Testing extended theories of gravity via black hole photon rings

Abstract As a unique observational feature in strong gravitational fields, photon rings around black holes provide a method to evaluate general relativity and alternative gravity theories. This research delves into the optical characteristics of stationary, spherically symmetric black holes. These black holes follow the Konoplya–Zhidenko deformation rule in arbitrary gravity theories. This black hole is characterized by three independent parameters $$(\varepsilon ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$a_2,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$b_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> to describe its divergence from the Schwarzschild geometry. This paper derives the geodesic equation and effective potential function from the Lagrangian, determines the photon sphere radius (unstable orbit), and analyzes the null geodesics using the reverse ray-tracing technique. This research finds that the effects of $$a_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and $$b_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> on photon orbital dynamics exhibit observational degeneracy, while $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> significantly governs photon capture characteristics. As $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> increases, the radius of the photon sphere $$r_{\text {ph}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mtext>ph</mml:mtext> </mml:msub> </mml:math> and the critical impact parameter $$b_{\text {ph}},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mtext>ph</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and the innermost stable circular orbit radius $$r_{\text {isco}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mtext>isco</mml:mtext> </mml:msub> </mml:math> all increase. The event horizon $$ r_{\text {h}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mtext>h</mml:mtext> </mml:msub> </mml:math> corresponds to that of the Schwarzschild black hole, while the impact parameter range for the lens and photon rings is reduced. Black hole shadow and photon ring analyses across three emission models show that increasing $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> shifts the peak rightward while enlarging the photon ring radius. The closer $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> is to zero, the more the results approach the Schwarzschild case, the more the results approach the Schwarzschild case. Additionally, by combining EHT observational data on the shadow diameters of M87 and Sgr A*, we imposed constraints on the correlation parameter $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> in the theoretical model(at the confidence level anchored by $$d_{\text {sh}}^{(M87^*)},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>d</mml:mi> <mml:mrow> <mml:mtext>sh</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:msup> <mml:mn>87</mml:mn> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> the parameter $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> is confined to the interval $$-0.09 \lesssim \varepsilon \lesssim 0.19.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>0.09</mml:mn> <mml:mo>≲</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>≲</mml:mo> <mml:mn>0.19</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> For $$d_{\text {sh}}^{(Sgr A^*)},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup>