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“ "Prediction is difficult, especially the future .” — Niels Bohr

Fundamental Cosmology: 5. The Equation of State. “ "Prediction is difficult, especially the future .” — Niels Bohr. Deriving the necessary components of The Einstein Field Equation Spacetime and the Energy within it are symbiotic The Einstein equation describes this relationship.

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“ "Prediction is difficult, especially the future .” — Niels Bohr

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  1. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Fundamental Cosmology: 5.The Equation of State “"Prediction is difficult, especially the future.” — Niels Bohr

  2. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 • Deriving the necessary components of The Einstein Field Equation • Spacetime and the Energy within it are symbiotic • The Einstein equation describes this relationship The Robertson-Walker Metric defines the geometry of the Universe The Friedmann Equations describe the evolution of the Universe Fluid Equation 5.1: The Equation of State • The story so far

  3. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 NOT INDEPENDENT !! 5.1: The Equation of State • Want to study the evolution of our Universe - but • 2 independent equations but 3 unknowns • unknowns • Scale factor,R(t) • 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) )

  4. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.1: The Equation of State • Consider the Universe as a perfect fluid • The Equation of State is given by; or w = dimensionless constant We will discover • Matter w  0 • Radiation w = 1/3 • Cosmological Constant w = -1 • (Incompressible Fluid w = -1) • (Dark Energy w = -1/3)

  5. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Total pressure is some of components Fluid Equation integrating Equation of State 5.1: The Equation of State • The evolution of the energy density of the universe

  6. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 1 2 =-P 2 3 = e actually implied by Tki;k=0 • Assume Dust: • P = 0 • e = rc2 3 Result ! Assume Radiation: 3 5.2: The Equation of State in GR • Einstein equations 本当にやりたいかな〜〜? • - energy density P - Pressure

  7. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 1 Follows Ideal Gas Law Can derive from F=ma; 2 1 2 5.3: Types of Pressure • MATTER (Dust)  non-relativistic ideal gas • P = pressure • V = volume • n = number of moles • M = molar mass • R = gas constant = 8.31J.mol-1K-1 • T = temperature • N = number of particles • k = Boltzman const. = 1.38e-23JK-1= NA k • NA=Avagadros Number = 6.022e23mol-1 • r = density • = mean particle mass v = particle speed

  8. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.2: Types of Pressure • MATTER (Dust)  non-relativistic ideal gas

  9. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Photon number density energy spectrum Energy density distribution Intensity Einstein 2 Can derive (from ) 1 2 5.3: Types of Pressure • RADIATION  relativistic massless particles 1 using P = pressure E = energy A = area n = number density of photons m = particle mass p = momentum T = temperature l = wavelength k = Boltzman constant h =planck constant r = density c = speed of light I = Intensity

  10. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.3: Types of Pressure • RADIATION  relativistic massless particles

  11. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Poisson equation for Gravitational Potential 5.2: Types of Pressure • 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 • initially static universe will contract • initially expanding universe will • expand forever • reach maximum size then contract Static -> a=0 (F=constant) For a static universe

  12. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Fluid Equation 5.3: Types of Pressure • COSMOLOGICAL CONSTANT  Vacuum Energy?

  13. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 • Matter w  0 • Radiation w = 1/3 • Cosmological Constant w = -1 5.3: Types of Pressure • Summary

  14. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 The Hubble Parameter (from lecture 2.5) Hubble Constant Hubble Time Hubble Distance 5.4: Definition of Cosmological Parameters • The Hubble Constant Ho

  15. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Friedmann Equation (L=0) /R2 THE CRITICAL DENSITY ~ 5x10-27kg m-3 For a Flat Universe (k=0) Define 2 2 1 1 What’s this ? THE DENSITY PARAMETER • W>1  k>0 • W<1  k<0 • W=1 k=0 W decides geometry of the Universe !! 5.4: Definition of Cosmological Parameters • The Density Parameter W この話に後で戻る

  16. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 /R(to) What’sqo qo= THE DECCELERATION PARAMETER Ho and qo are mathmatical parameters (no physics!!) Universe is decelerating (relative velocity between 2 points is decreasing) Universe is accelerating (relative velocity between 2 points is increasing) 5.4: Definition of Cosmological Parameters • The Deceleration Parameter q Expand SCALE FACTOR R(t) as Taylor Series around the present time to

  17. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Friedmann Equation • if L=0  W=2q • if k=0 3W=2(q+1) 5.4: Definition of Cosmological Parameters • The Deceleration Parameter q Acceleration Equation

  18. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Friedmann Equation Acceleration Equation Rewrite Friedmann eqn. as; Matter Cosmological Constant Curvature 5.4: Definition of Cosmological Parameters • The Cosmological Constant L • acceleration equation, L opposite sign to G& r (gravity) • Acts as “negative pressure” or “anti gravity” • Accelerates the expansion of the Universe (decelerate if L<0)

  19. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Lというのは？？ Wm- associated with real particles WL- associated with virtual particles • Quantum Mechanics: zero point to energy density of the vacuum ? • Particle/antiparticle pairs continually created and annihilated 5.2: Types of Pressure • The Cosmological Constant L 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 thevacuum) • New scalar field (Quintessence) Vacuum Energy ? • Prediction from Quantum Mechanics = rL~1095kg m-3 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 Wm

  20. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 r<rcWo<1 Open (hyperbolic) space r>rcWo>1 Closed (spherical) space r=rcWo=1 Flat space 5.5: Dependence of Geometry on W L=0 • W decides the fate of the Universe

  21. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 R open W=0 open W<1 closed W=1 closed W>1 t W<1 : low density, expands forever Unfortunately, Universe not that simple W=0 : no matter, expands forever W=1 : expands forever gradually slowing Galaxy Evolution W>1 : expand to maximum and then re-contract 5.5: Dependence of Geometry on W • W - What does it all mean ? Evolution of universes

  22. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 lg (r) lg (r) lg (R) Slope -3 Slope 2/3 Slope -2 lg(R) lg(t) lg(t) 5.6: Types of Universe • Matter only (k=0) Friedmann equation integrating

  23. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 lg (r) lg (r) lg (R) Slope -4 Slope 1/2 Slope -2 lg(R) lg(t) lg(t) 5.6: Types of Universe • Radiation only (k=0) Friedmann equation integrating

  24. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 g g Smallt R larget t 5.6: Types of Universe • Matter only (k = -1) Friedmann equation

  25. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 c2 Acceleration Equation R t 0 t 5.6: Types of Universe • Matter only (k = +1) Friedmann equation Expansion  Contraction (Oscillation) Big Bang Big Crunch

  26. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 Assuming rr & rm independent  both terms must seperately =0 Fluid Equation rm the present BUT, there was a time lg (r) Radiation Dominated Era Radiation era Matter era rr Matter Dominated Era lg(R) 5.6: Types of Universe • Matter and radiation r(R) At the present: rr0.001rm

  27. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 rr the present lg (r) rm Radiation era Matter era lg(t) rm the present lg (r) Radiation era Matter era rr lg(R) 5.6: Types of Universe • Matter and radiation r(t) Matter dominated Radiation dominated

  28. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.7: Evolution of the Cosmological Parameters • Evolution of the Cosmological Parameters H(t), W(t), q(t) using We can show, These relationships are general for all cosmologies

  29. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 for a matter dominated universe for a radiation dominated universe The Hubble Parameter Measure age of Universe Measure the density of the Universe The Density Parameter The Decceleration Parameter Measure acceleration of expansion of the Universe The Cosmological Constant The Vacuum Energy of the Universe 5.8: SUMMARY • Where are we now ? Shown that Introduced:

  30. Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.8: SUMMARY 終 Fundamental Cosmology 5. The Equation of State Fundamental Cosmology 6. Cosmological World Models 次：

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