  If during inflation the inflaton couples to a "fossil" field, some new
scalar, vector, or tensor field, it typically induces a scalar-scalar-fossil
bispectrum. Even if the fossil field leaves no direct physical trace after
inflation, it gives rise to correlations between different Fourier modes of the
curvature or, equivalently, a nonzero curvature trispectrum, but without a
curvature bispectrum. Here we quantify the effects of a fossil field on the
cosmic microwave background (CMB) temperature fluctuations in terms of bipolar
spherical harmonics (BiPoSHs). The effects of vector and tensor fossils can be
distinguished geometrically from those of scalars through the parity of the
BiPoSHs they induce. However, the two-dimensional nature of the CMB sky does
not allow vectors to be distinguished geometrically from tensors. We estimate
the detectability of a signal in terms of the scalar-scalar-fossil coupling for
scalar, vector, and tensor fossils, assuming a local-type coupling. We comment
on a divergence that arises in the quadrupolar BiPoSH from the
scalar-scalar-tensor correlation in single-field slow-roll inflation.
