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

Abstract

Soft limits of N-point correlation functions, in which one wavenumber ismuch smaller than the others, play a special role inconstraining 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 I2(ϕ)F2 interaction, namelyTζ(k1,k2,k3,k4)≡∑ndn[Pn(k̂1⋅k̂3)+Pn(k̂1⋅k̂12)+Pn(k̂3⋅k̂12)]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 thesides, ki. The coefficient of the isotropic part, d0, isequal to τNL/6 discussed in the literature. A I2(ϕ)F2 interaction generates d2 = 2d0which is, in turn, related to the quadrupole modulation parameter of thepower spectrum, g*, as d2 ≈ 14|g*|N2 with N ≈ 60. We showthat d0 and d2 can be equally well-constrained: theexpected 68% CL error bars on these coefficients from acosmic-variance-limited experiment measuring temperature anisotropy ofthe cosmic microwave background up to ℓmax = 2000 are δd2 ≈ 4δd0 = 105. Therefore, we can reach |g*| = 10−3by measuring the angle-dependent trispectrum. The current upper limit on τNL from the Planck temperature maps yields |g*| < 0.02 (95% CL).