Spherical harmonic ($Y_{\ell }^m$)

A spherical harmonic is an eigenfunction of the Laplacian on the unit sphere:

The angular dipole is the degree-one spherical harmonic component. A real angular dipole is a function $\mathbf{d}\cdot \hat{\mathbf{n}}$ for a fixed vector $\mathbf{d}$. Direction-dependent observer velocity and gravitational endpoint terms can therefore be discarded from higher-multipole calculations when they have this form.

The angular monopole of a function on the sphere is its constant spherical-harmonic component, equal to its angular average. Subtracting it removes the mean temperature of a cosmic microwave background sky map without altering higher angular multipoles.

For an angular spherical harmonic $Y$ of degree $n\ge 1$, the radial, tangential gradient and tangential curl families can be normalized as $\mathbf{P}=\hat{\mathbf{r}}Y$, $\mathbf{B}=r\nabla Y/\sqrt{n(n+1)}$ and $\mathbf{C}=-r\hat{\mathbf{r}}\times \nabla Y/\sqrt{n(n+1)}$. They depend only on direction. At $n=0$ only the radial family is defined by these formulas.

A solid harmonic extends a spherical harmonic to a harmonic function in space with either the regular radial power $r^n$ or the irregular radial power $r^{-n-1}$. In three dimensions these are the two separated radial solutions of the Laplace equation at angular degree $n$.

An irregular solid harmonic is harmonic on punctured three-dimensional Euclidean space and generally singular at the origin. The Kelvin transform maps a degree-$n$ regular solid harmonic to a constant multiple of its irregular counterpart.

A degree-$n$ homogeneous polynomial satisfies Euler homogeneous function theorem $\mathbf{r}\cdot \nabla H_n=n{H}_n$. The product rule gives $\Delta (r^s{H}_n)=r^s\Delta H_n+s(2n+s+1)r^{s-2}{H}_n$ in three dimensions. Taking $s=-(2n+1)$ gives the displayed identity and the equivalence of harmonicity on $r>0$. The origin is excluded; even $1/r$ has a distributional Dirac delta function source there.

For the positive Laplace-Beltrami operator on the unit sphere $S^d$, with $d\ge 1$, the eigenvalues are $n(n+d-1)$ for integers $n\ge 0$. Their eigenspaces consist of restrictions of degree-$n$ harmonic polynomials and have dimensions $\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)-\left(\genfrac{}{}{0ex}{}{n+d-2}{d}\right)$, with the second term zero for $n<2$. The Laplacian in polar coordinates gives the eigenvalue; Fischer decomposition and the Stone-Weierstrass theorem prove completeness.

The integral of three spherical harmonics on the sphere. In the conventional complex basis it is real and equals a product of two Wigner 3j symbols and $\sqrt{(2{\ell }_1+1)(2{\ell }_2+1)(2{\ell }_3+1)/(4\pi )}$. It vanishes unless the multipoles satisfy a triangle condition, ${\ell }_1+{\ell }_2+{\ell }_3$ is even, and $m_1+m_2+m_3=0$. It carries the angular geometry of the reduced CMB bispectrum.

A regular solid harmonic is $r^l{Y}_l^m(\theta ,\varphi )$: it is homogeneous of degree $l$ and solves the Laplace equation. Thus $\mathbf{r}\cdot \nabla F=lF$ and ${\nabla }^{2}F=0$. These identities turn angular spherical harmonic expansions of displacement, density and Newtonian gravitational potential into radial equations.

For orthonormal complex spherical harmonics, $P_{\ell }(\hat{\mathbf{n}}\cdot \hat{\mathbf{m}})=4\pi {\sum }_{m=-\ell }^{\ell }{Y}_{\ell m}(\hat{\mathbf{n}})Y_{\ell m}^{\ast }(\hat{\mathbf{m}})/(2\ell +1)$. This converts an angle between two directions into a separable sum. Together with Orthogonality of Legendre polynomials, it projects rotationally invariant scattering kernels onto angular multipoles.

A spin-weight-$s$ field on an oriented sphere acquires a phase under rotation of its local tangent-frame basis. Spin-weighted spherical harmonics form the angular basis for such fields. The polarization combinations $Q\pm iU$ carry opposite spin weights of magnitude two; ordinary scalar functions have spin weight zero.

The angular basis ${}_s{Y}_{\ell m}$ for a spin-weight-$s$ field generalizes ordinary spherical harmonics. It has $\ell \ge |s|$. For scalar-source CMB polarization about a wavevector, only $m=0$ survives and ${}_{\pm 2}{Y}_{20}\propto {\sin }^2\theta$.

On a spin-$s$ field $f$, the spin-raising and lowering operations are $ðf=-{\sin }^s\theta ({\partial }_{\theta }+i\csc \theta {\partial }_{\phi })({\sin }^{-s}\theta f)$ and $\overline{ð}f=-{\sin }^{-s}\theta ({\partial }_{\theta }-i\csc \theta {\partial }_{\phi })({\sin }^s\theta f)$. The intermediate spin changes after each application. For an axisymmetric scalar $f(\mu )$, either operation applied twice yields $(1-{\mu }^2)f^{\prime \prime }(\mu )$.

Here $\mu =\cos \theta$ and $a_{\ell ,m}=\sqrt{({\ell }^2-m^2)/[(2\ell +1)(2\ell -1)]}$. This follows by combining the associated Legendre function recurrence $(2\ell +1)\mu P_{\ell }^m=(\ell -m+1)P_{\ell +1}^m+(\ell +m)P_{\ell -1}^m$ with the normalized spherical harmonic constants. It gives the adjacent-multipole couplings in a photon Boltzmann hierarchy.

The Rayleigh plane-wave expansion iswhere $j_{\ell }$ is a spherical Bessel function.