Time-domain analysis of a dynamically tuned signal recycled interferometer for the detection of chirp gravitational waves from coalescing compact binaries

In this article, we study a particular method of detection of chirp signals from coalescing compact binary stars—the so-called dynamical tuning, i.e., amplification of the signal via tracking of its instantaneous frequency by the tuning of a signal-recycled detector. The motion of the signal-recycling mirror, the position of which defines the tuning of the detector, causes nonstationarity of the detector. The dynamically tuned detector can be simulated in a quasistationary approximation if the mirror position, amplitude, and frequency of a chirp signal are changing slowly. A time-domain consideration developed for signal-recycled interferometers, in particular GEO 600, describes the signal and noise evolution in the more general case of a purely nonstationary detector. We prove that the shot noise from the dark port and optical losses remains white in this case. The analysis of the transient effects shows that during the perfect tracking of the chirp frequency only transients from fast amplitude changes arise because the transients from changes of the detector tuning and signal frequency completely cancel each other. The slow change of the amplitude in this case establishes a so-called virtually stationary detection, meaning the signal fields at the detector hold their stationary values at each instance of time, corresponding to the instantaneous parameters of the gravitational wave and of the detector. The signal-to-noise-ratio gain from the implementation of dynamical tuning, calculated in this paper, is $\sim 17$ for a shot noise-limited GEO 600-like detector and $\sim 7$ for a detector with both shot and displacement noise.

Figure 1

Scheme of the considered GW detector. The notations are presented in Table 1.

Figure 2

The quantum noise of broadband and narrow band detector configurations. The quantum noise of a quasistationary dynamical tuning (the points of optical resonance in curves corresponding to each tuning) is also presented to compare with the mirror displacement noise.

Figure 3

The quasistationary approximations of dynamical tuning for the detector with the full quantum noise (with radiation pressure) and with the shot noise only.

Figure 4

The typical impulse response of the considered detector with a constant detuning $f_{\text{tun}}=100\mathrm{Hz}$: (a) in the response decay-time scale and (b) in the single round-trip time scale ($\sim 8\mu \mathrm{s}$).

Figure 5

Two orthogonal quadratures of the auxiliary impulse response (real and imaginary parts) to the vacuum quantum oscillations, injected from the dark port: (a) on the large amplitude scale, depicting the direct reflection of the input pulse from the SRM; (b) on the small amplitude scale, representing the output envelope from the oscillations and decay (in the rotated frame) of the amplitude pulse as it propagating inside the SRC; and (c) on the short time scale, picturing the round-trip time and the discrete nature of the impulse function.

Figure 6

Two quadratures of the auxiliary impulse response (real and imaginary parts), depicted on the decay time scale, to the vacuum quantum oscillations, injected from an equivalent end mirror. The difference between the east and the north end mirrors is insignificant on the plot scale.

Figure 7

The typical transients of the considered detector on the stepwise change of (a) $X(t)$, (b) $f(t)$, and (c) ${\delta }_{\mathrm{f}}(t)$.

Figure 8

The transients of the considered detector on the combinations of stepwise changes of (a) $f(t)$ and ${\delta }_{\mathrm{f}}$; (b) $X(t)$, $f(t)$, and ${\delta }_{\mathrm{f}}(t)$; and (c) $X(t)$ and ${\delta }_{\mathrm{f}}(t)$.

Figure 9

(a) The gravitational wave signal from a $5+5$ Solar mass spinless black hole binary. (b) The instantaneous frequency of this signal. (c) The output for the resonantly tracked detection of this signal, calculated using three models: numerical simulation, a transformed envelope, and a quasistationary approximation.

Figure 10

The theoretical noise budget of the current GEO 600 configuration.

Figure 11

Comparison of the output signal envelopes from the dynamical tuning with respect to the reference stationary detection. The detected GW is from a $5+5$ Solar mass spinless black hole binary (see Fig. 9).

Figure 12

The SNR gain from implementation of dynamical tuning into a shot-noise-limited detector that worked in a stationary broadband regime. The influence of the nonstationary dynamical effects on it becomes apparent in comparison with the depicted result of a quasistationary approximation.

Figure 13

The ratio of SNRs for dynamical tuning in a shot-noise-limited and in a displacement-noise-limited detector.

Figure 14

The SNR gain from the implementation of narrow-banded dynamical tuning into a broadband detector with both shot and displacement noise. Dynamical tuning is assumed to remove the influence of shot noise and to make the detector displacement noise-limited.

Figure 15

Scheme of the simplest Fabry–Perot cavity. $T$ and $T_{\mathrm{f}}$ are the amplitude transmittances of the input and of the end mirrors. $R$ and $R_{\mathrm{f}}$ are corresponding reflectivity coefficients. a–f are the electromagnetic field amplitudes in corresponding places. $L$ is the length of the cavity, resonant to laser frequency.