Third Year Astrophysics 2008 Acknowledgements Pictures from Ned Wright’s Home Page Pictures from...

Third Year Astrophysics2008

• Acknowledgements

• Pictures from Ned Wright’s Home Page

• Pictures from Charlie Lineweaver’s paper.

• Other material from many web sources, many following leads from WMAP web page.

Einstein’s Theory of Gravity

• General Theory of Relativity (GR)– published by Einstein in 1916– GR describes all gravitational systems

including the entire universe.

– T: Stress -energy tensor. Describes sources of curvature

– G:Einstein curvature tensor describes space-time geometry

• Connections between mass and space– mass creates curvature in space– space curvature tells masses how to move

Albert Einstein

€

T =c 4

8πG⋅G

In absence of external forces objects follow free fall trajectories that are geodesics -(shortest paths) in space time

Solutions for Expanding Universes• Cosmological principle: that we see

what every other observer sees

• Expansion/contraction are generic to homogeneous and isotropic solutions

• Static solutions do exist

– requires repulsive force to offset gravitational attraction

– Einstein associated this repulsive force with the parameter

• Hubble discovered expansion

– a bad day for Einstein

– Einstein came close to predicting that the universe is expanding

A static universe requires repulsive forceto exactly offset the gravitational attractionwhich attempts to contract space.

Slice through 2D, model universe onsurface of sphere.

Repulsive force

Gravity

Evolution of Expanding Debris Cloud• Evolution determined by balance of expansion and gravitational attraction• Equation for debris cloud the same as for universe as a whole• Consider isolated object exploding far out in space

• debris flies off in all directions at high velocity dependent on energy of the explosion

• gravitational attraction of the pieces slows the expansion• balance of expansion velocity and gravity determines fate of

debris cloud–if debris mass is too low, not enough gravity to halt expansion

: expansion of debris cloud continues to slow, but never stops–if debris mass is high enough then gravitational attraction

eventually halts expansion and cloud collapses to big crunch–for a particular debris mass- the critical mass- there will be

just enough gravitational attraction to eventually halt expansion.

–One dimensional solution identical to 3D case.

Friedmann Equation

• Hubble Law: v=Hr• Re-express: dr/dt = H(t)r• Define scale factor a(t)=r(t)/r(t0)• Then 1/a da/dt = H(t)

• In GR interpret as mass-energy density

• Interpret k as curvature parameter

• 1/2v2 - GM(r)/r = k. const

• For k=0, KE + PE = 0

•Solve equation of motion for unit mass of the universe (F=ma)

•First find

•By inspection static universe is impossible unless =0 or another term is present to cancel the G term.

Integrate above equation to obtain the Friedmann Equation

€

1

a

d2a

dt 2+

4πGρ

3= 0

€

˙ a 2

a2−

8πGρ

3=

−kc 2

a2€

v 2

2−

G

r⋅

4πr3ρ

3=

−kc 2r2

2a2

Critical density

• From Friedmann Equation

• solve for value of for which k=0

• Answer:

• This is the critical density crit for which = 1

= /crit

• We will return to these concepts later on.

€

˙ a 2

a2−

8πGρ

3=

−kc 2

a2

€

=3

8πG

˙ a 2

a2=

3

8πGHo

2

Einstein’s Blunder

• To force his equations to have a static solution Einstein introduced the cosmological constant:

• Friedmann equation modified:

• Modern Hubble data implies existence of such a term: acceleration of expansion is observed consistent with a cosmological constant. We call it Dark Energy

• Is Dark Energy a modification of GR or an independent property of space?

€

T =c 4

8πG⋅(G + Λg )

€

˙ a 2

a2−

8πGρ

3=

Λ

3−

kc 2

a2

Einstein:”the greatest blunder of my life”

No blunder!

Hubble Law

• Hubble’s orignal data 1929: v=H0D

– Expansion rate 464kms-1Mpc-1

• Modern data from Type 1A supernovae

– Expansion rate 64kms-1Mpc-1

• Redshift z:

• Redshift in Special relativity:

• Hence v= cz + higher order terms

• High order terms in cosmology depend on GR and on structure of universe

• Only a linear Hubble Law is consistent with the Cosmological Principle, also known as Copernican Principle or Principle of mediocrity.

• Think about a quadratic Hubble Law

• Hubble Law provides a universal reference frame

QuickTime™ and a decompressor

are needed to see this picture.



€

1+ z =λ obs

λ emit

€

1+ z =1+ v

c1− v

c

Galaxy Spectroscopy to Measure Hubble Flow

• Spectra of a nearby star and a distant galaxy

– Star is nearby, approximately at rest

– Galaxy is distant, traveling away from us at 12,000 km/s

Wavelength In

ten

sity

Sp

ectrum

courtesy B

ob K

irshn

er

Calcium

Magnesium

Sodium

Galaxy Spectrum

Stellar Spectrum

• Spectra of nearby and distant galaxies

– Nearby galaxy travels at 261 km/s

– Distant galaxy travels at 6,400 km/s

Universe is Homogeneous and Isotropic on Large Scale

• Good approximation: homogeneous on 100Mpc scale at few percent level

• Isotropic as seen by COBE, WMAP etc at 10-5 level

• Absolute reference frame easily measured in the CMB. (Image of universe at age 380,000 years), z=1200

• Our speed is 370kms-1 relative to the universal reference frame.

• Isotropic to about .001% from CMB data.









Speed:dipoleanisotropy

Low mp anisotropy

seen by COBE

Distances in Cosmology• Homogeneous and isotropic universe has measurable age.

• Age must be defined on a surface of constant proper time since the big bang.

• Proper time depends on observer velocity, so time t in cosmology is proper time for comoving observers.

• Homogeneity and isotropy means we can simplify to a 2-D space-time diagram.

• Hubble law v=HD true for all D, even v>c.

• Distance and velocity require careful interpretation:

– a) consider two close spaced but distant objects A and B. Separation DA-DB that we measure must be the same distance that A or B would measure at the same proper time t0 since the big bang.

– b) Determine Dnow for a distant galaxy by adding a set of local measurements all made at same time t0 .(negligible expansion during mesurement).

• Conformal time is a convenient time variable obtained by dividing proper time intervals by the scale factor of the universe, giving an expanded time axis for the past and a finite value for proper time t=infinity when scale size becomes infinite.

• Co-moving (conformal) distance is a distance that remains constant for objects that are subject only to the Hubble flow: Co-moving distance D/scale size.

Space Time Diagrams

• Locally space time diagram is a 45 degree cone (slope 1 light year per year) and finite rest mass particle trajectories must always fall within the cone.

• Light cone defines regions within which can be causally connected.

• The particle horizon is a sphere around an observer defining the maximum distance object to which the observer can be causally connected.

• Particle horizon is always smaller than the event horizon which is the infinite time particle horizon.



Hubble Law and Space-time diagrams

• Linear Hubble law invariant to location of observer: all locations see the same law





Recession velocity exceeds the speed of light where light cone becomes vertical

Time

Space

Spacetime diagram for Low Density Universe

• Uniform expansion (low density universe)



Past light cone 45 degrees 1 light year per year, v=c

Expansion rate exceeds c: photon velocity zero

Expansion speed exceeds c (photons receeding)

Cosmological Spacetime vs Special Relativistic

Spacetime

Changing to SR coordinates converts spacetime diagram to one with hyperbolic surfaces of constant proper time.

Conical past light cone

Hubble law distance =infinity, SR distance ct0/2.

In cosmological coordinates:

Hubble:v=H0Dnow

Dnow=(c/H0) ln(1+z)

1+z = exp(v/c)

In Special Relativity





Surfaces of constant proper time

€

1+ z =1+ v

c1− v

c

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

12

Worldlines of co-moving objects

Our worldline All events we currently observe

Photon velocity effectively falls to zero

Current distance to particle horizon

Photons initially receeding make transition to approaching

Co-moving distance

Example of Change of Representation



End Week 1

Hubble Law Interpretation

• Hubble sphere DHS=c/H: distance that recession velocity exceeds c.

• Hubble sphere is not a horizon: objects can cross DHS

• In concordance CDM cosmology, objects with z>1.46 are receding faster than c.

• This does not violate Special Relativity because

– Motion not in any observers inertial frame

– No observer overtakes a light beam

– All observers measure c to have the same value locally

• Hubble law data confirms that the red shift is NOT a special relativistic Doppler effect.

• Any Hubble Law expansion predicts superluminal expansion for sufficiently distant objects.

• SEE Expanding Confusion by TM Davis and Charlie Lineweaver Astro-ph

Redshift z

•Particle horizon: maximum distance a particle (light) can travel since t=0. The distance to the particle horizon is not given by ct0 because the universe expanded. The distance is roughly 3ct0

•Event horizon: distance light can travel from time t to t=infinity.

If redshifts were purely relativistic supernova brightness would follow lower curve

Measuring Distance

• Angular size distance DA:

• Luminosity Distance DL:

€

DA = sizeθ

€

Flux =Luminosity

4πDL2

Luminosity Distance, Angular Size Distance and Light Travel Time Distance



Einstein-de Sitter Universe: critical density matter only

Empty: no decelleration

LCDM: Dark energy 72%, matter and dark matter 28%

Scale factor and Critical Density

Velocity dDnow/dt is proportional to Dnow, so distance between a pair of co-moving objects increases by factor (1 + Hdt) during time interval dt.

Hence distance to co-moving galaxy G is DG(t) = a(t). DG(t0)

where DG(t0) is distance Dnow to galaxy Gnow and a(t) is a universal scale factor.

Dynamics of universe can be calculated by considering an object at distance D(t) = a(t) D0

Gravitational acceleration due to spherical ball, radius D(t): g = - GM/D(t)2

where M = (4/3)D(t)3(t) . Mass within D(t) independent of time, mass outside has zero effect.

M has an escape velocity. Remember v = HD. Escape velocity = (2GM/D)1/2

Setting v = escape velocity, H2D2 = 2G(4/3)D2

Hence crit = 3H2/8G

For H ~ 70km/s/Mpc, crit ~ 6 protons/ cubic meter or 1011Msun/cubicMpc

Ratio : crit =



0=0

0=1

0=2

v=1

m=0.27,

V=0.73 Big crunch in 80 Gyr

possible universes

Mean distance between galaxies

today

fainter

redshift

M = 1

Time

Closed M > 1

Open M < 1

M = 0

- 14 - 9 - 7

billion years

Perlmutter 1993

Flatness problem• If 0 > 1, universe will eventually stop expanding, H will drop to

zero, crit will drop to zero and will become infinite.

• If 0 < 1, actual density reduces faster than crit , so falls to zero.

• Value of unstable with time.



Density of universe 1ns after big bang: tiny change has drastic consequences.

Fine tuning 2 parts in 1024 at 1ns

1 part in 1059 at Planck time

Spatial CurvatureSpatial curvature depends on value of crit =

Note dark energy can prevent collapse for



Geometry

Parallel Lines Converge

Parallel Lines Remain Parallel

Parallel Lines DivergeOpen Space

<crit

0<1

Flat Space=crit

0=1

Closed Space>crit

0>1

Figures from Universe by William Kaufmann

Friedmann Equation Web Pages

• http://www.jb.man.ac.uk/~jpl/cosmo/index.html

• Create your own universes at:

– http://www.jb.man.ac.uk/~jpl/cosmo/friedman.html

– Nice animation allows you to choose present values of m and and see how they evolve from the big bang to the far future, while simultaneously seeing the universe scale size changing.

• http://scienceworld.wolfram.com/physics/FriedmannsEquation.html

http://www.jb.man.ac.uk/~jpl/cosmo/index.html





http://www.jb.man.ac.uk/~jpl/cosmo/friedman.html





Space-time diagram for critical density universe =1

• Critical density space-time diagram: gravitational decelleration causes curvature

• Scale size increases as t2/3

• All observers see same space time diagram: transformation to another observer

• Transformation is not Lorentz, not Galilean.

• Every coordinate system is a distorted representation







Transform to conformal spacetime diagram

Co-moving distance:

Divide Dnow by a(t)

Straight world lines

Conformal time:

Divide time by a(t)

Straight world lines and straight light cone







Example of Change of Representation



Horizon problem and the CMB

Cosmic background radiation:

radiation from surface of last

scattering at z>~1000.

Scale size a(t) ~ 10-3 a(t0).

Since a(t) ~ t2/3, t at surface

of last scattering ~ 3. 10-5t0

~380,000 yrs.

CMB temperature uniformity

~10-4…yet no causal connection



Temperature here determined by events within this light cone

Temperature here determined by events within this light cone

No causal connection

Inflation: solution to flatness and horizon problems

Why should vacuum have zero energy density?

Virtual particle-antiparticle pairs

Expected vacuum energy:1 particle per Compton

wavelength volume c3 = (h/mc)3.

Density =m/c3 = m4c3/h3

This gives vacuum= 1013 [M/proton mass]4 gm/cc

For Planck Mass Mp = (hc/2G)1/2 ,vac =1091 gm/cc

Observed vacuum energy < 10-29 gm/cc

Hence vacuum energy is 10120 times too big



Dark Energy

• Weak dark energy is needed today – to stretch the age of the universe to match stellar ages– To explain supernova redshifts– To explain observed redshift distributions of galaxies

• Strong dark energy is needed in the early universe to explain the flatness and horizon problems

• Somehow the strong dark energy must switch off very quickly.• Weak dark energy is a vacuum energy and its role expands as

the volume of space expands….it has negligible effect in high density universe.

Dark Energy and Negative Pressure

• Simplest explanation for inflation and observed cosmic acceleration is that they are both manifestations of the same phenomenon associated with the quantum vacuum.

• Einstein introduced as a means of creating a static solution.• Consider volume radius R: gravitational accelleration at edge is

where 3P/c2 is the gravity due to the energy density of the vacuum• To make a static solution, g=0, P must be negative.• Negative pressure (eg internal pressure in a stretched rubber band) has

positive energy density which partially compensates for the negative gravitational effect of negative pressure.– Set vacuum= 0.5matter.

Then total = 1.5 matter, and P=0.5matter/c2 .Then

so that g=0.• In an expanding universe the solution is an unstable “point solution”

– If universe expands 1% matter density reduces 3% but dark energy density stays constant and the balance of dark energy and matter is lost.

•

€

g =GM

R2=

4π

3G ρ +

3P

c 2

⎛

⎝ ⎜

⎞

⎠ ⎟R

€

+ 3P

c 2

⎛

⎝ ⎜

⎞

⎠ ⎟= 0,

Inflation: Derivation of Expansion Law

Start with Friedmann equation

Ignore matter (matter did not exist at this time) and curvature term k (not an obvious approximation but because we end up with huge expansion, this is ok.)

Solution is

Where H = (8/3 G )1/2

If is constant, then so is H. Universe expands exponentially.

For inflation to work this process must be cut off…otherwise we would have an empty universe

€

a(t) ~ eHt

Reference:Gary Watson Astro-ph/0005003 Brown University



Inflationary Expansion

• For large vacuum energy Friedmann Equation solution is a(t) = exp(H(t-t0))

• Inflation flattens the curvature of the universe

• Flatness ok if inflation lasts 100 doublings: 1030 fold growth

• Horizon ok because future light cone is expanded into a huge region

• 1030 implies <~1mm expanded to size of universe.

• Expanded future lightcone QuickTime™ and a

decompressorare needed to see this picture.

Quantum Fluctuations

• Tiny scale fluctuations expanded by inflation

• Extra-horizon scale fluctutions in CMB are imprint of inflation

• (Acoustic peaks are causal processes in the ionisation epoch before recombination)

• Total fluctuation observed integrates all effects from all times, which predicts equal fluctuation power on all angular scales.

• This agrees with CMB data



Future lightcones of fluctuation events

Fluctuation lightcones as seen on the sky

time

Inflaton PotentialsToy Model For Original Inflation. In this model of inflation the inflaton finds itself trapped in a false minimum. It is freed from this minimum when tunneling is allowed to occur resulting in a first order phase transition in the early universe.When tunnelling completed, inflation stops.





Toy Model For New Inflation - When the temperature of the universe decreases to the critical temperature Tc, the scalar field potential experiences a second order phase transition. This makes the `true' vacuum state available to , and inflation stops.

Toy Model For Chaotic Inflation - The inflaton finds itself displaced from the true vacuum and proceeds to `roll' back. Inflation takes place while the inflaton is displaced, finishes when it has reached the true vacuum.

Surprising facts about dark energy

• During inflation strong dark energy density was ~1071 gm/cm3 • Today dark energy comprises 73%, in 10 billion years it will be 96%. Ten billion years ago it was 9%. Why do we live at a time when dark energy was in transition. This violates the temporal cosmological principle but this need not be surprising.• If dark energy was zero, then it would always be zero…present era would not be special.

• Dark Energy should modify planetary orbits since it alters the apparent acceleration around any mass distribution in space. Solar system measurements set limits of vac ~10-18 gm/cm3, 11 orders away from being interesting!

The Angular Power Spectrum

• CMB temperature anisotropy is very close to Gaussian; thus, its spherical harmonic transform, alm, is also Gaussian.

• Since alm is Gaussian, the power spectrum:

completely specifies statistical properties of CMB.

*lmlml aaC =



Spherical Harmonics

• Orthogonal set of solutions to Laplace’s Equation in Spherical Polar Coordinates

• Solutions are products of trig functions and Legendre Functions.

• Any spherical map can be expanded as a series of spherical harmonics







Lowest Spherical harmonic Basis Functions

Order

0,0

(0,1), (1,0)

etc QuickTime™ and a decompressor


WMAP 3-yr Power Spectrum

Low multipole data consistent with equal power on all scales. Higher moment and peak at l=200 due to acoustic resonance of early universe. See later for more detail.

What CMB MeasuresA

mpl

itud

e of

tem

pera

ture

flu

ctua

tion

s at

a g

iven

sca

le, l

400 80020040 10010Multipole moment l~ Small scalesLarge scales

Ang.Diam. Distance

Baryon-to-photon Ratio

Mat-to-Radiation Ratio

ISW

Clustering and Structure Formation

• Inflation imprints quantum fluctuations on all scales

• Degree scale fluctuations are causally linked and grow by acoustic resonance in the plasma prior to recombination

• Fluctuations (few parts in 105) have gravitational potential energy depth ~ 3. 1011g-meters (equivalent to a valley 3. 1011 meters deep on the surface of the earth!)

• Gravity acts to drive clustering, structure formation

• Pressure on the plasma prevents adequate clustering unless there is much more non-interacting matter: dark matter

• March 2008 WMAP data gives data shown here.



QuickTime™ and aYUV420 codec decompressor


Galaxies in Redshift Space (2dF survey)



Dark Energy and Cosmic Expansion History

• See paper Cosmic Expansion History ..E V Linder Astro-ph 0507263 and Cosmic Structure Growth and Dark EnergyAstro-ph 0305286

• Dark energy influences the growth of structure which can be detected as an anisotropy in the galaxy two point correlation function in red shift space

• Nature 06555 Guzzo et al: Test of Cosmic Acceleration

• http://map.gsfc.nasa.gov/• http://background.uchicago.edu/~whu/• http://background.uchicago.edu/~whu/physics/tour.html• http://www.sciencedirect.com/science?

_ob=ArticleURL&_udi=B6VNJ-4M877RV-2&_user=554529&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000028118&_version=1&_urlVersion=0&_userid=554529&md5=257778da9d66f321cab08a3388bc45a4

CMB Physics

http://map.gsfc.nasa.gov/

http://background.uchicago.edu/~whu/

http://background.uchicago.edu/~whu/physics/tour.html

Physics of CMB Anisotropy

• SOLVE GENERAL RELATIVISTIC BOLTZMANN EQUATIONS TO THE FIRST SOLVE GENERAL RELATIVISTIC BOLTZMANN EQUATIONS TO THE FIRST ORDER IN PERTURBATIONSORDER IN PERTURBATIONS

Radiation Dominated Universe

Friedmann Equation

Separate density into radiation term plus matter term

Because radiation term exceeds matter term at small a, ignore matter term.

€

˙ a 2

a2−

8πGρ

3=

−kc 2

a2

€

=0,m

a03

a3+ ρ 0,rad

a04

a4

€

˙ a 2 −8πGρ 0a0

4

3a2= −kc 2

In young universe kc2 is negligible. Ignore this term.

Using

Then

Expansion rate tends to infinity as density goes to infinity

€

0a04 = ρa4

€

˙ a

a=

8πGρ

3

⎡ ⎣ ⎢

⎤ ⎦ ⎥

1

2

Temperature Evolution

• Radiation density= energy density/c2

• Energy density = 4/c.T4

• Hence

• But aT=constant since kT~hf~1/ Photons stretched same as universe. a~1/T~.

• Hence€

˙ a

a=

8πGρ

3

⎡ ⎣ ⎢

⎤ ⎦ ⎥

1

2=

8πG.4σT 4

3c 3

⎡

⎣ ⎢

⎤

⎦ ⎥

1

2

€

˙ a

a=

˙ T

T= T 2 32πGσ

3c 3( )1

2

Integrate wrt time

T=1.5 x 10-10 t-1/2

Since aT=constant=a0T0

=1.8 x 10-10 t1/2

Scale size relative to today

€

T t( ) =3c 3

128πGσ

⎛

⎝ ⎜

⎞

⎠ ⎟

14

⋅ t− 1

2

€

a(t) =128πGσT0

4

3c 3

⎛

⎝ ⎜

⎞

⎠ ⎟

1

4

⋅ t1

2

Assignment 1: deadline end of week 3

1. CosmologyA) Explain the concepts of angular size distance, luminosity distance and red shift

distance in cosmologyB) Use the Javascript cosmology calculator on Ned Wright’s homepage to make plots

of angular size distance and luminosity distance as a function of red shift for sources from z=0 to z=100. Use the latest cosmological parameters from the WMAP mission. Take care in choosing axes.

C) Discuss why these quantities do not obey the laws of Euclidean geometry.

2. Scale Size of Universe Plot the scale size of the universe as a function of time in four epochs

a) inflationary epochb) the radiation dominated erac) the matter dominated erad) the dark energy dominated era

Literature Review: Review one of the latest 5 year results papers from the WMAP mission, or any recent letter on cosmology on Astro-ph. Please Include a copy of the letter you reviewed to assist in marking.

Assignment 2: deadline end of week 5

1. Cosmic Microwave Background• Explain the meaning of spherical harmonic decomposition of an imageb) Explain the significance of the peaks in the spherical harmonic

deccomposition of the WMAP datac) Bonus marks challenge: Use spherical harmonic decomposition in

Mathematica or Matlab to decompose a photograph. Present the photograph and its decomposition. (you can get full marks without this question but this will confirm your understanding)

2. Supernovaea) Explain the difference between Type 1a supernovae and other supernova

types.b) Explain the nuclear physics leading to gravitational collapse.c) Investigate the physics of the detonation.

3. Literature reviewFind a letter relevant to either question 1 or 2 and write a I page review

following the Literature Review guidelines. Please Include a copy of the letter you reviewed to assist in marking.

Assignment 3 deadline 1 week after last lecture

1. Degenerate mattera) Use 5 or 6 bullet points to outline the derivation of the Chandrasekar limit of white dwarfsb) What factors influence the limiting mass of white dwarfs and neutron stars. 1 paragraph or 5-6 bullet points

2. Binary Neutron StarsCompare the gravitational wave luminosity of a neutron star-neutron star binary in a circular orbit of 50km separation with the electromagnetic luminosity of the system if one star has a temperature of 105K and the other has a temperature of 106K.

3. Accreting black hole binaryA 10 solar mass black hole increases its mass by 0.1Msun over 10 years by accretion from a 20Msun O supergiant in a 1 year circular orbit. How much gravitational potential energy has been lost. If about the same amount of energy is radiated from the system through an accretiondisk anf jet, make an order of magnitude estimate of its average electromagnetic luminosity in Watts. (this is a simple question requiring careful handling of big numbers and simple physics)

Literature review: Find a recent letter on gamma ray bursts and write a 1 page review. Please Include a copy of the letter you reviewed to assist in marking.

Third Year Astrophysics 2008 Acknowledgements Pictures from Ned Wright’s Home Page Pictures from...

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