10282020 Chris Pearson Fundamental Cosmology 3 Newtonian Cosmology

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY Fundamental Cosmology: 3. Newtonian Cosmology “ If I have seen further than others, it is by standing upon the shoulders of giants. ” Sir Isaac Newton (1642 -1727) b. Woolsthorpe, England 1

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 1: Newtonian Cosmology o Newton Laws of Motion • Bodies move with constant velocity unless acted on by outside force i. e. move in straight lines • The rate of change of momentum of a body is proportional to its’ acceleration • For every action there is an equal and opposite reaction The property that defines how strongly it is attracted by gravity Equivalence Principle = The property that defines how strongly something resists any force Galileo: acceleration independent of mass 2

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 1: Newtonian Cosmology o Gravity Dominates Strong Nuclear Electromagnetic GUT STRENGTH ELECTROWEAK t=10 -12 s T=1015 K E=102 Gev Weak Nuclear t=10 -35 s T=1027 K E=1015 Gev TOE t=10 -43 s T=1031 K E=1019 Gev Gravity Range (m) Coupling (J. m) mass Strong Nuclear 10 -15 10 -27 1. 4 Me. V Electromagnetic 10 -28 0 Weak Nuclear 10 -18 10 -31 1 Gev Gravity 10 -64 0 Gravity is weak but • mediator is massless • infinite range • only one charge On large scales - Gravity dominates 3

10/28/2020 • Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 1: Newtonian Cosmology Can we derive the cosmological equations from purely Newtonian dynamics ? ? The Cosmological Principle At any single epoch, the Universe appears Homogeneous and Isotropic to all Fundamental Observers v(r) v(r’) The Hubble Parameter r’ r and a O v(a) O’ -v(a) Where; Ro = R(to) such that r=ro at t=to R(t) is the SCALE FACTOR All distances scaled by factor R(t) with increasing time with simple isotropic expansion 4

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 2: Newtonian (Dynamic) Derivation of Friedmann Equations • Simple derivation from dynamical arguments Newton: For a given observer at position , r, within a sphere. • Only mass interior to sphere affects the observer. • The mass exterior to sphere does not exert a force on the interior. Like a Faraday Cage c. f. GR BIRKHOFF’s THEOREM: • The gravitational field within a spherical cavity embedded within an infinite medium is zero. Consider a particle P, on the edge of an isotropically expanding sphere ACCELERATION EQUATION 5

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 2: Newtonian (Dynamic) Derivation of Friedmann Equations • The Friedmann Equations Let FRIEDMANN EQUATION ACCELERATION EQUATION 2 cf = kc 2 ACCELERATION EQUATION 6

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 3: Newtonian (Energetic) Derivation of Friedmann Equations • Derivation from Energetic arguments : Friedmann Equation Analogous assumption: Ignore matter outside sphere of radius r(t) in expanding homogeneous universe Conservation of Energy Kinetic Energy Potential Energy Friedmann Equation 7

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 3: Newtonian (Energetic) Derivation of Friedmann Equations • Dynamics in the Newtonian Universe Evolution of universes R U>0 U=0 J U>0: RHS >0 for all time J U<0: RHS will become negative at some value of J U=0: RHS >0 for all time U<0 t U~ -kc 2 8

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 3: Newtonian (Energetic) Derivation of Friedmann Equations • The Fluid Equation First Law of Thermodynamics Homogeneity no bulk heat flow (adiabatic) d. Q=d. S=0 THE FLUID EQUATION 9

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 3: Newtonian (Energetic) Derivation of Friedmann Equations • The Acceleration Equation Friedmann Equation Fluid Equation Acceleration Equation • For any positive Energy Density the Universe is deccelerating • Pressure is always positive for baryons, photons, neutrinos, even WIMPS • but if P<-e/3 ? ? ? 10

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 4: Summary • Newtonian Cosmology • Can derive cosmological equations from a purely Newtonian Perspective Friedmann Equation Acceleration Equation Fluid Equation However it’s a bit 怪しい, e. g. , • Finite sphere embedded in Universe - violates HOMOGENEITY - special place ! • Preferred direction towards centre of sphere - violates ISOTROPY • Euclidean flat Universe is assumed 11

10/28/2020 Chris Pearson : Fundamental Cosmology 3: Newtonian Cosmology ISAS -2003 NEWTONIAN COSMOLOGY 3. 4: Summary Fundamental Cosmology 3. Newtonian Cosmology 終 Fundamental Cosmology 4. General Relativistic Cosmology 次： 12