Cosmology Winter School 6/12/2011 !

Inflationary universe!-!

dynamics of the early universe!

Jean-Philippe UZAN!

Lecture 3:!

Motivation for inflation

Motivation for inflation - solve the standard big-bang problems (flatness, horizon)

The origin of the flatness problem is clear:

During the cosmological evolution aH decreases

Naturalness problem.

Motivation for inflation

Problem with structure formation: initial conditions seem « acausal ».

Simplest way to solve the problems

Motivation for inflation - solve the standard big-bang problems (flatness, horizon)

The origin of the flatness problem is clear:

During the cosmological evolution aH decreases

Inflation = primordial phase of accelerated expansion

Assume there is a primordial phase during which aH increases

What do we need?

Inflation Standard hot big bang ?

Number of e-folds:

If we assume H constant (to be justified later) then

To solve the flatness problem, one needs at least

Horizon problem!

Horizon problem!

Conformal representation!

Standard FL universe (no !)

Conformal representation!

Standard FL universe (no !) Standard FL universe (with !)

Conformal representation!

Standard FL universe (no !)

FL universe with an intermediate stage of inflation.

Simplest solution!

We need to find a type of matter that alows for !+3P<0.

Simplest solution!

We need to find a type of matter that alows for !+3P<0.

The simplest idea is a cosmological constant: P = - !. If it dominates at early time, it always dominates the matter content.

Simplest solution!

We need to find a type of matter that alows for !+3P<0.

The simplest idea is a cosmological constant: P = - !. If it dominates at early time, it always dominates the matter content.

Old inflation idea: phase transition to have a cosmological constant acting only during a finite time.

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De Sitter spacetime!

In the case of a pure cosmological constant, the spacetime has a de Sitter geometry

H being a constant.

It is a simple exercise to determine the coformal time

so that the metric is

Inflation Hot big-bang

«"Solution"» for the origin of structures

Inflation Hot big-bang

Set initial conditions

here

Super-Hubble Sub-Hubble Sub-Hubble

During inflation, by construction so the comoving Hubble radius is decreasing.

Scalar field cosmology !

For a scalar field, the matter action is

so that the stress-energy tensor takes the form

from which we deduce that the energy density and pressure are

The Einstein equations are thus

to which we need to add the Klein-Gordon equation

[G]=M-2

[#]=M [V]=M4

Scalar field cosmology !

The Einstein equations are thus

to which we need to add the Klein-Gordon equation

We deduce that:

-!If K=0, then H is decreasing during inflation.

- The equation of the scalar field interpolates between -1 when the kinetic term is small compared to the potential term (slow-roll) and +1.

Conformal time!

Using the conformal time, the Friedmann equations take the form

while the Klein-Gordon equation becomes

Slow roll regime: idea!

We have seen that an accelerated expansion, with an almost de Sitter phase is possible if the field is slow-rolling.

Let us assume that

The Friedmann equations imply that

from which we deduce that

Now:

So the slow roll conditions can be fulfilled if the potential is flat enough

Slow-roll parameters!

Consider the 3 parameters

They can be rewritten as

The Friedmann equations take the form

The number of e-fold is then given by

Slow-roll formalism!

The time derivatives of the slow-roll parameters are

First order approximation:

- we work at first order in $ and %.

- & is then second order since

!- in $ and % can be assumed constant.

Expression in terms of the potential:

Explicit example: massive field!

Let us consider the simplest potential

In slow-roll, the Klein-Gordon and Friedmann equation are

The slow-roll parameters are easily found to be

The slow-roll conditions are satified at large field.

Explicit example: massive field!

Let us consider the simplest potential

In slow-roll, the Klein-Gordon and Friedmann equation are

During slow-roll, these equations can be integrated as

and the number of e-fold is

Explicit example: massive field!

Exponential

expansion

Reheating

Phase space analysis & sensitivity to IC

Inflationary attractor =

slow-roll solution

Chaotic initial conditions !The description we have used is typically valid up to

This implies that the maximum value for the scalar field is

and the maximum number of e-fold

Numerically:

A typical observer is in a zone with a large number of e-fold: chaotic IC.

Type of potentials!

There is a huge litterature on inflationary potentials.

Conclusions!

Inflation allows to solve the standard problems of the big bang.

It may offer the possibility to adress the initial conditions for structure formation.

The main class of models is single field slow-roll inflation.

Slow-roll regime is an attractor of the dynamics and allow to derive generic predictions of inflation IF the number of e-fold is large enough.

The chaotic initial conditions make us conclude that the number of e-fold is expected to be much larger than the minimum required value.

It should not be a surprise then to observe such a flat and homogeneous universe.