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Use of delta N formalism - Difficulties in generating large local-type non-Gaussianity during inflation -

Takahiro Tanaka, Teruaki Suyama, Shuichiro Yokoyama

TL;DR

This work analyzes the potential for large local-type non-Gaussianity to arise from super-horizon evolution during inflation using the $\delta N$ formalism. It derives how $f_{NL}$ depends on derivatives of the e-folding number with respect to multi-field initial conditions and discusses slow-roll suppression versus possible enhancements from the horizon-scale propagator $\Lambda^i_j$. By examining the double-inflation, hybrid inflation, and Kadota–Stewart models, the authors argue that achieving sizable $f_{NL}$ during inflation generally requires fine-tuning or unnatural initial conditions, whereas large non-Gaussianity at the end or after inflation is comparatively easier. They also highlight limitations of the standard $\delta N$ approach when small-scale perturbations are important and stress the ongoing issue of natural initial conditions in inflationary model-building.

Abstract

We discuss generation of non-Gaussianity in density perturbation through the super-horizon evolution during inflation by using the so-called $δN$ formalism. We first provide a general formula for the non-linearity parameter generated during inflation. We find that it is proportional to the slow-roll parameters, multiplied by the model dependent factors that may enhance the non-gaussianity to the observable ranges. Then we discuss three typical examples to illustrate how difficult to generate sizable non-Gaussianity through the super-horizon evolution. First example is the double inflation model, which shows that temporal violation of slow roll conditions is not enough for the generation of non-Gaussianity. Second example is the ordinary hybrid inflation model, which illustrates the importance of taking into account perturbations on small scales. Finally, we discuss Kadota-Stewart model. This model gives an example in which we have to choose rather unnatural initial conditions even if large non-Gaussianity can be generated.

Use of delta N formalism - Difficulties in generating large local-type non-Gaussianity during inflation -

TL;DR

This work analyzes the potential for large local-type non-Gaussianity to arise from super-horizon evolution during inflation using the formalism. It derives how depends on derivatives of the e-folding number with respect to multi-field initial conditions and discusses slow-roll suppression versus possible enhancements from the horizon-scale propagator . By examining the double-inflation, hybrid inflation, and Kadota–Stewart models, the authors argue that achieving sizable during inflation generally requires fine-tuning or unnatural initial conditions, whereas large non-Gaussianity at the end or after inflation is comparatively easier. They also highlight limitations of the standard approach when small-scale perturbations are important and stress the ongoing issue of natural initial conditions in inflationary model-building.

Abstract

We discuss generation of non-Gaussianity in density perturbation through the super-horizon evolution during inflation by using the so-called formalism. We first provide a general formula for the non-linearity parameter generated during inflation. We find that it is proportional to the slow-roll parameters, multiplied by the model dependent factors that may enhance the non-gaussianity to the observable ranges. Then we discuss three typical examples to illustrate how difficult to generate sizable non-Gaussianity through the super-horizon evolution. First example is the double inflation model, which shows that temporal violation of slow roll conditions is not enough for the generation of non-Gaussianity. Second example is the ordinary hybrid inflation model, which illustrates the importance of taking into account perturbations on small scales. Finally, we discuss Kadota-Stewart model. This model gives an example in which we have to choose rather unnatural initial conditions even if large non-Gaussianity can be generated.

Paper Structure

This paper contains 13 sections, 45 equations, 3 figures.

Table of Contents

  1. Introduction: Success of inflation
  2. Density perturbation
  3. Further steps from observations
  4. Basic feature of generation of non-Gaussianity of local-type.
  5. Local non-Gaussianity
  6. Super-horizon dynamics
  7. Non-Gaussianity produced at the end of or after inflation
  8. Non-Gaussianity produced during inflation
  9. Is there any model which produces large non-Gaussianity during inflation?
  10. Temporal slow-roll violation is not sufficient
  11. Simple hybrid inflation does not give large non-Gaussianity
  12. Modular inflation
  13. Conclusion

Figures (3)

  • Figure 1: Schematic picture to explain $\delta N$ formalism.
  • Figure 2: A background trajectory in double inflation model. Taken from Ref. Yokoyama:2007dw.
  • Figure 3: Contour plot of the potential of Kadota-Stewart model and a typical trajectory.