as 4022 cosmology 1 as 4022: cosmology hs zhao online notes: star-hz4/cos/cos.html...

AS 4022 Cosmology 1

AS 4022: Cosmology

HS Zhao

Online notes:

star-www.st-and.ac.uk/~hz4/cos/cos.html

star-www.st-and.ac.uk/~kdh/cos/cos.html

Final Note in LibrarySummary sheet of key results (from John Peacock)

take your own notes (including blackboard lectures)

AS 4022 Cosmology 2

Observable Space-Time and Bands

• See What is out there? In all Energy bands

– Pupil Galileo’s Lens 8m telescopes square km arrays– Radio, Infrared optical X-ray, Gamma-Ray (spectrum)

– COBE satellites Ground Underground DM detector

• Know How were we created? XYZ & T ?– Us, CNO in Life, Sun, Milky Way, … further and further first galaxy first star first Helium first quark – Now Billion years ago first second quantum origin

AS 4022 Cosmology 3

The Visible Cosmos: a hierarchy of structure and motion

• “Cosmos in a computer”

AS 4022 Cosmology 4

Observe A Hierarchical Universe

• Planets – moving around stars;

• Stars grouped together, – moving in a slow dance around the center of galaxies.

AS 4022 Cosmology 5

• Galaxies themselves– some 100 billion of them in the observable universe—

– form galaxy clusters bound by gravity as they journey through the void.

• But the largest structures of all are superclusters, – each containing thousands of galaxies

– and stretching many hundreds of millions of light years.

– are arranged in filament or sheet-like structures,

– between which are gigantic voids of seemingly empty space.

AS 4022 Cosmology 6

• The Milky Way and Andromeda galaxies, – along with about fifteen or sixteen smaller galaxies,

– form what's known as the Local Group of galaxies.

• The Local Group – sits near the outer edge of a supercluster, the Virgo cluster.

– the Milky Way and Andromeda are moving toward each other,

– the Local Group is falling into the middle of the Virgo cluster, and

• the entire Virgo cluster itself, – is speeding toward a mass

– known only as "The Great Attractor."

Cosmic Village

AS 4022 Cosmology 7

Introducing Gravity and DM (Key players)

• These structures and their movements– can't be explained purely by the expansion of the universe

• must be guided by the gravitational pull of matter.

• Visible matter is not enough

• one more player into our hierarchical scenario:

• dark matter.

AS 4022 Cosmology 8

Cosmologists hope to answer these questions:

• How old is the universe? H0

• Why was it so smooth? P(k), inflation

•

• How did structures emerge from smooth? N-body

• How did galaxies form? Hydro

• Will the universe expand forever? Omega, Lamda

• Or will it collapse upon itself like a bubble?

AS 4022 Cosmology 9

1st main concept in cosmology

• Cosmological Redshift

AS 4022 Cosmology 10

Stretch of photon wavelength in expanding space

• Emitted with intrinsic wavelength λ0 from Galaxy A at time t<tnow in smaller universe R(t) < Rnow

Received at Galaxy B now (tnow ) with λ • λ / λ0 = Rnow /R(t) = 1+z(t) > 1


1st main concept: Cosmological Redshift

• The space/universe is expanding, – Galaxies (pegs on grid points) are receding from each other

• As a photon travels through space, its wavelength becomes stretched gradually with time.– Photons wave-packets are like links between grid points

• This redshift is defined by:

1

o

o

o

z

z


• E.g. Consider a quasar with redshift z=2. Since the time the light left the quasar the universe has expanded by a factor of 1+z=3. At the epoch when the light left the quasar,

– What was the distance between us and Virgo (presently 15Mpc)?

– What was the CMB temperature then (presently 3K)?

1 (wavelength)( )

(expansion factor)( )

( )(Photon Blackbody T 1/ , ?)

now

now

now

zt

R

R t

T twhy

T


Lec 2: Cosmic Timeline

• Past Now


Trafalgar Square

London Jan 1

Set your watches 0h:0m:0s

Fundamental observers

H

H

HH

H

H

H

H

A comic explanation for cosmic expansion …


3 mins later

Homogeneous Isotropic Universe

He

He

Walking ↔ E levating ↔ E arth R adius Stretching R t


A1

A2

A3

B1

B2

B3

R(t)d

Feb 14 t=45 days later

dl2= [ R t dχ ]2[ R t sin χdφ ]2A1−B2

d

C1 C2 C3

D1

D2 D3


2nd Concept: metric of 1+2D universe

• Analogy of a network of civilization living on an expanding star (red giant).

– What is fixed (angular coordinates of the grid points)

– what is changing (distance).


Analogy: a network on a expanding sphere

.

Angle χ1

Expanding Radius R(t)1

23

4

1

3 2

4Angle φ1

Fundamental observers 1,2,3,4 with

Fixed angular (co-moving) coordinates (χ,φ)

on expanding spheres their distances are given by

Metric at cosmic time t ds2 = c2 dt2-dl2,

dl2 = R2(t) (dχ2 + sin2 χ dφ2)


3rd Concept: The Energy density of Universe

• The Universe is made up of three things:– VACUUM

– MATTER

– PHOTONS (radiation fields)

• The total energy density of the universe is made up of the sum of the energy density of these three components.

• From t=0 to t=109 years the universe has expanded by R(t).

ε t = ε vac ε matter ε rad


Eq. of State for Expansion & analogy of baking bread

• Vacuum~air holes in bread

• Matter ~nuts in bread

• Photons ~words painted

• Verify expansion doesn’t change Nhole, Nproton, Nphoton

– No Change with rest energy of a proton, changes energy of a photon

λ

▲►▼◄

λ λ

▲►▼◄ λλ


• VACUUM ENERGY:

• MATTER:

• RADIATION:number of photons Nph = constant

ε t = ρeff t c2

ε t c 2 = ρeff t

3constant Evac R

3 constant, constantR m

⇒ n ph≈N ph

R3

4

Wavelength stretches : ~

hc 1Photons:E h ~

1~ ~ph ph

R

Rhc

nR


• The total energy density is given by:

ε∝ ε vac ε matter ε ph

¿ R0

¿ R−3¿ R−4

log

R

Radiation Dominated

Matter Dominated Vacuum

Dominated

n=-4

n=-3n=0


Key Points

• Scaling Relation among – Redshift: z, – expansion factor: R

– Distance between galaxies– Temperature of CMB: T

– Wavelength of CMB photons: lambda

• Metric of an expanding 2D+time universe– Fundamental observers

– Galaxies on grid points with fixed angular coordinates

• Energy density in – vacuum, matter, photon– How they evolve with R or z

• If confused, recall the analogies of – balloon, bread, a network on red giant star, microwave oven


TopicsTheoretical and Observational

• Universe of uniform density– Metrics ds, Scale R(t) and Redshift

– EoS for mix of vacuum, photon, matter

• Thermal history– Nucleosynthesis

– He/D/H

• Structure formation– Growth of linear perturbation

– Origin of perturbations

– Relation to CMB

Hongsheng.Zhao (hz4)

• Quest of H0 /Omega (obs.)– Applications of expansion models

– Distances Ladders

– (GL, SZ)

– SNe surveys

– Cosmic Background fromCOBE/MAP/PLANCK etc


Acronyms in Cosmology

• Cosmic Background Radiation (CBR)– Or CMB (microwave because of present temperature 3K)

– Argue about 105 photons fit in a 10cmx10cmx10cm microwave oven. [Hint: 3kT = h c / λ ]

• CDM/WIMPs: Cold Dark Matter, weakly-interact massive particles

– At time DM decoupled from photons, T ~ 1014K, kT ~ 0.1 mc^2

– Argue that dark particles were

– non-relativistic (v/c << 1), hence “cold”.

– Massive (m >> mproton =1 GeV)


Acronyms and Physics Behind

• DL: Distance Ladder– Estimate the distance of a galaxy of size 1 kpc and angular size

1 arcsec? [About 0.6 109 light years]

• GL: Gravitational Lensing– Show that a light ray grazing a spherical galaxy of 1010 Msun at

typical b=1 kpc scale will be bent ~4GM/bc2 radian ~1 arcsec

– It is a distance ladder

• SZ: Sunyaev-Zeldovich effect – A cloud of 1kev thermal electrons scattering a 3K microwave

photon generally boost the latter’s energy by 1kev/500kev=0.2%

– This skews the blackbody CMB, moving low-energy photons to high-energy; effect is proportional to electron column density.


• the energy density of universe now consists roughly

– Equal amount of vacuum and matter,

– 1/10 of the matter is ordinary protons, rest in dark matter particles of 10Gev

– Argue dark-particle-to-proton ratio ~ 1

– Photons (3K ~10-4ev) make up only 10-4 part of total energy density of universe (which is ~ proton rest mass energy density)

– Argue photon-to-proton ratio ~ 10-4 GeV/(10-4ev) ~ 109


Brief History of Universe• Inflation

– Quantum fluctuations of a tiny region

– Expanded exponentially

• Radiation cools with expansion T ~ 1/R ~t-2/n

– He and D are produced (lower energy than H)

– Ionized H turns neutral (recombination)

– Photon decouple (path no longer scattered by electrons)

• Dark Matter Era– Slight overdensity in Matter can collapse/cool.

– Neutral transparent gas

• Lighthouses (Galaxies and Quasars) form– UV photons re-ionize H

– Larger Scale (Clusters of galaxies) form


What have we learned?

• Concepts of Thermal history of universe– Decoupling

– Last scattering

– Dark Matter era

– Compton scattering

– Gravitational lensing

– Distance Ladder

• Photon-to-baryon ratio >>1

• If confused, recall the analogy of – Crystalization from comic soup,

– Last scattering photons escape from the photosphere of the sun


The rate of expansion of Universe

• Consider a sphere of radius r=R(t) χ,

• If energy density inside is ρ c2

Total effective mass inside is M = 4 πρ r3 /3

• Consider a test mass m on this expanding sphere,

• For Test mass its Kin.Energy + Pot.E. = const E m (dr/dt)2/2 – G m M/r = cst (dR/dt)2/2 - 4 πG ρ R2/3 = cstcst>0, cst=0, cst<0

(dR/dt)2/2 = 4 πG (ρ + ρcur) R2/3

where cst is absorbed by ρcur ~ R(-2)


Typical solutions of expansion rate

H2=(dR/dt)2/R2=8πG (ρcur+ ρm + ρr + ρv )/3

Assume domination by a component ρ ~ R-n

• Argue also H = (2/n) t-1 ~ t-1. Important thing is scaling!


Lec 4 Feb 22

A powerful scaling relation (approximate):

t -2 ~ H2=(dR/dt)2/R2

~ (ρcur+ ρm + ρr + ρv ) ~ R-n ~(1+z)n ~ T n


Where are we heading?

Next few lectures will cover a few chapters of – Malcolm S. Longair’s “Galaxy Formation” [Library Short Loan]

• Chpt 1: Introduction

• Chpt 2: Metrics, Energy density and Expansion

• Chpt 9-10: Thermal History


Thermal Schedule of Universe [chpt 9-10]• At very early times, photons are typically energetic enough that they interact

strongly with matter so the whole universe sits at a temperature dictated by the radiation.

• The energy state of matter changes as a function of its temperature and so a number of key events in the history of the universe happen according to a schedule dictated by the temperature-time relation.

• Crudely (1+z)~1/R ~ (T/3) ~109 (t/100s)(-2/n) ~ 1000 (t/0.3Myr)-2/n, H~1/t

• n~4 during radiation domination

1012 109 106 103 1 1+z

T(K)

1010

103

Neutrinos decouple

Recombination

After this Barrier photons free-stream in universe

Radiation Matter

p p ~ 10−6 se−e ~ 1s

He D ~100s

Myr


A summary: Evolution of Number Densitiesof , P, e,

e e

A A γ γ

Num Density

Now

1210 910 310 ο

R

R

3

ο ο

N R

N R

v v

910

PP

P

e e

e

P

H+H

Protons condense at kT~0.1mp c2

Electrons freeze-out at kT~0.1me c2

All particles relativistic

Neutrinos decouple while relativistic


A busy schedule for the universe

• Universe crystalizes with a sophisticated schedule, much more confusing than simple expansion!

– Because of many bosonic/fermionic players changing balance

– Various phase transitions, numbers NOT conserved unless the chain of reaction is broken!

– p + p- <-> (baryongenesis)

– e + e+ <-> , v + e <-> v + e (neutrino decouple)

– n < p + e- + v, p + n < D + (BBN)

– H+ + e- < H + + e <-> + e (recombination)

• Here we will try to single out some rules of thumb. – We will caution where the formulae are not valid, exceptions.

– You are not required to reproduce many details, but might be asked for general ideas.


What is meant Particle-Freeze-Out?

• Freeze-out of equilibrium means NO LONGER in thermal equilibrium, means insulation.

• Freeze-out temperature means a species of particles have the SAME TEMPERATURE as radiation up to this point, then they bifurcate.

• Decouple = switch off = the chain is broken = Freeze-out


A general history of a massive particle

• Initially mass doesn’t matter in very hot universe

• relativistic, dense – frequent collisions with other species to be in thermal

equilibrium and cools with photon bath.

– Photon numbers (approximately) conserved, so is the number of relativistic massive particles


energy distribution in the photon bath

dN

dh

cKT

910

# hardest photons

hv25c chv KT


Initially zero chemical potential (~ Chain is on, equilibrium with photon)

• The number density of photon or massive particles is :

• Where we count the number of particles occupied in momentum space and g is the degeneracy factor. Assuming zero cost to annihilate/decay/recreate.

n=g

h3∫0

∞ d 4π3

p3exp E /kT ±1

+ for Fermions

- for Bosons

E=c2 p2mc2 2≈cp relativistic cp >> mc 2

≈mc212

p2

mnon relativistic cp mc2


• As kT cools, particles go from

• From Ultrarelativistic limit. (kT>>mc2)

particles behave as if they were massless

• To Non relativistic limit ( mc2/kT > 10 , i.e., kT<< 0.1mc2) Here we can neglect the 1 in the occupancy number

3 23

30

4~

(2 ) 1y

kT g y dyn n T

c e

∫

2 2

23 3

22 23

0

4(2 ) ~

(2 )

mc mcykT kTg

n e mkT e y dy n T e

∫


When does freeze-out happen?

• Happens when KT cools 10-20 times below mc2, run out of photons to create the particles

– Non-relativisitic decoupling

• Except for neutrinos


particles of energy Ec=hvc unbound by high energy tail of photon bath

dN

dh

cKT

cIf run short of hard photon to unbind => "Freeze-out" => KT25

chv

910

# hardest photons

~ # baryons

hv25c chv KT


Rule 1. Competition of two processes

• Interactions keeps equilibrium: – E.g., a particle A might undergo the annihilation reaction:

• depends on cross-section and speed v. & most importantly – the number density n of photons ( falls as t(-6/n) , Why? Hint R~t(-2/n) )

• What insulates: the increasing gap of space between particles due to Hubble expansion H~ t-1.

• Question: which process dominates at small time? Which process falls slower?

A A γ γ


• Rule 2. Survive of the weakest

• While in equilibrium, nA/nph ~ exp (Heavier is rarer)• When the reverse reaction rate A is slower than Hubble

expansion rate H(z) , the abundance ratio is frozen NA/Nph ~1/(A) /Tfreeze

• Question: why frozen while nA , nph both drop as T3 ~ R-3.

A ~ nph/(A) , if m ~ Tfreeze

N A

N ph

mc2

kTFreeze out

A LOW (v) smallest interaction, early freeze-out while relativistic

A HIGH later freeze-out at lower T


Effects of freeze-out

• Number of particles change (reduce) in this phase transition,

– (photons increase only slightly)

• Transparent to photons or neutrinos or some other particles

• This defines a “last scattering surface” where optical depth to future drops below unity.


Number density of non-relativistic particles to

relativistic photons

• Reduction factor ~ exp(- mc2/kT, which drop sharply with cooler temperature.

• Non-relativistic particles (relic) become *much rarer* by exp(-) as universe cools below mc2/

– So rare that infrequent collisions can no longer maintain

coupled-equilibrium.

– So Decouple = switch off = the chain is broken = Freeze-out


After freeze-out

• Particle numbers become conserved again.

• Simple expansion.– number density falls with expanding volume of universe, but

Ratio to photons kept constant.


Small Collision cross-section

• Decouple non-relativisticly once kT<mc2 . Number density ratio to photon drops steeply with cooling exp(- mc2/kT). – wimps (Cold DM) etc. decouple (stop creating/annihilating)

while non-relativistic. Abundance of CDM ~ 1/ A

• Tc~109K NUCLEOSYNTHESIS (100s)

• Tc~5000K RECOMBINATION (0.3 Myrs) (z=1000)


For example,

• Antiprotons freeze-out t=(1000)-6 sec,

• Why earlier than positrons freeze-out t=1sec ?– Hint: anti-proton is ~1000 times heavier than positron.

– Hence factor of 1000 hotter in freeze-out temperature

• Proton density falls as R-3 now, conserving

numbers

• Why it falls exponentially exp(-) earlier on– where mc2/kT~ R.

– Hint: their numbers were in chemical equilibrium, but not conserved earlier on.


smallest Collision cross-section

• neutrinos (Hot DM) decouple from electrons (due to very weak interaction) while still hot (relativistic 0.5 Mev ~ kT >mc2 ~ 0.02-2 eV)

•

• Presently there are 3 x 113 neutrinos and 452 CMB photons per cm3 . Details depend on– Neutrinos have 3 species of spin-1/2 fermions while photons are

1 species of spin-1 bosons

– Neutrinos are a wee bit colder, 1.95K vs. 2.7K for photons [during freeze-out of electron-positions, more photons created]


Evolution of Sound Speed

Expand a box of fluid

t

cRx P t

cRycRz

2sSound Speed C

/ vol,

/ ( vol )

3c c cVol R t x y z

3R t

/ R/ R


Radiation Matter

Where fluid density t r m

2

Fluid pressure t3 rc m

mKT Matter number

densityRandom motion energy

Non-RelativisticIDEAL GAS4

rNote R 3

m R 21Neglect mKT c

Coupled radiation-baryon relativistic fluid

Show C2s = c2/3 /(1+Q) , Q = (3 ρm) /(4 ρr) , Cs drops

– from c/sqrt(3) at radiation-dominated era

– to c/sqrt(5.25) at matter-radiation equality


Coupled Photon-Baryon Fluid

Keep electrons hot Te ~ Tr until redshift z1 + z

Tr 1500 500

Compton-scatter3

2KTe

electrons in bath

hv

-e

hv

KTγ


Temperature and Sound Speed of Decoupled Baryonic Gas

Until reionization z ~ 10 by stars quasars

R

TTe

After decoupling (z<500),

Cs ~ 6 (1+z) m/s because

dP

dX

dP

dX

Te ∞ Cs2 ∞ R-2

21+zTe 1500 ×

500~ K

3 3 invarient phase space volumexd P d

1 1So: P x- R 2 23

22 emT R



Where are we heading?

• Sound speed of gas before/after decoupling

Topics Next:

• Growth of [chpt 11 bankruptcy of uniform universe]– Density Perturbations (how galaxies form)

– peculiar velocity (how galaxies move and merge)

• CMB fluctuations (temperature variation in CMB)

• Inflation (origin of perturbations)


Peculiar Motion

• The motion of a galaxy has two parts:

v=ddt

[ R t θ t ]

= R t . θR t θ t Proper length vector

Uniform expansion vo Peculiar motion v


Damping of peculiar motion (in the absence of overdensity)

•

• Generally peculiar velocity drops with expansion.

• Similar to the drop of (non-relativistic) sound speed with expansion

2 *( ) constant~"Angular Momentum"R R R

δv=R t xc=constant

R t


Non-linear Collapse of an Overdense Sphere

• An overdense sphere is a very useful non linear model as it behaves in exactly the same way as a closed sub-universe.

• The density perturbations need not be a uniform sphere: any spherically symmetric perturbation will clearly evolve at a given radius in the same way as a uniform sphere containing the same amount of mass.

b

ρb


R, R1

t

Rmax

Rmax/2 virialize

log

logt

t-2

Background density changes this way

2

1

6b Gt


Gradual Growth of perturbation

2 42

2 3

(mainly radiation )3 1

8 (mainly matter )

Perturbations Grow!

R Rc

G R R R

Verify δ changes by a factor of 10 between z=10 and z=100? And a factor of 100 between z=105 and z=106?


Equations governing Fluid Motion

2

2

4 (Poissons Equation)

1 d ln. (Mass Conservation)

dt

dvln (Equation of motion)

dt s

G

dv

dt

c

��

∇ Pρ

since ∂ P=c s2∂ ρ


Decompose into unperturbed + perturbed

• Let

• We define the Fractional Density Perturbation:

( ) exp( ),

| | 2 / , where ( )

o

c

c c

t ik x

k R t

k x k x

o

o c c

o

v v v R R

x t = R t χ c


• Motion driven by gravity:

due to an overdensity:

• Gravity and overdensity by Poisson’s equation:

• Continuity equation:

Peculiar motion δv and peculiar gravity g1 both scale with δ and are in the same direction.

g o t g 1 θ , t

( ) (1 ( , ))ot t

1 4 og G

( , )d

v tdt

The over density will

rise if there is an inflow of matter


THE equation for structure formation

• In matter domination

• Equation becomes

∂2 δ

∂ t 2 2RR

∂ δ∂ t

= 4πGρ o c s2 ∇ 2 δ

Gravity has the tendency to make the density perturbation grow exponentially.

Pressure makes it oscillate

−cs2 k2


• Each eq. is similar to a forced spring

F

m

d2 x

dt 2 =Fm−ω2 x− μ

dxdt

d2 x

dt 2 μdxdt

ω2 x=F t m

Term due to friction

(Displacement for Harmonic Oscillator)

x

t

Restoring


e.g., Nearly Empty Pressure-less Universe

2

2

0

~ 0

2 10, ( )

constant

no growth

RH R t

t t t R t

t


What have we learned? Where are we heading?

• OverDensity grows as – R (matter) or R2 (radiation)

• Peculiar velocity points towards overdensities

• Topics Next: Jeans instability


Case III: Relativistic (photon) Fluid

• equation governing the growth of perturbations being:

• Oscillation solution happens on small scale 2π/k = λ<λJ

• On larger scale, growth as

⇒d2 δ

dt 2 2Hdδdt

=δ .32 πGρ3

− k 2 c s2

2 for length scale ~J st R c t

1/t21/t


Lec 8

• What have we learned: [chpt 11.4]– Conditions of gravitational collapse (=growth)

– Stable oscillation (no collapse) within sound horizon if pressure-dominated

• Where are we heading:– Cosmic Microwave Background [chpt 15.4]

– As an application of Jeans instability

– Inflation in the Early Universe [chpt 20.3]


Theory of CMB Fluctuations

• Linear theory of structure growth predicts that the perturbations:

will follow a set of coupled Harmonic Oscillator equations.

δ D in dark matter δρD

ρD

δ B in baryonsδρB

ρB

δ r in radiation δρr

ρ rδ r=

34

δ r=δnγ

nγ

Or


• The solution of the Harmonic Oscillator [within sound horizon] is:

• Amplitude is sinusoidal function of k cs t – if k=constant and oscillate with t

– or t=constant and oscillate with k.

δ t = A 1 cos kc s t A 2 sin kc s t A 3


• We don’t observe the baryon overdensity directly

• -- what we actually observe is temperature fluctuations.

• The driving force is due to dark matter over densities.

• The observed temperature is:

δ B

ΔTT

=Δn γ

3nγ

=δ B

3=δ R

3

nγ ~ R−3∝T 3

εγ ~ nγ kT∝T 4

ΔTT obs

=δ B

3

ψ

c2

Effect due to having to climb out of gravitational well


• The observed temperature also depends on how fast the Baryon Fluid is moving.

Velocity Field ∇ v=−dδB

dt

ΔTT obs

=δ B

3

ψ

c2±vc

Doppler Term


Inflation in Early Universe [chtp 20.3]

• Problems with normal expansion theory (n=2,3,4):– What is the state of the universe at t0? Pure E&M field

(radiation) or exotic scalar field?

– Why is the initial universe so precisely flat?

– What makes the universe homogeneous/similar in opposite directions of horizon?

• Solutions: Inflation, i.e., n=0 or n<2– Maybe the horizon can be pushed to infinity?

– Maybe there is no horizon?

– Maybe everything was in Causal contact at early times?

Consider universe goes through a phase with

( ) ~ ( )

( ) ~ q=2/n

n

q

t R t

R t t where


x sun x

2χ

Horizon

22( ) (0)

~ ~ 0 at 0( ) (0)

nK Kn

z RR t

z R

Why are these two galaxies so similar without communicating yet?

Why is the curvature term so small (universe so flat) at early universe if radiation dominates n=4 >2?



• What determines the patterns of CMB at last scattering– Analogy as patterns of fine sands on a drum at last hit.

• The need for inflation to– Bring different regions in contact

– Create a flat universe naturally.


Inflationary Physics

• Involve quantum theory to z~1032 and perhaps a scalar field (x,t) with energy density

2-n1

2 ( ) ~ R(t) , where n<<1

fluctuate between neighbouring points [A,B]

while *slowly* rolling down to ground state

dV

dt

V()

finish

Ground state


Inflation broadens Horizon

• Light signal travelling with speed c on an expanding sphere R(t), e.g., a fake universe R(t)=1lightyr ( t/1yr )q

– Emitted from time ti

– By time t=1yr will spread across (co-moving coordinate) angle xc

i i

1 1 1 1

qt t

1

Horizon in co-moving coordinates

(1 )cdt cdt =

R(t) t (1 )

1Normally is finite if q=2/n<1

(1 )

(e.g., n=3 matter-dominate or n=4 photon-dominate)

( 1)INFLATION phase

( 1)

q qi

c

c

qi

c

tx

q

xq

tx

q

∫ ∫

i

i

can be very large for very small t if q=2/n>1

(e.g., t 0.01, 2, 99 , Inflation allows we see everywhere)cq x


Inflation dilutes the effect of initial curvature of universe

2

i

i

( )( )~ 0 (for n<2) sometime after R>>R

( ) ( )

( )even if initially the universe is curvature-dominated 1

( )

E.g.

( )If a toy universe starts with 0.1 inflates from t

( )

n

K iK

i i

K i

i

K i

i

RR R

R R R

R

R

R

R

-40f=10 sec to t =1sec with n=1,

and then expand normally with n=4 to t=1 year,

SHOW at this time the universe is far from curvature-dominated.


Exotic Pressure drives Inflation2 3

3

2

2

2

2

( )

( )

( ) 2 if ~

3 3 3=>

P/ c =(n-3)/3

Inflation 2 requires exotic (negative) pressure,

define w=P/ c , then w = (n-3)/3<0,

Verify negligble pressure for cosmic dust (

n

d c RP

d R

P d R nR

c RdR

n

2

2

matter),

Verify for radiation P= c / 3

Verify for vaccum P=- c


What Have we learned?

• How to calculate Horizon.

• The basic concepts and merits of inflation

• Pressure of various kinds (radiation, vacuum, matter)


List of keys• Scaling relations among

– Redshift z, wavelength, temperature, cosmic time, energy density, number density, sound speed

– Definition formulae for pressure, sound speed, horizon

– Metrics in simple 2D universe.

• Describe in words the concepts of – Fundamental observers

– thermal decoupling

– Common temperature before,

– Fixed number to photon ratio after

– Hot and Cold DM.

– gravitational growth.

– Over-density,

– direction of peculiar motion driven by over-density, but damped by expansion

– pressure support vs. grav. collapse


Lecture 3

Metrics for Curved Geometry


Cosmological Observations in a Curved and Evolving Universe

Non-Euclidian geometries:( positive / negative curvature )

Evolving geometries:

( expanding / accelerating / decelerating )

Time-Redshift-Distance relations


Non-Euclidean Geometry

Curved 3-D Spaces

How Does Curvature affect Distance Measurements ?


Is our Universe Curved?

Curvature: + 0 --

Sum of angles of triangle:

> 180o = 180o < 180o

Circumference of circle:

< 2 r = 2 r > 2 r

Parallel lines: converge remain parallel diverge

Size: finite infinite infinite

Edge: no no no

Closed Flat Open


Flat Space: Euclidean Geometry

Cartesian coordinates :

1D : dl 2 dx2

2 D : dl 2 dx2 dy2

3 D : dl2 dx 2 dy 2 dz 2

4 D : dl 2 dw2 dx 2 dy 2 dz2dx

dz

dy

dl

Metric tensor : coordinates - > distance

dl2 ( dx dy dz ) 1 0 0

0 1 0

0 0 1

dx

dy

dz

Summation convention :

dl2 gij dx i dx j i

j

gij dx i dx j

Orthogonal coordinates <--> diagonal metric

gxx gyy gz z 1

gxy gxz gyz 0

symmetric : g i j g j i


Polar Coordinates

Radial coordinate r, angles , ,,...

1 D : dl 2 dr 2

2 D : dl2 dr 2 r 2 d 2

3 D : dl 2 dr 2 r 2 d 2 sin 2 d 2

4 D : dl 2 dr 2 r 2 d 2 sin 2 d 2 sin 2 d2

dl2 dr2 r2 d 2 generic angle : d2 d 2 sin2 d 2 ...

dl2 ( dr d d ) 1 0 0

0 r2 0

0 0 r 2 sin 2

dr

dd

dr

dr

dl

gr r ? gr ?

g ?

g ?

g ?


Using the Metric

dl2 dr2 r2 d 2 sin2 d 2

dl2 ( dr d d ) 1 0 0

0 r2 0

0 0 r 2 sin 2

dr

dd

dlr grr dr dr, dl g d r d, dl ?

Radial Distance : D dlr∫ grr dr0

r

∫ dr0

r

∫ r

Circumference :C dl∫ g d0

2

∫ r d0

2

∫ = 2 r

Area : A dAr∫ dlr dl∫ grr dr g d0

2

∫0

r

∫ dr0

r

∫ r d0

2

∫ r 2

Note : dx dy r drd∫∫

dr

dr

dl

Same result using metric for any choice of coordinates.


Embedded Spheres

1 D : R 2 x 2 0 - D 2 points

2 D : R2 x 2 y2 1- D circle

3 D : R 2 x2 y 2 z 2 2 - D surface of 3 - sphere

4 D : R2 x 2 y2 z2 w 2 3 - D surface of 4 - sphere ?

R = radius of curvature


Metric for 3-D surface of 4-D sphere

4 sphere : R2 x 2 y 2 z2 w2

i.e. R2 r2 w2 with r2 x 2 y2 z2 .

02 r dr 2w dw dw2 r dr

w

2

r2 dr 2

R2 r 2

dl2 dw2 dr 2 r2d2 4 - space metric

r2 dr2

R2 r2 dr2 r2d 2 confined to R2 r2 w2

dl2 dr2

1 r R 2 r 2d 2 d 2 d 2 sin 2 d

Metric for a 3 - D space with constant curvature radius R

R

w

r


Non-Euclidean Metricsopen flat

closed

k 1, 0,1 ( open, flat, closed )

dl2 dr 2

1 k r /R 2 r 2d 2

dimensionless radial coordinates :

u r /R Sk

dl2 R2 du2

1 k u 2 u2d 2

R2 d 2 Sk2 d 2

S 1( ) sinh() , S0() , S1( )sin()


DR

w

rCircumference

metric :

dl2 dr2

1 k r R 2 r 2 d 2

radial distance ( for k 1 ) :

Ddr

1 k r R 2R sin 1 r R

0

r

∫

circumference :

C r d0

2

∫ 2 r

"circumferencial" distance : r C

2R Sk (D /R)R Sk ()

If k = +1, coordinate r breaks down for r R


Circumference

metric :

dl2 R2 d 2 Sk2 d 2

radial distance :

D g d ∫ R d0

∫ R

circumference :

C g d∫ R Sk ( ) d0

2

∫ 2 R Sk( )

2 DSk ( )

DR

w

r

Same result for any choice of coordinates.


Lecture 4

Space-Time Metric


Minkowski Spacetime Metric

ds2 c 2dt 2 dl 2

d 2 dt 2 dl2

c 2 dt 2 1 1c2

dldt

2

Null intervals light cone: v = c, ds2 = 0

Time-like intervals: ds2 < 0, d2 > 0 Inside light cone. Causally connected.

Space-like intervals: ds2 > 0 , d2 < 0

Outside light cone. Causally disconnected.

d -ds2 /c 2

= dt 1 -v2

c 2 0

Photons arrive from our past light cone.

World line of massive particle at rest.

Proper time (moving clock):


Robertson-Walker metricuniformly curved, evolving spacetime

ds2 c 2dt 2 R 2 (t) d 2 Sk2 d 2

c 2dt 2 R 2 (t)du2

1 k u2 u2 d 2

c 2dt 2 a2 (t)dr2

1 k r R0 2 r 2 d2

d 2 d 2 sin2 d 2

a( t)R(t) / R0

R0 R(t0 )

Sk ()

sin (k 1) closed

(k 0) flat

sinh (k 1) open

DR

w

r

radial distance D(t) R(t ) circumference 2 r(t) r(t) a(t) r R(t ) u R(t) S k


Redshift and Time Dilation

Light rays are null geodessics :

ds2 R2(t) d 2 c 2 dt 2 0

d c dtR(t)

c dt

R(t)

et

et et

∫ c dt

R(t)et

et

ot∫

c dt

R(t)

et et

ot∫ c dt

R(t)ot

ot ot

∫

te

R(te )

to

R(to)

R(to)R(te )

to

te

o

e

1 z

Observed wavelengths and time intervals

appear "stretched" by a factor x 1 z R0 R(t).

t

t 0 t 0

te

t0

t e t e


Fidos and co-moving coordinates

Distance varies in time:

D( t) R(t )

“Co-moving” coordinates

or D0 R0

D(t)

t

D

t

“Fiducial observers” (Fidos)

Labels the Fidos


Coordinate Systems


Angular Diameter Distance• radial distance

– now ( when photon received ):

– when photon emitted:

• angular size– Fraction of circumference when photon was emitted:

• angular diameter distance

D 0 R ( t 0 ) R 0

DA lr(te )R(te ) Sk ()

R(te )R0

R0 Sk()

R0 Sk()

1 z

r(t0 )1 z

r0

1 z

0

te

to

D 0

D e

De R(t e ) R(te ) R0

R0

D0

1 z

2

l

2 r(te )

l

Circumference was smaller by factor x=1+z.

Souces look larger/closer.


Luminosity Distance

– Luminosity ( erg s-1 )

– area of photon sphere ( when photons observed ):

– redshift:

– time dilation: lower photon arrival rate

– observed flux ( erg cm-2 s-1 )

• Luminosity distance

F N h 0

A0 t0

L

4 r02 1 z

2 L

4 DL2

DL 1 z r0 (1 z) R0 Sk ( )

A0 4 r02 4 R0

2Sk2()

0 e (1 z)

0 e 1 z

L N h e

te

t 0 t e (1 z )

0

te

t0

t e

t 0

Sources look fainter/farther.


Lecture 5

Time - Redshift - Distance Relationships

General Relativity:

Geodesics


Robertson-Walker metricuniformly curved, evolving spacetime

ds2 c 2dt 2 R 2 (t) d 2 Sk2 d 2

c 2dt 2 R 2 (t)du2

1 k u2 u2 d 2

c 2dt 2 a2 (t)dr2

1 k r R0 2 r 2 d2

d 2 d 2 sin2 d 2

a( t)R(t) / R0

R0 R(t0 )

Sk ()

sin (k 1) closed

(k 0) flat

sinh (k 1) open

DR

r

radial distance D(t) R(t ) circumference 2 r(t) r(t) a(t) r R(t ) u R(t) S k


• We observe the redshift :

• Hence we know the expansion factor:

• When was the light emitted?• How far away was the source?

• How do these depend on cosmological parameters?

Time and Distance vs Redshift

D(t, ) R(t) DA r0 () 1 z

r(t, ) R(t) Sk ( ) DL r0 ( ) 1 z

x 1 z 0

(t0 )(t)

R(t0 )R(t)

R0

R(t)

H 0 M

t(z) ?

(z) ?

z 0

0

0

1 observed,

0 emitted (rest)

DR

r


Time -- Redshift relation

x 1 z R0

Rdxdt

R0

R2

dRdt

R0

R

R R

Hubble parameter : H R R

x H(x)

dt dx

x H (x)

dz1 z H (z)

Memorise this derivation!


Lecture 6

General Relativity: Field Equations

Dynamics of the Universe:

R(t) = ?H(x) = ?

Friedmann Equation


Einstein Field Equations

G R 12

R g

8 Gc 2 T g

g spacetime metric ( ds2 g dx dx )

G Einstein tensor (spacetime curvature)

R Ricci curvature tensor

R Ricci curvature scalar

G Netwon's gravitational constant

T energy - momentum tensor

cosmological constant


Homogeneity and Isotropy

homogeneous

not isotropic

isotropic

not homogeneous

For cosmology, assume Universe is Homogeneous.

Simplifies the equations. : )


Homogeneous perfect fluid

G 8 G

c 2

c 2 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

c2 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

R 2 8 G

3

R2 k c2

R 4 G3c 2 c 2 3p R

3

R

---> Friedmann equations :

Einstein field equations:

p pressure density

momentum

energy

Note: energy density and pressure decelerate, accelerates.


Local Conservation of Energy

d energy work

d c 2R3 p d R3

c 2R3 c 2 (3 R2 R ) p (3 R2 R )

3 pc2

R R

p p() equation of state

Friedmann 1 : R 2 8 G

3 R2

3

R2 k c 2

2 R R 8 G

3R2 2 R R

3

2 R R

R 8 G

3

R2

2 R R

3

R

R 4 G

3

3 pc 2

R

3

R Friedmann 2


Newtonian Analogy

R 2 8 G

3

R 2 k c2

Friedmann equation:

E m

2R 2

G M m

R M

43

R 3

R 2 8 G

3 R 2

2E

m

same equation if

8 G,

2E

m k c2

m

R


Newtonian Analogy

E m

2R 2

G M m

R

Vesc 2 G M

R

E 0 V Vesc R E 0 V Vesc R E 0 V Vesc R 0

V 0

V 0


Density - Evolution - Geometry

Open k = -1

Flat k = 0

Closed k = +1

c

c

c

R(t)

t


escape velocity :

Vesc2

2 G MR

2 GR

4 R3 3

8 G R2 3

Hubble expansion :

V R H 0 R

critical density :

Vesc

V

2

8 G 3 H 0

2 c

c 3 H 0

2

8 G

Critical Density• Derive using Newtonian analogy:

R


Lecture 7

Dynamics of the Universe

Solutions to the Friedmann Equation for R(t)


Hubble Parameter Evolution -- H(z)

H 2 R R

2

8 G

3

3

k c 2

R2

H 2

H02R x4 M x 3

k c 2

H02R0

2x2

evaluate at x = 1 10 k c2

H 0

2R0

2

x 1 z R0 R

c 3 H0

2

8 G

M M

c

, R R

c

c

3 H0

2

0 M R

H 2

H02 R x 4 M x 3 (1 0 ) x 2

R0 c

H0

k0 1

k 1 0 1

k 0 0 1

k 1 0 1

Dimensionless Friedmann Equation:

Curvature Radius today:

Density determines Geometry


Possible Universes

H 0 70km/s

Mpc

M ~ 0.3

~ 0.7

R ~ 810 5

1.0

Empty

CriticalCycloid

Vacuum Dominated

Sub-Critical


Empty Universe (Milne)

R 2 8 G

3

R2 k c2

Set 0, 0. Then R 2 k c2

k 1 ( negative curvature )

R c, R c t

H R

R

1

t

age : t0 R0

c

1

H 0

Negative curvature drives rapid expansion/flattening

t

R

Vacuum Dominated

Empty

Critical

CycloidSub-Critical


Hubble Parameter Evolution -- H(z)

H 2 R R

2

8 G

3

3

k c 2

R2

H(x)H 0

2

=R x 4 +M x3 + k c 2

H02R0

2 x2

evaluate at x = 1 10 k c2

H02R0

2

x 1 z R0 R

c 3 H0

2

8 G

M M

c

, R R

c

c

3 H0

2

0 M R

H(x)H0

2

= (x) + 1-0 x 2

H(x) H0 R x 4 +M x 3 + (1 0 ) x2


Look-Back Time and Age

ddt

x 1 z R0

R

dt

dxx H (x)

look - back time : age :

t(z) dtt

t0

∫ dx

x H(x)1

1z

∫ t0 t(z )

1

H0

dx

x R x 4 M x3 (1 0 ) x 21

1z

∫


Radial Distance

D0 (z) R0 R0

R(t)c dt

t e

t 0

∫ x c dxx H (x)1

1z

∫

c

H0

dx

R x 4 M x 3 (1 0 ) x21

1z

∫

DR

r

c dt

R(t)te

t0

∫

dt dx

x H(x)

x 1 z R0

R(t)

0

te

to

D 0


Lecture 8

Observational Cosmology

Parameters of Our Universe

The Concordance Model


Time and Distance vs Redshift

d

dtx 1 z

R0

R

dt

dx

x H(x)

Friedmann : H(x) H 0 M x3 (1 0) x 2

0 M ( R = 0 )

look - back time :

t(z) dtte

t0

∫ dx

x H(x)1

1z

∫ 1

H0

dx

x M x 3 (1 0) x 21

1z

∫

radial distance :

D0(z) R0 R0

R(t)c dt

te

t0

∫ x c dx

x H(x)1

1z

∫ c

H 0

dx

M x 3 (1 0 ) x21

1z

∫

circumferencial distance : r0 R0 Sk () R0 c

H0

k0 1

1/ 2

angular diameter distance : DA r0 (1 z )

luminosity distance : DL (1 z ) r0

DR

r


Angular Diameter Distance

M

M 1

0


“Concordance Model” Parameters

H0 100 hkm/sMpc

70km/sMpc

h 0.7

R 4.210 5 h 2 8.4 10 5 (CMB photons neutrinos)

B

~ 0.02h 2 ~ 0.04 (baryons)

M

~ 0.3 (Dark Matter )

~ 0.7 (Dark Energy)

0 R M 1.0 Flat Geometry


Our (Crazy?) Universe

H0 70km/s

Mpc

M ~ 0.3

~ 0.7

R ~ 810 5

0 1.0

Empty

CriticalCycloid

Vacuum Dominated

Sub-Critical

accelerating

decelerating


“Concordance” Model

1. Supernova Hubble Diagram

2. Galaxy Counts M/L ratios

M ~ 0.3

3. Flat Geometry

( inflation, CMB fluctuations)

0 M 1

concordance model

H0 72 M 0.3 0.7

2

1

3

Three main constraints:


HST Key Project

Freedman, et al. 2001 ApJ 553, 47.

1-10 Mpcskm7372 H


Hubble time and radius

Hubble constant :

H R R

H 0 R R

0

100 hkm/sMpc

70km/sMpc

Hubble time :

tH 1

H0

1010h 1 yr 14109 yr

~ age of Universe

Hubble radius :

RH c

H0

3000 h 1 Mpc 4109pc

distance light travels in a Hubble time.

~ distance to the Horizon.


Age vs Hubble time

01 Ht H

Age = t0

H =R R

H =R R t t0

deceleration decreases age

acceleration increases age

e.g. matter dominated

R t 2 / 3 R 23

Rt

H R R

23

1t

t0 23

1H 0

23

tH


Age Constraints

• Nuclear decay ( U, Th -> Pb )– Decay times for (232Th,235U,238U) = (20.3, 1.02, 6.45) Gyr

– 3.7 Gyr = oldest Earth rocks

– 4.57 Gyr = meteorites

– ~10 Gyr = time since supernova produced U, Th

– ( 235U / 238U = 1.3 --> 0.33, 232Th / 238U = 1.7 --> 2.3 )

• Stellar evolution– 13-17 Gyr = oldest globular clusters

• White dwarf cooling– ~13 Gyr = coolest white dwarfs in M4


Nuclear Decay Chronology

• P=parent D=daughter S=stable isotope of D

• Chemical fractionation changes P/S but not D/S:

• Samples have same D0 / S0 various P0 / S0

• P decays to D:

P(t) P0 e t /

D(t) D0 P0 1 e t / S(t ) S0

D0 P( t) e t / 1

D(t)S(t)

D0

S0

P(t)S0

e t / 1

P/S

D/S

D0/S0

Observed slope e t / 1

gives age t / , typically to ~ 1%

e.g. concentrate D+S in crystals


Globular Cluster Ages


Coolest White Dwarfs

Hansen et al. 2002 ApJ 574,155

12.70.7 Gyr

White dwarf cooling ages --> star formation at z > 5.

Cooling times have been measured using “ZZ Ceti” oscillation period changes.


Age Crisis (~1995)

H0 t0 dx

x M x 3 (1 0) x 21

∫

observations :

H0 72 8 km s 1 Mpc 1

t0 14 2 Gyr old globular clusters

H0 t0 1.0 0.15

H0 t0 23

for M , = (1,0)

Globular clusters older than the Universe ? Inconsistent with critical-density matter-only model :

Strong theoretical prejudice for inflation. Doubts about stellar evolution therory (e.g. convection).

0 1


Lecture 9

Observational Cosmology

Discovery of “Dark Energy”


Deceleration parameter

q R RR 2

R

R H 2

q0 R RR 2

0

R

R H 2

0

a( t)R(t)R0

1 H0 t t0 q0

2H0

2 t t0 2 ...

a H a a q H 2a

Dimensionless measure of the

deceleration of the Universe

q0 > 0 => deceleration

q0 = 0 => coasting at constant velocity

q0 < 0 => acceleration

t - t0

R

q0 > 0

q0 < 0


Deceleration parameter

q R RR 2

R

R H 2q0

R RR 2

0

R

R H 2

0

Friedmann momentum equation :

R 4 G3

3 pc 2

R

3R

R H0

2R 4 G

3H 02 1 3w

3H0

2

, p 0 decelerate, 0 accelerates

Equation of state : p wi i c2

i

wR 13

wM 0 w 1

q0 R

R H 2

0

1 3 wi

2

i

i

R M

2

M

1

0

1 / 2

q0 1 / 2

t - t0

R


Deceleration Parameter

q0 R RR 2

0

M

2

Measure q0 via :

1 . DA(z)

( e.g. radio jet lengths )

2. DL(z)

( curvature of Hubble Diagram )

+1

0

q0 = -1

acceleration

deceleration

Matter decelerates

Vacuum (Dark) Energy accelerates

Critical density matter-only --> q0=1/2.


Observable Distances

angular diameter distance :

=l

DA

DA r0

(1 z )

c zH0

1q0 3

2z ...

luminosity distance :

F =L

4 DL2

DL r0 (1 z ) c zH0

11 q0

2z ...

deceleration parameter :

q0 M

2

Verify these low-z expansions.


Kellerman 1993

1993 - Angular Size of Radio Jets

Deceleration

as expected for

But, are radio jets

standard rods ?

l

DA(z )

q0 ~ 0 .5

( M , ) (1, 0 )

( M , ) (0 .3, 0 .7 )

Also compatible with Concordance Model.

M


Hubble Diagram

m M 5 logDL (z)Mpc

25

A K(z)

m apparent mag

M absolute mag

A extinction (dust in galaxies)

K(z) K correction

( accounts for redshift of spectra

relative to observed bandpass )

DL(z) c zH0

11 q0

2z ...

slope = +5

vertical shift --> H0

curvature --> q0

log ( c z )

ap

pa

ren

t ma

g

deceleration ( q0 > 0 )

acceleration ( q0 < 0 )

faint

bright


Finding faint Supernovae

Observe 106 galaxies.

Again, 3 weeks later.

Find “new stars”.

Measure lightcurves.

Take spectra.

( Only rare Type Ia Supernovae work ).


Hi-Z Supernova Spectra

SN II --- hydrogen lines

(collapse and rebound of the core of a massive star)

SN I --- no hydrogen lines

(no H-rich envelope surrounding the core)

SN Ia --- best known standard candles

(implosion of 1.4 Msun white dwarf, probably due to accretion in a mass-transfer binary system).

HH


Calibrating “Standard Bombs”

1. Brighter ones decline more slowly.

2. Time runs slower by factor (1+z).

AFTER correcting:Constant peak brightness MB = -19.7

Observed peak magnitude:m = M + 5 log (d/Mpc) + 25gives the distance! Time ==>

Absolute m

agnitude M

B ==

>


SN Ia at z ~ 0.8 are ~25% fainter than expected

Acceleration ( ! ? )

1. Bad Observations?

-- 2 independent teams agree

1. Dust ?

-- corrected using reddening

2. Stellar populations ?

-- earlier generation of stars

-- lower metalicity

3. Lensing?

-- some brighter, some fainter

-- effect small at z ~ 0.8

Reiss et al. 1998

Perlmutter et al. 1998


1998 cosmology revolution

Acceleration ( ! ? )

matter-only models ruled out

cosmological constant > 0

“Dark Energy”

if 0 M 1

then M ~ 0.3 ~ 0.7


HST Supernova SurveysHST surveys to find SN Ia beyond z = 1Tonry et al. 2004.


25 HST SN 1a beyond z = 1Reiss et al. 2007.

SNAP = SuperNova Acceleration Probe

1.5m wide-angle multi-colour space telescope --- 1000 SN 1a

(Not Yet Funded)

Most distant Supernova SN 2007ff z =1.75


Lecture 10

Checking the Distance Ladder:

Sunyaev-Zeldovich Effect

Gravitational Lensing


“Concordance” Model

1. Supernova Hubble Diagram

2. Galaxy Counts M/L ratios

M ~ 0.3

3. Flat Geometry

( inflation, CMB fluctuations)

0 M 1

concordance model

H0 72 M 0.3 0.7

2

1

3


HST Key Project

Freedman, et al. 2001 ApJ 553, 47.

1-10 Mpcskm7372 H


Galaxy Clusters arefilled with hot X-ray gas

optical (galaxies) X-ray (hot gas)


Gravitational Lensing• Luminous arcs

in clusters of galaxies


Gravitational Lensing

multiple images

of background galaxy

lensed by the cluster


The Lensed Galaxy


Newtonian Bend Angle

vertical acceleration gy G M

r 2

br

G Mb2 gmax

time to pass t 2 b /Vx

vertical velocity Vy gy dt∫ gmaxt G M

b2

2 bVx

2 G MbVx

bend angle Vy

Vx

2 G M

bVx2

2 G Mbc 2

M

br

t

gy


Focal Length of Gravitational Lens

Einstein' s bend angle 4 G Mb c 2

Focal length : f b

b 2 c 2

4 G M

b

f


Spherical Aberration

Einstein' s bend angle 4 G Mb c 2

Focal length : f b

b 2 c 2

4 G M


Observer’s view:

Lensing by a point mass

Two distorted/magnified images of background source

Light from background source deflected by lens mass1

2

Einstein ring

1 2


Einstein Ring Radius

Geometric optics :

1

DS DL

1

DL

1f

4 G Mc2 b2

Einstein Ring Radius :

bRE 4 G M

c2

DL DS DL DS

E RE

DL

M

1011.1Msun

1/ 2DL DS /DLS

Gpc

1/ 2

arcsec

SourceLens

D S

D L


Lensing by a Point Mass

2 images

opposite sides of lens

major image outside ring

minor image inside ring

net magnification

(sum of 2 images)

vs time


Off-Axis Lensing Geometry

angular diameter distances from redshifts : zL, zS

impact parameter : b DL source offset : DS S DS DLS

bend angle : S DS

DLS

4 G M( b)

c2 b

S

LD LSD

source

lensobserver

SD

b


Lensing by an extended mass distribution

( )4 G M ( )

c2 DL

DS

DLS

S

Usually gives 3 images,

can be 5, 7, ...

If M known, measure image angles and solve for DLDS/DLS

C B

AS

3 images on sky

A

B

C

S

Lens equation:


Quasars Lensed by Galaxies


Masses from Einstein Rings

E RE

DL

4 G M

c 2

DLS

DL DS

1 / 2

E

arcsec

M

1011 Msun

1/ 2DLDS /DLS

Gpc

1 / 2

M

1011 M sun

DL DS /DLS

Gpc

E

arcsec

2

Use redshifts, zL, zS , for the angular diameter distances.

Or, if mass known, e.g M V 2R

G, then gives D

Mass usually less certain than distance,

so use theta and D to calculate M.

Perfect alignment gives an Einstein Ring


H0 from Time Delays

light travel time delay :

c t DL2 b2

1 / 2

DLS2 b2

1/ 2

DL DLS

DL 1 2

1/ 2 1 DLS 1

DL

DLS

2

1 /2

1

DL

2

21

DL

DLS

c zL

H0

2

21

zL

zS zL

c

H 0

2

2

zL zS

zS zL

measure ( images ), zL , zS ( spectra )

and t ( delay from lightcurves of images ).

LD LSD

b

b

DLS

DL

DLS


Time Delay MeasurementLight curves of the images show a shift in time.

Hjorth et al. 2003.

146 days


But, no simple lenses.Almost always several galaxies involved.

Prevents very accurate distance measurements.


Dark Matter

Galaxy CountsRedshift Surveys

Galaxy Rotation CurvesCluster Dynamics

Gravitational Lenses

M ~ 0.3

b 0.04

2

1

3


Mass Density by Direct Counting• Add up the mass of all the galaxies per unit volume

– Volume calculation as in Tutorial problem.

• Need representative volume > 100 Mpc.

• Can’t see faintest galaxies at large distance. Use local Luminosity Functions to include fainter ones.

• Mass/Light ratio depends on type of galaxy.

• Dark Matter needed to bind Galaxies and Galaxy Clusters dominates the normal matter (baryons).

• Hot x-ray gas dominates the baryon mass of Galaxy Clusters.


2dF galaxy redshift survey

z = 0.3

z = 0

Galaxy Redshift Surveys

Large Scale Structure:

Empty voids

~50Mpc.

Galaxies are in 1. Walls between voids.

2. Filaments where walls intersect.

3. Clusters where filaments intersect.

Like Soap Bubbles !


Cluster Masses from X-ray Gas

hydrostatic equilibrium :

dPdr

g G M ( r)r2

gas law :

P k T

mH

X - ray emission from gas gives : T(r), ne (r) (r),P(r)

M ( r) r 2

G(r)dPdr

Coma Cluster:

M(gas)~M(stars)~3x1013 Msun

often M(gas) > M(stars)M/L~100-200


Cluster Masses from X-ray Gas

gasstars

total mass

T~108K

g ~ 3x10-8 cm s-2

M ~ 1014 Msun


Masses from Gravitational Lensing

E RE

DL

4 G M

c 2

DLS

DL DS

1 / 2

M

1011 M sun

DL DS /DLS

Gpc

E

arcsec

2

Use redshifts, zL, zS ,

for the angular diameter distances.

General agreement with Virial Masses.


Evidence for Dark Matter ? Galaxies: ( r ~ 20 Kpc )

Flat Rotation Curves V ~ 200 km/s

Galaxy Clusters: ( r ~ 200 Kpc )

Galaxy velocities V ~ 1000 km/s

X-ray Gas T ~ 108 K

Giant Arcs

X-ray Optical


Or …. Has General Relativity Failed ?

~4% Normal Matter ~22% “Dark Matter” ? ~74% “Dark Energy” ?

Can Alternative Gravity Models fit all the data without 2 miracles ? ( Dark Matter,

Dark Energy )


MOND and TeVeS

g gN gN a0

gN a0 1 / 2

gN a0

MOdified Newtonian Dynamics:

Milgrom 1983 …

Covariant metric gravity theory that

reduces to MOND in weak-field low-velocity limit.

V 2 g r GM /r gN a0

G M a0 1 / 2

gN a0

Bekenstein 2004 …

Tensor Vector Scalar:

a0 ~ 2 10 8 cm s 2

MOND gives flat rotation curves V( r ) ~ const and Tully-Fischer : V4 ~ M

MOND acceleration parameter:


Cosmic Microwave Background

Flat Geometry

0 M

1.0

2

1

3


1965 -- Penzias + Wilson

Bell Labs telecommunications engineers find excess microwave noise from the sky.

~1% of thermal ( T ~ 300o K ) noise ---> T ~ 3o K

Afterglow of the Big Bang

CMB = Cosmic Microwave Background

Confirms a forgotten 1948 prediction by Gamow.

Nobel Prize -> P+W


Recombination Epoch ( z~1100 )ionised plasma --> neutral gas

• Redshift z > 1100

• Temp T > 3000 K

• H ionised

• electron -- photon Thompson scattering

• z < 1100

• T < 3000 K

• H recombined

• almost no electrons

• neutral atoms

• photons set free

e - scattering optical depth

(z) z

1080

13

thin surface of last scattering


NASA 1992 - COBECOsmic Background Explorer


COBE spectrum of CMB

A perfect Blackbody !

No spectral lines -- strong test of Big Bang. Expansion preserves the blackbody spectrum.

T(z) = T0 (1+z) T0 ~ 3000 K z ~ 1100


TT

~ 10 5

T = 2.728 K

Dipole anisotropy

Our velocity:

Milky Way sources

+ anisotropies

Vc

TT

10 3

Almost isotropic

V 6 0 0 k m /s

Cosmic Microwave Background


COBE - tiny ripples

510~

TT Resolution ~ 7o


Tiny Ripples at Redshift 1100

TT

4

~ 10 5 at z 1100

Ripples are :

relics of the Big Bang

initial quantum fluctuations expanded by early inflation

the seeds of later galaxy/cluster formation.

standard yardsticks for measuring curvature

( and other cosmology parameters )


1999 - Boomerang in Antarctica

Baloon Observations Of Millimetric Extragalactic Radiation ANisotropy and Geophysics


Boomerang in Antarctica


Boomerang’s Baloon


Boomerang’s Stratospheric Flight Track

Altitude 37 km

10 days


Resolution ~ 0.3o


Boomerang Map

Some point sources

Note preferred angular scale

~ 1o


Spherical Harmonics

spherical harmonics

TT

, al mYl m

l,m

,

angular power spectrum

Cl al m

2average l m l

dimensionless power spectrum

l l1 Cl d T /T

2

d ln l

angular scale : l

180o

l

m cycles in longitude

l - m nodes in latitude

Fit temperature map with a series of


Supernovae +

CMB ripples

Pre-WMAP constraints

From BOOMERANG

and MAXIMA

circa 2002


WMAP

NASA 2001...

Wilkinson

Microwave

Anisotropy

Probe


~ 1o

COBE

1994

WMAP

2004

CMB Anisotropies

TT

~ 10 5

Snapshot of Universe at z = 1100 Seeds that later form galaxies.


2003 -- WMAP Power SpectrumSpergel et al. 2003

ApJSup 148,175.

180o

l

l1 2201


Sound Horizon at z = 1100

0

zR 11 00

Rt

0t

0

LS

DA

~ 0.8o

3ccS

L S c S t R ~ 100 kpc

Standard Ruler :


Angular scale --> Geometry



distance travelled by a sound wave

cS dt

expand each step by factor R(tR )/R(t) :

LS(tR ) R(tR )cS dtR(t)

0

t R

∫

R0

1 z

x

R0

cS dx

x H(x)1z

∫

cS

( 1 z )

dx

H(x)1z

∫

cS

( 1 z ) H0

dx

x 4 R x 3 M 1 0 x 21z

∫

c S

(1 z ) H0

dx

x4 R x 3 M1z

∫

x1 z R0

R(t)

dt dx

x H(x)

sound speed

cS c

3

recombination at z = 1100

keep 2 largest terms.

H( x ) from Friedmann Eqn.

dt = - dx / x H( x ) R( t ) = R0 / x



LS(tR ) cS

(1 z )

dx

H(x)1z

∫ cS

(1 z ) H0

dx

x4 R x 3 M1z

∫

cS

( 1 z ) H0 R

dx

x3(x x0)1z

∫ x0 M

R

3500M

0.3

cS

( 1 z ) H0 R

2

x0

1x0

x

1z

2cS

( 1 z ) H0 M x0

1x 0

1 z 1

cS

c

3

c

H0

2 4.6 1 1100 30.33500

3.4 10 5 c

H0

1100.7

h

0.3

M

1/2

kpc

Expands by factor 1 + z = 1100 to ~120 Mpc today.


Angular Scale measures 0

sound horizon : angular diameter distance :

LS(z)1

1 z

cS dx

H(x)1z

∫ DA (z) R0 SK

1 z

c dt

R(t )t

t0

∫ c

R0

dx

H(x)1

1z

∫angular scale

LS (z)

DA (z)

cS dx

H(x)1z

∫

R0 Sk

cR0

dxH(x)

1

1z

∫

Angular scale depends mainly on the curvature.

Gives ~ 0.8o for flat geometry,

0 = M + =10.8o

1.0o

0.6o


Precision Cosmology

Energy Dark 04.073.0

Matter Dark 04.027.0

baryons 004.0044.0

flat 02.002.1

expanding 371

M

h

t0 13.70.2 109 yr now

t 180 220

80106 yr z 20 10

5 reionisation

tR 3791103 yr zR 1090 1 recombination

( From the WMAP 1-year data analysis)


Dark Energy ? Vacuum energy?

Bubble Cosmology?

Dark Matter ? Large-Scale Structure

Galaxy Rotation CurvesCluster Dynamics

Gravitational Lenses

MACHOs? --- No WIMPs? --- Maybe

Modified Gravity ?MOND , TeVeS

M 0.26

b 0.04

0 .74

vac ~ 10120

as 4022 cosmology 1 as 4022: cosmology hs zhao online notes: star-hz4/cos/cos.html...

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