Investigating the effects of gravitational lensing by Hu-Sawicki $$\varvec{f(R)}$$ gravity black holes

Abstract In this work, gravitational lensing in the weak and strong field limits is investigated for black hole spacetime within the framework of Hu–Sawicki f ( R ) gravity. We employ the Ishihara et al. approach for weak lensing and adopt Bozza’s method for strong lensing to explore the impact of Hu–Sawicki model parameters on lensing phenomenon. The deflection angles are computed and analyzed in both the field limits. Our investigation in the weak as well as the strong lensing reveals that in the case of Hu–Sawicki black holes, photons exhibit divergence at smaller impact parameters for different values of the model parameters compared to the Schwarzschild scenario and the photon experiences negative deflection angle when impact parameter moves towards the larger impact parameter values. Additionally, by calculating strong lensing coefficients we study their behavior with model parameters. The strong lensing key observables associated with the lensing effect viz. the angular position $$\vartheta _{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϑ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> , angular separation s and relative magnification $$r_\text {mag}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mtext>mag</mml:mtext> </mml:msub> </mml:math> are estimated numerically by extending the analysis to supermassive black holes $$\text {SgrA}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>SgrA</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> and $$\text {M87}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>M87</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> and analyzed their behavior concerning the parameters for each black hole. The analysis shows that $$\text {SgrA}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>SgrA</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> demonstrates larger values of $$\vartheta _{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϑ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> and s relative to $$\text {M87}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>M87</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> .