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Relativistic Corrections to the Formation Rate of Extreme Mass-Ratio Inspirals

Chen Feng, Yong Tang

Abstract

Extreme mass-ratio inspirals (EMRIs) are long-duration gravitational-wave sources in which a compact object gradually spirals into a massive black hole. Their formation is governed by the interplay between stochastic angular-momentum diffusion driven by two-body relaxation and the dissipative evolution caused by gravitational-wave emission, with the loss-cone boundary deciding whether an object undergoes an inspiral or a direct plunge. Building on this physical picture, we construct a relativistically self-consistent analytic framework for estimating EMRI event rates. In Schwarzschild spacetime, we generalize the standard loss-cone angular momentum to an energy-dependent quantity and revise the plunge pericenter by using the minimum stable radius derived from general relativity. Relative to the Newtonian treatment, we show that incorporating these relativistic effects increases the predicted EMRI rates by roughly a factor of 8. This enhancement becomes more pronounced for shallower stellar density profiles and is insensitive to the mass of the central massive black hole, which emphasizes that relativistic effects are essential for EMRI rate estimations that are relevant for space-based gravitational-wave detectors, such as LISA and Taiji.

Relativistic Corrections to the Formation Rate of Extreme Mass-Ratio Inspirals

Abstract

Extreme mass-ratio inspirals (EMRIs) are long-duration gravitational-wave sources in which a compact object gradually spirals into a massive black hole. Their formation is governed by the interplay between stochastic angular-momentum diffusion driven by two-body relaxation and the dissipative evolution caused by gravitational-wave emission, with the loss-cone boundary deciding whether an object undergoes an inspiral or a direct plunge. Building on this physical picture, we construct a relativistically self-consistent analytic framework for estimating EMRI event rates. In Schwarzschild spacetime, we generalize the standard loss-cone angular momentum to an energy-dependent quantity and revise the plunge pericenter by using the minimum stable radius derived from general relativity. Relative to the Newtonian treatment, we show that incorporating these relativistic effects increases the predicted EMRI rates by roughly a factor of 8. This enhancement becomes more pronounced for shallower stellar density profiles and is insensitive to the mass of the central massive black hole, which emphasizes that relativistic effects are essential for EMRI rate estimations that are relevant for space-based gravitational-wave detectors, such as LISA and Taiji.
Paper Structure (10 sections, 30 equations, 2 figures, 1 table)

This paper contains 10 sections, 30 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The event rate of EMRIs
  3. The formation of EMRIs
  4. EMRIs event rate
  5. General Relativistic Treatment of Loss-Cone Boundary
  6. Loss-cone angular momentum in Schwarzschild spacetime
  7. Mapping from semi-major axis to energy-dependent $L_{\mathrm{lc}}$
  8. GR-corrected plunge pericenter
  9. Effect on the EMRI event rate
  10. Conclusion

Figures (2)

  • Figure 1: Effect-dominated region and loss-cone curve in the $(1-e)$--$a$ plane. The blue, orange, and gray regions correspond to the relaxation-dominated, GW-emission-dominated, and loss-cone regions, respectively. The solid line indicates $t_{\mathrm{gw}} = t_{\mathrm{rlx}}$, whereas the dashed line represents the loss-cone condition $a(1 - e) = r_{\mathrm{p};\, \mathrm{lc}}$.
  • Figure 2: Comparison of two loss-cone curves in the $(1-e)$-$a$ plane. The black dashed line denotes the case $a(1-e)=8GM_{\mathrm{BH}}$, while the red solid line denotes the Schwarzschild-spacetime correction. The blue solid line indicates where the characteristic timescale of gravitational-wave emission equals the relaxation timescale for the evolution of periapsis. The circle- and star-shaped intersection points denote the values of $a_{\mathrm{cri}}$ corresponding to the two loss-cone curves, respectively.