Fundamental Cosmology: 5.The Equation of State Fundamental

Home , Perfect fluid

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

1 8pG Deriving the necessary components of The Einstein Field Equation 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)

† 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 ¯

† 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 ¯

† † † 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 †

† 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 †

† 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

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

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

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. as; 8pGr W = Matter kc 2 †m 3H 2 1 Wm + WL - = 2 2 L R H W = Cosmological Constant L 3H 2 † 2 Wm + WL - Wk =1 kc W = Curvature k R2H 2 † † 18

† † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55..22:: TTyypespes ooff PresPresssuurere • The Cosmological Constant L Lというのは？？ Candidates (Need component with constant energy density as Universe expands/contracts) • A constant of integration in General Relativity • Another (anti) gravitational constant • Zero-point for the energy density in quantum theory (energy density of the vacuum) • New scalar field (Quintessence)

Vacuum Energy ? Wm - associated with real particles • Quantum Mechanics: zero point to energy density of the vacuum ? WL - associated with virtual particles E t h • Particle/antiparticle pairs continually created and annihilated D D £ 2 95 -3 ‹ • Prediction from Quantum Mechanics = rL~10 kg m 120 orders of magnitude too high ! † “Quintessence” - The Fifth Element • Rolling homogeneous scalar field behaving like a decaying cosmological constant (i.e. NOT CONSTANT ) • Eventually attain the true vacuum energy (energy zero point)

• Strange that at this epoch is small but >0 WL ª Wm 19 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.5.5:: DDependependenceence ofof GGeometryeometry onon WW 2 • W decides the fate of the Universe kc L=0 = W -1 H 2R2

r=rc Wo=1 Flat space

r>rc Wo>1 Closed (spherical) space

20 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.5.5:: DDependependenceence ofof GGeometryeometry onon WW Integral Source Counts at 60mm • W - What does it all mean ? 8 SPICA (2.3mJy)

R Evolution of universes 4 ASTRO-F (20mJy) open W=0 open W<1 2 closed W=1

umber / sq. deg) 0 IRAS counts lg (N Omega=0 -2 Omega=0.1 Omega=1 Omega=2 closed W>1 -4

-10 -8 -6 -4 -2 0 2 1mJy 1mJy 1Jy t lg (Flux) {Jy} W<1 : low density, expands forever W=0 : no matter, expands forever Unfortunately, W=1 : expands forever gradually slowing Universe not that simple

W>1 : expand to maximum and then re-contract Galaxy Evolution 21 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.6.6:: TTyyppeess ooff UUnniviveersrsee -3 Ê R ˆ • Matter only (k=0) -3 rmatter µ R r = r0Á ˜ Ë Ro ¯ Friedmann equation • • 2 3 1/ 2 R 8 G 2 • Ê 3 ˆ p r † R 8pGroRo -3 8pGroRo -1/ 2 2 = 2 = R R = Á ˜ R R 3 R 3 † Ë 3 ¯

1/ 2 2 / 3 R Ê 8pGr R 3 ˆ t R µ t integrating R1/ 2dR = Á o o ˜ dt Ú 3 Ú -2 † † 0 Ë ¯† o r µ t 2 2 H = ﬁ t = H-1 ª13Gyr 3t 0 3 o ) ) r † r R) ( (

( † lg lg lg

Slope -3 Slope 2/3 † † Slope -2

lg(R) lg(t) lg(t) 22 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.6.6:: TTyyppeess ooff UUnniviveersrsee -4 Ê R ˆ • Radiation only (k=0) -4 rradiation µ R r = r0Á ˜ Ë Ro ¯ Friedmann equation • • 2 4 1/ 2 R 8 G 2 • Ê 3 ˆ p r † R 8pGroRo -4 8pGroRo -1 2 = 2 = R R = Á ˜ R R 3 R 3 † Ë 3 ¯

1/ 2 1/ 2 R Ê 8pGr R 3 ˆ t R µ t integrating RdR = Á o o ˜ dt Ú 3 Ú -2 † † 0 Ë ¯ †o r µ t 1 1 H = ﬁ t = H-1 ª 9.7Gyr 2t 0 2 o ) ) r † r ( R) (

( † lg lg lg

Slope -4 Slope 1/2 † † Slope -2

lg(R) lg(t) lg(t) 23 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.6.6:: TTyyppeess ooff UUnniviveersrsee -3 -3 Ê R ˆ • Matter only (k = -1) rmatter µ R r = r0Á ˜ Ë Ro ¯ Friedmann equation • • Ê 3 ˆ 2 2 2 8pGroRo -1 2 • R 8pGr kc † 2 = - R = Á ˜ R + c R > 0 " t R2 3 R2 Ë 3 † ¯

2 / 3 Small t R µ t g † g † † R R˙ Æ c † 1 large t Æ 0 R R µ±t Æ • t 24

† 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.6.6:: TTyyppeess ooff UUnniviveersrsee -3 -3 Ê R ˆ • Matter only (k = +1) rmatter µ R r = r0Á ˜ Ë Ro ¯ Friedmann equation • 3 2 2 • Ê 8pGr R ˆ • R 8pGr kc R2 o o R-1 c 2 2 = - = Á ˜ - $ Rmax where R = 0 2 2 † Ë 3 ¯ R 3 R † 8pGr R 3 • o o 2 3R 8pGr R 3 R R = o o † † max †3 c 2 † 2 Acceleration •• 4pGr Ê •• ˆ † c R = - R Á R < 0"R˜ Equation 3 Ë ¯ † 0 t Expansion ﬂ Contraction (Oscillation) R Big Bang ﬂ Big Crunch †

t 25 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.6.6:: TTyyppeess ooff UUnniviveersrsee -3 Ê R ˆ r µ R-3 r = r w = 0 • Matter and radiation (R) matter m m,0Á ˜ r Ë Ro ¯ Ê ˆ- 4 -4 R 1 rradiation µ R rr = rr,0Á ˜ w = Ë Ro ¯ 3 R˙ † r˙ + 3 (1+ w)r = 0 r Æ r + r Fluid Equation w R w m r † Assuming r & r independent 1 ∂ 3 1 ∂ 4 r m r R + r R = 0 ﬁ both terms must seperately =0 R3 ∂t ( m ) R4 ∂t ( r ) †

At the present: rr ª 0.001rm

) rm † ro,r r , R R ( BUT, there was a time rm = rr c = o the r lg o,m present R < Rc ﬁ rr > rm Radiation Dominated Era Radiation Matter R > Rc ﬁ rm > rr Matter Dominated Era † era era rr

R << Rc ﬁ all universes are radiation dominated lg(R)

† 26

† 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.6.6:: TTyyppeess ooff UUnniviveersrsee • Matter and radiation r(t)

) rm Radiation Matter r ( the dominated dominated lg present R(t) µ t1/ 2 µ t 2/3 Radiation Matter era era rr -3 -3/ 2 -2 lg(R) rm (µ R ) µ t µ t r r the -4 -2 -8/3 present † R ) rr µ µ t µ t

r ( ) (

lg r † m Radiation Matter era era

lg(t) † 27 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 55.7.7:: EEvolutionvolution ofof ththee CCosmosmologicaologicall PaPararammeetetersrs • Evolution of the Cosmological Parameters H(t), W(t), q(t)

• -3 2 2 Ê ˆ L Ê W ˆ 2Ê W ˆ R 8 G kc R H 2 q H o q 2 p r r = roÁ ˜ = Á - ˜ = o Á - o ˜ using = H = - R 3 Ë 2 ¯ Ë 2 ¯ R2 3 R2 Ë o ¯

Ï 2 3¸ 2 2 Wo Ê 3Wo ˆÊ Ro ˆ Ê Ro ˆ H(t) = †H o Ì - qo + Á1 †+ qo - ˜Á ˜ + WoÁ ˜ ˝ † We can show, Ó 2 Ë 2 ¯Ë R ¯ Ë R ¯ ˛ H(t)2 H 2 f Ro = o ( R ) 3 3 Ê R ˆ W Ê Ro ˆ o o Á -1˜ + q WoÁ ˜ Ë ( R ) ¯ o Ë R ¯ q(t) = 2 W(t) = Ro Ro f f R † ( R ) ( ) Ï ¸ -1/ 2 t 1Ô W Ê 3W ˆ Ê W ˆ R 2Ô R H t o o o q 1 o q o d o o o = = Ú Ì R -Á - o - ˜ + Á - o˜ R ˝ R † t 0 †o Ë 2 ¯ Ë 2 ¯( ) ( ) o ÓÔ ( R ) ˛Ô These relationships are general for all cosmologies 28 † 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.8:8: SUSUMMARMMARYY • Where are we now ? Shown that Ê ˆ- 3 -3 R for a matter dominated universe rmatter µ R rm = rm,0Á ˜ w = 0 Ë Ro ¯ Ê ˆ- 4 for a radiation dominated universe -4 R 1 rradiation µ R rr = rr,0Á ˜ w = Ë Ro ¯ 3 Introduced: † R˙ The Hubble Parameter H = Measure age of Universe † R 8pGr r The Density Parameter W = 2 = Measure the density of the Universe 3H rc † R˙˙ R The Decceleration Parameter q = - Measure acceleration of expansion of the Universe R˙ 2 †

L 2Ê W ˆ The Cosmological Constant = H Á - q˜ The Vacuum Energy of the Universe † 3 Ë 2 ¯ 29

† 11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 THE EQUATION OF STATE 5.5.8:8: SUSUMMARMMARYY

FundFundamentamentalal CosCosmologymology 55.. TThhee EEqquuaattioionn ooff SSttaattee 終終終

FundFundamentamentalal CosCosmologymology 次次次：：：6.6. CCosmologicaosmologicall WWorldorld MMododelsels