Palatini quadratic gravity: spontaneous breaking of gauged scale symmetry and inflation

Abstract

Abstract We study quadratic gravity $$R^2+R_{[\mu \nu ]}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>R</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:math> in the Palatini formalism where the connection and the metric are independent. This action has a gauged scale symmetry (also known as Weyl gauge symmetry) of Weyl gauge field $$v_\mu = (\tilde{\Gamma }_\mu -\Gamma _\mu )/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mover><mml:mi>Γ</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mi>μ</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Γ</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> , with $$\tilde{\Gamma }_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>Γ</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mi>μ</mml:mi></mml:msub></mml:math> ( $$\Gamma _\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Γ</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> ) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $$R_{[\mu \nu ]}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:math> term acting as a gauge kinetic term for $$v_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> . We show that this theory has an elegant spontaneous breaking of gauged scale symmetry and mass generation in the absence of matter, where the necessary scalar field ( $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> ) is not added ad-hoc to this purpose but is “extracted” from the $$R^2$$ <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. The gauge field becomes massive by absorbing the derivative term $$\partial _\mu \ln \phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>∂</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>ln</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow></mml:math> of the Stueckelberg field (“dilaton”). In the broken phase one finds the Einstein–Proca action of $$v_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> of mass proportional to the Planck scale $$M\sim \langle \phi \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>M</mml:mi><mml:mo>∼</mml:mo><mml:mo>⟨</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>⟩</mml:mo></mml:mrow></mml:math> , and a positive cosmological constant. Below this scale $$v_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math> decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. These results remain valid in the presence of non-minimally coupled scalar field (Higgs-like) with Palatini connection and the potential is computed. In this case the theory gives successful inflation and a specific prediction for the tensor-to-scalar ratio $$0.007\le r\le 0.01$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0.007</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo>≤</mml:mo><mml:mn>0.01</mml:mn></mml:mrow></mml:math> for current 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> (at $$95\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>95</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math> CL) and $$N=60$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>60</mml:mn></mml:mrow></mml:math> efolds. This value of r is mildly larger than in inflation in Weyl quadratic gravity of similar symmetry, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test such theories by future CMB experiments.

Topics & Concepts