Frequency-domain waveform approximants capturing Doppler shifts

Gravitational-wave astrophysics has only just begun, and as current detectors are upgraded and new detectors are built, many new, albeit faint, features in the signals will become accessible. One such feature is the presence of time-dependent Doppler shifts, generated by the acceleration of the center of mass of the gravitational-wave emitting system. We here develop a generic method that takes a frequency-domain, gravitational-wave model devoid of Doppler shifts and introduces modifications that incorporate them. Building upon a perturbative expansion that assumes the Doppler-shift velocity is small relative to the speed of light, the method consists of the inclusion of a single term in the Fourier phase and two terms in the Fourier amplitude. We validate the method through matches between waveforms with a Doppler shift in the time domain and waveforms constructed with our method for two toy problems: constant accelerations induced by a distant third body and Gaussian accelerations that resemble a kick profile. We find mismatches below $\sim {10}^{-6}$ for all of the astrophysically relevant cases considered and that improve further at smaller velocities. The work presented here will allow for the use of future detectors to extract new, faint features in the signal from the noise.

Figure 1

Schematic representation of the calculation presented in this paper. The blue arrows indicate the technical steps carried out in Sec. 2. The red arrow indicates the frequency-domain method we develop to modify a preexisting frequency-domain waveform approximant to include Doppler shifts.

Figure 2

Gravitational-wave strains in the time-domain for an equal mass $m_1=m_2=10M_{\odot }$ nonspinning system as viewed from (i) an inertial frame at rest with respect to the binary (black curve), (ii) an accelerating frame with $a={10}^4{a}_0$ computed exactly in the time domain (blue curve; TD), and (iii) the same accelerating frame waveform computed with the frequency-domain expressions developed in this paper (red curve; FD). The accelerated frame is chosen such that it coincides with the rest frame at the instant of merger. Accelerated and inertial waveforms, therefore, are in phase near merger and drift out of phase in the early inspiral (this can be seen most clearly in the inset plots). The bottom panel shows the (wrapped) phase difference between different pairs of waveforms. The black curve shows the phase difference between the accelerated TD waveform and the inertial frame waveform; observe that they dephase by just over one complete cycle in the 100 seconds before merger. The red curve shows the phase difference between the TD and FD methods of computing the accelerated waveform; the fast FD method dephases from the exact TD method by less than 0.1 radians over this time interval.

Figure 3

Mismatch between the inertial and accelerated waveforms (blue dashed curve), and between the accelerated TD and accelerated FD waveforms (solid red curve) as a function of the magnitude of the acceleration or the accumulated velocity. Mismatches are computed using the ET-D noise PSD [37] starting from 5 Hz, for which the signal duration is $T=240\mathrm{s}$. During this time, the accelerated frame accumulates a total change in velocity of $v=aT$ relative to the inertial frame as reported on the upper $x$-axis. Even at large values of the acceleration, the fast frequency-domain approximation shows excellent agreement with the exact time-domain method: mismatches are less than ${10}^{-2}$ in all cases and improve rapidly as the acceleration decreases to more astrophysically realistic values.

Figure 4

Plotted in black in the main panel is the normal, “unkicked” waveform for an equal mass $m_1=m_2=10M_{\odot }$ nonspinning waveform. The two colored curves show the same waveform but with an artificial recoil kick of $v_k=0.1c$ (unphysically large for testing purposes) applied at merger computed exactly in the time-domain (blue; TD) and the frequency-domain approximation developed in this paper (red; FD). The kick is only imparted for a period about $\sigma =10(m_1+m_2)$ near merger. The kicked and unkicked waveforms are therefore in phase in the early inspiral and drift out of phase during the merger and ringdown phase. The two methods of calculating the kicked waveform signal (TD and FD) are in excellent agreement even for such a large value of the kick velocity. The difference between the two is barely visible in the ringdown signal in the right hand inset plot.

Figure 5

Mismatch between the kicked and inertial waveforms (blue dashed curve), and between the kicked TD and kicked FD waveforms (solid red curve) as a function of the magnitude of the recoil kick velocity. Mismatches were computed using the LIGO noise PSD [41] starting from 30 Hz. Even at large values of the recoil velocity, our frequency-domain recipe shows excellent agreement with the exact time-domain method. Mismatches improve exponentially as the velocity decreases to more astrophysically realistic values.