1. 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

2. 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

3. 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) )

4. 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)

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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)

17. 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)

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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：