Chapter 23

1. Chapter 23 Cosmology (the study of the big and the small)

2. Fundamental Particles Primer (the small) • Hadrons – interact via strong force and have internal structure (made of quarks) • Baryons – massive (ex. proton and neutron) • composite fermions • Mesons – less massive • composite bosons • Quark combination for: • Proton = uud • Neutron = udd • Leptons – do not interact by strong force but obey Pauli exclusion (ex. electron and neutrino) Photons – massless with energy E = hc/λ Quark properties

3. The Cosmological Principle (the big): the universe is homogeneous and isotropic on sufficiently large scales The universe looks pretty much like this everywhere – “walls” and “voids” are present but no larger structures are seen…. It follows that the Universe has no “edge” or center. But is the Universe the same at all times?

4. Recall universal expansion: recession velocity = Ho x distance (Hubble Law) The cosmological principle does not imply that the Universe is constant at all times. This was once thought to be the case - Steady State Universe Universal expansion points to a beginning of the Universe and implies that the Universe is changing over time… The Hot Big Bang model – Universe has expanded from an initial hot and dense state to its current cooler and lower-density state

5. First Cosmological Observation: The night sky is dark! (Olbers’s Paradox) • Every line of sight would eventually hit a star in an infinite and eternal Universe and the sky would be always bright •  Number of stars increases by r2 for each shell • brightness decreases by r2 for each shell • Brightness per shell is a constant! • The fact that the sky is not uniformly bright indicates that either • the Universe has a finite size • and/or • the Universe has a finite age

6. Second Cosmological Observation: Hubble Law Indicates homogeneous, isotropic expansion where r(t) = a(t) ro ro = r(to) is the separation between two points at the current time to a(t) is a dimensionless scale factor Distance between two points increases with velocity Which takes form of the Hubble law v(t) = H(t) r(t) where H(t) is the Hubble parameter and Ho = H(to) is the Hubble Constant Thus, the primary resolution to Olbers’s paradox is that the Universe has a finite age (stars beyond the horizon distance ro ~ c/Ho are invisible to us because their light hasn’t had time to reach us).

7. Third Cosmological Observation: Cosmic Microwave Background • At the time of the Big Bang, the Universe was very HOT – emitting gamma-rays – the hot dense universe radiates as a blackbody. • This radiation has been redshifted by the expansion of the Universe so that the peak of the radiation is now at radio/microwave wavelengths. • Current temperature is To = 2.727 K

8. Third Cosmological Observation: Cosmic Microwave Background The Universe was opaque to light at the time of the Big Bang (T>104 K) due to high particle density, ionization, and scattering When cooled to ~3000 K during expansion, recombination of atoms allowed photons to pass without as much scattering or absorbing. At current temp (To = 2.7 K), CMB is cooled by a factor of 1100. T ~ 1/a(t)  temperature is inversely proportional to the scale factor λ ~ a(t)  wavelength is directly proportional to the scale factor

9. Discovery of the CMB • First observed (inadvertently) in 1965 by Arno Penzias and Robert Wilson at the Bell Telephone Laboratories in Murray Hill, NJ • Detected excess noise in a radio receiver peak emission at BB temp = 3 degrees • In parallel, researchers at Princeton were preparing an experiment to find the CMB. • When they heard about the Bell Labs result they immediately realized that the CMB had been found • The result was a pair of papers in the Physical Review: one by Penzias and Wilson detailing the observations, and one by Dicke, Peebles, Roll, and Wilkinson giving the cosmological interpretation. • Penzias and Wilson shared the 1978 Nobel prize in physics • Almost immediately after its detection, the Steady State theory was dead

10. These three cosmological observations agree with the concept of the Hot Big Bang model – Universe has expanded from an initial hot and dense state to its current cooler and lower-density state …but how exactly does the Universe expand?

11. Newtonian Gravity in Cosmology What can we learn about the evolution of an expanding Universe by applying Newtonian gravity? Assume Isotropy Universe is spherically symmetric from any point – spherical volume evolves under its own influence and homogeneity (r) = constant For a test mass moving on the surface of a sphere with mass M at position r (d2r/dt2) = -GM/r(t)2 (23.27) Multiply each side by dr/dt and integrate over time to get ½ (dr/dt)2 = GM/r(t) + k (23.28) Where k is an integration constant. Now let the sphere have any old mass and radius M = (4π/3) ρ(t) r(t)3 & use r(t) = a(t) ro (23.31) (23.32) Divide each side by (roa)2/2 to get H(t)2 = Friedmann Equation

12. ½ (dr/dt)2 = GM/r(t) + k(23.28) k > 0 Right hand side of equation is always positive Left hand side is always greater than zero Expansion continues forever! Open or Unbound Universe k < 0 Right hand side is zero when r reaches some maximum rmax = GM/(-k) After this, the Universe starts to collapse Closed or Bound Universe k = 0 The Universe expands at an ever decreasing rate (dr/dt)  0 as t  infinity Borderline Universe or Marginally Bound

13. (23.32) H(t)2 = For a given value of H(t) there is a critical mass density for which k=0, where gravity is just sufficient to halt the expansion ρc = 3H(t)2/(8πG) At the present time: Ho = 70 km/s/Mpc, then ρc,o ≈ 1.4 x 1011 Msun/Mpc3 (about one H atom in 200 L volume of space) The fate of the Universe depends on its density… High density = enough matter to gravitationally halt expansion and cause gravitational collapse Low density = not enough gravitational attraction to stop expansion…it goes on forever Except in the case of a positive Cosmological Constant – introduced by Einstein in context of General Relativity

14. Cosmology and General Relativity Einstein’s principle of equivalence led to the understanding that space-time is curved in the presence of a gravitational field. The mass/energy density of the Universe tells us about the geometry of space-time. Curvature of space (in 2-d) can be described in terms of angles/areas of a triangle. If the 3 angles on the triangles drawn below are α, β, γ then α + β + γ = 180° + (κA/rc2) where κ is the curvature constant, A is the areas of the triangle and rc is the radius of curvature κ = +1 Bound/Closed Universe Sphere  finite area, no edge/boundary κ = -1 Open Universe Hyperboloid  infinite area, no edge/boundary κ = 0 Flat/Marginally Bound/Critical Universe Plane  infinite area, no edge/boundary

15. How is space curved? Deviations of α + β + γ from 180 degrees are tiny unless the area of the triangle is comparable to r2c,o. If the Universe is curved, the radius of curvature must be close to the Hubble distance, c/Ho ~ 4300 Mpc. To see why, consider looking at a galaxy with diameter D a distance d away. If κ = 0, α = D/d (small angle formula) If κ > 0, α > D/d and mass/energy content acts like a magnifying lens when d = π rc,o, galaxy size would fill the sky – not observed! galaxy would also be seen at d+2Co, d+32Co, etc. where Co = 2πrc,o If κ < 0, α < D/d and Universe acts like demagnifying lens Objects at d much greater than rc will be exponentially tiny Since galaxies are resolved in angular size to distances comparable to Hubble distance, and the above effects are not seen, we conclude radius of curvature is comparable to or larger than Hubble distance. Universe consistent with flat.

16. Metrics of Space-time Compute distances between 2 events or objects in 4-d space-time Distances in 3-d space are given by the metric (omitting parentheses) Including time and switching to spherical coordinate system gives Minkowski Metric used in SR For an expanding, flat Universe we use the Robertson-Walker Metric Where a is the scale factor of the Universe and r, θ, φ are the comoving coordinates of a point in space. If the expansion of the universe is homogeneous and isotropic, comoving coordinates are constant with time. The photons from distant galaxies follow null geodesics (where dl=0).

17. Proper distance length of geodesic between them when the time and scale factor are fixed (assume flat Universe with θ and φ constant) Since proper distance is impossible to measure, we measure photons from the galaxy emitted at time te < to, remembering that photons follow null geodesics. Relating this to something we can observe… Redshift – which tells us scale factor at te Typo in book! Should be λo/λe = a(to)/a(te)

18. In GR, functional form of a(t), curvature constant κ and rc,o are determined by the Field Equations. These equations link curvature to energy density and pressure at each point in space-time through the relativistic Friedmann Equation Main differences with Newtonian form: mass density ρ energy density u (photons have energy and contribute as well as massive particles) k  κ (positively/negatively curved space rather than open/closed Universe) Addition of Λ/3 term on right – Cosmological Constant (with units 1/time2) This term adds a constant (with time) energy density to the Universe 

19. The “checkered” past of the Cosmological Constant Newton understood the nature of gravity on a Universe that contains matter. He postulated that either 1) The Universe is infinite in volume and mass (keeping it from gravitational collapse). 2) The Universe is expanding fast enough to overcome the attractive force of gravity. 3) The Universe is collapsing (either #2 or #3 imply a beginning of the Universe). Since Newton believed the universe was eternal and unchanging, he believed that the universe was therefore infinite. Solving Einstein’s field equations resulted in the constant of integration Λ. This energy density could balance the mass energy density and allow for a static Universe that contains matter – seemed a great finding to support the idea of a Steady-State Universe! However, once Hubble’s observations of an expanding Universe were discovered in 1920, Einstein realized a non-zero Λ was unnecessary (#2 above) and called it his greatest blunder. In fact, both Hubble and Einstein were correct – the Universe is expanding and changing which does point to a beginning (BB), but this does not exclude the possibility of a non-zero cosmological constant and, in fact, current estimates predict that it is non-zero.

20. Rewrite Friedmann equation in terms of energy density components ur = radiation density (from relativistic particles like photons) um = matter density (non-relativistic like protons, electrons, WIMPS) uΛ = lambda density (vacuum density from cosmological constant) Flatness of Universe implies ur + um + uΛ = uc (critical density) Components of Universe often given in terms of a dimensionless Density Parameter Ω(t) = u(t)/uc(t) where Ω<1 is a negatively curved Universe and Ω>1 is positively curved. Thus, knowing how the Universe expands with time requires knowing how much energy density is in radiation, matter and Λ today and how radiation and matter density evolve with time.