Fundamental Cosmology: 5.The Equation of State 02/01/2019 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

5.1: The Equation of State The story so far 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.1: The Equation of State The story so far 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

02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 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) NOT INDEPENDENT !! Need an equation of state Relate the Pressure, P(t) to the density, r(t) (or energy density e(t) )

w = dimensionless constant 02/01/2019 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)

Total pressure is some of components 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.1: The Equation of State The evolution of the energy density of the universe Total pressure is some of components Fluid Equation integrating Equation of State

5.2: The Equation of State in GR 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.2: The Equation of State in GR Einstein equations 本当にやりたいかな〜〜? 1 2 =-P - energy density P - Pressure = e 2 actually implied by Tki;k=0 3 Assume Dust: P = 0 e = rc2 3 Result ! Assume Radiation: 3

5.3: Types of Pressure MATTER (Dust)  non-relativistic ideal gas 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.3: Types of Pressure MATTER (Dust)  non-relativistic ideal gas Follows Ideal Gas Law 1 Can derive from F=ma; 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 2 1 2

5.2: Types of Pressure MATTER (Dust)  non-relativistic ideal gas 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.2: Types of Pressure MATTER (Dust)  non-relativistic ideal gas

5.3: Types of Pressure RADIATION  relativistic massless particles 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.3: Types of Pressure RADIATION  relativistic massless particles Photon number density energy spectrum Energy density distribution Intensity 1 using 1 2 Can derive (from ) 2 Einstein 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

5.3: Types of Pressure RADIATION  relativistic massless particles 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.3: Types of Pressure RADIATION  relativistic massless particles

COSMOLOGICAL CONSTANTって 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 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 Poisson equation for Gravitational Potential Static -> a=0 (F=constant) For a static universe

5.3: Types of Pressure COSMOLOGICAL CONSTANT  Vacuum Energy? Fluid 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.3: Types of Pressure COSMOLOGICAL CONSTANT  Vacuum Energy? Fluid Equation

5.3: Types of Pressure Matter w  0 Radiation w = 1/3 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.3: Types of Pressure Summary Matter w  0 Radiation w = 1/3 Cosmological Constant w = -1

5.4: Definition of Cosmological Parameters 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.4: Definition of Cosmological Parameters The Hubble Constant Ho The Hubble Parameter (from lecture 2.5) Hubble Constant Hubble Time Hubble Distance

5.4: Definition of Cosmological Parameters 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.4: Definition of Cosmological Parameters The Density Parameter W /R2 1 Friedmann Equation (L=0) For a Flat Universe (k=0) THE CRITICAL DENSITY ~ 5x10-27kg m-3 What’s this ? THE DENSITY PARAMETER Define 2 1 2 W>1  k>0 W<1  k<0 W=1 k=0 W decides geometry of the Universe !! この話に後で戻る

5.4: Definition of Cosmological Parameters 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.4: Definition of Cosmological Parameters The Deceleration Parameter q Expand SCALE FACTOR R(t) as Taylor Series around the present time to /R(to) What’s qo 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 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.4: Definition of Cosmological Parameters The Deceleration Parameter q Acceleration Equation Friedmann Equation if L=0  W=2q if k=0  3W=2(q+1)

5.4: Definition of Cosmological Parameters 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.4: Definition of Cosmological Parameters The Cosmological Constant L Acceleration Equation Friedmann Equation 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) Rewrite Friedmann eqn. as; Matter Cosmological Constant Curvature

5.2: Types of Pressure The Cosmological Constant L Lというのは？？ 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.2: Types of Pressure 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 WL - associated with virtual particles Quantum Mechanics: zero point to energy density of the vacuum ? Particle/antiparticle pairs continually created and annihilated 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

5.5: Dependence of Geometry on W 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.5: Dependence of Geometry on W W decides the fate of the Universe L=0 r<rc Wo<1 Open (hyperbolic) space Flat space r=rc Wo=1 r>rc Wo>1 Closed (spherical) space

5.5: Dependence of Geometry on W 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.5: Dependence of Geometry on W W - What does it all mean ? open W<1 W<1 : low density, expands forever closed W=1 W=1 : expands forever gradually slowing closed W>1 W>1 : expand to maximum and then re-contract R t Evolution of universes open W=0 W=0 : no matter, expands forever Unfortunately, Universe not that simple Galaxy Evolution

5.6: Types of Universe Matter only (k=0) Friedmann equation 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.6: Types of Universe Matter only (k=0) Friedmann equation integrating lg (r) lg(R) Slope -3 lg (R) lg(t) Slope 2/3 lg(t) lg (r) Slope -2

5.6: Types of Universe Radiation only (k=0) Friedmann equation 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.6: Types of Universe Radiation only (k=0) Friedmann equation integrating lg (r) lg(R) Slope -4 lg(t) lg (R) Slope 1/2 lg(t) lg (r) Slope -2

5.6: Types of Universe Matter only (k = -1) Friedmann equation Small t 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.6: Types of Universe Matter only (k = -1) Friedmann equation Small t R t g large t

Expansion  Contraction (Oscillation) 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.6: Types of Universe Matter only (k = +1) Friedmann equation c2 t Acceleration Equation Expansion  Contraction (Oscillation) Big Bang  Big Crunch R t

Radiation Dominated Era 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.6: Types of Universe Matter and radiation r(R) Fluid Equation Assuming rr & rm independent  both terms must seperately =0 lg (r) lg(R) Radiation era Matter the present rm rr At the present: rr  0.001rm BUT, there was a time Radiation Dominated Era Matter Dominated Era

5.6: Types of Universe rm rr rr rm Matter and radiation r(t) Radiation 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.6: Types of Universe Matter and radiation r(t) lg (r) lg(R) Radiation era Matter the present rm rr Radiation dominated Matter dominated lg(t) lg (r) rm rr Radiation era Matter the present

5.7: Evolution of the Cosmological Parameters 02/01/2019 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

5.8: SUMMARY Where are we now ? Shown that 02/01/2019 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003 5.8: SUMMARY Where are we now ? Shown that for a matter dominated universe for a radiation dominated universe Introduced: The Hubble Parameter Measure age of Universe The Density Parameter Measure the density of the Universe The Decceleration Parameter Measure acceleration of expansion of the Universe The Cosmological Constant The Vacuum Energy of the Universe

終 次： 5.8: SUMMARY Fundamental Cosmology 5. The Equation of State 02/01/2019 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：