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!
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.
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