Gravitational wave radiation from periodic orbits around a noncommutative-inspired black hole surrounded by quintessence

<title>Abstract</title> <p> We study the gravitational-wave (GW) emission from periodic orbits of a massive test particle around a noncommutative-inspired black hole surrounded by a quintessence (NCiBHSQ) field with equation-of-state parameter <italic>ω</italic> _{<italic>q</italic>} = −2/3. The geometry is governed by two parameters: the quintessence field strength parameter p and the noncommutative parameter Θ, which deform the spacetime in complementary radial domains. Using the Hamiltonian formalism, we derive the equations of motion and analyze the effective potential, the specific energy and angular momentum of circular orbits, and the innermost stable and innermost bound circular orbits (ISCO and IBCO). We classify timelike periodic orbits by the zoom-whirl taxonomy with three topological integers (z, w, v) and compute representative GW waveforms in both the plus ( <italic>h</italic> _{<italic>+</italic>} ) and cross ( <italic>h</italic> _{<italic>×</italic>} ) polarizations within the kludge approximation. We find that the linear term restricts the existence of bound periodic orbits to a narrow band of <italic>p</italic> , and that both <italic>p</italic> and Θ leave measurable imprints on the orbital energies and on the emitted waveforms: increasing Θ shifts the orbital structure inward, while a more negative <italic>p</italic> raises the energy of periodic orbits. The characteristic strains fall in the millihertz band accessible to space-based detectors such as LISA, suggesting that future GW missions could probe or constrain noncommutative and quintessence corrections in the strong-field regime. </p>