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 φ is coupled to a vector field through a I ²(φ)F² interaction, namely T_ζ(k ₁,k₂,k₃,k₄)≡∑ _ndn [P_n (_1ċ₃)+P _n (_1ċ₁₂)+P_n ( _3ċ₁₂)]P_ζ (k₁)P _ζ (k₃)P_ζ (k₁₂)+(23perm), where P_n denotes the Legendre polynomials. This shape is enhanced when the wavenumbers of the diagonals of the quadrilateral are much smaller than the sides, k_i. The coefficient of the isotropic part, d₀, is equal to τ_NL/6 discussed in the literature. A I ²(φ)F² interaction generates d₂ = 2d ₀ 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 ℓ_max = 2000 are δd₂ 4δd₀ = 105. Therefore, we can reach |g_*| = 10^-3 by measuring the angle-dependent trispectrum. The current upper limit on τ_NL from the Planck temperature maps yields |g_*| < 0.02 (95% CL).