Starobinsky inflation in emergent gravity

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Abstract

Inflation is the main paradigm for explaining the homogeneity and isotropy of the observed universe. In view of its agreement with the Planck CMB data, we studied in this work Starobinsky inflation in models of induced gravity. Induced gravity theories are attractive because they arise from matter loops and have the potential to qualify as natural.

Introduction

General Relativity (GR) can be considered one of the most successful theories of physics. It has withstood many trials firmly so far [1]. However, this does not mean that it is without flaws or that it is complete. The enigmatic horizon and flatness problems [2], [3] indicate that a correction is needed to GR. The idea of inflation [4], [5], whose constraints are well defined [6], however, solves these problems neatly. Inflation can be introduced to GR as a correction to Einstein–Hilbert(E–H) action and is generally studied on f(R) theories [7], [8]. This model is equivalent to scalar–tensor theories, which introduce a new scalar field generally known as the “inflaton”.

The R+R2 model, namely Starobinsky Inflation [9], provides us with the desired framework to inflation. It creates a single field that rolls slowly. This model can be seen as an f(R) theory whose function can be written as f(R)=R+R2(6m2) using west coast (+,,,) metric sign convention. The constant in mass dimension m represents the mass of the scalar field, domination of the term R2 over R causes the inflationary dynamics to appear while in the low curvature limit the linear term R ends the inflation.

While inflation aims to solve the aforementioned problems, merging the Standard Model (SM) and gravity is also an important problem of modern physics [10]. This unification can be partly introduced by Induced Gravity [11], [12], which was proposed by Sakharov by proposing gravity not as a fundamental but rather as an emergent phenomena that can be induced from one-loop effects of Quantum Field Theory(QFT).

Also, a novel idea called Symmergent Gravity(SG) [13], [14], [15], [16] incorporates gravity into the SM, restoring broken gauge symmetry to an extent. Just as in Sakharov’s Induced Gravity, the flatness of the space–time elasticizes in an emergent way. Contrary to Sakharov’s idea, the Ultraviolet(UV)-Electroweak gap is mapped to an affine curvature [17], [18], [19], [20]. Hence this symmetry restoring emergent gravity, Symmergent Gravity might be a solution to the unification problem.

While gravitational deductions have been made from both ideas in the past, in this paper we relate these ideas to the Starobinsky model.

Section snippets

Sakharov’s induced gravity

One of the early ideas about the emergence of gravity from QFT is Sakharov’s Induced Gravity. It assumes a Lorentzian Manifold where geometry exists as a classical background. From these assumptions, it brings out that gravity does not intrinsically exist but rather emerges from one-loop coefficients in QFT. The action proposed by Sakharov includes terms that resemble cosmological constant, linear, and quadratic curvature. Particularly the quadratic curvature can be used to obtain an

Symmergent gravity

“Symmergent” or explicitly “Gauge Symmetry Restoring Emergent” gravity is a novel framework that proposes to restore broken symmetry while incorporating GR to SM. E–H action which is an R theory obtained properly as has been demonstrated comprehensively before [13], [14], [15], [16]. SG also proposes solutions to the charge and colour breaking and the hierarchy problem. In this section, however, we expand on the idea of symmergence to obtain an R+R2 Starobinsky model. Let us review the idea of

Conclusion

Inflation has been the most successful idea so far to explain the homogeneity and isotropy of the observed universe. This work is a modest attempt to study inflation in both Sakharov’s induced gravity and SG.

Sakharov’s Induced gravity proposes an action that contains linear and quadratic curvature terms that emerge from one loop QFT. This emergent action can be mapped to the Starobinsky model, which is an R+R2 model. To get a higher-order curvature model, classical contribution to the

CRediT authorship contribution statement

İlim İrfan Çimdiker: Methodology, Analysis, Writing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This project is supported in part by TÜBİTAK, Turkey grant 118F387. The author is grateful to Durmuş Ali Demir for guidance and helpful discussions.

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