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11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
1
THE EQUATION OF STATE
“"Prediction is difficult, especially the future.”!!!—!Niels Bohr
““"Prediction is difficult, especially the future"Prediction is difficult, especially the future..””!!!!!!—!—!Niels Niels BohrBohr
Fundamental Cosmology: 5.The Equation of StateFundamental Cosmology: 5.Fundamental Cosmology: 5.The Equation of StateThe Equation of State
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.1: The Equation of State5.1: The Equation of State• The story so far
†
Gik = Rkl -12
gikR =8pGc 4 T ik
Deriving the necessary components of The Einstein Field Equation• Spacetime and the Energy within it are symbiotic• The Einstein equation describes this relationship
†
dS2 = c 2dt 2 - R2(t) dr2
1- kr2 + r2(dq 2 + sin2 qdf 2)Ê
Ë Á
ˆ
¯ ˜
The Robertson-Walker Metric defines thegeometry of the Universe
†
R2•
=8pGr
3R2 - kc 2 +
LR2
3Ê
Ë Á
ˆ
¯ ˜
†
R••
= -4pGr
3R +
LR3
Ê
Ë Á
ˆ
¯ ˜
The Friedmann Equations describe the evolution of the Universe
FluidEquation
†
˙ e + 3˙ R R
(e + P) = 0
†
˙ r + 3˙ R R
(r +Pc 2 ) = 0
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.1: The Equation of State5.1: The Equation of State• Want to study the evolution of our Universe - but• 2 independent equations but 3 unknowns
†
R2•
=8pGr
3R2 - kc 2 +
LR2
3Ê
Ë Á
ˆ
¯ ˜
†
R••
= -4pGr
3R +
LR3
Ê
Ë Á
ˆ
¯ ˜ †
˙ r + 3˙ R R
(r +Pc 2 ) = 0
NOT INDEPENDENT !!
Need an equation of state
Relate the Pressure, P(t) to the density, r(t) (or energy density e(t) )
unknowns• Scale factor, R(t)• Pressure, P(t)• Density, r(t)
†
P = P(r)
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.1: The Equation of State5.1: The Equation of State• Consider the Universe as a perfect fluid• The Equation of State is given by;
†
P = wrc 2 = we
†
w = P /rc 2 = P /eor
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)
w = dimensionless constant
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.1: The Equation of State5.1: The Equation of State• The evolution of the energy density of the universe
†
P = Pww
 = wrwc 2
w
ÂTotal pressure is some of components
†
˙ r w + 3˙ R R
(rw +Pw
c 2 ) = ˙ r w + 3˙ R R
(1+ w)rw = 0
fidrw
rw
= -3(1+ w) dRR
Fluid Equation
integrating
†
drw
rwrow
rw
Ú = -3(1+ w) dRRRo
R
Ú
†
r‹fiE= mc 2
e
†
rw = row
RRo
Ê
Ë Á
ˆ
¯ ˜
-3(1+w )
Equation of State
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.2: The Equation of State in 5.2: The Equation of State in GRGR• Einstein equations
= e
=-P
†
2 S••
S+
S2•
+ kc 2
S2 =8pG3c 2 T1
1 =8pG3c 2 T2
2 =8pG3c 2 T3
3
S2•
+ kc 2
S2 =8pG3c 2 T0
0
1
2
†
∂dt
2
†
d(eR3)dS
+ 3PR2 = 0
actually implied by Tki;k=0
3
†
d(rR3)dS
= 0 fi r = roRo
RÊ
Ë Á
ˆ
¯ ˜
3
fi T00 = roc
2 Ro
RÊ
Ë Á
ˆ
¯ ˜
3
, T11 = 0
e - energy densityP - Pressure
Assume Dust:• P = 0• e = rc2
3
†
T 00 = e
†
T11 = T 22 = T 22 =13
e
Assume Radiation:
3
†
d(rR4 )dS
= 0 fi r = roRo
RÊ
Ë Á
ˆ
¯ ˜
4
本当にやりたいかな~~?
Result !
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.3: Types of Pressure5.3: Types of Pressure• MATTER (Dust)fi non-relativistic ideal gas
P = pressureV = volumen = number of molesM = molar massR = gas constant = 8.31J.mol-1K-1
T = temperatureN = number of particlesk = Boltzman const. = 1.38e-23JK-1= NA kNA = Avagadros Number = 6.022e23mol-1
r = densitym= mean particle massv = particle speed
Can derive from F=ma;
†
PV =13
nM v 2__
†
w ªv 2__
3c 2 <<1
Follows Ideal Gas Law
†
PV = nRT = NkT fi P =rm
kT 1
†
NkT =13
nM v 2__
fi kT =mv 2
__
3
2
†
P = wrc 2
1 2
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.2: Types of Pressure5.2: Types of Pressure• MATTER (Dust)fi non-relativistic ideal gas
†
w = P /rc 2 ª 0 fi P = 0
†
rw = row
RRo
Ê
Ë Á
ˆ
¯ ˜
-3(1+w )
†
rmatter µ R-3
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.3: Types of Pressure5.3: Types of Pressure• RADIATIONfi relativistic massless particles
P = pressureE = energyA = arean = number density of photonsm = particle massp = momentumT = temperaturel = wavelengthk = Boltzman constanth = planck constantr = densityc = speed of lightI = Intensity
†
P = wrc 2
†
w =13
1
1 2
Can derive (from )
†
F = ma =dr p dt
P =FA
†
P =e3
=13
rc 2
using
†
ng (E)dE =8p
hc( )3E 2dE
eE / kT -1
Photon number density energy spectrum
†
e(l)dl =8phc
l5dl
ehc / lkT -1
Energy density distribution
†
e(l) =4pc
I(l)
Intensity
2
†
E =r p c
Einstein
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.3: Types of Pressure5.3: Types of Pressure• RADIATIONfi relativistic massless particles
†
w = P /rc 2 ª13
fi P =13
rc 2
†
rw = row
RRo
Ê
Ë Á
ˆ
¯ ˜
-3(1+w )
†
rradiation µ R-4
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.2: Types of Pressure5.2: 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
†
r =—2F4pG
= 0
†
4pGr = —2F + L
†
—2F = 4pGr
a = -—F
Poisson equation for Gravitational Potential
Static -> a=0 (F=constant)
Gravity
t initially static universe will contractt initially expanding universe will
• expand forever• reach maximum size then contract
For a static universe
†
L = 4pGr = constant
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.3: Types of Pressure5.3: Types of Pressure• COSMOLOGICAL CONSTANTfi Vacuum Energy?
†
PL = -rc 2
†
L = 4pGr = constant fi ˙ r = 0
†
˙ r + 3˙ R R
(r +Pc 2 ) = 0Fluid
Equation
†
rw = row
RRo
Ê
Ë Á
ˆ
¯ ˜
-3(1+w )
†
rL µ R0 = constant†
w = P /rc 2 = -1
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.3: Types of Pressure5.3: Types of Pressure• Summary
ß Cosmological Constant w = -1
†
rL µ R0 = constant
ß Radiation w = 1/3
†
rradiation µ R-4
ß Matter w ª 0
†
rmatter µ R-3
†
rw = row
RRo
Ê
Ë Á
ˆ
¯ ˜
-3(1+w )
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.4: Definition of Cosmological Parameters5.4: Definition of Cosmological Parameters• The Hubble Constant Ho
†
H(t) =˙ R R
where R = R(t)The Hubble Parameter
(from lecture 2.5)
†
H0 = H(t0)
H0 =100h km s-1 Mpc-1 h =H0
100
Hubble Constant
†
t 0 ≡1/H0
t 0 = 9.8 ¥109 h-1 yr = 3.09 ¥1017 h-1sHubble Time
†
dH ≡ c /H0
dH = 3000h-1Mpc = 9.26 ¥1025 h-1mHubble Distance
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.4: Definition of Cosmological Parameters5.4: Definition of Cosmological Parameters• The Density Parameter W
Friedmann Equation (L=0)
†
R2•
=8pGr
3R2 - kc 2
†
r = rc =3H 2
8pGFor a Flat Universe (k=0)
THE CRITICAL DENSITY~ 5x10-27kg m-3
THE CRITICAL DENSITY~ 5x10-27kg m-3
What’s this ?
†
R2
R2
•
=8pGr
3-
kc 2
R2 = H 2/R2
1
THE DENSITY PARAMETERTHE DENSITY PARAMETER
Define
†
W =rrc
=8pGr3H 2
2
†
kc 2
H 2R2 = W -11
2 • W>1 ˝ k>0• W<1 ˝ k<0• W=1˝ k=0
W decides geometryof the Universe !!
W decides geometryof the Universe !!
この話に後で戻る
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.4: Definition of Cosmological Parameters5.4: Definition of Cosmological Parameters• The Deceleration Parameter qExpand SCALE FACTOR R(t) as Taylor Series around the present time to
†
R(t) = R(to) + ˙ R to(t - to) +
˙ ̇ R 2 to
(t - to)2 + .....
What’s qo/R(to)
†
R(t) ª1 + Ho(t - to) -qo
2Ho
2(t - to)2
†
Ho =˙ R R to
, H =˙ R R
†
qo = -˙ ̇ R R˙ R 2 to
, q = -˙ ̇ R R˙ R 2
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)
†
q < 0 fi ˙ ̇ R > 0
qo = THE DECCELERATION PARAMETERqo = THE DECCELERATION PARAMETER
Ho and qo are mathmatical parameters (no physics!!)
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.4: Definition of Cosmological Parameters5.4: Definition of Cosmological Parameters• The Deceleration Parameter q
†
R2•
R2 =8pGr
3-
kc 2
R2 +L3
Friedmann Equation
†
H =˙ R R
†
q = -˙ ̇ R R˙ R 2
†
W =8pGr3H 2
†
L3
= H 2 W2
- qÊ
Ë Á
ˆ
¯ ˜
†
R••
= -4pGr
3R +
LR3
Acceleration Equation
†
kc 2 = H 2R2 3W2
- q -1Ê
Ë Á
ˆ
¯ ˜ = Ho
2Ro2 3Wo
2- qo -1
Ê
Ë Á
ˆ
¯ ˜
†
HoRo
c=
k3Wo
2- qo -1
• if L=0 ‡ W=2q
• if k=0 ‡ 3W=2(q+1)
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.4: Definition of Cosmological Parameters5.4: Definition of Cosmological Parameters• The Cosmological Constant L
†
R2•
R2 = H 2 =8pGr
3-
kc 2
R2 +L3
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)†
R••
= -4pGr
3R +
LR3
Acceleration Equation
†
Wm + WL -1=kc 2
R2H 2
†
Wm + WL - Wk =1
Rewrite Friedmann eqn. as;†
Wm =8pGr3H 2
†
WL =L
3H 2
†
Wm =8pGr3H 2
†
Wk =kc 2
R2H 2
Matter
Cosmological Constant
Curvature
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
Lというのは??
5.2: Types of Pressure5.2: 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 the vacuum)• New scalar field (Quintessence)
Vacuum Energy ?
• 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
†
DE Dt £h
2• 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
Wm - associated with real particlesWL - associated with virtual particles• Quantum Mechanics: zero point to energy density of the vacuum ?
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.5: Dependence of Geometry on 5.5: Dependence of Geometry on WW• W decides the fate of the Universe
r>rc Wo>1Closed (spherical) space
Flat spacer=rc Wo=1
r<rc Wo<1Open (hyperbolic) space
†
kc 2
H 2R2 = W -1 L=0
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.5: Dependence of Geometry on 5.5: Dependence of Geometry on WW• W - What does it all mean ?
Evolution of universes
Unfortunately, Universe not that simpleGalaxy Evolution
-4
-2
0
2
4
6
8
-10 -8 -6 -4 -2 0 2
Integral Source Counts at 60mm
IRAS countsOmega=0Omega=0.1Omega=1Omega=2
lg (N
umbe
r / s
q. d
eg)
lg (Flux) {Jy}1mJy1mJy 1Jy
SPICA (2.3mJy)
ASTRO-F (20mJy)R
t
open W=0
W=0 : no matter, expands forever
open W<1
W<1 : low density, expands forever
closed W>1
W>1 : expand to maximum and then re-contract
closed W=1
W=1 : expands forever gradually slowing
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.6: Types of Universe5.6: Types of Universe• Matter only (k=0)
†
rmatter µ R-3
†
r = r0RRo
Ê
Ë Á
ˆ
¯ ˜
-3
†
R2•
R2 =8pGr
3
Friedmann equation
†
R2•
R2 =8pGroRo
3
3R-3
†
R•
=8pGroRo
3
3Ê
Ë Á
ˆ
¯ ˜
1/ 2
R-1/ 2
†
R1/ 2dR0
R
Ú =8pGroRo
3
3Ê
Ë Á
ˆ
¯ ˜
1/ 2
dto
t
Ú
†
R µ t 2 / 3
†
r µ t-2
†
H =23t
fi t0 =23
Ho-1 ª13Gyr
integrating
lg (R
)
lg(t)
Slope 2/3
lg (r
)
lg(R)
Slope -3
lg(t)
lg (r
)
Slope -2
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.6: Types of Universe5.6: Types of Universe• Radiation only (k=0)
†
rradiation µ R-4
†
r = r0RRo
Ê
Ë Á
ˆ
¯ ˜
-4
†
R2•
R2 =8pGr
3
Friedmann equation
†
R2•
R2 =8pGroRo
4
3R-4
†
R•
=8pGroRo
3
3Ê
Ë Á
ˆ
¯ ˜
1/ 2
R-1
†
RdR0
R
Ú =8pGroRo
3
3Ê
Ë Á
ˆ
¯ ˜
1/ 2
dto
t
Ú
†
R µ t1/ 2
†
r µ t-2
†
H =12t
fi t0 =12
Ho-1 ª 9.7Gyr
integrating
lg(t)
lg (R
)
Slope 1/2
lg (r
)
lg(R)
Slope -4
lg(t)
lg (r
)
Slope -2
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.6: Types of Universe5.6: Types of Universe• Matter only (k = -1)
†
rmatter µ R-3
†
r = r0RRo
Ê
Ë Á
ˆ
¯ ˜
-3
†
R2•
R2 =8pGr
3-
kc 2
R2
Friedmann equation
†
R2•
> 0 " t
R
t
gg†
R2•
=8pGroRo
3
3Ê
Ë Á
ˆ
¯ ˜ R-1 + c 2
†
˙ R Æ c1R
Æ 0
R µ±t Æ •
large t
Small t
†
R µ t 2 / 3
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.6: Types of Universe5.6: Types of Universe• Matter only (k = +1)
†
rmatter µ R-3
†
r = r0RRo
Ê
Ë Á
ˆ
¯ ˜
-3
†
R2•
R2 =8pGr
3-
kc 2
R2
Friedmann equation
†
$ Rmax where R2•
= 0
†
R2•
=8pGroRo
3
3Ê
Ë Á
ˆ
¯ ˜ R-1 - c 2
†
Rmax =8pGroRo
3
3c 2
c2
0
†
8pGroRo3
3R
t
†
R2•
†
R••
= -4pGr
3R R
••
< 0"RÊ Ë Á
ˆ ¯ ˜
AccelerationEquation
Expansion fl Contraction (Oscillation)Big Bang fl Big Crunch
R
t
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.6: Types of Universe5.6: Types of Universe• Matter and radiation r(R)
†
rmatter µ R-3 rm = rm,0RRo
Ê
Ë Á
ˆ
¯ ˜
-3
w = 0
†
rradiation µ R-4 rr = rr,0RRo
Ê
Ë Á
ˆ
¯ ˜
-4
w =13
†
˙ r w + 3˙ R R
(1+ w)rw = 0 r Æ rm + rrFluid Equation
†
1R3
∂∂t
rmR3( ) +1
R4∂∂t
rrR4( ) = 0
Assuming rr & rm independentfi both terms must seperately =0
lg (r
)
lg(R)
Radiationera
Matterera
thepresent
rm
rr
At the present: rr ª 0.001rm
BUT, there was a time
†
rm = rr, Rc =ro,r
ro,m
Ro
†
R < Rc fi rr > rm
R > Rc fi rm > rr
Radiation Dominated Era
Matter Dominated Era
†
R << Rc fi all universes are radiation dominated
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.6: Types of Universe5.6: Types of Universe• Matter and radiation r(t)
lg (r
)
lg(R)
Radiationera
Matterera
thepresent
rm
rr
lg(t)
lg ( r
)
rm
rr
Radiationera
Matterera
thepresent
†
R t( ) 2/1tµ 3/2tµ
†
rm µ R-3( ) 2/3-µ t 2-µ t
†
rr µ R-4( ) 2-µ t 3/8-µ t
Radiationdominated
Matterdominated
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.7: Evolution of the Cosmological Parameters5.7: Evolution of the Cosmological Parameters• Evolution of the Cosmological Parameters H(t), W(t), q(t)
†
L3
= H 2 W2
- qÊ
Ë Á
ˆ
¯ ˜ = Ho
2 Wo
2- qo
Ê
Ë Á
ˆ
¯ ˜
†
R2•
R2 = H 2 =8pGr
3-
kc 2
R2
†
r = roRRo
Ê
Ë Á
ˆ
¯ ˜
-3
We can show,
†
H(t)2 = Ho2 Wo
2- qo + 1+ qo -
3Wo
2Ê
Ë Á
ˆ
¯ ˜
Ro
RÊ
Ë Á
ˆ
¯ ˜
2
+ WoRo
RÊ
Ë Á
ˆ
¯ ˜
3Ï Ì Ó
¸ ˝ ˛
H(t)2 = Ho2 f Ro
R( )
†
W(t) =Wo
Ro
RÊ
Ë Á
ˆ
¯ ˜
3
f RoR( )
†
q(t) =
Wo
2RoR( )
3-1Ê
Ë Á ˆ
¯ ˜ + qo
f RoR( )
†
Hoto =to
t o
=WoRoR( )
-3Wo
2- qo -1
Ê
Ë Á
ˆ
¯ ˜ +
Wo
2- qo
Ê
Ë Á
ˆ
¯ ˜
RoR( )
2Ï Ì Ô
Ó Ô
¸ ˝ Ô
˛ Ô 0
1Ú
-1/ 2
d RoR( )
using
These relationships are general for all cosmologies
11/2/03 Chris Pearson : Fundamental Cosmology 5: The Equation of State ISAS -2003
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THE EQUATION OF STATE
5.8: SUMMARY5.8: SUMMARY• Where are we now ?Shown that
for a matter dominated universe
†
rmatter µ R-3 rm = rm,0RRo
Ê
Ë Á
ˆ
¯ ˜
-3
w = 0
†
rradiation µ R-4 rr = rr,0RRo
Ê
Ë Á
ˆ
¯ ˜
-4
w =13
for a radiation dominated universe
Introduced:
†
H =˙ R R
The Hubble Parameter Measure age of Universe
†
W =8pGr3H 2 =
rrc
The Density Parameter Measure the density of the Universe
†
q = -˙ ̇ R R˙ R 2
The Decceleration Parameter Measure acceleration of expansion of the Universe
†
L3
= H 2 W2
- qÊ
Ë Á
ˆ
¯ ˜ The Cosmological Constant The Vacuum Energy of the Universe
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THE EQUATION OF STATE
5.8: SUMMARY5.8: SUMMARY
Fundamental CosmologyFundamental Cosmology5. The Equation of State5. The Equation of State 終終終
次:次:次:Fundamental CosmologyFundamental Cosmology
6. Cosmological World Models6. Cosmological World Models