1. Fundamental Cosmology: 6.Cosmological World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
Fundamental Cosmology: 6.Cosmological World Models
“ This is the way the World ends,
Not with a Bang, But a whimper ”
T.S. Elliot

2. 6.1: Cosmological World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.1: Cosmological World Models
What describes a universe ?
We want to classify the various cosmological models t R
W<1
W<1 Open universe expands forever
W=0
W=0 Open universe expands forever
W=1
W=1 Closed universe limiting case
W>1
W>1 Closed universe collapses
Matter
Cosmological Constant
Curvature 1 2 3
from Friedmann eqn.
Defined the density parameter W

3. 6.1: Cosmological World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.1: Cosmological World Models
L=0 World Models
Lets think about life without Lambda

4. 6.2: Curvature Dominated World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.2: Curvature Dominated World Models
Friedmann
Equation
The Milne Universe
Special relativistic Universe
negliable matter / radiation : r~0, Wm<<1
No Cosmological Constant : L=0, WL=0
Curvature, k=-1
Universe expands uniformly and monotomically : Rt R t
Age: to=Ho-1
Useful model for
Universes with Wm<<1
open Universes at late times

5. 6.3: Flat World Models General Flat Models W=1
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.3: Flat World Models
General Flat Models
Friedmann
Equation
Flat, k=0 universe
Assume only single dominant component
r= ro(Ro/R)3(1-w)
W=1
Solution
Age of Universe
For spatially flat universe :
Universes with w>-1/3 - Universe is younger than the Hubble Time
Universes with w<-1/3 - Universe is older than the Hubble Time

6. 6.3: Flat World Models The Einstein De Sitter Universe W=1 R t
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.3: Flat World Models
The Einstein De Sitter Universe
Friedmann
Equation
Flat, k=0 universe
Matter dominated r= ro(Ro/R)3
No Cosmological Constant : L=0, WL=0
W=1
Universe expands uniformly and monotomically
but at an ever decreasing rate: R t
W=0
W=1
Ho-1 to
t=0
Age: to=(2/3Ho )
Radiation dominated r= ro(Ro/R)4
Radiation dominated to=(1/2Ho )
Until relatively recently, the EdeS Universe was the
“most favoured model”

7. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
Density
Parameter 3
Curvature Density
Curvature
& Matter 4 5
Matter + Curvature
Friedmann
Equation 1 2
Matter
Dominated 1 2 6 3 4 6 5
Equation for the evolution of the scale factor independent of the explicit curvature

8. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
Friedmann
Equation
The Einstein Lemaitre Closed Model
Closed, k=+1 universe
Matter dominated r= ro(Ro/R)3
No Cosmological Constant : L=0, WL=0
W>1
The Scale Factor has parametric Solutions:
Cycloid Parametization:
R(q), t(q) are characteristic of a cycloid parameterization
cos(q), sin(q) are Circular Parametric Functions
q=0  t=0
q=p  dR/dt=0  Rmax
Universe will contract when q=2p  t
For the case of W=2, the Universe will be at half lifetime at maximum expansion

9. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
The Einstein Lemaitre Closed Model
Closed, k=+1 universe
No Cosmological Constant : L=0, WL=0
W>1
Age:
W=0
Models normalized at tangent to Milne Universe at present time
Ho-1 R t to
W=2
High W Age universe decreases (start point gets closer to Origin)
W=4
W=10
Universe evolves faster for higher values of W

10. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
The Einstein Lemaitre open Model
Friedmann
Equation
Open, k=-1 universe
Matter dominated r= ro(Ro/R)3
No Cosmological Constant : L=0, WL=0
W<1
The Scale Factor has parametric Solutions:
Hyperbolic Parametization:
R(f), t(f) are characteristic of a hyperbola parameterization
cosh(f), sinh(f) are Hyperbolic Parametric Functions
f=0  t=0
f  R 
Universe will become similar and similar to the Milne Model (W=0) as t

11. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
The Einstein Lemaitre open Model
Open, k=-1 universe
No Cosmological Constant : L=0, WL=0
W<1
Age:
W=0
W=0.2
Models normalized at tangent to Milne Universe at present time R t
Ho-1 to
W=0.4
Low W  Age universe increases (start point gets farther from origin) Oldest universe is Milne Universe
Universe evolves faster for lower values of W

12. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
Summary 4 8 6 2 1
0.5
1.0
Age (Ho-1) W0
Einstein
De Sitter
Open Cosmologies
Closed Cosmologies

13. 6.4: Matter Curvature World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.4: Matter Curvature World Models
Summary
Universe
Type
Parameters
Fate
Topology
R / t k W r q L
Milne
Special Relativistic
Open -1
Expand forever
Friedmann - Lemaitre open
Hyperbolic
<1
< rc
< 0.5
Expand Forever
Einstein De Sitter
Flat
Closed =1
= rc
= 0.5
t 
R 
Friedmann - Lemaitre
closed
Spherical 1
>1
> rc
> 0.5
Re-contract
Big Crunch

14. Lets think about life with Lambda
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
L  0 World Models
Lets think about life with Lambda

15. L 6.5: L World Models Living with Lambda L < 0 universes
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
Living with Lambda 1
The Friedmann Equations
including Cosmological Constant 2 L
Modifies gravity at large distances
Repulsive Force (L>0)
Repulsion proportional to distance (from acceleration eqn.)
For a static universe
There exists a critical size RC (=RE) where Friedmann eqns =0
Consider the following scenarios
The Einstein Static Universe
L < 0 universes
L > 0 universes
k < 0, k=0
L > LC
L ~ LC
L < LC 1 2

16. The is possibility of a Static Universe with
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
The Einstein Static Universe
(k=+1, r>0, L>0)
For a static universe
The is possibility of a Static Universe with
R=Rc=RE, L=Lc for all t t R RE
Einstein Static Model
original assumed solution to field equations
Problem:
no big bang
no redshift

17. When R=RC universe contracts
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
Oscillating Models
(L < 0)
To ensure a real
When R=RC universe contracts t R RC
L < 0
Universe is Oscillatory
Oscillatory independent of k

18. Monotomically expanding Universe, at large R
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
The De Sitter Universe
(k=0, r=0, L>0)
For k  0
Monotomically expanding Universe, at large R
Special Case k = 0, r=0
De Sitter Model t R RC
L < 0
De Sitter L > 0
Does have a Big Bang
But is infinitely old

19. Monotomically expanding Universe, at large R  De Sitter Universe
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
k=+1 , L>0 World Models
(k=+1, L> LC)
For k=+1, L> LC
Monotomically expanding Universe, at large R  De Sitter Universe

20. 6.5: L World Models R t (k=+1, L ~ LC= LC+e) 3 Models RC
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
k=+1 , L>0 Eddington Lemaitre Models
(k=+1, L ~ LC= LC+e) 3 Models
Einstein Static Model
Special Case  t R RC
L = LC (EL1)
L = LC (EL2) 1
Eddington Lemaitre 1 (EL1) :
Big Bang
R  Einstein Static Universe as t 
k=+1, R= finite, can see around the universe !!
Ghost Milky Way
(normally light doesn’t have time to make this journey inside the horizon) 2
Eddington Lemaitre 2 (EL2) :
Expands gradually from Einstein Static universe from t =-
Becomes exponential
No Big Bang (infinitely old) : But there exists a maximum redshift ~Ro/Rc

21. 6.5: L World Models R t (k=+1, L ~ LC= LC+e) 3 Models RC
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
k=+1 , L>0 Lemaitre Models
(k=+1, L ~ LC= LC+e) 3 Models
Lemaitre Models :
Long Period of Coasting at R~Rc
Repulsion & Attraction in balance
Finally repulsion wins and universe expands
Lemaitre Models permit ages longer than the Hubble Time (Ho-1) 3 t R RC
L = LC +e
Long Coast period  Concentration of objects at a particular redshift (1+z=Ro/Rcoast) (c.f. QSO at z=2)

22. R>R2 : Bounce Solution
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
k=+1 , L>0 Oscillatory and Bounce Models
(k=+1, 0<L < LC)
2 sets of solutions for 0<L < LC
separated by R1, R2 (R1<R2) for which no real solutions exist
No solution R1<R<R2 because (dR/dt)2<0 t R R2
R<R1 : Oscillatory Solution
R>R2 : Bounce Solution R1 1
R<R1 : Oscillatory Solution
Universe expands to a maximum size
Contracts to Big Crunch 2
R>R2 : Bounce Solution
Initial Contraction from finite R (universe is infinitely old)
Bounce under Cosmic repulsion  there exists a maximum redshift ~Ro/Rmin
Expands monotomically

23. 6.5: L World Models L Summary of L models k Name Dynamics Evolution
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
Summary of L models L k
Name
Dynamics
Evolution
<0
 k
Oscillatory (1st kind)
contract back to R=0 (oscillatory)
>0 0
monotomically expanding
De Sitter LC 1
Einstein Static
Static  t at R=RE with L= Lc
>LC
LC+e
Eddington Lemaitre (EL1)
Big Bang  Einstein Static universe
Eddington Lemaitre (EL2)
expand from Einstein Static 
Lemaitre
Long coasting period at R=RE
0<L < LC
Oscillatory (2nd kind)
Universe bounces at RB

24. 6.5: L World Models WL,0 k=0 k=+1 k=-1 Wm,0 Summary of L models 1 2 -1
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
Summary of L models 1 2 -1 3
WL,0
Wm,0
BOUNCE
MODEL
COAST MODEL
COLD DEATH
k=0
BIG CRUNCH
k=+1
k=-1

25. 6.5: L World Models R R1 R2 Summary of L models L >LC L = LC
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.5: L World Models
Summary of L models
De Sitter
Einstein Static
Eddington Lemaitre 1 (EL1)
Eddington Lemaitre 2 (EL2)
Lemaitre
Oscillatory (1st kind)
Oscillatory (2nd kind) - Bounce R
L = LC R1 R2
L <LC
L >LC
L <0 models all have a “big crunch”
L >0 models depenent on k
Expansion to  if k 0 : L becomes dominant
k>0 and L > 0  multiple solutions.
Our Universe…….?

26. 6.6: Alternative Cosmologies
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Alternative Cosmologies
変な宇宙論
There are a lot of strange theories out there !

27. 6.6: Alternative Cosmologies
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Alternative Cosmologies
Steady State Cosmology
Bondi & Gold 1948 (Narliker, Hoyle)
1948 Ho-1 = to < age of Galaxies
注意 Steady State  Static
Recall:
PERFECT COSMOLOGICAL PRINCIPAL
The Universe appears Homogeneous & Isotropic to all Fundamental Observers At All Times
Density of Matter = constant
 continuous creation of matter at steady rate / volume

28. 6.6: Alternative Cosmologies
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Alternative Cosmologies
Steady State Cosmology
The metric given by :
 Metric for De Sitter Model
Curvature : 3D Gaussian (k/R(t)2)  dependent on t if k0
Additional term in General Relativity field Equations
 Creation of matter!!
~ 10x mass found in galaxies  Intergalactic Hydrogen at creation rate ~10-44 kg/m3/s
Problems:
Magnitude-Redshift Relation
 qo=-1  De Sitter Model
qo=-1 not consistent with observation Moreover no evolution is permitted
Galaxy Source Counts
Corresponding N(S) slope flatter < -1.5
Inconsistent with observation
2.7K Cosmic Microwave Background
~~~ No explanation

29. 6.6: Alternative Cosmologies
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Alternative Cosmologies
Changing Gravitational Constant
Milne, Dirac, Jordan (Brans & Dicke, Hoyle & Narliker)
G decreases with time
e.g. Earth’s Continents fitted together as Pangea G as t continents drift apart.
Stars LG7 G as t  stars brighter in the past.
Earth is moving away from the Sun if G as t  Tt9n/4 inconsistent with Earth history
G(t)  Perturbations in moon & planet orbits (constraints (dG/dt)/G<3x10-11 yr-1 )
Light Elements Abundance (dG/dt)/G<3x10-12 yr-1

30. 6.6: Alternative Cosmologies
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Alternative Cosmologies
Changing Gravitational Constant
Brans & Dicke Cosmology
Variation on the variation of G Theory
As well as the Gravitational Tensor field there is an additional Scalar field G(t)
L=0, Mach Principle G-1~Sm/rc2
= coupling constant between scalar field and the geometry Such that Grt2 = constant
Observational limits and theoretical expections for D/H versus . The one (light shading) and 2 (dark shading) sigma observational uncertainties for D/H and are shown. They do not appear as ellipses due to the linear scale in D/H but logarithmic uncertainties from the observations. The BBN predictions are shown as the solid curves where the width is the 3% theoretical uncertainties. Three different
values of GBBN/G0 are shown. Copi et al. Astroph/
Diracs original 1937 theory w=-2/3
nucleosynthesis  w>100
 Analysis of lunar data for Nordtvedt effect  w>29  dG/dt)/G<10-12 yr-1

31. 6.6: Alternative Cosmologies
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Alternative Cosmologies
Other Cosmological Theories
Anisotropic Cosmologies
Anisotropic Cosmologies :
Universe is homogeneous and isotropic on the largest scales (CMB)
Obviously anisotropic on smaller scales  Clusters
Quiescent Cosmology
Universe is smooth except for inevitable statistical fluctuations that grow
Chaotic Cosmology (Misner)
Whatever the initial conditions, the Universe would evolve to what we observe today
Misner - neutrinos damp out initial anisotropies
Zeldovich - rapidly changing gravitational fields after Planck time ( s)  creation of particle pairs at expense of gravitational energy
But: initial fluctuations HAVE been observed and explainations are available !

32. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Kind of Universe do we live in then ?
Lets think about
Our Universe

33. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 1: Supernova Cosmology Project
Type Ia supernovae : Absolute luminosity depends on decay time  "standard candles”
Apparent magnitude (a measure of distance)
Redshifts (recession velocity).
Different cosmologies - different curves.
accelerating
empty
critical

34. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 1: Supernova Cosmology Project

35. 6.6: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 2: Hubble Key Project
Mould et al. 2000; Freedman et al. 2000
H0 = 716 km s-1 Mpc-1  t0 = 1.3  1010 yr

36. 6.6: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.6: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 2: Hubble Key Project
H0 = 716 km s-1 Mpc-1  t0 = 1.3  1010 yr

37. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 3: WMAP
Red - warm
Blue - cool
Wilkinson Microwave Anisotropy Probe (2001 at L2)
Detailed full-sky map of the oldest light in Universe.
It is a "baby picture" of the 380,000yr old Universe
fundemental
1st harmonic
Temperature fluctuations over angular scales in CMB correspond to variations in matter/radiation density
Temperature fluctuations imprinted on CMB at surface of last scattering
Largest scales ~ sonic horizon at surface of last scattering
Flat universe this scale is roughly 1 degree (l=180)
Relative heights and locations of these peaks  signatures of properties of the gas at this time

38. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 3: WMAP
WMAP - fingerprint of our Universe
Flat Universe - sonic horizon ~ 1sq. Deg. (l=180)
Open Universe - photons move on faster diverging pathes => angular scale is smaller for a given size
Peak moves to smaller angular scales (larger values of l)

39. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 3: WMAP
WMAP maps and geometry
Scale Factor (Size)
time t0 Wm WL 2 1
0.3
0.7
W>1
W=1
W<1

40. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Evidence 4: WMAP +SDSS
Tegmark et al. 2003

41. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
What Universe do we live in ?
Wtot = 1.0
WL= 0.7
Wm=0.3
Wb=0.02
H0=72 km s-1 Mpc-1
k=0, L>0
Concordance Model
Approximately Flat (k=0)
CMB measurements
WL=
Type Ia supernovae
There is also evidence that Wm~0.3
Structure formation, clusters
H0=72 km s-1 Mpc-1
Cepheid distances HST key program
Currently matter dominated

42. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
The Evolution of the Concordance Model - The Evolution of Our Universe
L>0, k = 0
 Monotonic expansion
t Universe  De Sitter Universe
Early times Universe is decelerating
Later times L dominates Universe accelerates

43. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
The Evolution of the Concordance Model - The Evolution of Our Universe
Matter
Dominated
Dark Energy
The here and now
lg(R)
lg(t)
tr=m
tm=L t0
Why do we live at a special epoch ??

44. 6.7: Our Universe - The Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.7: Our Universe - The Concordance Model
The Evolution of the Concordance Model - The Evolution of Our Universe

45. 6.8: SUMMARY Summary L = 0 Models L  0 Models Concordance Model
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.8: SUMMARY
Summary
Used the Friedmann Equations to derive Cosmological Models depending on the density W
Have discovered a large family of cosmological World Models
W=0 t R
W<1
W=1
W>1
L = 0 Models
L  0 Models
lg(R)
Concordance Model
lg(t)
Wtot = 1.0
WL= 0.7
Wm=0.3
Wb=0.02
H0=72 km s-1 Mpc-1
k=0, L>0
Parameters of
Concordance Model

46. 終 次： 6.8: SUMMARY Fundamental Cosmology 6. Cosmological World Models
20/09/2018
Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003
6.8: SUMMARY
Summary 終
Fundamental Cosmology
6. Cosmological World Models
Fundamental Cosmology
7. Big Bang Cosmology 次：