Fossilized Gravitational Wave Relic and Primordial Clocks

TL;DR

The paper explores fossilized gravitational wave relics as a cross-check of inflationary dynamics by allowing non-Bunch-Davies initial states for both scalar and tensor perturbations, which induces an off-diagonal component in the scalar power spectrum through long-short mode coupling. It derives the scalar-scalar-tensor bispectrum with generalized Bogoliubov coefficients, showing enhanced squeezed-limit signals that can be observable and introducing minimum-variance estimators for tensor fossils; the detectability depends on the Bogoliubov amplitudes, phases, and the scale range probed. A broader framework for scalar-scalar-fossil correlations is presented, including a squeezed-limit parameterization and a detectability inequality that highlights when a fossil signal becomes observable, along with a discussion of quadrupolar anisotropy sourced by superhorizon tensor modes. The results indicate that, under plausible pre-inflationary dynamics, the fossil signatures could be within reach of near-future surveys, offering a complementary probe to the tensor power spectrum and non-Gaussianity measurements for understanding the inflationary background and possible additional clocks.

Abstract

If long wavelength primordial tensor modes are coupled to short wavelength scalar modes, the scalar curvature two-point function will have an off-diagonal component. This `fossil' remnant is a signature of a mode coupling that cannot be achieved in single clock inflation. Any constraint on its presence allows a cross check of the relationship between the dynamical generation of the fluctuations and the evolution of the inflationary background. We use the example of non-Bunch Davies initial states for the tensor and scalar modes to demonstrate that physically reasonable fossils, consistent with current data, can be observable in the near future. We illustrate how the fossil off-diagonal power spectrum is a complementary probe to the squeezed limit bispectra of the scalar and tensor sectors individually. We also quantify the relation between the observable signal and the squeezed limit bispectrum for a general scalar-scalar-fossil coupling, and note the effect of superhorizon tensor modes on the anisotropy in scalar modes.

Figures (2)

• Figure 1: Contour plot of the minimum survey size $k_{\text{max}}/k_{\text{min}}$ needed to detect the effect of primordial tensor fluctuations through off-diagonal contributions to the scalar power spectrum, in terms of scalar and tensor Bogoliubov parameters $\beta^{(s)},\beta^{(t)}$, for different values of the tensor-to-scalar ratio $r$ and non-Bunch-Davies phases $\Theta^{(s)},\Theta^{(t)}$. The scales $k_{\text{min}}$ and $k_{\text{max}}$ are the longest and shortest observable scales (at which scalar modes are excited). The dashed line at $\beta^{(s)}=0.1$ indicates the back-reaction bound $\beta^{(s)}\lesssim0.1$ (this does not apply in the $\Theta^{(s)}=\pi$ case Flauger:2013hra). The dark shaded region is ruled out by the Planck constraint $f^{\rm NBD2}_{\rm NL}=0.2\pm0.4$. For $\Theta^{(s),(t)}=\pi$ the power spectra decrease rather than increase with $\beta^{(s),(t)}$, leading to different behavior.
• Figure 2: Contour plot of $k_{\text{max}}/k_{\text{min}}$ for a $3\sigma$ detection of the gravitational fossil in off-diagonal contributions to the scalar power spectrum, in terms of the tensor-to-scalar ratio $r$ and scalar-scalar-tensor bispectrum amplitude $f_{\zeta\zeta\gamma}$, assuming a stronger-than-local squeezed limit ($m_S=1$, $m_L=-1$ in Eq. \ref{['squeezedinequality']}). The signal is proportional to $f_{\zeta\zeta\gamma}^2 A_{\gamma}$. We set $r=A_{\gamma}/(2.2\times 10^{-9})$ and take $n_t\simeq0$.