Shadows and gravitational weak lensing by the Schwarzschild black hole in the string cloud background with quintessential field* *G. Mustafa acknowledges the Grant No. ZC304022919 to support his Postdoctoral Fellowship at Zhejiang Normal University. F.A. acknowledges the support of Inha University in Tashkent and research work has been supported by the Visitor Research Fellowship at Zhejiang Normal University. This research

Abstract

In this study, we observe that, in the presence of the string cloud parameter a and the quintessence parameter γ, with the equation of state parameter , the radius of the shadow of the Schwarzschild black hole increases as compared with that in the pure Schwarzschild black hole case. The existence of both quintessential dark energy and the cloud of strings increases the shadow size; hence, the strength of the gravitational field around the Schwarzschild black hole increases. Using the data collected by the Event Horizon Telescope (EHT) collaboration for M87* and Sgr A*, we obtain upper bounds on the values of a and γ. Further, we see the effects of a and γ on the rate of emission energy for the Schwarzschild black hole. We notice that the rate of emission energy is higher in the presence of clouds of strings and quintessence. Moreover, we study the weak deflection angle using the Gauss-Bonnet theorem. We show the influence of a and γ on the weak deflection angle. We notice that both a and γ increase the deflection angle α.