Extreme-mass ratio inspirals in Schwarzschild - de Sitter spacetime I: Weak-field orbits

Abstract

The inspiral of a compact object into a black hole is a key source of low-frequency gravitational waves for future space-based detectors like LISA. While models of this process have advanced, they typically focus on asymptotically flat spacetimes. In this paper, we explore how the absence of asymptotic flatness affects the slow, adiabatic orbital evolution due to radiation reaction. This lack of asymptotic flatness can arise from external environments or an expanding universe. Using the Schwarzschild-de Sitter (SdS) spacetime, where the deviation from flatness is governed by the cosmological constant, we study bound orbits characterized by their semi-latus rectum $p$ and eccentricity $e$. We calculate how the cosmological constant shifts the separatrix between bound and plunging orbits and alters the relationship between the binary's binding energy, angular momentum, and orbital parameters. Assuming the orbital timescale is much shorter than the inspiral timescale, we apply a modified quadrupole formula to examine the impact of a small positive cosmological constant on the orbital evolution in the weak-field limit. We find that the cosmological constant accelerates the decrease in eccentricity, reducing inspiral plunge times, which could influence event rate estimates for space-based detectors.

Figures (18)

• Figure 1: (a) Constant $p$ and $e$ curves in SdS $(E,L)$ space with black lines as constant $p$ curves increasing to the right and red lines as constant $e$ curves increasing upwards. (b) Constant $E$ (black) and $L$ curves (red) in $(p,e)$ space in SdS spacetime. The dashed curves are the separatrices (equation \ref{['eq:separatrix']} and equation \ref{['eq:scattersep']}) of bound orbits or the curves where the determinant of the Jacobian of transformation from $(p,e)\rightarrow(E,L)$ becomes zero.
• Figure 2: The separatrix equation $P(p,e,\lambda)$ (equation \ref{['eq:separatrix']}) in $(p,e)$ space. In (a) the contours have varying $\lambda$ with the filled region as the allowable parameter space for stable bound orbits. Introducing a $\lambda>0$ makes most eccentric orbits unbound. We see in the red and violet curve that a sufficiently high cosmological constant would also make orbits with high separation (larger orbits) also unbound. In (b) we show a continuous variation of $\lambda$ onto the phase space $(p,e)$ wherein the marginally bound orbits are the points lying on the surface.
• Figure 3: Similar contour plot and 3D plot to Figure \ref{['fig:separatrix']} but with the scattering separatrix equation $S(p,e,\lambda)$ (equation \ref{['eq:scattersep']}).
• Figure 4: Potential shapes, $V(r)$ vs $r$, (black curves in the right with dashed line as the orbital energy, $E$) of selected orbital parameters (black dots) in $(p,e)$ space. The intersection of orange and blue region contains the space of all bound orbits. The orange boundary (orange dashed line) marks the point at which orbits are marginally bound and may scatter away to the cosmological horizon while the blue boundary (blue dashed line) marks the points at which orbits are also marginally bound but may plunge to the black hole. The complement of the colored regions are unbound and may either plunge or scatter away.
• Figure 5: Gravitational flux (vectors) tangent to the $e=0$ curve (black) in the $(E,L)$ space. We see here as discussed in Section \ref{['sec:sec3.1']} that far from the OSCO (inset, low $(E,L)$), circularity is maintained (vectors lie along the black curve) but as for orbits with separation near the OSCO, the orbits will cross the scattering separatrix (yellow) and become unbound with the vectors crossing the yellow dashed line.