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Home , Cosmological principle, Homogeneity (physics), Isotropy

11/2/03 Chris Pearson : Fundamental 2: and 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 • 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 (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 : 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 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 . • 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 “”. • 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 are the way they are as seen by us. (STRONG) There could be many different , 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 + 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 / ) : 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 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 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= 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 vv == HH(t)r(t)r v(a) v13= H(t)r3 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 LLaaww Cosmological Principle: Hubble’s Law & the

注意: Hubble Constant is really Hubble Parameter 1 dR(t) Rewrite the Hubble Parameter H(t) = R(t) dt dr 1 dR(t) Such that = r dt R(t) dt R(t) Solution † r µ R(t) fi r = ro Ro

Where; R†o = R(to) such that r=ro at t=to R(t) is the SCALE FACTOR 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 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.6:6: SuSummammaryry

• 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 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.6:6: SuSummammaryry

FundFundamentamentalal CosCosmologymology 2.2. HomogeneityHomogeneity && IsotropyIsotropy 終終終

FundFundamentamentalal CosCosmologymology 次次次：：： 33.. NNeewwttoonianiann CCoossmmoolologgyy 18