The trans-Planckian censorship conjecture (TCC) forbids quantum fluctuation modes of sub-Planckian wavelength from ever being stretched beyond the Hubble horizon, which caps the duration and energy scale of any single inflationary stage and, for standard post-inflation histories, pushes the inflation scale below about 10910^9 GeV, making primordial gravitational waves unobservable and the initial conditions of inflation severely fine-tuned. Torabian (2020) showed, entirely at the effective equation-of-state level, that a "breathing comoving Hubble" history — a chain of short inflationary stages beginning near the Planck era that prepare initial conditions for the CMB-observable stage, followed by further short late inflations that collectively accommodate the present horizon — can satisfy the TCC at every stage while touching the Planck 2018 bound on the inflationary energy scale and even permitting eternal inflation; the paper closes by stating that no particular model was presented. The open problem is the construction of an explicit microphysical model — scalar-field potentials together with the graceful-exit (end-of-inflation transition) and reheating dynamics between stages — that reproduces the required sequence of Hubble rates and inter-stage equations of state with the TCC verified on the actual dynamical trajectory at every stage, together with a check that the claimed initial-condition alleviation holds and that the non-observable stages do not jeopardize the success of the CMB observations. No explicit realization of the breathing scenario itself has been constructed; the problem remains open.

Background

Cosmic inflation is an early phase of accelerated expansion proposed to solve the flatness and horizon problems of the standard hot big bang model: during inflation the comoving Hubble horizon (aH)−1(aH)^{-1} (with aa the scale factor and HH the Hubble rate) shrinks, so causal physics can smooth a patch that later grows to encompass the observable universe. A stage of inflation lasting NN e-folds expands lengths by eNe^N. The equation-of-state (EoS) parameter w=p/ρw=p/\rho of the dominant energy component controls the interplay: acceleration requires w<−1/3w<-1/3, and between inflationary stages a matter-like (w=0w=0) or radiation-like (w=1/3w=1/3) epoch makes the comoving horizon expand again — the alternation of shrinking and growing horizon that the source calls a "breathing comoving Hubble". The initial-condition problem of inflation is that, at fixed inflation scale, the smooth patch entering inflation must already span many causally disconnected Hubble volumes at the Planck time. Eternal inflation is the regime in which quantum fluctuations of the inflaton dominate its classical rolling in some regions, so that inflation, once begun, never ends everywhere and populates a landscape of vacua (a multiverse).

The trans-Planckian censorship conjecture (TCC) is a swampland-condition proposal by Bedroya and Vafa: the swampland is the set of effective field theories (EFTs) that cannot be completed by a consistent theory of quantum gravity, and the TCC asserts that in any consistent cosmology, sub-Planckian quantum fluctuation modes must remain quantum — no mode whose wavelength starts below the Planck length may be stretched beyond the Hubble horizon and classicalize. For an inflationary stage this gives the bound eNi<mPl/Hend,ie^{N_i} < m_{\rm Pl}/H_{{\rm end},i}, where mPlm_{\rm Pl} is the Planck mass and Hend,iH_{{\rm end},i} the Hubble rate at the end of stage ii. Bedroya, Brandenberger, Loverde and Vafa showed that for a single inflationary stage followed by a standard thermal history this caps the inflation energy scale below roughly 10910^9 GeV, forcing the generalized slow-roll parameter ϵ\epsilon below 10−3110^{-31}, predicting a negligible tensor-to-scalar ratio rr in primordial gravitational waves, and requiring severe fine-tuning of initial conditions. Torabian's companion work showed that non-thermal histories and multiple inflationary stages can raise the allowed scale of the observable inflation up to about 101410^{14} GeV, touching the Planck 2018 bound — the upper limit on the inflationary energy scale implied by the Planck 2018 CMB measurement of the tensor-to-scalar ratio rr (the relative amplitude of primordial gravitational waves to density perturbations).

In the breathing comoving Hubble scenario, the universe undergoes a sequence of inflationary stages separated by non-inflationary epochs: early "pre-inflations" starting immediately after the Planck era, the observable inflation producing the anisotropies of the cosmic microwave background (CMB), and "late inflations" after it. The TCC restricts each stage separately; for matter-dominated inter-stage epochs (w=0w=0) a representative solution has Hubble rates H1≈0.5 mPlH_1 \approx 0.5\,m_{\rm Pl}, H2>0.1 mPlH_2 > 0.1\,m_{\rm Pl}, H3>10−4 mPlH_3 > 10^{-4}\,m_{\rm Pl} with the TCC bounding the e-fold numbers to N1<0.5N_1 < 0.5, N2<2.2N_2 < 2.2, N3<8.8N_3 < 8.8 — so the first stages last less than a single e-fold, and a handful of pre-inflation stages totaling a few e-folds suffice to set the initial conditions of the observable stage without fine-tuning. The late inflations, satisfying their own TCC bounds, collectively accommodate the present horizon; fewer stages are needed when the inter-stage EoS approaches the critical value w=−1/3w = -1/3 (marginal acceleration, realized by power-law inflation a∝tpa \propto t^p driven by an exponential potential). Within this framework the source argues that eternal inflation is allowed and consistent with observation, evading earlier swampland-based arguments against it. The source's only observational requirement on the non-observable stages is that the succeeding inflations "must not jeopardize the success of CMB observations"; the modes of the pre-inflationary stages themselves cannot be observed.

Crucially, the entire construction is parametric: each stage is described only by its effective EoS parameter and Hubble rate, and the paper states verbatim that it "did not present any particular model". A microphysical realization would have to specify the degrees of freedom and dynamics — e.g. scalar fields with concrete potentials, their graceful exit from each stage, and reheating into the required inter-stage EoS — while keeping the expansion history close enough to the parametric one that the stage-by-stage TCC bounds and the horizon bookkeeping remain valid. This is nontrivial: the early stages sit at curvatures near the Planck scale where the EFT treatment is itself questionable, and Kamali and Brandenberger, studying the closely related mechanism of a marginally accelerating post-inflation epoch, noted that constructing an actual model in which it happens "appears to be unnatural".

Problem Statement

Construct an explicit microphysical model that realizes the breathing comoving Hubble multi-inflation scenario and demonstrate that it satisfies the trans-Planckian censorship conjecture at every stage. Concretely, the required answer must deliver, within a single field-theoretic framework (for example one or more scalar fields, possibly coupled as in modified gravity):

  1. A complete specification of the model — the action, the scalar potentials with their parameter values, and the reheating or transition dynamics that connect consecutive inflationary stages — such that integrating the full background equations yields the breathing pattern: a first inflationary stage starting near the Planck era with Hubble rate of order mPlm_{\rm Pl}, a sequence of pre-inflation stages with monotonically descending Hubble rates down to the CMB-observable stage (as in the source's representative solution H1≈0.5 mPlH_1 \approx 0.5\,m_{\rm Pl}, H2>0.1 mPlH_2 > 0.1\,m_{\rm Pl}, H3>10−4 mPlH_3 > 10^{-4}\,m_{\rm Pl}), then the observable inflation, then late inflation stages, with non-inflationary inter-stage epochs of the required equation-of-state (EoS) parameter value (w≈0w \approx 0, w≈1/3w \approx 1/3, or near the critical marginal value w≈−1/3w \approx -1/3) during which the comoving Hubble horizon expands.

  2. Verification of the TCC bound eNi<mPl/Hend,ie^{N_i} < m_{\rm Pl}/H_{{\rm end},i} — where NiN_i is the e-fold number of stage ii, Hend,iH_{{\rm end},i} its Hubble rate at the end of that stage, and mPlm_{\rm Pl} the Planck mass — at every inflationary stage using the actual dynamical trajectory of the model rather than the effective-EoS parametrization, together with the horizon bookkeeping showing that the present horizon is accommodated (the cumulative product formula of the source) and that the initial-condition fine-tuning of the observable stage is genuinely alleviated by the pre-inflation chain.

  3. A demonstration that the CMB observables produced by the observable stage remain compatible with observational constraints, and that the non-observable early and late stages do not jeopardize the success of those CMB observations — the only consistency condition the source imposes on the succeeding inflationary stages.

The overarching objective uniting these deliverables is a demonstrated microphysical realization of the breathing comoving Hubble scenario, not merely a parametric feasibility analysis. Embedding such a model in a UV-complete theory of quantum gravity (e.g. a string compactification) remains a legitimate open-ended extension, but it does not by itself bound completion, since the swampland status of quasi-de Sitter stages is unsettled; the completion boundary of the problem stated here is the explicit model with all-stage TCC verification and the CMB-consistency checks required above.

References

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