Bridging inflation and reheating: chiral gravitational waves from aHz to GHz

Technical Summary: Bridging Inflation and Reheating: Chiral Gravitational Waves from aHz to GHz

Problem Statement
While General Relativity conserves parity, various extensions of gravity theories (such as Chern-Simons modified gravity and parity-violating extensions of teleparallel gravity) predict parity violation (PV) in the gravitational sector. Such violations manifest as birefringence in gravitational waves (GWs), where left- and right-handed polarization modes evolve differently. Previous studies have largely focused on GWs generated during inflation or specific reheating scenarios in isolation. However, a comprehensive analysis of chiral GW signals spanning the entire evolution from inflation through reheating, particularly within the context of teleparallel gravity extensions, remains underexplored. Furthermore, the constraints imposed by the backreaction of amplified GWs on the energy scale of PV terms and the reheating history require rigorous investigation.

Methodology
The authors investigate linear chiral gravitational waves generated by a parity-violating term coupled to the inflaton, specifically the operator $\dot{\phi}\epsilon_{ijk}h_{il}\partial_j h^l_k$, which arises naturally in PV extensions of teleparallel gravity. The study employs the following methodological framework:

1. Theoretical Formulation: The quadratic action for chiral GWs is derived, leading to a modified equation of motion (EoM) for the mode functions $h^A_k$ (where $A=R,L$ denotes right and left helicities). The EoM includes a parity-violating term proportional to $\dot{\phi}/\Lambda$, where $\Lambda$ is the energy scale of the PV term.
2. Inflationary Phase: Assuming standard single-field slow-roll inflation with an E-model $\alpha$-attractor potential, the authors solve the EoM analytically using the Bunch-Davies vacuum and de Sitter approximation. The solution involves Kummer functions of the second kind. The power spectra for each helicity are calculated at horizon crossing.
3. Reheating Phase: The dynamics are analyzed during the reheating epoch where the inflaton oscillates around the potential minimum. The authors employ both numerical integration of the full EoM and analytical WKB approximations (utilizing Airy functions near turning points) to study tachyonic instability. This instability occurs on subhorizon scales when the effective frequency squared becomes negative, leading to exponential amplification of specific modes.
4. Observables and Constraints: The study calculates the resulting Cosmic Microwave Background (CMB) angular power spectra ($BB$, $EB$, $TB$) and the present-day GW energy spectrum ($\Omega_{GW,0}$). A critical constraint is applied: the energy density of the enhanced GWs must not exceed 10% of the total energy density ($\rho_{GW}/3H^2M_{Pl}^2 \lesssim 0.1$) to avoid significant backreaction effects that would alter the background expansion.

Key Contributions

• Unified Evolution: This work provides the first analysis of the full evolution of chiral GWs from inflation through reheating within the specific framework of teleparallel gravity extensions.
• Frequency Bridging: The authors demonstrate that the resulting chiral GW signals span a broad frequency range, extending from the aHz (accessible to CMB experiments) to the GHz range (accessible to future high-frequency detectors).
• Reheating Dynamics: The paper details how tachyonic instability during the reheating phase leads to fully circularly polarized GWs with enhanced amplitudes, distinct from the inflationary generation mechanism.
• Constraint Analysis: By imposing the no-backreaction condition, the authors derive lower bounds on the energy scale $\Lambda$ of the PV term and establish constraints on the reheating history (specifically the e-folding number $N_{reh}$).

Results

• Inflationary Signatures: During inflation, the PV term reduces the sound horizon for right-handed GWs, causing them to exit the horizon earlier and with larger amplitudes than left-handed GWs. This results in a non-zero chirality parameter $\Pi = \tanh(\pi\eta_*)$ and a blue-tilted tensor spectral index. At CMB scales, these signals induce non-vanishing $EB$ and $TB$ correlations alongside the $BB$ spectrum. The authors note that for specific parameters (e.g., $\alpha=10$), these signals could be detectable by the LiteBIRD mission.
• Reheating Signatures: During reheating, subhorizon modes undergo tachyonic instability, leading to a "bump-like" spectrum in the GW energy density. The amplification is highly dependent on the wavenumber $k$ and the inflaton's velocity. The resulting spectrum in the MHz–GHz range is characterized by fully right-handed polarization and distinct peak structures.
• Spectral Features: The GW energy spectra exhibit oscillatory features (dips and peaks) arising from the alternating amplification of polarization modes during inflaton oscillations. These features are explained via the WKB approximation and the interference of Bogoliubov coefficients.
• Constraints: The requirement to avoid backreaction imposes a lower bound on $\Lambda$ (e.g., $M_{Pl}/\Lambda \approx 8.34$ to $13.9$ for the tested cases). This bound is found to be more stringent than limits derived from Big Bang Nucleosynthesis (BBN) for the considered parameters.

Significance and Claims
The paper claims that multi-frequency GW experiments offer a unique probe of parity violation and early Universe physics. By combining CMB observations (specifically $BB$, $EB$, and $TB$ spectra) with future high-frequency GW detection concepts (such as resonant cavities), it is possible to simultaneously constrain the PV coupling, the inflationary potential, and the reheating history.

The authors modestly note that current experimental sensitivities in the MHz–GHz range are not yet sufficient to detect these signals, positioning them as targets for future detection concepts. They also clarify that their analysis focuses on the no-backreaction regime; investigating the regime where backreaction is significant (corresponding to lower $\Lambda$) is left for future work. Additionally, the study does not focus on low-frequency GWs ($f < \text{MHz}$) as they remain suppressed below the sensitivity of planned experiments like SKA, LISA, and the Einstein Telescope due to CMB amplitude constraints.

The findings highlight the potential of chiral GWs as a distinctive signature of parity-violating gravity, differentiating them from other high-frequency GW backgrounds (e.g., those from preheating scalar or gauge fields) through their unique polarization state and spectral profile.