A brief history of cosmology

1. Basic concepts spatial extent finite (with edges) finite (unbounded) infinite our location Earth at centre Sun at centre solar system near centre solar system far from centre no centre past and future both finite(creation, future destruction) both infinite(no beginning, no end) finite past, infinite future dynamics static expanding cyclic A brief history of cosmology PHY306

2. Early ideas: astronomy • Clearly understood concepts in Greek and Hellenistic astronomy • shape and size of the Earth (Eratosthenes, BC 276-197) • size and distance of the Moon (Aristarchos, BC 310-230) • Sun is much larger than Earth (Aristarchos) • exact value was wrong by a large factor: method sound in principle, impossible in practice! • Ideas raised but not generally accepted • Earth rotates on its axis (Heraclides, BC 387-312) • Sun-centred solar system (Aristarchos) PHY306

3. Early ideas: cosmology • Aristotle/Ptolemy • Earth-centred, finite, eternal, static • Aristarchos/Copernicus • Sun-centred, finite, eternal, static • At this time, little observational evidence for Sun-centred system! PHY306

4. Renaissance • Birth of modern science • scientific method • Galileo • better observations • Tycho, Galileo • development of mathematical analysis • Kepler, Galileo, Newton • Newtonian cosmology PHY306

5. Newtonian Cosmology • Newton’s Philosophiae Naturalis Principia Mathematica, 1687 • Newtonian gravity, F = GMm/r2, and second law, F = ma • Approximate size of solar system (Cassini, 1672) • from parallax of Mars • Finite speed of light (Ole Rømer, 1676) • from timing of Jupiter’smoons • No distances to stars • No galaxies PHY306

6. Newtonian Cosmology • Newton assumed a static universe • Problem: unstable unless completely homogeneous • Consider mass m on edge of sphereof mass M and radius r • mass outside sphere does notcontribute (if sphericallysymmetric) • mass inside behaves like central point mass • if there exists an overdense region,everything will fall into it PHY306

7. Olbers’ Paradox • Named for Wilhelm Olbers, but known to Kepler and Halley • Consider spherical shell of radius r and thickness dr • Number of stars in this shell is 4πr2n dr, where n is number density of stars • Light from each star is L/4πr2, therefore light from shell is nL dr, independent of r • therefore, in infinite universe, night sky should be infinitely bright (or at least as bright as typical stellar surface – stars themselves block light from behind them) • Why is the sky dark at night? PHY306

8. Light is absorbed by intervening dust suggested by Olbers doesn’t work: dust will heat up over time until it reaches the same temperature as the stars that illuminate it (I’m not sure 17th century astronomers would have realised this) Universe has finite size suggested by Kepler this works (integral is truncated at finite r) but now Newtonian universe will definitely collapse Universe has finite age equivalent to finite size if speed of light finite light from stars more than ct distant has not had time to reach us (currently accepted explanation) Universe is expanding effective temperature of distant starlight is redshifted down this effect not known until 19th century (does work, but does not dominate (for stars) in current models) Resolution(s) Olbers + Newton could have led to prediction of expanding/contracting universe PHY306

9. Further developments • James Bradley, 1728: aberration • proves that the Earth orbits the Sun • also allowed Bradley to calculate the speed of light to an accuracy of better than 1% • Friedrich Bessel, 1838: parallax • distances of nearby stars • a discovery whose time had come: 3 good measurements in the same year by 3 independent people, after 2000 years of searching! • Michelson and Morley, 1887: no aether drift • the speed of light does not depend on the Earth’s motion PHY306

10. We know speed of light distance to nearby stars the Earth is at least several million years old Our toolkit includes Newtonian mechanics Newtonian gravity Maxwell’s electromagnetism We don’t know galaxies exist the universe is expanding the Earth is several billion years old We are worried about conflict between geology and physics regarding age of Earth about to be resolved lack of aether drift State of Play ~1900 PHY306

11. Relativity • Principle of relativity • not a new idea! • Basic concepts of special relativity • …an idea whose time had come… • Basic concepts of general relativity • a genuinely new idea • Implications for cosmology PHY306

12. Relativity • “If the Earth moves,why don’t we get leftbehind?” • Relativity of motion(Galileo) • velocities are measured relative to given frame • moving observer only sees velocity difference • no absolute state of rest (cf. Newton’s first law) • uniformly moving observer equivalent to static PHY306

13. Principle of relativity physical laws hold for all observers in inertial frames inertial frame = one in rest or uniform motion consider observer B moving at vx relative to A xB = xA – vxt yB = yA; zB = zA; tB = tA VB = dxB/dtB = VA – vx aB = dVB/dtB = aA Using this Newton’s laws of motion OK, same acceleration Newton’s law of gravity OK, same acceleration Maxwell’s equations of electromagnetism c = 1/√μ0ε0 – not frame dependent but c = speed of light – frame dependent problem! Relativity PHY306

14. Michelson-Morley experiment • interferometer measures phase shift between two arms • if motion of Earth affects value of c, expect time-dependent shift • no significant shift found PHY306

15. Basics of special relativity • Assume speed of light constant in all inertial frames • “Einstein clock” in which light reflects from parallel mirrors • time between clicks tA = 2d/c • time between clicks tB = 2dB/c • but dB = √(d2 + ¼v2tB2) • so tA2 = tB2(1 – β2) where β = v/c • moving clock seen to tick more slowly, by factor γ = (1 – β2)−1/2 • note: if we sit on clock B, we see clock A tick more slowly stationary clock A d moving clock B dB vt PHY306

16. Basics of special relativity • Lorentz transformation • xB = γ(xA – βctA); yB = yA; zB = zA; ctB = γ(ctA – βxA) • mixes up space and time coordinates  spacetime • time dilation: moving clocks tick more slowly • Lorentz contraction: moving object appears shorter • all inertial observers see same speed of light c • spacetime interval ds2 = c2dt2 – dx2 – dy2 – dz2 same for all inertial observers • same for energy and momentum: EB = γ(EA – βcpxA); cpxB = γ(cpxA – βEA); cpyB = cpyA; cpzB = cpzA; • interval here is invariant mass m2c4 = E2 – c2p2 PHY306

17. The light cone • For any observer, spacetime is divided into: • the observer’s past: ds2 > 0, t < 0 • these events can influence observer • the observer’s future: ds2 > 0, t > 0 • observer can influencethese events • the light cone: ds2 = 0 • path of light to/fromobserver • “elsewhere”: ds2 < 0 • no causal contact PHY306

18. Basics of general relativity astronaut in inertial frame astronaut in freefall frame falling freely in a gravitational field “looks like” inertial frame PHY306

19. Basics of general relativity astronaut in accelerating frame astronaut under gravity gravity looks like acceleration (gravity appears to be a “kinematic force”) PHY306

20. Basics of general relativity • (Weak) Principle of Equivalence • gravitational acceleration same for all bodies • as with kinematic forces such as centrifugal force • gravitational mass  inertial mass • experimentally verified to high accuracy • gravitational field locally indistinguishable from acceleration • light bends in gravitational field • but light takes shortest possible pathbetween two points (Fermat) • spacetime must be curved by gravity PHY306

21. Light bent by gravity • First test of general relativity, 1919 • Sir Arthur Eddington photographs stars near Sun during total eclipse, Sobral, Brazil • results appear to support Einstein (but large error bars!) PHY306 photos from National Maritime Museum, Greenwich

22. Light bent by gravity lensed galaxy member of lensing cluster PHY306

23. Conclusions • If we assume • physical laws same for all inertial observers • i.e. speed of light same for all inertial observers • gravity behaves like a kinematic (or fictitious) force • i.e. gravitational mass = inertial mass • then we conclude • absolute space and time replaced by observer-dependent spacetime • light trajectories are bent in gravitational field • gravitational field creates a curved spacetime PHY306

24. Curved spacetime and implications for cosmology • General Relativity implies spacetime is curved in the presence of matter • since universe contains matter, might expect overall curvature (as well as local “gravity wells”) • how does this affect measurements of large-scale distances? • what are the implications for cosmology? PHY306

25. Curved spacetime • Two-dimensional curved space: surface of sphere • distance between (r,θ) and (r+dr,θ+dθ) given by • ds2 = dr2 + R2sin2(r/R)dθ2 • r = distance from poleθ = angle from meridianR = radius of sphere • positive curvature • “Saddle” (negative curvature) • ds2 = dr2 + R2sinh2(r/R)dθ2 • (2D surface of constant negative curvature can’t really be constructed in 3D space) Nick Strobel’s Astronomy Notes PHY306

26. 3D curved spacetime • Robertson-Walker metric • ds2 = −c2dt2 + a2(t)[dx2/(1 – kx2/R2) + x2(dθ2+sin2θ dφ2)] • note sign change from our previous definition of ds2! • a(t) is an overall scale factor allowing for expansion or contraction (a(t0) ≡ 1) • x is called a comoving coordinate (unchanged by overall expansion or contraction) • k defines sign of curvature (k = ±1 or 0),R is radius of curvature • path of photon has ds2 = 0, as before PHY306

27. Implications for cosmology • comoving proper distance (dt = 0) between origin and object at coordinate x: • for k = +1 this gives r = R sin−1(x/R), i.e. r ≤ 2πR • finite but unbounded universe, cf. sphere • for k = −1 we get r = R sinh−1(x/R), and for k = 0, r = x • infinite universe, cf. saddle • for x << R all values of k give r ≈ x • any spacetime looks flat on small enough scales • this is independent of a • it’s a comoving distance PHY306

28. Implications for cosmology • cosmological redshift: change variables in RW metric from x to r: • ds2 = −c2dt2 + a2(t)[dr2 + x(r)2dΩ2] • for light ds = 0, so c2dt2 = a2(t)dr2, i.e. c dt/a(t) = dr(assuming beam directed radially) • suppose wave crest emitted at time te and observed at to first wave crest next wave crest PHY306

29. Implications for cosmology • Thenbut if λ << c/H0, a(t) is almost constant over this integral, so we can write i.e. PHY306

30. Implications for cosmology • So expanding universe produces redshift z, where • Note: • z can have any value from 0 to ∞ • z is a measure of te • often interpretz using relativistic Doppler shift formula but note that this is misleading: the object is not changing its local coordinates PHY306

31. Implications for cosmology • Conclusions • in general relativity universe can be infinite (if k = −1 or 0) or finite but unbounded (if k = +1) • universe can expand or contract (if overall scale factor a(t) is not constant) • if universe expands or contracts, radiation emitted by a comoving source will appear redshifted or blueshifted respectively PHY306