Mohammad Bagher Jahani Poshteh^*

Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 and Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran

Robert B. Mann^†

Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 and Perimeter Institute, 31 Caroline St. N., Waterloo, Ontario, Canada, N2L 2Y5

^*mb.jahani@iasbs.ac.ir
^†rbmann@uwaterloo.ca

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Issue

Vol. 99, Iss. 2 — 15 January 2019

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Images

Figure 1
The lens diagram: As the light ray pass the black hole it deflects by an angle $\hat{α}$ . Those rays which pass closer to the black hole have a larger deflection angle. If $\hat{α} > 2 π$ , the corresponding light ray winds the black hole at least once before reaching the observer. These rays would make the relativistic images. Here $S$ , $I$ , $O$ , and $L$ stand, respectively, for the source, image, observer, and the lens, which is a black hole in our study. $β$ is the actual angular position for the source with respect to the line of sight to the black hole. $θ$ is the angular position of the image. $D_{d}$ and $D_{d s}$ represent, respectively, the distance from lens to observer and from lens to the source.
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Figure 2
Finding the source position: Top: Image positions as a function of the angular source position $β$ in GR (dotted, red curve) and ECG (solid, blue curve) with $D = 0.5$ . Those lines with positive slope correspond to the primary image position $θ_{p}$ and those with negative slope to the secondary image position $| θ_{s} |$ . The horizontal dashed black lines indicate the position of the primary (upper line) and secondary (lower line) images in a particular observation. Each of these horizontal lines crosses both the dotted red and solid blue lines; $β$ is determined by finding a common intersection point for the two curves, illustrated by the vertical dashed black line. We have used Sgr A* as the lens with $M_{Sgr A *} = 5.94 \times 10^{9} m$ and $D_{d} = 2.43 \times 10^{20} m$ , and have taken $λ / M_{Sgr A *}^{4} \approx 1.76 \times 10^{- 4}$ . Bottom: Difference between the differential time delay in GR with $D = 0.49972$ , ${\bar{t}}_{d, GR}$ , and that in ECG with $D = 0.5$ , $t_{d, ECG}$ .
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Figure 3
Deviation of primary image angular position in ECG from GR for Sgr A*: The deviation increase with angular source position $β$ . The black dashed line is for the case $D = 0.5$ and the red line is for $D = 0.05$ . It is obvious that, for a fixed lens-observer distance, the deviation of ECG results for angular positions of primary images from that of GR is larger for sources further away from the lens.
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Figure 4
Deviation of differential time delay in ECG from GR for Sgr A*: The differential time delay $t_{d} = τ_{s} - τ_{p}$ deviate from its corresponding value in GR if ECG governs the strong gravitational field around the black hole. The deviation increases with angular source position $β$ . The black dashed line is for the case $D = 0.5$ and the red line is for $D = 0.05$ . Here we see that, for a fixed lens-observer distance, the deviation of ECG results for the differential time delay from that of GR is larger for sources if the source is closer to the lens.
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Figure 5
Deviation of primary image angular position in ECG from GR for different SMBHs: The ratio $(θ_{p, GR} - θ_{p, ECG}) / \bar{M}$ increases as $\bar{M}$ decreases. Here $\bar{M} = M / M_{Sgr A *}$ , where $M$ is the mass of the SMBH from Table 6. We have taken $D = 0.5$ . The dots refer to the numerical results of Table 7 for the 14 SMBHs, and the solid curve is the interpolation between the points.
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Figure 6
Deviation of differential time delay in ECG from GR for different SMBHs: The ratio $(t_{d, GR} - t_{d, ECG}) / \bar{M}$ increases as ${\bar{D}}_{d} / \bar{M}$ increases. Here ${\bar{D}}_{d} = D_{d} / D_{d, Sgr A *}$ and $\bar{M} = M / M_{Sgr A *}$ , where $D_{d}$ and $M$ are the mass and distance of the SMBH from Table 6. We have taken $D = 0.5$ . The dots refer to the numerical results of Table 7 for the 14 SMBHs, and the solid curve is the interpolation between the points.
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