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2.Homogeneity & Isotropy Fundamental 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY FuFundndaammententaall CCoosmsmoolologygy:: 22..HomHomogeneityogeneity && IsotropyIsotropy “When I trace at my pleasure the windings to and fro of the heavenly bodies, I no longer touch the earth with my feet: I stand in the presence of Zeus himself and take my fill of ambrosia, food of the gods.”!! Ptolemy (90-168(90-168 BC) 1 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.1:1: AncientAncient CosCosmologymology • Egyptian Cosmology (3000B.C.) Nun Atum Ra 結婚 Shu (地球) Tefnut (座空天国) 双子 Geb Nut 2 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.1:1: AncientAncient CosCosmologymology • Ancient Greek Cosmology (700B.C.) {from Hesoid’s Theogony} Chaos Gaea (地球) 結婚 Gaea (地球) Uranus (天国) 結婚 Rhea Cronus (天国) The Titans Zeus The Olympians 3 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.1:1: AncientAncient CosCosmologymology • Aristotle and the Ptolemaic Geocentric Universe • Ancient Greeks : Aristotle (384-322B.C.) - first Cosmological model. • Stars fixed on a celestial sphere which rotated about the spherical Earth every 24 hours • Planets, the Sun and the Moon, moved in the ether between the Earth and stars. • Heavens composed of 55 concentric, crystalline spheres to which the celestial objects were attached and rotated at different velocities (w = constant for a given sphere) • Ptolemy (90-168A.D.) (using work of Hipparchus 2 A.D.) • Ptolemy's great system Almagest. • Perfect motion should be in circles, so the stars and planets, being heavenly objects, moved in circles. • Planets appear to periodically loop back upon themselves (retrograde motion), epicycles had to be introduced -> planets moved in circles upon circles about the fixed Earth. 4 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2: The Development of Modern Cosmology •• TheThe CopernicanCopernican HeliocentricHeliocentric UniverseUniverse Copernicus (1473-1543) • Revolution in Astronomy •De Revolutionibus orbitum caelestium (On the revolution of the celestial spheres) • The Universe does not revolve around the Earth! Circa 1514 Copernicus distributed a short, hand written book - The Little Commentary containing 7 axioms 1.There is no one centre in the Universe. 2.The Earth's centre is not the centre of the Universe. 3.The centre of the Universe is near the sun. 4.The distance from the Earth to the sun is imperceptible compared with the distance to the stars. 5.The rotation of the Earth accounts for the apparent daily rotation of the stars. 6.The apparent annual cycle of movements of the sun is caused by the Earth revolution around it. 7.The apparent retrograde motion of the planets is caused by the motion of the Earth from which one observes. Copernican Cosmological Principle : THE EARTH DOES NOT OCCUPY A SPECIAL PLACE IN THE UNIVERSE 5 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2: The Development of Modern Cosmology •• FromFrom thenthen untiluntil nownow • c.168年: Ptolemy’s Geocentric Universe. • c.1514年: Copernicus Proposes Heliocentric Universe. • c.1588年: Tcycho Brahe argues against Heliocentric model due to lack of observed Stellar Parallax. • c.1609年: Kepler orbital laws : Key to Copernican Heliocentric model - Planets move in ellipses not circles. • c.1610年: Galileo discovers moons orbiting Jupiter : Death blow for the Ptolemy’s Geocentric model. • c.1687年: Newton publishes Philosophiae naturalis principia mathematic. • c.1838年: Bessel measures parallax of 61 Cygni showing it lies far beyond solar system. • c.1915年: Einsteins Theory of Relativity. • c.1917年: Friedmann formulates equations for expanding Universe. • c.1929年: Hubble shows the nebulae are extragalactic in origin and moving away from us. • c.1948年: Gamov postulates the initial singularity. • c.1948年: Hoyle coins the phrase “Big Bang”. • c.1964年: Penzias & Wilson discover the Cosmic Microwave Background (kills steady state model). 6 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 22..33:: TThhee CCoossmmoolologicgicaall PrincPrincipleiple DEFINITIONS : • Fundamental Observer : Someone at rest with respect to the rest of the Universe in their locality. Universe ~ smooth fluid ~ substratum -> Fundamental observers are co-moving with it. • Homogeneity : Same picture of the Universe at any time is seen by all Fundamental Observers. No preferred locations i.e. measure the same amounts of scalar quantities such as mass, density, temperature. • Isotropy : The Universe looks the same in all directions to a Fundamental Observer. No preferred Directions 7 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 22..33:: TThhee CCoossmmoolologicgicaall PrincPrincipleiple •• Copernican Cosmological Principle The Earth does not occupy a special place in the Universe •• The Cosmological Principle At any single epoch, the Universe appears Homogeneous and Isotropic to all Fundamental Observers •• Perfect Cosmological Principle The Universe appears Homogeneous and Isotropic to all Fundamental Observers AT ALL TIMES •• Anthropic Cosmological Principle (WEAK) The conditions necessary for sentient life will only exist in a Universe where the laws of physics are the way they are as seen by us. (STRONG) There could be many different universes, or regions in a single Universe, where the laws of physics are different. 8 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.4:2.4: HomogeneitHomogeneityy andand IsotIsotropyropy Examples homogeneous inhomogeneous homogeneous (on scales>strip) but anisotropic isotropic but inhomogeneous isotropic (about centre) anisotropic 9 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.4:2.4: HomogeneitHomogeneityy andand IsotIsotropyropy Isotropy + Copernican Principle implies Homogeneity Castor sees Isotropic Universe fx = fa = fy x Castor b † a y Pollux Pollux sees Isotropic Universe fx = fb = fy 10 † 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.4:2.4: HomogeneitHomogeneityy andand IsotIsotropyropy Cosmological Principle (Homogeneity) implies a cosmic time Since the Universe appears the same to all fundamental observers at any given time, All observers see the same sequence of events=> they can all synchronize their watches to some event which occurs in the history of the Universe, thereafter all the watches measure the same cosmological time 11 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.4:2.4: HomogeneitHomogeneityy andand IsotIsotropyropy Is the real Universe really isotropic ?? Matter: Small scales : Highly anisotropic Large scales > 100Mpc (Clusters / Superclusters) : fairly isotropic Radio Sources: isotropic to a few percent Radiation: CMB - Isotropic to 1 part in 105, 0.003%, 2mK 12 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.4:2.4: HomogeneitHomogeneityy andand IsotIsotropyropy Is the real Universe really homogeneous ?? For population of uniformly distributed objects of constant number density h, flux S, luminosity L; The number of objects per steradian out to some radius r, N = hr3 /3 1/ 2 We can measure the slope of galaxy counts!! Ê Lˆ S = L /r2 fi r = Á ˜ Ë S ¯ 3 / 2 Ê hP ˆ -3 / 2 N(S) = Á ˜ S Ë 3 ¯ log10 N µ1.5log10 S or m = A - 2.5log10 S log10 N = B + 0.6m unfortunately galaxies evolve (A & B are constants) 13 † 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.5:2.5: HHuubbbbleless LLaaww The Recession of the galaxies lo - le Cosmological Doppler Effect z = z= redshift le • 1912: Vesto Silpher (Lowell) measured blue shift from M31 • By 1925 measured ~ 40 galaxies, all redshifted except local group • 1929 Hubble relates redshift to distance (Cepheids) † Linear Relation cz=Hor HUBBLE’s LAW km/s/Mpc Original Estimate Ho = 500 km/s/Mpc (severely underestimated distances to galaxies Modern Estimate Ho = 70 km/s/Mpc 14 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.5:5: HubbleHubble’’ss LLaaww Cosmological Principle: Derivation of Hubble’s Law How can Universe be homogeneous if all galaxies are running away from us !!?? ¿ r’ = r - a (r,r’,a = VECTORS) ¡ v’(r’) = v(r) - v(a) ¬ v’(r’) = v’(r - a) v(r) v(r’) Homogeneity => O & O’ see same events v’(r - a) = v(r - a) r’ v(r - a) = v(r) - v(a) r General Solution Ê h11 h12 h13 ˆÊ r1 ˆ 3 Á ˜Á ˜ v (r,t) h(t) r or h h h r i =  ij j Á 21 22 23 ˜Á 2˜ j=1 Á ˜Á ˜ h h h r h(t) is fn(t) Ë 31 32 33 ¯Ë 3¯ Isotropy => matrix is rotationally invariant => h =0 for i≠j a ij O’ and h11 = h22 = h33 = constant = H(t) So; v = H(t)r O † 1 1 v = H(t)r -v(a) 2 2 v = H(t)r v = H(t)r v(a) v13= H(t)r3 Velocity of any co-moving particle is either zero (H=0) or moving radially away (H>0) or toward us (H<0) with a velocity proportional to the distance : i.e. HUBBLES LAW 15 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.5:5: HubbleHubble’’ss
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