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 f is coupled to a vector field through a I-2 (phi) F-2 interaction, namely T-zeta(k(1), k(2), k(3), k(4)) equivalent to Sigma(n) d(n) [P-n ((k) over cap (1).(k) over cap (3)) + P-n((k) over cap (1).(k) over cap (12)) + P-n((k) over cap (3).(k) over cap (12))] P-zeta((k) over cap (1)) P-zeta((k) over cap (3)) P-zeta((k) over cap (12)) + (23 perm), 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, d(0), is equal to tau(NL)/6 discussed in the literature. A I-2 (f) F-2 interaction generates d(2) = 2d(0) which is, in turn, related to the quadrupole modulation parameter of the power spectrum, g*, as delta d(2) approximate to 14|g*|N-2 with N approximate to 60. We show that d(0) and d 2 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 delta d(2) approximate to 4 delta d(0) = 105. Therefore, we can reach vertical bar g*vertical bar = 10(-3) by measuring the angle-dependent trispectrum. The current upper limit on t N L from the P l a n c k temperature maps yields vertical bar g*vertical bar < 0.02 (95% CL).