Investigating the interaction of a Cosmic String with an Accreting Black Hole

Rotating black holes when attached to a cosmic string have their rotational energy extracted leading to a change in its spin and mass. The spin of a black hole can be measured using various methods for an accreting black hole in an X-ray binary system. Accretion disks around black holes have an innermost stable circular orbit (ISCO) whose location is directly dependent on spin and mass of the black hole. The orbit’s location changes as the black hole’s spin changes and hence can be a method to detect the presence of cosmic strings. This study investigates this change and suggests the ejection of accretion material as black hole spin approaches maximum for prograde motion and material falling into the black hole for retrograde motion, regardless of the presence of cosmic string. However, in the presence of cosmic string, the spin-up process due to accretion is found out to be slower, even with high accretion rates and is detectable. There is a transition phase that occurs as the black hole approaches maximum spin, where even small changes in spin result in significant changes in the ISCO’s position. Accreting black holes attached to a large string never reach this transition phase and this absence serves as potential evidence for the existence of a cosmic string.

💡 Research Summary

The paper investigates how a cosmic string attached to a rotating, accreting stellar‑mass black hole influences the black hole’s spin evolution and the radius of the innermost stable circular orbit (ISCO). Starting from the standard Kerr expression for ISCO, the authors differentiate with respect to time to obtain a general formula for (\dot R_{\rm ISCO}) that depends on both the mass‑change rate and the spin‑change rate. They model the mass change as (\dot M = \dot M_{\rm acc} - \dot M_{\rm str}), where (\dot M_{\rm acc}) is the accretion inflow (including a spin‑dependent radiative efficiency) and (\dot M_{\rm str}) is a loss term proportional to the dimensionless string tension (\mu) (Eq. 14).

Numerical experiments are performed for a 10 (M_\odot) black hole accreting at (10^{-8},M_\odot,{\rm yr}^{-1}). Two string‑tension regimes are explored: a “small” tension (\mu = 10^{-40}) (giving (\dot M_{\rm str}\sim10^{-29},M_\odot,{\rm yr}^{-1})) and a “large” tension (\mu = 10^{-29}) (giving (\dot M_{\rm str}\sim10^{-18},M_\odot,{\rm yr}^{-1})). Spin‑up rates are set to (\dot\alpha = 10^{-7},{\rm yr}^{-1}) for the small‑tension case and (\dot\alpha = 10^{-10},{\rm yr}^{-1}) for the large‑tension case, reflecting the expectation that a strong string extracts angular momentum more efficiently.

The results show that, irrespective of a string, (\dot R_{\rm ISCO}) diverges as the dimensionless spin parameter (\alpha) approaches its extremal limits ((\alpha\to\pm1)). For prograde spins ((0<\alpha<1)), as (\alpha) nears unity the ISCO moves inward dramatically; the excess angular momentum is then expelled as winds, while for retrograde spins ((-1<\alpha<0)) the ISCO recedes to infinity, causing material to plunge directly into the hole. A “transition phase” near maximal spin is identified where tiny changes in (\alpha) produce large shifts in ISCO radius, offering a potentially observable signature.

Crucially, in the large‑tension scenario the spin saturates at a value (\alpha_{\rm sat}\sim \dot M_{\rm acc}/\mu) well below unity, so the system never reaches the transition phase. Consequently, the characteristic rapid ISCO variation is absent, which the authors argue could serve as indirect evidence for a massive cosmic string attached to the black hole.

The paper proposes observational strategies based on shifts in the iron‑K(\alpha) line profile and multi‑temperature blackbody modeling of the accretion disk spectrum to detect the ISCO movement. It also acknowledges several simplifications: the string is treated as a static, infinitely long Nambu–Goto line with no oscillations; back‑reaction of the disk plasma on the string, jet‑string interactions, and detailed general‑relativistic magnetohydrodynamic effects are omitted.

In summary, the study presents a novel theoretical link between cosmic‑string tension and observable ISCO dynamics in low‑mass X‑ray binaries, highlighting the transition phase as a promising diagnostic. While the analytical framework is clear, the quantitative estimates rely on highly idealized assumptions, and realistic detection of the predicted effects—especially for very low string tensions—will require more sophisticated modeling and high‑resolution X‑ray spectroscopy. Future work should incorporate full GRMHD simulations and explore the impact of string dynamics on jet formation and disk turbulence.