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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. 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 $I^2 \left( ϕ\right) F^2$ interaction, namely $T_ζ({\bf k}_1, {\bf k}_2, {\bf k}_3, {\bf k}_4) \equiv \sum_n d_n [ P_n(\hat{\bf k}_1 \cdot \hat{\bf k}_3) + P_n(\hat{\bf k}_1 \cdot \hat{\bf k}_{12}) + P_n(\hat{\bf k}_3 \cdot \hat{\bf k}_{12}) ] P_ζ(k_1) P_ζ(k_3) P_ζ(k_{12}) + (23~{\rm perm})$, 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, ${\bf k}_i$. The coefficient of the isotropic part, $d_0$, is equal to $τ_{\rm NL}/6$ discussed in the literature. A $I^2 \left( ϕ\right) F^2$ interaction generates $d_2 = 2 d_0$ which is, in turn, related to the quadrupole modulation parameter of the power spectrum, $g_*$, as $d_2 \approx 14 |g_*| N^2$ with $N \approx 60$. 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 $\ell_{\rm max}=2000$ are $δd_2 \approx 4 δd_0 = 105$. Therefore, we can reach $|g_*|=10^{-3}$ by measuring the angle-dependent trispectrum. The current upper limit on $τ_{\rm NL}$ from the ${\it Planck}$ temperature maps yields $|g_*|<0.02$ ($95 \%$ CL).