Slow-rotating black holes with potential in dynamical Chern-Simons modified gravitational theory

The Chern-Simons amended gravity theory appears as a low-energy effective of string theory. includes an anomaly-cancelation correction to the Einstein-Hilbert action. expression consists product $\varphi R \tilde $ Pontryagin density $R with scalar field $\varphi$, where latter is considered background (dynamical construction or non-dynamical construction). Many different solutions Einstein's general relativity continue be valid in theories. Kerr metric is, however, exceptional case that raised search for rotating black hole solutions. We generalize solution presented Phys. Rev. D \textbf{77}, 064007 (2008) by allowing potential $V$ have non-vanishing value and we discuss three cases potential, $V=\mathrm{const.}$, $V\propto \varphi$, \varphi^2$ cases. present study presents, first time, novel prescribing holes frame dynamical formulation gravity, include previously derived derive slow-rotation limit, write parameter expansion $\varepsilon$. These are axisymmetric stationary they give distortion dipole field. Moreover, investigate rectification behaves inverse fourth order radial distance from center when \varphi$. This suggests any meaningful limits weak-field experiments could passed.