Seeking Inflation Fossils in the Cosmic Microwave Background

TL;DR

The paper presents a comprehensive framework to search for inflation fossils (scalar, vector, or tensor fields coupled to the inflaton) via CMB BiPoSHs, connecting primordial cross-mode correlations to observable four-point (trispectrum) signatures on the sky. By adopting the total-angular-momentum (TAM) formalism and a local-type coupling, it derives how fossil fields imprint even- and odd-parity BiPoSHs, and constructs minimal-variance quadratic estimators for the fossil amplitudes, encapsulated in the reduced parameter $ ext{A}^Z_h= ext{P}^Z_h( ext{B}^Z_h)^2$. Numerical forecasts for a local-type fossil bispectrum with a scale-free fossil spectrum indicate Planck-level data could detect these signals, with odd-parity BiPoSHs providing clean probes of vector/tensor fossils and allowing discrimination from lensing backgrounds. The work also offers a real-space interpretation of the modulation effects and discusses an infrared divergence in quadrupolar signals for scale-invariant spectra, highlighting future opportunities to leverage polarization and three-dimensional data for stronger constraints.

Abstract

If during inflation the inflaton couples to a "fossil" field, some new scalar, vector, or tensor field, it typically induces a scalar-scalar-fossil bispectrum. Even if the fossil field leaves no direct physical trace after inflation, it gives rise to correlations between different Fourier modes of the curvature or, equivalently, a nonzero curvature trispectrum, but without a curvature bispectrum. Here we quantify the effects of a fossil field on the cosmic microwave background (CMB) temperature fluctuations in terms of bipolar spherical harmonics (BiPoSHs). The effects of vector and tensor fossils can be distinguished geometrically from those of scalars through the parity of the BiPoSHs they induce. However, the two-dimensional nature of the CMB sky does not allow vectors to be distinguished geometrically from tensors. We estimate the detectability of a signal in terms of the scalar-scalar-fossil coupling for scalar, vector, and tensor fossils, assuming a local-type coupling. We comment on a divergence that arises in the quadrupolar BiPoSH from the scalar-scalar-tensor correlation in single-field slow-roll inflation.

Figures (5)

• Figure 1: Primordial correlations of scalar perturbation (solid vectors) induced by the fossil field (dashed vectors) in Fourier space: (a) For a given realization of the fossil field, two different scalar-perturbation modes are correlated with each other. (b) For a stochastic background of fossil fields, a connected trispectrum of scalar perturbations is induced.
• Figure 2: Examples of coefficient functions $F^{J,\alpha}_{l_1 l_2}(K)$ with $\Delta l=l_2-l_1$. Note that we plot $-iF^{J,\alpha}_{l_1 l_2}(K)$ for $J=$ odd since $F^{J,\alpha}_{l_1 l_2}(K)$ is imaginary in that case. (a) Top left panel: $F^{J,\alpha}_{l_1 l_2}$ of different polarization types $\alpha=L,VE,VB,TE,TB$ have different peak locations. Parity-even $L,VE,TE$ modes contribute infrared divergence to the quadrupolar $J=2$ BiPoSHs. Parity-odd $VB,TB$ modes are not relevant to this infrared divergence. (b) Top right panel: as $J$ increases, extrema and nodes of $F^{J,\alpha}_{l_1 l_2}$ are systematically shifted to smaller scales, and the amplitudes decrease as well. (c) Bottom left panel: increasing the CMB multipole $l_1$ rescales the amplitude of $F^{J,\alpha}_{l_1 l_2}$, but extrema and nodes are marginally shifted. (d) Bottom right panel: the amplitude varies for different values of $\Delta l$, and the locations of extrema and nodes change slightly.
• Figure 3: Predicted $3\sigma$ sensitivity for the fossil-field reduced amplitude $\mathcal{A}^{Z}_h$ as a function of the maximum multipole $l_{\mathrm{max}}$. Only BiPoSHs with $J\leqslant 5$ are considered, as they dominate the signal. For the quadrupolar signal $J=2$ from $L,VE,TE$ modes, we cut off the infrared divergence at $K=5\times10^{-6}\mathrm{Mpc}^{-1}$. We relate our results to the primordial trispectrum parameter $\tau_{NL}$ in the local model of non-Gaussianity through $\mathcal{A}^L_h=5.3\times 10^{-23}\tau_{NL}$. We show the WMAP 5-year constraint (horizontal solid), and the constraint from the first Planck data release (horizontal dashed).
• Figure 4: Local distortion of hot/cold spots as a result of modulation from tensor harmonics. Intrinsic (dashed) and distorted (solid) isothermal contours are shown. (a) longitudinal harmonic $Y^{L}_{(JM)ab}(\mathbf{\hat{n}})$; (b) vectorial harmonics $Y^{VE,VB}_{(JM)ab}(\mathbf{\hat{n}})$; (c) tensorial harmonics $Y^{TE,TB}_{(JM)ab}(\mathbf{\hat{n}})$.
• Figure 5: Modulation of isothermal contours by different tensor harmonics: (a) intrinsic contours without modulation; (b) longitudinal type; (c) vectorial $E$-type; (d) vectorial $B$-type; (e) tensorial $E$-type; (f) tensorial $B$-type.