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

Abstract

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 $\varphi$ is coupled to a vector field through a $I^2\left(\varphi \right)F^2$ interaction, namely $T_{\zeta }({\mathbf{k}}_1,{\mathbf{k}}_2,{\mathbf{k}}_3,{\mathbf{k}}_4)\equiv {\sum }_n{d}_n[P_n({\hat{\mathbf{k}}}_1\cdot {\hat{\mathbf{k}}}_3)+P_n({\hat{\mathbf{k}}}_1\cdot {\hat{\mathbf{k}}}_{12})+P_n({\hat{\mathbf{k}}}_3\cdot {\hat{\mathbf{k}}}_{12})]P_{\zeta }(k_1)P_{\zeta }(k_3)P_{\zeta }(k_{12})+(23\mathrm{perm})$, 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, ${\mathbf{k}}_i$. The coefficient of the isotropic part, $d_0$, is equal to ${\tau }_{\mathrm{NL}}/6$ discussed in the literature. A $I^2\left(\varphi \right)F^2$ interaction generates $d_2=2d_0$ which is, in turn, related to the quadrupole modulation parameter of the power spectrum, $g_{\ast }$, as $d_2\approx 14|g_{\ast }|N^2$ 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 ${\ell }_{\max }=2000$ are $\delta d_2\approx 4\delta d_0=105$. Therefore, we can reach $|g_{\ast }|=10^{-3}$ by measuring the angle-dependent trispectrum. The current upper limit on ${\tau }_{\mathrm{NL}}$ from the $Planck$ temperature maps yields $|g_{\ast }|<0.02$ ($95\%$ CL).

Comments: 15 pages, 5 figures