The LISA Optimal Sensitivity

TL;DR

The paper tackles maximizing LISA's gravitational-wave sensitivity by exploiting its multiple, laser-noise–free interferometric observables. It formulates an optimization over the four TDI generators (α, β, γ, ζ), showing that the optimal SNR is achieved by a linear combination that effectively uses only three observables (α, β, γ) through a Rayleigh-principle eigenproblem involving the noise covariance ${\bf C}$ and signal matrix ${\bf A}$. The resulting optimal SNR is ${SNR_{\eta}^2}_{\rm opt.} = \int_{f_l}^{f_u} {\bf x^{(s)*}_i (\bf C^{-1})_{ij} {\bf x^{(s)}_j}} df$, and the approach yields an orthogonal three-mode decomposition into A, E, T modes. Application to sinusoidal sources in an equal-arm LISA configuration shows a sensitivity improvement by a factor of about $\sqrt{2}$ in the long-wavelength band, increasing to beyond $\sqrt{3}$ in higher-frequency bands, with the T mode contributing notably at certain frequencies. These results reinforce viewing LISA as a network of detectors and have practical implications for the inverse problem of locating sources and estimating their amplitudes.

Abstract

The multiple Doppler readouts available on the Laser Interferometer Space Antenna (LISA) permit simultaneous formation of several interferometric observables. All these observables are independent of laser frequency fluctuations and have different couplings to gravitational waves and to the various LISA instrumental noises. Within the functional space of interferometric combinations LISA will be able to synthesize, we have identified a triplet of interferometric combinations that show optimally combined sensitivity. As an application of the method, we computed the sensitivity improvement for sinusoidal sources in the nominal, equal-arm LISA configuration. In the part of the Fourier band where the period of the wave is longer than the typical light travel-time across LISA, the sensitivity gain over a single Michelson interferometer is equal to $\sqrt{2}$. In the mid-band region, where the LISA Michelson combination has its best sensitivity, the improvement over the Michelson sensitivity is slightly better than $\sqrt{2}$, and the frequency band of best sensitivity is broadened. For frequencies greater than the reciprocal of the light travel-time, the sensitivity improvement is oscillatory and $\sim \sqrt{3}$, but can be greater than $\sqrt{3}$ near frequencies that are integer multiples of the inverse of the one-way light travel-time in the LISA arm.

Figures (5)

• Figure 1: Schematic LISA configuration. Each spacecraft is equidistant from the point O, in the plane of the spacecraft. Unit vectors $\hat{n}_i$ point between spacecraft pairs with the indicated orientation. At each vertex spacecraft there are two optical benches (denoted 1, $1^*$, etc.), as indicated.
• Figure 2: Schematic diagram of proof-masses-plus-optical-benches for a LISA spacecraft. The left-hand bench reads out the Doppler signals $y_{31}$ and $z_{31}$. The right hand bench analogously reads out $y_{21}$ and $z_{21}$. The random velocities of the two proof masses and two optical benches are indicated (lower case $\vec{v}_i$ for the proof masses, upper case $\vec{V}_i$ for the optical benches.)
• Figure 3: The LISA Michelson sensitivity curve (SNR = 5) and the sensitivity curve for the optimal combination of the data, both as a function of Fourier frequency. The integration time is equal to one year, and LISA is assumed to have a nominal armlength $L = 16.67 {\rm sec}$.
• Figure 4: The optimal SNR divided by the SNR of a single Michelson interferometer, as a function of the Fourier frequency $f$. The sensitivity gain in the low-frequency band is equal to $\sqrt{2}$, while it can get larger than $2$ at selected frequencies in the high-frequency region of the accessible band. The integration time has been assumed to be one year, and the proof mass and optical path noise spectra are the nominal ones. See the main body of the paper for a quantitative discussion of this point.
• Figure 5: The SNRs of the three combinations, ($A, E, T$), and their sum as a function of the Fourier frequency $f$. The SNRs of $A$ and $E$ are equal over the entire frequency band. The SNR of $T$ is significantly smaller than the other two in the low part of the frequency band, while is comparable to (and at times larger than) the SNR of the other two in the high-frequency region. See text for a complete discussion.