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Ultraviolet completion of the inflationary paradigm

Leonardo Modesto, Lorenzo Orlando

Abstract

After an exhaustive introduction highlighting the strengths and weaknesses of the non-local models proposed so far as ultraviolet completions of the Starobinsky theory, we propose a new nonlocal completion of a general $f(R)$ theory (in the Einstein's frame) suitable for driving inflation in the early universe consistently with observations. The nonlocal theory shares with $f(R)$ the same background solutions and the same equations of motion for perturbations at linear and nonlinear level. Therefore, the classical cosmological observables are not affected by the nonlocal operators needed for the quantum completion. Our construction applies to any local action written in the Einstein's frame, but we will provide the details only for two explicit examples: the Starobinsky model and a general $f(R)$ theory. The new model overcomes the incompatibility of renormalizability and stability present in the previous proposals. Since the nonlocal theory is at least super-renormalizable, but can also be finite depending on the details of the model, this work shows the consistency of the inflationary paradigm with a well-defined quantum theory of gravity at high energy. We could rephrase the latter statement saying that the success of the $f(R)$ theories in cosmology can be traced back to the existence of an ultraviolet completion that preserves all the classical features. The inflationary paradigm survives, or it is actually insensitive to quantum gravity, because it is an exact solution of quantum gravity, up to perturbative corrections.

Ultraviolet completion of the inflationary paradigm

Abstract

After an exhaustive introduction highlighting the strengths and weaknesses of the non-local models proposed so far as ultraviolet completions of the Starobinsky theory, we propose a new nonlocal completion of a general theory (in the Einstein's frame) suitable for driving inflation in the early universe consistently with observations. The nonlocal theory shares with the same background solutions and the same equations of motion for perturbations at linear and nonlinear level. Therefore, the classical cosmological observables are not affected by the nonlocal operators needed for the quantum completion. Our construction applies to any local action written in the Einstein's frame, but we will provide the details only for two explicit examples: the Starobinsky model and a general theory. The new model overcomes the incompatibility of renormalizability and stability present in the previous proposals. Since the nonlocal theory is at least super-renormalizable, but can also be finite depending on the details of the model, this work shows the consistency of the inflationary paradigm with a well-defined quantum theory of gravity at high energy. We could rephrase the latter statement saying that the success of the theories in cosmology can be traced back to the existence of an ultraviolet completion that preserves all the classical features. The inflationary paradigm survives, or it is actually insensitive to quantum gravity, because it is an exact solution of quantum gravity, up to perturbative corrections.
Paper Structure (14 sections, 136 equations, 1 figure, 1 table)

This paper contains 14 sections, 136 equations, 1 figure, 1 table.

Table of Contents

  1. The Starobinsky theory: pros and cons
  2. A non-local completion of the Starobinsky theory
  3. Power-counting analysis and the renormalizability problem
  4. Instabilities in a super-renormalizable non-local Starobinsky model
  5. A super-renormalizable and stable non-local Starobinsky model
  6. Ultraviolet divergences
  7. Renormalizability
  8. Finite number of counterterms
  9. Achieving finiteness
  10. Conclusions
  11. Stability problems around a conformally flat background
  12. Non-renormalizability of nonlocal $f(R)$ in the Jordan frame
  13. Dimensional regularization and cut-off scheme
  14. Unified nonlocal theory of gravity and matter

Figures (1)

  • Figure 1: Comparison between the stability bound region (blue region), which is dictated by \ref{['Kosce']}, and the location of some unstable zeroes located in the red region of \ref{['perturb_EOM_simpl']}. The plot is in the adimensional variable $z$ for $\Lambda_{*}=M_{\mathrm{p}}$. We used a form factor with asymptotic polynomial of degree $n=1$, and the angular region $\omega^{\epsilon}_0$ in \ref{['unstable']} with $|z|>10^{-7}$. We recall that the red region is extended upon $|z|=\infty$, in order for it to be a singularity neighborhood.