Dark Energy

• How far away something is gets complicated at high z• How far it is now? How far it was then?

How far light travelled? How distant it looks?• The metric is

• We want to find the distance now d0 corresponding to some red shift z• Light will be coming to us along a constant and curve• Light follows lightlike curves – that is, ds2 = 0

DistanceHow Light Travels

Dark Energy

22 2 2 2 2 2 2 2sinds c dt a d f d d

0d d

2 2 2 20 c dt a d d cdt a

• Useful, as before, to define x as the relative size of universe compared to now• And therefore• Now, we previously found, if all we have is matter, then• Dividing these, we have• Integrate both sides• The distance is a0

• Can we understand this formula?• For small z, approximate integrand as 1• Rearrange and this is Hubble’s Law• But the full formula includes corrections

Finding the Distance to an Objectd cdt a

0

11

axa z

0

d cdt a x

0 1dx H xdt

1

0

0 1cHd

dx a x x

1

110 0 1 2 2z

dxa cHx x x

111

0 0 1 zd cH x

10

111

cHz

1

0cH z

0 0cz H d

0d

22 2 2 2 2 2sinds a t d f d d

How does the presence of matter () change things?• 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

Understanding This Formula Qualitatively

1

110 0 0 1 2 2z

dxd a cHx x x

Last time:• We assumed the universe only contained matter• Anything whose density falls as a–3

• We found d0, the current distance to an objects, as a function of red-shift z

• We actually don’t measure the current distance d0

• We get distances from standard candles• We don’t measure d0, we measure the luminosity distance dL:

Luminosity Distance:

The distances d0 and dL differ in two ways:• The universe is curved• Actual area of sphere is not 4d2

• The object has a large redshift 1+z• Frequency decreases by 1/(1+z)• Each photon has energy

decreased by 1/(1+z)• Rate at which photons are

received decreased by 1/(1+z)

4LLd

b

0

sinh if 11 1 if 1

sin if 1Ld z d

2204A a f 22 2

04 d f

22 2 2 2 2 2sinds a t d f d d 1

110 0 0 1 2 2z

dxd a cHx x x

• Measure luminosity distance dL as a function of redshift• Compare to model for different values of :

Type Ia Supernovae and Deceleration

m = 0.0, = 0.0

m = 0.3, = 0.0

m = 1.0, = 0.0

m = 0.3, = 0.7

• No model worked!• Radiation only made it worse• At modest z, it looked like the

universe used to be expanding quicker!

Dark Energy2 2

2 2

83

a kcGa a

• Friedmann Equation has density of stuff in the universe

• Multiply by a2

• To match supernova data, we need this velocity increasing today• To have speed increasing, must have a2 increasing with time• Any matter which satisfies this constraint will be called dark energy

2 2 283a G a kc

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• 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

• It was found that the data could be well fit with:

?u

0.30.7

m

Dark Energy vs. the Cosmological Constant• When Einstein first introduced his

general theory of relativity, he firsthad what we now call Einstein’s equations:

• He realized that this would describe a universethat was either contracting or expanding• Hubble had not yet discovered Hubble’s Law

• To ‘fix’ it, he introduced another term, whichincluded the cosmological constant

• Vacuum energy density, instead, introducesa new term in the stress-energy tensor:

• The two are equivalent if you set • There is no point in arguing which one is right

4

8 GG Tc

4

8 GG g Tc

vac(vac)T g

vac4

8 Gc

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: an

• Including all data, experimental result:

Assumption 2: Suppose we assume only that it is constant:• Including all data, experimental result:

Assumption 3: Suppose we demand that it is constant, and tot = 1:• Including all data, experimental result• This model is called the CDM model• It has become the standard model for cosmology• stands for the vacuum energy density• CDM stands for cold dark matter• “Cold” is a term we will clarify later

0.06 0.16n

tot 1.0023 0.0055

0.0486 0.00100.2589 0.00570.6911 0.0062

b

d

Note: normally given as w, where

n = –3(1+w)

The CDM Model

0.1 1.2 3.6

25.9

69.2

Neutrinos

Stars

Gas, Dust

Dark Matter

Dark Energy

Composition of the Universe

t0 = 13.8 Gyr

= 1

0 67.8 0.9 km/s/MpcH

Age of the Universe, Round 3 (1)2 2

832 2

a kcGa a

28

0 03 i iG H Assume:• 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:

• Let x = a/a0:

280 03

280 03

2 2 20 0 1

m

m m

m

G H

G H

kc a H

3280 03

2803

22 2 20 01

m mG H a a

G H

kc a H a a

2

3 220 0 02 1m m

a H a a a aa

2

2 3 202 1m m

x H x xx

20 1m m

dx H x xdt

11

0 0 20 1m m

dxt Hx x

• As m goes up, t0 goes down• As goes up, t0 goes up• If m + = 1, then

• If = 0.691, then

• Compare oldest stars 13 1 Gyr• Best estimate: No Age Problem

11

0 0 20 1m m

dxt Hx x

11

0 0

2 tanh3

t H

10 00.956t H 13.8 Gyr

0 13.80 0.02 Gyrt

Age of the Universe, Round 3 (2)

What Dominates the Universe?2 2

2 2

83

a kcGa a

2 2

2 283 m r

kGa

caa

What dominates it now?• Matter: m = 0.31 (significant)• Radiation r = 10–4 (tiny)• Dark Energy: = 0.69 (dominant)• Curvature kc2/a2: 1 – tot < 0.01 (small)

These change with time for two reasons:• Scaling – universe scale factor a

changes• Conversion: one type turns into another• Stars cause matter radiation

How do each of these scale?• Matter:m a–3

• Radiation: r a–4

• Dark Energy: 1• Curvature: a–2

When were each of these dominant?

Future

Near PastDistant Past

Never

At any time, except maybe now, we can ignore curvature ( = 1)

2

2

83

a Ga

The Future of the Universe2

2

83

a Ga

• Starting soon, we can ignore all stuff except vacuum energy density

803

20

ii

GH

2832

a Ga

2

0H803 G 0

a Ha

0da H adt

0expa t H t

Universe grows exponentially• The time for one e-folding of growth is• This is longer than current age of universe

• On the long time scale we are “alone”• Except for Local group

• In a few 100 Gyr, rest of universe will look empty• Red shifted to invisibility

1 1/20 17.3 Gyrt H

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

0.16If n is small but negative:• Universe expands as a power law, not exponential• Functionally the sameIf 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