11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE FuFundndaammententaall CCoosmsmoolologygy:: 55..TheThe EquationEquation ofof StateState ““"P"Predictionrediction isis difficult,difficult, espespeciallyecially ththee futurefuture..””!!!!!! ——!!NielsNiels BohrBohr 1 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.1:1: TheThe EquatEquationion ofof StStatatee • The story so far Deriving the necessary components of The Einstein Field Equation 1 8pG ik • Spacetime and the Energy within it are symbiotic Gik = Rkl - gikR = 4 T 2 c • The Einstein equation describes this relationship 2 2 2 2 2 Ê dr 2 2 2 2 ˆ The Robertson-Walker Metric defines the dS = c dt - R (t)Á + r (dq + sin qdf )˜ Ë 1- kr2 ¯ geometry of the Universe † •• 4pGr Ê LRˆ R = - R Á + ˜ 3 Ë 3 ¯ † The Friedmann Equations describe • 2 the evolution of the Universe 2 8pGr 2 2 Ê LR ˆ R = R - kc Á + ˜ 3 Ë 3 ¯ † R˙ R˙ P e˙ + 3 (e + P) = 0 r˙ + 3 (r + ) = 0 Fluid R R c 2 Equation † 2 † † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.1:1: TheThe EquatEquationion ofof StStatatee • Want to study the evolution of our Universe - but • 2 independent equations but 3 unknowns • 2 ˙ 2 8pGr 2 2 Ê LR ˆ R P R = R - kc Á + ˜ r˙ + 3 (r + ) = 0 3 Ë 3 ¯ R c 2 •• 4pGr Ê LRˆ † R = - R Á + ˜ unknowns 3† Ë 3 ¯ • Scale factor, R(t) NOT INDEPENDENT !! • Pressure, P(t) • Density, r(t) † Need an equation of state Relate the Pressure, P(t) to the density, r(t) (or energy density e(t) ) P = P(r) 3 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.1:1: TheThe EquatEquationion ofof StStatatee • Consider the Universe as a perfect fluid • The Equation of State is given by; 2 2 P = wrc = we or w = P /rc = P /e w = dimensionless constant We will discover † ß Matter w ª 0 † ß Radiation w = 1/3 ß Cosmological Constant w = -1 ß (Incompressible Fluid w = -1) ß (Dark Energy w = -1/3) 4 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.1:1: TheThe EquatEquationion ofof StStatatee 2 • The evolution of the energy density of the universe E= mc r‹fie Total pressure is some of components 2 P = ÂPw = Âwrwc w w R˙ P R˙ † Fluid Equation r˙ + 3 (r + w ) = r˙ + 3 (1+ w)r = 0 w R w c 2 w R w dr dR fi w = -3(1+ w) † rw R rw dr R dR integrating w = -3(1+ w) Ú r Ú R r ow w Ro † -3(1+w ) Ê R ˆ Equation of State rw = row Á ˜ † Ë Ro ¯ 5 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..22:: TThhee EEqquuaattiioonn ooff StStaattee iinn GGRR • Einstein equations 本当にやりたいかな~~? •• • 2 2 1 S S + kc 8pG 1 8pG 2 8pG 3 2 + 2 = 2 T1 = 2 T2 = 2 T3 e - energy density S S 3c 3c 3c P - Pressure • 2 2 2 S + kc 8pG = T 0 S 2 3c 2 0 =-P ∂ 2 d(eR3) + 3PR2 = 0 3 dt dS = e k † actually implied by T i;k=0 Assume Dust: 3 Ê ˆ 3 Ê ˆ 3 • P = 0 d(rR ) Ro 0 2 Ro 1 3 = 0 fi r = roÁ ˜ fi T0 = r†oc Á ˜ , T1 = 0 • e = rc2 dS Ë R†¯ Ë R ¯ Result ! Assume Radiation: 4 1 d(rR4 ) Ê R ˆ 11 † 22 22 00 0 o T = T = T = e T = e 3 = fi r = roÁ ˜ 3 dS Ë R ¯ 6 † † † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..33:: TTyypespes ooff PresPresssuurere • MATTER (Dust) fi non-relativistic ideal gas r 1 Follows Ideal Gas Law PV = nRT = NkT fi P = kT m __ 1 2 Can derive from F=ma; PV = nM v 3 P = pressure V = volume † n = number of moles M = molar mass R = gas constant = 8.31J.mol-1K-1 2 † __ T = temperature __ 2 1 2 mv N = number of particles NkT = nM v fi kT = -1 k = Boltzman const. = 1.38e-23JK = NA k 1 2 3 3 -1 NA = Avagadros Number = 6.022e23mol r = density __ m= mean particle mass 2 P = wrc 2 v v = particle speed † w ª <<1 3c 2 7 † † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..22:: TTyypespes ooff PresPresssuurere • MATTER (Dust) fi non-relativistic ideal gas w = P /rc 2 ª 0 fi P = 0 -3(1+w ) Ê R ˆ rw = row Á ˜ † Ë Ro ¯ -3 rmatter µ R † 8 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..33:: TTyypespes ooff PresPresssuurere • RADIATION fi relativistic massless particles Photon number density energy spectrum Energy density distribution Intensity 8p E 2dE 8phc dl 4p 1 using ng (E)dE = 3 E / kT e(l)dl = e(l) = I(l) (hc) e -1 l5 ehc / lkT -1 c 1 2 Einsteinr E = p c 2 dpr F = ma = † † † dt e 1 2 Can derive (from ) P = pressure F P = = rc P = 3 3 E = energy A A = area n = number density of photons † m = particle mass † p = momentum T = temperature l = wavelength † k = Boltzman constant 2 1 h = planck constant P = wrc w = r = density 3 c = speed of light I = Intensity 9 † † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..33:: TTyypespes ooff PresPresssuurere • RADIATION fi relativistic massless particles 1 1 w = P /rc 2 ª fi P = rc 2 3 3 -3(1+w ) Ê R ˆ rw = ro Á ˜ w R † Ë o ¯ -4 rradiation µ R † 10 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..22:: TTyypespes ooff PresPresssuurere • COSMOLOGICAL CONSTANT COSMOLOGICAL CONSTANTって • A Bit of History •Einstein’s Universe : Matter and Radiation • no CMB so Ematter>>Eradiation => Pressure=0 • Galaxies still thought as nebula, i.e. Our Universe = Our Galaxy • Stars moving randomly (toward & away from us) => Universe neither expanding nor contracting • Universe is STATIC !! • But r>0, P~0 Universe must be either expanding or contracting Gravity Poisson equation for Gravitational Potential t 2 2 initially static universe will contract — F = 4pGr — F t initially expanding universe will Static -> a=0 r = = 0 (F=constant) 4 G • expand forever a = -—F p • reach maximum size then contract 2 For a static universe 4pGr = — F + L L = 4pGr = constant † † 11 † † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..33:: TTyypespes ooff PresPresssuurere • COSMOLOGICAL CONSTANT fi Vacuum Energy? L = 4pGr = constant fi r˙ = 0 P = -rc 2 R˙ P L Fluid r˙ + 3 (r + ) = 0 Equation R c 2 † 3(1 ) † Ê ˆ- +w 2 R w = P /rc = -1 rw = row Á ˜ † Ë Ro ¯ † R0 constant rL µ †= 12 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..33:: TTyypespes ooff PresPresssuurere -3(1+w ) • Summary Ê R ˆ rw = row Á ˜ Ë Ro ¯ -3 ß Matter w ª 0 rmatter µ R † -4 ß Radiation w = 1/3 rradiation µ R † ß 0 Cosmological Constant w = -1 rL µ R = constant † 13 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.4:4: DefinitDefinitionion ofof CosmologicalCosmological ParametParametersers • The Hubble Constant Ho The Hubble Parameter R˙ (from lecture 2.5) H(t) = where R = R(t) R Hubble Constant H0 = H(t0 ) H H =100h km s-1 Mpc-1 h = 0 † 0 100 Hubble Time t 0 ≡1/H0 9 -1 17 -1 t 0 = 9.8 ¥10 h yr = 3.09 ¥10 h s † Hubble Distance dH ≡ c /H0 d 3000h-1Mpc 9.26 1025 h-1m H = = ¥ 14 † † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.4:4: DefinitDefinitionion ofof CosmologicalCosmological ParametParametersers • The Density Parameter W 1 • • Friedmann 2 8pGr 2 2 2 2 R R kc /R2 R 8pGr kc 2 Equation (L=0) = - = - = H 3 R2 3 R2 2 3H THE CRITICAL DENSITY What’s this ? For a Flat Universe (k=0) r = r = c 8 G ~ 5x10-27kg m-3 † p † Define r 8pGr 2 THE DENSITY PARAMETER W = = 2 † rc 3H 2 • W>1 ˝ k>0 kc 2 W decides geometry 2 2 = W -1† • W<1 ˝ k<0 1 H R of the Universe !! • W=1˝ k=0 この話に後で戻る 15 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.4:4: DefinitDefinitionion ofof CosmologicalCosmological ParametParametersers • The Deceleration Parameter q Expand SCALE FACTOR R(t) as Taylor Series around the present time to R˙˙ R(t) = R(t ) + R˙ (t - t ) + (t - t )2 + ..... o to o 2 to o /R(to) q R(t) ª1 + H (t - t ) - o H 2 (t - t )2 What’s qo o o 2 o o † R˙ R˙ H = , H = o R to R q = THE DECCELERATION PARAMETER † R˙˙ R R˙˙ R o q = - , q = - o 2 to 2 H and q are mathmatical parameters (no physics!!) R˙ R˙ o o † q > 0 fi R˙˙ < 0 Universe is decelerating (relative velocity between 2 points is decreasing) † q < 0 fi R˙˙ > 0 Universe is accelerating (relative velocity between 2 points is increasing) † 16 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.4:4: DefinitDefinitionion ofof CosmologicalCosmological ParametParametersers • The Deceleration Parameter q 8pGr W = Acceleration Equation 3H 2 Friedmann Equation • •• 4pGr LR R˙ 2 2 R = - R + H = R 8pGr kc L R = - + 3 3 R2 3 R2 3 † R˙˙ R q = - R˙ 2 † † † L Ê W ˆ Ê 3W ˆ 2 2Ê 3W ˆ = H 2Á - q˜ † kc 2 = H 2R2Á - q -1˜ = H R Á o - q -1˜ 3 Ë 2 ¯ Ë 2 ¯ o o Ë 2 o ¯ H R k o o = • if L=0 ‡ W=2q 3W c o - q -1 † † 2 o • if k=0 ‡ 3W=2(q+1) 17 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.4:4: DefinitDefinitionion ofof CosmologicalCosmological ParametParametersers • The Cosmological Constant L Acceleration Equation Friedmann Equation • •• 4pGr LR R2 8pGr kc 2 L R = - R + = H 2 = - + 3 3 R2 3 R2 3 • acceleration equation, L opposite sign to G& r (gravity) 8pGr • Acts as “negative pressure” or “anti gravity” Wm = 2 † 3H • Accelerates the expansion of the Universe† (decelerate if L<0) Rewrite Friedmann eqn.