Transcript Lecture 22 Cosmological Models

Lecture 22
Cosmological Models
ASTR 340
Fall 2006
Dennis Papadopoulos
Chapter 11
Newtonian Universe
Send r->, k is twice the kinetic
energy per unit mass remaining when
the sphere expanded to infinite size
Fates of Expanding Universe
FINITE SPHERE
V 2  2GM s / R  k
k  2 E
Explore R->
2GM s
R 
 2 E
R
2
1. E <0, negative energy per unit mass; expansion stops and re-collapses
2. E =0, zero net energy; exactly the velocity required to expand forever
but velocity tends to zero as t and R go to infinity
3. E> 0, positive energy per unit mass; keeps expanding forever; reaches
infinity with some velocity to spare
BIG LEAP -> CONSIDER SPHERE THE UNIVERSE
4
M s   R3 
3
R
V
R
t
GM
V
R g  2s
t
R
What happens
when R-> ?
4
R    G R
3
8
R 2   G  R 2  2 E
3
Standard Model
From Newtonian to GR
The Friedmann Equation
4
R    G R
3
8
2
R   G  R 2  2 E
3
8
R 2   G  R 2  kc 2
3
Robertson Walker (RW)
metric : k=0, +1, -1
In Friedmann’s equation R is the scale factor rather than the radius of an
arbitrary sphere
Gravity of mass and energy of the Universe acts on space time scale factor
much as the gravity of mass inside a uniform sphere acts on its radius
and
E replaced by curvature constant. Term retains significance as an energy at
infinity but it is tied to the overall geometry of space
Standard Model Simplifications
8
R   G  R 2  kc 2
3
2
4
R    G R
3
To solve we need to know how mass-energy density changes with time.
If only mass R3=constant.
Now need relativistic equation of mass-energy conservation and
equation of state i.e E =f(m).
Notice that here mR3 =constant but ER4=constant. Why the extra R?
Mainly photons left out of Big-Bang. Red-shifting due to expansion
reduces energy density per unit volume faster than 1/R3
Photons dominant early in Universe are negligible source of space time
curvature compared with mass to day.
All models decelerate R  0
Also now dR/dt>0 expansion.
For all models R=0 at some time. R=0 at t=0. Density -> infinity, and kc2
term negligible at early times. Great simplification.
Fate of Universe-Standard Model
While early time independent of curvature factor ultimate fate critically
dependant on value of k, since mass-energy term decreases as 1/R.
Fate of Universe in Newtonian form depended on value of E . In
Friedmann Universe it depends on value of curvature k.
All models begin with a BANG but only the spherical ends with BANG
while the other two end with a whimper.
Theoretical Observables
• Friedmann equation describes evolution of scale factor R(t) in the
Robertson-Walker metric. i.e. universe isotropic and homogeneous.
• Solution for a choice of  and k is a model of the Universe and gives
R(t)
• We cannot observe R(t) directly. What else can we observe to check
whether model predictions fit observations?
• Need to find observable quantities derived from R(t).
Enter Hubble
H  v/l  R/ R
Since R and its rate are functions of time H function of time. NOT
CONSTANT. Constant only at a particular time. Now given symbol H0
8
R 2   G  R 2  kc 2
3
R2
8
kc 2
2
 H   G  2
2
R
3
R
Time evolution equation for H(t)
Replaces scale factor R by
measurable quantities H,  and
spatial geometry
Observing Standard Model
Average mass density critical parameter why?
kc2
2 8 G 0
 H0 (
 1)
2
2
R0
3H 0
Measurement of H0 and 0 give
curvature constant k
Explore equation:
1. Empty universe 0, k negative hyperbolic universe, expand forever
2. Flat or require matter or energy.
3. k=0 -> critical density
8 G  0
 M  1
2
3H 0
3H 02
c 
8 G
M 
0
c
Critical Density
8 G 0
 M  1
2
3H 0
2
0
3H
c 
8 G
0
M 
c
• If H0 100 km s-1 Mpc-1 critical
density is 2x10-26 kg/m3 or 10
Hydrogen atoms per cubic meter of
space
• Scales as H2
• 50 km/sec Mpc gives ¼ density
• Current value of 72 km/sec Mpc
gives critical density 10-26 kg/m3
•  M=1 gives boundary between
open hyperbolic universes and
closed, finite, spherical universe
• In a flat universe  is constant
otherwise it changes with cosmic
time
Deceleration Parameter q
R0
R  H  M ( )  kc 2
R
2
2
0
R
q
RH 2
1
q  M
2
Deceleration Parameter. Now q0 . All
standard models decelerate q.0. Need
cosmological constant to change it
For standard models specification of q determines
geometry of space and therefore specific model
Summary - Definitions