Fundamental Cosmology: 2.Homogeneity & Isotropy

Fundamental Cosmology: 2.Homogeneity & Isotropy 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 BC) 1

2.1: Ancient Cosmology Egyptian Cosmology (3000B.C.) Nun Atum Ra Shu ( ) Tefnut ( ) Geb Nut 2

2.1: Ancient Cosmology Ancient Greek Cosmology (700B.C.) {from Hesoid s s Theogony} Chaos Gaea ( ) Gaea ( ) Uranus ( ) Rhea Cronus ( ) The Titans Zeus The Olympians 3

2.1: Ancient Cosmology 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

2.2: The Development of Modern Cosmology The Copernican Heliocentric Universe 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

2.2: The Development of Modern Cosmology From then until now c.168 : Ptolemy s Geocentric Universe. c.1514 : Copernicus Proposes Heliocentric Universe. c.1588 : Tcycho Brahe argues against Heliocentric model due to lack of observed Stellar Parallax. c.1609 : Kepler orbital laws : Key to Copernican Heliocentric model - Planets move in ellipses not circles. c.1610 : Galileo discovers moons orbiting Jupiter : Death blow for the Ptolemy s Geocentric model. c.1687 : Newton publishes Philosophiae naturalis principia mathematic. c.1838 : Bessel measures parallax of 61 Cygni showing it lies far beyond solar system. c.1915 : Einsteins Theory of Relativity. c.1917 : Friedmann formulates equations for expanding Universe. c.1929 : Hubble shows the nebulae are extragalactic in origin and moving away from us. c.1948 : Gamov postulates the initial singularity. c.1948 : Hoyle coins the phrase Big Bang. c.1964 : Penzias & Wilson discover the Cosmic Microwave Background (kills steady state model). 6

2.3: The Cosmological Principle 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

2.3: The Cosmological Principle 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

Examples 2.4: Homogeneity and Isotropy homogeneous inhomogeneous homogeneous (on scales>strip) but anisotropic isotropic but inhomogeneous isotropic (about centre) anisotropic 9

2.4: Homogeneity and Isotropy Isotropy + Copernican Principle implies Homogeneity Castor sees Isotropic Universe f x = f a = f y x Castor b a y Pollux Pollux sees Isotropic Universe f x = f b = f y 10

2.4: Homogeneity and Isotropy 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

2.4: Homogeneity and Isotropy 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 10 5, 0.003%, 2mK 12

2.4: Homogeneity and Isotropy 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 = hr 3 /3 Ê S = L /r 2 fi r = L ˆ Á Ë S Ê N(S) = hp 3 / 2 ˆ Á S -3 / 2 Ë 3 1/ 2 We can measure the slope of galaxy counts!! log 10 N µ1.5log 10 S or m = A - 2.5log 10 S log 10 N = B + 0.6m (A & B are constants) unfortunately galaxies evolve 13

2.5: Hubbles Law The Recession of the galaxies Cosmological Doppler Effect z= redshift z = l o - l e l e 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=h o r 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

2.5: Hubble s s Law Cosmological Principle: Derivation of Hubble s s Law How can Universe be homogeneous if all galaxies are running away from us!!?? v(r) v(r ) r = r - a (r,r,a = VECTORS) v (r ) = v(r) - v(a) v (r ) = v (r - a) Homogeneity => O & O see same events v (r - a) = v(r - a) O r v(a) a r -v(a) O General Solution v i (r,t) = v(r - a) = v(r) - v(a) 3 Âh(t) ij r j or j=1 Ê h 11 h 12 h 13 ˆ Ê Á Á h 21 h 22 h 23 Á Á Ë h 31 h 32 h 33 Ë h(t) is fn(t) Isotropy => matrix is rotationally invariant => h ij =0 for i j and h 11 = h 22 = h 33 = constant = H(t) So; v 1 = H(t)r 1 v 2 = H(t)r 2 v 13 = H(t)r 3 v = H(t)r r 1 r 2 r 3 ˆ 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

2.5: Hubble s s Law Cosmological Principle: Hubble s s Law & the Scale Factor Rewrite the Hubble Parameter H(t) = 1 R(t) Solution Hubble Constant is really Hubble Parameter Such that R(t) is the SCALE FACTOR dr dt = 1 R(t) Where; R o = R(t o ) such that r=r o at t=t o dr(t) dt dr(t) dt r µ R(t) fi r = R(t) R o r r o All distances are scaled up by this factor R(t) with increasing time with simple isotropic expansion (note: Volume V(t)µ R(t) 3 16

2.6: Summary Over the last 2 millennia our place in the Universe has become less and less important Geocentric - Heliocentric - isotropic homogeneous expanding Universe The Cosmological Principle is a very simple but very powerful tool for Cosmology Unfortunately the evolution of galaxies makes it very difficult to prove homogeneity The Universe itself is isotropic on the largest scales Hubbles Law follows simply from the Cosmological Principle and its assumptions about Isotropy and Homogeneity The Universe can be thought of a set of fixed coordinates that are scaled by some scale factor as a function of time. 17

2.6: Summary Fundamental Cosmology 2. Homogeneity & Isotropy Fundamental Cosmology 3. Newtonian Cosmology 18