Signatures of anisotropic sources in the trispectrum of the cosmic microwave background

Soft limits of N-point correlation functions, in which one wavenumber is much smaller than the others, play a special role in constraining the physics of inflation. Anisotropic sources such as a vector field during inflation generate distinct angular dependence in all these correlators, and introduce a fix privileged direction in our sky. In this paper we focus on the four-point correlator (the trispectrum T). We adopt a parametrization motivated by models in which the inflaton phgr is coupled to a vector field through a I2(phgr)F2 interaction, namely Tζ(k1,k2,k3,k4)≡∑ndn[Pn(hat k1⋅hat k3)+Pn(hat k1⋅hat k12)+Pn(hat k3⋅hat k12)]Pζ(k1)Pζ(k3)Pζ(k12)+(23perm), where Pn denotes the Legendre polynomials. This shape is enhanced when the wavenumbers of the diagonals of the quadrilateral are much smaller than the sides, ki. The coefficient of the isotropic part, d0, is equal to τNL/6 discussed in the literature. A I2(phgr)F2 interaction generates d2 = 2d0 which is, in turn, related to the quadrupole modulation parameter of the power spectrum, g*, as d2 ≈ 14|g*|N2 with N ≈ 60. We show that d0 and d2 can be equally well-constrained: the expected 68% CL error bars on these coefficients from a cosmic-variance-limited experiment measuring temperature anisotropy of the cosmic microwave background up to ℓmax = 2000 are δd2 ≈ 4 δd0 = 105. Therefore, we can reach |g*| = 10−3 by measuring the angle-dependent trispectrum. The current upper limit on τNL from the Planck temperature maps yields |g*| < 0.02 (95% CL).