Higher order modes

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This section will give some mathematical notes on how a generic spherical harmonic decomposition for a frequency-domain GW works.

A time-domain wave \(h(t)\) is expressed as [ABB+11]:

\[ h(t) = h_+(t) - i h_\times (t) = \frac{M}{d_L} \sum_{\ell \geq 2} \sum_{|m| \leq \ell} H_{\ell m}(t) {}^{(-2)}Y_{\ell m} \]

Here, we are using \(G = c = 1\) units (and thus neglecting a \(G/c^2\) factor multiplying \(M/d_L\) to make it dimensionless).

The polarizations \(h_+(t)\) and \(h_\times(t)\) are both real-valued, so knowing \(h(t)\) allows us to recover them both.

The reason for this parametrization is that it relates to the asymptotic Weyl scalar \(\Psi _4\) by

\[ \Psi _4 = \ddot{h}_+ - i \ddot{h}_ \times \,; \]

for more details see Ajith et al. [ABB+11]. The expansion coefficients \(H_{\ell m}\) are defined by an integral in the form

\[ H_{\ell m} = \frac{d_L}{M} \int \mathrm{d}\Omega {}^{(-2)}Y^*_{\ell m} (h_+ - i h_ \times ) \]

One can see from this that separating out the mass dependence in the definition of \(H_{\ell m}\) is an arbitrary choice, we could just as well have defined \(h \sim d_L^{-1} \sum_{\ell m} \widetilde{H} Y\).

Another arbitrary choice is setting \(h = h_+ - i h_\times\) as opposed to \(h = h_+ + i h_\times\) — switching between the two is equivalent to changing the sign of the phase.

Spherical harmonics

The spin-weighted spherical harmonics are functions of the orientation of the source, parametrized as \(\iota\) (inclination angle, between the observation direction and the source angular momentum) and \(\phi_0\) (initial phase of the source’s rotation). The formulas given here are again from Ajith et al. [ABB+11], if anything does not make sense check there.

Explicitly, they are given as:

\[{}^{(-s)}Y_{\ell m} = (-1)^s \sqrt{\frac{2 \ell + 1}{4 \pi }} d^\ell_{m, s} (\iota ) e^{im \phi_0 } \]

where \(d^\ell_{m, s} (\iota )\) is called a Wigner \(d\)-function, and is given by

\[ d^\ell_{m, s} (\iota ) = \sum _{k=k_1 }^{k_2 } \frac{(-1)^k \sqrt{(\ell+m)! (\ell-m)! (\ell+s)! (\ell-s)!}}{(\ell+m-k)! (\ell-s-k)! k! (k+s-m)!} \left(\cos (\iota / 2)\right)^{2 \ell + m -s - 2k} \left(\sin (\iota / 2)\right)^{2 k + s - m}\,, \]

where \(k_1 = \max(0, m-s)\) and \(k_2 = \min (\ell + m, \ell - s)\).

Note that in the GW case the label for the harmonics \(Y\) is \(-2\), but the parameter \(s\) in the \(d\)-function is equal to \(+2\). For reference, we give the two most useful harmonics:

\[ {}^{(-2)}Y_{2 \pm 2} = \sqrt{ \frac{5}{64 \pi }} (1 \pm \cos \iota )^2 e^{\pm 2i \phi }\,. \]

Identities

These harmonics are an orthogonal basis:

\[ \int \mathrm{d}\Omega {}^{(-2)}Y_{\ell m} {}^{(-2)}Y_{\ell' m'} = \delta _{\ell \ell'} \delta _{m m'} \]

Also, they satisfy:

\[ {}^sY_{\ell m} = (-1)^{s+m} Y^*_{\ell -m} \]

The time-domain 22 wave

We derive the time-domain expression for the \(\ell=2\), \(|m| = 2\) harmonic, which is the most commonly used approximation for a gravitational wave.

According to the expression we gave earlier, we will have

\[\begin{split} \begin{align} h(t) &= h_+(t) - i h_\times (t) = \frac{M}{d_L} \sum_{\ell = 2} \sum_{|m| = 2} H_{\ell m}(t) {}^{(-2)}Y_{\ell m} \\ &= \frac{M}{d_L} \left( H_{22}(t) {}^{(-2)}Y_{22} + H_{2-2}(t) {}^{(-2)}Y_{2-2}\right) \\ &= \frac{M}{d_L} \sqrt{ \frac{5}{64 \pi }} \left( H_{22}(t) (1 + \cos \iota )^2 e^{2 i \phi_0 } + H_{2-2}(t) (1 - \cos \iota )^2 e^{-2 i \phi_0 }\right) \end{align} \end{split}\]

Now, we can make use of the fact that, thanks to symmetry under reflection across the orbital plane, we have \(H_{\ell m} = (-1)^\ell H_{\ell -m}^*\), which in this case reduces to \(H_{22} = H_{2-2}^*\) (see section II.D in [OBM+20]). Therefore, if we define \(\widetilde{H}_{22} = H_{22} e^{2i \phi_0 }\), we will have

\[\begin{split} \begin{align} h(t) &= \frac{M}{d_L} \sqrt{ \frac{5}{64 \pi }} \left( \widetilde{H}_{22} (t) (1 + \cos \iota )^2 + \widetilde{H}_{22}^* (t) (1 - \cos \iota )^2 \right) \\ &= \frac{M}{d_L} \sqrt{ \frac{5}{64 \pi }} \left( 2 \Re \widetilde{H}_{22} (t) (1 + \cos^2 \iota ) + 4 i \Im \widetilde{H}_{22} (t) \cos \iota \right) \\ &= h_+ - i h_\times \end{align} \end{split}\]

At this point, we can identify the real and imaginary components, as well as expressing \(\widetilde{H}_{22} = H_{22} e^{2 i \phi_0 } = A_{22}(t) e^{i \phi_{22}(t) + 2i \phi_0 }\):

\[\begin{split} \begin{align} h_+ &= \frac{4 M}{d_L} \sqrt{ \frac{5}{64 \pi }} A_{22}(t) \frac{1 + \cos^2 \iota }{2} \cos(\phi_{22}(t) + 2 \phi_0) \\ h_\times &= - \frac{4 M}{d_L} \sqrt{ \frac{5}{64 \pi }} A_{22}(t) \cos \iota \sin(\phi_{22}(t) + 2 \phi_0 ) \end{align} \end{split}\]

This is the same expression we get in the quadrupole, Newtonian approximation — see, for example, equations 4.3 in Maggiore [Mag07], as long as we reabsorb the coefficients into the amplitude.

Frequency-domain waves

If we wish to work in the frequency domain, things get slightly more complicated since we cannot assume that \(h_+\) and \(h_\times\) are real-valued anymore — their Fourier transforms will not be.

Most discussions about how to go from the frequency-domain modes \(H_{\ell m} (f)\) to the polarizations \(\widetilde{h}_{+, \times } (f)\) (Khan et al. [KOCH20], appendix E in García-Quirós et al. [GarciaQuirosCH+20]) also discuss the issue of performing a time-dependent rotation to move from the precessing case to the non-precessing one.

We shall write the expressions without the rotation matrices, one may refer to those papers for the general case.

The frequency-domain polarizations are the Fourier transforms of the real and imaginary parts of the waveform \(h(t)\):

\[ \widetilde{h}_+ (f) = \text{FT}[\Re h(t)] = \frac{1}{2} \left( \widetilde{h}(f) + \widetilde{h}^*(-f)\right) \]

\[ \widetilde{h}_\times (f) = \text{FT}[\Im h(t)] = \frac{i}{2} \left( \widetilde{h}(f) - \widetilde{h}^*(-f)\right) \]

where we used the facts that \(\Re h = (h + h^*) / 2\), \(\Im h = (h - h^*) / 2\), and \(\text{FT}[h(t)^*](f) = \text{FT}[h(t)]^* (-f)\).

The aforementioned relation \(\widetilde{h}_{\ell m} =(-1)^\ell \widetilde{h}^* _{\ell -m} (-f)\) allows us to simplify the summation we get when substuting \(h(f)\) and \(h^*(-f)\) with their expression in terms of the Fourier transforms of the modes, \(\widetilde{H}_{\ell m}\):

\[\begin{split} \begin{align} \widetilde{h}_+ (f) &= \frac{1}{2} \left( \widetilde{h}(f) + \widetilde{h}^*(-f)\right) \\ &= \frac{1}{2} \frac{M}{d_L} \sum _{\ell \geq 2} \sum _{|m|\leq \ell} \left( \widetilde{H}_{\ell m} (f) {}^{(-2)}Y_{\ell m} + \widetilde{H}^*_{\ell m} (-f) {}^{(-2)}Y_{\ell m} \right) \\ &\approx \frac{1}{2} \frac{M}{d_L} \sum _{\ell \geq 2} \sum _{1 < m \leq \ell} \left( \widetilde{H}_{\ell m} (f) {}^{(-2)}Y_{\ell m} + \widetilde{H}^*_{\ell m} (-f) {}^{(-2)}Y_{\ell m} + \widetilde{H}_{\ell -m} (f) {}^{(-2)}Y_{\ell -m} + \widetilde{H}^*_{\ell -m} (-f) {}^{(-2)}Y_{\ell - m} \right) \\ &= \frac{1}{2} \frac{M}{d_L} \sum _{\ell \geq 2} \sum _{1 < m \leq \ell} \left( \widetilde{H}_{\ell m} (f) {}^{(-2)}Y_{\ell m} + (-1)^\ell\widetilde{H}_{\ell -m} (f) {}^{(-2)}Y_{\ell m} + \widetilde{H}_{\ell -m} (f) {}^{(-2)}Y_{\ell -m} + (-1)^\ell \widetilde{H}_{\ell m} (f) {}^{(-2)}Y_{\ell - m} \right) \\ &= \frac{1}{2} \frac{M}{d_L} \sum _{\ell \geq 2} \sum _{1 < m \leq \ell} \left( \widetilde{H}_{\ell m} (f) \left( {}^{(-2)}Y_{\ell m} + (-1)^\ell {}^{(-2)}Y_{\ell - m} \right) + \widetilde{H}_{\ell -m} (f) \left( (-1)^\ell {}^{(-2)}Y_{\ell m} + {}^{(-2)}Y_{\ell -m} \right) \right) \\ &\approx \frac{1}{2} \frac{M}{d_L} \sum _{\ell \geq 2} \sum _{1 < m \leq \ell} \left( \widetilde{H}_{\ell m} (f) \left( {}^{(-2)}Y_{\ell m} + (-1)^\ell {}^{(-2)}Y_{\ell - m} \right) \right) \end{align} \end{split}\]

In the last step, we have approximated the contribution of the \(\widetilde{H}_{\ell -m}\) modes as zero. This is because we are working with positive frequencies, and modes with negative (positive) \(m\) computed at positive (negative) frequency are negligible.

Also, we have been neglecting the \(m = 0\) term, which is also typically small.

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A similar computation leads to

\[ h_\times(f) = - \frac{i}{2} \frac{M}{d_L} \sum _{\ell \geq 2} \sum _{0<m \leq \ell} \widetilde{H}_{\ell m} (f) \left( {}^{(-2)}Y_{\ell m} - (-1)^\ell {}^{(-2)}Y_{\ell - m} \right) \]

These are the final expressions we need, since they express the frequency-domain polarizations \(h_+(f)\) and \(h_\times(f)\) as a function of the frequency-domain modes \(\widetilde{H}_{\ell m} (f)\).