Scale-independent R 2 inflation

2020

We study quadratic gravity $$R^2+R_{[\mu \nu ]}^2$$ R 2 + R [ μ ν ] 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 $$v_\mu = (\tilde{\Gamma }_\mu -\Gamma _\mu )/2$$ v μ = ( Γ ~ μ - Γ μ ) / 2 , with $$\tilde{\Gamma }_\mu $$ Γ ~ μ ($$\Gamma _\mu $$ Γ μ ) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $$R_{[\mu \nu ]}^2$$ R [ μ ν ] 2 term acting as a gauge kinetic term for $$v_\mu $$ v μ . 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 ($$\phi $$ ϕ ) is not added ad-hoc to this purpose but is “extracted” from the $$R^2$$ R 2 term. The gauge field becomes massive by absorbing the derivative term $$\partial _\mu \ln \phi $$ ∂ μ ln ϕ of the Stueckelberg field (“dilaton”). In the broken phase one finds the Einstein–Proca action of $$v_\mu $$ v μ of mass proportional to the Planck scale $$M\sim \langle \phi \rangle $$ M ∼ ⟨ ϕ ⟩ , and a positive cosmological constant. Below this scale $$v_\mu $$ v μ 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 $$0.007\le r\le 0.01$$ 0.007 ≤ r ≤ 0.01 for current spectral index $$n_s$$ n s (at $$95\%$$ 95 % CL) and $$N=60$$ 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.

Section: Palatini R 2 Inflationmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 77%

See 1 more Smart Citation

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

2020

“…The SMW model can have successful inflaton. The Higgs potential in (43) can drive inflation as discussed in [30,31,58]. But who "ordered" the Higgs in the early Universe?…”

Section: Inflationmentioning

confidence: 99%

Standard Model in Weyl conformal geometry

2022

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal with no new fields beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry D(1) (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of D(1) by a geometric Stueckelberg mechanism in which the Weyl gauge field ($$\omega _\mu $$ ω μ ) acquires mass by “absorbing” the spin-zero mode of the $${\tilde{R}}^2$$ R ~ 2 term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The EW scale is proportional to the vev of the Stueckelberg field. The Higgs field ($$\sigma $$ σ ) has direct couplings to the Weyl gauge field ($$\sigma ^2\omega _\mu \omega ^\mu $$ σ 2 ω μ ω μ ). The SM fermions only acquire such couplings for non-vanishing kinetic mixing of the gauge fields of $$D(1)\times U(1)_Y$$ D ( 1 ) × U ( 1 ) Y . If this mixing is present, part of the mass of Z boson is not due to the usual Higgs mechanism, but to its mixing with massive $$\omega _\mu $$ ω μ . Precision measurements of Z mass then set lower bounds on the mass of $$\omega _\mu $$ ω μ which can be light (few TeV). In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio r on the spectral index $$n_s$$ n s is similar to that in Starobinsky inflation but mildly shifted to lower r by the Higgs non-minimal coupling to Weyl geometry.

“…We can now use Lagrangian ( 16) and potential V (ϕ) of ( 17) to study inflation with ϕ as the inflaton and compare its predictions for the Weyl (γ = 1/4) and Palatini (γ = 1) cases. For a previous study of inflation in the Weyl case, see 9 [7,76]. Lagrangian ( 16) describes a single scalar field in Einstein gravity and the usual formalism for a single-field inflation can be used.…”

Section: Weyl Versus Palatinimentioning

confidence: 99%

Gauging scale symmetry and inflation: Weyl versus Palatini gravity

2021

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 ($$w_\mu $$ w μ ) 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 $$w_\mu $$ w μ ), Planck scale and metricity emerge in the broken phase after $$w_\mu $$ w μ acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ($$\phi _1$$ ϕ 1 ), 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 $$R^2$$ R 2 term, both theories have a small tensor-to-scalar ratio ($$r\sim 10^{-3}$$ r ∼ 10 - 3 ), larger in Palatini case. For a fixed spectral index $$n_s$$ n s , reducing the non-minimal coupling ($$\xi _1$$ ξ 1 ) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough $$\xi _1\le 10^{-3}$$ ξ 1 ≤ 10 - 3 , unlike the Palatini version, Weyl theory gives a dependence $$r(n_s)$$ r ( n s ) similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections.