Signatures of anisotropic sources in the squeezed-limit bispectrum of the cosmic microwave background

TL;DR

This work introduces an angle-dependent signature in the squeezed-limit CMB bispectrum, encapsulated by $B_\zeta(k_1,k_2,k_3) = 2 \sum_L c_L P_L(\hat{\mathbf{k}}_1 \cdot \hat{\mathbf{k}}_3) P_\zeta(k_1)P_\zeta(k_3)$, linking $c_0$ to local-type non-Gaussianity and elevating higher-$L$ coefficients as probes of anisotropic sources such as primordial magnetic fields, the $I^2(\phi)F^2$ coupling, and solid inflation. The paper derives flat-sky and full-sky CMB bispectrum expressions, and provides Fisher-forecasted uncertainties showing $\delta c_0 \simeq 4.4$, $\delta c_1 \simeq 61$, and $\delta c_2 \simeq 13$ for a cosmic-variance-limited experiment to $\ell_{\max}=2000$, with $c_0$ largely uncorrelated with $c_1$ and $c_2$, while $c_1$ and $c_2$ are correlated. It also discusses consistency relations with higher-spin fields, noting that the original Suyama-Yamaguchi inequality does not apply to anisotropic cases, though a generalized bound holds. Overall, the study proposes a new observational window into inflationary physics and the role of vector fields or nontrivial rotational symmetry in the early universe.

Abstract

The bispectrum of primordial curvature perturbations in the squeezed configuration, in which one wavenumber, $k_3$, is much smaller than the other two, $k_3\ll k_1\approx k_2$, plays a special role in constraining the physics of inflation. In this paper we study a new phenomenological signature in the squeezed-limit bispectrum: namely, the amplitude of the squeezed-limit bispectrum depends on an angle between ${\bf k}_1$ and ${\bf k}_3$ such that $B_ζ(k_1, k_2, k_3) \to 2 \sum_L c_L P_L(\hat{\bf k}_1 \cdot \hat{\bf k}_3) P_ζ(k_1)P_ζ(k_3)$, where $P_L$ are the Legendre polynomials. While $c_0$ is related to the usual local-form $f_{\rm NL}$ parameter as $c_0=6f_{\rm NL}/5$, the higher-multipole coefficients, $c_1$, $c_2$, etc., have not been constrained by the data. Primordial curvature perturbations sourced by large-scale magnetic fields generate non-vanishing $c_0$, $c_1$, and $c_2$. Inflation models whose action contains a term like $I(φ)^2 F^2$ generate $c_2=c_0/2$. A recently proposed "solid inflation" model generates $c_2\gg c_0$. A cosmic-variance-limited experiment measuring temperature anisotropy of the cosmic microwave background up to $\ell_{\rm max}=2000$ is able to measure these coefficients down to $δc_0=4.4$, $δc_1=61$, and $δc_2=13$ (68% CL). We also find that $c_0$ and $c_1$, and $c_0$ and $c_2$, are nearly uncorrelated. Measurements of these coefficients will open up a new window into the physics of inflation such as the existence of vector fields during inflation or non-trivial symmetry structure of inflaton fields. Finally, we show that the original form of the Suyama-Yamaguchi inequality does not apply to the case involving higher-spin fields, but a generalized form does.

Figures (6)

• Figure 1: Absolute values of the shape function of $L=0$, $(k_1 k_2 k_3)^2 S_0$ (top left panel), that of $L=1$, $(k_1 k_2 k_3)^2 S_1$ (top right panel), and that of $L=2$, $(k_1 k_2 k_3)^2 S_2$ (bottom panel). We restrict the plot range to $k_3 \leq k_2 \leq k_1$ and $|k_1 - k_2| \leq k_3 \leq k_1 + k_2$ for symmetry and the triangular condition. The shape of $L=2$ peaks at the squeezed configuration, $k_3/k_1\ll 1$ and $k_2/k_1\approx 1$, in the same way as that of $L=0$ whereas the shape of $L=1$ is suppressed at the squeezed configuration. While the shape function of $L=0$ has positive values for all $k_2 / k_1$ and $k_3 / k_1$, those of $L = 1$ and $2$ have negative values except in the flattened configurations, $k_2/k_1 + k_3/k_1 \approx 1$.
• Figure 2: Absolute values of the equilateral CMB temperature reduced bispectra, $|b_{\ell\ell\ell}^L|$, for $L=0$ (solid), $L=1$ (long-dashed), and $L=2$ (short-dashed).
• Figure 3: Same as figure \ref{['fig:SSS_squ']}, but for the squeezed triangles, $|b_{\ell_1\ell_2\ell_3}^L|$, with $\ell_1 = \ell_2 = 200$, as a function of $\ell_3$.
• Figure 4: Ratios of the expected error bars, $\delta c_L/\delta c_0$ ($L=1$ and 2), as a function of the maximum multipoles in the sum, $\ell_{\rm max}$. The solid and short-dashed lines show the exact results for $L=1$ and $L=2$, respectively, while the long-dashed and dotted lines show the corresponding Sachs--Wolfe approximations for $L=1$ and $L=2$, respectively. We find that the Sachs--Wolfe approximations trace the overall behavior of the exact calculations well.
• Figure 5: Absolute values of the CMB temperature reduced bispectra. The solid, long-dashed and short-dashed lines show the full-sky results for $L=0$, 1, and 2, respectively, while the plus, cross, and star symbols show the simplified flat-sky results from eq. (\ref{['eq:flatskybispectrum']}) for $L=0$, 1, and 2, respectively. The square symbols show the original form of the flat-sky result for $L=2$ from eq. (\ref{['eq:originalflat']}) before further approximation. (Top panel) Equilateral triangles, $|b_{\ell\ell\ell}^L|$. (Bottom panel) Squeezed triangles, $|b_{\ell_1\ell_2\ell_3}^L|$, with $\ell_1 = \ell_2 = 200$, as a function of $\ell_3$.