Hello. So far, we've found out how to compute
different cosmological models, but what good are they? Basic goal of
cosmology is to figure out in what model universe do we live and you will recall
that they are basically distinguished by their history of the expansion rate.
How does the scale factor changes as a function of time? If we can figure out
which curve of those we live on, we know, we'll know about cosmological
parameters. The expansion factor, R(t), is simply
related to red shift, that is an observable quantity and that's
an easy part. The other axis is the time axis.
Now, unfortunately, distant galaxies do not carry gigantic clocks on them, so we,
it's very hard to figure out what is the look back time between us and some
distant point in a way that can be measured.
So instead of that, what we do is, we do, we transform coordinates.
Instead of the look back time, we can use distance which is simply time multiplied
by the speed of light. And distance is in principle, we can
measure. So we flip this diagram, and instead of
expansion R(t), we use the red shift, which is an observable quantity.
And instead of the time, we use a distance, which we can figure, figure out
how to measure in some way. So centrally, all cosmological tests boil
down to this. We have to somehow measure a set of
distances to points of the function of redshift and because the whole thing just
scales with Hubble constant, we only need to determine the shape of that curve.
So let's figure out how to measure distances in cosmology.
A convenient unit of the distance is Hubble distance, which is simply speed of
the light divided by the hubble constant. We call it hubble constant, it has
dimensions one over time, and obviously, one over Hubble constant
is called the Hubble time. In the units of Hubble constant of 70
kilometers per second per megaparsec, which is close to its actual measured
value, Hubble length is about 4.3 gigaparsecs and Hubble time is a little
shy of 14 billion years, which is actually pretty close to the
actual age of years. Now, at low redshifts, the expansion is
linear. Hubble's law applies and the distance is
simply redshift times the Hubble distance,
but the further out we go, the realistic effects come into play and things are a
little more complicated. Generally speaking, we can compute the
comoving distance, distance in coordinates that do expand
with the expanding space. As follows, we can integrate the element
of redshift divided by the function, which is really Hubble constant at that
given time and which is given here. You may actually remember similiar
expression inside a Friedmann equation. You'll see three terms,
the density of matter multiplied by the cube of one plus redshift. Well, that's
probably because density of the matter scales as the cube of the expansion,
then there is a curvature term and there is a constant term, which corresponds to
cosmological constant. There is actually one more term missing
which is the relativistic matter, which would be omega of radiation times one
plus redshift to the fourth power. But that's only important in the early
universe, so it's generally neglected in this computation.
Now, the really useful quantity in computing physical parameters, is not
comoving distance per say, but a quantity that is adjusted for the, for the
curvature of the space, which is called the transverse comoving distance,
and it's in the formula for that is given here.
In the case of a flat universe, those are the same.
In the case of positively or negatively curved ones, there is a crtuch.
Now remember, omega curvature, so called curvature density does not really
correspond to density of anything. It is just a quantity that makes the sum
of all densities equal to one, always. So, omega curvature is really one minus
the omega of matter, omega radiation, and omega of cosmological constant.
So how can we derive this? We'll start with Robertson-Walker metric,
the basis of all relativistic cosmological models.
Since, only the radial coordinate matters by large, we can then simplify that get
rid of all the, both the angular terms and just have the radial term.
So take square root of both sides and integrate.
Generally speaking, this integral cannot be solved analytically.
It has to be solved numerically. In some special cases, such, as when the
density of the vacuum, cosmology constant is equal to zero, there are analytical
solutions. In that case, the formula for the
distance is shown here. It's expressed in the older units of q
naught, which remember is deceleration parameter,
and in the absence of cosmological constant, it is equal to the density
parameter divided by two. Now, for universe with non-zero
cosmological constant, things are little more complicated and generally speaking,
infigural has to be computed numerically. So here is a plot of distances for
several different cosmological models expressed in the units of the Hubble
distance as a function of redshift. It's in units of Hubble distance because
the whole thing scales directly, linearly, proportionately to Hubble
constant, and we can separate measurement of the scale of the universe through
Hubble's constant, in the shape of the universe if you will, which is the outer
cosmological parameters. But how do we relate this to the things
that are actually measured? We can measure, roughly speaking, two
kinds of things, brightness, from some source far away or
its angular size. In cases where we can use brightness, you
can imagine using source of a standard luminosity,
also called standard candle, viewed at different distances and using the
relativistic version of universe square law to find out how far it is. So in
simple, Euclidean, non-relativistic universe, the flux would be luminosity
divide, divided by the area of the sphere with a radius from here to there.
In the case of an expanding universe, there are two terms of one plus redshift
that come into play. One of them accounts for the energy loss
of photons, because their wavelength is stretched as one plus redshift. But the
thing is moving far away from us and fast, and so, we have to account for the
relativistic time dilation. The clocks tick slower there by a factor
of one plus z from our point of view, and so, the rate of the photon emission is
affected by exactly the same factor, so we have 1 + ^2 in denominator.
In other words, in an expanding relativistic universe, objects appear
dimmer than they would if the universe wasn't expanding.
So for convenience, luminosity distance is defined as the actual radial distance
times one plus z. So when you take the square of it, you
recover a familiar universe square of formula.
That is if we can measure the total luminosity, but, we usually measure
specific flux, which is power per unit wavelength or unit frequency.
In that case, we recover one power of 1 + z, because the angstroms are stretched by
exactly same factor or the hertz are also stretched by one of the factor.
So in case of specific fluxes there are, no two powers of 1 + z,
they're one. So here is a plot of luminosity distance
in several cosmological models, again, expressed in units of Hubble
length as a function of redshift. You see, the typical redshifts of
interest in modern cosmology, it's of the order of ten Hubble lengths.
The other kind of thing that we can measure is the angular size of things.
Imagine if you had standard gigantic ruler that you can view a different size,
different [INAUDIBLE] from us, then its angular diameter will tell us
how far we are for a given cosmology. In a simple Euclidean non-expanding
space, that angle will be the size of the ruler divided by the distance.
That is as far as comoving coordinates are concerned,
but, what if the object is fixed in proper coordinates,
say like a galaxy? Galaxies do not expand with expanding
space at any given redshift. Proper size is larger than the comoving
size, because comoving coordinates have expanded since then and it's larger by
exactly power of 1 + z. So here we have the opposite scenario.
The objects in an expanding universe are actually bigger than they would be in a
non-expanding case. And we divide the distance by 1 + z so that we can use the,
a familiar formula and that's called the angular diameter distance.
Here is a plot of angular diameter distance as a function of redshift for
several popular cosmological models. You will notice that, in many of them,
there is a maximum. At some distance from us, things no
longer appear smaller, this is specific to relativistic
cosmology. In a Euclidean space,
the further away something is, the smaller it's going to look,
always. Here, past certain depth, things actually
start getting larger again. Finally, we can imagine measuring volumes
out to some redshift. If say, universe was populated by
particles such as galaxies or alignment of a clouds,
and we can count, then their number per unit volume is something that will be
also dependent on the expansion rate. So the volume element,
its increment, radial increment as a function of redshift is given by this
formula. Generally speaking, that has to be always
evaluated numerically. And the total volume integrated from here
to some redshift of z is given by these formulas.
So here are plots of the volume element as a function of redshift for several
models. As you can see, those curves can differ
very substantially at high redshifts, and that's why this is potentially an
interesting comological test. Here as well as in the other plots, all
of the curves are really close together at very low redshifts, when things that
are asymptotically Euclidean. So that's the distances, what about the
look-back time? Whereas we cannot directly measure it as
a function of redshift. It is a useful quantity when considering things like
galaxy evolution. The time elapsed, since some redshift of
z, which is the look-back time is given by the same formula as we've seen before,
except corrected by 1 + z. So then, again, this also has to be
evaluated numerically, except for some special cases, like with zero
cosmological constant, when an elliptical solution does exist.
If we're to in, integrate this all the way out to infinity redshift, that is to
the Big Bang, we will get the total age of the universe.
So here is the look-back time as the function of redshift or alternatively,
age of the universe as a function of redshift measured in units of the Hubble
time for a variety of different models. They all behave qualitatively in similar
fashion, but the curves are separate. So this is what will lead us into
cosmological tests. We're trying to determine which R(t)
curve we live on. This is really measured, done by
measuring redshifts and some form of distance.
But, take these curves, and pinch them together when the slope is the same.
In other words, when they have the same Hubble constant.
Well then, the intercept on the time axis would be different for different curves.
Cosmological models with high densities and or zero cosmological constant will
tend to have smaller distances, smaller look-back times, and smaller volume, than
models with lower densities and positive cosmological constant.
Thus, if you pinch these curves at the same value, Hubble constant, which is the
same slope, and see where the intercepts are on the time axis, that the lower
density or positive cosmological constant models will tend to be larger and both
longer duration. So the objects in lower density models,
or, models with positive cosmological constant will appear smaller and fainter
than they would in the same models with high density in all cosmological
constant. So next, we will start looking into
measuring the scale of the universe or the Hubble constant.