The standard cosmological model ΛCDM explains neither the physical nature of dark energy nor the connection between the early inflationary stage and the late acceleration. The idea of scale symmetry, which forbids dimensionful parameters in the fundamental action, offers an elegant solution: all masses and scales arise dynamically. Combined with unimodular gravity, where the cosmological constant is not a parameter but an integration constant, a natural arena emerges for unifying inflation and dark energy via a single 'chameleon' field.
The authors construct an action for scale-invariant R² gravity with a scalar field and a unimodular constraint. Dynamics on the inflationary attractor are investigated analytically and numerically, where the conserved Noether current fixes the trajectory in field space, effectively 'freezing' the Goldstone direction. Then the post-inflationary rolling toward the minimum and the incorporation of the unimodular potential are considered. The evolution during the hot Universe era and late acceleration are calculated numerically, taking into account constraints on the cosmic microwave background (Planck, BICEP/Keck) and supernovae (DES).
The main result is a consistency relation linking inflationary observables (spectral index ns and tensor-to-scalar ratio r) to the current equation of state of dark energy w0. For a representative set of parameters, we obtain ns≈0.958, r≈0.00287, and the equation of state parameters in the CPL parameterization: w0≈−0.992, wa≈−0.011. This corresponds to a slowly 'thawing' field that remained frozen for almost the entire history of the Universe and only recently began to roll. The model automatically excludes a tracker regime during the matter era and guarantees an accelerated thawing mode across the entire region of inflationary viability.
The model predicts a tiny but fundamentally measurable deviation from a cosmological constant. The tight correlation between early and late cosmological parameters turns future high-precision measurements (DESI, Euclid, Rubin Observatory) into a test of a single symmetry structure. Even a slight improvement in the precision of ns and r could confirm or refute such a mechanism.
Further development of the model may include accounting for quantum corrections, investigation of the radiative stability of the Minkowski branch, as well as extension to higher curvature terms compatible with scale symmetry. Also necessary is detailed modeling of post-inflationary reheating and connection to the Standard Model.
The results are important for observational cosmology, theories of modified gravity, and fundamental physics, uniting the two epochs of accelerated expansion into a single picture.
Immediate next steps: inclusion of non-perturbative reheating and explicit decay channels into Standard Model particles, analysis of the stability of the Goldstone flat direction against quantum effects.
The paper directly addresses the cosmological constant problem (why vacuum energy is so small) and the origin of dark energy, proposing a dynamical mechanism instead of anthropic tuning.
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What the work does
This work constructs an R² cosmology with scale invariance and unimodular gravity, where the Goldstone direction is 'frozen' on the inflationary attractor and later drives the present acceleration. Numerically, correlations are obtained between inflationary parameters and the dark energy equation of state.
Where its strength is
The tight link between early inflation and late dark energy through a single symmetry mechanism yields a concrete prediction testable with upcoming surveys.
Where it could go next
Worth trying
Why this may matter
If the connection between inflation and dark energy is confirmed, it would point to a deep symmetry structure unifying the entire cosmological history.
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The Goldstone boson of scale symmetry remains strictly massless at tree level, but unimodular gravity endows it with a potential of the order of the current dark energy density — about 10⁻¹²⁰ of the Planck density. This is one of the smallest known numbers in physics.
The paper includes an ironic footnote: 'That’s not how the Force works!'—a reference to Han Solo from Star Wars. The authors explain that, despite being massless, the Goldstone boson does not produce a fifth force that could be observed in the laboratory.