[Paper Review] Is there an effect of a nontrivial $c_T$ during inflation?

Yong Cai, Yutong Wang+1 more|arXiv (Cornell University)|Oct 27, 2015

Cosmology and Gravitation Theories5 citations

TL;DR

This paper investigates the observable effects of a nontrivial propagation speed $c_T$ for primordial gravitational waves during inflation. Using numerical simulations in a disformal frame where $c_T$ is set to unity, the authors show that oscillatory features in the GW power spectrum persist because the $c_T$-dependence is encoded in the time-varying Hubble parameter, preserving observable imprints despite frame redefinition. The key result is that nontrivial $c_T(t)$ evolution induces measurable oscillations and blue-tilted spectra in the primordial GW power spectrum, offering a potential probe of modified gravity beyond general relativity.

ABSTRACT

Recently, we have shown that the propagation speed $c_T$ of the primordial gravitational waves (GWs) might be nontrivially varying during inflation, which could induce local oscillations in the power spectrum of primordial GWs. In this paper, we numerically confirm that, although with a disformal redefinition of the metric the nontrivial $c_T$ may be set as unity, the power spectrum in the frame with $c_T=1$ is completely the same as that in the original disformal frame, i.e., the oscillating shape in the power spectrum is still reserved, since here the effect of $c_T$ is actually encoded in the nontrivially varying Hubble parameter. In addition, we also clarify how to obtain a blue-tilted GWs spectrum by imposing a rapidly decreasing $c_T$ during inflation.

Motivation & Objective

• To investigate whether a time-varying propagation speed $c_T$ of primordial gravitational waves during inflation can leave observable imprints in the CMB B-mode spectrum.
• To resolve the apparent contradiction that $c_T$ can be set to unity via disformal redefinition yet still affect the GW power spectrum.
• To explore how rapidly varying $c_T(t)$ can generate a blue-tilted tensor power spectrum while maintaining slow-roll conditions in the physical frame.
• To clarify the role of $c_T$ in modified gravity models and its implications for future gravitational wave detection experiments.

Proposed method

• The study uses the effective field theory of inflation with a disformal metric redefinition to transform the action into a frame where $c_T = 1$.
• The quadratic action for tensor modes is derived in the disformal frame, with $c_T(t)$ explicitly appearing in the kinetic term: $S^{(2)} = \int d\tau d^3x \frac{M_p^2 a^2}{8} \left[ \frac{1}{c_T^2} \left( \frac{d\gamma_{ij}}{d\tau} \right)^2 - (\nabla \gamma_{ij})^2 \right]$.
• The power spectrum of primordial GWs is computed numerically in both the original disformal frame and the $c_T=1$ Einstein frame to compare results.
• The Hubble parameter $\tilde{H}$ in the $c_T=1$ frame is shown to inherit the $c_T$-dependence, preserving oscillatory features in the GW spectrum.
• The authors construct explicit models with $c_T(\alpha) = 1 - c_{T*} e^{-D(\alpha - \alpha_*)^2}$ and $c_T(\alpha) = 1 + \frac{c_{T*}}{1 + (\alpha - \alpha_*)^n} \sin[B(\alpha - \alpha_*)]$ to simulate varying $c_T$.
• Energy density spectra $\Omega_{gw}(f)$ are computed to assess detectability by LIGO/Virgo and space-based detectors like BBO or DECIGO.

Experimental results

Research questions

• RQ1 Can a nontrivial, time-varying $c_T$ during inflation produce observable oscillations in the primordial gravitational wave power spectrum?
• RQ2 Why does the power spectrum remain sensitive to $c_T$ even after a disformal redefinition that sets $c_T = 1$?
• RQ3 How can a rapidly decreasing $c_T(t)$ lead to a blue-tilted GW power spectrum without violating the null energy condition?
• RQ4 What is the physical origin of the preserved oscillatory features in the $c_T=1$ frame, and how is the $c_T$-dependence encoded?
• RQ5 Can such features be detected by future CMB B-mode or ground-based and space-based GW experiments?

Key findings

• The power spectrum of primordial gravitational waves remains identical in both the original disformal frame and the $c_T=1$ frame, confirming frame-independence of physical observables.
• Oscillatory features in the GW power spectrum are preserved in the $c_T=1$ frame because the nontrivial time evolution of $c_T$ is encoded in the effective Hubble parameter $\tilde{H}$.
• A rapidly decreasing $c_T(t)$ can generate a strongly blue-tilted tensor power spectrum while maintaining a nearly scale-invariant scalar spectrum with a slight red tilt.
• The model with $c_T(\alpha) = 1 - 0.9 e^{-0.2(\alpha - 9)^2}$ produces a significant enhancement in the stochastic GW background at LIGO/Virgo and BBO/DECIGO frequencies.
• The oscillatory features induced by $c_T(t)$ may leave measurable imprints in the CMB B-mode polarization spectrum, particularly around the recombination peak.
• The results imply that $c_T$-dependent effects in modified gravity can be probed through high-precision CMB and GW observations, even if $c_T$ is formally set to unity in a new frame.