The Inflationary Universe

Max CamenzindIMPRS Cosmology SS2009

Day 6/1

Two Major Cosmological DiscoveriesTwo Major Cosmological Discoveries

• (i) The new-born universe experienced rapid acceleration (called Inflation)

• (ii) A new (slow) stage of acceleration started 5 billion years ago (Dark Energy)

Two Major questions:Two Major questions: How did the Universe start, How did the Universe start, and how it is going to end?and how it is going to end?

Topics

• What is Inflation ?• On Inflation History• Problems of the Standard Model• How to describe Inflation ? – The Inflaton

Field and Slow-Roll Conditions.• Fluctuations in the Inflaton field.• Quantisation … Universal Fluctuation Spectrum

What is Inflation ?

• In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized exponential expansion of the universe at the end of the grand unification epoch, 10-36 seconds after the Big Bang, driven by a negative-pressure vacuum energy density. The term "inflation" is also used to refer to the hypothesis that inflation occurred, to the theory of inflation, or to the inflationary epoch.

Idea of Inflationary UniverseInflationary Universe

1030

Inflation - Mechanisms• As a direct consequence of this expansion, all of the

observable universe originated in a small causally connected region. Inflation answers the classic conundrum of the big bang cosmology: why does the universe appear flat, homogeneous and isotropic in accordance with the cosmological principle when one would expect, on the basis of the physics of the big bang, a highly curved, heterogeneous universe? Inflation also explains the origin of the large-scale structure of the cosmos. Quantum fluctuations in the microscopic inflationary region, magnified to cosmic size, become the seeds for the growth of structure in the universe (see galaxy formation and evolution and structure formation)

Inflation - History

• Inflation was proposed by Alexei Starobinski (1979/80) in the Soviet Union, and simultaneously by Alan Guth (1980/81) in the United States. Guth's mechanism is different from Starobinski's, and requires a modification to allow for a graceful exit from inflation. This modification was provided independently by Andrei Linde, and by Andreas Albrecht and Paul Steinhardt.

Inflation Pioneers

• Proposed by Guth in 1981 to solve:– Horizon problem– Flatness problem

• Basic idea: universe undergoes exponential expansion in early history

Andrei LindeStanford

Alan GuthMIT

Theories of Inflation over the Years1980

2000

1990

-inflation Old Inflation

New Inflation Chaotic inflation

Double Inflation Extended inflation

DBI inflation

Super-natural Inflation

Hybrid inflation

SUGRA inflation

SUSY F-term inflation SUSY D-term

inflation

SUSY P-term inflation

Brane inflation

K-flationN-flation

Warped Brane inflation

inflation

Power-law inflation

Tachyon inflationRacetrack inflation

Assisted inflation

Roulette inflation Kahler moduli/axion

Natural inflation

Problems in Standard Model

• The Standard Big-Bang Model has many deep problems:

• Flatness Problem: The flatness problem (also known as the oldness problem) is an observational problem associated with a FRW model.

• Causality Problem: The causlaity or horizon problem results from the premise that information cannot travel faster than light.

• Monopole Problem: Grand unification theories predicted topological defects in space that would manifest as magnetic monopoles.

• General Scale Problem: ~ µm is no natural scale

The Flatness Problem

The Horizon Problem

• When we look at the CMB it comes from 46 billion comoving light years away. However when the light was emitted the universe was much younger (300,000 years old). In that time light would have only reached as far as the smaller circles. The two points indicated on the diagram would not have been able to contact each other because their spheres of causality do not overlap.

Horizon Problem

Big Bang

Decoupling

We observer

The Horizon Problemin Conformal Time

The Relic Problems• The Gravitino: The gravitino is the supersymmetric

partner of the graviton, as predicted by theories combining general relativity and supersymmetry; i.e. supergravity theories. If it exists it is a fermion of spin 3/2. Thay later, after they decay later, after BBN, would ruin BBN.

• The Monopoles: In physics, a magnetic monopole is a hypothetical particle that is a magnet with only one pole (see Maxwell's equations for more on magnetic poles). In more technical terms, it would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably Grand Unified Theories and superstring theories, which predict their existence.

• Other topological defects (Strings etc)

Inflaton DynamicsThe simplest scenario features a single scalar field moving in a potential V(φ). Many apparently more complicated scenarios can be reduced to this.

( )( )

φφφ

φφπ

ddVH

VGaaH

−=+

+=≡

3

38 2

21

2

22

φ

V(φ)

The slow-Roll Approximation

( )( )

φφφ

φφπ

ddVH

VGH

−=+

+=

3

38 2

212

These equations can only be solved exactly for a few choices of potential, for example an exponential potential

Ordinarily the equations can then be solved analytically. Conveniently, the condition for inflation to occur is almost precisely the same as that for validity of the slow-roll approximation.

However, usually sufficiently accurate results can be obtained by using the slow-roll approximation.

2;)exp()( 2 <∝ λλ φφV

The Inflaton Field

Equation State

The Inflaton Field

Ex: Slow-Roll

Field >> Planck value !

Inflation Conditions

Models of Inflation

• A potential V(φ)• A way to end inflation, e.g. if slow-roll

condition is no longer valid Reheating• or when extra physics enters: hybrid

inflation.

Amount of Inflation

Flow lines for the Universe

Universe starts at (Ωmatter,ΩΛ)=(1,0) and moves to attractor point at (0,1) (de Sitter) – which curve are we on??

Ωmatter

ΩΛ

This side Universe closed

This side Universe open

Why do we need inflation?Why do we need inflation?

• What was before the Big Bang?• Why is our universe so homogeneoushomogeneous (better

than 1 part in 10000) ?• Why is it isotropicisotropic (the same in all directions)?• Why all of its parts started expanding

simultaneously?• Why it is flatflat? Why parallel lines do not intersect?

Why it contains so many particles? Why there are so many people in this auditorium?

Problems of the standard Big Bang theory:Problems of the standard Big Bang theory:

Gut’s New InflationGut’s New Inflation

V

Ideas fromGUT phasetransitions

Inflation as a theory of a harmonic oscillatorInflation as a theory of a harmonic oscillator

Eternal Inflation

• Einstein:

• Klein-Gordon:

Equations of motion:Equations of motion:

Compare with equation for the harmonic oscillator with friction:

Pha

se P

lot f

orC

haot

ic In

flatio

n

Attractors

Logic of Inflation:Logic of Inflation:Large φ large H large friction

field φ moves very slowly, so that its potential energy for a long time remains nearly constant

No need for false vacuum, supercooling, phase transitions, etc.

Inflation makes the Universe flat, Inflation makes the Universe flat, homogeneous and isotropichomogeneous and isotropic

In this simple model the universe typically grows 1030 times during inflation.

Now we can see just a tiny part of the universe of size ct = 1010 light yrs. That is why the universe looks homogeneous, isotropic, and flat.

Hybrid InflationHybrid Inflation

String Theory LandscapeString Theory Landscape

Perhaps 10Perhaps 10100100 - 10 - 1010001000 different minimadifferent minima

Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…

Lerche, Lust, Schellekens 1987Lerche, Lust, Schellekens 1987

Example: Racetrack InflationExample: Racetrack Inflation

waterfall from the saddle point

Example: SUSY LandscapeExample: SUSY Landscape

V

SU(5) SU(3)xSU(2)xU(1)SU(4)xU(1)

Weinberg 1982: Supersymmetry forbids tunneling from SU(5) to SU(3)xSU(2)XU(1). This implied that we cannot break SU(5) symmetry.

A.L. 1983: Inflation solves this problem. Inflationary fluctuations bring us to each of the three minima. Inflation make each of the parts of the universe exponentially big. We can live only in the SU(3)xSU(2)xU(1) minimum.

Supersymmetric SU(5)

Self-Reproducing Inflationary UniverseSelf-Reproducing Inflationary Universe

A. Linde

Linear theory (coordinate approach)

• Perturbed Friedmann universe

curvature perturbation

xi = const.

Σ(t)

Σ(t+dt)dτ

• proper time along xi = const.: (1 )d dtAτ = +

• curvature perturbation on Σ(t): ψ ( ) ψ∆−= 23 4

aR

ds² = -(1 + 2A) dt² + a²(t) (1 – 2Ψ) δij dxi dxj

Quantum Fluctuations in Φ

Inhomogeneous spacetime in Newtonian gauge (vanishing stress)(see Sect. on Perturbations)

Einstein‘s equations imply ~ Klein-Gordon equation with mass-term:

Adiabatic fluctuations dS = 0

-

Klein-Gordonequation with mass given bym² = -z‘‘/z Can bequantized similar

For K = 0 and cS = c Inflaton field is the source for metric perturbations.

Make a rescaling, such that 1st order derivative disappears:

Quantisation of Φ

Short wavelength limit

Long wavelength limit

Sim

ple

Scal

ar F

ield

in

de

Sitt

er

Mode leavingthe Horizon isfrozen in.

Potential Barrier and Power Spec

Power Spectrum

Conventional Slow-Roll Approx

Solution by Hankel functions

Wav

elen

gths

are

sim

ply

stre

tche

d in

Exp

ansio

n

QuantumFluctuationsin Pot Φ

Power Spectrum – Chaotic Inflation

• Parameters:• Chaotic

Inflation• m² = 1.9x10-12 MP² Φ(0) = 16.8 MP

dΦ(0) = -0.1 MP/s

N = 57.65

MP² = 1/8πG

Andreas Heinen – Thesis 2005

Fini

te V

olum

eSu

ppre

ssio

n?

QG

ravi

tyC

utof

f ?

Tensor Power Spectrum - GWaves

Andreas Heinen – Thesis 2005

Power Spectrum – Running Spec Index

Andreas Heinen – Thesis 2005

Running spectral index:

Andreas Heinen – Thesis 2005

Power Spectrum – Running Spec Index

Andreas Heinen – Thesis 2005

Power Spectrum – Spectral Ratio

Power Spectrum – Chaotic Inflation

Andreas Heinen – Thesis 2005

Power Spectrum – Quartic Potential

Andreas Heinen – Thesis 2005

• Parameters:• Quartic

Inflation∀ λ = 1.75 x 10-13

Φ(0) = 24 MP

dΦ(0) = - 1 MP/s

N = 60.58

MP² = 1/8πG

Power Spectrum – Quartic Potential

Andreas Heinen – Thesis 2005

Power Spectrum – Quartic Inflation

Andreas Heinen – Thesis 2005

Preheating in Inflation

Mathiey Equation

Stab

ility

Reg

ion

s in

Mat

hie

u E

quat

ion

Frolov 2009

Fluctuation in Density

Frolov 2009

Fluctuation in Psi

Lognormal Distribution

Frolov 2009

Connection QGravity ~ Inflation

The Trans-Planckian “Problem”

• What was the physical size of cosmological scales contributing to the CMB today before inflation?

• This depends on the number of e-folds of inflation. Most models give more than the minimum of 60’ish e-folds.

• Generically, those scales begin at sizes less than the Planck scale! Certainly, we should expect these scales to encompass new physics thresholds.

• Does new physics stretch as well?

space

H–1(t)

rhor(t)

time

inflation ends

MPl

Summary• Inflation solves Flatness problem, Horizon

problem & many other aspects: N > 55.• Inflation also provides source for

perturbations on the Friedmann background by means of quantum fluctuations in the very Early Universe Φ ~ 10-5.

• These perturbations are frozen in, once they are stretched by expansion beyond the horizon.

• Power spectrum and spectral index will depend on inflation model.