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Weyl $R^2$ inflation with an emergent Planck scale
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abstract
We study inflation in Weyl gravity. The original Weyl quadratic gravity, based on Weyl conformal geometry, is a theory invariant under Weyl symmetry of (gauged) local scale transformations. In this theory Planck scale ($M$) emerges as the scale where this symmetry is broken spontaneously by a geometric Stueckelberg mechanism, to Einstein-Proca action for the Weyl "photon" (of mass near $M$). With this action as a "low energy" broken phase of Weyl gravity, century-old criticisms of the latter (due to non-metricity) are avoided. In this context, inflation with field values above $M$ is natural, since this is just a phase transition scale from Weyl gravity (geometry) to Einstein gravity (Riemannian geometry), where the massive Weyl photon decouples. We show that inflation in Weyl gravity coupled to a scalar field has results close to those in Starobinsky model (recovered for vanishing non-minimal coupling), with a mildly smaller tensor-to-scalar ratio ($r$). Weyl gravity predicts a specific, narrow range $0.00257 \leq r\leq 0.00303$, for a spectral index $n_s$ within experimental bounds at $68\%$CL and e-folds number $N=60$. This range of values will soon be reached by CMB experiments and provides a test of Weyl gravity. Unlike in the Starobinsky model, the prediction for $(r, n_s)$ is not affected by unknown higher dimensional curvature operators (suppressed by some large mass scale) since these are forbidden by the Weyl gauge symmetry.
Forward citations
Cited by 7 Pith papers
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World-Line Actions in Weyl Geometry
A dimensionless Weyl-invariant additive world-line action exists for time-like particles in Weyl geometry, but proper time cannot be defined until scale symmetry breaks.
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Weyl gauge symmetry at LIGO-Virgo-KAGRA
In Weyl quadratic gravity linearized around de Sitter space, gravitational waves carry two tensor modes plus two transverse vector modes from the Weyl gauge field, with no scalar modes.
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Weyl-invariant Einstein-Cartan gravity with a heavy ALP: Higgs Inflation and $\alpha$-attractors
In a Weyl-invariant Einstein-Cartan gravity theory with the SM Higgs and a heavy gravitational ALP, tuning two nonminimal couplings reproduces metric Higgs inflation and α-attractor-like inflation with ns≈1−2/N and r≈12/N^2.
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Primordial Gravitational Waves in Quadratic Gravity
In quadratic gravity, the inflationary tensor power spectrum is suppressed by (1 + 2H_*^2/m_gh^2)^{-1}, which restores the standard consistency relation r = -8n_t.
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The Goldstone Awakens: Unimodular dark energy in scale-invariant $R^2$ gravity
In a scale-invariant R² + unimodular-gravity model, the Goldstone boson of scale symmetry becomes thawing dark energy, with its equation of state fixed by the same coupling that sets the inflationary spectral tilt.
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The fall and the rise of Weyl gauge theory
Weyl quadratic gauge gravity is anomaly-free, breaks via Stueckelberg to Einstein-Hilbert plus positive Lambda, and is the leading order of a regulator-free WDBI action that can include the SM.
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Atomic clocks and gravitational waves as probes of non-metricity
The paper claims existing gravitational-wave data already bound Weyl non-metricity, α²ω̄0<10⁻⁶⁹ GeV, via backreaction of a Planck-scale Weyl field, but a dropped kinetic term numerically exceeds the assumed sensitivity.
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