New Developments and Perspectives inGeneral Relativity and Cosmology

Thesis

Dennis SmootUniversity of Illinois

Chicago, IL

Table of Contents Chapter 1 The Classical Einstein-Hilbert Action• 1.1 Structure of the Einstein-Hilbert Lagrangian • 1.2 The Relation between the Bulk and Surface Terms • 1.3 Thermodynamic Derivation of the Einstein-Hilbert Action • 1.4 Thermodynamic Interpretation of the Einstein Field Equations • 1.5 Curvature Chapter 2 Statistical Mechanics of Gravitating Systems Chapter 3 The Quantum Mechanical Perspective • 3.1 Hawking Radiation • 3.2 Unruh Effect Chapter 4 Relativistic Formalism • 4.1 Generalized Actions and Entropy Functionals • 4.2 Gravitational Energy Densities in the Universe

Chapter 6 A History of the Early Universe • 5.1 Introduction

5.1.1 Newtonian Cosmology • 5.2 The Radiation Era

5.2.1 Relativistic Cosmology 5.2.2 The Planck Epoch 5.2.3 The String Epoch 5.2.4 The GUT Epoch 5.2.5 The Inflation Epoch 5.2.6 The Electroweak Epoch 5.2.7 The Parton Epoch 5.2.8 The Hadron Epoch 5.2.9 The Lepton Epoch 5.2.10 The Nuclear Epoch

• 5.3 The Matter Era5.3.1 The Atomic Epoch

• 5.4 The Vacuum Era• 5.5 Further Thermodynamics of the Early Universe• 5.6 Free Energy in the Universe Chapter 7 Discussion and Conclusion • 6.1 The first part: Gravitation• 6.2 The second part: History of the Universe

Louis Kauffman, ProfessorDepartment of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at Chicago

A. Lewis Licht, Professor EmeritusDepartment of PhysicsUniversity of Illinois at Chicago

Advisors

Special Relativity Einstein 1905

General Relativity Einstein 1915

Goal: Quantum Gravity

Quantum Mechanics

Quantum Field Theory

Loop Gravity Ashtekar 1988

Strings

History

Modern Theories Standard Model of Particles

Quantum Field Theory in Curved SpaceTime

Quantum Gravity nonrenormalizable

First Part. Gravitation and General Relativity

Schrodinger, Heisenberg, Born 1925

Dirac, Feynman

Can Gravity be united, combined with the other Forces

What are the (basic) Degrees of Freedom

What is the underlying Mechanism

What are the relevant Observations, Experiments

Thanu Padamanabhan

Professor and Dean of Core Academic Programs of Inter-University Center for Astronomy and Astrophysics (IUCAA) at Pune , India

15-20 books

More than 200 research papers

General Relativity videos on Web

Relation between Bulk and Surface terms

R g R R R

R

x x

1( )

2

g ggg

x x x

41-

16 EHA R gd x

G

( ) ( ) [ ] bulk surR g g g g g L L

,( )bulk

surf

LL g

g

EHR L

,( ) ( )bul EH

EHk

bulk bu EHlk

L LL L g L L

g

Conventional Action

Surface term is ignored, subtracted off in modified Lagrangian.

Lagrangian is covariant (under diffeomorphisms), but neither the bulk nor the surface are.bulkL

As just shown there is a differential relation between the bulk and surface Lagrangians.

In inertial coördinates and . 0bulkL 0surfL

The conclusion is that the degrees of freedom (dof) of Gravity are located on the boundary of a Region and not in the Bulk.

surfL

Thermodynamic Derivation of the Einstein-Hilbert Action

Suppose the only known is the relation between the bulk and surface terms that was quadratic in the first derivatives of the metric. Then it is possible to determine .

bulkL

EHL

Can be done in some simpler more tractable cases, e.g. static or stationary SpaceTimes.

Require assumptions that the Entropy S is the 2-surface area and that

P

1const

Α

dS

dA

Then it can be shown that .grav EHL L R

where is the 2-surface area. A

Thermodynamic Interpretation of Einstein’s Equations

Again this is done in tractable cases, but turns out in all cases that can be calculated.

Static spherically symmetric metric 2 2 1 2 2 2 2 2( ) ( ) ( sin )ds f r dt f r dr r d d

The Einstein Equations (EE) reduce to 2

1 1(1 )f f r f r

r r ò

The equations are solved using the boundary conditions of the horizon giving

21 1 1( )

2 2 2Ba a a ò where ( )B f a and ( ) 0f a is the horizon.

Multiplying by da

2 31 1 1 44 ( )

4 4 2 8 3 r

r

Bd a d a T a d a

T PdS dE dV

T is the Stress-Energy tensor

Statistical Mechanics of Gravitating Systems

( ) 3

0

1( ) exp [ ( )]

16 EA g

E Eg M g M

Z e d d x g R f r

ò ò

2 2 1 2 2( ) ( )ds f r dt f r dr dL

( ) 0 ( )f a f a B

g is Density of States

t it

Z(β) is the Partition Function Σ is a Path Integral Sum

is Euclidean extension of RER

20

1( ) exp (4 ) ( ) exp[ ( ) ( ) ]

2

4

aZ Z a S a E a

1FE S E

T

M is the microcanonical ensemble

E-H Action is FE of spacetime

Hawking Radiation: a BH radiates in a thermal spectrum

Quantum Mechanics Law Thermodynamics BH Mechanics

0 T = const in equilibrium const on BH surface

18

2 0 0

3 0 unobtainable finitely 0 unobtainable finitely

dE TdS PdV dM dAG

dS dA

T

1. Schwarzschild metric

2. Hamilton-Jacobi equation

( , ) exp(   ( , ))t r i t r 3. Wave Function

4. Fourier decompose

( , )t r t

2 0g A A m

2 2 2 22 1(1 )

21

Mds dt dr dL

Mrr

( , ) ( )2

i tdt r f e

5. Solve for Power Spectrum of ( )f 2

8

1( )

( 1)Mf

e

( , )t r

6. The latter is a Planckian thermal spectrum with1

8T

M

Unruh Effect: a vacuum wrst an accelerated observer will contain a thermal spectrum of particles.

Quantum Mechanics cont’d

Vacuum state a massless scalar field, φ(u,w)= φ(û,ŵ) in light cone coördinates (u,w)

Flat Minkowski space with coördinates (t, x), accelerated observer coördinates (τ, ξ)

Action eom

Solutions in both coördinates

Fourier Transform and compare coëfficients: Bogoliubov transformations relating the creation and annihilation operators

Observer particle number operator is

Power spectrum is a thermal spectrum with temperature

2, ,

1[ ]

2S g d xg

2 2

( , ) 0, ( , ) 0u w u wu w u w

Particular solution in both coördinates

( , ) ( ) ( )u w A u B w ( , ) ( ) ( )u w A u B w

1/20

1ˆ ˆ ˆ ˆ ˆ( , )(2 ) 2

i u i u i w i wdu w e a e a e a e a

1/20

1 ˆ ˆ ˆ ˆˆ( , )(2 ) 2

i u i u i w i wdu w e b e b e b e b

ˆ ˆN̂ b b

2

0 ˆ ˆˆ 0 0 ( , )M MN b b d F

1( , ) exp ln

2 2

i iF

a a a a a a

2

2

1 1( , )

21a

Fae

2

aT

0

ˆ ˆ ˆ( , ) ( , )b d F a F a

Relativistic Formalism Generalized Actions and Entropy Functionals

Dark Energy Fluid, unclustered, nonclumping, negative pressure Equation of state

Problem: Cosmological Constant

Neither the EH Action

nor the EE1

82

R Rg GT

4

6

1

1A d x gR

are invariant under a shift by a constant in the Lagrangians

But the matter Lagrangian and the matter eom are invariant under a shift by a constant

Padmanabhan: Neither the DE nor the CC problem will be solved until these discrepancies are resolved.

Related facts: separation of bulk and surface terms, differential relation between bulk and surface terms, noncovariance of terms, neglect of surface term, bulk term = 0 in inertial frame, light cone (sphere) structure, the horizons of accelerated observers, horizons of black holes, entanglement across horizons or tunneling through horizons.

1p ˆ

Relativistic Formalism, cont’d Modified eom 2 0E T n n

For all null vectors n

2 proofs: 1. Using conserved Noether current

2. Using “Elastic” Entropy functional ( ) 4D

VS n d x g P n n T n n

Now the Action and the eom are invariant under a shift by a constant

“On shell” (eom are satisfied) Entropy Functional above is entropy of horizon agreeing with Wald

2 ( 2)

11

14 (Area) + corrections

4| K D DH nnn H

S mc d x

L

Lagrangian special case: Lanczos-Lovelock lagrangians

1 2 2 3 4 2 1 21 2

1 2 2 1 2 3 4 2 1 2

......

12

16n n n

n n n

a a a b b b bb bnn b b b a a a a a aR R R

L1 n nnc

L L

But the eom contain an arbitrary constant integration Λ

116 8

2P R L T

LP

R

0P

2J L E v

It can be shown LL Lagrangians are the unique lagrangians constructed from the metric and curvature tensors , satisfying the PE, and containing no derivatives of the metric above 2nd order.

0g n n n n

Gravitational Energy Densities in the Universe

Relativistic formalism, cont’d• CC decouples from dof in Actions

• Set gauge = 0

• Then DE gravity is surface not bulk

Early Universe phase transitions 4T T L

2810GUTL mm 1510WeakL mm

Casimir effect too small

Conclusion: Gravity ignores Bulk Energy.

Detectors respond to fluctuations

( ) ( ( ))IL m x 0 ( ) ( ) 0x y

L T 0 ( ) ( ) 0T x T y

Unruh-Dewitt:

Gravity:

1/ .1obs DEL mm

910L mmˆ

Inflation: energy fluctuations couple to gravity

Analysis of length scales shows surface dimensions have correct energy values2

4P

L

L

Conclusion: CC or DE is quantum fluctuations of Universe’s boundary.

Analysis of some Padmanabhan Results

The differential relation between the bulk and surface terms

The Göckeler and Schücker Gauge Formulation of General Relativity

1[ , ] ( )

32a b

GS b aS RG

Uab ab ab a cb

cD R d D R d

D is the exterior covariant derivative, d the differential, ω the connection (potential) gl4 1-form, the wedge product, R the curvature gl4 2-form, and the s are an oriented basis of the cotangent space, they are 4 valued 1-forms, is the Hodge duality operator, a,b = 0…3, the metric is gab raises and lowers indices and has signature +---, and the integration domain.

Also D T d

1a ai dx

To reduce the SGS[,] to SEH[dx,] assumptions of GR • s are holonomic, a = dxa

• T = 0, vanishing torsion• Rab

cd Rabab = R, scalar

• the gauge is fixed, i.e. particular coördinates are chosen

1[ , ] ( )

32a b

GS b aS RG

U

( ) ( )b b ia iag

1

2( ) detb i bb ii c dia ia bicd mng dx dx g g g dx dx g  

a jab jbR R g

  ( )a b ab c d ab c db a bacd abcdR R dx dx g R dx dx g

1

2ab ab r s

rsR R 14·

2ab ab a b

abR R dx dx ababR R

41 1[ , ] [ , ]

16 16a b c d

GS abcd EHS R dx dx dx dx g gdx R S dxG G

U U

[ ]1

4!a b c d a b c ddx dx dx dx dx dx dx dx

4a b c dabcddx dx dx dx d x where

and

ab ab ab a kbkD R d

Padmanabhan’s differential Bulk-Surface relation1

[ , ] ( )32

a bGS b aS R

G

U ab ab ab a cb

cD R d

Substituting

Splits into 2 terms

1[ , ] ( )

32bb aa c d

GS ab ai b baci

dS d gg gG

U

 

1[ , ]

32 i aa bb c d aa bb c d

GS ai b abcd ab abcdS gg g d gg gG

U U

Transforming to exact differential

( )

( ) ( ) ( ) ( )

ab c dabcd

ab c d ab c d ab c d ab c dabcd abcd abcd abcd

d g

d g d g d g d g

1

32

1

32

[ , ]

2 ( ) ( )

( )

GS

a ib c d ab c d ab c di abcd abcd abcd

ab c dabcd

G

G

S

g d g d g

d g

U U

U

Substituting

1

32

1

32

[ , ]

2 ( ) ( )

( )

GS

a ib c d ab c d ab c di abcd abcd abcd

ab c dabcd

G

G

S

g d g d g

d g

U U

U

Padmanabhan’s differential Bulk-Surface relation, cont’d

Stoke’s Theorem dA A U U Last term above is a Surface term

Now if there is a differential relation between the bulk and surface

 ab a ib

id

d C 0 TDg dg g g 1

2ij

ijd g gg dg 1a a

i dx

1

32

1

32

[ , ]

2 ( ) ( )

GS

a ib c d ab c d ab c di abcd abcd abcd

ab c dabcd

G

G

S

g d g d g

g

U U

U

2R d

Padmanabhan’s differential Bulk-Surface relation, cont’d

This is a condition on d

This is true irregardless of whether assumptions are made for d or d-g. These results follow only from the GS gauge formulation of GR.

1

32

1

32

[ , ]

2 ( ) ( )

( )

GS

a ib c d ab c d ab c di abcd abcd abcd

ab c dabcd

G

G

S

g d g d g

d g

U U

U

Again if there is such a differential bulk-surface relation then the bulk and surface terms contain the same information, one being the differential of the other.

Check if bulk and surface terms are generally covariant;check conditions when bulk and surface terms=0

Reduction of above GS differential bulk-surface relation to Padmanabhan’s termsIt has already been shown above that SGS[,] reduces to SEH[dx,]

Reiterating 0 TDg dg g g k kij ik j i kj ij jidg g g

1

2ij

ijd g gg dg iid g g differential geometric

GR 0 T d  a a c

cd

1

32

1

32

[ , ]

2 ( ) ( )

( )

GS

a ib c d ab c d ab c di abcd abcd abcd

ab c dabcd

G

G

S

g d g d g

d g

U U

U

 

[ , ]

12

32

1( )

32

GS

a ib c d ab c d i ab c d ii abcd i abcd i abcd

ab c dabcd

S

g g gG

d gG

U U

U

Substituting

This is at the level of GR except it has not been contracted.

Reduction of above GS differential bulk-surface relation to Padmanabhan’s terms, cont’d

 

[ , ]

12

32

1( )

32

GS

a ib c d ab c d i ab c d ii abcd i abcd i abcd

ab c dabcd

S

g g gG

d gG

U U

U

Applying Stokes Theorem and setting a a ic ci

       

1[ , ]

32

2

1

32

GS

a ib j k c d ab d j k c i ab i j k c dij k abcd j ik abcd j ik abcd

ab i c di abcd

SG

g g g

gG

U U U

U

Claim: The respective terms will reduce to Padmanabhan’s bulk and surface terms

ik m l l mil km ik lmgg ck m ik c

km ikg g g bulk: surface:

Reduction of above GS differential bulk-surface relation to Padmanabhan’s terms, cont’d

bulk: ik m l l mil km ik lmgg surface: ck m ik c

km ikg g g

Pad’s bulk:

a k k j kb j

a ib j k c d ij c dij k abcd ij k kjcdg g

2nd term in Pad’s bulk:

except there is 2 of them

   ab i j k c d j k c dj ik ab

a k kj ib jcd j ik kjcdg

           2 2ab d j k c i j k c ij ik abcd j ik k

a k kj ib j d i jcdg g

cancels one of the previous terms

ab i c di abcd

ai i c d ib i c di aicd i ibcd

ai ib i c di i aicd

g

g g

g

U

U U

U

ck m ik ckm ikg g g

ik m lil kmgg

ik l mik lmgg

Pad surface:

1st term: ij ij i

ij k k ij nk ijk l i mk k jn k ki m lj k n i il mkg g

Implications of Bulk Surface Relations

Then

  ab ab ab a cbcD R d   a a a a c

cD T d

 0 a a a ccT d    

1

2 a c a a b c

c bcd C

     

1 1

2 2 a a b a b

c bc cbC C    

   

2

1 1 12·

2 2 2

ab ab a cb a cbc c

a i cb j a cb i jci j ci j

R d

C C C C

This expresses the curvature R in terms of d.

1

2 ab ab r s

rsR R  ab a cbij ci jR C C

Cabc antisymmetric

Case i. Within GR

Implications of Bulk Surface Relations, cont’d

Case ii. Outside GR Now T = 0

If the bulk-surface relation holds 2 R

T d T d

1

2 T d R

Places a condition on the torsion, T

Some observations indicate the Torsion 10-14 in order of magnitude

Padmanabhan and Paranjape’s Entropy Functional based on Elasticity

[ ] 4DS d x g P T V ( )a a ax x x

This becomes [ ] 4a b

abS P d d V

Writing out in full 1 2

1 2

1 1

( 2)! 2!D

D

a b a acd a bab ab cda aP d d P

D

ò

Interchanging indices

1

1

1

! ò D

D

a aa aD

ò

d i ix  d a a a bb b  a a c

b bc

1

2!P ab ab c d

cdP

d a a bb

e aa

1 2

1 2

1 1

( 2)! 2!P

ò D

D

a aabab cd cda aP

D

ea

1 D 2

1 D 2

a aa b b a c d aa bb b a ab abcd aba a

1 1R ( ) R ( ) R gg g

(D 2)! 2!\

ò

1 D 2

1 D 2

1 D 2

1 D 2

a acd a bab cda a

a acd a bab cda a

1 1g R

(D 2)! 2!

1 1g R

(D 2)! 2!

ò

ò

basis vectors

Goal: History of Matter

Radiation Era Matter Era

• Strings

Goal: Degrees of freedom, mechanisms; Products, Relics Eras and Epochs

Second Part. A History of the Early Universe

Vacuum Era

• Planck Epoch

• String Epoch• GUT Epoch

• Inflation Epoch• Electroweak Epoch• Parton Epoch

• Hadron Epoch

• Lepton Epoch

• Nuclear Epoch

• Atomic Epoch

Newtonian Gravity,General Relativity

Hubble expansion, 1925

homogeneous and isotropic Universe, 100 Mpc

Robertson-Walker metric, 1935 2 2 2 2 2 2 2 22

1( ) ( sin )

1ds dt a t dr r d d

kr

big bang nucleosynthesis (BBN), Gamov & Alpher, 1948

cosmic microwave background radiation (CMBR), Penzias & Wilson, 1965

general relativity (GR), Einstein, 1915

standard model of particles (SMP), 1970

Theory and Observations

Basic Equations

Robertson-Walker(RW) metric2 2 2 2 2 2 2 22

1( ) ( sin )

1ds dt a t dr r d d

kr

18

2G R g R GT Einstein equations

( ), ( ), ( ), ( )T dia t p t p t p t Stress-Energy tensor

Friedmann-Lemaitre(FL) equations

2

2

8

3

a k G

a

2

28 2

a a kGp

a a

expansion parameter k curvature parameter

RW metric conformal to Minkowski metricwhen

2 2 2 2 2 2 2 2( ) ( sin )ds a t d dr r d d 0k

( )a t

Radiation Era Planck EpochTheory of Everything(TOE) or Superunification Epoch

Planck time 445

5.391 10PP

l Gt s

c c

Planck length 353

1.616 10P

Gl m

c

8 19 22.177 10 1.221 10 /P

cm kg GeV c

G

Planck mass

Fundamental Constants

5 232

21.417 10P

PB B

c m cT K

Gk k

Planck temperature

82.998 10 /c m s 34 161.055 10 6.582 10Js eV s

23 51.381 10 / 8.617 10 /Bk J K eV K

Gravitation, General Relativity

Statistical Mechanics, Thermodynamics

Special Relativity Quantum Mechanics

2 26 211 1 1

2.4 10 0.326 10GeV K

t s sT Tg g

Time-Temperature relation

11 3 26.673 10 /G m kg s

is dofg

reaction rate of gravitons at this time, Binetruy

2

~P

TH

m

common theme

Planck Epoch, cont’d

52 5

4~ ~ grav

P

TG T

m

P gravT m H

A Reaction rates

H Expansion rate

Products, Relics annihilation, bound, decoupled, go out of equilibrium, no longer created

so gravitons decouple, go out of equilibrium, and form presumably CGBR; not detected

e e examples p e H relativistic nonrelativistic

GUT Epoch

String Epoch pre-Big Bang FutureD-branes, D dimensional objects

43 3910 10s t s 16 1910 10GeV E GeV 29 3210 10K T K

dof

The first use of group theory, enlarged groups

Extends gauge theory to very high energies

Lagrangian formulation

SUSY prediction: LSP good candidate for DM; CERN

Hierarchy of couplings explained by screening and antiscreening (asymptotic freedom) of unified couplings

Gauge hierarchy

7(5) 24 3 24 6 3 (64 3) 282

8SU 7

282 192 114 573.758

SUSYGUT (10) ?SO

19100 10EW Pm GeV em G V

Inflation Epoch

Resolves BB problems:

Flatness, why the Universe is so flat or so close to the critical density

Horizon, why is the Universe so homogeneous when the regions are too far apart to be in causal contact.

Monopoles, unobserved prediction of GUTs; diluted by expansion

Tiny scale-invariant fluctuations, perturbations for later gravitational collapse

0k

2138

c HG

Old Inflation; new Inflation reheating; inflaton decay

Lasted for with33~ 10 s 39 3310 10s t s and 2710T K

Electroweak Epoch 33 12 14 15 2710 10 200 10 10 10s t s GeV E GeV K T K ˆ ˆ ˆ

SMP, at low end

SMP complete for ordinary physics, but incomplete

Separate couplings, , no Higgs, masses = 0, ∞ EW corrections, Higgs mass also divergent, fine tuning of Higgs mass

Or Supersymmetric Epoch

(2) (1) (1)L Y EMSU U U

Strong force separates at beginning, Weak force separates at end

New Physics

SUSY graded Lie algebra of bosons and fermions in [ , ], { , }

SUSY only nontrivial extension of spacetime symmetries, largest spacetime symmetry of S-Matrix

Symmetry breaking: kinetic: like Higgs OR dynamic: like SUSY, bosons, fermions; or form bound systems (recent CERN )

p p

Electroweak Epoch, cont’d

Haag-Lopuszanski-Sohnius(HLS) Theorem. The largest symmetry of an interacting, unitary field theory is the direct product of a (possibly very large) gauge symmetry, a Lorentz invariance, and a (possibly extended) supersymmetry

SUSY is also incomplete

SUSY evades CM Theorem being a Lie SuperAlgebra

Local SUSY implies graviton, mSUGRA; fewer parameters, more predictive, dynamical SB

SUSY resolves Higgs mass problems, preserves hierarchy, MSSM coupling, Desert, DM

SUSY not observed: degenerate masses, LSP, CERN; is a broken symmetry

‘soft’ symmetry breaking at TeV scale

Coleman-Mandula(CM) Thm. All possible Lie Algebra symmetries of the S-matrix under general assumptionscan only be a direct product of the Poincare algebra and an internal symmetry algebra.

the pattern of fermion masses and mixings, the replication of generations, the origin of CP violation. Causes new problems: baryon and lepton number, supersymmetric parameter μ M∼ Weak, origin of SUSY breaking, why are SUSY breaking parameters 1015 < LP, CP violation of SUSY so small.

Parton Epoch 5 12 12 1510 10 200 200 10 10s t s MeV E GeV K T K ˆ ˆ

3 3

3

( )

1( , )

21

AA E A

T tA

gf t d d

e

p

p p p

thermal distribution function for the Bose-Einstein(B-E) or Fermi-Dirac(F-D) equilibrium distributions

4 known forces of today

all fundamental particles acquire masses via Higgs mechanism

dof = 106.75, the SMP dof

heavy and light particles initially in equilibrium

later heaviest particles condense out 0, , , , , , ,t t H W Z b b etc

at dof: next slide

again: particle condensation (creation); structure, order decomposition (annihilation)

much better understood physics: , partons; however no cosmological evidence ,W Z

12 6 17 310 10 10 /aveT K t s kg m

Parton Epoch, cont’d

• thick blue: flat

• gray: linear

• thin blue: exponential

Matter-Antimatter asymmetry

Very weakly understood particle physics and no

cosmological observational evidence:

Baryogenesis

Leptogenesis

Hadron Epoch 5 3 11 1210 10 10 200 10 10s t s MeV E MeV K T K

Universe: “hot”; high entropy; ratio of photons to baryons 910B

n

n

Parton confinement

Leptogenesis

Moments

Special cases

bosons , fermions

1/22 2

32

( )2 exp[( ) / ] 1

k km

E m Egn f d dE

E T

• number density

1/22 2 2

32

( )2 exp[( ) / ] 1

k km

E m EgE f d dE

E T

• energy density

2

3 3

3/22 2

2

1 1( ) ( ) ( )

3 3

2 exp[( ) / ] 1

k k k k k k

m

kp v f d f d

E

E m EgdE

E T

• pressure

highly relativistic, Tm, nondegenerate, T nonrelativistic, Tm

• Coupled

• Decoupled

highly relativistic, TDm nonrelativistic, TDm

3 3 3 3 31 2

3 3 3 3 31 2

2 24 41 2 1 2 1 21 2 1 2

3(2 ) 2 (2 ) 2 (2 ) 2 (2 ) 2 (2 ) 2

(2 ) ( ) [ (1 )(1 ) (1 )(1 ) ]

b c d

b c d

b c d b c b cb c b c

dn a d p d p d p d p d pn

dt a E E E E E

p p p p p f f f f M f f f f M

Hadron Epoch, cont’d

Boltzmann Equation derivation

i. Louisville and Collision Operator ii. geodesic equation

iii. Γs from RW metric

dpp p

dt

0 2 0 2 0 2 211 22 33 sin ( )

1

aar aar ar

k

v. Boltzmann Equation 3

3( ) ( , )2

ii i

gn t d p f p t

iv. number density

GR

[ ] [ ]f fL C

1 2 3b c d

2 2 2 2 2 2 2 22

1( ) ( sin )

1ds dt a t dr r d d

kr

numerical data simplifying assumptions

Boltzmann equation [ ] [ ]f fL C

Classical and SR · ·d dx dp

Ldt x dt p dt t

v px p

Lepton Epoch, ,e e & 3 species of ν 7 7

2 2 2 3 2 10.758 8

g

heavier particles condensing out

lepton-antilepton annihilation;--lepton residue

neutrinos decouple • initially at same T as γ

• but as T↓ T of γ ↑2e e

1/34

11

T

T

• dof argument, assuming constant S, entropy

•

• CBNR2.73 1.95T K T K

• not yet detectable, like the gravitons, CBGR

• can also consider μ, τ, the other ν

3 10 1110 1 1 10 10 10s t s MeV E MeV K T K

Nuclear Epoch 9 101 2min 0.1 1 10 10s t MeV E MeV K T K

71 3 3.63

8g , 1

7~n

p

n

n

binding energies, MeV

D T He-3 He-4

2.22 6.97 7.72 28.3

Synthesis 0.1MeV

1 1010B

n

n

Neutron decay, exponential

Earliest, most rigorous, best understood

High entropy

Weak reactions cannot maintain equilibrium

Nucleosynthesis starts: T ~ .07-.08 MeV, and neutron fraction/total nucleons = 1/6.

Matter Era Atomic Epoch

Matter expansion overtakes radiation expansion

Formation of neutral H, He

300,000 400,000 3000yr T K ˆ

For T < 0.1MeV the main constituents H and He-4 nuclei, and the decoupled ν ,e

2 2 2He e He

Recombination → Combination

H e H

Photons no longer EM interacting, decouple, form CMBR

Once again the latter is delayed due to high Entropy of Universe; mechanism known, phenomena can be calculated

Vacuum Era

Dark Energy, ~ 70%

Free Energy

Today, 13.9 Gyr after the Big Bang

Further Thermodynamics of the Early Universe

Dark Matter, 24-25%Concerns:

Kinetics: e.g. Boltzmann Eqn, reaction rates vs expansion rates

Dynamics: e.g. boson fermion cancellation in corrections to Higgs mass, not put in ‘by hand’, energetic and preferable.

primordial vacuum-inflaton fluctuations seed gravitational collapse to form nebula,stars,galaxies

Symmetry and Order Order and Control parameters

Equilibrium; Phase transitions

• today: nucleons • past: + in Planck epoch, - all other times

ConclusionsFirst part: Gravitation and GR

PE • geometric

• kinetic

• curvature

QG: dof of gravity: gravitons; CGBR

Rindler observers • accelerated frames

• horizons

Relativity, Thermodynamics, QM

• temperature, entropy of horizons

E-H Action and EE derived in variety of ways; independent of metric gαβ (neither is Tαβ), metric not varied

Law Thermodynamics BH Mechanics

0 T = const in equilibrium const on BH surface

18

2 0 0

3 0 unobtainable finitely 0 unobtainable finitely

dE TdS PdV dM dAG

dS dA

T

Relation between bulk surface terms in Action

Thermodynamic derivation of E-H Action

Thermodynamic interpretation of EE

E-H Action is FE of spacetime

Hawking radiation: BH radiate in thermal spectrum

Unruh effect: accelerated observers find vacuum is thermal particle spectrum

Conclusions, cont’dSecond part: Early history of the Universe

The SMP & Lagrangian gauge theory generalizations form the basis of the SMC

3 Eras: Radiation, Matter, and Vacuum

(Near) equilibrium and nonequilibrium (phase) transitions due to expansion

The Universe is also expanding and its composition depends on a comparison between the rates of expansion and reaction

Gauge hierarchy explained by screening and antiscreening

Symmetry, symmetry breaking, (phase) transitions, order (bound states)