Third Year Astrophysics 2008

1. 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.

2. 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 Inabsence of external forces objects follow free fall trajectories that are geodesics -(shortest paths) in space time

3. Solutions for Expanding Universes Repulsive force Slice through 2D, model universe on surface of sphere. • 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 L • Hubble discovered expansion • a bad day for Einstein • Einstein came close to predicting that the universe is expanding Gravity A static universe requires repulsive force to exactly offset the gravitational attraction which attempts to contract space.

4. 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.

5. Friedmann Equation • Solve equation of motion for unit mass of the universe (F=ma) • First find • By inspection static universe is impossible unless r=0 or another term is present to cancel the G term. • Integrate above equation to obtain the 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 r as mass-energy density • Interpret k as curvature parameter • 1/2v2 - GM(r)/r = k. const • For k=0, KE + PE = 0

6. Critical density • From Friedmann Equation • solve for value of r for which k=0 • Answer: • This is the critical density rcrit for which W = 1 • W = r/rcrit • We will return to these concepts later on.

7. 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? Einstein:”the greatest blunder of my life” No blunder!

8. 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

9. Galaxy Spectroscopy to Measure Hubble Flow Stellar Spectrum • 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 Intensity Spectrum courtesy Bob Kirshner Sodium Magnesium Galaxy Spectrum Calcium Wavelength l • Spectra of nearby and distant galaxies • Nearby galaxy travels at 261 km/s • Distant galaxy travels at 6,400 km/s

10. 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. Low mp anisotropy seen by COBE Speed:dipole anisotropy

11. 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.

12. 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.

13. Hubble Law and Space-time diagrams Time Space • 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

14. Spacetime diagram for Low Density Universe Past light cone 45 degrees 1 light year per year, v=c • Uniform expansion (low density universe) Expansion rate exceeds c: photon velocity zero Expansion speed exceeds c (photons receeding)

15. Cosmological Spacetime vs Special Relativistic Spacetime Surfaces of constant proper time 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

16. Our worldline All events we currently observe Worldlines of co-moving objects Photon velocity effectively falls to zero Photons initially receeding make transition to approaching Current distance to particle horizon Co-moving distance

18. Example of Change of Representation

19. End Week 1

20. Hubble Law Interpretation • 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 distanceis roughly 3ct0 • Event horizon: distance light can travel from time t to t=infinity. • 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

21. If redshifts were purely relativistic supernova brightness would follow lower curve

22. Measuring Distance • Angular size distance DA: • Luminosity Distance DL:

23. 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%

24. 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 = 3H2/8G For H ~ 70km/s/Mpc, crit ~ 6 protons/ cubic meter or 1011Msun/cubicMpc Ratio : crit = 

25. possible universes m=0.27, V=0.73 Big crunch in 80 Gyr v=1 0=1 0=0 0=2

26. M = 0 Perlmutter 1993 Open M < 1 M = 1 Closed M > 1 fainter redshift - 14 - 9 - 7 today billion years Mean distance between galaxies Time

27. 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

28. Spatial Curvature Spatial curvature depends on value of crit =     Note dark energy can prevent collapse for

29. Geometry Parallel Lines Converge Closed Space r>rcrit W0>1 Parallel Lines Remain Parallel Flat Space r=rcrit W0=1 Open Space r<rcrit W0<1 Parallel Lines Diverge Figures from Universe by William Kaufmann

30. 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

31. 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

32. 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

33. Example of Change of Representation

34. 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

35. 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/2G)1/2 ,vac =1091 gm/cc Observed vacuum energy < 10-29 gm/cc Hence vacuum energy is 10120 times too big

36. 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.

37. 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= 0.5matter. Then total = 1.5 matter, and P=0.5matter/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.

38. Reference:Gary Watson Astro-ph/0005003 Brown University 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

39. 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

40. Quantum Fluctuations time • 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 Fluctuation lightcones as seen on the sky Future lightcones of fluctuation events

41. Inflaton Potentials Toy 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.

42. 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!

43. 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.

44. 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

45. Lowest Spherical harmonic Basis Functions Order 0,0 (0,1), (1,0) etc

46. 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.

47. What CMB Measures Ang.Diam. Distance ISW Baryon-to-photon Ratio Mat-to-Radiation Ratio Amplitude of temperature fluctuations at a given scale, l~p/q 10 40 100 200 400 800 Multipole momentl~p/q Large scales Small scales

48. 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.