Abstract
We study quadratic gravity R^2+R_{[mu nu ]}^2 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, with tilde{Gamma }_mu (Gamma _mu ) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the R_{[mu nu ]}^2 term acting as a gauge kinetic term for v_mu . 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 ) is not added ad-hoc to this purpose but is “extracted” from the R^2 term. The gauge field becomes massive by absorbing the derivative term partial _mu ln phi of the Stueckelberg field (“dilaton”). In the broken phase one finds the Einstein–Proca action of v_mu of mass proportional to the Planck scale Msim langle phi rangle , and a positive cosmological constant. Below this scale v_mu 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 0.007le rle 0.01 for current spectral index n_s (at 95% CL) and N=60 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.
Highlights
At a fundamental level gravity may be regarded as a theory of connections
The question remains, if such general actions in the EP formalism and in the absence of matter can recover dynamically the Levi-Civita connection and Einstein gravity. This would be similar to the original Weyl quadratic gravity theory [22,23,24,25] as we showed recently in [26,27]
For Einstein action in the EP approach one finds that the connection is equal to the Levi-Civita connection; Einstein gravity is recovered, so the metric and EP approaches are equivalent
Summary
At a fundamental level gravity may be regarded as a theory of connections. An example is the “Palatini approach” to gravity due to Einstein [1,2], hereafter called EP approach [3,4,5]. The question remains, if such general actions in the EP formalism and in the absence of matter can recover dynamically the Levi-Civita connection and Einstein gravity If true, this would be similar to the original Weyl quadratic gravity theory [22,23,24,25] as we showed recently in [26,27]. The EP quadratic gravity is a gauged scale invariant theory broken à la Stueckelberg, even in the absence of matter, to an Einstein–Proca action with a positive cosmological constant and a potential for the scalar fields – if present. This answers the main goal of the paper.
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