Observable signatures of Black Holes with Hernquist Dark Matter Halo having a cloud of strings: From Geodesics to Shadow

We present a comprehensive theoretical investigation of a novel black hole (BH) spacetime: a Schwarzschild BH embedded in a Hernquist-type dark matter halo (HDMH) and surrounded by a cloud of cosmic strings (CSs) -- collectively termed the Schwarzschild-HDMH with CS (SHDMHCS) configuration. By analyzing the spacetime geometry, we explore how key parameters -- the core radius and halo density of the dark matter, along with the string tension -- affect the geodesic motion of both massless and massive particles. Our results reveal that the combined influence of HDMH and CSs modifies the effective potentials for null and timelike geodesics, leading to distinct dynamical behavior compared to standard Schwarzschild geometry. We perform a perturbative analysis for scalar (spin-0), electromagnetic (spin-1), and Dirac (spin-1/2) fields, deriving the associated effective potentials and showing how both the dark halo and CSs alter field propagation and potential barriers. The shadow of the BH is studied in detail: we derive analytical expressions for photon sphere and shadow radii, finding that CSs tend to enlarge the shadow, while HDMH properties tend to shrink it. The combined effects of these parameters significantly influence the shadow's shape and size, producing potentially observable signatures. Our results establish that the SHDMHCS configuration yields distinct observational imprints detectable by present and forthcoming astrophysical instruments. This framework provides new tools for probing exotic matter distributions via gravitational wave observations, orbital dynamics, and high-resolution black hole imaging, offering a pathway to distinguish such configurations from simpler BH models in realistic environments.