Distance
•How far away something is gets complicated at high z
•How far it is now? How far it was then?
dt
1
1
 H0
How far light travelled? How distant it looks?
dx
 x 1 
•Let’s call r the distance to an object now
a
•It used to be closer, because everything was closer, by factor of a/a0
x
a0
•Naively, since it is light traveling, we have dr dt  c
•This is wrong, because universe used to be smaller dr ca0 c


•Universe used to be smaller by a/a0, so
dt
a
x
1
c
dx
c dt
1
c
dx
r   dt  
dx  cH 0 
r
x
x dx
x  x 1 
H 0 1z 1 x 2   x  x 2


Distance (2)
Can we understand this formula qualitatively?
•For z small, approximate integrand as 1
•This is simply Hubble’s Law (with corrections)
What is the effect of ?
•Quantity (x – x2) is always positive
•As you increase , you increase denominator
•This decreases the integrand
•This makes the distance smaller
c
r
H0
c
r
H0
Can we understand this effect qualitatively? Deceleration:
•As you increase , you have more gravity
•Universe decelerates more quickly
•Universe has less time since fixed red shift z
•Shorter distance for fixed z
1

dx
1 z 
1
1

dx 
1 z 1
cz
r
H0

x2   x  x2

c 
1 
1 

H0  1  z 
v  H 0r
Luminosity Distance:
•Actual results are more complicated than this
•We get distances from standard candles
•We don’t measure r, we measure the luminosity distance dL:
L
dL 
4 b
There are two things different about dL :
1
c
dx
•The universe is curved
r

H
2
0 1 z 1
x2   x  x2
•Actual area of sphere is not 4r
•The object has a large redshift 1+z
•Frequency decreases by 1/(1+z)

•Each photon has energy
sinh H 0 r 1   c
1  z 
if   1
decreased by 1/(1+z)

H0 1  c
•Rate at which photons are

dL  
if   1
received decreased by 1/(1+z)
1  z  r

sin H 0 r   1 c

if   1
 1  z 
H0  1 c







Type Ia Supernovae and Deceleration
•Measure luminosity distance dL as a function of redshift
•Compare to model for different values of :
•No model worked!
•Radiation only made it worse
•At modest z, it looked like the
universe used to be expanding
quicker!
m = 0.0,  = 0.0
m = 0.3,  = 0.7
m = 0.3,  = 0.0
m = 1.0,  = 0.0
Dark Energy
a2 8
kc 2
 3  G  2
2
a
a
•Friedmann Equation has density  of stuff in the universe
•Multiply by a2
•To match supernova data, we need this velocity increasing today 2 8
2
2
a


G

a

kc
3
•To have speed increasing, must have a2 increasing with time
•Any matter which satisfies this constraint will be called dark energy
Matter: (atoms, dark matter, neutrinos?)
•As universe expands, number of atoms decreases as a-3
•So  ~ a-3
Radiation: (light, neutrinos?, gravitons?)
•As universe expands, density of stuff decreases as a-3
•Also, each photon gets red shifted to lower energy by factor a-1
•So  ~ a-4
Dark Energy: What is it?
Vacuum Energy Density, aka Cosmological Constant:
•According to particle physics, empty space has stuff constantly appearing and
disappearing
•This means empty space has energy associated with it:
•And therefore mass density
•Unfortunately, particle physicists aren’t very helpful with calculating how much
energy density
u  ?
•The density of the vacuum doesn’t change as universe expands
• ~ 1, independent of a
•This contributes another term, the vacuum term to the energy density
•Its contribution is labeled 
 m  0.3
•It was found that the data could be well fit with:
   0.7
Dark Energy: What does the experiment say?
It depends on what assumptions we use:
Assumption 1: Suppose we assume only that it behaves as a power law:   rn
•Including all data, experimental result:
Note: normally
n  0.06  0.16
given as w, where
Assumption 2: Suppose we assume only that it is constant:
n = - 3 (1 + w)
•Including all data, experimental result:
tot  1.0023  0.0055
Assumption 3: Suppose we demand that it is constant, and tot = 1:
•Including all data, experimental result
b  0.0456  0.0016
•This model is called the CDM model
•It has become the standard model for cosmology d  0.227  0.014
• stands for the vacuum energy density
  0.728  0.016
•CDM stands for cold dark matter
•“cold” is a term we will clarify later
Age of the Universe, Round 3 (page 1)
  m  
a2 8
kc 2 83  G 0i  H 02 i
 3  G  2
2
Assume:
a
a
•Universe has only matter and dark energy
•Dark energy density is constant
•Matter scales as a-3
•kc2/a2 scales as a-2, of course
•Substitute in:
8
3
 G 0 m  H 02  m
8
3
 G 0   H 02  
 kc 2 a02  H 02 1   m    
2
a
3
2
2 


H

a
a



1




a
a





0
m
0

m

0


a2
8
3
 G m  H m  a0 a 
8
3
 G   H 02
2
0
3
 kc a  H 1    a0 a 
2
2
2
0
2
•Let x = a/a0:
t0  H
x
2
3
2



H

x



1




x

0  m

m

2

x
dx
 H 0 m x   x 2  1  m  
dt
2
1
0
1

0
dx
m x   x 2  1   m   
Age of the Universe, Round 3 (page 2)
t0  H
1
0
1

0
dx
m x   x 2  1   m   
•As m goes up, t0 goes down
•As  goes up, t0 goes up
•If m +  = 1, then
1
2
tanh

1
t0  H 0
3 
•If  = 0.728, then
t0  0.991H 01  13.8 Gyr
•Compare oldest stars 13 Gyr
•Best estimate:
No Age Problem
t0  13.75  0.11 Gyr
What dominates the Universe?
2
2
a
kc
8
 3  G  2
2
a
a
a2 8
kc 2
 3  G  m  r     2
2
a
a
What dominates it now?
These change with time for two reasons:
•Scaling – universe scale factor a changes
•Matter: m = 0.27 (significant)
•Conversion: one type turns into another
•Radiation r = 10-4 (tiny)
•Stars cause matter  radiation
•Dark Energy:  = 0.73 (dominant)
•Curvature kc2/a2: 1 - tot < 0.01 (small)
How do each of these scale? When were each of these dominant?
Near Past
•Matter:
m  a-3
At any time, except maybe now,
Distant Past we can ignore curvature ( = 1)
•Radiation: r  a-4
•Dark Energy:   1
Future
2
a
8
•Curvature:
 a-2

Never
3  G
2
a
The Future of the Universe
a2 8
 3  G
2
a
i 
8
3
 G i 0
H 02
•Starting soon, we can ignore all stuff except vacuum energy density
a2 8
2
a
da
8


G



H


G


 H 0 
 H 0  a

0
0
3
a2 3
a
dt


a  t   exp H 0t 

Universe grows exponentially
1 1/ 2
t

H
 16.3 Gyr
•The time for one e-folding of growth is

0 
•This is longer than current age of universe
•On the long time scale we are “alone”
•Except for Local group
•Maybe Virgo supercluster?
•In a few 100 Gyr, rest of universe will look empty
•Red shifted to invisibility

Alternate Futures?
•Previous slide assumes that the dark energy is:
•Vacuum energy density (no scaling)
•Eternal (no decay) [this one probably true]
We know very little about dark energy:
•If it scales as   an, then n = - 0.06  0.16
If n is small but negative:
•Universe expands as a power law, not exponential
•Functionally the same
If n is small but positive:
•As universe expands faster and faster,  gets bigger and bigger
•The faster it goes, the faster it accelerates
•In finite time,   an, becomes infinite
•H becomes infinite
•All objects in the universe get ripped apart
•“The Big Rip”
•At least 100 Gyr in the future