1. Introduction: Big-Bang Cosmology

2. Basic Assumptions Principle of Relativity:
The laws of nature are the same everywhere and at all times
The Cosmological principle:
The universe is homogeneous and isotropic
Space time is simply connected, can be filled with comoving observers (CO).
Each CO performs local measurements of distance and time in it’s own frame of reference, locally flat. No global inertial frame.
Cosmic time: synchronized clocks of COs in space at every given time.

3. North Galactic Hemisphere
Lick Survey
1M galaxies
isotropy→homogeneity
North Galactic Hemisphere

4. Microwave Anisotropy Probe
February 2003, 2004
WMAP
Science breakthrough of the year
δT/T~10-5
isotropy→homogeneity

5. Homogeneous Universe:
Flat (Euclidean), therefore open, infinite
Curved and closed, finite

6. Three-dimensional Two dimensional One-dimensional
Open
Closed
???
123

7. Olbers Paradox L R f  L/R2 R N  nR2dR dR The simplest assumptions:
Homogeneity and isotropy
Euclidean geometry
Infinite space
A static universe

8. The Universe is evolving in time

9. Baby galaxies in early universe
z=2

10. the universe is expanding

11. Discovery of the Expansion
1929
Discovery of the Expansion

12. Doppler Shift
receding source
approaching source

13. Red-shift
distance
wavelength

14. Hubble Expansion:
V =H R
Distance R
Velocity V
Acceleration

15. Hubble Expansion
V = H R
velocity
Hubble constant
distance

16. V = H R
A special center?

17. V = H R
A special center?
Other Origin

18. Prediction from homogeneity: The Hubble Law
V=HR
time t2 ax t1 x x
distance

19. distance
galaxy 2
galaxy 1
here
now
time
t0 13.7 Gyr
The Big Bang

20. היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות?
היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות? ? ?
היכן המפץ

21. היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות?
היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות?
כל הנקודות מתלכדות לנקודה אחת
מפץ באינסוף נקודות

22. The Big Bang model
The Steady State model
The Steady State model

23. Cosmic Microwave Background Radiation
1965

24. COBE 1992 Nobel Prize 2006 to Smoot and Mather
Plank black-body spectrum
I=energy flux per unit area, solid angle, and frequency interval
Spectrum of the CMB

25. Homogeneity and Isotropy: Robertson-Walker Metric

26. Metric Distance B A Metric distance Hubble expansion in curved space:
local Hubble law
t=t1
universal expansion factor
comoving distance

27. Metric Coordinate system: x1, x2 (2d example)
In a small neighborhood (locally flat)
Line element:
The metric:
Specifies the geometry uniquely Exact form depends on the choice of coordinates
Orthogonal coordinates: gij=0 for i≠j

28. Coordinates: cartezian spherical
Example: E3
Coordinates: cartezian spherical
interval
angular distance
cartezian spherical
Example: a 2D sphere embedded in E3 : r=const.
Area:

29. The Metric of a Homogeneous and Isotropic Universe (Robertson-Walker)
t=const.
In comoving spherical coordinates
Isotropy:
Three solutions:
Space-time interval

30. k=0 flat space (E3)
infinite volume

31. k=+1 a closed space
For visualization: a 3D sphere embedded in E4: (w, x, y, z)
a 3D sphere
the embedding is defined by the transformation:
consistent with w x y
To visualize plot subspace
2D sphere in w,x,y,z=0 a u φ
u=const. is a sphere of comoving radius u
A grows for 0 < u < π/2 and decreases for π/2 < u < π

32. k=-1 an open space

33. Homogeneity and Isotropy  Robertson-Walker Metric
expansion factor
comoving radius
angular area

34. de Broglie wavelength (particles or photons):
Redshift
Radial ray
local Hubble
local flat
nearby observers along a light path, separated by δr:
For Black-Body radiation, Planck’s spectrum:
Also for free massive particles:
de Broglie wavelength (particles or photons):
like photons:
useful: conformal time

35. Horizon limit exists Example: EdS (k=0, Λ=0) Causality problem:
Our Horizon is not causally connected: what is the origin of the isotropy?

36. Friedman’s Equation and its solutions

37. Newtonian Gravity shell
3 types of solutions, depending on the sign of E
H=0 at maximum expansion, possible only if E<0
independent of r because lhs is

38. The Friedman Equation Einstein’s equations For the isotropic RW metric
Newton’s gravity: space fixed, external force determining motions
Einstein’s equations
left side of E’s eq. is the most general function of g and its 1st and 2nd time derivatives that reduces to Newton’s equation
Gravity is an intrinsic property of space-time. geometry <-> energy density. Particles move on geodesics (local straight lines) determined by the local curvature.
For the isotropic RW metric
Stress-energy tensor
Einstein’s tensor
Add λ
energy conservation
eq. of motion
conservation of number of photons
mass conservation
A differential equation for a(t)

39. Solutions of Friedman eq. (matter era)
radiation era
acceleration
expansion
forever
conformal time
collapse
critical density
big bang
here & now

40. Light travel in a closed universe
A photon is emitted at the origin (ue=0) right after the big-bang (te=0)
photon:
conformal time
π/ π
north pole u south pole 2π
3π/2 π
π/2 η
big bang
big crunch t

41. H and t
General:
Carrol, Press, Turner 1992, Ann Rev A&A 30, 499

42. Solutions with a Cosmological Constant
k=0 t a
k=-1
same asymptotic behavior as k=0
3rd-order polynomial – one real root exists t a

43. Solutions with a Cosmological Constant (cont.)
a solution for every a
Lematre
k=+1 (closed)
a critical value:
the rhs at as to be >0
a double root at
static expanding to inflation Eddington-Lematre t a
Einstein’s static universe (unstable)
static ac
big-bang expanding to static

44. Solutions with a Cosmological Constant (cont.)
k=+1 (closed)
a critical value:
cos. const. wins t a a2
for a small and large a1
mass attraction wins
for a1<a<a2 – no solution t a
only attraction

45. kinetic potential curvature vacuum
Friedman Equation
kinetic potential curvature vacuum
Carrol, Press, Turner 1992, Ann Rev A&A 30, 499
two free parameters
closed/open
decelerate/accelerate
Flat: k=0 a 1

46. Dark Matter and Dark Energy
vacuum – repulsion? 1
decelerate
accelerate
m /2 - =0
open
closed
flat
m + =1 
bound
unbound
=0 1 2 m
mass - attraction

47. Cosmological Constant: Newtonian Analog
vacuum energy density
force per mass on shell
vs.
potential
vs.
in Einstein’s eqs.

48. Acceleration, pressure, energy density
FRW:
 p dV
Energy change by work
(2)
Construct a static model:
de Sitter:
Quintessence
for inflation need
to exceed

49. Future SN Cosmology Project

50. Dark Energy
acceleration
WMAP_1 
WMAP_3