The Inflationary Universe

Max CamenzindModern Cosmology Nov 2010 @ Day 6/2

Quark-GluonPlasma

Big Bangt = 0

• 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?

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

Why do we Need Inflation ?

Two New Cosmological DiscoveriesTwo New 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?

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?

• What is Inflation ?• On Inflation History: 1981 Alan Guth, …• 5 Problems of the Standard Model• How to describe Inflation ? – The Inflaton

Field and Slow-Roll Conditions.• Fluctuations in the Inflaton field.• Evolution of the Perturbations. Universal Fluctuation Spectrum

Topics

• In physical cosmology, cosmic inflation, cosmological inflation or just inflation is an exponential expansion of the Universe at the end of the grand unification epoch, ~10-38 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.

What is Inflation ?

1030 Universe withoutInflation

Idea of Inflationary Universe

• 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 - Mechanisms

• 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 – HistoryThe Pioneers of Inflation

• QCD phase transition T = 175 MeVQCD phase transition T = 175 MeV• Electroweak phase transition T = 150 GeVElectroweak phase transition T = 150 GeV baryogenesisbaryogenesis ??• GUT phase transition(s) ? T = 10GUT phase transition(s) ? T = 101616 GeV GeV monopoles,cosmic strings ?monopoles,cosmic strings ?• ““Inflation” T = 10Inflation” T = 101515 GeV GeV primordial density fluctuations !primordial density fluctuations ! primordial magnetic fields ?primordial magnetic fields ?

Cosmological Phase Transitions

Order ofOrder ofthethePhasePhaseTransitionTransition

Temperature Temperature dependence ofdependence oforder parameterorder parameter(magnetisation)(magnetisation)

GuthGuth’’s Scenario 1981: GUT Phase Transitions Scenario 1981: GUT Phase Transition

FigFig. . A. Albrecht and P. Steinhardt, Phys. Rev. D, A. Albrecht and P. Steinhardt, Phys. Rev. D, 4848, 1220 (1981)., 1220 (1981).

““False VacuumFalse Vacuum””

Universe lingersUniverse lingersin the so-called in the so-called metastablemetastable

with energy with energy densitydensity

As the universe As the universe cools, the true cools, the true minimum gets minimum gets deeper.deeper.At T~Ts, the At T~Ts, the universe tunnels universe tunnels to the to the ““True True VacuumVacuum””..

What went wrong with GuthWhat went wrong with Guth’’s Theory?s Theory?

GuthGuth’’s phase transition scenario surely conforms to these s phase transition scenario surely conforms to these requirements. But there was a serious problem.requirements. But there was a serious problem.

• The bubbles of true vacuum will form at The bubbles of true vacuum will form at various times in various placesvarious times in various places in the false in the false vacuum.vacuum.• They will have great difficulty merging They will have great difficulty merging because the because the space separating themspace separating them will still will still be be expanding exponentiallyexpanding exponentially..• The phase The phase transition will never be transition will never be completedcompleted even if the bubbles grow at the even if the bubbles grow at the speed of light.speed of light.

Guth & Weinberg (1983)Guth & Weinberg (1983)

What came after GuthWhat came after Guth’’s Theory?s Theory?

•New Inflation (Linde 1982, Albrecht & Steinhardt 1982): New Inflation (Linde 1982, Albrecht & Steinhardt 1982):

Roll down a flat potential.Roll down a flat potential.

•Chaotic Inflation (Linde 1983)Chaotic Inflation (Linde 1983)

Slide through a viscous medium.Slide through a viscous medium.

Chaotic Inflation 1983Chaotic Inflation 1983Simple!!Simple!!

No supercooling/tunneling from No supercooling/tunneling from false vacuum. No plateau.false vacuum. No plateau.No thermal equilibrium!No thermal equilibrium!In 10In 10-35-35 s, a Plank size region s, a Plank size region blows up to ~ cms ! blows up to ~ cms ! Nonsense! Nonsense!

Initial Universe may be thought of as having chaotically distributed values Initial Universe may be thought of as having chaotically distributed values of field. Inflation took place only where was large. of field. Inflation took place only where was large. At the end of inflation, the field oscillates and decays into particle-pairs, which At the end of inflation, the field oscillates and decays into particle-pairs, which interact and thermalize at some temperature. Standard Big Bang takes over from interact and thermalize at some temperature. Standard Big Bang takes over from here.here.

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φ

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

• Basic idea: Universe undergoes exponential expansion in early history

Andrei LindeStanford

Alan GuthMIT

Inflation Pioneers

1980

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

Theories of Inflation 2010

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

• Scale Problem: ~ µm is no natural scale for RH.• Flatness Problem: The flatness problem is an

observational problem associated with a FRW model: Why is Ωk ~ 0 ?

• 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.

• Structure Problem: Structure formation not explained no natural mechanism!

5 Problems in Standard BigBang

?

The Scale Problem

Flatness Problem: Why is Universe so flat? a multi-component universe satisfies (see Friedmann Equ)

and, neglecting Λ,

therefore during radiation dominated era |1 – Ω(t)| ∝ a2

during matter dominated era |1 – Ω(t)| ∝ a if |1 – Ω0| < 0.02 (WMAP), then at CMB emission |1 – Ω| < 0.00002

we have a fine tuning problem!

( ) )()(

1)()(

)(1 220

20

20

22

2

tatHH

RtatHkct Ω−=−=Ω−

3m0

4r0

2

0

)(aaH

tH Ω+Ω=

Outstanding Problems

• 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 (380,000 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.

On the Horizon Problem

Big Bang

Decoupling

We observer

The Horizon Problemin Conformal Coordinates

ds² = a²(η) [ - c²dη² + dr² + r² ( dθ² + sin²θ dφ²) ]

The Monopole Problem Big issue in early 1980s

Grand Unified Theories of particle physics → at high energies the strong, electromagnetic and weak forces are unified.

the symmetry between strong and electroweak forces ‘breaks’ at an energy of ~1015 GeV (T ~ 1028 K, t ~ 10−36 s)

– this is a phase transition similar to freezing of water.– expect to form ‘topological defects’ (like defects in crystals).– point defects act as magnetic monopoles and have mass

~1015 GeV/c2 (10−12 kg).– expect one per horizon volume at t ~ 10−36 s, i.e. a number

density of 1082 m−3 at 10−36 s.– result: Universe would be today completely dominated by

monopoles !

Outstanding Problems

• There is no natural mechanism to produce the fluctuations seen in CMB, in form of Temp flucts.

On the Structure Problem

All five problems are solved, if the Universe expands very rapidly at some time tinf where 10 38 − s < tinf << tBBN

Monopole concentration diluted by expansion factor;

Exponential increase of radius of curvature;Visible universe expands from one causally

connected region.

this is inflation.inflation.Alan Guth and

Andrei Linde, 1981

Inflation solves the Problems

Inflation and the horizonInflation and the horizon Assume large positive

cosmological constant Λ acting from tinf to tend

then for tinf < t < tend a(t) = a(tinf) exp[Hi(t – tinf)] Hi = ( ⅓Λ)1/2

if Λ large a can increase by many orders of magnitude in a very short time

Exponential inflation is the usual assumption but a power law a = ainf(t/tinf)n

works if n > 1

0.0001

100

1E+08

1E+14

1E+20

1E+26

1E+32

1E+38

1E+44

1E+50

1E+56

1.E-40 1.E-34 1.E-28 1.E-22 1.E-16

t (s)

a(t)

with inflation

without inflation

horizon

27

Inflation and FlatnessInflation and Flatness We had

for matter-dominated universe 1 – Ω ∝ a for cosmological constant H is constant, so 1 – Ω ∝ a−2

Assume at start of inflation |1 – Ω| ~ 1

Now |1 – Ω| ~ 0.06 at matter-radiation equality

|1 – Ω| ~ 2×10 5− , t ~ 50000 yr at end of inflation |1 – Ω| ~ 10 50−

so need to inflate by 1025 = e58

( ) )()(

1)()(

)(1 220

20

20

22

2

tatHH

RtatHkct Ω−=−=Ω−

What powers Inflation?What powers Inflation?

We need Hinf(tend – tinf) 58≥ if tend ~ 10 34− s and tinf ~ 10 36− s, Hinf ~ 6 × 1035 s 1−

this implies Λ ~ 1072 s 2−

energy density εΛ ~ 6 × 1097 J m 3− ~ 4 × 10104 TeV m 3−

cf. current value of Λ ~ 10−35 s−2, εΛ ~ 10−9 J m−3 ~ 0.004 TeV m−3

We also need an equation of state with negative pressure → accelerating expansion needs P < 0

cosmological constant Λ has ε = −P

( )PcG

aa 3

34

2 +−= επ

Inflation and Particle PhysicsInflation and Particle Physics At very high energies particle

physicists expect that all forces will become unified this introduces new particles some take the form of scalar fields

φ with equation of state

if this looks like Λ

ToE

GUT gravity

strong electro- weak

electro- mag.

weak

1 TeV

1012 TeV

1016 TeV

)(2

1

)(2

1

23

23

ϕϕ

ϕϕε

ϕ

ϕ

Uc

P

Uc

−=

+=

)(2 32 ϕϕ Uc < <

Inflation with Scalar FieldInflation with Scalar Field Need potential U with broad nearly flat plateau

near Higgs field (inflaton field) φ = 0: metastable false vacuum inflation as φ moves very slowly away from 0 stops at drop to minimum

(true vacuum) decay of inflaton field at this

point reheats universe, producing photons, quarks etc.(but not monopoles – too heavy)

equivalent to latent heat of a phase transition.

U

φ

The 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 Inflation Dynamics

( )( )

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 Slow-Roll Approximation

EoS for theInflaton Field

Field >> Planck value !

The InflatonField

Slow-Roll Aprox

Initial Conditions

• A potential V(φ) must be given, probably from the GUT phase.

• A way to end inflation, e.g. if slow-roll condition is no longer valid.

Reheating phase, because the inflationary Universe is adiabatically cooling down! Or warm Inflation.

• or when extra physics enters: hybrid inflation.

Models of Inflation

Amount of InflationMinimum N ~ 57 required!

V

Ideas fromGUT phasetransitions

Alan Guth’s New Inflation

Eternal Inflation

A.Linde: Inflation as Harmonic Oscillator

• Einstein:

• Klein-Gordon:

Compare with equation for the harmonic oscillator with friction:

Coupled Equations of Motion

Logic of InflationLogic of InflationLarge 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.

Coupled Equations of Motion

AttractorsOscillationsaround Minimum

Phas

e Pl

ot fo

rC

haot

ic In

flatio

nInitialCondition

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 Inflation: 2 Fields

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

SU(4)xU(1)

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

Uncertainty Principle means that in quantum mechanics vacuum constantly produces temporary particle-antiparticle pairs: minute density fluctuations; inflation blows these up to

macroscopic size; seeds for structure formation;

Expect spectrum of fluctuations tobe approximately scale invariant possible test of inflation idea?

Inflation and Structure

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

The “Trans-Planckian” Problem

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) c² dt² + a²(t) (1 – 2Ψ) δij dxi dxj

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

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

Adiabatic fluctuations dS = 0

-

Quantum Fluctuations in Φ

Klein-Gordonequation with mass given bym² = -z‘‘/z (Barrier) 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:

Wave Equation in Φ ∼ u

Short wavelength limit

Long wavelength limit

Quantization of Φ

Mode leavingthe Horizon isfrozen in.

log(ckη)

The Mode SolutionSi

mpl

e Sc

alar

Fie

ldin

a d

eSitt

er In

flatio

n

u k /

a

Horizon

Mode freezing

k = 2π/λ

InsideHorizon

OutsideHorizon

Power Spectrum

Potential Barrier and Power Spec

Solution by Hankel functions

Convential Slow-Roll Aprox

• 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 ?

Power Spectrum – Chaotic Inflation

Andreas Heinen Thesis 2005

Power Spectrum – Grav. Waves

Andreas Heinen Thesis 2005

Running spectral index:

Power Spectrum - Running Spectral Index

Spectral Index – Grav. Waves

Andreas Heinen Thesis 2005

Andreas Heinen Thesis 2005

Ratio R = Grav. Waves / Scalar

Andreas Heinen Thesis 2005

• Parameters:• Quartic

Inflation: ~ λΦ4

∀ λ = 1.75 x 10-13

Φ(0) = 24 MP

dΦ(0) = - 1 MP/s

N = 60.58

MP² = 1/8πG

Spectral Index – Quartic Potential

Andreas Heinen Thesis 2005

Spectral Index – Quartic Potential

• While the Universe is inflating, its contents is cold. But eventually, inflation has to end and the field driving the inflation must decay, depositing energy into high-energy particles. This process, known as reheating, “boils” the vacuum and starts the thermal history of the Universe with the hot Big Bang. As the Universe continues to cool down, it could undergo more phase transitions, which would happen at symmetry breaking points of the theory. Very little is known about the fundamental physics at these energy scales, and cosmological observations could be our only source of information for the foreseeable future. No photons reach us directly from this epoch, as the Universe is filled with hot plasma and is opaque until recombination.

Reheating the Universe

Reheating the Universe

Horizon and Comoving Scale

DEFROST: Frolov 2010

~ 105 LP

Horizon size (red):(i) ~ constant during Inflation;(ii) turns at end of Inflation to radiation- dominance;(iii) dH = 2ct ~ a² in rad-dominance.

Perturbations leave horizon and re-enter later on.

Wav

elen

gths

are

sim

ply

stre

tche

d in

Exp

ansio

n

QuantumFluctuationsin Potential Φgenerated inInflation

Inflationary Universe

DEFROST: Frolov 2008

Fluctuations in Density / Sim

Inflaton mass m = 5x10-6 MPl ~ 1013 GeV/c²

Box size:L = 10/m

Frolov 2008

Fluctuations in Φ

Inflation – Connection to Quantum Gravity ?

Inflation scenario predicts: Universe should be very close to flat; No causality problem; CMB should be isotropic, with small scale invariant

perturbations – seeds for cosmic web and galaxies; Monopole number density unobservably low.

Inflation scenario does not predict: current near-equality of Ωm and ΩΛ

matter-antimatter asymmetry. Underlying particle physics very difficult to test

energy scale is much too high for accelerators !

Inflation - Summary

Summary• Inflation solves Flatness problem, Horizon

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

perturbations on the Friedmann backgrd. 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. Later on, they will enter the horizon

• Power spectrum and spectral index will depend on inflation model, but nS ~ 0.962.

Inflation - Summary