Let us now talk about what's often called the cosmic concordance.
What's really meant by that is an agreement between all various methods,
converging to the same values of cosmological parameters.
Which, needless to say, never happened in the past.
Usually, this is expressed in the diagram that plots omega matter on x axis and
omega vacuum or omega lambda on the y axis.
On that plane, flat universe is a straight line.
a slope of -45 degrees. Because.
The 2 always have to add up to 1 for a flat universe.
Now, there are also models in which there is no big band, where cosmological
constant is so large that it prevents the initial singularity.
So, in this diagram we can plot error contours of different measurements.
Cosmic microbackground 1 is an error elilipse nicely alligned with the flat
universe line. The, supernovae Hubble diagram, is almost
orthogonal to it. We can also make dynamical measurements,
of omega mass, from clusters of galaxies, or from, by, barionic acoustic
constellations, and those tend to be almost vertical.
Mercifully, they all seem to converge and intersect at one place and that plat is
on the flat universe line. This is where omega matter is a little
less than 0.3 and omega vacuum is low more than 0.7.
So remember if we look at supernova alone.
They suggested there is a omega vacuum or cosmological constant.
And, but it, by themselves alone do not tell you that universe is also flat.
For the micro background alone we can certainly tell that universe by itself is
flat but not how the omega matter and omega lambda are distributed although One
can get probability distributions for those.
If we then add measurements from large scale structure which probe the dynamics
then altogether they intersect in the same place.
And so the point is, that various methods of measurement.
Completely different methods of measurement converge to the same thing,
which is what gives us some faith that indeed these are correct interpretations,
that we do have finally measurements of the cosmological parameters.
Note, however, that in most tests. There is a degeneracy between parameters.
You can trade one for the other a little bit in the particular direction in this
parameter space, so in cosmic micro background, that's fairly obvious that
there are all these elongated error ellipsis.
And generally speaking, for any one of these tests, you get probability dance
the contours in some sort of parameter space.
And there is a best fit father, but that need not be the be the actual best
answer. And somehow, and product of this
probability, clouds, has to really converge to the most likely over all
value of the parameters. Generally this is done through basen/g
statistical approach, especially for the micro background measurements.
It involves a lot of computations and Monte Carlo simulations, but it's a
process that's well understood. Also if you can declare values of some
parameters now, it immediately narrows down probability distributions for
others. For example you can take value of hubble
constant just derived from hubble key project, and fix that and then get better
answers for the omegas, So, here is an example of probability distributions for
various parameters from a a paper by Tegmark et al that includes all of these
data together. And you can see that, for most of them.
What's obtained is not a sharp value of a parameter, but rather an extended
distribution. You can then project these probability
density distributions in various parameter space planes.
And, for example, here is one that shows Hubble constant expressed as little h
versus omega of matter. The big red regions are regions that are
excluded using WMAP observations alone. Then the other sources of measurement
like Hubble Space Telescope and Zone intersect and you find out that indeed,
all of the probability distributions peak out at one common value.
It's not a sharp value for sure, but it's a much narrower distribution than
otherwise would be. And here is our concordance plot now
shown a little differently. The red regions again are excluded by the
WMAP measurements alone. The orange regions by there in crude
analysis, if they add Measurements from large scale structure from Sloan Digital
sky survey. Then that goes further down and then
finally they also use supernova narrows the arrow's ellipse right around a
magical set of values, of .7 and .3. Here is another one where Hubble Constant
is plotted against total omega. Total omega seems to converge to unity in
the resist flat with Hubble Constant again having the same range of values as
we've seen before. Or one can ask the question, in terms of
the age of the universe and omegas and here we have plot that shows error
contrast of this, so. Different colors, remember, respond to
introduction of evermore obsvervational constraints.
Red, from just original w map data alone, orange for better analysis, yellow,
including large scale structure and finally, using supernovae.
Now recall the equation of state paraeter w for pure cosmological constant that is
exactly minus one. Series the plot of error counters of w
against omega and there is a fairly broad distribution thats allowed but subsequent
measurements got that narrowed down to minus one.
Another quantity that can be estimated is the number of neutrino species.
intrestingly enough, because neutrino do contribute to energy lasting universe in
a similar way that protons do so here is a gamble that concordance plug with as
shown earlier, various methods together. Make estimate circle's more larger
parameters sharper, and they always do all converge to the same set of values.
Alpha omega matter, a little less than 0.03 and omega vacuum or omega landa, a
little more than 0.07. A great deal was made out of this as the
discovery of dark energy, but. This is perhaps overstating things a bit.
Cosmological constants been with us for a long time now and astronomers have
considered it seriously to explain more or less the same problems.
And here's for example a title page from a paper in nature by two famous
astronomers, James Gunn, and Beatrice Tinsley.
They introduce cosmological constant for more or less the same reasons,
astronomers invoked it later in 1990's to reconcile what, otherwise would be
discrepant measurements of Hubble Constant and the density.
Here is a simple concordance diagram I myself made as a graduate student of 30
years before the supernova results were announced.
We didn't have a supernova a Hubble diagram then.
But one could use lines of equal age of these models and choose the set that's
roughly what globular clusters gave you and that turns what out works just as
well as supernova. So that too pointed out that if you
wanted. A flat universe say, demanded by the
inflation theory, then, you must accept solution where by, density is really only
around 0.3, and, cosmological constant is around 0.7.
So to recap, our best guess universe parameters today, is that the age is
about 13.7 billion years. The Hubble constant is just a tad over 70
kilometers per second per megaparsec. The density of variance, normal matter as
we know it, is around 4% or 4 1/2%. That the total matter density including
the dark matter, which is not barionic, is about 0.27 and the energy density
corresponding to the dark energy possibly cosmological constant it makes up the
rest of about 0.73. So let's just look at 3 key densities,
the total density again expressed as fractional critical density which is
close to 1. The matter density and the baryonic
matter density. Interestingly enough, if we just add up
all the luminous matter that we see in galaxies, stars, radio emission from gas
and so on. That only adds to like half a percent of
the critical density. Much less than the total number of
variants that we know. And so because 0.5% is less than about
4%, there has to be some sort of missing baryonic matter, and that was an
interesting problem in of itself, and because 0.04 is less than 0.27, there has
to be a non-baryonic component of matter. Which is the dark matter.
And because .27 is less than 1, there has to be dark energy in order to fill up the
rest. So to recap the cosmological tests.
They always based on comparing some kind of measure of distance versus redshift.
And in order to do this we use test particles of different kinds.
If we have objects of ostensibly co-luminosity, those are standard
candles. If we have objects of ostensible same
size, those are standard rulers. And if we have a standard population that
fills up the universe, say, galaxy clusters, then we have standard
population. In addition to these cosmological
observations, we also obtain measurements of the omega matter from local dynamics
or large scale structure. We can then combine that with independent
measurements of Hubble constant, through the distance ladder process we talked
about earlier. Or ages of globular clusters that can
find age of the universe. And so, even though there are many
couplings and degeneracies for any given method between different parameters, all
of them converge to the same set of values.
Because of this we believe that we have a good picture now.
Many different methods, completely different kinds of measurements,
completely different kinds of physics. All converts to the same thing and this
is why we think that concordance cosmology is now fairly well-established.
Next time, we'll start talking about the early universe.