Under the presence of anisotropic sources in the inflationary era , the trispectrum of the primordial curvature perturbation has a very specific angular dependence between each wavevector that is distinguishable from the one encountered when only scalar fields are present , characterized by an angular dependence described by Legendre polynomials .
We examine the imprints left by curvature trispectra on the TT \mu bispectrum , generated by the correlation between temperature anisotropies ( T ) and chemical potential spectral distortions ( \mu ) of the Cosmic Microwave Background ( CMB ) .
Due to the angular dependence of the primordial signal , the corresponding TT \mu bispectrum strongly differs in shape from TT \mu sourced by the usual g _ { NL } or \tau _ { NL } local trispectra , enabling us to obtain an unbiased estimation .
From a Fisher matrix analysis , we find that , in a cosmic-variance-limited ( CVL ) survey of TT \mu , a minimum detectable value of the quadrupolar Legendre coefficient is d _ { 2 } \sim 0.01 , which is 4 orders of magnitude better than the best value attainable from the TTTT CMB trispectrum .
In the case of an anisotropic inflationary model with a f ( \phi ) F ^ { 2 } interaction ( coupling the inflaton field \phi with a vector kinetic term F ^ { 2 } ) , the size of the curvature trispectrum is related to that of quadrupolar power spectrum asymmetry , g _ { * } .
In this case , a CVL measurement of TT \mu makes it possible to measure g _ { * } down to 10 ^ { -3 } .