Higher-order spin effects in the amplitude and phase of gravitational waveforms emitted by inspiraling compact binaries: Ready-to-use gravitational waveforms

We provide ready-to-use time-domain gravitational waveforms for spinning compact binaries with precession effects through 1.5 post-Newtonian (PN) order in amplitude, and compute their mode decomposition using spin-weighted $-2$ spherical harmonics. In the presence of precession, the gravitational-wave modes $(\ell ,m)$ contain harmonics originating from combinations of the orbital frequency and precession frequencies. We find that the gravitational radiation from binary systems with large mass asymmetry and large inclination angle can be distributed among several modes. For example, during the last stages of inspiral, for some maximally spinning configurations, the amplitude of the (2, 0) and (2, 1) modes can be comparable to the amplitude of the (2, 2) mode. If the mass ratio is not too extreme, the $\ell =3$ and $\ell =4$ modes are generally 1 or 2 orders of magnitude smaller than the $\ell =2$ modes. Restricting ourselves to spinning, nonprecessing compact binaries, we apply the stationary-phase approximation and derive the frequency-domain gravitational waveforms including spin-orbit and spin(1)-spin(2) effects through 1.5PN and 2PN order, respectively, in amplitude, and 2.5PN order in phase. Since spin effects in the amplitude through 2PN order affect only the first and second harmonics of the orbital phase, they do not extend the mass reach of gravitational-wave detectors. However, they can interfere with other harmonics and lower or raise the signal-to-noise ratio depending on the spin orientation. These ready-to-use waveforms could be employed in the data analysis of the spinning, inspiraling binaries as well as in comparison studies at the interface between analytical and numerical relativity.

Figure 1

We show (i) our source frame defined by the orthonormal basis $(\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}})$, (ii) the instantaneous orbital plane which is described by the orthonormal basis $({\widehat{\mathbf{x}}}_{\mathrm{L}},{\widehat{\mathbf{y}}}_{\mathrm{L}},{\widehat{\mathbf{L}}}_N)$, (iii) the polarization triad $(\widehat{\mathbf{N}},\widehat{\mathbf{P}},\widehat{\mathbf{Q}})$, and (iv) the direction of the total angular momentum at initial time ${\mathbf{J}}_0$. Dashed lines show projections into the $x\mathrm{\text{−}}y$ plane.

Figure 2

The left panel shows the inclination angle of the orbital angular momentum relative to the total angular momentum, $\iota$, as a function of $2M\dot{\Phi }$ for binaries with mass ratios $1:1$ and $4:1$, having initial spin orientations relative to the orbital angular momentum: ${\mathrm{\text{Spin}}}_{\mathrm{A}}=\{{\theta }_1=\pi /2,{\varphi }_1=0,{\theta }_2=\pi /2,{\varphi }_2=\pi /2\}$ and ${\mathrm{\text{Spin}}}_{\mathrm{B}}=\{{\theta }_1=\pi /6,{\varphi }_1=\pi /4,{\theta }_2=\pi /6,{\varphi }_2=\pi \}$. The right panel compares the modulus of $h_{22}$ for the two precessing spin configurations with the nonspinning, aligned and antialigned cases for equal masses. The computations use waveforms accurate to 1.5PN in amplitude and phase evolved with the precession equations at 1.5PN order [see Eqs. (3.11, 3.18, 3.20, 3.21)]. Note that these plots (and those of Figs. 3, 4) begin at $2M\dot{\Phi }=0.02$ which is approximately where the dominant second harmonic from a binary of total mass $16M_{\odot }$ enters the LIGO band at 40 Hz and where the second harmonic from a binary of total mass $6.5\times {10}^6{M}_{\odot }$ enters the LISA band at ${10}^{-4}\mathrm{Hz}$.

Figure 3

We plot the modulus of the $\ell =2$ modes for mass ratios $1:1$ (left panel) and $4:1$ (right panel) with the spin configurations described in Fig. 2. The computations use waveforms accurate to 1.5PN order in amplitude and phase evolved with precession equations at 1.5PN order. The dashed lines are the larger $\iota$ configuration (${\mathrm{\text{Spin}}}_{\mathrm{A}}$) and the solid lines are the smaller $\iota$ configuration (${\mathrm{\text{Spin}}}_{\mathrm{B}}$). We see that as $\iota$ increases, the modulus of ${\widehat{h}}_{22}$ decreases while the modulus of the other $\ell =2$ modes increases. This effect becomes more pronounced when the mass ratio is more extreme.

Figure 4

We plot the modulus of the $\ell =3$ modes (left panel) and $\ell =4$ modes (right panel) for equal masses with the spin configurations described in Fig. 2. The computations use waveforms accurate to 1.5PN order in both amplitude and phase evolved by means of the 1.5PN precession equations. The dashed lines refer to the larger $\iota$ configuration (${\mathrm{\text{Spin}}}_{\mathrm{A}}$), the solid lines to the smaller iota configuration (${\mathrm{\text{Spin}}}_{\mathrm{B}}$). We see a redistribution of power among the modes similar to Fig. 3. As $\iota$ increases, the largest modes for the nonprecessing cases ($|{\widehat{h}}_{32}|$ and $|{\widehat{h}}_{44}|$) become smaller, while the other modes become larger.

Figure 5

We compare the power spectra computed with the Newtonian amplitude waveform (red dashed line) and the 2.5PN waveform with 1.5PN SO and 2PN SS effects included (blue, continuous line). In the left panel we consider a typical source for LISA, a binary with total mass $({10}^6+{10}^5)M_{\odot }$, and spins maximal and aligned with the orbital angular momentum. In the right panel we consider a typical source for Advanced LIGO, a binary of total mass $(30+30)M_{\odot }$ with spins ${\chi }_1=1$, ${\chi }_2=0.5$ aligned with the orbital angular momentum. Note that the $k$th harmonic ends at $k{F}_{\mathrm{LSO}}$, and these frequencies are marked by the vertical dashed lines on the graph. The spectrum of the 2.5PN waveform is much simpler in the equal-mass case than unequal mass case because in the former case all nonspinning odd harmonics are suppressed.

Figure 6

For a binary of total mass $(30+30)M_{\odot }$ with spins ${\chi }_1=1$, ${\chi }_2=0.5$ aligned with the orbital angular momentum (the same binary of the right panel of Fig. 5), we show the power spectra up to $2F_{\mathrm{LSO}}$. We plot the power spectrum for the waveform through 2.5PN order with no spin corrections (cyan solid line) and with SO corrections through 1.5PN (that is, 1PN and 1.5PN) and SS corrections at 2PN order (dark-blue solid line). We also plot power spectra for the waveform with Newtonian amplitude (red dashed line), Newtonian amplitude plus the 1PN SO correction (black dotted line), Newtonian amplitude plus SO effects through 1.5PN (green, dot-dashed line), and Newtonian amplitude plus SO corrections through 1.5PN and the 2PN SS correction (magenta, double-dot-dashed line). The 1.5PN SO and 2PN SS effects raise and lower the power in the dominant harmonic while the 1PN SO effect merely changes the modulation pattern up to its cutoff frequency of $F_{\mathrm{LSO}}$. Vertical dashed lines mark the frequencies $F_{\mathrm{LSO}}$ and $2F_{\mathrm{LSO}}$.