First-order corrections to equatorial periodic orbits around rapidly rotating black holes in dynamical Chern--Simons gravity

We calculate first-order corrections to the energy, radial period, and circumferential turning radii of equatorial periodic orbits around rapidly rotating black holes in dynamical Chern--Simons (dCS) gravity. Using an analytic fit to the metric correction, we match each orbit to a Kerr reference at fixed dimensionless spin, dimensionless specific angular momentum, and rational periodicity. The period correction combines a direct contribution at fixed energy with compensation from the energy adjustment that preserves periodicity. Their competition produces sign changes across orbital families and reference paths. In a two-parameter scan, negative period corrections are dominated by energy compensation for sampled prograde orbits and by the direct contribution for sampled retrograde orbits. We locate C_T=0 boundaries, where the coefficient of the first-order fractional period correction vanishes. Circumferential shifts can differ in sign from coordinate shifts. Numerical kludge waveforms at leading quadrupole order and their discrete spectra connect these corrections to phase accumulation and harmonic frequency shifts. At fixed dimensionless orbital parameters and physical mass of the small body, this phase contribution can be reproduced to first order by rescaling the Kerr mass. For the C_T=0 orbit examined here, this contribution is suppressed at equal black hole masses, while corrections to the waveform and spectral amplitudes remain nonzero.