Parity Violation of Gravitons in the CMB Bispectrum

TL;DR

The paper addresses parity violation in primordial graviton non-Gaussianities and their imprint on the CMB bispectrum. It computes the graviton bispectra from parity-even $W^3$ and parity-odd $\widetilde{W}W^2$ with a time-dependent coupling $f(\tau)=(\tau/\tau_*)^A$, using the in-in formalism, and maps them to CMB bispectra via spin-harmonic decomposition and transfer functions. A key finding is that the parity-odd contribution can be nonzero in exact de Sitter for $A\neq 0$ and can be comparable to the parity-even piece for certain $A$, with distinct observational signatures arising in even vs. odd sums of multipoles. The work yields a large-scale amplitude estimate $|b_{\ell\ell\ell}| \sim \ell^{-4} (GeV/\Lambda)^2 (r/0.1)^4$, leading to a bound $\Lambda \gtrsim 3\times 10^6$ GeV for $r=0.1$, thereby providing a concrete observational probe of parity violation in the graviton sector and constraints on higher-derivative gravity scales; it also highlights the importance of including odd-sum configurations for robust detection.

Abstract

We investigate the cosmic microwave background (CMB) bispectra of the intensity (temperature) and polarization modes induced by the graviton non-Gaussianities, which arise from the parity-conserving and parity-violating Weyl cubic terms with time-dependent coupling. By considering the time-dependent coupling, we find that even in the exact de Sitter space time, the parity violation still appears in the three-point function of the primordial gravitational waves and could become large. Through the estimation of the CMB bispectra, we demonstrate that the signals generated from the parity-conserving and parity-violating terms appear in completely different configurations of multipoles. For example, the parity-conserving non-Gaussianity induces the nonzero CMB temperature bispectrum in the configuration with $\sum_{n=1}^3 \ell_n = {\rm even}$ and, while due to the parity-violating non-Gaussianity, the CMB temperature bispectrum also appears for $\sum_{n=1}^3 \ell_n = {\rm odd}$. This signal is just good evidence of the parity violation in the non-Gaussianity of primordial gravitational waves. We find that the shape of this non-Gaussianity is similar to the so-called equilateral one and the amplitudes of these spectra at large scale are roughly estimated as $|b_{\ell \ell \ell}| \sim \ell^{-4} \times 3.2 \times 10^{-2} ({\rm GeV} / Λ)^2 (r / 0.1)^4$, where $Λ$ is an energy scale that sets the magnitude of the Weyl cubic terms (higher derivative corrections) and $r$ is a tensor-to-scalar ratio. Taking the limit for the nonlinearity parameter of the equilateral type as $f_{\rm NL}^{\rm eq} < 300$, we can obtain a bound as $Λ\gtrsim 3 \times 10^6 {\rm GeV}$, assuming $r=0.1$.

Figures (3)

• Figure 1: Shape of $k_1^2 k_2^2 k_3^2 S_{A}$ for $A = -1/2$ (top left figure), $0$ (top right one), $1/2$ (bottom left one), and $1$ (bottom right one) as the function of $k_2 / k_1$ and $k_3 / k_1$.
• Figure 2: Absolute values of the CMB $III, IIB, IBB$, and $BBB$ spectra induced by $W^3$ and $\widetilde{W}W^2$ for $A = -1/2, 0, 1/2$, and $1$. We set that three multipoles have identical values as $\ell_1 - 2 = \ell_2 - 1 = \ell_3$. The left figures show the spectra not vanishing for $\sum_{n=1}^3 \ell_n = {\rm even}$ (parity-even mode) and the right ones present the spectra for $\sum_{n=1}^3 \ell_n = {\rm odd}$ (parity-odd mode). Here, we fix the parameters as $\Lambda = 3 \times 10^6 {\rm GeV}, r = 0.1$, and $\tau_* = -k_*^{-1} = - 14 {\rm Gpc}$, and other cosmological parameters are fixed as the mean values limited from the WMAP $7$-yr dataKomatsu:2010fb.
• Figure 3: (color online) Absolute value of the CMB $III$ spectra generated from $W^3$ for $A = -1/2$ (red solid line), $0$ (green dashed one), and $1/2$ (blue dotted one) described in Fig. \ref{['fig:TTT_A_difl']}, and generated from the equilateral-type non-Gaussianity given by Eq. (\ref{['eq:cmb_bis_scal']}) with $f_{\rm NL}^{\rm eq} = 300$ (magenta dot-dashed one). We set that three multipoles have identical values as $\ell_1 = \ell_2 = \ell_3 \equiv \ell$. Here, we fix the parameters as the same values mentioned in Fig. \ref{['fig:TTT_A_difl']}.