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Relativistic Cosmology 1. Introduction & history of ideas 1500-1700 Stars Solar System, sun as a star: Copernicus (1473-1543) Galileo (1564-1642), Kepler (1571-1630), Newton (1642-1727) … Steady State: prevention of gravitational collapse …..Newton's gaffe (missing Gauss Theorem) Isotropy and Homogeneity assumed by Leibnitz (1646-1714) to apply to star distribution, infinite universe 1700-1850 Our galaxy as a rotating disc John Lambert (1728-1777). Galaxies Thomas Wright (1711-1786), Immanuel Kant (1724-1804), Nebulae = Galaxies, homogeneity & isotropy at Galaxy distribution level William + John Herschel (1738-1822): …. no nebulae seen in region of galactic disc ?! …Richard Proctor (1837-1888) ... because dust obscures them. Collapse prevented by angular momentum: Laplace (1749-1827) THESE ARE NOT A COMPLETE SET OF LECTURE NOTES. USE THIS SPACE! YOU WILL CERTAINLY NEED TO ADD DIAGRAMS AND OTHER ANNOTATIONS EXPLANATIONS NOTE OF ILLUSTRATIONS COMMENTS AS TO DETAILS NOT EXAMINABLE 1850-1918 Relativity Finite universe without boundaries: Riemann (1826-1866) curved space Ernst Mach (1838-1916) 1883: all motion relative Einstein 1905 Special Relativity, 1915: General Relativity with cosmological constant (quote Einstein…) see: http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/General_relativity.html 1919-1939 Expanding Universe better telescopes + benefit of spectroscopy: Harold Shapley 1919 determined size of our galaxy and our position in it from the distribution of globular star clusters Edwin Hubble (1889-1953) 1925 first determined distances to nebulae such as M31, proving they were separate galaxies. Hubble 1929: Hubble's Law: the universe is expanding Shift of ideas from Steady State (but not without a fight - Fred Hoyle) towards Big Bang 1945-1995 Big Bang Physics 1964: microwave technology (inc cryogenics): CMBR 4K background radiation - direct relic of Big Bang, redshifted x 104 Cosmology <-> Stellar Physics: Abundance of Helium <-> Big Bang nucleosynthesis Age of Universe approx 10 14 Gyr (Stellar Physics gave way) 1995- Quantitative Cosmology ‘Standard Model’ 2003 Hubble Parameter H= 70 +/- 4 km/s Mpc-1 Age of Universe 13.70.3 G yr Structure in CMBR Spatial Curvature 0 4% Regular Matter (well understood - baryons) 23% Dark Matter (stuff we don’t understand) 73% Dark Energy (aka Cosmological Constant – little idea) 2004: There remain dissenting voices! Galaxy clustering claimed more consistent with Dark Energy replaced by Dark Matter. 2. Size, Expansion and Age Cosmological Distances 1 pc = 3.261 ly = 3 1016 m 1.3 pc Sun to alpha Centauri 0.3 kpc thickness of Milky Way 12 kpc radius of Milky Way 0.8 Mpc distance to M31 50 Mpc superclusters 100++ Mpc largest filaments, walls 4000 Mpc Hubble Radius Age of Universe Heavy isotope relative abundances Especially 238U : 235U, 10-20 Gyr (including astrophysics) White dwarf stars: t0 > 12.7 Gyr (+/-0.7) Globular Cluster ages: CMBR : 11-16 Gyr 13.7 +/- 0.3 Gyr Hubble Parameter (“Hubble Constant”) v = H(t) r H0 = H(now) Hubble radius (where v c ) rH c H H0 = 100 h km/s Mpc-1 Hubble Telescope 1995, Cepheid plots: h = 0.7 +/- 10% H= 70 +/-7 km/s Mpc-1 Red-shift measurement: f emitted observed 1 v / c 1 z f observer emitted 1 v2 / c2 Cosmology interpretation: expansion of Universe between emission and observation Linearity of Hubble’s Law 1. Experimental fact 2. Independence of Observer The expansion law should look the same from all (co-expanding) observers. This uniquely selects a linear law. (Blackboard…). 3. Relation to the Scale Factor Cosmological expansion interpretation: observed S0 1 z emitted S (t emitted ) S(t) = Scale Factor of Universe at time t; S0 = value now Expansion rate H (t ) S (t ) S (t ) and H 0 S0 S 0 Deceleration Parameter S(t ) q(t ) S (t ) H (t ) 2 and q0 S0 S0 H 0 3. The Metric of Space and Time Euclidean space Metric d 2 dx 2 dy 2 dz 2 (dx, dy, dz ) coordinate separations between two nearby points d = distance between them: its value is independent of coordinate system used even if the expression looks different. For example we can express the same metric in spherical polars as d 2 dr 2 r 2 d 2 r 2 sin 2 d 2 where we have changed the coordinates but not the geometry they describe. Curved Space (constant curvature) The two-sphere is the two dimensional spherical surface of the three (dimensional)-ball x2 y2 z 2 S 2 2 2 2 2 x y z S . Viewed as a two dimensional surface at constant radius in three dimensions, its metric can be given as d 2 S 2 (d 2 sin 2 d 2 ) where S is just the (fixed) radius. However we can also ask what this looks like from the point of view of someone living in the surface (without awareness of more than two dimensions). From the pole ( 0 ), ‘radial’ distance away is given by integrating d at constant leading to ‘radius’ S . One can also trace out a circumference of points at that distance, and the length of this is given by integrating d w.r.t. at constant giving C 2S sin 2S sin S which is not quite the same as 2 but agrees as 0 . S This mismatch fundamental to the space being curved. The 2 C 3 curvature is defined from the deficit as K lim 0 3 giving K 1 S 2 in this case (and for higher dimensional spheres also). History and convenience have led to a slightly different notation being widely favoured: define a different ‘radius’ from the circumference by C r S , 2 where sin and d cos d 1 2 d . Then you can express the metric of the 2-sphere as d 2 2 2 d S ( 2 d 2 ) 2 1 k where k 1 and is just circumference . 2S Now we have two ways to generalise. possibilities for k . The first is other k 0 is flat space, that is Euclidean space, slightly disguised by what amounts to plane polar coordinates. Restoring S r and setting k 0 gives d 2 dr 2 r 2 d 2 . k 1 is a hyperbolic space, the surface x 2 y 2 z 2 S 2 , where circumference grows faster than radial distance as you go out. (picture) The other generalisation is to three dimensional spaces (and higher if you like!), which just amounts to adding more angular coordinates. As we look out from point in three dimensions the direction “in the sky” is characterised by two angles and the corresponding metric is given by d 2 d 2 S 2 2 d 2 sin 2 d 2 2 1 k [The angle is now just angle from North and NOT to be confused with our usage above for sin 1 . That confusion is the price we pay for having understood the two sphere from its embedding in three flat dimensions.] The 3-sphere, k 1 , is a very serious candidate for the geometry of space in our universe, and it would mean that the further away you look, , the more: a circumference drawn around at that distance would be smaller than 2 and the area at that distance would be less than 4 2 . Ultimately when you reached our ‘antipode’ both shrink to zero, and you have discovered there is only a finite volume to the Universe (at fixed cosmological time – see later). Locally it is hard to tell which value of k we have. In terms of r S the metric is given by dr 2 d 2 r 2 d 2 sin 2 d 2 2 1 Kr so it is the curvature K k 2 which we need to detect, which is S 2 hard to do for Kr 1 . The furthest structure we can see today is that of the CMBR, and its structure implies that KrH 2 0.02 . (2003) Here rH c is the Hubble radius (next lecture): the result means H that looking so far out as to be seeing back to the beginning of time, we can scarcely see any curvature of space. The Universe looks flat. Special Relativity: Minkowski space-time Metric ds 2 c 2 d 2 d 2 c 2 dt 2 dx 2 dy 2 dz 2 (dt , dx, dy, dz ) time and space coordinate separations between two events ds = space-time interval between events: this is independent of frame of reference & coordinate system. Depending on the sign of ds 2 it is convenient to interpret it as Time-like separations: 0 ds 2 c 2 d 2 : d = proper time (interval) between events: the time difference in a frame of reference where they occur at the same place. Space-like separations: 0 ds 2 d 2 : d = proper distance between events: the distance between them in a frame of reference where they occur at the same time ds 0 corresponds to null or lightlike separation. Two events connected (or connectable) by a light signal have this separation. GR Cosmology: The Robertson-Walker Metric d 2 ds 2 c 2 dt 2 S (t )2 2 d 2 2 sin 2 d 2 2 1 k This metric (1934) can be constructed from the assumptions that: the universe is isotropic (in space) the universe is homogeneous (in space) spacetime is locally Minkowski-like and some ideas about time (to follow). It is clearly a combination of Minkowski-like time signature with a uniformly curved space, whose scale factor S (t ) varies with time but not space. The spatial curvature k varies with time, in S (t ) 2 magnitude but not in sign. The Cosmic Time coordinate used is constructed so that for a "Fundamental Observer" at fixed , , , ds 2 c 2 dt 2 , hence the time coordinate is proper time for such an observer. All the standard approaches to large-scale Relativistic Cosmology approximate galaxies as points at fixed , , . Then evidently the spatial separation between them at equal cosmic time simply inflates with the scale factor S (t ) , giving the Hubble flow (below). From the point of view of large-scale cosmology, the remaining problem is how S (t ) varies with time. Our Particle Horizon How much of the Universe today could have influenced us in the past? This can be finite back to the beginning of time, even for universes without positive curvature. The useful way to express the answer is as a value of comoving radius p (away from us), and this is determined by the condition that a light ray could have propagated from (time, space)= (t min , p ) to (t 0 ,0) , remembering the convention that t 0 means time now. d 2 0 all 1 k 2 along the ray path, which conveniently leads to the integral For a light ray, we must have ds 2 c 2 dt 2 S (t ) 2 t0 c dt S (t ) p d . 1 k 2 We will see later that for simple cosmologies we expect S (t ) t where the exponent obeys <1, then the time integral is finite. This then leads to a finite particle horizon in the flat and negatively curved cases. t min 0 In the positively curved case the "radial" integral has a maximum value sin 1 (1) = / 2 , and it is possible that we might see light which has traversed the Universe! Searches for this have been made for quasars, the negative result setting S 0 400h 1 Mpc. (only about 13% of the Hubble Radius)