We now turn to cosmological tests, which in some sense, are at the very heart of
cosmology. First, let's make some general
considerations. The goal of cosmological tests is to
determine the global geometry, and therefore ultimate fate of the universe,
which particular universe we live in. And since the behavior of different
cosmological models is expressed as expansion curve R of T, we somehow have
to compare that with the observations. So we assume a family of models, such as
solutions of the Friedmann equation, or Friedmann-Lemaitre models.
There are others, but these are pretty generic, and they are characterized by a
number of cosmological parameters that we defined earlier They create a whole
family of these curves of R of T, and somehow, we have to find out on which one
we are on. we can observe only in the past, but from
that, we can predict what the universe will do in the future as well.
We integrate the equations, producing observable quantities, such as distances,
and then we compare that to the observations.
The basis of all cosmological tests is as follows.
We have R of T curve. R of T is relatively easily observable as a
redshift. The look-back time, T, is not. So instead
of that, we multiply that with speed of light, converting it into a distance.
And distances, at least the relative distances, we can measure, at least in
principle. So we somehow invert this diagram. So
instead of R of T versus time, we plot a distance of some sort, versus redshift.
Note that the whole thing does not depend on the exact value of the Hubble
constant, that just scales the whole diagram up or down, but the curvature,
the shape of the curves, does not depend on it.
Let's see. What is the generic behavior that we
expect? Consider just the models where the matter
dominates. If there is more matter, there is more
gravity and so the, the, the expansion slows faster.
It will always be faster at the beginning, slowing down, depending on how
much gravity is there to slow it down. And so, if there is more gravity, or
higher density, it will reach a certain value of Hubble constant sooner.
So if you take two models, one of high density, one of low density and pinch
them at the same value of the Hubble constant, which is what defines today,
Hubble time, well, then, the one with higher density
actually spends shorter time getting there, and its intercept on the time axis
is shorter. So models with the high density would
tend to have shorter life times, therefore smaller distances.
Therefore, things would look brighter and bigger than they would be in models with
low density. Adding cosmoligical constant can boost the effect of gravity, or
counterveil it, if it's a positive cosmoligical constant that essentially
acts as anti-gravity. And so, low density and or positive cosmlogical constant work
in the same way. High density and or negative cosmological
constant will work the other way. There are several types of cosmological
tests. Ultimately we have to measure distances
somehow. The first one was introduced by Hubble,
and that's Hubble diagram. The slope of Hubble diagram is the Hubble
constant. Deviations of the trend with redshift are
dependent on cosmological parameters. So its the curvature of the Hubble
diagram that is really a cosmological test.
In order to perform this test, we need a family of sources that are bright enough
to be seen far away, and that have constant intrinsic luminosity or standard
candles. The other classical cosmological test is
the angular diameter test. There, if we had a family of objects big
enough to see far away, say clusters of galaxies, and of constant intrinsic size,
standard rulers, we can then perform angular diameter test.
So this is where luminosity distance and angular diameter distance come in.
Finally we could see, in principle, how the volume of the universe is changing as
a function of time. And so if we had a universe populated uniformly with some
sort of test particles, say, galaxies, and if we can count them as a function of
redshift, or alternatively, as a function of flux, because flux would get dimmer
with redshift, then we could constrain the models in which we are.
So that is the basis of the source counts, and that is another type of
cosmological test that people do. And in principle, if we could somehow
estimate the look-back time to a family of objects, say, galaxies, through the
age of their stellar populations, we could perform direct tests of look-back
time versus redshift. But that turns out to be extremely
difficult and model dependent, and essentially, has never been done.
What can be done is to measure the local matter density of the universe, using
dynamics of galaxies or clusters of galaxies near us.
That produces measurement of the omega matter parameter, a fraction of the
critical density in regular matter, regardless of all of the distant
measurements. And finally, if you can somehow
independently measure Hubble constant and age of the universe, as we discussed
earlier, then that combination constrains a combination of omega matter and omega
lack in density of cosmological constant, but does not determine it uniquely.
And that's all it can do. So cosmological tests where developed, at least in
principle, immediately after the discovery of the expansion of the
universe, thus the Hubble diagram bears his name, and Hubble and his successors,
notably Alan Sandage, tried to put them in practice.
There was a lot of work done one this from the 1950s through 1970s at Mount
Palomar and elsewhere. Various types of luminous, and or large
objects, we use in lieu of standard candles, or, standard rulers.
Most notably, Sandage and collaborators, and others, have worked on the use of the
brightest cluster galaxies, which tend to be giant elliptical
galaxies of standard candles. Standard rulers were things like typical
sizes of galaxy clusters or radio sources, and all of these objects evolved
in some way or other. Galaxies are composed of stars. stars
evolve, their brightness will change in time collectively. Galaxies merge.
Radio sources expand. Clusters collapse. And so, there were no really standard
candles or standard rulers. So galaxy evolution, or other forms of
evolution, basically prevented us from doing cosmological tests as originally
envisioned. The revival of this came in the 1990s
through Supernova Hubble diagram, and also through a revival of the angular
diameter test through cosmic micro-background.
This, essentially, completely revived the subject, and the original goal of
cosmology, that, to determine what kind of universe we
live in, was finally achieved, at the level of precision that people, in prior
decades, just couldn't even dream about. But there are some general issues to be
aware of. We are always observing some sort of
sources, say, whether they're supernovae, or
galaxies. It doesn't matter.
Our observations have a flux limit. We usually cannot see fainter than some
given threshold, and so if you look deeper and deeper, at some point you are
going to start losing objects that are just below the detection threshold.
So if you try to fit a model curve only to those that you do see, you're going to
get a biased result. If this is understood, and if you can
somehow make a statistical correction for it, that's fine.
But generally, you don't know which part of the population are you missing, and so
that can cause significant bias. Likewise, if you are measuring angular
diameter test, there is usually a limit to angular resolution, maybe because of
the seeing through the atmosphere, and so you're not going to measure anything
smaller than a certain level, which will bias your result in that way as well.
So lets now consider Hubble diagram. There, in classical rendering, we compare
magnitudes, which, remember, are inversely
proprotional to the log of the flux, so a higher magnitude means further away,
versus redshift. And, different models will have different
curves in that diagram. In the past, things like brightest
cluster galaxies were considered, but more modern the renderings include
supernovae or even gamma ray bursts. In principle, anything else that you can
convincingly state can be standardized to a given luminosity can be used.
But before you do this, there is an important correction that usually has to
be made. We observe, typically, in a given filter,
a red filter, say, or a visual filter and so on.
And here are the spectra of galaxies of different types.
The reddish band indicates roughly where the rest frame filter, like a red filter,
would cover the spectrum, as observed in the universe.
But, if you observe them say, at redshift of 1, then the wavelength shifts by a
factor of 2, and so does the width of the filter.
So you're sampling a completely different part of the spectrum of the galaxy, and
not in quite the same width of the, of the measurement.
That has to be compensated for, because galaxies do have different spectra, and
different types will have different amount of correction.
If you take the ratio, if you integrate the flux over the two filters, the rest
frame 1 and, its red shifted version, take the ratio of that, that is called
the K-correction, usually expressed in magnitudes.
And since most galaxies tend to be redder, redder than bluer, K-corrections
are positive. So here is a set of curves that were
computed for non-evolving galaxies, galaxies that are observed here and now.
How much would you have to correct, brightness in a given filter, if you were
to put them at some redshift? So that way, you, uniform, make a uniform
set of measurement, comparing apples and apples,
and you can say, if this wasn't an evolving population, then, this is what I
should see to that given redshift. It doesn't have to be galaxies, you can
do the exact same exercise for supernovae.
So next time, we will see how we actually use Hubble diagram as a cosmological
tool.