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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.