Shadows and gravitational weak lensing by the Schwarzschild black hole in the string cloud background with quintessential field

In this work, we observe that in the presence of the string cloud parameter $a$ and the quintessence parameter $\gamma$, with the equation of state parameter ${\omega }_q=-2\mathrm{/}3$ the radius of the shadow of the Schwarzschild black hole increases as compared with the pure Schwarzschild black hole case. The existence of both quintessential dark energy and cloud of strings magnify the shadow size and 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 the M87* and Sgr A*, we obtain upper bounds on the values of the parameters $a$ and $\gamma$. Further, we see the effects of the parameters $a$ and $\gamma$ 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 string and quintessence. Moreover, we study the weak deflection angle using the Gauss-Bonnet theorem. We show the influence of the cloud of string parameter $a$ and the quintessential parameter $\gamma$ on the weak deflection angle. We notice that both the parameters $a$ and $\gamma$ increase the deflection angle $\alpha$.