Gauging scale symmetry and inflation: Weyl versus Palatini gravity
D. M. Ghilencea
Abstract
Abstract We present a comparative study of inflation in two theories of quadratic gravity with gauged scale symmetry: (1) the original Weyl quadratic gravity and (2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ( $$w_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>w</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> ) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric origin and part of the quadratic action. The Einstein-Proca action (of $$w_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>w</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> ), Planck scale and metricity emerge in the broken phase after $$w_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>w</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ( $$\phi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> ), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field rescaling. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their $$R^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> term, both theories have a small tensor-to-scalar ratio ( $$r\sim 10^{-3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>r</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> ), larger in Palatini case. For a fixed spectral index $$n_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math> , reducing the non-minimal coupling ( $$\xi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ξ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> ) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough $$\xi _1\le 10^{-3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ξ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> , unlike the Palatini version, Weyl theory gives a dependence $$r(n_s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>r</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections.