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Chapter 5: Cosmology Part 1: the Large Scale Homogeneous Universe Some Important Things for Life About (Homogeneous) Cosmology

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Chapter 5: Cosmology Part 1: the Large Scale Homogeneous Universe Some Important Things for Life About (Homogeneous) Cosmology Chapter 5: Cosmology Part 1: the large scale homogeneous Universe Some important things for Life about (homogeneous) cosmology • The Universe as we know it is old, but had a finite beginning 13.8 Gyr ago, • The (observable) Universe is large, but is finite in size, • Baryonic material is a minority component of matter in the Universe, but is not zero, • The Universe is expanding and is not (currently) in thermal equilibrium, with a characteristic temperature that is very low (2.7K), • The Universe will continue to expand for a long time in the future, • The average density of atoms in the Universe is very low (of order 1 m-3), • The primordial composition after the Big Bang was only H and 4He (+ tiny amounts of 7Li) Hubble 1929 Expansion of the Universe Hubble (1929) redshift-distance relation The “redshift” is the λ − λ shift of spectral features z = obs em λ in distant galaxies to em longer wavelengths v 1 + v Modern SN1a data Relativistic Doppler z c 1 z for z 1 = v − ⇒ = << effect (but see later) 1− c c H z = 0 d v = H d Hubble’s Law c 0 v µ d is a signature of an expanding “space”, but note: • This does not imply we are at a “center” • The scatter is due to the “peculiar motions” of individual galaxies due to local gravitational effects – typically 200 kms-1 3K Cosmic Microwave Background (CMB) dominates the radiation content of the Universe CMB has Planck Black Body spectrum at 2.728 ± 0.002 K, with exquisite precision. But, matter and radiation are not in thermal equilibrium today (there is no matter at 2.73 K!). Photon to baryon ratio in the universe is 9 high: ng/nB ~ 2×10 . Also, the CMB radiation is isotropic at the level of 10-3 (dipole term due to our own 600 kms-1 motion w.r.t. CMB frame, otherwise isotropic at 10-5). The extragalactic background light The last scattering surface of the CMB At very early times, matter and radiation were indeed in thermal equilibrium via creation/annihilation reactions of particle/anti-particle pairs. Prior to 300,000 years after Big Bang, all matter was ionized plasma (T > 4500 K) and the Universe was effectively “opaque” due to scattering of light off charged particles. After this epoch, matter was generally neutral (at least for a while) and the Universe was “transparent”, as it still is today. Since the CMB has been around since the beginning, the visual impression is of being located at the center of a clearing in a fog bank. We “see” the “Last Scattering Surface” as a section through the Universe at an immense distance, and as it was 3 x 105 yr after Big Bang. p + opaque The Big Bang transparent US Out in space and back in Appearance of time “inside of fog bank” Isotropy of the CMB is really a statement about the homogeneity of the Universe 300,000 yr after Big Bang. Roberston-Walker metric for expanding isotropic homogeneous Universe The Minkowski metric in Special Relativity is: ds2 = c2dt 2 − dl 2 3 3 The most general metric will be of the ds2 = g dxidx j ≡ g dxidx j form: ij ∑∑ ij i=0 j=0 About our choice of co-ordinates: (a) It is convenient to define time as a cosmic time t: t is defined as the proper time of the “fundamental observers” who see an isotropic CMB and Hubble flow (and are thus “at rest” with respect to the Universe as a whole). (b) It is convenient to define a polar co-ordinate system centered on us with two angles (q,f = position on sky) and a radial coordinate. (c) Also, it is convenient to split the radial coordinate into a fixed (for a fundamental observer) “co-moving” position w and a time-varying cosmic scale factor R(t). q and f of any fundamental observer are anyway also constant with time. 3 3 2 i j i j ds = gijdx dx ≡ ∑∑gijdx dx i=0 j=0 Simple symmetry arguments then allow us to show that the metric for any homogeneous and isotropic, but possibly expanding, Universe must have the form of the Robertson-Walker metric. & dr 2 # ds2 c2d R2 ( ) $ r 2d 2 ! = τ − τ $ 2 2 + γ ! %1− kr / A " Conventional 2 2 2 2 2 2 angular ds = c dτ − R (τ)(dω + Sk (ω)dγ ) coordinates q,f “Cosmic time” Comoving radial Effects of the curvature of coordinate w a t = constant surface, Scale factor of with A = (comoving) expanding Universe radius of curvature Again, simple symmetry arguments require that Sk(w) has one of only two forms, which both approach the straightward Sk(w) = w as the radius of curvature A tends to infinity: We measure the curvature and find ⎛ω ⎞ ⎛ω ⎞ Sk (ω) = Asin⎜ ⎟ or Asinh⎜ ⎟ that it is very small, i.e. A is very ⎝ A ⎠ ⎝ A ⎠ large. This 2-d analogue of our 3-d Universe illustrates • Isotropic expansion with v µ d • Patterns and geometry/topology remain constant: “comoving radius of curvature” doesn’t change. • Our “observable region” (horizon = light travel distance in the age of Universe) is likely to be finite. • We can measure the curvature in different ways. We know that we live in a Universe with A very much larger than the “horizon”. Note: the volume of Universe (i.e. the number of galaxies) will also be finite if it is (uniformly) positively curved, but will be formally infinite if it is (uniformly) negatively curved. The volume will definitely be “very large” if it is time more or less flat in our observable Maps of galaxies (fixed in comoving region as it is. space) at different t Some straightforward consequences of the RW metric The redshift of galaxies in terms of the change R(τ observed ) in scale factor R. (1+ z) = R(τ emitted ) Hubble’s “constant” (better: Hubble’s “parameter”) R H = is the normalized first derivative of R(t) R The “deceleration parameter” is the normalized RR second derivative of R(t) q = − R 2 Behaviour of a black body radiation field RTbb = constant The last equation gives us a natural explanation of the 2.7K microwave background: The Universe was in thermal equilibrium at very early times, in a -1 hot dense state. The black body Tbb will simply decline as R even after thermal equilibrium is lost. The CMB is the relic radiation of this hot dense phase. The Friedmann equation Solution of General Relativity field equations for homogenous isotropic Universe gives the Friedmann equation linking the dynamics of the expansion R(t), the r(R) “equation of state”, and the curvature c2/A2 2 2 8πG 2 kc & Λ 2 # R = ρR − 2 + R 3 A %$ 3 "! R - scale factor from the Robertson Walker metric R! -derivative w.r.t. cosmic time t ρ - density in whatever form (non-relativistic matter, radiation, others?) kc2 - curvature as in the Robertson Walker metric: A = comoving radius of curvature (fixed for all time) and k = ± 1 just to fix the sign. A2 Λ - Einstein’s “cosmological constant” (note that this is entirely equivalent to a r = constant term, also known as a “false vacuum”, i.e. a form of “Dark energy”) To solve for R(t) we must know the (constant) curvature and r(R), which will depend on the form of the density. What are the possibilities for r(R)? -3 rM µ R ”cold’ matter, either normal baryonic matter or “dark matter” -4 rg µ R radiation, or relativistic matter rL = constant “false vacuum” … and possibly others Note: the different r(R) dependences mean that different components may dominate at different R. We can also can imagine a given component changing from one R dependence to another, e.g. relativistic particles lose energy à non-relativistic. Dynamical effects on R(t) It is easy to solve Friedmann equation if the density is dominated by one single component and the curvature term is negligible: -3 2/3 rM µ R R µ t Decelerated expansion (as you’d expect) -4 ½ rg µ R R µ t Decelerated expansion (as you’d expect) Ht rL = constant R µ e (Exponentially) accelerated expansion under gravity!! ρi It is convenient to parameterize the density ri of different Ωi = components i in terms of a “critical density” that is given at any ρc epoch by the expansion rate, H, and gravitational constant, G, 3H 2 and call this W . We can then have a “total” W = SW . ρc = i i 8πG The Friedmann Equation then can be re-arranged in different ways 2 € ⎛ ⎞ kc2 1 ⎛ c ⎞ 1 Ω −1 = Ω −1 = = k⎜ ⎟ ⎜∑ i ⎟ 2 ! 2 2 ⎝ i ⎠ A R ⎝ H ⎠ (RA) • c/H is of order the size of the horizon, so |W-1| tells us the ratio of the horizon scale to the (physical) radius of curvature RA, i.e. it gives the “importance” of the curvature within our horizon. We know that in our Universe at the moment W ~ 1. • Note that if the expansion decelerates, then R-dot decreases, |W-1| increases, and W diverges away from unity, with curvature becoming more important, and vice versa: if the expansion accelerates then W converges towards unity. For illustration, the following slides look at three cases that were, 25 years ago, thought to be all more or less consistent with observations. Wm = 1: A flat “Einstein-de Sitter” universe, in which the density is dominated by a critical density of matter. This was preferred theoretically in the 1990’s.
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