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Advancements in Teleparallel Theories for Inflation: Insights from the 2012 Cross-Strait Meeting

The 2012 Cross-Strait Meeting on Particle Physics and Cosmology focused on teleparallel theories of gravity and their implications for inflation. Presented by Yi-Peng Wu of National Tsing Hua University, this work explores the unified field theory for gravitation and electromagnetism via teleparallel geometry. Key discussions included higher-order teleparallel theory, the single-field inflationary model, and the equivalence of teleparallel and general relativity. This research paves the way for new observational insights in cosmology and particle physics.

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Advancements in Teleparallel Theories for Inflation: Insights from the 2012 Cross-Strait Meeting

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  1. 2012 Cross-Strait Meeting on Particle Physics and Cosmology Yi-Peng Wu Department of Physics, National TsingHua University May 10th @ Chongquing primordial fluctuations within tele-parallelism Collaborate with Prof. Chao-QiangGeng

  2. Introduction • teleparallel geometry • teleparallel description of General Relativity • higher-order teleparallel theory • Tele-parallel Theories for Inflation • the single field inflationary model • inflation driven by higher-order teleparallel theory Out-line

  3. The origin: an unified field theory for gravitation and electromagnetism Nowadays: theories for gravity; geometrized by purely torsion Tele-parallel geometry (1916) gravity (1928) electromagnetism

  4. Tele-parallel geometry curvature Riemann Minkowski

  5. New General Relativity (Hayashi & Shirafuji 1979) Teleparallelequivelance of General Relativity (Maluf 1993) Tele-parallel description of GR → first order Lagrangian; second order field equation 16 components (Einstein equation)

  6. The Lagrangian density of TEGR “T” differs from the Ricci scalar “R” only by a total divergence. The field equation doesn’t determinate the entire dynamic field Tele-parallel description of gr Local Lorentz symmetry Local Lorentz transformations

  7. An ad hoc generalization of TEGR inspired from f(R) theories Local Lorentz symmetry is broken Higher-order teleparallel theory extra degrees of freedom !! (Li, Sotiriou & Barrow 2011) second order field equation covariant representation

  8. 2012 Cross-Strait Meeting on Particle Physics and Cosmology Tele-parallel theories for inflation

  9. Models of inflation with observables in the framework of teleparallel geometry Re-exam the “equivalence” of TEGR via the minimal coupling single field inflationary model Can higher-order TEGR theories for inflation be available? Tele-parallel theories for inflation

  10. Is the “equivalence” between single field inflationary models trivial ? • Classify the teleparallel geometry: • eA0 : (i) time-like (ii) space-like (iii) null Tele-parallel theories for inflation ADM formalism

  11. Teleparallel geometry with time-like eA0 Fix the time and spatial reparametrizations Tele-parallel theories for inflation 3- curvature extrinsic curvature ζ and γ are first order quantities The results can be generalized to teleparallel geometry with space-like eA0 expected results!!

  12. Driven inflation by modified teleparallel gravity opens up the study of f(T) theories Conformal transformation in f(T) theories: Tele-parallel theories for inflation (Ferraro & Fiorini 2007) still non-minimally coupled

  13. Before calculations…. • (i) we need more assumptions to reduce the degrees of freedom for the flat FRW choice • (ii) the unitary field gauge δφ = 0 ? • (iii) the equivalence between conformal frames? Tele-parallel theories for inflation They are all Lorentz violated!!! (Faraoni & Nadeau 2007)

  14. Linear perturbations concerning the same degrees of freedom as in GR Tele-parallel theories for inflation (Chen et. al. 2011)

  15. If the gravitational field is protected by gauge invariance (at least in the energy scale before horizon crossing) : • Extra dofs induced by Lorentz violation are supressed • The previous parametrization is applied • A single-field theory with second-order field equations: Tele-parallel theories for inflation (Kobayashi, Yamaguchi & Yokoyama 2011) quadratic actions background eqs. tensor perturbations unchanged

  16. The single field inflationary model for TEGR, indistinguishable from the standard inflation results, can be realized in the certain classes of teleparallel geometry. It is possible to carry out observables from higher-order TEGR inflation if the gravitational field is characterized by local gauge invariance. summary Thank you !

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