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Introduction to Cosmology 1

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Introduction to Cosmology 1 /ntroduction to Cosmology 1 True progress in cosmology began in the 20th century: ● General Relativity ● Powerful tools of observational astronomy Two critical observations: 1. Hubble s Law (1#29): Measure both the distance and ra&ial velocity of a set of galaxies. Radial velocity is easy to find) by measuring the re&shift z and using the first-order Doppler shift: +istance is much harder to measure. ,eed to use “cosmic distance la&&er.) with uncertainties at each step. "1 Mpc = 3.086 < 1028 cm$ Hubble s Law: v = H d 0 Recent &etermination (Riess et al. 2019) ApJ, 876, 85): H = 74.03 : 1.42 km s*1 Mpc*1 0 Hubble constant is often expressed as H = 100 h km s*1 Mpc*1 0 2 2. The cosmic microwave background (0%B): Discovered by Penzias & @ilson in 1#67. =lackbody radiation with T = 2.725 K observed (with very little variation) on all directions on the sky. Hubble and CO=C F/R2E plots are from Ned Wright s cosmology tutorial "www.astro.ucla.edu/GwrightFcosmolog.htm$ 9 8 ,eed to introduce a tremendous simplification to make the theoretical study of the universe as a whole feasible: The cosmological principle: the universe is spatially homogeneous and isotropic The CP is clearly ridiculous on small distance scales, but galaxy surveys (e.g., the Sloan Digital Sky Survey) suggest that homogeneity applies on scales ~10# lyr. /sotropy: CMB and Hubble expansion (as well as galaxy distribution) /sotropy for us plus Copernican principle 1H isotropy everywhere 1H homogeneity @e ll assume the CP in constructing cosmological models and hope that the results are also relevant to the real universe. This hope can be put to the test! 7 60)000 S+EE galaxies Dig from Freedman & Turner (2003, Rev Mod Phys, 57, 1499$ The CP severely constrains the metric: 6 a) Each point in space is characterize& by physical and metric conditions (e.g., energy density, velocity, Riemann tensor). CP =H it must be possible to de(ne a time coord such that at any given instant, these conditions are i&entical everywhere in the universe 1H the existence of a preferred cosmic time b) The only possible type of relative motion btwn points in the universe is an expansion or contraction of the universe as a whole. Btherwise, homogeneity would be spoile&. ,ote: obviously) this &oesn't apply on small scalesI c) Fundamental observer: an observer located at some point in the universe and at rest wrt the local material. Homogeneity reJuires that proper time ticks at the same rate for each of these observers. 2lso, it's possible for them to synchronize their clocks. @e will take our cosmic time coord to be the synchronize& proper time of the fun&amental observers. 5 Eo far, we know the metric can be written: f "t) is a function of time and dl2 is a spatial metric, independent of time CP also severely constrains dl2. ,ow consider just the spatial geometry: 2dopting geodesic coords, the metric is that of Cuclidean 3-space: g = ij ij /sotropy of space 1H the components of R cannot &epend on how we hijk orient the axes of our geodesic coord system. Otherwise, the curvature would have a directionality in space. 1H R is expressible in terms of tensors whose components don't hijk change on spatial rotations. The only such tensor is . ij 4 1H (in each term, h pairs with either i, j, or kL the remaining 2 indices then pair) R = - R =H k = - k and k 1 0 hijk hikj 1 2 1H Dor arbitrary coor&s: 1H R = -2 k g and R = -6 k ij ij The spatial metric dl2 must satisfy the above &iff eqn for g , which ij has only one free parameter; k must be the same everywhere. # @e know that a 2-sphere has constant curvature everywhere, so let s look at a 3-sphere (living in Euclidean 4-space$: 1H This metric satis(es the equation for g with k = a*2 ij 10 +e(ne = r/a and factor out an a2. 1H a can be a function of time, while ) ) and are constant for a fundamental observer. Euch coor& systems are known as “comoving”. These coords for a point in the universe do not change, even as the universe expan&s or contracts. With rather than r, k = 1 rather than k = a*2. 2lternatively, with = sin (0 ≤ < $: 11 Total volume of the space is: 1 22a9 "(nite, as expected in analogy with the 2-sphere$ The metric for the 9*sphere is with k = 1. @hen k = 0 or k = -1, this metric also satis(es the reJuirement on g . The resulting spaces are said to have positive ij "negative$ curvature when k = O1 (k = -1) and are flat when k = 0. 12 Clearly) k = 0 corresponds to Euclidean 3-space, which is infinite and open. The lack of spatial curvature does not exclude it from consideration) given that there is matter in the universe. Even if the spatial part is always flat, there can still be spacetime curvature. k = -1 is an infinite) open space. With = sinh , The resulting spacetime metrics are with k = ±1 or 0. These are known as the “Robertson-Walker” metrics, and they are the only ones consistent with the CP. Qsing only the CP (no GR at all!), we ve found metrics with just one 13 free parameter (k$ and one function of time, a"t$, known as the “scale factor”. ,ote that the metric is the same everywhere in the universe at a given cosmological time. To fin&a "t$) we need to solve Cinstein's eJuations. Cinstein's equations involve the stress tensor. @hat types of material constituents are possibleR 0P 1H only perfect fluids, in which the only stress is isotropic pressure P 1H T 1 diag (*g P , c2 ) 2 ii "c is the relativistic energy &ensity) Dor idealized non*relativistic particles (like galaxies$, P = 0. Dor idealized relativistic particles (like photons$, P = c2/ 9. 18 1. Baryons: proton (mc2 1 938.3 MeS$ an& neutron (mc2 = 939.6 MeS$ electron "mc2 = 0.511 MeS$, though not technically a baryon, is included here. Today, baryons are predominantly non-relativistic. 2. Radiation: photon has mc2 = 0; E 1 h 3. Neutrinos: small but non->ero mass; relativistic (sometimes included with the radiation) 4. Other (&ark matter...) 15 Eolution of Einstein's equations: (dots denote diff wrt t$ (the Frie&mann eqn) (the acceleration eqn) 1H a static universe, with a constant, is not possibleI The acceleration is negative because of the gravitational force. If the material has pressure, then this increases the deceleration, because in GR there is gravitation associated with pressure. Eince the universe is homogeneous, there is no force associated with pressure gradients. 16 Cinstein obtained this result shortly after introducing GR, over a decade before Hubble s work. At that time, there was a strong bias for a static universe (a = const). So, Einstein added a new term to his equation of GR, introducing a new fundamental constant with units of inverse area "the -cosmological constant”$: This is equivalent to adding a new, ubiquitous source component with 1H a perfect fluid with Dollowing Hubble s discovery) Einstein called the intro&uction of the cosmological constant his greatest blunder, but there is evidence today that the expansion is accelerating, so we ll keep . The cosmological eqns become: 17 EP 11.1-2 1H static universe is possible. yields a net acceleration because the acceleration associated with the negative pressure term exceeds the deceleration associate& with the density term. H 0 =H k = 1 and > 0 "Cinstein universe$ The require& value of is so small that it wouldn't affect the results of any of the solar system tests of GR. But: the equilibrium is unstableI 18 0an we find a physical interpretation for the cosmological constant? /t could arise as a vacuum energy &ensity, e.g. as a quantum-mechanical >ero-point energy. Simple attempts to quantify this yiel& a value of which is vastly larger than observationally allowed; known as the -cosmological constant problem.” Other types of “Puid” with negative pressure, but without constant density (in time) are possible "e.g., -quintessence.). %ultiply the Friedmann eqn by a9: ,ow differentiate: %ult 2n& cosmological eqn by : Eubtract eqns: 19 0ompare to the 1st Law of Thermodynamics for a perfect fluid and an adiabatic process (since the CP prohibits heat flow): 1H This results automatically in GR since Einstein's eqns imply T = 0. L The Hubble !aw revisite& Cosmological redshifts are due to the expansion of space, not to the motion of galaxies within space. The redshift arises not from the Doppler effect, but because the wavelength is stretched in proportion to a"t): Eubscripts 0 and e refer to to&ay and the time of emission. 20 2t small &istances, we can Taylor expand a"t): (for small t – t $ 0 e "d = distance to the galaxy) +oppler 1H v = H d (Hubble s Law$ shift: 0 The Hubble parameter ; it varies with time. /ts value today is called the Hubble constant, H . 0 +eviations from Hubble Law at large redshift tell us about the acceleration or deceleration of the expansion. Using Type Ia S,e, Riess et al. (1998, 23, 116, 1009) and Perlmutter et al. "1999, Ap3, 517, 565) found that the expansion is accelerating! +ifferential Hubble plot m 0.7 0.3 0 0.3 0 1 Points above the &ashed curve indicate acceleration. "from Dreedman & Turner 2003) 21 Consider a universe consisting of a combination of matter and radiation, and perhaps with a cosmological constant.
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