Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4

Structures in the early Universe

Particle Astrophysics chapter 8Lecture 4

Structures lect 4 22014-15

Structures lect 4 3

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

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Structures lect 4 4

PART 1:PROBLEMS IN STANDARD COSMOLOGICAL MODEL

•problems in Standard Model of Cosmology related to initial conditions required: horizon problemflatness problem

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Structures lect 4 5

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 radiation

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Lecture 1

Structures lect 4 6




small amount of photons and neutrinos Is presently accelerating

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Lecture 1

Q: Which mechanism caused the exponential inflation?A: scalar inflaton field

Exponential inflation takes care of horizon and flatness problemsParts 1, 2, 3

Structures lect 4 7




small amount of photons and neutrinos Is presently accelerating

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Lecture 1

Structures observed in galaxy surveys and anisotropies in CMB observations:

Q: What seeded these structures?A: quantum fluctuations in inflaton field

Parts 4, 5, 6

Structures lect 4 8

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 a static universe of age t photon emitted at time t=0 would travel

• Would give as horizon distance today the Hubble radius

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STATICHD ct

v=c

t < <14 Gyr

0 0 4.2STATICHD t ct Gpc

Structures lect 4 9

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 Ωm=0.24, ΩΛ=0.76

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0

00

0H

t R t

R tD t cdt

00

12

R t

t

t

R t

0 02HD t ct

00

23

R t

t

t

R t

0 03HD t ct

0 03.3

~ 14HD t ct

Gpc

Lecture 1

Structures lect 4 10

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

• Angle subtented today by horizon size at decoupling is about 1°

• We expect that there was thermal equilibrium only within this horizon

• Why is CMB uniform (up to factor 10-5) over much larger angles, i.e. over full sky?

• Answer : short exponential inflation before decoupling

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Horizon at time of decoupling

• Age of universe at matter-radiation decoupling

• Optical horizon at decoupling

• Expanded to • angle subtended today in

flat matter dominated universe

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54 10dect y

2dec decdecHD t ct

1100decz

0 2 1dec decdecHD 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)

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2

2 21k c

tR t H t

nR t t 1 ~t t

2

31 ~t t

61 ~ 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

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?todayGUT scale

Big BangPlanck scale

34 10t s44 10t s


The solution is INFLATION

Horizon problem

Flatness problem

Non-homogenous universe

Questions?


PART 2EXPONENTIAL EXPANSION

The need for an exponential expansion in the very early universe

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Structures lect 4 17© Rubakov

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Exponential expansion inflation phase

Steady state expansionRadiation dominatedDescribed by SM of cosmology


When does inflation happen?

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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• Caused by a scalar inflaton field

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16~ 10kT GeVGUT34~ 10 sect

GUT


Constant energy density & exponential expansion

• Assume that at given moment vacuum energy dominates over all other terms – and is given by cosmological constant

• A constant energy density causes an exponential expansion• For instance between t1 and t2 in flat universe

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2

1

12H t -tR=e

R

8tot G


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

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26HD ~ ~ 10GUT GUTt ct m

17 260 0 0~ 4 10 ~ 4 10static

HR t D t ct c s m

34

0 17

1210

~4 10GUT

sR t R t

s

~ 1EXPECTEDGUTR 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

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261 10R m

2 2626

1

R 110

R start inflation 10

GUT m

m 60e 2 1 60H t t

2

1

2 1-H t tRe

R At least

60 e-folds are needed


horizon problem is solvedWhole space was in causal contact and thermal equilibrium

before inflation Some points got disconnected during exponential

expansion They entered into our horizon again at later stages

• → horizon problem solved

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© 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 curvatureis even closer to 1

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2

2 2

1 1

1kc

OR t H t

52 341 10 at 10t s


Conclusion so far

• Constant energy density yields exponential expansion• This solves horizon and flatness problems• Which mechanism causes the exponential expansion?

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Questions?


PART 3INFLATION SCENARIOS

Guth (1981): need extremely rapid exponential inflation by huge factor as preliminary stage of Big Bang

-> Scalar inflaton field

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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 : ϕ(t)

• ‘inflaton field’ ϕ describes a scalar particle with mass m• Dynamics is analogue to ball rolling from hill• Scalar has kinetic and potential energy

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Chaotic inflation scenario (Linde, 1982)

• Inflation starts at different field values in different locations• Inflaton field is displaced by some arbitrary mechanism, probably

quantum fluctuations

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• Total energy density associated with scalar field

• During inflation the potential V(ϕ) is only slowly varying with ϕ – slow roll approximation

• Kinetic energy can be neglected

© 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

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2 ~~ Vc cst

2 ~H cst

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 = reheating2014-15

Frictional force due to expansion


Reheating and particle creation

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• Inflation ends when field ϕ reaches minimum of potential

• Transformation of potential energy to kinetic energy = reheating

• Scalar field decays in particles

© Lineweaver

V(Φ

)

Φ

inflation reheating


Summary

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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


Questions?


PART 4PRIMORDIAL FLUCTUATIONS

Quantum fluctuations in inflaton field as seed for present-day structures: CMB and galaxies

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© Lineweaver

V(Φ

)

Φ


quantum fluctuations in inflaton field

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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

• Introduce quantum fluctuations in inflaton field Δϕ• This yields fluctuations in inflation end time Δt• Inflation lasts longer in some locations – different ‘bubbles’

- reheating will take place at different times• Hence: variations of energy density in space• Production of photons and baryons• Adiabatic fluctuations in energy density

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Kinetic Energy


Perturbations in the cosmic fluid

• density fluctuations modify locally the space-time metric• and therefore also the curvature of space• Modification of space-time metric yields change in

gravitation potential Φ

• Particles and radiation created during reheating will ‘fall’ in gravitational potential wells

• Fluctuations are adiabatic: same for photons and particles

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x


Gravitational potential fluctuations

• gravitational potential Φ due to energy density ρspherical symmetric case

• 2 cases– On scale of horizon rhor = 1/H

– Potential change due to fluctuation Δρ over arbitrary small scale λ

• Fractional perturbation in gravitational potential at certain location

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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• Amplitudes of fluctuations are ‘frozen’ and will not depend

on space and time

• rms amplitude of fluctuation

• Amplitude is independent of horizon size – independent of epoch -> scale-invariant

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~ cst

2

rms

2 2H

2 ~H cst


Observations

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CMB rms

lum

pysm

ooth

λ (Mpc)


Questions?


PART 5GROWTH OF STRUCTURE

Clustering of particles at end of inflation Evolution of fluctuations during radiation dominated eraEvolution of fluctuations during matter dominated eraDamping effects

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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?

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Gravitational collapse of gas cloud

• when gas cloud condenses gravitational potential energy is transformed in kinetic (heat) energy of gas particles

• Hydrostatic equilibrium when pressure of heated gas = inward gravitational pressure

• Cloud condenses when

• Example: cloud of molecular H with mass = 10.000 M and T=20K : rcrit = 0.35 kpc

• For smaller scale fluctuations in cosmic fluid: use Jeans criterium

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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 ρ and sound speed vs

• Sound wave in gas = pressure wave• Sound velocity in an ideal gas 2014-15

12

J G

sv

5

3p

V

CP

C

sv

x


Jeans length = critical dimension

• Compare cloud (density fluctuation) size L with Jeans length

• when L << λJ : no condensation

• When L >> λJ : cloud condenses around the density perturbations

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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 (equ 5.47 in chap 5)

• Density fluctuations inside horizon survive – fluctuations continue to grow in amplitude as R(t)

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5~

3sP Pv

2

3

cP

1 12 232

9svG

Jλ ct HD t ct


Evolution of fluctuations in matter era

• After decoupling: dominated by non-relativistic gas• 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

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12

52

52 10

3mat H

kTc M c

sv

ideal gas5 5 kT

3 3sP

mv

12

J G

sv


Damping by photons

• Fluctuations are most probably adiabatic: baryon and photon densities fluctuate together

• Photons travel at light velocity - have most of time 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)

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• 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

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CMB - 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

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C(θ) = correlation between temperature fluctuations in 2 directions (n,m) separated by angle θ average over all pairs with given θ


CMB - Angular spectrum of anisotropies 2

• Expand in Legendre polynomials, integrate over ϕ

• Cℓ describes intensity of correlations

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12 1 cos

4 ll

C l P

lC


Physics of Young universeLarge wavelengths

Physics of primordial fluctuationsShort wavelengths

Silk damping

1° = horizon at decoupling

WMAP result


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

2014-15


Examen

• Book Perkins 2nd edition chapters 5, 6, 7, 8, + slides• Delen die niet gekend moeten zijn wordt opgestuurd per

email• ‘Open book’ examen

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Questions?

Documents

Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4