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 $\phi$ is coupled to
a vector field through a $I^2 \left( \phi \right) F^2$ interaction, namely
$T_{\zeta}({\bf k}_1, {\bf k}_2, {\bf k}_3, {\bf k}_4) \equiv \sum_n d_n [
P_n(\hat{\bf k}_1 \cdot \hat{\bf k}_3) + P_n(\hat{\bf k}_1 \cdot \hat{\bf
k}_{12}) + P_n(\hat{\bf k}_3 \cdot \hat{\bf k}_{12}) ] P_{\zeta}(k_1)
P_{\zeta}(k_3) P_\zeta(k_{12}) + (23~{\rm perm})$, 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 $\tau_{\rm NL}/6$
discussed in the literature. A $I^2 \left( \phi \right) F^2$ 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 $\delta d_2 \approx
4 \delta d_0 = 105$. Therefore, we can reach $|g_*|=10^{-3}$ by measuring the
angle-dependent trispectrum. The current upper limit on $\tau_{\rm NL}$ from
the ${\it Planck}$ temperature maps yields $|g_*|<0.02$ ($95 \%$ CL).
