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.
