Quantum Primordial Standard Clocks

TL;DR

This work proposes quantum fluctuations of massive fields as universal primordial standard clocks that encode the time evolution $a(t)$ of the early universe in the shape of non-Gaussianities, enabling model-independent discrimination between inflation and its alternatives. It develops a general framework for massive field fluctuations, identifies relativistic and classical regimes, and shows that clock signals arise as oscillatory features in the squeezed bispectrum, with distinct behavior in inflation and contracting scenarios. The authors derive exact and approximate results for inflation, including non-time-ordered and time-ordered integrals, and demonstrate Boltzmann suppression for very heavy clocks, while extending the analysis to $0<p<1$ alternative backgrounds. They emphasize that clock amplitudes are highly model-dependent and discuss prospects for observability and template-based data analyses. Overall, quantum primordial standard clocks broaden the toolbox for probing early-universe dynamics beyond tensor modes and sharp features.

Abstract

In this paper, we point out and study a generic type of signals existing in the primordial universe models, which can be used to model-independently distinguish the inflation scenario from alternatives. These signals are generated by massive fields that function as standard clocks. The role of massive fields as standard clocks has been realized in previous works. Although the existence of such massive fields is generic, the previous realizations require sharp features to classically excite the oscillations of the massive clock fields. Here, we point out that the quantum fluctuations of massive fields can actually serve the same purpose as the standard clocks. We show that they are also able to directly record the defining property of the scenario type, namely, the scale factor of the primordial universe as a function of time a(t), but through shape-dependent oscillatory features in non-Gaussianities. Since quantum fluctuating massive fields exist in any realistic primordial universe models, these quantum primordial standard clock signals are present in any inflation models, and should exist quite generally in alternative-to-inflation scenarios as well. However, the amplitude of such signals is very model-dependent.

Figures (6)

• Figure 1: These figures illustrate the conditions for a massive field to be qualified as a quantum standard clock field in different scenarios. We qualitatively sketch the evolution of four scales (the constant mass of the massive field $m$, the physical wavenumber of the massive field $k_{\rm clock}/a$, the mass of the horizon scale $m_{\rm horizon}$, and the physical wavenumber of the massless curvature scalar mode $k_{\rm curvature}/a$) as functions of the conformal time $\tau$ or real time $t$ in the inflation, fast (or slow) contraction, and slow expansion scenarios, respectively. Two different cases of $m$ values are shown in some scenarios. The thick lines indicate the classical regime during which the massive field can be used as a standard clock. When the dashed lines intersect with the thick lines, the resonance between the curvature scalar mode and the clock field happens, and in all cases we find $k_{\rm clock} < k_{\rm curvature}$. At the resonance, $k_{\rm curvature}/a$ is always at the sub-horizon scales, $k_{\rm clock}/a$ can be at either the sub- or super-horizon scales.
• Figure 2: An example of three-point function. The solid lines represent the curvature scalar mode, the dashed line represents the massive clock field.
• Figure 3: Examples of clock signals in four different scenarios: inflation ($p=20$, $m/m_{{\rm h},k_3}=5$), fast contraction ($p=2/3$, $m/m_{{\rm h},k_3}=0.5$), slow expansion ($p=-0.2$, $m/m_{{\rm h},k_3}=2000$), and slow contraction ($p=0.2$, $m/m_{{\rm h},k_3}=5\times 10^{-5}$). Also, see the last paragraph of Sec. \ref{['Sec:QuantumSC']} and footnote \ref{['Footnote:Boltzmann_envelop']} for some comments on this figure.
• Figure 4: We demonstrate that both the squeezed-limit approximation \ref{['eq:I2F2_squeezed']} and the large $\mu$ limit exact result \ref{['eq:large_mu_limit']} give good approximations to the full ${\cal I}_{\rm 2F2}$. The vertical axis is the corresponding contribution to the shape function, and the unit of $S_{\rm contrib}$ is $2c_2c_3/(\epsilon H M_{\rm P}^2)$. We used the isosceles configuration $k_1=k_2$.
• Figure 5: A comparison between the squeezed limit approximation, \ref{['eq:IHankel_squeezed']}$+$\ref{['eq:I2F2_squeezed']}, and the full numerical calculation of the bispectrum where the massive field is the long mode. The isosceles configuration $k_1=k_2$ is used. The vertical axis is the corresponding contribution to the shape function, and the unit of $S_{\rm contrib}$ is $2c_2c_3/(\epsilon H M_{\rm P}^2)$. The black solid line denotes the full analytical result, and the green dashed line denotes the Hankel part separately. The blue dots line represents the numerical calculation. One observes that for small $\mu$, the Hankel part gives dominate contribution. For larger $\mu$, though the Hankel part still contribute all the clock signals, the ${}_2F_2$ part contributes a significant shift of the amplitude.