Survival of primordial standard clock signals through the cosmological bounce in alternatives to inflation

The primordial standard clock program uses oscillations of massive fields during the primordial phase to imprint clock signals — scale-dependent oscillatory corrections to the density power spectrum whose phase pattern $\sin(2\beta\tilde k^{1/p}+\varphi)$ directly records the expansion history $a(t)\propto t^p$ — providing a model-independent discriminator between inflation and alternatives to inflation such as ekpyrotic contraction. This audit examines the question flagged as open by Chen, Loeb and Xianyu (Phys. Rev. Lett. 122, 121301 (2019)): whether clock-signal-carrying density perturbations from a contracting phase survive the nonsingular cosmological bounce that connects contraction to the hot big bang, and whether the bounce modifies the mapping between the observed wave-number-space features and the pre-bounce $a(t)$. The source paper argues robustness only by assuming the bounce acts as a wave-number-independent rescaling; no later work computes the propagation of an oscillatory clock component through a concrete bounce. The most advanced follow-up (Quintin, Chen and Ebadi, JCAP 09 (2024) 026) computes classical clock signals numerically during slow ekpyrotic contraction and explicitly defers the through-bounce evolution to future work; the adjacent bounce-perturbation literature evolves near-scale-invariant or sourced spectra through concrete bounces but never an oscillatory clock component, while entropy-sector growth and potentially k-dependent entropy-to-curvature conversion are known to be nontrivial. The audited problem — evolve clock-signal-carrying perturbations through at least one concrete nonsingular bounce, quantify wave-number-dependent distortions of the clock pattern, and state when the clock reading of $p$ is trustworthy, or demonstrate generic destruction with the mechanism identified — remains open.

Background

Cosmic inflation — an early phase of quasi-exponential expansion — and its alternatives agree on the observed near-scale-invariant spectrum of primordial density perturbations, but they differ drastically in the expansion history of the very early universe, encoded in the scale factor $a(t)$ as a function of physical time $t$. In alternatives such as ekpyrotic or matter-contraction scenarios, a phase of slow contraction (for ekpyrosis, $a(t)\propto t^p$ with $0<p<1/3$) is followed by a bounce that connects to the hot big bang; a nonsingular bounce is a regular, non-singular completion of this transition, realized for example in ghost-condensate or Galileon models (effective scalar-field theories whose derivative self-interactions permit a brief, stable violation of the null energy condition, the energy condition that otherwise makes a regular bounce impossible) or in Cuscuton gravity (a scalar modification of general relativity that adds no new propagating degree of freedom on cosmological backgrounds). Because standard observables such as the spectral index, tensor fraction, and non-Gaussianity constrain $a(t)$ only indirectly, Chen, Loeb and Xianyu (2019) analyzed the 'primordial standard clock': any realistic model embedded in a larger ultraviolet-complete framework contains massive fields heavier than the horizon energy scale, and such a field oscillates like a harmonic oscillator during the primordial phase. Each oscillation is a tick that imprints, on each fluctuation mode as it crosses the relevant physical scale, a phase shift; the accumulated phases appear as clock signals — scale-dependent oscillatory corrections to the power spectrum of the comoving curvature perturbation $\zeta$ (the standard gauge-invariant primordial fluctuation variable, conserved for adiabatic modes on super-horizon scales). For a power-law phase $a(t)\propto t^p$ the correction takes the form $\Delta\langle\zeta^2\rangle\propto|c_+c_-^*|\sin(2\beta\tilde k^{1/p}+\varphi)$, where $\tilde k$ is the ratio of the comoving wave number to a reference wave number, $\beta$ is a $p$-dependent constant, and $c_\pm$ fix the occupation of positive- and negative-frequency states of the massive field. While the amplitude is model-dependent (set by couplings and the initial state), the phase pattern is not: the exponent $1/p$ in the oscillation frequency directly measures $a(t)$, and it differs qualitatively from the log-periodic signals of near-exponential inflation (in the exact exponential limit the wave-number-dependent signal cancels in the power spectrum and survives only in the shape of non-Gaussianities). A measurement of the $\tilde k^{1/p}$ pattern would therefore identify the scenario type, and even the exponent $p$, in a model-independent way. The paper also forecasted detectability with a survey like LSST (the Legacy Survey of Space and Time, a next-generation galaxy redshift survey) and argued that inflationary feature models reproducing the same pattern would be highly tuned. The open issue audited here sits behind all of this: in every alternative scenario the observed perturbations must pass from the contracting phase through the bounce, and the clock signals were computed only during contraction. The paper acknowledges verbatim: 'How density perturbations in contraction scenarios survive the bounce remains an open question. However, since the most important pattern of the clock signal is unchanged under a universal rescaling of all wave numbers or the amplitude, the information about a(t) in the clock signal most likely remains unchanged under whatever bounce mechanism is at work.' This robustness argument implicitly assumes the bounce acts on the signal-carrying modes at most as a wave-number-independent rescaling — an assumption that has not been computed for the oscillatory subleading component carrying the clock reading in any concrete nonsingular bouncing model. The concern is sharpened by two known facts: in ekpyrotic realizations the clock-carrying excitations live in the entropy (isocurvature) sector — fluctuations in the direction orthogonal to the background field-space trajectory, which later convert to curvature perturbations — and entropy perturbations can grow significantly through a bounce and convert with an efficiency that can depend on wave number (Fertig, Lehners, Mallwitz and Wilson-Ewing 2016); and bounces can imprint their own wave-number-dependent features on the scalar spectrum (Raveendran and Sriramkumar 2019; Ben-Dayan and Thattarampilly 2024).

Problem Statement

Determine whether, and under what conditions, clock-signal-carrying density perturbations from a contracting phase survive a concrete nonsingular cosmological bounce with the clock information intact, as posed by Chen, Loeb and Xianyu (2019). Concretely: in at least one fully specified nonsingular bouncing realization of a contraction alternative to inflation — for example, the state-of-the-art slowly contracting ekpyrotic background with massive spectator clock fields of Quintin, Chen and Ebadi (2024), completed by a concrete bounce mechanism such as the ghost-condensate/Galileon bounce or the Cuscuton bounce — evolve the clock-signal-carrying perturbations (the oscillatory massive-field components together with the coupled entropy and curvature perturbations, including the entropy-to-curvature conversion where the scenario requires it) from the end of contraction through the bounce into the radiation-dominated era. Extract the post-bounce oscillatory component of the curvature power spectrum and quantify any wave-number-dependent distortions of the clock pattern: the frequency scaling $\tilde k^{1/p}$, the amplitude, and the phase. Then state the conditions — or demonstrate that there are none beyond the wave-number-independent rescalings assumed in the source paper — under which the clock reading of the exponent $p$, and hence of the pre-bounce $a(t)$, is trustworthy. A demonstration that the bounce generically destroys or distorts the clock signal, with the mechanism identified, would resolve the question equally. Conclusions of this universal kind — that there are no such conditions, or that the bounce generically destroys the signal — answer the source paper's claim for 'whatever bounce mechanism is at work' and therefore require a generic argument across bounce mechanisms, such as a parametrized model-independent treatment of the transfer or multiple worked bounce models showing a common conclusion; a computation in a single bounce model establishes the model-conditional answer for that model.

References

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