# Recent Advances in Inflation

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## Abstract

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## 1. Introduction

## 2. Brief Overview of Inflation

Inflation is an era of an abrupt accelerated expansion that took place in a very early period after the beginning of the Universe.

#### 2.1. The Shortcomings of the Hot Big Bang Cosmology

**unstable**one, more specifically. This indicates that if the strong energy condition is satisfied, when $\mathrm{w}$ deviates a bit from that unstable equilibrium point, for $\mathrm{w}>-1/3$ with ${\mathsf{\Omega}}_{K}>0$, the parameter ${\mathsf{\Omega}}_{K}$ keeps growing and with ${\mathsf{\Omega}}_{K}<0$, ${\mathsf{\Omega}}_{K}$ keeps decreasing away from zero. From experimental data and observations of large-scale structures and the CMB, it is concluded that $-0.0179<{\mathsf{\Omega}}_{K}<0.0081$ (95% C.L.). This means that the presently observed universe is very flat, and therefore it used to be even flatter in the past, with the value of parameter ${\mathsf{\Omega}}_{K}$ reaching exceptionally small values, ${\mathsf{\Omega}}_{K}\sim 0$. This could be the case if we assumed that $K=0$ precisely for the universe in its very early initial state. However, it seems quite peculiar to have such a precise value of K with no explanation as to why this would be the case.

#### 2.1.1. The Horizon Problem

**particle horizon $\mathcal{R}$**, which is the distance that light could have traveled since the beginning when $a=0$ and regions separated by distance more that the particle horizon are “causally disconnected”, meaning they can never communicate with each other. The particle horizon is also called the

**comoving horizon**and is given by the following relation:

**comoving Hubble radius**and if some regions are separated by a distance greater than the coming Hubble radius, they cannot communicate with each other at that time. By this definition, for a universe dominated with a fluid with an equation-of-state parameter w, the Hubble radius is

#### 2.1.2. The Flatness Problem

#### 2.1.3. The Primordial Relics Problem

**grand unified theories**(GUT) or string theory or some Standard Model extensions. In a GUT scenario, the universe used to have a temperature of the order of the GUT temperature, which is about ${T}_{GUT}\sim {10}^{28}\phantom{\rule{3.33333pt}{0ex}}K$, and the electromagnetic, weak and strong forces were unified. According to GUT, the universe went through a phase transition when the temperature of the universe dropped below the ${T}_{GUT}$, and during that transition, there was a production of primordial relics (e.g., domain walls, magnetic monopoles or other topological defects), which are described as point-like topological defects in the scheme of GUT. So at the time of their creation, the number and energy density of primordial relics, like magnetic monopoles, should have been large but still smaller than the ones for radiation during the GUT period. Thus, the universe in that early stage was still radiation-dominated. Later on, as those relics should have been quite large, they would have quickly become non-relativistic since ${\rho}_{M}\propto {a}^{-3}$ for magnetic monopoles (massive particles) and ${\rho}_{r}\propto {a}^{-4}$ for radiation; thus, they would have dominated over radiation and ordinary matter until the present time. However, from the observations, there is no evidence for the existence of such primordial relics and definitely no signs that they dominate the universe, with research setting an upper limit to their number density today of ${n}_{M}\left({t}_{0}\right)\sim {10}^{-19}$ cm${}^{-3}$. This very small number density and the lack of these kinds of observations today in the universe compared to the predictions of the standard Big Bang scenario along with particle physics is the primordial relics problem.