Structures in the early Universe - iihe.ac.becdeclerc/astroparticles/... · early universe •How come that curvature was so finely tuned to Ω=1 in very early universe? •Friedman

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



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



• ingredients:








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



• ingredients:








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


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


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


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


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


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)


2

2 21

k ct

R t H t

nR t t1 ~t t

231 ~t t

161 ~ 10

531 ~ 10


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


? today GUT scale

Big Bang Planck scale

34 10t s44 10t s


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


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


2

1

12H t -tR= e

R

8tot

G

2

2 8

3 3

R GH

R

When does inflation happen?


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


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


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


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


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



© 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


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?


PART 3 INFLATION SCENARIOS

Scalar inflaton field


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


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


• 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


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


• 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


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


• 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



PART 4 PRIMORDIAL FLUCTUATIONS

Quantum fluctuations in early universe as seed for present-day structures: CMB and galaxies


quantum fluctuations in inflaton field


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


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


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


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


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’


~ 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


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?


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


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)


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


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)


• 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


Observations


CMB

rms

lum

py

smo

oth

λ (Mpc)

PART 6 OBSERVATIONS IN CMB

From density contrast to temperature anisotropy

Angular power spectrum of anisotropies


CMB temperature map


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


1

3Adiab

T

T


© 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


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


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

….


Fit of Cℓ as function of ℓ yields cosmological parameters

200 degrees


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


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


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


Documents

Structures in the early Universe - iihe.ac.becdeclerc/astroparticles/... · early universe •How come that curvature was so finely tuned to Ω=1 in very early universe? •Friedman