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Structures in the early Universe
Particle Astrophysics chapter 8
Lecture 4
overview
• Part 1: problems in Standard Model of Cosmology: horizon and flatness problems – presence of structures
• Part 2 : Need for an exponential expansion in the early universe
• Part 3 : Inflation scenarios – scalar inflaton field
• Part 4 : Primordial fluctuations in inflaton field
• Part 5 : Growth of density fluctuations during radiation and matter dominated eras
• Part 6 : CMB observations related to primordial fluctuations
2012-13 Structures in early Universe 2
PART 1: PROBLEMS IN STANDARD COSMOLOGICAL MODEL
•problems in Standard Model of Cosmology related to initial conditions required:
horizon problem
flatness problem
2012-13 Structures in early Universe 3
2012-13 Structures in early Universe 4
The ΛCDM cosmological model
• Concordance model of cosmology – in agreement with all observations = Standard Model of Big Bang cosmology
• ingredients:
Universe = homogeneous and isotropic on large scales
Universe is expanding with time dependent rate
Started from hot Big Bang, followed by short inflation period
Is essentially flat over large distances
Made up of baryons, cold dark matter and a constant dark energy + small amount of photons and neutrinos
Is presently accelerating
2012-13 Expanding Universe 5
Lecture 1
The ΛCDM cosmological model
• Concordance model of cosmology – in agreement with all observations = Standard Model of Big Bang cosmology
• ingredients:
Universe = homogeneous and isotropic on large scales
Universe is expanding with time dependent rate
Started from hot Big Bang, followed by short inflation period
Is essentially flat over large distances
Made up of baryons, cold dark matter and a constant dark energy + small amount of photons and neutrinos
Is presently accelerating
2012-13 Expanding Universe 6
Lecture 1
Q: Which mechanism caused the exponential inflation? A: scalar inflaton field
Exponential inflation takes care of horizon and flatness problems
Parts 1, 2, 3
The ΛCDM cosmological model
• Concordance model of cosmology – in agreement with all observations = Standard Model of Big Bang cosmology
• ingredients:
Universe = homogeneous and isotropic on large scales
Universe is expanding with time dependent rate
Started from hot Big Bang, followed by short inflation period
Is essentially flat over large distances
Made up of baryons, cold dark matter and a constant dark energy + small amount of photons and neutrinos
Is presently accelerating
2012-13 Expanding Universe 7
Lecture 1
Structures observed in galaxy surveys and anisotropies in CMB observations:
Q: What seeded these structures? A: quantum fluctuations in primordial universe
Parts 4, 5, 6
Horizon in static universe
• Particle horizon = distance over which one can observe a particle by exchange of a light signal
• Particle and observer are causally connected
• In flat universe with k=0 : on very large scale light travels nearly in straight line
• In a static universe of age t photon emitted at time t=0 would travel
• Would give as horizon distance today the Hubble radius
2012-13 Structures in early Universe 8
STATICHD ct
v=c
t < 14 Gyr
0 0 4.2STATICHD t ct Gpc
Horizon in expanding universe
• In expanding universe, particle horizon for observer at t0
• Expanding, flat, radiation dominated universe
• Expanding, flat, matter dominated universe
• expanding, flat, with m=0.24, Λ=0.76
2012-13 Structures in early Universe 9
0
0
0
0
H
t R t
R tD t cdt
00
12
R t
t
t
R t0 02HD t ct
00
23
R t
t
t
R t0 03HD t ct
0 03.3
~ 14
HD t ct
Gpc
Lecture 1
Horizon problem
• At time of radiation-matter decoupling the particle horizon was much smaller than now
• Causal connection between photons was only possible within this horizon
• We expect that there was thermal equilibrium only within this horizon
• Angle subtented today by horizon size at decoupling is about 1
• Why is CMB uniform (up to factor 10-5) over much larger angles, i.e. over full sky?
• Answer : short exponential inflation before decoupling
2012-13 Structures in early Universe 10
Horizon at time of decoupling
• Age of universe at matter-radiation decoupling
• Optical horizon at decoupling
• This distance measured today
• angle subtended today in
flat matter dominated universe
2012-13 Structures in early Universe 11
54 10dect y
0
0 0
2 11
3
dec dec
dec
H dec
decHD t ct z
D t c t t
2dec dec
decHD t ct
1100decz
0 2 1dec dec
decHD t ct z
2012-13 Structures in early Universe 12
Flatness problem 1
• universe is flat today, but was much closer to being flat in early universe
• How come that curvature was so finely tuned to Ω=1 in very early universe?
• Friedman equation rewritten
radiation domination
matter domination
• At time of decoupling
• At GUT time (10-34s)
2012-13 Structures in early Universe 13
2
2 21
k ct
R t H t
nR t t1 ~t t
231 ~t t
161 ~ 10
531 ~ 10
2012-13 Structures in early Universe 14
Flatness problem 2
• How could Ω have been so closely tuned to 1 in early universe?
• Standard Cosmology Model does not contain a mechanisme to explain the extreme flatness
• The solution is INFLATION
2012-13 Structures in early Universe 15
? today GUT scale
Big Bang Planck scale
34 10t s44 10t s
2012-13 Structures in early Universe 16
The solution is INFLATION
Horizon problem
Flatness problem
Non-homogenous universe
PART 2 EXPONENTIAL EXPANSION
The need for an exponential expansion in the very early universe
2012-13 Structures in early Universe 17
Structures in early Universe 18 © Rubakov
2012-13
Exponential expansion inflation phase
Steady state expansion Radiation dominated Described by SM of cosmology
Constant energy density & exponential expansion
• Assume that at Planck scale the energy density is given by a cosmological constant
• A constant energy density causes an exponential expansion
• For instance between t1 and t2 in flat universe
2012-13 Structures in early Universe 19
2
1
12H t -tR= e
R
8tot
G
2
2 8
3 3
R GH
R
When does inflation happen?
2012-13 Structures in early Universe 20
Inflation period
When does inflation happen?
• Grand Unification at GUT energy scale
• GUT time
• couplings of
– Electromagnetic
– Weak
– Strong
interactions are the same
• At end of GUT era : spontaneous symmetry breaking into electroweak and strong interactions
• Next, reheating and production of particles and radiation
• Start of Big Bang expansion with radiation domination
2012-13 Structures in early Universe 21
16~10kT GeVGUT
34~ 10 sectGUT
How much inflation is needed? 1
• Calculate backwards size of universe at GUT scale
• after GUT time universe is dominated by radiation
• If today
• Then we expect
• On the other hand horizon distance at GUT time was
2012-13 Structures in early Universe 22
0 0
12
GUT GUTR t t
R t t
12R t t
26
HD ~ ~ 10GUT GUTt ct m
17 26
0 0 0~ 4 10 ~ 4 10static
HR t D t ct c s m
34
0 17
1210
~4 10
GUT
sR t R t
s
~ 1EXPECTED
GUTR t m
2012-13 Structures in early Universe 23
How much inflation is needed? 2
• It is postulated that before inflation the universe radius was smaller than the horizon distance at GUT time
• Size of universe before inflation
• Inflation needs to increase size of universe by factor
2012-13 Structures in early Universe 24
26
1 10R m
2 26
26
1
R 110
R start inflation 10
GUT m
m
60e2 1 60H t t
2
1
2 1-H t tRe
R
At least 60 e-folds are needed
horizon problem is solved
Whole space was in causal contact and thermal equilibrium before inflation
Some points got disconnected during exponential expansion
entered into our horizon again at later stages
• → horizon problem solved
2012-13 Structures in early Universe 25
2012-13 Structures in early Universe 26
© Scientific American Feb 2004
Flatness problem is solved
• If curvature term at start of inflation is roughly 1
• Then at the end of inflation the curvature
is even closer to 1
2012-13 Structures in early Universe 27
2
2 2
1 1
1kc
OR t H t
2
2 2
22
2 2
1
kc
R t H
kc
R t H
2
2
1
1 22H t -t - 52
1R
= e ∼101R
52 341 10 at 10t s
Conclusion so far
• Constant energy density yields exponential expansion
• This solves horizon and flatness problems
• What causes the exponential inflation?
2012-13 Structures in early Universe 28
PART 3 INFLATION SCENARIOS
Scalar inflaton field
2012-13 Structures in early Universe 29
Introduce a scalar inflaton field
• Physical mechanism underlying inflation is unknown
• Postulate by Guth (1981): inflation is caused by an unstable scalar field present in the very early universe
• dominates during GUT era (kT≈ 1016 GeV) – spatially homogeneous – depends only on time : (t)
• ‘inflaton field’ describes a scalar particle with mass m
• Dynamics is analogue to ball rolling from hill
• Scalar has kinetic and potential energy
2012-13 Structures in early Universe 30
Chaotic inflation scenario (Linde, 1982) • Inflation starts at different field values in different locations
• Inflaton field is displaced from true minimum by some arbitrary mechanism, probably quantum fluctuations
2012-13 Structures in early Universe 31
• Total energy density associated with scalar field
• During inflation the potential V() is only slowly varying with – slow roll approximation
• Kinetic energy can be neglected
2 21
2c V
© Lineweaver
V(Φ
)
Φ
inflation reheating
Slow roll leads to exponential expansion
• Total energy in universe = potential energy of field
• Friedman equation for flat universe becomes
2012-13 Structures in early Universe 32
2 ~~ Vc cst
2 ~H cst
22
2c V
2
2
2
2
8
3
8
3 PL
H
GRH
R
c
MV
Exponential expansion
Reheating at the end of inflation
• Lagrangian energy of inflaton field
• Mechanics (Euler-Lagrange) : equation of motion of inflaton field
•
• = oscillator/pendulum motion with damping
• field oscilllates around minimum
• stops oscillating = reheating 2012-13 Structures in early Universe 33
23
2L T V R V
3 0dV
dH t
Frictional force due to expansion
Reheating and particle creation
2012-13 Structures in early Universe 34
• Inflation ends when field reaches minimum of potential
• Transformation of potential energy to kinetic energy = reheating
• Scalar field decays in particles
© Lineweaver
V(Φ
)
Φ
inflation reheating
Summary
2012-13 Structures in early Universe 35
Planck era
GUT era
• When the field reaches the minimum, potential energy is transformed in kinetic energy
• This is reheating phase
• Relativistic particles are created
• Expansion is now radiation dominated
• Hot Big Bang evolution starts
t
R
kT
Chaotic inflation scenario (Linde, 1982) • Inflation starts at different field values in different locations
• Inflaton field is displaced from true minimum by some arbitrary mechanism, probably quantum fluctuations
2012-13 Structures in early Universe 36
• Total energy density associated with scalar field
• During inflation the potential V() is only slowly varying with – slow roll approximation
• Kinetic energy can be neglected
2 21
2c V
© Lineweaver
V(Φ
)
Φ
inflation reheating
2012-13 Structures in early Universe 37
2012-13 Structures in early Universe 38
PART 4 PRIMORDIAL FLUCTUATIONS
Quantum fluctuations in early universe as seed for present-day structures: CMB and galaxies
2012-13 Structures in early Universe 39
quantum fluctuations in inflaton field
2012-13 Structures in early Universe 40
Field starting in B needs more time to reach minimum than field starting in A
Reaches minimum Δt later
Quantum fluctuations set randomly starting value in A or B
Potential should vary slowly
Quantum fluctuations as seed 1
• Introduce quantum fluctuations in inflaton field
• This yields fluctuations in inflation end time Δt
• Inflation will end at different times in different locations of universe
• Hence : Fluctuations in kinetic energy at end of inflation
• Hence different density of matter at
different locations
• And different temperatures
2012-13 Structures in early Universe 41
t
2ckinE T
Potential Energy
Kinetic Energy
Quantum fluctuations as seed 2
• Due to rapid expansion, particles and anti-particles are inflated away from each other
• One of (anti-)particles moves out of horizon – no causal contact with other (anti-)particle – no annihilation
• Energy balance is restored during reheating
• Fluctuations can be treated as classical perturbations & superposition of waves
2012-13 Structures in early Universe 42
Perturbations in the cosmic fluid
• density fluctuations modify the energy-momentum tensor
• and therefore also the curvature of space
• Change in curvature of space-time yields change in gravitation potential
• Particles and radiation created during reheating will ‘fall’ in gravitational potential wells
• These perturbations remain as such till matter-radiation decoupling – are imprinted in CMB
2012-13 Structures in early Universe 43
x
5~ 10T
T
Gravitational potential fluctuations
• gravitational potential Φ due to energy density ρ
spherical symmetric case
• 2 cases
– Global ‘background’ potential from ‘average’ field within horizon
– Potential change due to fluctuation Δρ over arbitrary scale λ
• Fractional perturbation in gravitational potential at certain location
2012-13 Structures in early Universe 44
22
3
G rr
2
2
3
G
H
22
3
G
2 2H
Spectrum of fluctuations
• During inflation Universe is effectively in stationary state
• Fluctuation amplitudes will not depend on space and time
• Define rms amplitude of fluctuation
• Amplitude is independent of horizon size – independent of epoch at which it enters horizon – is ‘scale-invariant’
2012-13 Structures in early Universe 45
~ cst
2
2 1
2
rms
2 2H
2 ~H cst
PART 5 GROWTH OF STRUCTURE
Clustering of particles at end of inflation
Evolution of fluctuations during radiation dominated era
Evolution of fluctuations during matter dominated era
Damping effects
2012-13 Structures in early Universe 46
Introduction
• We assume that the structures observed today in the CMB originate from tiny fluctuations in the cosmic fluid during the inflationary phase
• These perturbations grow in the expanding universe
• Will the primordial fluctuations of matter density survive the expansion?
2012-13 Structures in early Universe 47
Jeans length = critical dimension
• After reheating matter condensates around primordial density fluctuations
• universe = gas of relativistic particles
• Consider a cloud of gas around fluctuation Δρ
• When will this lead to condensation?
• Compare cloud size L to Jeans length λJ
Depends on density and sound speed vs
• Sound wave in gas = pressure wave
• Sound velocity in an ideal gas 2012-13 Structures in early Universe 48
12
JG
sv
5
3
p
V
CP
Csv
x
Jeans length = critical dimension
• Compare cloud size L with Jeans length
• when L << λJ : sound wave travel time < gravitational collaps time – no condensation
• When L >> λJ : sound cannot travel fast enough from one side to the other
• cloud condenses around the density perturbations
• Start of structure formation
2012-13 Structures in early Universe 49
s
Ltv
Evolution of fluctuations in radiation era
• After inflation : photons and relativistic particles
• Radiation dominated till decoupling at z ≈ 1100
• Velocity of sound is relativistic
• horizon distance ~ Jeans length : no condensation of matter
• Density fluctuations inside horizon survive – fluctuations continue to grow in amplitude as R(t)
2012-13 Structures in early Universe 50
3s
cv5
~3
s
P Pv
2
3
cP
1 12 232
9sv
GJλ ctHD t ct
Evolution of fluctuations in matter era
• After decoupling: dominated by non-relativistic matter
• Neutral atoms (H) are formed – sound velocity drops
• Jeans length becomes smaller
– Horizon grows
– therefore faster gravitational collapse
– Matter structures can grow
• Cold dark matter plays vital role in growth of structures
2012-13 Structures in early Universe 51
12
5
2
52 10
3mat H
kT
c M csv
ideal gas5 5 kT
3 3s
P
mv
12
JG
sv
Damping by photons
• Fluctuations are most probably adiabatic: baryon and photon densities fluctuate together
• Photons have nearly light velocity - have higher energy density than baryons and dark matter
• If time to stream away from denser region of size λ is shorter than life of universe → move to regions of low density
• Result : reduction of density contrast → diffusion damping (Silk damping)
2012-13 Structures in early Universe 52
• Neutrinos = relativistic hot dark matter
• In early universe: same number of neutrinos as photons
• Have only weak interactions – no interaction with matter as soon as kT below 3 MeV
• Stream away from denser regions = collisionless damping
• Velocity close to c : stream up to horizon – new perturbations coming into horizon are not able to grow – ‘iron out’ fluctuations
Damping by neutrinos
2012-13 Structures in early Universe 53
Observations
2012-13 Structures in early Universe 54
CMB
rms
lum
py
smo
oth
λ (Mpc)
PART 6 OBSERVATIONS IN CMB
From density contrast to temperature anisotropy
Angular power spectrum of anisotropies
2012-13 Structures in early Universe 55
CMB temperature map
2012-13 Structures in early Universe 56
3dipole 10T O mKT
510T O µKT
Contrast enhanced to make 10-5 variations visible!
Small angle anisotropies
• Adiabatic fluctuations at end of radiation domination : photons & matter fluctuate together
• Photons keep imprint of density contrast
• horizon at decoupling is seen now within an angle of 1
• temperature correlations at angles < 1 are due to primordial density fluctuations
• Seen as acoustic peaks in CMB angular power distribution
• Peaks should all have same amplitude – but: Silk damping by photons
2012-13 Structures in early Universe 57
1
3Adiab
T
T
2012-13 Structures in early Universe 58
© Scientific American Feb 2004
2
2 1
rmsx
Angular spectrum of anisotropies 1
• CMB map = map of temperatures after subtraction of dipole and galactic emission → extra-galactic photons
• Statistical analysis of the temperature variations in CMB map – multipole analysis
• In given direction n measure difference with average of 2.7K
2012-13 Structures in early Universe 59
0T n T n T
C(θ) = correlation between
temperature fluctuations in 2
directions (n,m) separated by
angle θ
average over all pairs with
given θ
Angular spectrum of anisotropies 2
• Expand in Legendre polynomials, integrate over
• Cℓ describes intensity of correlations
• Temperature fluctuations ΔT are superposition of fluctuations at different scales λ
• Decompose in spherical harmonics
2012-13 Structures in early Universe 60
00
,,
l
lm lm
l m l
Ta Y
T
2 21
2 1lm lm
m
a al
lC
12 1 cos
4l
l
C l PlC
Angular spectrum of CMB anisotropies
• Coefficients Cℓ = angular power spectrum - measure level of structure found at angular separation of
• Large ℓ means small angles
• ℓ = 200 corresponds to 1 = horizon at decoupling as seen today
• Amplitudes seen today depend on: primordial density fluctuations
Baryon/photon ratio
Amount of dark matter
….
2012-13 Structures in early Universe 61
Fit of Cℓ as function of ℓ yields cosmological parameters
200 degrees
2012-13 Structures in early Universe 62
Physics of Young universe Large wavelengths
Physics of primordial fluctuations Short wavelengths
Silk damping
1° = horizon at decoupling
WMAP result
Position and height of peaks
• Position and heigth of acoustic peaks depend on curvature, matter and energy content and other cosmological parameters
2012-13 Structures in early Universe 63
Flat universe Dependence on baryon content Dependence on curvature
overview
• Part 1: problems in Standard Model of Cosmology: horizon and flatness problems – presence of structures
• Part 2 : Need for an exponential expansion in the early universe
• Part 3 : Inflation scenarios – scalar inflaton field
• Part 4 : Primordial fluctuations in inflaton field
• Part 5 : Growth of density fluctuations during radiation and matter dominated eras
• Part 6 : CMB observations related to primordial fluctuations
2012-13 Structures in early Universe 64
Examen
• Book Perkins 2nd edition chapters 5, 6, 7, 8, + slides
• Few examples worked out : ex5.1, ex5.3, ex6.1, ex7.2
• ‘Open book’ examination
2012-13 Structures in early Universe 65