## Thesis Summary

Following the University of Tehran regulations, I wrote my master's thesis in Persian. Therefore, not everyone can read it. On this page, I have provided an English translation of my thesis's abstract, introduction, and table of contents. This is to give you a general view of what I studied during my master's program.**Note**: My thesis is mainly a review of the following papers:

- Maldacena, Juan. "Non-Gaussian features of primordial fluctuations in single field inflationary models." (arXiv:astro-ph/0210603v5)
- Seery, David, and James E. Lidsey. "Primordial non-Gaussianities in single-field inflation." (arXiv:astro-ph/0503692v2)
- Cheung, Clifford, et al. "The effective field theory of inflation." (arXiv:0709.0293)
- Martin, Jerome, and L. Sriramkumar. "The scalar bi-spectrum in the Starobinsky model: The equilateral case." (arXiv:1109.5838v1)
- Martin, Jerome, Hayato Motohashi, and Teruaki Suyama. "Ultra slow-roll inflation and the non-Gaussianity consistency relation." (arXiv:1211.0083v1)
- Cai, Yi-Fu, et al. "Revisiting non-Gaussianity from non-attractor inflation models." (arXiv:1712.09998v2)
- Chen, Xingang, and Yi Wang. "Quasi-single field inflation and non-Gaussianities." (arXiv:0911.3380v4)

### Abstract

Inflation has become the leading paradigm of the early universe. However, the detailed dynamics of inflation are still a mystery. A major theme in cosmology is building inflationary models and comparing their predictions with experimental data. Primordial non-Gaussianity is widely known for its strength in discriminating different inflationary models or alternative scenarios. In this work, we introduce some practical methods for evaluating the non-Gaussianity. Then, we use these methods to study the non-Gaussianity in some cosmological inflationary models. We start by introducing the simplest inflation scenarios as a toy model. We continue by discussing models that allow us to get significant levels of non-gaussianity and violation of Maldacena's consistency relation; This relation proves that non-Gaussianities are small in any single field inflationary model, and hence, studies of non-Gaussianity will not resolve the degeneracy between models. We finish our thesis by presenting a two-field scenario as an example of models that produce large local non-Gaussianity while satisfying the consistency relation.

### Introduction

The **Big Bang theory** successfully explains the "blackbody spectrum" of the cosmic microwave background radiation and the origin of the light elements. Nevertheless, it leaves three fundamental questions unanswered:

- Why is the geometry of the universe nearly flat?
- Why is the large-scale structure of the universe homogeneous and isotropic?
- Why have magnetic monopoles never been observed?

The **Inflation theory** developed around 1980 to explain these puzzles with the standard big bang theory, in which the universe gradually expands throughout its history. Inflation has become the leading paradigm of the early universe due to its precise predictions. However, the detailed dynamic of inflation is still a mystery. Therefore, A major theme in cosmology is to build inflationary models and compare their predictions with experimental data.

Scientists use the concepts like **primordial non-Gaussianity**, **primordial gravitational waves**, and **primorial features** to probe the early universe. Professor Xingang Chen at Harvard University points out three questions that primordial features and non-Gaussianity might answer:

- Was the primordial universe inflationary or non-inflationary?
- What were the details of the inflation model and the inflationary dynamics?
- What were the particle contents of the primordial universe?

In this thesis, we only focus on the second question above; We describe the inflation model building and methods to evaluate the non-Gaussianity in each model. With great advancements in our theoretical and observational techniques, our understanding of the inflationary paradigm has developed significantly. However, the nature of inflation is still hidden from us; we do not know which fields are responsible for the accelerated expansion, the Lagrangian of these fields is yet to be determined, and inflation theory must be distinguished from the alternatives.

The Lagrangian for the **simplest inflation model** contains one scalar field with a canonical kinetic term. This model successfully predicts the scale-invariance fluctuations and agrees well with the CMB temperature observations. The non-Gaussianity amplitude in this model is of order of the slow roll parameter, which satisfies Maldacena's consistency relation. **Consistency relation** proves that non-Gaussianities are small in any single field inflationary model; hence, studies of non-Gaussianity will not resolve the degeneracy between models.

Studies of the inflation models with standard kinetic terms are indeed instructive. However, it is not always the case; in models descending from supergravity or superstring, it is generally expected that corrections to the kinetic term will arise. To study such cases, we can consider an inflation model with a kinetic term in the general form. These models are called **P(X) theories**, which reproduce the results of standard models in a particular case. These classes of models, again, predict a small among of non-Gaussianity and satisfy the consistency relation. However, P(X) theories are not yet the most general action for single field models. In order to write the most general action for describing the cosmological perturbation that unifies all the single field models we need to use a more fundamental theory - the effective field theory. In the context of the **EFT of inflation**, we look at perturbations as goldstone modes produced as a result of spontaneous symmetry breaking.

Another powerful inflationary scenario is the **Starobinsky model**. The Starobinsky model consists of a linear inflaton potential with a sudden change in the slope. The change in the slope causes a brief period of departure from slow-roll, which in turn could produce a large among of non-Gaussianity. These features in the spectrum are known to lead to a better fit to the data.

The **importance of consistency relation** relies on the fact that deviations from it might be detected in future experiments, allowing us to rule out all single field inflationary models. However, not all the single field models follow the consistency relation. **Ultra slow-roll** (USR) inflation has long been used to challenge the non-Gaussianity consistency relation. The canonical USR model is constructed by assuming that the inflaton's potential is almost constant. This model leads to an order one slow-roll parameter, which will result in large non-Gaussianity and violation of consistency relation. The experimental data prove neither USR nor the consistency relation. Therefore, we can not rule out either of them. However, by studying the reasons for violation of consistency relation in USR we can improve our understanding of this relation.

To study the ultra slow-roll inflation, one should consider that **this scenario is incomplete by itself** and should be followed by a phase of slow-roll attractor (URS is only stable for a few efolds and having a second phase is necessary to have 60 efold of accelerating expansion). Therefore, USR consists of at least three phases: the USR phase, the transition phase, and the slow-roll phase. The transition phase here can be defined in different ways, and each kind will have a different impact on the non-Gaussianity parameter. Studying different types of transitions can reveal rich information about the USR and consistency relation.

As we said before, consistency relation states that every single field inflation model leads to a small among of non-Gaussianity. Therefore, it seems logical that one search for the large non-Gaussianity in multi-field models. **Quasi-Single Field** is a two-field inflationary scenario in which large non-Gaussianity is produced while the consistency relation stays satisfied. In this scenario, we consider a non-flat path for the inflation field. The flat inflation path is tilted by the effect of a heavy isocurvature field (carvaton have a mass at least of order the Hubble parameter H). If the inflaton decouples from the isocurvatons or the isocurvaton mass are all much larger than O(H), quasi-single field inflation makes the same prediction as the single field inflation. It can be shown that these massive isocurvatons can have important effects on density perturbations.

### Table of Contents

Following is the table of contents of my master's thesis. Please note that this list includes titles of chapters, sections, and sub-sections, while titles of sub-sub-sections and appendixes are removed.

### Chapter 1: Primordial Non-Gaussianity

- Why Non-Gaussianity
- Statistics
- Power Spectrum and Correlation Function
- Gaussian Random Fields
- Wick's Theorem
- The simplest form of Non-Gaussianity
- Sources of Non-Gaussianity

- Shape and Amplitude of Non-Gaussianity
- In-In Formalism and Correlation Function
- In-In Formalism
- Mode Functions and Vacuum
- Contractions
- Three forms

- Slow-roll single-field inflation
- ADM Formalism
- Minimally coupled Scalar Field
- The Quadratic Action
- The Cubic Action
- The three point function

- Consistency Condition
- δN formalism

### Chapter 2: General Single Field Models

- Non-standard kinetic terms: P(x) theories
- The background model
- The Quadratic Action
- The Cubic Action

- The effective field theory of inflation
- Spontaneous symmetry breaking
- Unitary Gauge
- Most general action in unitary gauge
- Recovering gauge invariance and decoupling limit

### Chapter 3: Non-Gaussianity in the Starobinsky Model

- Background evolution
- The scalar power spectrum
- The dominant contribution to the bi-spectrum
- The three point function
- Evaluating the dominant contribution

- The sub-dominant contributions to the bi-spectrum
- The contribution due to the second term
- The contribution due to the first and the third terms
- The contribution due to the fifth and the sixth terms
- The contribution due to the field redefinition

- Amplitude of Non-Gaussianity
- Can Non-Gaussianity parameter be large in the Starobinsky model?
- The hierarchy of contributions to the bi-spectrum

### Chapter 4: Ultra Slow-Roll Inflation and Consistency Relation

- Ultra Slow-Roll Inflation
- Power Spectrum and Non-Gaussianity
- Transition to slow-roll phase

### Chapter 5: Quasi-Single Field Inflation and Non-Gaussianities

- Quasi-single field inflation
- Lagrangian and mode functions
- Two gauges
- Power spectrum and spectral index
- Bispectra