11/2/03 chris pearson : fundamental cosmology 5: the equation …cpp/teaching/cosmology/... ·...

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


2


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


3


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)


4


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


5


5.1: The Equation of State5.1: The Equation of State• The evolution of the energy density of the universe

†

P = Pww

Â = 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


6


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 !


7


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


8


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


9


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


10


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


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


12


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


13


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 )


14


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


15


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


16


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!!)


17


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)


18


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


19


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 ?


20


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


21


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


22


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


23


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


24


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


25


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


26


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


27


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


28


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


29


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


30


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

11/2/03 chris pearson : fundamental cosmology 5: the equation …cpp/teaching/cosmology/... ·...

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