Cosmology: Finite or Infinite Universe? Relativity & Cosmological Models

1. Cosmology • Finite or infinite Universe? • Relativity • Cosmologicam models • The Big Bang model • The anthropic principle

2. Finite or infinite Universe? Newton: tries to apply his universal gravitation theory to the Universe as a whole • If finite Universe → everything should collapse towards the center Unless the Universe is rotating – But with respect to what? • If infiniteUniverse → different ways to tackle the problem lead to different solutions → Newton turned to something else Gausstheorem → the only possible infinite and uniform Universe is an empty Universe Rem: but pursuing his line of reasoning, Newton might have predicted the expansion of the Universe!!!

3. Finite or infinite Universe? - 2 The Olbers paradox Why is the sky dark at night? If the Universe is infinite and homogeneous → all lines of sight should encounter the surface of a star → the sky should be as bright as the surface of the Sun Olbers tried to invoke absorption of light (by dust) Conservation of energy → Olbers’solution does not work as dust would heat up until it radiates as much as the stars Heinrich Olbers (1758-1840)

4. Relativity Maxwell equations and invariance 1864: Maxwell publishes his equations of electromagnetism → electromagnetic waves propagating in vacuum at speed c ≈ 300 000 km/s Problem: the Maxwell equations are not invariant under the Galileo transformation (change of inertial frame): where vR is the velocity (along the x axis) of inertial reference frame R′ with respect to inertial reference frame R J. C. Maxwell (1831-1879)

5. Relativity - 2 Eather and speed of light In which medium do electromagnetic waves propagate? Physicists postulated that they propagate thougt eather, a still unknown medium… 1887: Albert Michelson tries to detect the motion of Earth with respect to eather by measuring the speed of light in perpendicular directions vEarth/c≈ 10–4→ he builds a very accurate interferometer → negative result: c is the same in all directions

6. Relativity - 3 Special relativity Two problems, a single solution… 1905: Albert Einstein develops a theory based on the fundamental postulate that: Whatever the motions of the source and observer, the latter will always measure the same value for the speed of light in vacuum → c is a fundamental constant (c = 299 792 458 m/s) → theory of special relativity Albert Einstein (1879-1955)

7. 1) M h 2) R vΔtR Relativity - 4 Time dilation Consider an observer M in motion at a speed v with respect to another observer R at rest The two observers measure the time it takes for light to travel the same distance 1) 2)

8. Relativity - 5 Equivalence principle • In vacuum, the acceleration of a test body is independent of its mass (Galileo) → inertialmassMI = gravitationalmass MG (Newton) → the (local) effects of a gravitational field are equivalent to those of an acceleration of the observer’s reference frame (Einstein) → first step towards the theory of general relativity

9. Relativity - 6 General relativity Gravitation↔ curvature of4 dimensional space-time → geometric representation of gravitation • Newton: action at a distance by an unknown mechanism • Einstein: space-time deformation Bodies moving under the effect of gravitation follow geodesics of a curved space-time • Same results as Newton in weak gravitational fields • Departures growing with the field intensity

10. Relativity - 7 Some predictions of general relativity • Advance of Mercury’s perihelion • Curvature of light rays close to massive objects – eclipse of 1919 – gravitational mirages • Gravitational waves – binary pulsars: increase of orbital period – direct detection • Gravitational time dilation – measured on Earth + redshift on the surface of compact objects (Earth: 10–9 – white dwarf: 6% –neutron star: 30%)

11. Cosmological models General relativity allows studying the structure and evolution of the Universe as a whole → cosmology Geometry of space Space can have positive, negative or no curvature • Positive curvature: finite but unlimited space • Negative curvature: infinite space • No curvature: euclidean infinite space

12. Cosmological models - 2 The cosmological principle To be able, from observations of our portion of the Universe, to test models representing the Universe as a whole, we must adopt the hypothesis that our region is representative of the Universe → one assumes that any sufficiently large region of the Universe is representative of the Universe as a whole This is the cosmologicalprinciple (it is necessary to make cosmology a science) N.B. In practice, sufficiently large = larger than 500 millions LY

13. Cosmological models - 3 Friedmann-Lemaître equation • Consequence of the cosmological principle: The Universe is homogeneous at large scale • One also assumes it is isotropic at large scale → one obtains a simple from of Einstein’s equations of general relativity: Friedmann-Lemaître equation R = scale factor ρ = density of matter k = curvature parameter

14. Cosmological models - 4 Cosmological constant In 1917, Einstein realizes that his equations have no static solution The expansion of the Universe had not been discovered yet → he modified his equations by adding a terme containing a cosmologicalconstant Λ: By choosing an adequate value for Λ, one can obtain a static solution After the discovery of the expansion of the Universe, Einstein considered the cosmological constant as the biggest blunder of his career

15. Cosmological models - 5 Critical density Density parameter: • Ω0 > 1: spherical universe, closed • Ω0 = 1: parabolic (flat) universe, open Λ = 0 • Ω0 < 1: hyperbolic universe, open → matter density determines the fate of the Universe Best estimates: Ωm,0 ≈ 0.3 (m: visible + dark matter)

16. Cosmological models - 6 The Big Bang Universe in continuous expansion → if one goes back in time: • the scale factor R decreases • the density ρ increases → one reaches a point at which R → 0 and ρ→ ∞ → beginning (of the Universe, of space-time…) = Big Bang (Fred Hoyle, 1950s) Beginning → creation ??? → debate (more philosophical than scientific) Georges Lemaître

17. Cosmological models - 7 The steady state Universe The idea of a beginning disturbed some scientists → steady state theory (Gold, Bondi & Hoyle, 1948) based on the perfectcosmological principle: the Universe appears the same in all places and at all times However, galaxies move away from each other → continuouscreation of matter to keep a constant ρ (~ 1 H atom par m3 per billion years) Variant:quasi steady state (Hoyle, Burbidge & Narlikar, 1993): `minibangs´ Fred Hoyle

18. Cosmological models - 8 The cosmological microwave background (CMB) 1964: Penzias and Wilson try to measure the radio emission of the Milky Way → they discover an isotropic and non seasonal radiation → cannot come from the atmosphere, nor from the Milky way They get in touch with cosmologists Dicke and Peebles → interpreted as relic radiation from the first epochs of the Universe (CMB) (existence predicted by Gamov, black body spectrum by Dorochkevitch and Novikov) (success 1) Robert Wilson and Arno Penzias

19. Cosmological models - 9 Victory of the Big Bang The CMB is interpreted as a relic from an earlier stage of the Universe, much hotter (when matter was ionized, thus opaque), now cooled to 2.7 K due to the expansion of the Universe → knock down for steady state cosmology Their hypothesis to `save the day´: stellar radiation scattered by small `metallic sticks´ present in interstellar matter Problem: how to explain such a perfect isotropy? CMB spectrum (COBE)

20. The Big Bang model Creation of matter (t = 10−32 s ; T = 1026 K ; ρ = 1073 kg/m3) • Emergence of a `soup´ of quarks, electrons, photons, neutrinos • In principle, creation of particles – antiparticles pairs • How to explain that we only observe matter in the Universe? → one assumes an asymmetry: creation of 1 000 000 001 particles for 1 000 000 000 antiparticles (epicycle 1) • annihilation of all particle – antiparticle pairs → photons • the present-day matter is the tiny relic of that gigantic annihilation

21. The Big Bang model – 2 Formation of protons and neutrons (t = 10−4 s ; T = 1012 K ; ρ = 1017 kg/m3) • Quarks combine into nucleons • T is so high that transmutations proton → neutron balance transmutations neutron → proton → at start, number of neutrons Nn = number of protons Np • When T decreases, the energetically most favorable reaction dominates → the ratio Nn / Np goes down • When T = 1010 K, 4He becomes stable but still inaccessible since the intermediate stage 2H remains unstable → neutrons stay free and the ratio Nn / Np continues to decrease

22. The Big Bang model – 3 Formation of helium (t = 100 s ; T = 109 K ; ρ = 105 kg/m3) • 2H becomes stable → neutrons can bind with protons into 2H nuclei and are finally safe! • Then, 2H combine into 4He • At that time, the proportion is 1 neutron for 7 protons → 2 neutrons for 14 protons → one 4He nucleus for 12 1H nuclei → proportion in mass: 4/(4+12) = 25% → prediction confirmed by observations (success 2)

23. The Big Bang model – 4 End of primordial nucleosynthesis (t = 12 days ; T = 107 K ; ρ = 10−3 kg/m3) • Fusions 4He + 1H et 4He + 4He would give nuclei of atomic masses 5 and 8 • However, no stable nuclei have those masses → nucleosynthesis ends there (apart from a little 3He and 7Li) • In stars, that problem is circumvented by the triple α reaction • That reaction needs higher densities and longer times than available at that stage of the Big Bang → that solution is not available in cosmological nucleosynthesis

24. The Big Bang model – 5 Results of primordial nucleosynthesis • The abundances predicted by primordial nucleosynthesis models are sensitive to the baryonic matter density (protons + neutrons [+ e–]) → a single value of density must give rise to the observed cosmological abundances (success 3) • Everything fitted well until the WMAP results, which imply a 7Li abundance 2 to 3 times higher than observed in old stars → diffusion? (epicycle 2)

25. The Big Bang model – 6 The baryonic density To get the observed primordial abundance for the light elements, we need a well defined value of the baryonic density: Ωb,0 ≈ 0.05 – 0.06 This is higher than the value deduced from observations: Ωb,0(obs) ≈ 0.01 – 0.02 But lower than total mass estimates in galaxy clusters: Ωm,0 ≈ 0.3 → part of the dark matter (Ω0 ≈ 0.04) is baryonic (ordinary matter) But the largest part (Ω0 ≈ 0.24) would correspond to exoticmatter (ex:WIMPs) not discovered yet (epicycle 3)

26. The Big Bang model – 7 The cosmological constant Observations of high redshift supernovae suggest that the expansion of the Universe is accelerated → return of the cosmological constant: Λ0≈ 0.7 (interpreted as a kind od vacuum energy) (epicycle 4)

27. The Big Bang model – 8 Vacuum energy and expansion Why does the vacuum energy accelerate expansion? • The expansion rate depends on the mass–energy density • Expansion of space → the matter density decreases → the expansion rate progressively goes down • The energy density of vacuum is constant (does not decrease with expansion) → favors a constant expansion rate (when it dominates) → exponential expansion

28. • It sounds a bit like saying that, since the planetary orbits are close to circles, they must be perfect circles → is Plato back?… The Big Bang model – 9 The flat Universe • One would have: Ωm,0 ≈ 0.3 and Λ0≈ 0.7 → Ω0 + Λ0 ≈ 1 • Models with Ω + Λ = 1 are flat universes → mainly for philosophical reasons, most cosmologists think that, if curvature is close to zero, then it must be exactly zero: Ω0 + Λ0 ≈ 1 →Ω0 + Λ0 ≡ 1

29. The Big Bang model – 10 Inflation How to explain that the initial conditions select, among an infinity of models with any curvature, precisely the one with no curvature? 1981: Alan Guth proposes the theory of primordial inflation • Some theoreticians think that, at very high temperature (1026 K), the four fundamental forces are unified into a single one → there would be only one type of particle + a unifiedvacuum of energy density much larger than present vacuum This is this unified vacuum which would be responsible for the inflation phase (epicycle 5) Alan Guth

30. The Big Bang model – 11 Inflation • t < 10–33 s: photons dominate → fast expansion, progressively slowed down • t ~ 10–33 s: photon density < density of unified vacuum → inflation: very fast expansion, size of the Universe × ~1030 to1040 • t ~ 10–32 s: temperature goes below the unification value → photons dominate again → expansion slowed down again • During inflation, the huge expansion annihilates any pre-existing curvature → after inflation, the Universe is flat

31. The Big Bang model – 12 Origin of the cosmic microwave background • As long as T > 3000 K, matter is mainly ionized → its opacity is large (basically opaque) • When T < 3000 K, electrons and protons combine into hydrogen atoms → the opacity suddenly goes down → matter becomes transparent → photons propagate freely (decoupling) → their wavelength increases with the expansion of space → λ0 ~ 1000 λdecoupling → T0 ~ 1/1000 Tdecoupling ~ 3 K

32. The Big Bang model – 13 The age of the Universe The age of the Universe can be calculated from H0, Ω0 and Λ0 The age of the oldest stars in our Galaxy (globular clusters) amounts to ~ 13 billion years → any cosmological model predicting an age of the Universe < 13 × 109yrs is in conflict with stellar evolution models → the `new standard model´ (H0 = 72, Ω0 = 0.3, Λ0 = 0.7) is very close to the limit H0Ω0Λ0Age (109 yrs) 72 1.0 0.0 9.0 72 0.3 0.0 11.0 72 0.3 0.7 13.1 60 1.0 0.0 10.9 60 0.3 0.0 13.2 60 0.3 0.7 15.7

33. The anthropic principle Anthropic principle (greek anthropos = man) What consequences on the laws of physics can we deduce from the mere existence of mankind? Example: it is not a coincidence that the age of the Universe is a few billion years • if the Universe was much younger: Life and intelligence would not have enough time to develop → the Universe must be at least a few billion years old → this is the weakest version of the anthropic principle (`trivial´ version)

34. The anthropic principle – 2 Nucleosynthesis of nuclei heavier than helium • No stable nucleus of mass number 5 or 8 → primordial nucleosynthesis does not go over 4He → only stellar nucleosynthesis (triple α) allows to produce 12C 8Be highly unstable → generally disintegrates before 4He + 8Be → 12C But… That reaction is favored by the presence of an excited state of 12C whose energy nearly coincides with that of 4He + 8Be (resonance) The existence of that excited state at 7.68 MeV had been predicted by Fred Hoyle in 1953 from such considerations It was discovered soon afterwards by Dunbar, Pixley et al. (1953)

35. The strong anthropic principle The constants of physics have been adjusted to allow the existence of that excited state… … and, more generally, our existence! The anthropic principle – 3 The weak anthropic principle We exist, we are made of C, O,… → stellar nucleosynthesis could proceed further from 4He → the triple α reaction actually happens in stars → there exists an excited state of 12C at an energy close to 7.68 MeV

36. The anthropic principle – 4 Application of the weak anthropic principle The characteristic time for development of intelligent life is either: (1) much shorter than the Sun’s lifetime (2) of the same order of magnitude (3) much longer • If (1) we would probably have developed much earlier • (2) is an unlikely coincidence between totally unrelated phenomena* • (3) is thus the most likely hypothesis → there must be only few intelligent civilisations in the Galaxy * Might be the weak part of the argument

37. The anthropic principle – 5 Intelligent design There seems to be a number of coincidences linked to the values of the fundamental constants → this has led some to pretend that: the constants and laws of nature have been adjusted to allow our existence… … and even that the whole biological evolution with the appearance of complex structures, so well adjusted, could not result from chance… … but would be guided towards an aim (us, of course!) by some superior being… → intelligent design, pseudo scientific avatar of creationism

38. The anthropic principle – 6 Pertinence of intelligent design? 1. This is not a science, as a fundamental condition for any scientific theory is to be testable, thus refutable This is not the case of intelligent design since: whatever the result of an experiment, its proponents can argue that: `this is the will of the superior being´ → belief, not science!

39. The anthropic principle – 7 Pertinence of intelligent design? 2. Rather than searching for a rational explanation, one relies on a superior intelligence… QED… → this is just laziness, resignation of free thought 3. This is a complete lack of modesty since it is based on the belief that, if we cannot explain something now, it will never be explanable… Who do we think we are?

40. « Two things are infinite: the universe and human stupidity; and I’m not sure about the universe!  » Albert Einstein The anthropic principle – 8 Why intelligent design? • First mankind’s trauma (Copernicus): Earth is no more at the center of the Universe • Second trauma (Darwin): man is just an animal among others… and the (partially) random product of evolution → hard to admit for our ego!

41. The teacher… The anthropic principle – 9 THE END