Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

Total-angular-momentum (TAM) waves provide a set of basis functions for scalar, vector, and tensor fields that can be used in place of plane waves and that reflect the rotational symmetry of the spherical sky. Here we discuss three-point correlation functions, or bispectra in harmonic space, for scalar, vector, and tensor fields in terms of TAM waves. The Wigner-Eckart theorem dictates that the expectation value, assuming statistical isotropy, of the product of three TAM waves is the product of a Clebsch-Gordan coefficient (or Wigner-3j symbol) times a function only of the total-angular-momentum quantum numbers. Here we show how this works, and we provide explicit expressions relating the bispectra for TAM waves in terms of the more commonly used Fourier-space bispectra. This formalism will be useful to simplify calculations of projections of three-dimensional bispectra onto the spherical sky.

Figure 1

Geometry of the semiclassical picture: three classical angular-momentum vectors ${\mathbf{J}}_1$, ${\mathbf{J}}_2$, and ${\mathbf{J}}_3$ form a triangle and precess about the chosen $z$ axis, associated with the three TAM waves of wave numbers $k_1$, $k_2$, and $k_3$, respectively. The angle ${\theta }_{12}$ between ${\mathbf{J}}_1$ and ${\mathbf{J}}_2$, however, remains the same. The wave vectors ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$, which are not definitive for TAM waves, are perpendicular to angular momenta ${\mathbf{J}}_1$ and ${\mathbf{J}}_2$, respectively.