CFT driven cosmology and conformal higher spin fields

Conformal higher spin (CHS) field theory, which is a solid part of recent advanced checks of AdS/CFT correspondence, finds applications in cosmology. The hidden sector of weakly interacting CHS fields suggests a resolution of the hierarchy problem in the model of initial conditions for inflationary cosmology driven by a conformal field theory. These initial conditions are set by thermal garland-type cosmological instantons in the sub-Planckian energy range for the model of CHS fields with a large positive coefficient $\beta$ of the Gauss-Bonnet term in their total conformal anomaly and a large number of their polarizations $\mathbb{N}$. The upper bound of this range $M_P/\sqrt{\beta }$ is shown to be much lower than the gravitational cutoff $M_P/\sqrt{\mathbb{N}}$ which is defined by the requirement of smallness of the perturbatively nonrenormalizable graviton loop contributions. In this way we justify the approximation scheme in which the nonrenormalizable graviton sector is subject to effective field theory under this cutoff, whereas the renormalizable sector of multiple CHS fields is treated beyond perturbation theory and dynamically generates the bound on the inflation scale of the CFT cosmology $M_P/\sqrt{\beta }\ll M_P/\sqrt{\mathbb{N}}$. This confirms recent predictions for the origin of the Starobinsky $R^2$ and Higgs inflation models from the CHS cosmology, which occurs at the energy scale 3 or 4 orders of magnitude below the gravitational cutoff, $\sqrt{\mathbb{N}/\beta }\sim {10}^{-3}–{10}^{-4}$. We also consider cosmological models dominated by fermionic CHS fields with a negative $\beta$ and anomaly free models of infinite towers of CHS fields with $\beta =0$ and briefly discuss the status of unitarity in CHS models.

Figure 1

Transition from the density matrix to the statistical sum.

Figure 2

Origin of boundary conditions: in the transition to the statistical sum the pinching point $x$ goes over into two different points of $S^4$, $x\to x_{\pm }$, with equal field values.