Lecture 22Cosmological 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

2 2 /

2sV GM R k

k E

2 22sGM

R ER

FINITE SPHERE

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

3

2

4

3s

s

M R

RV R

tGMV

R gt R

2 2

4

38

23

R G R

R G R E

What happens when R-> ?

Explore R->

Standard Model

2 2 28

3R G R kc

From Newtonian to GR

The Friedmann Equation

2 2

4

38

23

R G R

R G R E

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

2 2 28

3R G R kc

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 =fm).

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 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.

4

3R G R

0R

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).

/ /H v l R R Enter Hubble

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

2 2 2

2 22

2 2

8

3

8

3

R G R kc

R kcH G

R 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?

22 002 2

0 0

8( 1)3

GkcH

R H

Measurement of H0 and give

curvature constant k

Explore equation:

1. Empty universe k negative hyperbolic universe, expand forever

2. Flat or require matter or energy.

3. k=0 -> critical density

020

20

0

81

3

3

8

M

c

Mc

G

H

H

G

Critical Density

020

20

0

81

3

3

8

M

c

Mc

G

H

H

G

• 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

2 2 200 ( )M

RR H kc

R

2

Rq

RH

Deceleration Parameter. Now q0 . All standard models decelerate q.0. Need cosmological constant to change it

1

2 Mq For standard models specification of q determines geometry of space and therefore specific model

Summary - Definitions