Jatila van der Veen and Philip LubinUniversity of California, Santa BarbaraJatila van der Veen and Philip LubinUniversity of California, Santa Barbara

The Standard Cosmological Model of the 20th century was founded on three observational pillars: the red shifts of galaxies, the abundances of light elements,

and the cosmic microwave background radiation.From an initial intensely hot, dense phase, in which matter was spontaneously

created out of radiation according to Master Einstein’s famous E=mc2, our Universe has been cooling and expanding. Hydrogen, helium, and lithium,

the lightest elements, were produced within the first 300 seconds of the infant universe. All other elements up through Iron-56 are produced in stars,

and those heavier than Iron in great stellar explosions called supernovae.In the beginning, radiation dominated over matter, but soon matter took over.

If the initial cosmic explosion had sufficient kinetic energy to balance the gravitational potential of all the matter, the universe would expand forever.

The current expansion rate of 50 km/sec/Mpc would asymptotically approach zero. Eventually the Universe would end its glorious existence

in emptiness and darkness, after the last star faded away... Such was the Standard Cosmological Model of the last century...

The Standard Cosmological Model of the 20th century was founded on three observational pillars: the red shifts of galaxies, the abundances of light elements,

and the cosmic microwave background radiation.From an initial intensely hot, dense phase, in which matter was spontaneously

created out of radiation according to Master Einstein’s famous E=mc2, our Universe has been cooling and expanding. Hydrogen, helium, and lithium,

the lightest elements, were produced within the first 300 seconds of the infant universe. All other elements up through Iron-56 are produced in stars,

and those heavier than Iron in great stellar explosions called supernovae.In the beginning, radiation dominated over matter, but soon matter took over.

If the initial cosmic explosion had sufficient kinetic energy to balance the gravitational potential of all the matter, the universe would expand forever.

The current expansion rate of 50 km/sec/Mpc would asymptotically approach zero. Eventually the Universe would end its glorious existence

in emptiness and darkness, after the last star faded away... Such was the Standard Cosmological Model of the last century...

The red shifts of galaxies discovered by Hubble in 1929 led to the conclusion that the universe is expanding .

Edwin Hubble λ0 λ’

nearby galaxy

distant galaxy

z=−

0

0'

λλλ Red shift

defined

z==− 25.400400500

This example

v(r) = H0r cvz ≈=∆

λλ Assuming a simple

Doppler shift … which led Hubble to his relationship:

The recession velocity of a galaxy is proportional to its distance from us.

H0 = Hubble’s constant

10 −==∆RRz

λλ

But the red shift really tells us about the stretching of the wavelengths of light due to the expansion of the Universe:

So H0 gives us the ratio of the present expansion rate to the present size of the Universe:

00 R

RH&

=rate of change of “size” of universe

“size” of universe “now”

H “now”

Because looking out in space = looking back in time

“now”

“then”

R0

R R0 = size of Universe “now” R = size of Universe when light left source

The value of H0 puts an upper limit on the age of the Universe:

In the Standard Model of the 20th Century,H0 = 50 km/sec/Mpc , putting the maximum age of the Universe at about 19-20 billion years, which is at least consistent with the ages of the oldest stars.

yrstt

H

MpckmH

10

1180

1200

1096.1

1sec1068.1

sec1024.350sec

50

×≈∴

≈×=

××=⋅

=

−−

−−

Of course, the expansion rate could not have been constant. It was expected that the expansion would have been fastest in the very beginning, slowed down as the universe cooled off, then sped up a bit as matter “took over” from radiation, and has been slowing down ever since...at least, sopeople thought in the last century... H0 is the expansion rate as we measure it today,from our perspective. H(t) is what we refer to as the Hubble Parameter – denoted as a function of time.

(The original value that Hubble himself estimated was about 10x too fast...)

rM m

v (r)

Let m be a little chunk ofmatter at radius r, moving with v(r) – thesame rate as the generalexpansion.

Imagine a spherical region of the universe,of radius r and mass M, homogeneous and isotropic, expanding withvelocity v(r) . If this region is expanding with the

Hubble “flow”, then we can say that

v(r) = Hr

ρπ

ρ3

34 r

VM=

=

If the Universe is matter dominated, and has the “critical density” necessary so that the kinetic energy of itsexpansion just balances the gravitational potential energy of all the mass it contains...

( )22 2 Hrr

GMv ==

21

38

⎥⎦⎤

⎢⎣⎡= c

GH ρπ

02

2

=−=R

GMmmvEtotalstarting with conservation of energy...

ρπ

ρ3

34 r

VM=

=

rearrange

substitute

get expressionfor H in terms ofcritical density

From the previous expression, one could – in principle – calculate the critical density of the universeby knowing the Hubble parameter... if all we had to worry about was the matter content .

21

38

⎥⎦⎤

⎢⎣⎡= c

GH ρπ

GH

c πρ

83 2

=

rearrange

02

2

≠−=R

GMmmvEtotal

But, if...That is, if the kinetic energy of expansion and the gravitational potential energy are not balanced...

Then, after a bit of rearranging, and using our assumption of a spherical chunk of expanding universe with density ρ...

ρπ

ρ3

34 r

VM=

=

We get this relationship:

382 2

02

ρπGHmREtotal −=

382 2

02

ρπGHmREtotal −=

continuing with thisline of reasoning...

RRH&

=0Substitute in the definition of H0

And use the first order “Einstein Equation” from General Relativity theory ...

2

2

38

RkG

RR

−=⎟⎟⎠

⎞⎜⎜⎝

⎛ ρπ& Where k is a curvature parameter...

Next substitute this expression for H0 2 into the equation at the top of this page...

we get...

ρπρπ3

83

8222

GRkG

mREtotal −−=

kmEtotal −=2

which is another way of saying that the total matter/energy content of the Universe determines the curvature of spacetime,

denoted by “-k”.

This leads to three possibilities for the geometry of space :

k < 0 |KE|>|PE| Et > 0

V(t)

t “open””

“flat””

“closed”

k = 0 |KE|=|PE|Et = 0

k > 0 |KE|

Einstein said this in his General Relativity theory, in a more complicated way. He predicted that the geometry of space could be completely specified if we knew the total mass and energy content of the Universe.

This is summarized in his famous equation:

µνµνµν πGTRgR 821 =−

= Left hand side = geometry of spacetime where “R” here is called the “Ricci tensor” and describes the shape of spacetime in x,y,z, and time

Right hand side = the matter and energy contained in the Universe, where “T” is the “stress-energy tensor” which describes the distribution of matter and energy in spacetime, in x,y,z, and time

Hmmm… but something was missing...

Any new theory that is developed in Physics has to agree with Newton’s Laws in the low-energy limit. According to Newton, gravity is always attractive, and has infinite range.

2rGMmFg −=

So... Einstein reasoned... if mass always attracts mass, and if (as was believed before Hubble) the Universe is static and unchanging, why had not everything attracted everything else by this time? Why is there a Universe?

To answer this, Einstein postulated a “cosmological constant”, or “antigravity” term, which he called “Lambda” ( Λ) to basically keep the Universe from collapsing into itself!

After Hubble announced his discovery in 1929, Einstein retractedhis cosmological constant, calling it “the biggest blunder of his career”.

So the idea of a lambda term in the equations faded into the background for the time being. It was not part of the Standard Cosmological Modelof the 20th Century, but neither was it totally forgotten….

Meanwhile...the Big Bang nucleosynthesis and abundances of light elements were predicted by Ralph Alpher in his 1948 Ph.D. thesis.

And in 1964 Arno Penzias and Robert Wilson, then at Bell Labs, noticed a small discrepancy in their microwave instruments that indicated an excess of radiation coming in from space. Not content to ignore it, they soon made one of the profound discoveries of the 20th century:..

They had found the after glow of the enormous heat left over from

the “BIG BANG”in the form of the

Cosmic Microwave Background.

They had found the after glow of the enormous heat left over from

the “BIG BANG”in the form of the

Cosmic Microwave Background.

Since the Cosmic Microwave Background radiation (CMB), fit a black body spectrum with a precision of 1 part in 104, the question became: If the universe started in some uniform hot, dense state, how did the structures that we observe today form?

So people started looking for small variations in the CMB. To make a long story short, the landmark measurement of the 20th century was the

COBE satellite, which measured variations in the microwave background of approximately 10 to 30 microKelvin, and an angular

resolution of about 20 degrees.

Red = warmer cyan = cooler above and below a background temperature of 2.726 K

So, the Standard Cosmological Model of the 20th Century favoredthe following values for the cosmological parameters:

H0 = 50 km/sec/Mpc; CMB black body temperature = 2.726K,,no cosmological constant, and “flat” geometry.

We define the dimensionless density parameter of the universe, “Omega”

1==Ωc

total

ρρ

Where ρtotal = total average density of the all the components of the universe, considered at

sufficiently large scales to look homogeneous.

The only problem was that the critical density needed to make the Standard Cosmological Model work was MORE than the density of matter that has been measured between galaxies, of approximately 1 proton per cubic meter...

GH

c πρ

83 2

=Using this expression for critical density for a flat, matter-dominated Universe,

with no cosmological constant,

and the using the value we derived for H0 , 1.68 x 10 -18 sec -1 ,

for the Standard Cosmological Model of the 20th century, and using G = 6.67 x 10-11 N-m2-kg - 2 ,

this value turns out to be about 5 x 10 –27 kg/m3

you can verify this yourself...

In the 1970’s people began to observe strange deviations from “normal Keplerian” rotation of spiral galaxies. Instead of the rotational velocity decreasing with distance from the central bulge, the rotation appeared to be constant outside the central bulge...

So unexpected were these observations, that the data were discounted as being anomalous...

approaching receding

Vera Rubin, of Carnegie Mellon Observatories is credited with offering the first explanation of DARK MATTER… stuff that gravitates but does not

shine as being responsible for the anomalous rotation velocities.

Gravitational lenses, also predicted by Einstein’s General Relativity Theory, were discovered in the 1990s, providing further evidence for ubiquitous dark matter.

So dark matter found its way into the Standard Cosmological Model, and had to be accounted for in the total density of the Universe... perhaps accounting for the discrepancy between the predicted value of critical density and the observed density...

1=Ω+Ω=Ω darkmatterbaryontotalAha! perhaps... And the first change in the Standard Model included the presence of dark matter, but we still don’t know quite what it is. It is definitely part of our universe, but is it just lots of ordinary matter that is just too cool to shine, or is it in the form of some background of exotic particles, that exist all around, which we have not yet detected, but which could be going through you right now? We don’t know yet...

Then, in the early ’90’s,Wendy Freedman’s team used the newly-repaired HST to measure H0 , using Cepheid variables in galaxies of the Virgo Cluster as

standard candles,

making the age of the Universe appear to be less than that of its oldest stars!

and reported a value for H0 of more than 70 km/sec/Mpc...

So in the mid 1990’s clever theorists started putting lambda back into their models, saying that Einstein’s “biggest blunder” was actually correct... which they claim to have suspected all along!

µνµνµν πGTRgR 821 =− Λ+We told ‘em so! The

cosmological constant is back!

But that did not necessarily mean that the universe was “open”...we could still have a flat universe if we include the contribution of this lambda..

1=Ω+Ω+Ω=Ω Λdarkmatterbaryontotal

mass/energy densityof all gravitating matter

energy density of “lambda” – Einstein’scosmological constant

drawing shamelessly plagiarized from Sean Carroll

drawing shamelessly plagiarized from Sean Carroll

We now have good reason to believe that most of the mass-energy of the Universe is in

some form that we can’t even see!

...AND we now have evidence that the Universe is Accelerating!

and that this acceleration phase began about 5 billion years ago ...

http://universeadventure.org/

We need a new cosmological model...So... The Standard Cosmological Model of the 21st Century

seems to be: * The mass/energy of the Universe is dominated by “lambda”which may be a cosmological constant, but may also be some

other form of “dark energy” which varies over time.

* The expansion rate is increasing, and this acceleration seems to have begun about 5 billion years ago.

* The age of the universe is 13.7 billion years.

* The geometry of the universe still appears to be flat as far as we can see back in time.

Three Observational Pillars of the New Lambda-dominated Cosmological model : Three Observational Pillars of the New Lambda-dominated Cosmological model :

• High – redshift type Ia Supernovae

•Accurately determined Power spectrum of the CMB

• Ages of Globular Clusters

What observations support the new model?

High - z Sne Accelerating universe

Distribution of temperature anisotropies in the CMB

Geometry of universe

Ages of Globular Clusters Age of the universe

And the Power Spectrum of the CMB anisotropies provides an independent way of confirming all the other observations,and constraining estimates on ALL the cosmological parameters!

LABS FOR A LAMBDA-DOMINATED UNIVERSE

Students use real, public domain data to understand how cosmologists derived the new cosmological model.

1. Students use light curves from high-Z type 1a supernovae to get absolute magnitudes, calculate red shift, and graph to find z at whichdeviation from straight line occurs.

2. Students perform photometry on globular cluster images taken with the HST it two different filter bands, calculate color indices, graph HR

diagram for cluster, and derive age from turn off point.

3. Students use CMBFAST to derive the CMB power spectrum from cosmological models and find the set of parameters that best fits the data.

#1: Type Ia Supernovae

from Supernova Cosmology Project at Lawrence Berkeley Lab

Rotating the line so that it has zero slope allows one to see the deviation more easily.

z = .5

Type 1a Supernovae are good Standard Candles

3 x 10 9 L < Lpeak < 5 x 10 9 L

Using our software, students can do photometry on a series of images of supernovae and determine the light curve .

light curves from Knop, et al., UC Berkeley

#2: Globular Clusters: the oldest stellar groups in our galactic neighborhood are at least 11 – 13 Gyr old,

which is consistent with the age derived from the CMB

Using our software, students can do B-V photometry on stars associated with the oldest globulars,

using HST images that are now in the public domain, to obtain an H-R diagram and find the age from the turn-off point.

Luminosity at turn off point gives approximation of age of cluster .

#3: The CMB – the oldest “light” we can measure Just for comparison:

The sky at 150 GHz(~2 mm)

This is what the sky over Antarctica would look like to you if you could “see” in the microwave region of the spectrum, instead of the visible portion, as shown below.

The sky at 545 THz(~550 nm)

Variations in the temperature of space

the shape of the Universe is FLAT!have precisely the angular size we would expect if

The Power Spectrum of the CMB constrains all the fundamental cosmological parameters

WMAP

BOOMERANG

Ωb Ωcdm ΩlambdaΩneutrinoH0 initial conditions,ionization, k, TCMB ...

Using CMBFAST now available ON LINE,in REAL TIME, students can model the

Composition and Geometry of the Universe http://lambda.gsfc.nasa.gov/cgi-bin/cmbfast_proc.pl

http://lambda.gsfc.nasa.gov/cgi-bin/cmbfast_proc.plhttp://lambda.gsfc.nasa.gov/cmbfast/output/cmb_11986705.fcl.tt.png

and compare their models to the REAL data

Labs for a Lambda-Dominated

Universe Jatila van der Veen , Philip Lubin, and assorted friends

to be published this summer :

Names you can trust. We’ve been in this business for 15 years ...

Supernova Cosmology Project and High Red Shift Supernova SearchCMBFAST by Matias Zaldariaga and Uros Seljak

Teaching about Cosmology, awesome booklet by Krauss and Starkmancosmology labs on line by van der Veen and Lubin

various and sundry articles by famous cosmologists, including Lawrence Krauss, Sean Carroll, Mad Max Tegmark,

Adam (“CJ”) Reiss, Mike Turner, and many others...Introduction to Cosmology by Barbara Ryden

Spacetime and Geometry –an Introduction to General Relativity by Sean Carroll

GR lecture notes from James Hartle and numerous illustrious visitors to UCSB/ KITP from whose talks I have plagiarized shamelessly.

Supernova Cosmology Project and High Red Shift Supernova SearchCMBFAST by Matias Zaldariaga and Uros Seljak

Teaching about Cosmology, awesome booklet by Krauss and Starkmancosmology labs on line by van der Veen and Lubin

various and sundry articles by famous cosmologists, including Lawrence Krauss, Sean Carroll, Mad Max Tegmark,

Adam (“CJ”) Reiss, Mike Turner, and many others...Introduction to Cosmology by Barbara Ryden

Spacetime and Geometry –an Introduction to General Relativity by Sean Carroll

GR lecture notes from James Hartle and numerous illustrious visitors to UCSB/ KITP from whose talks I have plagiarized shamelessly.

References