4D scale-dependent Schwarzschild-AdS/dS black holes: study of shadow and weak deflection angle and greybody bounding

Abstract We investigate the shadow, deflection angle, and the greybody bounding of a Schwarzschild-AdS/dS black hole in scale-dependent gravity. We used the EHT data to constrain the parameter $$\epsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> . We have found that within the $$1\sigma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> level of uncertainty, $$\epsilon M_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> ranges at $$10^{-11} - 10^{-16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> orders of magnitude for Sgr. A*, and $$10^{-11} - 10^{-17}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>17</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for M87*. Using these parameters, we explored how the shadow radius behaves as perceived by a static observer. Using the known parameters for Sgr. A* and M87*, we found different shadow cast behavior near the cosmological horizon even if the same scaling parameters were used. We explored the deflection angle $${\hat{\alpha }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> in the weak field limit for both massive and null particles, in addition to the effect of finite distances using the non-asymptotically flat version of the Gauss-Bonnet theorem. We have found that the $${\hat{\alpha }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> strongly depends on massive particles. Tests on $$\epsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> ’s effect using Sgr. A* is only along points of low impact parameters. For M87*, only the scaled BH in AdS type can be tested at high impact parameters at the expense of very low value for $${\hat{\alpha }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> . We study the rigorous bound on the greybody factor for scalar field emitted from black holes in the theory and show the bound on the transmission probability. The effects of the scale-dependent gravity on the greybody factors are investigated, and the results indicate that the bound on the greybody factor in this case is less than the bound for the Schwarzschild black hole.

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