Repulsive Gravity & the Cosmological Constant

basic physics covered in LLLB Chap 8
figures stolen from: Supernova Cosmology Project
High-z Supernovae Search
Ned Wright's Cosmology Tutorial

Outline

Observational Evidence
repulsive gravity in General Relativity
Inflation

Observational Evidence for Accelerating Expansion

Hubble showed that the Universe is expanding
But, is it speeding up or slowing down?

normal matter + gravity should make it slow down
but how do we know for sure?
improved "Hubble Diagram"
here's Hubble's original:

x-axis is distance, y-axis is velocity

How to do a better job

get a more accurate distance indicator
which can be seen at very large distances

Supernovae of type Ia

these are white dwarfs in binary systems which accrete mass from their partner
when M grows to ~ M_ch= 1.4M_sun, the carbon-oxygen white dwarf has a spectacular nuclear explosion that totally disrupts the star
probably always happens at the same mass
no outer layers to make things messy
the core collapse Supernovae (type II) are messy because of the outer layers of the massive star

making a Hubble diagram:

1st, find some supernovae at high redshift (here's an example from HST )

(note: HST is not actually used for a SN search - discoveries come from the ground)

confirm that the supernovae are of type Ia with a spectrum (usually with the Keck 10-m telescope)

use that same spectrum to measure the redshift

measure the brightness of the supernova as it brightens and then fades and compare to nearby supernovae like this one:

Here are some SN brightness measurements. The curves are the best fit SN light curve as determined from nearby supernovae

Now determine the distance to the SN using or

Plot d vs. v to get a Hubble diagram
here's one from the High-z Supernovae Search Team:

The "distance modulus" plotted on the y-axis is a peculiar distance unit used by astronomers. It is equal to 5*log(distance/10 pc) so the most distant SN on this plot is at about 7.6 billion pc.)

for equal time, here's one from the Supernova Cosmology Project:

both plots show redshift z vs. a logarithmic measure of brightness (and therefore distance)

So, what's the result?
Why is Saul Permuter (leader of the SN Cosmology Project) so happy?

It's not because his preliminary conclusion from the supernova video has been confirmed!
perhaps its because the SN result is Science Magazine's Breakthrough of the Year for 1998:

To be fair, this is a discovery by two groups. Here's Brian Schmidt, leader of the High-z SN Search:

Prof. Peter Garnavich is also a member of the High-z SN Search

So, what's going on here?

Take a look at the bottom panel of the High-Z Hubble diagram:

This plot has the brightnesses scaled to the expectation for an empty Universe

the white straight line represents an empty = 0, = 0 Universe

note that = + is the total density in ordinary matter

the magenta line curving upward represents a = 0, = 1 Universe

the SN appear to be slightly fainter than in the empty Universe case
expansion is accelerating

the green line curving downward represents a = 1, = 0 Universe

the SN appear to be slightly fainter than in the brighter Universe case
expansion is slowing down
this was the expected case a few years ago

Conclusion, the Universe seems to be that the expansion of the Universe is accelerating, but not as fast as the empty Universe

this is good, because the Universe isn't empty!

Here are the results from the SCP and the Hi-Z SN for a fit to the best values of & :

High-Z SN Search:

SN Cosmology Project:

Both groups prefer = 0.3 & = 0.7.

theory seems to prefer + = 1 (a flat Universe)

Repulsion in Einstein Theory of Gravity

How can we understand gravitational repulsion in Einstein's theory?

isn't gravity just an attractive force? (i.e. doesn't it just suck?)

Gravity doesn't just depend on mass, but it depends of velocity, too.

gravitational acceleration

or in our usual notation

the density, > 0 for any type of matter
the presure, p = 0 for normal slowly moving matter
and p/c² = /3 for radiation (matter moving at v=c)

for radiation + 3p/c² = 2, so radiation yields a stronger gravitational attraction than ordinary matter moving at v << c.

if we had negative pressure, we might get + 3p/c² < 0 which would give a positive acceleration.

What is negative pressure?

it pulls instead of pushes

stretch a rubber band and it pulls, so it has negative pressure

but p/c² << for a rubber band, so the modification of the gravitational force is tiny

when you let it go, the tension quickly converts into energy of motion which gives a positive pressure.

a hypothetical cosmic string is like a rubber band with a very high tension

stretch it and let it go, and it will oscillate at v ~ c.
it only pulls in 1 of 3 directions, so it has p/c² = -(1/3)

or + 3p/c² = 0, so a stationary cosmic string causes no gravitational acceleration

if you stretch it and let it go, it develops a high velocity which gives a positive contribution to the pressure

so an oscillating loop of cosmic string will give an attractive gravitational force.

To get gravitational repulsion, we need something with a very strong tension in 2 or 3 dimensions

3 dimensions is best, and stuff that has the maximal amount of tension in 3 dimensions is called "false vacuum"
"false vacuum" has p/c² = - or + 3p/c² = -2

this is the most negative pressure that is allowed by General Relativity

What does this negative pressure mean?

"false vacuum" has the peculiar property that the energy density is a constant

suppose you have a piston full of ordinary matter.

squeeze the piston and the density of matter increases
you must input a lot of energy to squeeze it because the pressure of the ordinary matter is pushing against you

now, suppose you have a piston full of "false vacuum" matter

you don't need to squeeze it because the negative pressure will cause it to squeeze itself
the "false vacuum" can produce energy - it will pull you with the piston if you are holding on
the "false vacuum's" mass energy is proportional to its volume (i.e. through E=mc²)
it is 100% efficient at convering mass to energy (compared to < 1% for nuclear reactions)

Why is it called false vacuum?

vacuum means essentially nothing
a false vacuum is like a vacuum (i.e. no oridinary matter or radiation)

but, it has this extra energy density
if space was filled with a false vacuum we couldn't easily tell
energy only comes out if we convert the false vacuum to something else