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

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

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

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

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

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

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 $\text{FT}h(t)^* = \text{FT}[h(t)]^* (-f)$.

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, $\widetilde{H}{\ell m}$:

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

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 $\widetilde{H}_{\ell m} (f)$.