Sensitivity curves for spaceborne gravitational wave interferometers

TL;DR

This work develops a rigorous framework to compute sensitivity curves for spaceborne gravitational wave interferometers by deriving an exact gravitational-wave transfer function averaged over direction and polarization. It corrects prior underestimations (notably by a factor of $\sqrt{3}$) and shows how to combine independent noise sources with the transfer function to yield $h_f$ and $S_h$ curves, demonstrated with the LISA design. The analysis identifies a knee near $0.01$ Hz set by the arm length and a floor governed by position noise, clarifying how interpretation differs for continuous, burst, and stochastic sources. By reconciling different methodologies in the literature and outlining a consistent mapping to various source classes, the paper provides a practical toolkit for evaluating and optimizing space-based GW observatories.

Abstract

To determine whether particular sources of gravitational radiation will be detectable by a specific gravitational wave detector, it is necessary to know the sensitivity limits of the instrument. These instrumental sensitivities are often depicted (after averaging over source position and polarization) by graphing the minimal values of the gravitational wave amplitude detectable by the instrument versus the frequency of the gravitational wave. This paper describes in detail how to compute such a sensitivity curve given a set of specifications for a spaceborne laser interferometer gravitational wave observatory. Minor errors in the prior literature are corrected, and the first (mostly) analytic calculation of the gravitational wave transfer function is presented. Example sensitivity curve calculations are presented for the proposed LISA interferometer. We find that previous treatments of LISA have underestimated its sensitivity by a factor of $\sqrt{3}$.

Figures (5)

• Figure 1: A typical configuration for a spaceborne gravitational wave interferometer. Three probes form an equilateral triangle, inscribed on the relative orbits of the spacecraft. A single Michelson interferometer is formed using any two legs of the triangle, and a second (non-independent) interferometer can be formed using the third arm of the configuration in conjunction with an arm already in use for the primary signal.
• Figure 2: The root spectral density of the noise in the LISA interferometer formed by adding the strain noises induced by acceleration noise and by total position noise in quadrature.
• Figure 3: The geometrical relationship of the interferometer to the propagation vector of a gravitational wave, used to conduct spatial averaging. The arms are designated by vectors $\bf{L_{1}}$ and $\bf{L_{2}}$ (solid black vectors), while the propagation vector of the gravitational wave is given by $\bf{k}$ (open white arrow). One arm of the interferometer is aligned along the polar axis of a 2-sphere, the other arm lying an angular distance $\gamma$ away long a line of constant longitude. The angles $\theta_{i}$ relate the vector $\bf{k}$ to the arms of the interferometer, and the angle $\epsilon$ is the inclination of the plane containing $\bf{k}$ and $\bf{L_{1}}$ to the plane of the interferometer.
• Figure 4: The transfer function $R(u)$ is shown as a function of the dimensionless variable $u = \omega \tau$. Note that it is roughly constant at low frequencies, and has a "knee" located at $u = \omega \tau \sim 1$.
• Figure 5: The sensitivity curves for the proposed LISA observatory is shown. The low frequency rise is due to acceleration noise in each of the systems. The high frequency rise is due to the "knee" in the transfer function at $f \simeq (2 \pi \tau)^{-1}$. The structure at high frequencies is a consequence of the high frequency structure in the gravitational wave transfer function.