4 Friedmann-Robertson-Walker Metric4.1 Concept Questions1. What does it mean that the Universe is expanding?2. Does the expansion affect the solar system or the Milky Way?3. How far out do you have to go before the expansion is evident?4. What is the Univer se expanding into?5. In what sense is the Hubble constant constant?6. Does our Universe have a center, and if so where is it?7. What evidence suggests that the Universe at large is homogeneous and isotropic?8. How can the CMB be construed as evidence for homogeneity and isotropy given thatit provides information only about a 2D surface on the sky?9. What is thermodynamic equilibrium? What evidence suggests that the early Universewas in thermodynamic equilibrium?10. What are cosmological parameters?11. What cosmological parameters can or cannot be measured from the power spectrumof fluctuations of the CMB?12. FRW Universes are characterized as closed, flat, or open. Does flat here mean the sameas flat Minkowski space?13. What is it that astronomers call dark matter?14. What is the primary evidence for the existence o f non-baryonic cold dark matter?15. How can astronomers detect dark matt er in galaxies or clusters of galaxies?16. How can cosmologists claim that the Universe is dominated by not one but two distinctkinds of mysterious mass-energy, dark matter and dark energy, neither of which hasbeen observed in the labora t ory?17. What key property or properties distinguish dark energy from dark matter?18. Does the Universe conserve entropy?19. Does the annihilation of electron-positron pairs into photons generate entropy in theearly Universe, as its temperature cools through 1 MeV?20. How does t he wavelength of light change with the expansion of the Universe?121. How does t he temperature of the CMB change with the expansion of the Universe?22. How does a blackbody (Planck) distribution change with the expansion of the Uni-verse? What about a non-relativistic distribution? What about a semi-relativisticdistribution?23. What is the hor izon of our Universe?24. What happens beyond the horizon of our Universe?25. What caused the Big Bang?26. What happened before the Big Bang?27. What will be the fate of the Universe?4.2 What’s important?1. The CMB indicates that the early (≈ 40 0 ,000 year old) Universe was (a) uniform to afew ×10−5, and (b) in thermodynamic equilibrium. This indicates thatthe Universe was once very simple .It is this simplicity that makes it possible to model the early Universe with some degreeof confidence.2. The power spectrum of fluctuations of the CMB has enabled precise measurements ofcosmological parameters, excepting the Hubble constant.3. There is a remarkable concordance of evidence from a broad range of astronomicalobservations — supernovae, big bang nucleosynthesis, the clustering of galaxies, theabundances of clusters of ga la xies, measurements of the Hubble constant from Cepheidvariables, the ages of the oldest stars.4. Observational evidence is consistent with the predictions of the theory of inflation inits simplest form — the expansion of the Universe, the spatial flatness of the Universe,the near uniformity of temperature fluctuations of the CMB (the horizon problem),the presence of acoustic peaks and troughs in the power spectrum of fluctuations ofthe CMB, the near power law shape of the power spectrum at large scales, its spectralindex (tilt), the gaussian distribution of fluctuations at large scales.5. What is non-baryonic dark matter?6. What is dark energy? What is its equation of state w ≡ p/ρ, and how does w evolvewith time?24.3 Observational basisIn the last decade, observations have converged on a Standard Model of Cosmology, a spa-tially flat universe dominated by dark energy and by non-baryonic dark matter.1. The Hubble diagram (distance versus redshift) of gala xies indicates that the Universeis expanding ( Hubble 1929).2. The Cosmic Microwave Background (CMB).• Near black body spectrum, with T0= 2.725 ± 0.001 K (Fixsen & Mather 2002).• Dipole ⇒ the solar system is moving at 365 km s−1through the CMB.• After dipole subtraction, the temperature o f the CMB over the sky is uniform toa few parts in 105.• The power spectrum of temperature T fluctuations shows a scale-invariant spec-trum at large scales, and prominent acoustic peaks at smaller scales. Allows mea-surement of the amplitude Asand tilt nsof primordial fluctuations, the curvaturedensity Ωk, and the proper densities Ωch2of non-baryonic cold dark matter andΩbh2of baryons. Does not measure Hubble constant h ≡ H0/(100 km s−1Mpc−1).• The power spectra of E and B polarization fluctuations, and the various crosspower spectra (only T -E should be non-vanishing).3. The Hubble diagram of Type Ia (thermonuclear) supernovae indicates that t he Universeis accelerating. This points to the dominance of gravitationally repulsive dark energy,with ΩΛ≈ 0 .7 5. The amount of dark energy is consistent with observations from theCMB indicating that the Universe is spatially flat, Ω ≈ 1, and observations from CMB,galaxy clustering, and clusters of galaxies indicating that the density in gravitationallyattractive matter is only Ωm≈ 0.2 5.4. Observed abundances of light elements H, D,3He, He, and Li are consistent with thepredictions o f big bang nucleosynthesis (BBN) provideed that Ωb≈ 0.04, in goodagreement with measurements from the CMB.5. The clustering of matter (dark and bright) shows a power spectrum in good agreementwith the Standard Model:• galaxies;• the Lyman alpha forest;• gravitational lensing.Historically, the principle evidence for non-baryonic cold dark matter is comparisonbetween the p ower spectra o f galaxies versus CMB. How can tiny fluctuations in t heCMB grow into the observed fluctuations in matter today in only the age of the Uni-verse? Answer: non-baryonic dark matter that begins to cluster before Recombination,when the CMB was released.6. The abundance of galaxy clusters as a function o f redshift.7. The ages o f the oldest stars, in globular clusters. The Hubble constant yields anestimate of the age of the Universe t hat is older with dark energy than without. Theages of the oldest stars agree with the age of the Universe with dark enery, but areolder than the Universe without dark energy.38. Ubiquitous evidence for dark matter, deduced from sizes and velocities (or in the caseof gravitational lensing, t he gravitational potential) o f various objects.• The