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

Maresuke Shiraishi; Eiichiro Komatsu; Marco Peloso

doi:10.1088/1475-7516/2014/04/027

paper

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

arXiv:1312.5221 · doi:10.1088/1475-7516/2014/04/027

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 $ϕ$ is coupled to a vector field through a $I^{2} (ϕ) F^{2}$ interaction, namely $T_{ζ} (k_{1}, k_{2}, k_{3}, k_{4}) \equiv \sum_{n} d_{n} [P_{n} (k^_{1} \cdot k^_{3}) + P_{n} (k^_{1} \cdot k^_{12}) + P_{n} (k^_{3} \cdot k^_{12})] P_{ζ} (k_{1}) P_{ζ} (k_{3}) P_{ζ} (k_{12}) + (23 perm)$ , 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 $τ_{NL} /6$ discussed in the literature. A $I^{2} (ϕ) F^{2}$ 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 $ℓ_{max} = 2000$ are $δ d_{2} \approx 4 δ d_{0} = 105$ . Therefore, we can reach $∣ g_{*} ∣ = 1 0^{- 3}$ by measuring the angle-dependent trispectrum. The current upper limit on $τ_{NL}$ from the $Planck$ temperature maps yields $∣ g_{*} ∣ < 0.02$ ( $95%$ CL).

15 pages, 5 figures

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

Abstract

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